Download Silicon quantum electronics

Silicon quantum electronics
Cover: Transport spectroscopy data of a single arsenic donor embedded in a
silicon field effect transistor, measured at 0.3 Kelvin. The signal is the
channel current versus gate voltage, with each trace recorded at a constant
(negative) source/drain bias voltage. The steps in current, such as in the red
trace, are due to the quantum mechanical excited states of the donor atom.
The red line, on the right side of the front cover, shows the signal from the
donor in the ionized state. Then, moving to the left, via the spine to the
back cover, the donor undergoes charge transitions from the ionized state to
the neutral state and finally to the singly charged state.
Silicon quantum electronics
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op donderdag 20 december 2012 om 12.30 uur
door
Jan Verduijn
natuurkundig ingenieur
geboren te Rotterdam, Nederland.
Dit proefschrift is goedgekeurd door de promotor:
prof. dr. S. Rogge
Samenstelling van de promotiecommissie:
Rector Magnificus
prof. dr. S. Rogge
prof. dr. H. W. M. Salemink
prof. dr. L. M. K. Vandersypen
prof. dr. P. M. Koenraad
dr. A. J. Ferguson
dr. F. A. Zwanenburg
voorzitter
University of New South Wales, promotor
Technische Universiteit Delft
Technische Universiteit Delft
Technische Universiteit Eindhoven
University of Cambridge
Universiteit Twente
prof. dr. H. W. Zandbergen
Technische Universiteit Delft, reservelid
Printed by: GVO drukkers & vormgevers B.V. | Ponsen & Looijen, Ede
Cover design: Jan Verduijn
Copyright © 2012 by J. Verduijn
ISBN 978-90-6464-613-3
An electronic version of this dissertation is available at
http://repository.tudelft.nl/.
Contents
1 Entering the quantum world
1.1 Improving devices . . . . . .
1.2 Quantum computation . . .
1.3 Designing particles . . . . . .
1.4 This thesis . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
2
4
6
7
2 Quantum electronics and transport
2.1 Devices and fabrication . . . . . . . . . . . .
2.2 Transport spectroscopy of single donors . .
2.2.1 Single donor transport . . . . . . . .
2.2.2 Single electron transport . . . . . . .
2.2.3 Interference effects and correlations
2.3 Single donors as qubits . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
9
10
14
14
15
18
22
3 Donor states in nanostructures
3.1 The hydrogenic model . . . . . . . . . . . .
3.2 Valley-orbit corrections . . . . . . . . . . .
3.3 Electric fields, interfaces and confinement
3.4 The two-electron state and spin filling . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
31
32
34
37
42
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Gate control of valley-orbit splittings in
semiconductor nanostructures
4.1 Valley-orbit splitting in quantum devices
4.2 Measuring the valley-orbit gap . . . . . . .
4.3 Donors and quantum dots . . . . . . . . .
4.4 Comparison to a model . . . . . . . . . . .
4.5 Influence of barrier and disorder . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . .
.
.
.
.
silicon metal-oxide.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Wave function control and mapping of a single
5.1 Access to single donors . . . . . . . . . . . . . . .
5.2 Tuning the electric field . . . . . . . . . . . . . . .
5.3 Valley structure of a confined donor . . . . . . .
5.4 Single donor amplitude spectroscopy . . . . . . .
5.5 Tunable confinement & valley population . . . .
5.6 Possible applications . . . . . . . . . . . . . . . . .
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . .
v
.
.
.
.
.
.
47
48
49
51
53
53
54
donor atom
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
59
60
60
62
63
64
66
67
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
6 Coherent transport through
6.1 Introduction . . . . . . . . .
6.2 Devices . . . . . . . . . . . .
6.3 Experimental results . . . .
6.4 Summary & conclusions . .
Contents
a double donor
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
system in
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
silicon
. . . .
. . . .
. . . .
. . . .
7 Non-local coupling of two donor-bound electrons in silicon
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Single donor transport . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Aharonov-Bohm effect in the Kondo regime . . . . . . . . . . .
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Sequential transport & the Kondo effect . . . . . . . .
7.4.2 Interfering Kondo channels . . . . . . . . . . . . . . . .
7.4.3 Phase coherence . . . . . . . . . . . . . . . . . . . . . . .
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
71
72
72
72
76
79
80
80
81
83
83
84
85
85
A Appendix A: The valley-orbit correction
89
B Appendix B: Supporting information to chapter 5
95
Conclusions & Outlook
101
Summary
105
Samenvatting
109
Curriculum Vitae
113
List of Publications
115
Afterword
119
1
Entering the quantum world
The invention of the transistor started a revolution that greatly improved the
quality of life for mankind. Scientists and researchers are continuously looking
for technologies that may eventually have a similar impact on our daily lives.
This chapter puts the work of this thesis in a historical perspective and presents
some of the recent developments. Hereby it is attempted to show how this
work contributes to the development of new technologies and the understanding
of fundamental laws of nature. This chapter concludes with a section that
provides and overview of the chapters that follow.
1
2
1.1
M
Entering the quantum world
Improving devices
iniaturization of logic circuits was first made possible with the invention of a working solid-state transistor. The invention of the transistor
was followed by the crucial development of a fabrication process for circuits
that integrated all components on a singe piece of material. This sparked
the developments that lead to present day, pcs, laptops, smart phones, game
consoles, etc. All of this took place at an unimaginable speed. The first demonstration of a working transistor was at Bell labs in 1947. This was followed
by the development of a practical implementation in the years thereafter, also
mainly at Bell labs [1]. The concept of a small fully electrical switch was
then combined with the idea to fabricate many individual devices in a circuit
on a single piece of material (chip), the integrated circuit. For technological
reasons silicon became the dominant material to make these circuits in these
years. Today, silicon is still used for the vast majority of integrated circuits.
Since the early 70s these developments have resulted in a doubling the
number of components within integrated circuits, compared to their size, about
every 18 months. Gordon Moore, a co-founder of Intel, a company that is
still one of largest producers of integrated circuits in the world, noted this
exponential trend in 1965, and therefore it is named after him as “Moore’s
Law”. Moore’s Law is an economic law and there is no physical reason why
it should be true. But the rate at which computers, smart phones and other
devices that rely on integrated circuits have improved is still exponential at
the moment. There are, however, physical limits to these improvements in the
conventional way. Laws of nature put bounds to this development and will
eventually bring it to an end.
a
FinFET
b
TFET
Figure 1.1: Cross-sectional transmission electron micrograph image of a FinFET and
TFET device, see text. a) The FinFET is for the first time used by Intel in their latest
processor because of the superior performance over conventional planar device designs. The
device studied in chapter 5 of this thesis is virtually identical to the shown device. Image adapted from: [2] b) A radically different physical effect is used for switching in the
TFET: tunneling of electrons. Image adapted from “IMEC Scientific Report 2011’’, url:
http://www.imec.be
1.1. Improving devices
3
To sustain the increase in computational power of integrated circuits, the
smallest feature size of the individual components in the circuit have been
shrunken in an exponential way as well [3]. In the early 70s the size of the
smallest features was 10 – 20 µm, whereas in Intel’s latest processors they are
22 nm. This is roughly a 10,000-fold decrease in about forty years. A decrease
in the size of the features increases the number of possible devices per unit
area present on a chip. This, in turn, increases the number of computational
operations per unit time. In addition, the speed of the devices increases and
dissipation per device decreases. Today, the main factor limiting scaling is
the dissipation per unit area on a chip. Therefore, especially in the last ten
to fifteen years, many new materials have been introduced to follow Moore’s
law and create reliably working devices at the same time. Before that time,
there had been no significant changes to the technology as developed in the
early days. The industry has also started to work actively on the development
of transistors with a non-conventional geometry, as well as devices that use
fundamentally different physical principles to operate. Trusting these measures will help them to keep improving performance, Intel has announced in
their latest R&D pipeline that they plan to introduce a technology with 5 nm
minimum feature size around 2020. At the moment however, there exists no
technology that can achieve this reduction in size and can be used in mass
production.
A successful new design that is radically different from the conventional
planar geometry is the FinFET (Fin Field Effect Transistor), Figure 1.1a.
This transistor design is used in Intel’s latest processors, introduced in early
2012. It basically consists of a thin wire fabricated on a substrate with a gate
that wraps around the wire to form a channel region. By applying a voltage on
this gate the electric current in the wire can be switched on and off. Because
of the three-dimensional design, which provides improved electrostatic control
over the channel compared to conventional planar FETs, the switching voltages
and on/off current ratio are both reduced. Therefore, the power dissipation
per device is reduced and the switching speed can be increased [4, 5]. The
physical principle on which this switching is based is, however, still the same
as in the early 70s.
A device that uses a radically different physical operation principle and is
currently considered a promising candidate for future devices, is the TFET
(Tunneling Field Effect Transistor), Figure 1.1b. This device relies on a quantum mechanical effect for its operation; tunneling of electrons. Just like a
FET, a voltage on the gate switches the current through the device on and off.
One advantage of using electron tunneling is that, in contrast to conventional
FETs, the voltage range over which the device switches is, in principle, not
limited by the temperature of the device. Therefore, lower switching voltages
can be used and, consequently the power dissipation in the device can be up
to a 100-fold reduced [6]. It should be noted here, that for the development
4
Entering the quantum world
of a practically useful TFET some challenges, such as a high enough contrast
between the on and off state, still need to be overcome. But many of these
challenges are currently being addressed by introducing new materials and
improving designs [6].
At the same time, scientist and researchers are exploring the use of new
materials and physical principles, not only to improve conventional computer
architectures, but there is also a search for radically different ways to perform
computations. A new paradigm for computation that has seen tremendous
progress towards a practical implementation in the last fifteen years is the
quantum computer [7].
1.2
Quantum computation
The development of a quantum computer is only partly driven by the neverending need for better devices, as it can probably not replace today’s computers. Quantum computers promise a huge increase in the computational
efficiency for certain problems that take a very long time to solve on a conventional computer, such as searching large databases and breaking encryption
codes. Besides this, a quantum computer will be able to perform quantum
simulations that are impossible to do on classical computers [8]. Quantum
simulations would, for example, help a researcher finding a new drug by designing a chemical reaction by an exact quantum simulation, instead of empirically having to find the reaction path. The power of the quantum computer
lies in the fact that it can perform computations in a highly parallel way. The
quantum mechanical principles on which such a computer is based allows to
encode information and perform computational operations on a large amount
of information in parallel. Instead of using a 0s and 1s, which naturally map
to the “off” and “on” state of a transistor, a quantum computer uses quantum
bits or qubits.
A qubit can be set up to be partly in a state ∣0⟩ and partly in a state ∣1⟩
at the same time. Furthermore, states of different qubits can be entangled
with each other. This means that the outcome of a measurement on one of the
entangled qubits will influence the outcome of a measurement on the other,
irrespective of how far the qubits are apart. There are detailed theoretical
ideas that show how these properties of the quantum world can be used in
a clever way [8]. The challenge for experimentalists is to demonstrate real
working implementations, which turns out to be not so easy.
In principle, many physical systems can be used as a qubit. Examples
that have been demonstrated to work are isolated atoms, photons and spins
in the solid state [7]. For all these systems the basic criteria that allow the
construction of a quantum computer from these qubits are not at all, or only
partly, met at the moment [10]. One of these criteria is that the qubit has
to hold the information as long as it takes to perform the computation. This
is a major challenge, because quantum information is very sensitive to a pro-
5
1.2. Quantum computation
Bias
SQUID
Qubit B
Cc = 15 fF
Qubit C
Qubit A
Coupler
Qubit D
100 µm
Figure 1.2: An example of a quantum device consisting of four flux qubits (A, B, C and
D) that can be entangled with the couplers. This device has been used to demonstrate
three-qubit entanglement. Image adapted from [9]
cess that causes loss of information, called decoherence. In other words, the
computation has to be finished well within the typical time scale on which decoherence takes place, i.e. the coherence time. Decoherence can be prevented
if the qubit is isolated from the outside world. But this is in contradiction with
the requirement of actually performing a computation, because this requires
access to the quantum states in which the information is encoded. Thus, this
links the states to the outside world and causes decoherence. Another challenge is to link qubits to each-other to build a network that can perform more
complex computational tasks than a single qubit. This requires a reliable way
of transportation of the quantum information from one stationary qubit to
another distant qubit.
Spins in the solid state have proven to maintain their quantum state for
an extremely long time, up to minutes have been observed [11]. Especially
spins in silicon are promising candidates. The main reason for this is that the
long coherence times of a naturally occurring system can be combined with
advanced fabrication techniques. Furthermore, it is possible to make silicon
very pure and thereby removing sources of distortion (decoherence) of the
quantum state. Another technological reason is that the years of development
to improve the performance of silicon devices has generated an enormous base
of knowledge from which can be drawn to create devices for quantum applications. At the same time, however, it is difficult to manipulate spins at a
speed that allow a sufficient number of operations to be performed before the
6
Entering the quantum world
information is lost by decoherence. Since it is a natural system, it is not easy
to change this property. A different approach is to design a device which can
mimic a spin state.
A spin is a naturally occurring property of elementary particles. An example of a device that simulates a spin state in the solid state is the superconducting flux qubit. This system consists of relatively large ring in which
the quantum information is encoded in clockwise and anti-clockwise running
currents. Fast manipulation of the ‘spin’ in this system is easier than for natural spins in the solid state, because of the freedom to engineer the qubits and
optimize of fast manipulation. Furthermore, it has been shown that they can
be coupled, see Figure 1.2. Their main limitation is the coherence time, which
is much shorter than natural spins in solids.
Recently, a new idea, that may help to overcome limitations of naturally
occurring spins as well as engineered spins, sparked the interest of many researchers: the topological quantum computer. The idea is to use a certain
type of quantum state that is robust to decoherence in a way similar to a knot
in a rope. Such a state can not be ‘unknotted’ by simply twisting and pulling.
A more complex operation is needed to unknot the rope or to change the knot
from one type into another. Driven by this idea, recently an important goal
has been achieved, discussed in the next section.
1.3
Designing particles
Engineered solid state devices can be used to design quantum mechanical degrees of freedom that are not readily available in nature, as well as tailor quantum systems to requirements for specific applications. Using the right mix of
materials and the right geometry, new quantum states can emerge under the
right conditions. Such a state can also be pictured as a particle, or more precisely, a quasi-particle. A quasi-particle is a composite particle build up out
of elementary particles (atoms and electrons) of which the device consists. An
intriguing example of a quasi-particle that is really and ‘engineered’ particle,
which has not been found as an elementary particle in nature, is the Majorana
Fermion [12], see Figure 1.3. A Majorana Fermion is the particle (or quantum
state) that can be used to ‘knit’ quantum states together, as explained in the
previous paragraph, but its discovery is also relevant to fundamental physics.
In 1937 the Italian physicist Ettore Majorana predicted the existence of a
certain type of particle from a then recently developed theory for fundamental
particles. This particle is now called a Majorana Fermion. Special about the
Majorana Fermion is that it is its own anti-particle. Furthermore, it has been
suggested that the knotted quantum states, that are particularly robust to
decoherence, can be made with it. Since the prediction by Majorana, physicists have searched for this particle in nature, but have found no sign of it
so far. Very recently, however, the first experimental evidence of a Majorana
Fermion was observed in a superconducting device [12], see Figure 1.3. Even
7
1.4. This thesis
a
b
4
S
3
2
S
N
1
1
11 μm
m
N
B
B
eV
*
2
3
4
*
Figure 1.3: a) A quasi-particle can be created with the right device design under certain
conditions. b) By combining a superconductor labeled (S) and normal conductor labeled
(N) in the same structure, a Majorana Fermion appears at each side denoted by the red “∗”.
Image adapted from [12]
though this experiment still needs to be confirmed, and the properties that
make the observation interesting from a fundamental point of view and for
quantum computation have not been probed directly yet, this experiment is
a nice illustration of how combining and structuring materials allow one to
create objects in solids that do not exist in nature.
1.4
This thesis
Silicon FinFET devices are in this thesis used to study quantum effects. The
main focus will be on devices with dopants embedded in them. A dopant is
a naturally occurring impurity that can hold a single electron and thereby
localizes a single spin. This makes it suitable to serve as a building block for
a quantum device. The main objective of this thesis is not to directly work on
the development of a physical implementation of a qubit in silicon, but rather
to aid the understanding of a physical system that has favorable properties for
the implementation of several proposed quantum device schemes.
Chapter 2 starts by discussing the fabrication and basic characteristics of
devices studied in this thesis. This is followed by an overview of the basic
theory of single electron transport and the effect correlations and interference
that play a role in the presented experiments. To illustrate the usefulness of
donors for quantum device applications, chapter 2 finishes with some examples of qubit implementations and coherent control. Chapter 3 discusses the
level spectrum of donors confined in silicon nano-structures by outlining an
effective mass theory that is used to calculate the level spectrum. In chapter
4, a detailed analysis of the quantum mechanical level spectrum of an electron localized near an interface and subject to an electric fields is presented.
By comparing devices with and without donors, the influence of the donors
and the electric field is studied. In chapter 5 this is further detailed and the
8
Entering the quantum world
experimental observation of a controlled transition of a electron from a donor
to a nearby interface is presented. Chapter 6 discusses an interference effect
associated with transport mediated by donors and in chapter 7 it is shown that
similar interference effects persists in the presence of many-body correlations.
References
[1] The Silicon Engine: A Timeline of Semiconductors in Computers, url:
http://www.computerhistory.org/semiconductor/timeline.html, accessed
on 13 September 2012.
[2] Roche, B. et al. A tunable, dual mode field-effect or single electron transistor. Appl. Phys. Lett. 100, 032107 (2012).
[3] Ferain, I., Colinge, C. A. & Colinge, J.-P. Multigate transistors as the
future of classical metal–oxide–semiconductor field-effect transistors. Nature 479, 310–316 (2011).
[4] Hu, C. et al. FinFET-a self-aligned double-gate MOSFET scalable to 20
nm. IEEE Transactions On Electron Devices 47, 2320–2325 (2000).
[5] Yang, F.-L. et al. 5nm-Gate Nanowire FinFET. In Digest of Technical
Papers, Symposium on VLSI Technology, 196–197 (IEEE, 2004).
[6] Ionescu, A. M. & Riel, H. Tunnel field-effect transistors as energy-efficient
electronic switches. Nature 479, 329–337 (2011).
[7] Buluta, I., Ashhab, S. & Nori, F. Natural and artificial atoms for quantum
computation. Reports on Progress in Physics 74, 104401 (2011).
[8] Nielsen, M. Quantum computation and quantum information (Cambridge
University Press, Cambridge New York, 2010).
[9] Neeley, M. et al. Generation of three-qubit entangled states using superconducting phase qubits. Nature 467, 570 (2010).
[10] DiVincenzo, D. P. The physical implementation of quantum computation.
Fortschritte der Physik 48, 771–783 (2000).
[11] Steger, M. et al. Quantum information storage for over 180s using donor
spins in a 28 Si “semiconductor vacuum”. Science 336, 1280–1283 (2012).
[12] Mourik, V. et al.
Signatures of majorana fermions in hybrid
superconductor-semiconductor nanowire devices. Science 336, 1003–1007
(2012).
2
Quantum electronics and transport
Silicon is a material that has some major advantages over other materials for
quantum electronics applications. First of all, due to the wide use of silicon
for conventional electronics applications, there is a large base of knowledge
available about the fabrication of devices as well as extremely high quality raw
material. Processing equipment and fabrication techniques for both electronic
and optical applications are readily available. Furthermore, there are some
more fundamental advantages to the choice for silicon, such as the possibility
to remove the nuclear spins from the silicon. These nuclear spins are disadvantageous for spin-based quantum applications because they cause the loss of
quantum information that is encoded in spin states. In this thesis transport
spectroscopy is used to study single donors in silicon. The transport is studied
in two ways: it is used as a spectroscopic tool to probe the electronic properties
of donors, and to study the transport of electrons.
In this chapter, the basics of electron transport at low temperature via quantum
states are outlined. First, the main features and fabrication of the devices that
serve as a platform are explained. Then the basic concepts of single electron
transport are discussed, followed by some examples of the effect of interference
and correlations on the transport. Finally, the properties of donors in the
context of donor-based quantum computation are highlighted.
Section 2.1 and 2.2 of this chapter have some overlap with the chapter titled “Single dopant
transport” in “Single-Atom Nanoelectronics” to be published 31st March 2013 by Pan Stanford Publishing, ISBN: 978-981-4316-31-6
9
10
2.1
W
Quantum electronics and transport
Devices and fabrication
ith single dopant transport spectroscopy at low temperature it has become possible to study the properties of individual dopants directly
[1–6]. This opportunity initially emerged as a natural consequence of the effort of the semiconductor industry to overcome the fundamental limitations
of classical device geometries. Very recently however, an atomic-scale fabrication technique using STM lithography on a surface passivated with atomic
hydrogen was successfully used to fabricate a single dopant transport device
[6]. Another important milestone in the study of single dopants is the read-out
and coherent manipulation of the electron spin bound to a single phosphorus
donor [7, 8]. In this thesis, the FinFET (Fin Field Effect Transistor) is used as
a platform to study single dopant transport, see Figure 2.1. This device was
developed by the semiconductor industry as an alternative to the conventional
planar field-effect transistor geometry. The fin-geometry of the device has
favorable properties that delay CMOS (Complementary Metal Oxide Semiconductor) scaling problems, such as short channel effects and drain induced
barrier lowering [9]. It has also enabled the investigation of the electron occupation of single donor atom [1], the excited state spectra [10], the binding
energy [5] and the effect of a nearby interface on the electron wave function.
[10–13].
a
z
gate
source
b
SiN Spacer
x
y
drain
200 nm
Poly-Si
Gate
SiO2 Gate Dielectric
Channel
source
drain
Buried SiO2
Si Substrate Wafer
Figure 2.1: a) SEM image of a FinFET device. b) A schematic source/drain cross-section
device.
The devices studied in this thesis are MOSFETs (Metal Oxide Semiconductor Field Effect Transistors) and have been fabricated at a CMOS foundry
on 200 mm wafers. The reason for this choice is two-fold. (i) It is more difficult to achieve the degree of control in fabrication processes in a smaller-scale
research cleanroom compared to a highly optimized and standardized foundry
2.1. Devices and fabrication
11
fabrication process. (ii) A large number of devices with small intentional variations in characteristics can be fabricated with high precision in a reproducible
way. There are, however, disadvantages to this choice as well. First of all,
it is often quite involved to modify individual fabrication steps as this may
require re-optimization of the process flow as a whole. Second, the turnover
time is relatively long. To fabricate one batch of wafers can take up to several
months, excluding the time spent on process development. Despite these difficulties, fabricating single dopant devices and single electron devices, even with
multiple gates, with a foundry-based CMOS process seems to be a promising
route toward large-scale integration of quantum devices [14]. The fabrication
process consists of many fabrication steps (>150) and therefore, instead of
describing the fabrication, the aspects important for single dopant transport
are outlined here. Also, some experimental techniques that give insight in the
mode of transport of nano-scale devices will be briefly discussed.
FinFET devices consist of a nano-wire that is etched out of a SOI film
(silicon on insulator film), see Figure 2.1b. The insulator consists of a buried
silicon dioxide that is about 150 nm thick for modern devices. Because of this
thin oxide, the substrate wafer can be used as a back gate at low temperature
[15]. A 5 nm thick silicon dioxide layer is electrically isolating the channel
from the polycrystalline silicon gate. This conservative choice in terms of gate
material, oxide material and thickness, was made to minimize gate leakage
and charge noise. For a nano-scale device with dimensions ≲ 60nm the gate
leakage is typically less than 10 pA with 1 Volt on the gates. The source and
drain contacts are formed by implanting the device with a high dose of arsenic
which is activated in a subsequent anneal process. During the source/drain
implantation, the channel region is protected by the gate and the silicon nitride
(SiN) spacers next to the gate, see Figure 2.1b. This results in an underlapping
structure with good gate control and low access resistance to the channel. The
term ‘underlap’ is used when the highly doped source and drain contacts do
not extend all the way under the gate, but there is a small portion of the
wire next to the gate that is left undoped (or has the same doping as the
channel). Especially at low temperature, the access regions determine the size
of the tunnel barrier to the part of the channel under the gate. Since the gate
control over the access regions is weak, large access regions will result in the
formation of a quantum dot (potential dip) centered between the source and
drain contacts that can confine electrons. Small access regions, on the other
hand, will result in a structure with a single tunnel barrier controlled by the
gate [16]. The latter structure is particularly useful to study single dopants.
To fabricate FinFET devices for single dopant transport, two strategies
have been shown to work, see Figure 2.2. One possibility is to make use of indiffusion of dopants from the source/drain contacts into the channel region. In
this case the transport is direct resonant tunneling from source via a dopant to
the drain [5] or mediated by a transport channel that is formed at the channel-
12
Quantum electronics and transport
a
c
Ionization energy (meV)
−108
100
−50
−24
0
0.7
0.6
50
0.5
Vd (mV)
b
g
s
0.3
0.2
50 nm
−50
0.1
d
300 nm
G ( e2/ h)
0.4
0
0
−100
g
−1.4
−1.2
−1.0
Vg (V)
−0.8
−0.6
−0.4
d
10−1
d
Ids ( µA)
s
electron energy
conduction band edge
Source Gate Drain
10−3
10−5
10−7
−2
30 nm
−1
0
Vg (V)
1
2
Figure 2.2: a) Stability diagram, showing Coulomb diamonds (white on this color scale)
that are distorted by the presence of a dopant in the barrier region. b) This dopant it not
carrying any current, but influences the Coulomb blockade of the quantum dot beneath the
gate. c) Direct transport can be achieved if the channel of the device is very short (see inset
of d) and is used, in this case, to determine the ionization energy. d) In these devices the
dopant is visible in the current versus gate voltage trace, even at room temperature. The
red and blue lines correspond to devices with dopants in the central region and the dashed
line to a device without. Images a) and b) are adapted from [17] and c) and d) from [5].
gate dielectric interface [1, 18]. Especially in devices with wide spacers, this
strategy very often leads to dopants that indirectly influence the transport.
This results in a modification of the tunnel barrier to the quantum dot formed
under the gate in a series configuration [19], or by modifying the spacing of the
Coulomb oscillations in a side-coupled configuration [17, 20], see for example
Figure 2.2a and b. Another strategy is to dope the silicon film before etching
the channel. This will result in randomly distributed dopants in the channel,
that can induce resonant tunneling to single dopants if the channel is short
enough. Fairly high doping concentrations will need to be used, because the
used devices typically have a gate length ≲ 60 nm. For example, a doping
concentration of 1 ⋅ 1017 cm−3 results in on average one dopant in a 22 × 22 × 22
nm3 cube and a concentration of 1 ⋅ 1018 cm−3 results in one dopant every
10 × 10 × 10 nm3 .
Low temperature sequential tunneling transport (explained in section 2.2)
13
2.1. Devices and fabrication
can be used to obtain information about the shape of the potential landscape.
This, however, probes only local properties as the transport will take place in
regions with the lowest potential. Instead, studying thermally assisted transport as a function of temperature has proven to be a very powerful method to
get insight in the various modes of operation of nano-scale FinFETs [18, 21, 22].
This technique has the advantage that it can be directly applied to a nanoscale FET device and no special test structures need to be used, as is very
often the case with other methods.
Eb [meV]
a
b
60
40
c
20
0
0
200
Vg [mV]
400
Figure 2.3: a) A thermionic measurement provides information about the barriers formed
in a device. b) The conductance versus gate voltage at small source/drain bias, c) and at
higher bias of a similar device, confirms the formation of residual barriers on both sides of
the gate. Images adapted from [16].
Thermally assisted transport, or thermionic transport, allows for the determination of the barrier height as well as the effective cross-section of the device
channel participating in the transport. With this technique, the barrier height
is almost directly related to temperature and gate voltage, and is therefore
straightforward to extract from experimental data alone. The active crosssection is more difficult to extract. Details of the band structure need to be
known to extract quantitative results. This is not straightforward, especially
for nano-scale devices, since the band structure is warped by the nano-scale
confinement [21]. Figure 2.3 shows an example of the barrier height extracted
by fitting conductance data versus gate voltage and temperature to an analytic
formula [18, 23]. This clearly demonstrates that increasing the gate voltage
first lowers the barriers in a linear fashion, red line in Figure 2.3a. But at larger
gate voltages residual barriers are formed that are more or less constant for
Vg > 0.1 V. These barriers are formed in the access regions under the spacers
which are only weakly influenced by the gate. Figure 2.3b and c confirm this
scenario. Clear Coulomb blockade due to electrons confined in the channel
under the gate is observed in this regime, see also section 2.2.
Much progress has been made in recent years with the fabrication of devices in CMOS processes as well as the understanding of microscopic device
properties. Single dopant transport and, very recently, also transport through
14
Quantum electronics and transport
a system of tunnel-coupled dopants [24], has become reasonably straightforward. Also, the fabrication of devices with multiple gates in which coupled
quantum dots can be formed has been demonstrated with this technology [25].
Beside low temperature transport, thermionic transport has proven to be a
useful tool to study nano-scale devices directly. In the next section transport
spectroscopy of single donors in FinFET devices will be discussed.
2.2
Transport spectroscopy of single donors
To explain the basic ideas of transport spectroscopy at low temperature, electronic transport through quantum dots is discussed. Electronic transport
through quantum dots is very similar to transport through dopants and many
concepts can be transferred to single dopant transport. There are, however,
some important differences, such as the nature of the confining potential, which
causes deviations from the constant interaction model, commonly used to understand transport through quantum dots. Therefore, before single electron
transport in general is explained, some of these differences are discussed. For a
more complete overview of transport in quantum dots please refer to Kouwenhoven et al. [26] or Hanson et al. [27].
2.2.1 Single donor transport
Single donor transport can draw on a large number of ideas developed in the
context of transport through quantum dots in metals and semi-conductors [28].
The most important difference, however, is the nature of the confining potential. A quantum dot is formed by gates that define the potential landscape
and therefore the size of the dot. Due to variations in the fabrication, each
quantum dot will therefore be different and the electrical characteristics vary.
Dopants, on the other hand, are all identical since the confining potential is
defined by the charged nucleus, see Figure 2.4. When the donor is ionized, it
is customary to denoted it as the D+ state, the neutral state of the system,
the nucleus with one electron bound to it, is called the D0 and the charged
two-electron state D− . A bulk donor cannot bind more than two electrons.
The imperfect screening of the nuclear charge by the first electron allows the
binding of a second electron [29] with a binding energy of a few meV, but
excited states are unbound with respect to the conduction band minimum.
This is another important difference between donors and quantum dots. Depending on the size and strength of the confining potential, quantum dots can
accommodate up to several thousands of electrons.
A consequence of the two-electron system being confined in a small region is
that the energy difference between the D0 and D− state, the charging energy, is
relatively large. This is advantageous for device applications involving the D0
state, because it makes this state relatively stable with respect to changes in the
local chemical potential that may be induced with a nearby gate. It also allows
such a device to operate at temperatures ≲4 K, while for quantum dots often
2.2. Transport spectroscopy of . . .
a
15
b
+
dopant
quantum dot
Figure 2.4: The confining potential and single particle excitation spectrum of a) a dopant
and b) a typical quantum dot.
lower temperatures are required. Another advantage for high temperature
operation is the large splitting between the ground state and the first excited
state, 11.7 meV for phosphorus and 21.1 meV for arsenic [30]. This large
splitting provides a well isolated ground state in which the electron spin can
be used to encode quantum information. First of all the quantum information
encoded in the spin of the ground state will not very easily be lost due to
thermal excitation to one of the excited states. In addition, a strong microwave
field at high frequencies can be applied to a spin in the orbital ground state
without inducing transitions to excited states.
It should be noted however, that if electrical addressing of single dopants
is required, gates need to be placed in close vicinity of the dopants. This
affects many of the above mentioned favorable characteristics. In chapter 3
the electronic structure of confined donors is discussed. In nano-structures
that are used to get access to single donors, electric fields and the presence of
metallic gates and dielectric interfaces generally decrease the splitting between
excited states, and may change the binding energy [12, 13].
2.2.2 Single electron transport
Single electron transport spectroscopy relies on the Coulomb blockade effect.
The Coulomb blockade effect is a natural consequence of the fact that electrons
repel each other because of their equal charge. When a region is connected
to metal contacts with a small enough conductance, this leads to quantized
charging. A small conductance in this context means smaller than the conductance quantum G0 = e2 /h = 38.6 µS or, equivalently, a resistance larger than
25.9 kΩ [28]. A three-terminal configuration is the simplest way to change
the potential independently from the chemical potential of the contacts. For
an accessible overview of single electron tunneling please refer to Thijssen and
van der Zant [31].
A simple model to describe single electron charging is the constant inter-
16
Quantum electronics and transport
action model. In this model the energy to charge the system with exactly one
additional electron is given by:
e2
(2.1)
C
where C is the total capacitive coupling to the charged region. The most significant contributions to the total capacitance are usually assumed to be the
geometric capacitance due to the presence of gates to change the electrostatic
potential and source and drain electrodes to allow charge transport, thus the
total capacitance is given by C = Cg + Cs + Cd , with Cs , Cd and Cg the source,
drain and gate capacitances, see Figure 2.5a. Note that the charging energy is
assumed to be independent of the number of electrons N within this approximation. An additional requirement for charge transport is that the source
and drain are tunnel coupled to the localized state. The electric gate is used
to change the potential of the confined region by applying a gate voltage Vg
and thereby increasing or decreasing the number of electrons of the ground
state. These electrons are supplied by the source and drain contacts. Besides
the capacitances, Cs,d , the source and drain contacts are characterized by a
tunnel coupling, Γs,d , see Figure 2.5b. Because of the capacitive coupling,
the potential of the source and drain contacts will also influence the potential
of the confined region. Therefore, this can also cause a change in the number of electrons. A stability diagram is a convenient way to plot transport
spectroscopy data, see Figure 2.5c. Electron transport is allowed in the grey
regions where electrons tunnel sequentially. In the diamond-shaped regions
(white) the number of electrons is stable and sequential electron tunneling is
forbidden due to Coulomb blockade.
EC =
a
b
Vsd
EC, Ex
Cs, Γs
Vsd
Cd, Γd
Cg
Vg
0
Id
N=0
N=1
N=2
Ex
EC
Vg
Figure 2.5: a) Circuit of a quantum dot or dopant (open circle) tunnel coupled to a source
and drain and just capacitively coupled to a gate. b) A stability diagram shows Coulomb
blockade and can be used to extract values for the charging energy EC (see text) and the
excited state energy Ex .
Sequential transport of electrons can be conveniently described using rate
2.2. Transport spectroscopy of . . .
17
equations. Sequential tunneling considers the individual tunnel events as being
uncorrelated, once the electron has tunneled across a barrier, it will not “remember” anything of its past. Within this framework, obtaining expressions
or simulating transport is straightforward using the master equation approach.
To calculate steady state transport (DC), the procedure is to first compute the
mean occupation of the localized state and then calculate the stationary current. The rate at which the occupation of a state coupled to two leads changes
is given by:
dρo
+ Γid,out )ρo
)ρe − (Γs,out
+ Γd,in
= ∑(Γs,in
i
i
i
dt
i
(2.2)
with Γin,out
the tunnel rate in and out of the ith state, and ρo and ρe are the
i
total probability of finding the system respectively occupied or empty. Once ρo
(= 1 − ρe ) is known, the current can be calculated by considering the transport
across one of the barriers on the left or right (s or d), see Figure 2.5a. The
current across the left barrier (source side), given by:
s,out
I(t) = Is (t) = e ∑ [Γs,in
ρo (t)]
i ρe (t) − Γi
(2.3)
i
with e one electron charge. Note that in a steady state situation this is equal
to the current across the right barrier (drain side). For simple systems in a
steady state, i.e. dρo /dt = 0, this can be done analytically. One useful example
is a simple system with Coulomb blockade and thermally occupied leads. In
this case, the current at small source/drain bias voltage, Vsd , and transport
through a single quantum state, develops resonance given by [32]:
I = Vsd
e2 Γs Γd
cosh−2 (αVg′ /2kT )
4kT Γs + Γd
(2.4)
with Vg′ the detuning with respect to the level and α the conversion factor
from gate voltage to energy given by eCg /C, also referred to as the capacitive
coupling. This equation is valid when a single quantum level is occupied and
the thermal energy, kT , is larger than the level spacing (explained below)
and eVsd , but much smaller than the charging energy EC . The characteristic
temperature dependence of the resonance height and the width can be used
to prove that the transport is mediated by a single quantum level only [33].
Equation 2.4 was derived under the assumption that only a single level (the
ground state) contributes to the transport. If excited states also take part,
single electron tunneling exhibits some qualitative differences [33].
Confinement of electrons in a small region results in discrete quantum
states with a well-defined energy. This is typically the case for dopant states
and small quantum dots in semiconductors. An electron tunneling onto such
a system will have to occupy one of these states. As a consequence of the discrete excitation spectrum, the sequential current shows structure as a function
18
Quantum electronics and transport
of the voltages on the contacts. When a quantum level becomes accessible
and contributes the current, the current changes in a stepwise manner. This
enables the determination of the excited state spectrum of localized states of
dopants or quantum dots. For that purpose, it is most of the time convenient
to represent the spectroscopy data as the differential conductance, G, which
is the derivative of the current with respect to the source/drain bias voltage,
dId /dVsd . Figure 2.5c schematically illustrates how the excited states show up
in the transport spectroscopy data of a single donor embedded in a nano-scale
transistor. At the finite experimental temperature, the thermal occupation of
the leads limits the resolution of transport spectroscopy. It broadens the transitions in current, or differential conductance resonances, by the characteristic
energy kT , because the thermal environment supplies or absorbs energy to or
from the tunneling electrons.
Up till now, only sequential tunneling has been discussed. In the model
of sequential tunneling, individual tunneling events are uncorrelated and follow Poissonian statistics. In general, this means that there is no coherence.
Even though the sequential tunneling model gives good insight, very often its
approximations are not sufficient to explain certain features of the transport.
In certain cases electrons can interfere in a way similar to a double slit interference experiment, or form collective states that mediate transport. In the
following section three examples of this will be discussed, the Fano effect for
electron transport, the Aharonov-Bohm effect and the Kondo effect.
2.2.3 Interference effects and correlations
Fano resonances can be observed even when interactions are not important.
In that case electrons interfere with themselves. The symmetric Breit-Wigner
or Lorentzian resonance can be considered a special case of a Fano resonance,
with a large so-called Fano factor [34], discussed in more detail below. Mesoscopic systems that are likely to exhibit a Fano-type resonance, must have a
transport channel with a rapidly varying phase and at least one other channel
with a slow phase variation. For example, an open channel with a side-coupled
localized state [35] or a system with one open channel in parallel with a resonant channel, see Figure 2.6, can show Fano-type transport. Theoretically,
Fano-like transport can be obtained if the interference between an open nonresonant channel and a resonant channel is considered, where tn and tr are the
non-resonant and resonant transmission amplitudes respectively,
2
∣tn + tr ∣2 = ∣βe−iθ + γ
iΓ/2
∣ .
iΓ/2 + (2.5)
Here, θ is the acquired phase, Γ is the decay rate of the resonance and the
detuning of the resonance from the Fermi energy, β and γ are the respective
transmission amplitudes of the non-resonant and resonant paths. The acquired
phase, θ, can be just fixed by the geometry or can be tunable by, for example,
2.2. Transport spectroscopy of . . .
19
the Aharonov-Bohm effect as discussed below. The decay rate depends on the
tunnel coupling of the localized state that induces the resonance (rapid phase
change). Depending on this coupling the degree of hybridization of localized
state with the continuum environment, broadens the resonance and decreases
the life time.
a
c
4
3
b
2
B
1
0
-5
-2.5
0
2.5
5
Figure 2.6: Examples of systems in which a Fano effect can be observed are a a side-coupled
localized state and b) a localized state embedded in an Aharonov-Bohm ring arrangement
coupled to leads. The black arrows indicate the possible paths (only to first order) from the
left to the right lead. c) A plot of the Fano-like behavior of Eq. 2.5 for different values of θ
with β = γ = Γ = 1.
The phase across the resonance changes rapidly by π, over a range ∼ Γ.
Note that, even though this has not been explicitly taken into account in Eq.
2.5, also electrons through the resonant path may acquire a geometrical phase.
This, however, results in just an overall phase offset and results in the same
resonance. Figure 2.6 shows plots of Eq. 2.5 for different values of θ. The most
commonly used form of the of Fano formula for a general complex parameter
q is:
∣ + qΓ/2∣2
(2.6)
∣t()∣2 =
(Γ/2)2 + 2
This is a phenomenological form and detailed modeling is in general required
to related q to physical properties of the system. Even though Eq. 2.5 cannot
be derived exactly from Eq. 2.6, the role of tan(θ) in Eq. 2.5 is similar to q
in the standard form Eq. 2.6. For a Fano effect it is important that electrons
preserve coherence while traversing the structure to interfere with themselves
or with other electrons. Therefore, the observation of a Fano resonance is
direct evidence that coherence is preserved. As mentioned before, apart from
the geometrical phase electrons acquire while traversing the structure, the
phase can be tuned by the Aharanov-Bohm (AB) effect [36].
20
Quantum electronics and transport
Electrons traversing a loop with magnetic field, B, applied perpendicular
to the loop, will acquire a phase which is proportional to the applied magnetic
field. The phase difference for electrons with the same start and end points
will have acquired a phase given by [37]:
e
A⋅B
∆ϕ = A ⋅ B ̵ = 2π
h
Φ0
(2.7)
where A is the area enclosed by both paths and Φ0 = e/h the flux quantum.
The conventional manifestation of the AB effect in a mesoscopic ring is the sinusoidal modulation of the current as a function of magnetic field. Embedding
a system, such as a dopant or quantum dot, in an AB ring geometry is one
of the most direct methods available to study coherent transport phenomena.
Such a system allows to monitor the phase of electrons as they interfere with
themselves or with other electrons. And local electrical gates on the embedded system allow for the investigation of the energy spectrum. Tuning the
magnetic field changes the phase through the AB effect, see from Eq. 2.7. Besides single electron coherence, coherence can also be carried by a correlated
many-body electron state. Many-body effects in electron transport arise at
low temperature due to interactions, as a result of a reduced disturbance by
the thermal environment. The Kondo effect, discussed below, is an example
of such a correlation effect mediated by virtual intermediate states.
Virtual intermediate states can mediate transport in the Coulomb blockade
region of dopants or quantum dots and generate considerable current. Intuitively, tunneling via virtual states can be understood as the electron allowing
to gain energy ∆E for a short time ∆t while still satisfying the Heisenberg
̵
uncertainty relation ∆E∆t ≳ h/2.
Here, the time ∆t is set by the tunnel rate
Γs,d ≈ 1/∆t and ∆E is the energy needed to occupy the virtual intermediate
state. In order to preserve energy, these processes involve simultaneous tunneling of multiple electrons, and the tunnel probability is therefore higher order
in the tunnel rate. For this reason these tunnel prosesses are called cotunneling processes. Only for systems that have a relatively large tunnel coupling
to both leads higher order tunneling processes can generate a net measurable
current. In the following, the main characteristics of a higher order tunneling
effect, the Kondo effect, will be briefly discussed. An overview of transport
regimes in the Coulomb blockage region is published by Pustilnik & Glazman
[38].
Traditionally, the Kondo effect was observed in experiments on metals with
magnetic impurities. It was found that the resistance unexpectedly increased
below a critical temperature [39]. In transport through quantum dots [40]
and donors [41] on the other hand, the Kondo effect enhances the transport
in the Coulomb blockade regime. It emerges as a resonance in differential
conductance around zero source/drain bias, see Figure 2.7. The Kondo effect
can be seen as the coherent addition of elastic cotunneling processes, see Figure
2.2. Transport spectroscopy of . . .
21
2.7a. These particluar tunneling processes involve a spin-flip and thus leave the
system in a different spin state. Collectively, these processes form a transport
channel due to the formation of a many-body state in which electrons in the
contacts screen the magnetic moment of a electron localized on the quantum
dot or donor and form a spin singlet state.
dI/dV (μS)
4
b
VSD (mV)
a
-6
-4
-2
0
2
4
6
3
2
1
360
380
400
VG (mV)
420
440
0
Figure 2.7: a) The Kondo can be seen as a many-electron spin singlet state that forms
between the localized electron spin and the spins in the leads. b) This can mediate transport
in the Coulomb blockade regime and shows up as a resonance at zero source/drain bias
indicated by the arrow. Image b adapted from [41]
Characteristics that are generally seen as the experimental prove of a Kondo
effect are [40]: (i) a resonance in differential conductance at zero bias, (ii) the
logarithmic increase of the zero bias conductance below the Kondo temperature
TK and (iii) the splitting of the zero bias resonance by Vsd = ±gµB B in a
magnetic field B. Because the energy difference equal to the Zeeman energy
has to be supplied at finite magnetic field, the Kondo resonance splits in two
resonances at finite bias. When the Zeeman energy exceeds the width of the
Kondo resonance, kTK , the current is quenched completely. Since the binding
energy of the spin singlet Kondo ground state is ∼ kTK , thermal fluctuations
will destroy the Kondo state for T > TK . As the temperature is lowered below
the Kondo temperature, coherence increases and the ground state forms until
it saturates as T → 0. This zero-temperature regime is called the unitary limit
of the Kondo effect.
For quantum device applications tunable interactions between states are
necessary to manipulate the quantum information that is encoded in them.
At the same time it is important that the coupling of the system to a thermal
environment is weak as this can cause the decoherence of the quantum state.
In the next section a brief overview of the usefulness of donors for quantum
device applications are given. To illustrate the current status of the field, some
examples of coherent manipulation of donor states are discussed.
22
2.3
Quantum electronics and transport
Single donors as qubits
A quantum bit, or qubit, takes advantage of the fact that quantum states can
be built from arbitrary superpositions of states that can be entangled. The
vast majority of information is nowadays stored in the form of classical bits,
0s and 1s, in some physical system. This is convenient because many systems
in nature can easily be made bi-stable and information can be processed using simple switches that switch ‘on’ and ‘off’. There is however a large base
of knowledge being build up on how a quantum system may be able to perform some tasks in a much more efficient way than ever can be done using
a classical system. Some first implementations have been demonstrated on a
small scale in various physical systems [42]. Promising candidate systems for
a practically useful quantum computer are for example real atoms or ions (in
the gas phase), superconducting devices, semi-conductor quantum dots and
donor spins in the solid state. Below, first a brief explanation of a qubit and
a prototypical implementation of a quantum computer as proposed in 1998 by
Kane will be given. Then two examples of coherent manipulation are highlighted; manipulation of donor spins, and coherent manipulation donor-bound
charge. Finally, an overview of work in the following chapters, which aids to
the advance of the implementation of a donor-based quantum computer, is
given. Please refer to the review by Buluta et al. [42] for a comprehensive
overview of physical implementations of qubits.
In a qubit information is stored in two basis states ∣0⟩ and ∣1⟩ by bringing the qubit in a superposition of these two states, α∣0⟩ + β∣1⟩. Here, the
normalization condition ∣α∣2 + ∣β∣2 = 1 must be satisfied at all times. Such
a state can be visualized in a convenient way by a unit vector directed to a
point on the surface of a sphere, called the Bloch sphere, see Figure 2.8a. By
performing rotations about the origin, for example about the x-, y- and z-axis,
any arbitrary linear combination can be made. A computational task can be
performed by initializing multiple qubits in some initial state, then performing
some rotations on individual qubits that may be dependent on the state of the
other qubits and eventually read off the outcome [43]. During this process a
fundamental requirement is that the coherence of the qubits is preserved [43].
Figure 2.8b shows the signal of coherent rotations of the nuclear spin associated with a phosphorus donor in a silicon crystal. Donors (in silicon) can be
used to build a quantum computer in so-called Kane quantum computers [46].
The physical implementation of the Kane quantum computer is schematically
illustrated in Figure 2.9. Kane envisioned that donor-bound electrons can be
used to mediate the interaction between the donor nuclear spins [46]. Using
electric gates to manipulate the donor-bound electrons located just beneath
the surface in combination with a global microwave field and a static magnetic
field, quantum computations can be performed, see Figure 2.9b. The electric
gates are used to switch the interactions between individual donors as well as
to control the interactions between the electron spin and the donor nuclear
23
2.3. Single donors as qubits
a
b
10
B1
M
|1
ESE Intensity (au)
|0
y
5
x
0
-5
in-phase (X)
quadrature (Y)
-10
0
100
200
300
Time (ns)
400
500
Figure 2.8: a) A point on the Bloch sphere is a convenient way to visualize arbitrary
rotations of the state of a qubit. Image adapted from [44]. b) Coherent rotations of the
phosphorus-31 nuclear spin embedded in a silicon crystal about the z-axis, as apparent from
the time dependence of the projections of the qubit state on the x- and y-axis. Image taken
from [45]
spin [44]. The read-out is performed by charge sensing using a spin-to-charge
conversion process [8, 47].
a
b Donor architecture
Donors
Si conduction band
Control ga tes
SiO 2
31 P
R ead-out
e–
|1
e–
31 P
|0
20 nm
e–
31 P
e–
31 P
e–
Si
31 P
Figure 2.9: a) The nuclear-electron spin system of a phosphorus donor (blue and red
arrows respectively) forms a natural basis for a qubit with extremely long coherence times.
b) Placing several dopants close to an interface to allow electrical manipulation of the donorbound electron is the prototypical architecture of a donor-based quantum computer. Images
adapted from [44]
A spin-1/2 system, which has only two eigenstates, such as the spin of a
single electron and the nuclear spin of phosphorus-31 atom, form a natural
well-defined state space for a qubit. Silicon has further the advantage that
it can be purified to one of the purest materials that exists, even on isotopic
level [48]. A typical wafer on which semiconductor devices are fabricated has a
purity better than 10−9 . Because the spin states are so well isolated, the spin
state will coherently evolve in a predictable manner for seconds for an electrons
24
Quantum electronics and transport
spin [49] and up to several minutes for a nuclear spin state of a donor [50].
To create well separated spin states, a static magnetic field is applied, see
Figure 2.9a. Using a resonant microwave excitation, interactions between the
four spin states of the electron-nucleus system can be switched on and off. By
switching the interaction on for only a certain amount of time and varying
the phase of the microwave field, coherent rotations on the Bloch sphere can
be performed. Morton et al. have demonstrated a nuclear spin memory in a
phosphorus doped silicon crystal using these techniques [51].
There is no complete consensus yet on what is required from a physical system to be suitable to be practically useful for quantum computation, but one
generally accepted requirement is that at least around 104 – 106 operation have
to be performed within the time the coherence of the complete system is preserved [52, 53]. This can be achieved by choosing a system with an extremely
long coherence time, such as donor spins, or by performing the rotations sufficiently fast. Next, a qubit is discussed where much faster manipulation of the
states is possible than for a donor-spin based qubit.
Using electrical pulses to tune the potential difference of two donors that
form a donor molecule, Dupont-Ferrier et al. show that the wave function of
a single electron can be manipulated [54], see Figure 2.10. The donors used
here are arsenic donors, a shallow donor in silicon, embedded in a nano-wire
transistor that allows electrical gating. When the donor ground-states are
tuned in resonance with each other, the one-electron ground state is a linear
combination of the states on the left and right donor, labeled ∣L⟩ and ∣R⟩. By
rapidly tuning the energy of state ∣L⟩ by a certain amount below ∣R⟩ and vice
versa, the charge density alternates between being located on the left and right,
while preserving the coherence between ∣L⟩ and ∣R⟩. Changing the amount of
detuning results in oscillations in the measured current through the system.
This principle is called Landau-Zener-Stuckelberg interferometry. Please refer
to [55] for a more complete overview of experiments and a detailed explanation
of the principle.
The main significance of the charge manipulation experiment by DupontFerrier et al. as discussed above, lies in the fact that these manipulation can,
in principle, be performed extremely fast by using shallow donors in silicon.
In contrast to the case of donor spins, where the weak coupling to the environment helps to preserve coherence but limits the manipulation speed, electrical
charge control can take advantage of a much stronger coupling. The price
to pay for the stronger coupling is that the coherence time is much shorter,
measured as only 0.3 ns for the system in Figure 2.10. Manipulation speeds
can in principle be much larger. Because the ground state is well isolated from
the first excited state for shallow donors in silicon, excitation pulses with harmonic components up to a frequency of ∼ 1 THz can be used. This frequency
corresponds to a splitting of ∼ 4 meV as observed in the experiment discussed
here. The manipulation speed exceeds the speed at which donor spins can
25
2.3. Single donors as qubits
a
b
μs
Excited state
Ground state
D
D
μd
Figure 2.10: a) A double donor system coupled to source and drain reservoirs can act a
two-level system that allows for extremely fast electrical manipulation. The excited state is
never occupied during the manipulation described in the text. b) Oscillations in the current
reveal that rapid manipulation of the donor levels results in the shift of the charge density
of the one-electron state from left to right and back as a function of detuning and pulse
height. Images adapted from: [54]
be manipulated by many orders of magnitude. It should be noted however
that despite the speed, only a couple of hundred rotations can be performed
within the coherence time and therefore this needs to extended, or instead, a
larger splitting is necessary. But encouraging prospects are offered by the fact
that, for example, the bulk splitting for phosphorus is known to be 11.67 meV
[30], and a value of 10 meV has been observed for a gated phosphorus donor
[24]. This should allow to use frequencies up to ∼ 12 THz. With the measured
coherence time of 0.3 ns, about 1,000 manipulations can be performed, still
below the 104 operations typically required for a useful qubit. Therefore, despite the fast manipulation, there is the need to extent the coherence time of
this system. This experiment is a nice illustration of fast coherent rotations
and provides a powerful tool to study the dynamics of the double donor twolevel system. However, swapping the states ∣L⟩ and ∣R⟩ only corresponds to a
rotation about x (or y), and is therefore by itself not directly useful to perform
arbitrary rotations on the Bloch sphere.
Relevance of the work in this thesis to quantum computation – Instead of
directly presenting qubit implementations, this thesis describes several results
that may add to the understanding of donor-based quantum device applications. Chapter 3 and 4 discuss how the one-electron D0 states (ground state
and excited states) of donors are modified when they are embedded in a nanostructure. The ability to gate donors, and thereby adiabatically tune the
electron wave function, is an essential aspect of a scalable quantum device
architecture. Chapter 5 is concerned with the tunability of the ground state,
26
Quantum electronics and transport
but focusses on a different aspect relevant for quantum computation: namely,
how the ground state is affected by the band structure of silicon and how
new quantum numbers occur for donor atoms in silicon as compared to an
atom in vacuum. Chapter 6 and 7 are concerned with coherent interactions of
donor-bound electrons with their environment. Exploring these interactions
may result in finding mechanisms useful for quantum computation as well as
a better understanding of effects that can cause decoherence when an isolated
system couples to the outside world.
References
[1] Sellier, H., Lansbergen, G. P., Caro, J. & Rogge, S. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett.
97, 206805 (2006).
[2] Calvet, L., Wheeler, R. & Reed, M. Observation of the linear stark effect
in a single acceptor in Si. Phys. Rev. Lett. 98, 096805 (2007).
[3] Calvet, L., Wernsdorfer, W., Snyder, J. & Reed, M. Transport spectroscopy of single pt impurities in silicon using schottky barrier mosfets.
J. Phys.: Condens. Matter 20, 374125 (2008).
[4] Tan, K. Y. et al. Transport spectroscopy of single phosphorus donors in
a silicon nanoscale transistor. Nano Lett. 10, 11–15 (2010).
[5] Pierre, M. et al. Single-donor ionization energies in a nanoscale CMOS
channel. Nature Nanotechnology 5, 133–137 (2010).
[6] Fuechsle, M. et al. A single-atom transistor. Nature Nanotechnology 7,
242 (2012).
[7] Morello, A. et al. Single-shot readout of an electron spin in silicon. Nature
467, 687 (2010).
[8] Pla, J. J. et al. A single-atom electron spin qubit in silicon. Nature 489,
541 (2012).
[9] Doyle, B. et al. High performance fully-depleted tri-gate CMOS transistors. Electron Device Letters, IEEE 24, 263–265 (2003).
[10] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[11] Rahman, R. et al. Orbital stark effect and quantum confinement transition of donors in silicon. Phys. Rev. B 80, 165314 (2009).
References
27
[12] Calderon, M. J. et al. Heterointerface effects on the charging energy of
the shallow D− ground state in silicon: Role of dielectric mismatch. Phys.
Rev. B 82, 075317 (2010).
[13] Rahman, R. et al. Electric field reduced charging energies and twoelectron bound excited states of single donors in silicon. Phys. Rev. B
84, 115428 (2011).
[14] Sanquer, M. et al. Single dopant and single electron effects in CMOS
devices. In Nazarov, A. et al. (eds.) Semiconductor-On-Insulator Materials for Nanoelectronics Applications, Engineering Materials, 251–263
(Springer Berlin Heidelberg, 2011).
[15] Roche, B. et al. A tunable, dual mode field-effect or single electron transistor. Appl. Phys. Lett. 100, 032107 (2012).
[16] Wacquez, R. et al. Single dopant impact on electrical characteristics of
SOI NMOSFETs with effective length down to 10nm. VLSI Technology
(VLSIT), 2010 Symposium on 193–194 (2010).
[17] Hofheinz, M. et al. Individual charge traps in silicon nanowires. The
European Physical Journal B-Condensed Matter and Complex Systems
54, 299–307 (2006).
[18] Sellier, H. et al. Subthreshold channels at the edges of nanoscale triplegate silicon transistors. Appl. Phys. Lett. 90, 073502 (2007).
[19] Golovach, V. et al. Single-dopant resonance in a single-electron transistor.
Phys. Rev. B 83, 075401 (2011).
[20] Pierre, M. et al. Background charges and quantum effects in quantum
dots transport spectroscopy. Eur. Phys. J. B 70, 475–481 (2009).
[21] Tettamanzi, G. C. et al. Thermionic emission as a tool to study transport
in undoped nfinfets. IEEE Electron Device Letters 31, 150–152 (2010).
[22] Tettamanzi, G. et al. Interface trap density metrology of state-ofthe-art undoped Si n-FinFETs. Electron Device Letters, IEEE DOI 10.1109/LED.2011.2106150 32, 440–442 (2011).
[23] Sze, S. & Ng, K. K. Physics of Semiconductor Devices (John Wiley &
Sons, 2007), 3rd edn.
[24] Roche, B. et al. Detection of a large valley-orbit splitting in silicon with
two-donor spectroscopy. Phys. Rev. Lett. 108, 206812 (2012).
[25] Pierre, M. et al. Compact silicon double and triple dots realized with only
two gates. Appl. Phys. Lett. 95, 242107 (2009).
28
Quantum electronics and transport
[26] Kouwenhoven, L., Austing, D. & Tarucha, S. Few-electron quantum dots.
Reports on Progress in Physics 64, 701 (2001).
[27] Hanson, R., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in
few-electron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).
[28] Kouwenhoven, L. et al. Electron transport in quantum dots. Mesoscopic
Electron Transport 345, 16–23 (1997).
[29] Norton, P. Formation of the upper hubbard band from negative-donor-ion
states in silicon. Phys. Rev. Lett. 37, 164 (1976).
[30] Ramdas, A. & Rodriguez, S. Spectroscopy of the solid-state analogues of
the hydrogen atom: donors and acceptors in semiconductors. Reports on
Progress in Physics 44, 1297 (1981).
[31] Thijssen, J. M. & Zant, H. V. D. Charge transport and single-electron effects in nanoscale systems. Phys. Status Solidi (b) 245, 1455–1470 (2008).
[32] Beenakker, C. Theory of coulomb-blockade oscillations in the conductance
of a quantum dot. Phys. Rev. B 44, 1646–1656 (1991).
[33] Foxman, E. et al. Crossover from single-level to multilevel transport in
artificial atoms. Phys. Rev. B 50, 14193–14199 (1994).
[34] Miroshnichenko, A., Flach, S. & Kivshar, Y. Fano resonances in nanoscale
structures. Rev. Mod. Phys. 82, 2257–2298 (2010).
[35] Sasaki, S., Tamura, H., Akazaki, T. & Fujisawa, T. Fano-Kondo interplay
in a side-coupled double quantum dot. Phys. Rev. Lett. 103, 266806
(2009).
[36] Yacoby, A., Heiblum, H., Mahalu, D. & Shtrikman, H. Coherence and
phase sensitive measurements in a quantum dot. Phys. Rev. Lett. 74,
4047–4050 (1995).
[37] Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in
the quantum theory. Physical Review 115, 485–491 (1959).
[38] Pustilnik, M. & Glazman, L. Kondo effect in quantum dots. J. Phys.:
Condens. Matter 16, R513 (2004).
[39] Kondo, J. Resistance minimum in dilute magnetic alloys. Prog. Theor.
Phys. 32, 37–49 (1964).
[40] Grobis, M., Rau, I., Potok, R. & Goldhaber-Gordon, D. The kondo
effect in mesoscopic quantum dots. Handbook of Magnetism and Advanced
Magnetic Materials (2007).
References
29
[41] Lansbergen, G. P. et al. Tunable kondo effect in a single donor atom.
Nano Lett. 10, 455–460 (2010).
[42] Buluta, I., Ashhab, S. & Nori, F. Natural and artificial atoms for quantum
computation. Reports on Progress in Physics 74, 104401 (2011).
[43] Nielsen, M. Quantum computation and quantum information (Cambridge
University Press, Cambridge New York, 2010).
[44] Morton, J., McCamey, D., Eriksson, M. & Lyon, S. Embracing the quantum limit in silicon computing. Nature 479, 345–353 (2011).
[45] Morton, J. J. L. A silicon-based cluster state quantum computer. arXiv
quant-ph (2009). 0905.4008v1.
[46] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[47] Elzerman, J. M. et al. Single-shot read-out of an individual electron spin
in a quantum dot. Nature 430, 431 (2004).
[48] Becker, P., Pohl, H.-J., Riemann, H. & Abrosimov, N. Enrichment of
silicon for a better kilogram. Phys. Status Solidi (a) 207, 49–66 (2010).
[49] Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in
high-purity silicon. Nature Materials 11, 143 (2011).
[50] Steger, M. et al. Quantum information storage for over 180s using donor
spins in a 28 Si “semiconductor vacuum”. Science 336, 1280–1283 (2012).
[51] Morton, J. J. L. et al. Solid-state quantum memory using the 31 P nuclear
spin. Nature 455, 1085 (2008).
[52] Shor, P. Scheme for reducing decoherence in quantum computer memory.
Phys. Rev. A 52, 2493–2496 (1995).
[53] Niwa, J., Matsumoto, K. & Imai, H. Simulating the effects of quantum
error-correction schemes. arXiv quant-ph (2002). quant-ph/0211071v1.
[54] Dupont-Ferrier, E. et al. Coupling and coherent electrical control of two
dopants in a silicon nanowire. arXiv cond-mat.mes-hall (2012). 1207.
1884v1.
[55] Shevchenko, S. N., Ashhab, S. & Nori, F. Landau–zener–stückelberg
interferometry. Physics Reports 492, 1–30 (2010).
3
Donor states in nanostructures
The effective mass model for donors, corrected for the influence of the band
structure of silicon, yields a reasonably accurate description and captures the
important physics. In this chapter an overview of a procedure to obtain the
donor states within this formalism is presented. This overview serves as an
insight in the rich and versatile structure of donor atoms in bulk silicon and
near interfaces. As an example, the consequences for the two-electron state
bound to a donor-interface system are discussed.
The results presented in section 3.4 of this chapter have been published in
Physical Review Letters 107, 136602, 2011
31
32
3.1
D
Donor states in nanostructures
The hydrogenic model
opant atoms in a crystal can accept or donate electrons to the host lattice
and thereby change the crystal’s electrical properties. Because silicon
has four valence electrons, atoms from group III in the periodic table (with
three valence electrons) that substitute a silicon atom are electron accepting
species. Group V (five valence electrons) atoms are donating species when they
substitute a silicon atom. An ionized dopant can be modeled as a point charge
in an environment with the dielectric constant of the host material. Most
electrical properties can be described by a model covering an energy range
around the conduction band minimum for donors and around the valence band
maximum for acceptors. The simplest model is a scaled hydrogen model [1].
This model turns out to be accurately predicting the binding energy of some
acceptor states, such as boron in silicon, but fails to produce the spectrum
of donor states accurately. The reason for this is that the conduction-band
minima, also called a conduction-band valleys, are located at finite crystal
̵
momentum, h∣k∣,
in the band diagram [2]. To obtain a better approximation
of the donor spectrum, the orbits of the hydrogen model in silicon can be
corrected by a so-called valley-orbit correction.
This chapter describes the scaled hydrogen model and the basic ideas of
the valley orbit as developed by Kohn and Lüttinger [1]. For a complete
overview of the assumptions underlying this effective mass approach please
refer to Pantelides and Sah [3] or for a comprehensive review about dopants
in silicon to Ramdas and Rodgriges [4].
Figure 3.1: The charge of an electron bound to a dopant is distributed over many atoms
of the lattice. Therefore, the dielectric function can be considered constant, and a hydrogenic effective mass model predicts binding energies that are in reasonable agreement with
experiment. On the right, the charge distribution in a cross-section over the donor is shown.
33
3.1. The hydrogenic model
The scaled hydrogen model is based on the assumptions that bound donor
electrons are delocalized over a large number of atoms of the host lattice.
Therefore their environment can be characterized by a constant dielectric function r and a constant effective mass m∗ , dictated by the curvature of the bands
at the conduction band minimum. Even though the hydrogenic model (without the valley-orbit correction) can also be used to calculate the binding energy
of acceptor states in silicon, this discussion will exclusively focus on the donor
state. Under the above approximations the following Schrödinger equation for
a donor at location r0 can be written down:
Hψ = Eψ
(3.1)
where the Hamiltonian, H, in cartesian coordinates is given by:
H =−
̵ 2 1 ∂2
1 ∂2
1 ∂2
e2
h
[
+
+
]
−
.
2 mx ∂x2 my ∂y 2 mz ∂z 2
4π∣r − r0 ∣
(3.2)
Here mx,y,z = m∗x,y,z me is the effective mass of the electron in the lattice with
me the free electron mass and m∗x,y,z takes the scaling and an anisotropy into
account. The dielectric function is included by the dielectric constant = r 0
with 0 the dielectric constant of vacuum. The silicon conduction band minima
have an anisotropic effective mass with a large mass in the lateral direction
and a small mass in the transversal one. For example at the ±x-valley minima
m∗y = m∗z = 0.191 and m∗x = 0.916. Simplifying, by assuming an isotropic
effective mass, this Schrödinger equation can be solved exactly, just like the
hydrogen atom. The energy levels are given by:
En = −
̵2
m∗ e4 1
h
1
=− ∗ 2 2
2
2
2
8 h n
2m aB n
(3.3)
where n = 1, 2, 3, .. is an integer and aB is the scaled Bohr radius given by:
aB =
h2
.
πm∗ e2
(3.4)
Using a constant value for the masses m∗ = m∗x = m∗y = m∗z = 0.3 and the low
temperature dielectric constant of silicon r = 11.4, a Bohr radius of aB = 2.0
nm and a ground state energy of 31.9 meV is found. This result implies that
all group V donor atoms would have the same binding energy which is known
not to be the case. For example, arsenic has a binding energy of 53.8 meV
with respect to the conduction band edge and phosphorus 45.6 meV, which is a
deviation from the effective mass value of 41% and 30% respectively. This difference can be explained as a break-down of the effective mass approximation
close to the donor nucleus.
34
Donor states in nanostructures
3.2
Valley-orbit corrections
The anisotropic effective mass is the only effect of the band structure that is
accounted for in the hydrogenic model. Further, the polarizability of the silicon
lattice, due to the charge of the donor-bound electron, is taken into account
by the dielectric constant. This is a simplified model that captures only part
of the physics. It turns out that the indirect bandgap nature of silicon has a
significant influence on the states. First of all, because the conduction band
minima in reciprocal space are located at non-zero momentum, (±k0 , 0, 0),
(0, ±k0 , 0) and (0, 0, ±k0 ), states near the bottom of the conduction band are
six-fold degenerate. The minima are located at k0 ≈ 0.85 ⋅ 2π/a, where a the
lattice spacing, equal to 0.543 nm. The effective mass solution, as discussed
in the previous section, is a single-valley approximation. There are six such
valleys, located at ∣k∣ = k0 . At the donor site, the six-fold degeneracy is lifted
because the donor breaks the symmetry of the lattice in such a way that states
in different valleys overlap in reciprocal space. This causes the degenerated
ground state to split into a non-degenerate singlet ground state, a doublet and
a triplet state. Below, a framework that can be used to include this effect in
an effective mass approximation is outlined.
a
b
5
kz
Energy [eV]
0
ky
−5
kx
−10
−15
Figure 3.2: a) The band structure of silicon in a cross-section along different symmetry
axis in the first Brillouin zone [2]. The red dot denotes the conduction band minimum at
k = k0 . b) Close to the minima, constant energy surfaces in each valley are ellipsoid-shaped.
The donor wave functions can be written as a linear combination of states
in the six valleys:
ψ(r) = ∑ cµ ϕµ (r)
(3.5)
µ
where the six conduction band valleys µ are labeled as {±x, ±y, ±z}, ϕµ (r) is
a state at the conduction band minimum and the coefficients are referred to
as valley coefficients, with the condition ∑µ ∣cµ ∣2 = 1. The valley state ϕµ (r)
can be written as:
ϕµ (r) = Fµ (r)uµ (r)e−ir⋅kµ .
(3.6)
35
3.2. Valley-orbit corrections
Here the factor uµ (r)e−ir⋅kµ is the Bloch wave function at the conduction band
minimum µ with momentum kµ [2] and Fµ (r) is the slowly varying envelope
function, which is identical for valley states along the same axis in reciprocal
space [1, 5]. The envelope functions satisfy a hydrogenic Schrödinger equation
similar to Eq. 3.1. For a bulk donor wave function the coefficients, cµ , can be
determined by group theory [1]. Because the donor state exists on the silicon
lattice, the wave function must have the same tetrahedral symmetry as the
crystal. As mentioned before, this splits the six-fold degenerate hydrogenic
ground-state into a singlet state
1
(cx , c−x , .., c−z ) = √ (1, 1, 1, 1, 1, 1),
6
(3.7)
a doublet state
(cx , c−x , .., c−z )
=
=
1
(1, 1, −1, −1, 0, 0)
2
1
√ (1, 1, 1, 1, −2, −2)
12
(3.8)
1
√ (1, −1, 0, 0, 0, 0)
2
1
√ (0, 0, 1, −1, 0, 0)
2
1
√ (0, 0, 0, 0, 1, −1).
2
(3.10)
(3.9)
and a triplet state
(cx , c−x , .., c−z )
=
=
=
(3.11)
(3.12)
A convenient way to include the splitting between the singlet, doublet
and triplet states, which captures most of the relevant physics, is to match
the states to experimental values with a perturbation to the hydrogenic wave
functions. The perturbation is included with the following Hamiltonian:
H = H0 + H ′
(3.13)
where eigenfunction of H0 are the hydrogenic wave functions given by for
example Eq. 3.2 and H ′ describes the perturbations. The correction to the
energy is now given by:
′
′
Eµ,ν
= ⟨ψν ∣Hµ,ν
∣ψµ ⟩
(3.14)
′
with ψµ and ψν the hydrogenic wave functions in the valleys µ and ν and Hµ,ν
′
defines the inter-valley coupling. Hµ,ν depends on the overlap of the wave
functions with the donor nucleus, as this is the region where the effective mass
approximation breaks down, see Appendix A for more details. This approach
36
Donor states in nanostructures
reproduces the splittings exactly and results in a much better agreement for
the binding energy. For phosphorus this correction results in a binding energy
of 42.0 meV, which is much closer to the experimental value (45.6 meV) than
the hydrogenic model, only about 8% off. To further improve on this, nonperturbative methods like for example a tight binding model with empirically
adjusted parameters can be used [6, 7].
Si:P
E
T2
H*
Si:As
E
T2
E = 5 meV
A1
A1
Figure 3.3: Experimental values of the energy splitting of the lowest six arsenic and
phosphorus donors states in silicon are drawn. The splittings are shown to scale. For
phosphorus the energy, with respect to the conduction band minimum, of the orbital singlet
ground state A1 , the triplet T2 and the E doublet state is 45.6, 33.9 and 32.6 meV, and for
arsenic 53.8, 32.7 and 31.3 meV respectively (energies from [4]). For comparison the six-fold
degenerate ground state of a scaled hydrogen model H∗ at 31.9 meV (Eq. 3.3) is shown, i.e.
the unperturbed ground state of H0 .
Physically, the experimentally found binding energy is larger than the binding energy calculated with the effective mass approximation, because the confining potential deviates from a simple Coulomb potential as given by Eq. 3.2
for ∣r − r0 ∣ ≲ aB . Near the nucleus the electron ‘feels’ the discrete nature of the
lattice and the donor nucleus is not perfectly screened by the silicon atoms.
In other words, the dielectric function cannot be regarded as a constant anymore, but is a function of the strength of the donor potential. This lowers the
potential as compared to a screened Coulomb potential. Since the different
linear combinations of hydrogenic states with s-like symmetry, as given above
(Eq. 3.7 to 3.12), have different densities at the nucleus, they will experience
′
different corrections, which were parametrized by Hµ,ν
in Eq. 3.14. The states
with p-like symmetry, on the other hand, are accurately described by the hy-
3.3. Electric fields, interfaces and . . .
37
drogenic model as the density at the nucleus is negligible. For phosphorus this
is the most significant effect. Which is perhaps not surprising because of the
fact that it is an atom from the same row in the periodic table and thus the
electronic configuration (minus one valence electron, i.e. P+ ) is very similar to
silicon. For heavier substitutional atoms however, effects such as strain in the
lattice around the nucleus play an important role in some case [4].
When a donor wave function is modified by electric fields or confinement effects, not only the energy of the uncorrected hydrogenic wave functions changes
but also the valley-orbit splitting is affected. Therefore, the spectra of donors
in nanostructures with features smaller than ∼ 40 nm are different from their
bulk counterparts. This is discussed in the next section.
3.3
Electric fields, interfaces and confinement
Electric fields applied to a donor in silicon, perturb the wave functions and can
eventually ionize it. Strictly speaking, any small electric field in a large bulk
silicon crystal renders the donor state unbound as the potential far away from
the donor is lower than the donors’ ground state energy. In reality however,
field assisted ionization will only occur when the rate of tunneling into this
lower potential region exceeds a certain critical value [8]. Here, the discussion
will be limited to donors in nanostructures, where the electron often experiences additional confinement and the effect of interfaces, besides the donor
potential.
A donor subject to an electric field in the z-direction can be modeled with
the following Hamiltonian:
H =−
̵ 2 1 ∂2
h
1 ∂2
1 ∂2
e2
[
+
+
]
−
+ eFz z.
2 mx ∂x2 my ∂y 2 mz ∂z 2
4π∣r − r0 ∣
(3.15)
Where the direction of the field is chosen along the z-axis and its potential with
respect to the conduction band at the position of the nucleus, r0 = (0, 0, z0 ).
Depending on the symmetry of the orbitals, the field either increases or decreases the binding energy. For example, states with a px,y -like symmetry will
be pushed up with increasing field, but a pz state will decrease in energy as
the wave function has a lobe in the direction of the field. As mentioned before, these perturbations to the wave functions will also have an effect on the
valley-orbit correction as they may increase or decrease the charge density on
the nucleus with respect to the bulk case.
Additional confinement by a box or potential well can prevent the donor
from ionizing in the presence of an electric field. Therefore, the effect of
a strong electric field on a donor in a nano-scale structure is qualitatively
different from the bulk case. Further, if the donor is within a few Bohr radii
from an interface, it hybridizes with an interface well that is induced by an
electric field. Even at zero electric field the presence of an interface is felt
differently by the different orbitals. This influence is similar to the effect of an
38
Donor states in nanostructures
electric field. These effects are mainly due to the effective mass anisotropy and
the different orientation of the valleys with respect to the field or confinement
direction. But, in addition, there also effects associated with the atomic scale
nature of the wave function. For example, an abrupt step in the potential
as present at an interface, breaks the symmetry at the atomic scale. If this
abruptness becomes comparable to the lattice spacing, a, wave functions with
density at the interface experience an additional correction to their energy.
Physically, the underlying mechanism is very similar to the mechanism that
causes the valley-orbit correction. And it can also be taken into account with
a perturbation in an effective mass framework.
Wave functions of confined electron states are built up from Bloch states
near the conduction band minimum according to Eq. 3.6. In silicon these wave
functions are waves with a wave vector ∣k∣ ≈ 0.85 ⋅ 2π/a = 2π/0.64 nm−1 . This
means that the wave function of the localized electron spatially varies over a
length scale comparable to the interatomic distances. Since the electrons are
confined within the crystal, the wave function has to go to zero at the edge
of the crystal. In addition, quantum mechanics requires that the derivative of
the wave function is continuous. These conditions impose restrictions to the
momentum, as well as the phase of the wave function, and lift the degeneracy
of the states. In a bulk crystal the energy scales associated with this are
negligible as the density at the interface is practically zero compared to total
density in the bulk. But when the spatial extent of a wave function that
is located near an interface approaches the atomic scale, a, this energy can
become appreciable. This effect is referred to as valley splitting.
The physics of electrons that are strongly confined in one dimension in
silicon can be understood with a one-dimensional model. In this context, the
wave functions of localized electrons in the ±x-valley minima at ±k0 can be
written as linear combinations:
√
ψ± (r) = Fx (r) [ux (r)eik0 x ± u−x (r)e−ik0 x ] / 2
(3.16)
where Fx (r) is the envelope function given by the global confining potential,
for example a one-dimensional well as in Figure 3.4. The zoom-in in Figure 3.4
illustrates how the mismatch of the wave functions in different valleys differs
because there is a phase difference between the oscillations modulating the
envelope wave function. Since only Fx (r) is an eigenfunction, the functions
ψ± (r) are no eigenfunctions in the effective mass approximation. Therefore,
this mixes the states and changes their energy. In a finite square, well as shown
in Figure 3.4a, the wave function ‘penetrates’ into the interface to a certain
extend. If the penetration depth of the wave function is ≲ a, the perturbation
to ψ+ (r) will be different from ψ− (r) and therefore the states split.
Applying a field across the well has a similar effect, it confines the electron
at the interface, see Figure 3.4b. This increases the charge density at the
interface and increases the splitting between ψ+ (r) and ψ− (r). It turns out
3.3. Electric fields, interfaces and . . .
a
39
b
Figure 3.4: Schematic plots of the wave function confined in a finite square well a) and
in the same well, but subject to an electric field b). Arrows in the zoom-in of a) indicate
the mismatch of the ψ± wave functions with respect to the envelope wave function. The
envelope function is indicated as a dashed line. The fast oscillations, with spatial period
2π/k0 , are phase shifted.
that the field dependence of this effect is in good approximation linear and
a first order perturbative correction suffices in most cases [9]. The energy of
wave functions in the coupled valleys ±µ is given by:
′
H±µ
=[
0
∆∗vs
∆vs
]
0
(3.17)
where ∆vs is the overlap between between valley states, and ∆∗vs its complex
conjugate. One way to see this, is that when the real-space wave function is
squeezed in the direction of the field in real space, it will delocalize in reciprocal
space. Valley states parallel to the field therefore overlap, while in the other
directions the wave function is unaffected. Therefore, for a field in the [100]
or equivalent crystal direction, this effect couples only parallel valleys. The
overlap between different valleys is in general a complex number and is given
by [9, 10]:
∆vs = ∣Vvs ∣e−iθ
(3.18)
with ∣Vvs ∣ a linear function of the electric field and the phase θ depends on
the details of the interface and the field as well [11]. The splitting between
the states, the valley splitting, equals 2∣∆vs ∣ in this case. The perturbed wave
functions are given by:
1
ψe,o = √ [∣ϕµ ⟩ ± e−iθ ∣ϕ−µ ⟩]
2
(3.19)
with ϕ±µ single valley states as before in Eq.√3.6. Note that the valley coeffi√
cients of these states are (cµ , c−µ ) = (1, e−iθ )/ 2 and (cµ , c−µ ) = (1, −e−iθ )/ 2
for ψe and ψo (“even” and “odd”) respectively. For a flat interface, with a high
40
Donor states in nanostructures
barrier compared to the energy of the states, θ ≈ π [12]. Besides the intrinsic
mismatch at the interface, θ also depends on a number of other effects such as
for example strain, interface mis-orientation and atomic scale disorder [12, 13].
Despite the different terminology, the physical origin of the interface-induced
valley-splitting is very similar to the valley-orbit splitting of donor wave functions described above. Both originate from the atomic-scale nature of the wave
functions in combination with the broken symmetry of the host lattice.
In chapter 2 the devices used to get electrical access to single donors in
this thesis are discussed. There, and in the following chapters, FinFETs with
typical dimensions ≲ 60 nm are used to study single dopants. With the current
nano-fabrication techniques it is possible to fabricate devices with dimensions
down to 5-10 nm. This confines the donor wave function, often in an asymmetric manner. In addition, donors in such structures are very likely to be subject
to electric fields up to tens of mV/nm [14]. Below, the effect of confinement
and electric field on donor states in silicon are discussed.
To illustrate the properties of donors in a nanostructures, the level spectrum of a phosphorus donor is calculated. This is done as a function of electric
field in the z-direction (Figure 3.5a) and as a function of the size of the nanostructure (Figure 3.5b). In both cases this confines the wave function in an
asymmetric manner.
To study the effect of confinement, the size of the box confining the donor
is varied in size in the z-direction, while the position of the donor is maintained
at the center of the box, r0 = (Lx , Ly , Lz )/2. For both calculations the lowest
20 states are shown. The box dimensions in the x- and y-direction are Lx =
Ly = 40 nm. For the electric field dependent calculations Lz is kept constant
at 20 nm. A field dependent valley splitting is included using a value of π/2 for
the phase of the inter-valley coupling, θ, see Eq. 3.18. Both level spectra have
been calculated within a framework similar to the one presented above, but
taking 10 single valley excited states into account instead of only the ground
state. Please refer to Appendix A for more details of this calculation. In the
following, the quantity P∣µ∣ = ∣c+µ ∣2 + ∣c−µ ∣2 will be referred to as the valley
population and is shown in Figure 3.5c and d for the computed ground states.
The valley composition of each state is characterized by a set of three numbers,
(Px , Py , Pz ).
First, the electric field dependence of the wave functions is discussed. Since
the donor in Figure 3.5a is quite close the interface, z0 = 6 nm, there is already
considerable splitting of the triplet and doublet states (between ∼ −30and ∼
−25 meV) at zero electric field due to the asymmetry in the confining potential.
As mentioned before, the effective mass within the valleys is larger in the lateral
direction and smaller in the transverse direction. Therefore, donor states with
components in the ±z valleys will be less affected by the confinement in the
z-direction than the ±x, ±y-valleys, see Eq. 3.7 to 3.12. Further, as the field
increases, the s-like states do not shift as much as the p-like states. Especially
3.3. Electric fields, interfaces and . . .
a
b
20 nm
0
Binding energy [meV]
-15
-40
-25
-60
Valley population [−]
c
1
-35
5
10
15
Electric field [mV/nm]
20
0
-45
0
d
Px = P y
Pz
1
10
20
Box size, (Lz) [nm]
30
Px = P y
Pz
0.5
0.5
0
r0=Lz/2
-5
-20
-80
0
Lz
5
r0=6 nm
41
5
10
15
Electric field [mV/nm]
20
0
0
10
20
Box size, (Lz) [nm]
30
Figure 3.5: a) The excited state spectrum of an electron confined by the potential of a
phosphorus donor within a nanoscale silicon box with the dimensions Lx = Ly = 40 nm and
Lz = 20 nm. The dependence of the energy spectrum is plot with respect to the conduction
band at the donor site. For some points the valley composition, (Px , Py , Pz ), is shown. b)
For a nano-structure that is reduced in size in the z-direction, the effect is similar to an
electric field and the states are split with respect to the bulk value. The valley population
of the ground state, corresponding to a) and b), is shown in c) and d). At large field and
strong confinement, the z-valleys dominate the ground state in both cases.
the pz state crosses the s-like ground state at the critical value of the field
determined by the distance to the interface z0 and the binding energy. Beyond
this field the wave function tends towards being purely build up of ±z-valley
states, as can be seen in Figure 3.5c.
Beyond a certain critical field, the spectrum of the box and the donor is
practically the same as a triangular well in silicon. This is especially true
for donors further away from the interface, for which the hybridization with
the interface-well states is weaker. At the highest shown field, the ground
state and the first excited state are only slightly split. This effect is the field
dependent valley splitting discussed above, parametrized by Eq. 3.18. It mixes
the ±z-valleys in this case and results in linear combinations similar to Eq.
3.19. Compared to the bulk situation, an electric field dramatically reduces
42
Donor states in nanostructures
the splitting between the ground state and first excited state. Section 2.3 of
chapter 2 discusses some of the consequences of this for donor-based device
applications.
Now, the effect of confinement of the donor-bound electron in realistic
nanostructures is discussed. Confining the donor in the z-direction more than
in x and y also results in splitting of the states. Figure 3.5b shows that especially the degenerate excited states E and T2 shift by a considerable amount
for Lz ≲ 20 nm. Overall, the binding energy of the states decreases. On first
sight, this seems qualitatively different from the effect of an electric field, but
this is very dependent on the chosen reference for the potential energy. The
confinement effect is also reflected by the valley population of the ground state,
shown in Figure 3.5d. Initially, at Lz = 30 nm, the valley population is equal
to the bulk value of 1/3 for all directions, but as the confinement increases,
the valley population shifts into the ±z-valleys. This symmetry breaking effect
is very similar to the electric-field effect shown in Figure 3.5a and c.
To conclude, it is clear from these examples that in nanostructures with any
dimension smaller than a few tens of nanometer, donor states are considerably
perturbed compared to the bulk case. In addition to this, electric fields, that
are likely to be present in these structures, perturb the wave function as well.
At the same time, the latter effect allows for tuning wave functions and changes
the valley composition, which is useful for donor-based device applications.
3.4
The two-electron state and spin filling
Valleys in an intrinsic bulk silicon crystal cause the conduction band minimum
to be six-fold degenerate. This has not only consequences for the energy of
localized states. It turns out that, especially for the spin filling in donors
and quantum dots, this results in some peculiar phenomena [15–17]. Below,
an example of a two-electron state that is bound to a donor subject to an
external electric field is shown as published by Lansbergen et al. [15].
Multi-electron states can be build up from single electron states. Especially when the electron-electron interactions can be described in a simple way
or when they are weak. This is a practical approach which provides intuitive insights. Furthermore, as electrons are fermions, the states have to be
anti-symmetric under the exchange of a pair of particles and therefore a nondegenerate single particle state cannot be occupied by more than one electron
at the same time, i.e. the Pauli exclusion principle. The way multi-particle
states are constructed is dictated by the single-particle binding energy of the
involved state and electron-electron interactions, i.e. Coulomb interactions
and exchange interaction. Exchange and Coulomb interactions are competing
energy scales in the problem of determining the filling and the total spin of
the system. For a state confined in a potential with a characteristic size l
the single electron level spacing is proportional to 1/l2 , whereas the Coulomband exchange interactions are proportional to 1/l. Therefore, when l is large,
3.4. The two-electron state and . . .
43
the levels will fill in such a way that the exchange energy gain is maximized.
So far, this is all very general, but it turns out that in silicon the exchange
interaction hardly works between electrons occupying different valley states
[17].
a
b
1
2
3
4
5
6
7
8
Figure 3.6: a) Spin filling of one-electron levels build up of even and odd valley combinations of the ±z-valleys. The splitting of the one-electron levels is not to scale. b) The
transition energies as determined theoretically and as experimentally obtained by transport
spectroscopy. The simulated spectrum is not complete, it only shows the lowest excited
states made up from the first three single electron levels with an energy ⩽ 11.2 meV. To
show the correspondence between the single particle states in the table and the spectra
blue/red/green color coding is used. The states marked grey are illustrating the effect of a
vanishing exchange interaction between electrons in different valleys. This figure is adapted
from [15].
The exchange interaction between different valleys is a fast oscillating function of position (with spatial frequency ∼ k0 ) and averages out over long distances compared to the lattice spacing, a [17]. The Coulomb interaction, on
the other hand, depends on how the charge is distributed on average and is
therefore equally effective between electrons occupying the same valley as when
they occupy different valleys. One consequence is that even if l is relatively
large, two states with equal spin but occupying different valleys are almost
degenerate. Another consequence of the vanishing exchange is that in a relatively small magnetic field, spins will align and result in a high total spin of
the state [16]. Even when inter-valley coupling is important, as in donors and
quantum dots, this is still true for the renormalized states ψe,o such as given
by Eq. 3.19.
As an example of the vanishing exchange, Figure 3.6 shows the transition
energies between a one- and two-electron excited-state spectrum a is measured
by transport spectroscopy. By measuring many FinFETs with donors in the
channel (see chapter 2), transport via a single donor is found in some devices.
In this specified example, the electrons are bound to a donor that was found to
be located at 4.7 nm depth and subject to an electric field of 23 mV/nm. The
44
Donor states in nanostructures
depth and electric field were determined from the comparison of the measured
one-electron excited spectrum to a large-scale tight binding model [15]. The
spectrum is dominated by the ±z-valley states at this electric field, renormalized by the inter-valley mixing, Figure 3.6a. The lowest two single-electron
levels are the even and odd combination of the lowest valley state, labeled
∣◻⟩ and ∣◇⟩ (Figure 3.6 shows the colored version of the symbols). The third
level is a combination of the first orbital excited state with an even valley
combination, labeled ∣△⟩.
From the data shown in the table in Figure 3.6b, it can be seen that the
exchange practically vanishes between different valleys “even” and “odd”. For
example the spin singlet in state 2 and the spin triplet in state 3, both marked
grey in Figure 3.6, are practically degenerate. Just like this pair, the singlet
and triplet 5 and 7 are almost degenerate, also marked grey. The states 6 and
8, on the other hand, show a very signifiant splitting. This is because they are
build up from the same “even” valley state, see Figure 3.6a and therefore the
exchange interaction works effectively between these states. These numbers
show that the valley index can be regarded as a good quantum number in this
system.
Even though the electrons in this example are spatially in the same location,
their spins hardly interact if they are located in different valley states. It should
be noted that the Coulomb interaction does work, irrespective of the valley
state the electrons are in, but this interaction is roughly the same for all states
and can therefore be ignored in the excited state spectrum.
References
[1] Kohn, W. & Luttinger, J. Theory of donor states in silicon. Physical
Review 98, 915–922 (1955).
[2] Kittel, C. Introduction to solid state physics (John Wiley & Sons, 2005),
8th edn.
[3] Pantelides, S. & Sah, C. Theory of localized states in semiconductors. I.
New results using an old method. Phys. Rev. B 10, 621–637 (1974).
[4] Ramdas, A. & Rodriguez, S. Spectroscopy of the solid-state analogues of
the hydrogen atom: donors and acceptors in semiconductors. Reports on
Progress in Physics 44, 1297 (1981).
[5] Koiller, B., Hu, X. & Sarma, S. D. Exchange in silicon-based quantum
computer architecture. Phys. Rev. Lett. 88, 027903 (2001).
[6] Boykin, T., Klimeck, G. & Oyafuso, F. Valence band effective-mass expressions in the sp3 d5 s∗ empirical tight-binding model applied to a Si and
Ge parametrization. Phys. Rev. B 69, 115201 (2004).
References
45
[7] Rahman, R. et al. Orbital stark effect and quantum confinement transition of donors in silicon. Phys. Rev. B 80, 165314 (2009).
[8] Smit, G., Rogge, S., Caro, J. & Klapwijk, T. Gate-induced ionization of
single dopant atoms. Phys. Rev. B 68, 193302 (2003).
[9] Sham, L. & Nakayama, M. Effective-mass approximation in the presence
of an interface. Phys. Rev. B 20, 734–747 (1979).
[10] Koiller, B., Hu, X. & Sarma, S. D. Strain effects on silicon donor exchange:
Quantum computer architecture considerations. Phys. Rev. B 66, 115201
(2002).
[11] Wu, Y. & Culcer, D. Coherent electrical rotations of valley states in Si
quantum dots using the phase of the valley-orbit coupling. Phys. Rev. B
86, 035321 (2012).
[12] Saraiva, A. et al. Intervalley coupling for interface-bound electrons in
silicon: An effective mass study. Phys. Rev. B 84, 155320 (2011).
[13] Saraiva, A., Calderón, M., Hu, X., Sarma, S. D. & Koiller, B. Physical
mechanisms of interface-mediated intervalley coupling in Si. Phys. Rev.
B 80, 081305 (2009).
[14] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[15] Lansbergen, G. et al. Lifetime-enhanced transport in silicon due to spin
and valley blockade. Phys. Rev. Lett. 107, 136602 (2011).
[16] Lim, W. H., Yang, C. H., Zwanenburg, F. A. & Dzurak, A. S. Spin
filling of valley–orbit states in a silicon quantum dot. Nanotechnology 22,
335704 (2011).
[17] Hada, Y. & Eto, M. Electronic states in silicon quantum dots: Multivalley
artificial atoms. Phys. Rev. B 68, 155322 (2003).
4
Gate control of valley-orbit
splittings in silicon
metal-oxide-semiconductor
nanostructures
R. Rahman, J. Verduijn, N. Kharche, G. P. Lansbergen,
G. Klimeck, L. C. L. Hollenberg, S. Rogge
An important challenge in silicon quantum electronics in the few electron
regime is the potentially small energy gap between the ground and excited orbital states in 3D quantum-confined nanostructures due to the multiple valley
degeneracies of the conduction band present in silicon. Understanding the
“valley-orbit” (VO) gap is essential for silicon qubits, as a large VO gap prevents leakage of the qubit states into a higher dimensional Hilbert space. The
VO gap varies considerably depending on quantum confinement, and can be
engineered by external electric fields. In this work we investigate VO splitting
experimentally and theoretically in a range of confinement regimes. We report
measurements of the VO splitting in silicon quantum dot and donor devices
through excited state transport spectroscopy. These results are underpinned
by large-scale atomistic tight-binding calculations involving over one million
atoms to compute VO splittings as functions of electric fields, donor depths,
and surface disorder. The results provide a comprehensive picture of the range
of VO splittings that can be achieved through quantum engineering.
This chapter has been published in Physical Review B 83, 195323, 2011
47
48
4.1
T
Gate control of valley-orbit . . .
Valley-orbit splitting in quantum devices
he ability to generate and manipulate three dimensionally confined quantum states in silicon down to the single electron regime is a much soughtafter goal both in semiconductor quantum computing (QC) and quantum electronics. Silicon has not only been the primary platform of the semiconductor
industry for over half a century, but in the quantum regime it also offers the
advantage of long spin coherence times [1] necessary for QC. Over a decade,
steady progress has been made towards realizing quantum dot (QD) [2, 3] and
donor [4–7] based single electron states to encode and process quantum information. However, non-trivial challenges towards quantum electronics in Si
arise from the existence of multiple conduction band (CB) valley degeneracies
in Si, which multiples the orbital degrees of freedom. Since qubits require a
two-level spin system well isolated in energy from other states, a critical goal is
to establish control over valley-orbit (VO) splitting by means of external perturbations and engineered quantum confinement. To date, a comprehensive
understanding of VO splitting in Si has been hindered by a lack of consistent
experimental data, with VO splittings measured over several orders of magnitude from µeV to meV [8–12]. Although a number of investigations are made
inroads towards a theoretical understanding of this problem [13–17], a unified
theory is still absent primarily due to the lack of realistic models of the region
between Si and an insulator. In this chapter, we present measurements of
the VO splittings in Si metal-oxide-semiconductor (MOS) nanostructures under various quantum confinement regimes engineered by internal and external
electric fields. Million atom tight-binding (TB) simulations of the various confinement regimes are performed to explore the range of VO splittings possible
in these structures, and to provide a unified theoretical underpinning of the
experimental data. With this combined experimental and theoretical approach
over a range of confinement regimes we are able to not only explain the range
of VO splittings observed here, but, in addition, understand how to control
the VO splitting in a range of device configurations for future applications.
In general, VO splittings are expected to depend critically on the details
of the confinement potential, interfacial disorder, barrier material, lattice miscuts, substrate orientation, strain, electric and magnetic fields [13–17]. Once
understood, this suggests the ability to engineer VO interaction externally,
and hence to directly tune the momentum space properties of Si.
Hall-bar experiments in strained Si quantum well with SiGe barrier have
reported VO splittings of the order of µeVs [8–10]. It was shown that a strong
vertical magnetic field, which reduces the lateral extent of the wavefunction
and hence the exposure to disorder, could be used to obtain a relatively larger
VO of 1.5 meV in the SiGe system [10]. Relatively little data exits for VO
splittings in Si MOS QDs near the few electron regime. In Ref [18], a VO
splitting of 0.76 meV was reported recently. A past measurement of VO in a
SIMOX device showed an unusually high value of 23 meV [12], which is yet
49
4.2. Measuring the valley-orbit gap
to be explained conclusively, although a recent work has given some plausible
arguments for the cause [19].
4.2
Measuring the valley-orbit gap
Figure 4.1a shows an scanning electron micrograph (SEM) image of a MOS
FinFET device used in the experiments reported here. A silicon nanowire connected to source and drain leads forms the channel of the transistor. A second
nanowire is deposited perpendicular to the channel as the gate electrode. A
thin nitrided oxide layer (1.4 nm equivalent oxide thickness) separates the gate
from the channel. The channel is 60 nm tall for all devices and 60 nm long,
unless stated otherwise. Due to the corner effect, the channel width does have
no influence on the localized states probed at low temperature [20].
(b) Coulomb regime
gate
drain
200 nm
SiO2
a)
source
Si
F
VO
VO
(d) Interface regime with lateral
Coulomb confinement
(e) Interface regime
Lateral V
Lateral V
VO
F
(c) Donor-interface
hybrid regime
F
VO
F
Figure 4.1: a) SEM image of a FinFET device used in the experiments. b)-e) Schematic of
the various confinement regimes showing the vertical confinement potential near the oxide
interface. b) is a donor bound Coulomb confinement regime at low E-field. c) is a hybrid
confinement regime between the donor and the interface well realized at higher E-fields
when the two wells are lined up in energies and are strongly tunnel coupled. d) is an
interfacial confinement regime realized at even higher fields, but laterally bound by the
donor Coulomb potential. e) is a QD-like confinement regime realized at strong E-fields
for device samples without any influence of the donor. The lateral confinement is provided
by the residual barriers in the access regions. The insets show schematics of the lateral
confinement potential in d) and e).
This device has been used to realize and probe different confinement regimes
in Si nanostructures. The measurements are based on excited state transport
spectroscopy of a localized electron in a single quantum dot or donor. Most
measurements utilize an open system and investigate the quantum-Hall effect
in high B-field, and then extrapolate the measured value to the low B-field
limit. Our method in essence provides an all-electrical means of measuring
valley-orbit splitting. As noted, VO splittings change significantly with the
applied electric field ranging from µeVs to tens of meVs. Hence, it is important to report the field at which VO measurements are done in experiments.
50
Gate control of valley-orbit . . .
Some device samples contained a single As donor in the channel due to diffusion from the leads, and could be used to probe donor states [21, 22]. Other
samples without donors could be used to realize QD type states.
To provide a comprehensive theoretical basis for the experimental results
we use a 10-band sp3 d5 s∗ nearest neighbor TB model involving over a million
atoms, detailed in Refs [23–25]. The method has been used successfully to
accurately model a number of experiments on Si nanostructures [15, 21, 24–
26]. A dangling bond passivated surface model has been used to represent
the interfacial boundary [27], except when modeling the SiO2 insulator layer
explicitly, as shown in Figure 4.4.
Figure 4.1b-e show the schematic of the different confinement regimes investigated in this work. The gate potential generates a triangular well near the
oxide interface. If a donor is present in the channel, an additional Coulomb
potential well forms on top of the triangular well at some distance from the
interface, making it possible to study different confinement regimes (Figure
4.1 caption) in a number of device samples.
First, we discuss the three confinement regimes of Figure 4.1b, c and d that
are influenced by the presence of a donor. In the experimental device, the As
donors are located less than 6 nm from the oxide interface and are subjected to
fields of tens of MV/m. This donor-interface configuration has been proposed
as an important system for implementing a donor-dot hybrid qubit [22, 28–30].
SiO2
20
Si
Interfacial
VO
Donor
VO
Depth
15
Depth: 2.7 nm
10
5
0
0
b)
Field
a)
Electric Field [MV/m]
Valley Orbit Splitting [meV]
25
20
30
Electric Field [MV/m]
50
40
30
(2.0)
(3.5)
(4.5)
(5.0)
20
(1.3)
(2.0)
10
Depth: 5.4 nm
10
Valley Orbit Splitting [meV]
5
10
15
20
40
50
0
3
3.5
4 4.5 5 5.5
Donor Depth [nm]
6
6.5
Figure 4.2: a) Tight-binding calculations of VO splitting in Si in the presence of a single
As donor as a function of field for two donor depths. The inset shows a 1D schematic of the
confining potential with the donor. b) VO splitting as a function of donor depth and electric
field obtained from tight-binding calculations. The white markers show the measured VO
splitting for six device samples extracted from the measurements of Ref [21]. The measured
VO energy in meV is shown in parenthesis beside each data point.
4.3. Donors and quantum dots
4.3
51
Donors and quantum dots
For bulk donors at zero fields, all six valleys contribute to VO splittings. Since
the central cell potential varies from one donor species to another, so does
the VO splitting [22]: in bulk the VO splitting of a bulk As donor is about
21 meV, compared to 12 meV for a P donor. If an electric field is applied
in the z direction, the weight of the wavefunction increases in the kz valleys
and diminishes in the others. This wavefunction redistribution in momentum
space causes a reduced VO splitting. Figure 4.2a shows the calculated VO
splitting as a function of the E-field for two different donor depths. At low
E-fields, the VO splitting is about 20 meV, comparable to the VO splitting of
a bulk As donor. As the E-field is increased, the donor states hybridize with
interface states, and VO splitting reduces gradually. At high enough E-fields,
the electron is pulled to the interface, reducing VO splitting to a few meV, as
expected of QD bound states at strong fields. Once the electron resides at the
interface, the VO splitting varies linearly with the field, as shown by the red
curve of a donor at 5.4 nm depth at fields above 30 MV/m. The blue curve
is for a donor at a shallower depth of 2.7 nm. The change in VO is smoother
because of stronger tunnel coupling between the two wells.
In Figure 4.2b, the VO splitting is plotted in color code as a function of
donor depth and electric field. At low fields, the donor bound Coulomb-like
regime of Figure 4.1b is realized, whereas at high fields, the states are mostly
interface bound, as in Figure 4.1d. The measured VO splittings for six devices
with donors at various depths and fields are mapped on this figure as white
markers. The data points sample out all three confinement regimes. The
measured VO data has been extracted from measurements of D0 excited state
spectroscopy as reported in Ref [21], where the donor depths and applied fields
were reported for various device samples. It is therefore possible to obtain a
whole range of VO splitting in single donor devices ranging from 20 meV to an
meV or even less. This means that VO splittings can be engineered through
donor implantation depths and applied E-fields.
In Figure 4.3, new measurements are shown for a device with a QD-like
confinement regime. In this regime, the electrons are confined at the gatechannel interface by the gate electric field, as described in Figure 4.1e. Unlike
the previous cases, where a dopant provided the confining potential, the lateral
confinement is provided by the residual barriers in the access regions of the
device on both sides of the channel [20]. Hence, these are more extended
interface bound states. The measured experimental data for transport through
the gate-field confined states is shown in Figure 4.3a. The low charging energy
of about 10 meV, determined by the height of the Coulomb blockade diamonds,
indicates there are no dopants present. Typical charging energies for dopant
bound states range from 30 to 50 meVs [21]. Furthermore, it was found that
spin filling is consistent with the first diamond corresponding to the the first
electron [31].
52
Gate control of valley-orbit . . .
0
Differential Conductance [μS]
5
10
15
20
b)
25
20
Gmax [μS]
Vb [mV]
2.5
0
90
1
400
3
10
1.5
Vg [mV]
415
2
-10
-20
250
c)
Energy w.r.t. GS [meV]
4
3.5
G [μS]
a)
15
300
350
Vg [mV]
400
1.5
0
0.2
0.4
1/T [K-1]
0.6
Expt. data
10
5
0
EVO(F)
5
10
15
20
25
30
Field [MV/m]
ΔE(L=38.5 nm)
35
40
ΔE(L=30.5 nm)
45
50
Figure 4.3: a) Conductance versus gate and bias voltage plot for a device without the
influence of a donor (confinement regime Figure 4.1e), showing blocked diamond region and
tunneling through the QD states, characteristic of Coulomb blockade. b) Inset: Measured
conductance data of the device as a function of gate voltage and temperature (T ), between
1.5 and 90 K. The temperature steps are not all equal, but are approximately 0.2 K for the
data point T < 10 K. Main plot: The peak maxima (extracted from the Inset) vs. 1/T ,
showing a cross-over from single level transport (linear regime) to classical transport at
higher T . c) Calculated spectrum of a MOS QD relative to the ground state, with lateral
dimensions of 30.5 nm × 38.5 nm, as a function of the vertical electric field, showing VO split
states, matched to the measured values. The black squares correspond to the experimentally
observed excited states (see main text).
To make sure there is no unobserved low lying state between the ground
and the first excited state in the stability diagram, we measured the temperature dependence of the low bias trace versus gate voltage. This method gives
us the position of the lowest excited state [32]. The inset of Figure 4.3b shows
the temperature dependence of the traces. In the temperature regime where
the peak conductance is a linear function of the inverse temperature, 1/T ,
there is transport through a single quantum state. Here the peak conductance
increases with decreasing temperature (Figure 4.3b). At higher temperatures,
multilevel based classical transport mechanisms dominate, and at even higher
temperatures, thermally activated transport dominates. The position of the
cross-over point between the quantum regime and classical regime is determined by the position of the first excited state. Here, we found VO to be ∼ 12
K, or ∼ 1 ± 0.1 meV.
4.4. Comparison to a model
4.4
53
Comparison to a model
A variety of factors such as orbital confinement, valley polarization, and VO
interaction compete to determine the electronic structure of Si QDs. We have
therefore performed TB simulations for this QD-like confinement regime as
shown in Figure 4.3c. Since the effective lateral dimensions and the microscopic
E-fields (F ) in the channel are not known, we considered a range of values for
these parameters, and obtained best agreement with the measured data for
lateral confinement lengths of Lx = 30.5 nm, Ly = 38.5 nm and an electric field
in the z-direction of F = 11.6 MV/m.
The simulated energy spectrum is shown in Figure 4.2c as a function of the
E-field. The energy of the excited states are plotted relative to the ground state
energy. The orbital states of the dot arising from the same valley configuration
appear as flat lines with the vertical E-field. Their energies are primarily
determined by the lateral dimensions of the box. The shorter the confinement
lengths, the higher are the orbital energies, consistent with a 2D particle in
a box model. The VO split states appear as tilted lines, showing a linear
dependence on the E-field. An 1 meV VO splitting is obtained at F = 11.6
MV/m, independent of the lateral dimensions. The lowest three measured
energy gaps extracted from the transport data are superimposed on this plot
as black squares. To check that the observed 1 meV state is not an orbital
excited state of the same valley configuration, we use a particle-in-a-box model
to estimate the confinement length. With the transversal effective mass of the
z-valley (∼ 0.19 times the free electron mass), we find it would correspond to
L = 77 nm if it were indeed an orbital excited state. Comparing with the
channel and the gate lengths of the device, the QD states are expected to have
confinement lengths of less than 60 nm. Therefore, the observed 1 meV state
is indeed a VO split state. Since the VO split states can cross the orbital
states of the dot at higher E-fields, the level arrangements in Si dots can vary
from one experiment to another depending on the E-field and effective lateral
confinement lengths. The first excited state energy gap of the QD changes
slope with E-field at this crossing point, which can be used as a reference
point for measuring VO and orbital splittings in experiments in which the
vertical field can be tuned.
4.5
Influence of barrier and disorder
VO splittings in realistic systems are likely to be influenced by the atomistic
details of the SiO2 − Si interface. We performed TB calculations to investigate
the role of interface disorder on VO splittings in MOS QDs. We have used
a virtual crystal (VC) 4 band sp3 TB model of SiO2 [33]. Since the oxide is
amorphous or highly disordered in reality, this is an approximation. However,
given that ab-initio studies have shown some crystalline structure in the oxide
near the interface [34] where the exponential tail of the QD wavefunctions
reside, this model is expected to provide a good qualitative picture.
54
Gate control of valley-orbit . . .
(b)
3
Lateral V
VO
2
1
0
10
SiO2
barrier
Regular
interface
F
Irregular
interface
Samples
Average
20
30
40
50
Electric Field [MV/m]
2 nm
(c)
SiO2
barrier
SiO2/Si
4
Si QD
25 nm
Si QD
2 nm 0.5 nm 25 nm
Interface
disorder
{
Valley Orbit Splitting [meV]
(a)
Figure 4.4: a) TB simulations of VO splitting as a function of the vertical field for an
ordered b) and disordered c) surface.
In Figure 4.4a, we included 2 nm width of a VC SiO2 in the simulation
domain along with Si. To simulate disorder, we have used an additional unit
cell of SiO2 at the the SiO2 −Si interface (Figure 4.3b), and replaced some of the
virtual SiO2 atoms with Si randomly. This in effect creates a local variation in
∆Ec , and an overall decrease in the average ∆Ec at the interface. Simulations
of 10 randomly generated samples show a slightly decreased VO splitting as
a function of field. The standard deviation of VO splittings is represented by
the error bars. Thus interface disorder does not have a significant effect in the
calculated VO splittings in this model. In the field regimes investigated, the
vertical E-field creates the most significant change in VO splittings.
4.6
Conclusions
We have investigated experimentally and theoretically the valley-orbit splitting
in Si MOS devices under a range of quantum confinement conditions and
provide a unified description of the large range of VO gaps observed in QDs and
in hybrid single donor states. Large-scale atomistic tight-binding simulations
confirm quantitatively the range of observed VO splittings, and shed light on
the role of interfaces, disorder, and electric fields. In Si MOS QD states, VO
splittings are shown to increase with the vertical E-field, and can be influenced
by atomistic disorder at low E-fields. Presence of single donors in these devices
can yield VO splittings from 20 meVs to sub meVs depending on the field. This
work enhances our understanding of VO interaction induced energy gaps in Si
nanostructures, a critical consideration for Si qubits.
Acknowledgments
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Corporation, for the United States Department of Energy under
Contract No. DEAC04- 94AL85000. NEMO-3D was initially developed at
JPL, Caltech under a contract with NASA. NCN/nanohub.org computational
References
55
resources were used. This research was conducted by the Australian Research
Council Centre of Excellence for Quantum Computation and Communication
Technology (project number CE110001029), NSA and ARO (contract number
W911NF-08-1-0527).
References
[1] Tyryshkin, A., Lyon, S., Astashkin, A. & Raitsimring, A. Electron spin
relaxation times of phosphorus donors in silicon. Phys. Rev. B 68, 193207
(2003).
[2] Loss, D. & DiVincenzo, D. Quantum computation with quantum dots.
Phys. Rev. A 57, 120–126 (1998).
[3] Friesen, M. et al. Practical design and simulation of silicon-based
quantum-dot qubits. Phys. Rev. B 67, 121301 (2003).
[4] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[5] Vrijen, R. et al. Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures. Phys. Rev. A 62, 012306
(2000).
[6] Hollenberg, L. et al. Charge-based quantum computing using single
donors in semiconductors. Phys. Rev. B 69, 113301 (2004).
[7] Hill, C. et al. Global control and fast solid-state donor electron spin
quantum computing. Phys. Rev. B 72, 045350 (2005).
[8] Wilde, M. et al. Direct measurements of the spin and valley splittings
in the magnetization of a Si/SiGe quantum well in tilted magnetic fields.
Phys. Rev. B 72, 165429 (2005).
[9] Lai, K. et al. Valley splitting of Si/SiGe heterostructures in tilted magnetic fields. Phys. Rev. B 73, 161301 (2006).
[10] Goswami, S. et al. Controllable valley splitting in silicon quantum devices.
Nature Physics 3, 41–45 (2007).
[11] Xiao, M., House, M. G. & Jiang, H. W. Measurement of the spin relaxation time of single electrons in a silicon metal-oxide-semiconductor-based
quantum dot. Phys. Rev. Lett. 104, 096801 (2010).
[12] Takashina, K., Ono, Y., Fujiwara, A., Takahashi, Y. & Hirayama, Y.
Valley polarization in Si(100) at zero magnetic field. Phys. Rev. Lett. 96,
236801 (2006).
56
Gate control of valley-orbit . . .
[13] Boykin, T. B. et al. Valley splitting in strained silicon quantum wells.
Appl. Phys. Lett. 84, 115 (2004).
[14] Friesen, M. & Coppersmith, S. N. Theory of valley-orbit coupling in a
Si/SiGe quantum dot. Phys. Rev. B 81, 115324 (2010).
[15] Kharche, N., Prada, M., Boykin, T. & Klimeck, G. Valley splitting in
strained silicon quantum wells modeled with 2○ miscuts, step disorder,
and alloy disorder. Appl. Phys. Lett. 90, 092109–3 (2007).
[16] Srinivasan, S., Klimeck, G. & Rokhinson, L. P. Valley splitting in Si
quantum dots embedded in SiGe. Appl. Phys. Lett. 93, 112102 (2008).
[17] Saraiva, A., Calderón, M., Hu, X., Sarma, S. D. & Koiller, B. Physical
mechanisms of interface-mediated intervalley coupling in Si. Phys. Rev.
B 80, 081305 (2009).
[18] Xiao, M., House, M. & Jiang, H. Parallel spin filling and energy spectroscopy in few-electron Si metal-on-semiconductor-based quantum dots.
Appl. Phys. Lett. 97, 032103–3 (2010).
[19] Saraiva, A., Koiller, B. & Friesen, M. Extended interface states enhance
valley splitting in Si/SiO2 . Phys. Rev. B 82, 245314 (2010).
[20] Sellier, H. et al. Subthreshold channels at the edges of nanoscale triplegate silicon transistors. Appl. Phys. Lett. 90, 073502 (2007).
[21] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[22] Rahman, R. et al. Orbital stark effect and quantum confinement transition of donors in silicon. Phys. Rev. B 80, 165314 (2009).
[23] Klimeck, G., Oyafuso, F., Boykin, T. B., Bowen, R. C. & von Allmen,
P. Development of a nanoelectronic 3-D (NEMO 3-D) simulator for multimillion atom simulations and its application to alloyed quantum dots.
CMES: Computer Modeling in Engineering & Sciences 3, 601–642 (2002).
[24] Klimeck, G. et al. Atomistic simulation of realistically sized nanodevices
using NEMO 3-D–part i: Models and benchmarks. Electron Devices,
IEEE Transactions on 54, 2079–2089 (2007).
[25] Rahman, R. et al. High precision quantum control of single donor spins
in silicon. Phys. Rev. Lett. 99, 036403 (2007).
[26] Lansbergen, G. et al. Lifetime-enhanced transport in silicon due to spin
and valley blockade. Phys. Rev. Lett. 107, 136602 (2011).
References
57
[27] Lee, S., Oyafuso, F., Allmen, P. V. & Klimeck, G. Boundary conditions for the electronic structure of finite-extent embedded semiconductor
nanostructures. Phys. Rev. B 69, 045316 (2004).
[28] Smit, G. D. J., Rogge, S., Caro, J. & Klapwijk, T. M. Stark effect in
shallow impurities in Si. Phys. Rev. B 70, 035206 (2004).
[29] Martins, A. S., Capaz, R. B. & Koiller, B. Electric-field control and
adiabatic evolution of shallow donor impurities in silicon. Phys. Rev. B
69, 085320 (2004).
[30] Calderón, M., Koiller, B., Hu, X. & Sarma, S. D. Quantum control of
donor electrons at the Si-SiO2 interface. Phys. Rev. Lett. 96, 096802
(2006).
[31] Sellier, H., Lansbergen, G. P., Caro, J. & Rogge, S. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett.
97, 206805 (2006).
[32] Foxman, E. et al. Crossover from single-level to multilevel transport in
artificial atoms. Phys. Rev. B 50, 14193–14199 (1994).
[33] Kim, S. G., Luisier, M., Paul, A., Boykin, T. & Klimeck, G. Full threedimensional quantum transport simulation of atomistic interface roughness in silicon nanowire fets. Electron Devices, IEEE Transactions on 58,
1371–1380 (2011).
[34] Pasquarello, A., Hybertsen, M. S. & Car, R. Interface structure between
silicon and its oxide by first-principles molecular dynamics. Nature 396,
58 (1998).
5
Wave function control and mapping
of a single donor atom
J. Verduijn, G.C. Tettamanzi, S. Rogge
Donors in semiconductors are attractive basic components for quantum device
applications. One reason for this is the long coherence time of nuclear and
electrons spins. Another important aspect is that the confining potential is
naturally formed a with large ground-state splitting. This makes all donors
identical and therefore paves the way to large-scale integration of donors in
a network that couples them. Furthermore, local electric gates can be used to
address individual donors. Here, we present an experiment that investigates the
nature of the ground state of a single phosphorus donor embedded in a nanoscale device while the electron wave function is manipulated by an electric
field. From the amplitude of the tunnel current we extract the shape of the
wave function and valley composition of the ground state. This shows that the
valley composition changes dramatically as the donor gradually ionizes and
hybridizes with an interface state formed by the electric field. Our results are
a step forward in the tunability of single donor states as necessary for quantum
device applications.
This chapter has been submitted for publication.
59
60
5.1
Wave function control and mapping . . .
T
Access to single donors
5.2
Tuning the electric field
hanks to their unique properties, single isolated dopants hold great promises
for quantum device applications. A famous example of a prototypical application is the dopant-based quantum computer [1, 2]. For this proposal long
electron spin and nuclear spin coherence times are important. Donors in silicon, in particular phosphorus, seem to be able to meet these prerequisites
because extending the coherence times can be achieved by drastically reducing number of nuclear spins in the silicon lattice [3]. Even though, by now,
many experiments show that access to single dopants has become reasonably
straightforward, see e.g. [4–8], control over tunnel coupling and adiabatic
manipulation of the donor-bound electrons are relatively unexplored aspects.
Further, quite extensively studied theoretically [9–11], the influence of the conduction band valley structure on the shuttling of electrons between the donor
and the interface, as required for some applications, has not been investigated
experimentally to date.
Here, we present experimental data of a single atom transistor which demonstrates that it is possible to smoothly tune the electron wave function from a
bulk-like dopant state to a interface-well state, via a molecular dopant-interface
state. In addition to the direct usefulness to donor-based devices, we show that
this provides a mechanism to tune the tunnel coupling to dopants. In the process of shuttling the electron, the valley population of the state changes in a
nontrivial, but controllable, way which we probe directly with current spectroscopy at low temperature. This mechanism achieves a goal similar to the
proposal of using the hyperfine interactions with silicon-29 nuclear spins to
map out the wave function by Park et al. [12]. Our experiment is sensitive to
the spatial extent of the wave function and we estimate the donors’ distance
to the interface from the size of the hybridization gap that is observed in the
transport data.
Nano-wire FETs with a few randomly placed donors in the channel are used as
a platform to study single dopants [4, 7, 13], see Figure 5.1. The device used
here consists of a nanowire etched from a 20 nm thick silicon film of a silicon on
insulator wafer (SOI). The gate, which is isolated from the channel by a 5 nm
thick silicon dioxide layer, protects the channel region during the high dose
source/drain implantation, see Figure 5.1a and b. To obtain single dopant
transport in a controllable way, we have pre-implanted the silicon film with
1⋅1017 cm−3 phosphorus. For a device with channel region of (40×60×20) nm3
this results in on average only 3-7 donors in the channel, where the dimensions
are the gate length, the channel width and channel height respectively. The
donor positions are random, therefore, the tunnel coupling to the source and
drain contacts of the dopants is random as well. We measure many devices
and select a device with a donor in a favorable location in the channel such
61
5.2. Tuning the electric field
that it carries sub-threshold resonant current and that it is sufficiently well
isolated from other donors in the channel. For additional electrostatic control
over the channel, the substrate wafer, below the 145 nm thick buried oxide, can
be used as a gate [14]. This additional degree of tunability allows us to control
the wave function of a single donor in the channel region while maintaining the
possibility to perform current spectroscopy on the dopant at the same time.
We define a coordinate system with the x-direction parallel to the current flow
and the z-direction perpendicular to the channel as well as the substrate, see
Figure 5.1b. For a certain range of top- and backgate voltages, where the
electric field in the channel is small, the donor is bulk-like. In the limit of
large electric field however, the donor is ionized and the electron is localized
at the interface, as schematically shown in Figure 5.1c and d.
100 nm
a
Top
gate
source
z
n++
top gate
x
y
n++
Buried SiO
Back gate
drain
c
y
x
b
2
d
tg
s
tg
d
s
d
Fz
Figure 5.1: Device structure of the single donor device. a) Scanning electron micrograph
of a device similar to the one studied in this chapter. The channel (blue) is a nano-wire
etched from a silicon film on a silicon dioxide layer. The gate (yellow) is wrapped around
the channel on three sides. b) A schematic cross-section of the device across the dashed line
in a). The source and drain contacts are doped with a high concentration of arsenic donors
to make them metallic. The topgate and backgate can be used to control the electric field
in the channel. c) Shows the wave function of an electron bound to a single phosphorus
donor in the center of the channel. d) By applying an appropriate gate voltage, and thereby
inducing an electric field in the z-direction, Fz , the donor can be ionized in a controlled way.
In the shown case, the electron is de-localized along the interface of the channel.
At low temperature (T = 4.2 K) we perform current spectroscopy on the
gated donor, see Figure 5.2. By applying a source/drain voltage, Vsd , of 0.2
mV, a current will flow from source to drain. The magnitude of the current depends on the voltages on the topgate, Vtg , and backgate, Vbg , and the presence
of localized electron states in the channel. At low gate voltages the current is
blocked by the barrier under the gate that exceeds the energy of the electrons
in the contacts. As the gate voltages are increased, a localized state in the
channel comes in resonance with the source/drain chemical potential, µs,d , and
a current can flow, see Figure 5.2b [4]. Note that we keep the source/drain
62
Wave function control and mapping . . .
bias small such that (µs,d ≈ eVs ≈ eVd ) < kB T at all times and only the ground
state is probed, here kB is the Boltzmann constant. When the localized state
is situated below µs,d the current is blocked again due to (Coulomb blockade)
[15], see also section 2.2. This mechanism allow us to measure a clean resonant tunnel current signal probing a single donor state in the channel of the
device. Figure 5.2a shows the measured drain current through the channel of
our device.
As we will argue below, in the region labeled B the current flows mediated
by a wave function localized at the donor nucleus, schematically shown in
Figure 5.2b. In region A and C the current flows through a wave function
localized in an interface well that is induced by the electric field, see Figure
5.2c. From the electrostatic coupling of the localized states to both gates,
we are able to extract a qualitative picture of the potential landscape of the
device. This analysis is presented in Appendix B. The arrow, at Vbg ≈ 2.5 V
for the first resonance, denotes the position where we find that the conduction
band is on average flat and thus the electric field at the donor location is
approximately zero. As shown in Appendix B, the flatband point is signaled
by a change of the slope of the resonant tunneling lines.
5.3
Valley structure of a confined donor
Silicon is an indirect band-gap material. This renders the conduction band
minimum six-fold degenerate with states at a finite momentum, k0 , in each ±x-,
±y- and ±z-direction in reciprocal space, where k0 ≈ 0.85⋅2π/a = 2π/0.64 nm−1 ,
see the familiar reciprocal space representation in Figure 5.2d. In addition, the
effective mass in each of these minima is anisotropic, in the lateral direction
the mass is 0.914me , and in the two transverse directions 0.191me [16], where
me is the free electron mass. When confined in one direction by an fieldinduced interface-well, this results in the splitting of the conduction band
states in a two-fold degenerate band at low energy and four-fold degenerate
band at higher energy, see inset Figure 5.2e. As a consequence, the ground
state populates only the two valleys in the confined direction, see Figure 5.2e.
For spherical confinement by a donor potential, on the other hand, this results
in a non-degenerate singlet ground state and triplet and doublet excited states
[17]. For the ground state singlet the valley population is equal for all valleys,
as shown in Figure 5.2d. As the applied electric field in our device induces a
well at the interface and pulls the electron wave function away from the donor,
the splitting of states varies dramatically and the ground state population of
the valley states evolves from donor-like to interface-well-like [10, 11, 18]. This
also changes the wave functions’ spatial extent, and thus the tunnel coupling
with the source and drain contacts. Therefore, these modifications to the wave
function can be probed by measuring the current.
5.4. Single donor amplitude . . .
a
63
c
b
Topgate Voltage [mV]
200
Vtg
Vtg
0
A
-200
5
Current [nA]
-400
E
-600
−10
B
kx
ky
mt
kx
C
0
0
5
10
Backgate Voltage [V]
μs,d
Vbg
z
d
2.5
−5
Vbg
μs,d
k0
e
ky
ml
kz
4 x |x|,|y|
6x
kx
kz
15
Bulk-like
2 x |z|
kz
Vbg
Interface well
Figure 5.2: Low temperature transport data and wave function manipulation in real- and
reciprocal space. a) The drain current as a function of backgate and topgate voltage data at
a source/drain bias of 0.2 meV. The white dashed line indicates the intrinsic device threshold
voltage and the green circles a single donor resonance. In region B, between the vertical
dashed lines, the resonance is induced by a bulk-like donor state and in the regions A and C
an interface-well-like one. Panel b) and c) schematically visualize how the real space electron
wave function is modified by and electric field. A one-dimensional cut of the wave function
and the conduction band structure is shown as a function of position between source and
drain. On the resonance the donor level aligns with the source/drain chemical potential,
µs,d . Panel d) and e) show a schematic reciprocal space representation corresponding to b)
and c). An electric field lifts the six-fold degeneracy, see inset of panel e). Therefore, low
energy wave functions localized at the interface are mainly built up from the two z-valleys.
The inset of d) shows a detail of the +z-valley, centered at k0 , with the transversal, mt , and
lateral ml effective mass indicated.
5.4
Single donor amplitude spectroscopy
The resonant tunnel current into the donor state at small source/drain bias
is proportional to the tunnel coupling of the contact states to the ground
state. As explained above, the backgate is used to tune the electric field
subject to the donor. At the same time, the topgate voltage is tuned such that
the ground state stays in resonance with the source/drain chemical potential,
see Figure 5.2b and c. Figure 5.3a shows the measured drain current as a
function of the applied backgate voltage extracted from a fit of a thermally
broadened resonance to the peak denoted by the green circles in Figure 5.2a.
The corresponding topgate voltage can be read from Figure 5.2a, see also the
section Methods in Appendix B. We assume a linear relation between the
backgate voltage and the electric field, because this is true in the absence of
free charge, which is what we expect to be the case in the sub-threshold region
at low temperature. In addition, this assumption is confirmed by the analysis
presented below and in Appendix B. This allows us to compare the measured
tunnel current to a model describing the field dependence of the wave function.
Due to the anisotropy in the effective mass, states in different valleys have
64
Wave function control and mapping . . .
a different spatial extent in the transport direction, which we have chosen as
the x-direction. This means that the zero-field wave function overlap with
states in the source and drain contacts is larger for the y- and z-valleys than
for the x-valley. Since the x-valley states have a larger effective mass in the xdirection and are more confined, therefore, the tunnel coupling with the source
and drain contact states is much smaller, see Figure 5.2. This situation changes
if an electric field is applied. Initially, the effect on the donor wave function is
to increase the spatial extent while the electron stays localized at the donor.
At a certain critical field however, the electron is pulled to the interface and
delocalizes along the interface. Figure 5.3c schematically illustrates this effect
for a donor at ∼ 5 Bohr radii from the interface, i.e. the Bohr radius for
phosphorus in silicon is ∼ 2nm. Since the tunnel coupling increases with an
increase of the spatial extent of the wave function, the delocalization of the
wave function can be observed in the tunnel current. We have developed a
model to quantify this effect and to perform a more detailed analysis of the
data.
5.5
Tunable confinement & valley population
We solve a single valley effective mass Schrödinger equation for a donor located
in the center of the channel on a grid. The channel is modeled as a 40 ×
40 × 20 nm box subject to a linear electric field in the z-direction. For the
moment, the influence of the Bloch wave functions is ignored and we obtain
an envelope wave function of the ground state [17]. We compute the tunnel
coupling to one of the contacts by integrating the envelope wave function over
a plane perpendicular to the transport direction (in the x-direction), similar
to Bardeen’s approximation, see the see also Appendix B for more details.
Figure 5.3a shows the result of the calculation. The curve is matched in two
different regimes: the high field regime and the low field regime, labeled i and ii
respectively. Around the flatband voltage, curve i matches the data very well.
This is a strong indication that the location of the donor in the z-direction
is indeed approximately the center of the channel, i.e. at 10 nm (∼ 5 Bohr
radii) from the interface. Furthermore, the tunneling amplitude is symmetric
around the flatband voltage (dashed vertical line), at a backgate voltage of
(2.50 ± 0.02) V. This leads us to conclude that the donor wave function is
closest to bulk-like at this point. At the flatband point the electron is tightly
bound to the donor nucleus since the electric field is approximately zero and
therefore the tunnel coupling is small. Note that these results are consistent
with the electrostatic analysis of the device as presented in Appendix B, which
independently predicts the same trend for the electric field. At larger backgate
voltage, however, the experimental data shows a shoulder at positive as well
as negative backgate voltage. These features turn out to be caused by the
non-trivial way in which the valley population shifts from the x- and y-valleys
to the z-valley as an electric field is applied.
5.5. Tunable confinement & valley . . .
ii
0
10
−1
10
i
−2
10
−3
10
A
−5
B
0
5
10
Backgate Voltage [V]
C
1
0.8
0.6
0.4
0.2
0
|x|-valleys, Fz = 0
|y|,|z|-valleys, Fz = 0
|z|-valley, Fz ≠ 0
lead states
10 meV
c
40 nm
Pz
Px = Py
bulk-like
data
model
fit
−5
10
0
5
Backgate Voltage [V]
d
Topgate Voltage [mV]
b
1
10
Valley Occupation [−]
Peak drain Current [nA]
a
65
−280
−320
−360
data
fit
states
10
11
Backgate Voltage [V]
Figure 5.3: Tunneling spectroscopy of a single donor and its valley composition. a) The
magnitude of the current at the resonance (denoted by the green circles in Figure 5.2a) as
a function of the backgate voltage. A light blue area is used to show the size of the 98%
confident bounds of the resonance height from the fit. The grey band at the bottom denotes
the region in which the electron is located on the donor, region B, or in the interface-well,
regions A and C. Even though the band is divided in discrete sections, the transition from
donor-like to interface-like is smooth. b) A fit allows to extract the valley composition of
the ground state (see main text). c) The gradual delocalization of the electron as a function
electric field can be detected in the tunnel current, because states in different valleys have a
different spatial extent. d) When the donor state crosses the interface-well state, a hybridized
state is formed. The dashed lines are the positions of the uncoupled states, i.e. with ∆ = 0,
see the section Methods in Appendix B.
66
Wave function control and mapping . . .
To allow further quantitate analysis of the data, we assume a phenomenological description of the valley population as a function of backgate voltage
(electric field). In this way we introduce some of the valley physics in the
model that we have ignored in our analysis so far. This enables the extraction
of the valley population from the tunnel current amplitude and translate the
shoulders at Vbg ∼ −2.4 V and Vbg ∼ 7.3 V in Figure 5.3a into a transition
of the valley population. Figure 5.3b shows the result of this fit. A positive
(or negative) applied field in the z-direction, moves the valley population for
the x- and y-valleys into the z-valleys. The reason for this is that due to the
anisotropic effective mass the electric field has a smaller effect on the valleys
parallel to the field (z-valleys) than the other ones, perpendicular to the field.
In Appendix B the analysis is outlined in detail. It should be noted here that
even though the exact shape of the obtained curve does depend on the details
the fitting procedure, the general trend is that the valley population of the zvalley increase when the magnitude of the electric field is made larger. These
results demonstrates the tunability of the valley population for a gated donor
in a single device.
At the backgate voltage where the interface-well state aligns with the donor
state, these states hybridize with each other and shift in energy. Since the energy shift can be detected in the position of the Coulomb blockade resonance
as a function of the top- and backgate voltage, this provides and independent
way to validate the donor-interface transition of the localized electron [19].
Figure 5.3d shows the peak position in topgate voltage as a function of backgate voltage in the vicinity of the hybridization point. The shift of the state
manifests itself clearly as one branch of an anti-crossing. Fitting a two-level
dispersion to this crossing results in a hybridization gap of (1.4 ± 0.2) meV,
see Appendix B for more details. Using the values published by Calderón et
al. [20] we find that this corresponds to a donor at ∼ 12 nm from the interface, approximately in the center of the channel, halfway between the top
and back side of the 20 nm thick channel. This result agrees well with the
estimate from the fitting of the tunneling current amplitude and is consistent
with the electrostatic analysis presented in the section Methods in Appendix
B. Furthermore, the amplitude of the tunnel current as shown in Figure 5.3d
is almost completely symmetric around the flatband voltage within region B.
This reflects the fact that the response of the donor wave function is symmetric
with respect to a sign change of the electric field, as expected when the donor
is located in the center of the channel.
5.6
Possible applications
The ability to tune the wave function of a single donor-bound electron is relevant in many different ways. It allows, for example, for tuning the hyperfine
coupling between the electron and the nuclear spin of the donor [21] and will
allow to tune the exchange interaction between electrons bound to spatially
5.7. Conclusions
67
separated donors [22, 23]. The former mechanism could be used to build a
scaleable quantum computer based on the electron and nuclear spin interaction. The results presented here are encouraging for the next step towards
a donor-based quantum computer and the investigation of the tunability of
interactions between two donors mediated by an interface state [1]. To achieve
this, control with multiple gates, similar to our experiment, is required [1]. Recently, very similar devices that allow for multi-gate control have been used to
measure resonant tunneling between two donors [24] as well as the fast coherent control of the charge density on two coupled donors [25]. With sufficiently
fast pulses (how fast this has to be depends on the details of the used scheme
and the coherence time) and the control over the wave function demonstrated
here, qubit gates on a two-donor system could be performed.
Further, it has been proposed to use the valley degree of freedom in silicon
double quantum dots to encode the states of a qubit that can be fully electrically controlled [26, 27]. Even though there is to date no equivalent proposal
for a donor-interface system such as studied here, it may be possible to use a
subset of the donor-interface states to encode and manipulate quantum information in a useful way. The excited states with the rich valley structure of
this system can readily be tuned in and out of resonance with each other using
an electric field [10]. Depending on the valley composition they can hybridize
or do not interact at all. Some explorations of these effects and the associated
physics are presented by Debernardi et al. [28, 29] and Baena et al. [11].
5.7
Conclusions
We demonstrated electrical control over the wave function of a single donor.
By adjusting the voltages on a togate and backgate, we are able to change the
electric field. This modifies the wave function and changes its spatial extend
and therefore the tunnel coupling to the source and drain contacts. Therefore,
this deformation of the wave function can be probed in the low temperature
tunnel current. Furthermore, we find that the valley composition of the donor
state changes as the electric field pulls the electron from the donor toward the
interface. The latter mechanism is useful for tuning the hyperfine interaction
between the electron spin and nuclear spin. The demonstrated control is a
step forward for the realization of single donor-based devices in general.
acknowledgments
The devices have been designed and fabricated by the AFSiD Project partners,
see http://www.afsid.eu. J.V. acknowledges Jan Mol for discussions and help
with the finite differences calculations. This work was supported by the EC
FP7 FET-proactive NanoICT project AFSiD (214989) and by the Australian
Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027). S.R. acknowledges
an ARC Future Fellowship. G.C.T. acknowledges financial support from the
68
Wave function control and mapping . . .
Australian Research Council via the Discovery Early Career Researcher Award
(DECRA) scheme, project ID DE120100702.
References
[1] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[2] Hollenberg, L. et al. Charge-based quantum computing using single
donors in semiconductors. Phys. Rev. B 69, 113301 (2004).
[3] Steger, M. et al. Quantum information storage for over 180s using donor
spins in a 28 Si “semiconductor vacuum”. Science 336, 1280–1283 (2012).
[4] Sellier, H., Lansbergen, G. P., Caro, J. & Rogge, S. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett.
97, 206805 (2006).
[5] Calvet, L., Wheeler, R. & Reed, M. Observation of the linear stark effect
in a single acceptor in Si. Phys. Rev. Lett. 98, 096805 (2007).
[6] Tan, K. Y. et al. Transport spectroscopy of single phosphorus donors in
a silicon nanoscale transistor. Nano Lett. 10, 11–15 (2010).
[7] Pierre, M. et al. Single-donor ionization energies in a nanoscale CMOS
channel. Nature Nanotechnology 5, 133–137 (2010).
[8] Morello, A. et al. Single-shot readout of an electron spin in silicon. Nature
467, 687 (2010).
[9] Koiller, B., Capaz, R., Hu, X. & Sarma, S. D. Shallow-donor wave functions and donor-pair exchange in silicon: Ab initio theory and floatingphase heitler-london approach. Phys. Rev. B 70, 115207 (2004).
[10] Rahman, R. et al. Orbital stark effect and quantum confinement transition of donors in silicon. Phys. Rev. B 80, 165314 (2009).
[11] Baena, A., Saraiva, A., Koiller, B. & Calderón, M. Impact of the valley degree of freedom on the control of donor electrons near a Si/SiO2
interface. Phys. Rev. B 86, 035317 (2012).
[12] Park, S., Rahman, R., Klimeck, G. & Hollenberg, L. Mapping donor
electron wave function deformations at a sub-bohr orbit resolution. Phys.
Rev. Lett. 103, 106802 (2009).
[13] Khalafalla, M. A. H., Ono, Y., Nishiguchi, K. & Fujiwara, A. Identification of single and coupled acceptors in silicon nano-field-effect transistors.
Appl. Phys. Lett. 91, 263513 (2007).
References
69
[14] Roche, B. et al. A tunable, dual mode field-effect or single electron transistor. Appl. Phys. Lett. 100, 032107 (2012).
[15] Beenakker, C. Theory of coulomb-blockade oscillations in the conductance
of a quantum dot. Phys. Rev. B 44, 1646–1656 (1991).
[16] Madelung, O. Semiconductors: data handbook (Springer, Berlin New
York, 2004).
[17] Kohn, W. & Luttinger, J. Theory of donor states in silicon. Physical
Review 98, 915–922 (1955).
[18] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[19] Hüttel, A., Ludwig, S., Lorenz, H., Eberl, K. & Kotthaus, J. Direct control
of the tunnel splitting in a one-electron double quantum dot. Phys. Rev.
B 72, 081310 (2005).
[20] Calderón, M., Koiller, B., Hu, X. & Sarma, S. D. Quantum control of
donor electrons at the Si-SiO2 interface. Phys. Rev. Lett. 96, 096802
(2006).
[21] Bradbury, F. et al. Stark tuning of donor electron spins in silicon. Phys.
Rev. Lett. 97, 176404 (2006).
[22] Koiller, B., Hu, X. & Sarma, S. D. Exchange in silicon-based quantum
computer architecture. Phys. Rev. Lett. 88, 027903 (2001).
[23] Wellard, C. & Hollenberg, L. Donor electron wave functions for phosphorus in silicon: Beyond effective-mass theory. Phys. Rev. B 72, 085202
(2005).
[24] Roche, B. et al. Detection of a large valley-orbit splitting in silicon with
two-donor spectroscopy. Phys. Rev. Lett. 108, 206812 (2012).
[25] Dupont-Ferrier, E. et al. Coupling and coherent electrical control of two
dopants in a silicon nanowire. arXiv cond-mat.mes-hall (2012). 1207.
1884v1.
[26] Culcer, D., Saraiva, A., Koiller, B., Hu, X. & Sarma, S. D. Valley-based
noise-resistant quantum computation using Si quantum dots. Phys. Rev.
Lett. 108, 126804 (2012).
[27] Wu, Y. & Culcer, D. Coherent electrical rotations of valley states in Si
quantum dots using the phase of the valley-orbit coupling. Phys. Rev. B
86, 035321 (2012).
70
Wave function control and mapping . . .
[28] Debernardi, A., Baldereschi, A. & Fanciulli, M. Computation of the Stark
effect in P impurity states in silicon. Phys. Rev. B 74, 035202 (2006).
[29] Debernardi, A. & Fanciulli, M. Stark effect of confined shallow levels in
phosphorus-doped silicon nanocrystals. Phys. Rev. B 81, 195302 (2010).
6
Coherent transport through a
double donor system in silicon
J. Verduijn, G.C. Tettamanzi, G.P. Lansbergen,
N. Collaert, S. Biesemans, S. Rogge
In this chapter we describe the observation of the interference of conduction
paths induced by two donors in a nano-scale silicon transistor, resulting in a
Fano resonance. This demonstrates the coherent exchange of electrons between
two donors. In addition, the phase difference between the two conduction paths
can be tuned by means of a magnetic field, in full analogy to the AharonovBohm effect. One of the crucial ingredients for donor-based quantum computation is phase coherent manipulation of electrons. The presented results
demonstrate a way to achieve this.
This chapter has been published in Applied Physics Letters 96, 072110, 2010
71
72
6.1
Coherent transport through a . . .
D
Introduction
6.2
Devices
opants gained attention in the past years due to their potential applicability in quantum computation architectures using the charge or spin
degree of freedom [1, 2]. In a bulk system, dopants provide long spin-coherence
times [3]. Furthermore, since all dopants are virtually the same in a bulk crystal, the 1/r potential landscape of a dopant is very reproducible. However, for
practical applications, the dopants need to be embedded in nanostructures allowing manipulation and readout of the quantum mechanical state [1, 2]. This
modifies their bulk properties significantly [4–6] and thus requires new experiments to probe quantum coherent electron exchange and electronic properties
such as the level spectrum. In this chapter, we study transport signatures that
provide information about the electronic coherence. In particular, we report
the observation of phase coherent transport of electrons through two physically
separated donors, resulting in Fano resonances at low temperature.
Our devices are three dimensional silicon field effect transistors (FinFETs) with
a boron-doped channel and a polycrystalline silicon gate wrapped around the
channel [7]. Few arsenic dopants (n-type) diffuse into the p-type channel from
the highly doped source/drain regions and modify the transport characteristics
[4]. Recently, it has been shown that the level spectrum of isolated dopants
can be determined by means of low temperature transport spectroscopy [5].
However, this work relies on statistics to find a single dopant in the transport,
and therefore there are also devices that exhibit multi-dopant transport. In
fact, transport occurs through a single dopant only in about 1 out of 7 devices
with a fixed gate length and channel height of 60 nm and channel widths
varying between 35 nm and 1 µm [5]. All other devices show multi-dopant
transport or no signatures of dopants at all [4]. The device we discuss in this
chapter has a gate length of 60 nm and the channel is 60 nm high and 60 nm
wide. The inset of Figure 6.1 shows an electron micrograph of such a device.
6.3
Experimental results
We measure the dc characteristics of our devices, namely the drain current,
I, and the differential conductance, G = dI/dVb , versus the gate voltage, Vg ,
and bias voltage, Vb , in a three-terminal configuration. The differential conductance is measured using a lock-in technique with a 50 µV sinusoidal ac
excitation at 89 Hz, superimposed on the dc bias component. These obtained
data can be plot in a stability diagram, a two-dimensional color-scale plot with
the gate voltage and bias voltage on the axes. From the stability diagram,
measured at low temperature (≲ 4.2 K), one can typically extract information
such as the level spectrum of the donor and the energy needed to add a second
electron to the system, the charging energy [5].
73
6.3. Experimental results
G [μS]
20
30
S
D
20
G
16
500 nm
V b [mV]
10
0
12
D0(1)
D0(2)
D-(1)
D-(2)
8
−10
4
−20
D+ ↔ D0
−30
360
410
460
D0 ↔ D510
0
560
V g [mV]
Figure 6.1: The inset shows an electron micrograph of the device with the source (S),
drain (D) and gate (G) indicated. The differential conductance of this device is measured in
a three-terminal configuration. These data are plotted in a differential conductance stability
diagram and reveal that the transport at 0.3 K is largely determined by a single arsenic
donor. Regions where transport occurs through the neutral D0 state and through the D−
state can be distinguished (indicated by the black dashed lines). In addition, we observe
0
−
transport features (indicated
a two narrow Fano lines in the vicinity of the D(2)
and D(2)
by arrows). Inside the Coulomb diamond there is a zero-bias feature visible which can be
attributed to a Kondo effect[8]
Figure 6.1 shows the differential conductance as a function of Vb and Vg .
We observe two triangular regions with a non-zero differential conductance due
to direct tunneling processes through donor states in the FinFET channel. The
0
−
corresponding resonances at Vb = 0 mV are denoted D(2)
and D(2)
in Figure
6.1 and Figure 6.2. At lower gate voltage (Vg ∼455 mV), direct transport
regions are visible, bound by black dashed lines in Figure 6.1. Here the donor
is alternating between the ionized (D+ ) and neutral state (D0 ) while electrons
traverse the donor one-by-one. At higher gate voltage (Vg ∼530 mV) a second
region is visible where the donor alternates between the D0 and the negatively
charged state (D− ). The diamond shaped area in between marks Coulomb
blockade of the donor with a fixed number of electrons. To investigate the
mode of transport we show conductance traces in Vg at zero Vb as a function
of temperature (Figure 6.2). Lowering the temperature from 75 K to 50 K
0
results in an increase of the resonance denoted by D(2)
, indicating that the
low temperature quantum transport regime (kB T < ∆E, with ∆E the level
spacing) is entered [9]. At base temperature (0.3 K) the maximum conductance
74
Coherent transport through a . . .
exceeds the room temperature value, approaching 0.67e2 /h. There are several
0
−
facts proving that the D(2)
/D(2)
resonances are due to an arsenic donor close
to the Si/SiO2 interface [4, 5, 7, 8]. First of all, the charging energy of 29 meV
is of the same order as reported earlier for similar devices [4, 5]. Secondly, the
spin filling, deduced from the energy shift in magnetic field, is in agreement
with earlier observations. Note that the first fact is providing strong evidence
against the resonances originating from an electrostatically defined quantum
dot state as reported in [7].
40
G [µS]
30
T = 300 K
T = 75 K
T = 50 K
T = 25 K
T = 0.3 K
2
1.5
1
D0(1)
D0(2)
0.5
D-(2)
0
410 420 430 440
20
D-(1)
10
0
300
350
400
450
Gate Voltage [mV]
500
550
Figure 6.2: A differential conductance trace at zero bias in gate voltage versus temperature
is plotted. We observe a strong increase of the height of the Coulomb oscillations with
0
decreasing temperature. The inset shows a zoom-in of the D(1)
resonance (same axis).
0
−
In addition to the clearly visible D(2)
and D(2)
features discussed above,
two much fainter resonances are visible, denoted by the arrows in Figure 6.1
0
−
and labeled as D(1)
/D(1)
. At high bias, Vb ≫ 0, these resonances develop in
faintly visible triangular regions, due to first order sequential tunneling (red
dashed lines), using these lines we find a charging energy of ∼35 meV. Further−
more, the resonance at Vg ∼ 510 mV (D(1)
) shows a linear shift towards higher
gate voltages of about 0.12 ± 0.02 meV/T when magnetic field, B, between 0 T
and 10 T in the direction of the channel is applied (Figure 6.3a, in agreement
with the expected theoretical value of 0.116 meV/T for transport through a
spin singlet state [4, 10], using a Landé g-factor of 2. Altogether, this makes us
confident that the origin of the resonances is a second donor. It must be noted
0
that the D(1)
resonance is too weak compared to the background to observe
any shift under magnetic field unambiguously.
−
A trace in Vg around the D(1)
-resonance (Figure 6.3c and 6.3d) reveals that
this resonance has a Fano line shape [11]. We suggest that the two conduction
paths, induced by the donors 1 and 2, add in a coherent way, resulting in
destructive or constructive interference, and in this way give rise to a Fano
resonance. Fano resonances have been observed in a wide range of physical
systems [12]. To observe this effect, a path with a constant or slowly varying
75
6.3. Experimental results
phase that interferes with a path with a rapid phase variation is required. In
our system the phase varies as a function of the energy difference between the
donor state and the chemical potential of the contacts [13]. The gate allows us
to tune the energy of the localized states and thereby, effectively, the transport
phase. The remainder of this chapter discusses the nature of the interference
in the device.
To gain more insight, we measure differential conductance traces in Vg
−
around the D(2)
Fano resonance while applying a magnetic field parallel to
the FinFET channel between 0 and 10 T (Figure 6.3)a. We observe that
the line shape changes as we increase the field and even changes symmetry
in an alternating fashion (Figure 6.3c and 6.3d). Note that this symmetry
change confirms that the resonance is due to a Fano effect and excludes the
possibility of the shape being induced by, for example, another nearby charge.
This symmetry transition indicates that the magnetic field tunes the phase
difference between (at least) two coherent conduction paths [14]. For this to
occur, the paths must be physically separated and form a closed loop in such a
way that a net magnetic flux can pierce the formed loop and modify the phase
difference by the Aharonov-Bohm (AB) effect [15]. Therefore, we conclude
that we probe two physically separated donors in a phase coherent way.
In order to make this effect more quantitative, we fit the traces in gate
voltage taken at magnetic fields between 0 T and 10 T to a phenomenological
Fano formula [11]
2
G () = GF
∣ + qΓ/2∣
.
2 + (Γ/2)2
Where , Γ and GF are the detuning of the resonance, the tunnel coupling
and a pre-factor respectively. The detuning can be related to the gate voltage
via the gate coupling α, defined as = α(Vg − Vg,0 ), where Vg,0 is the position
of the resonance. The gate coupling, α, can be obtained from the stability
diagram by dividing half the height of the Coulomb diamond by its width
[7]. We take the Fano parameter q = qx + iqy as a complex number [16]. The
argument of the Fano parameter, arg (q) = arctan (qy /qx ), varies between ±π/2
as a function of the magnetic field, as can be seen in Fig 6.3b. This reflects
the symmetry change of the resonance. Since the symmetry change in the
resonance is periodic in the flux quantum, Φ0 = h/e, by the nature of the AB
effect [14], we can determine the projected surface area of the loop formed by
the two current paths. Using a period of 6.5 T from the data (Figure 6.3b)
we find a surface area A ∼ 6.3 ⋅ 10−16 m2 . This corresponds to a circular loop
with a diameter of ∼28 nm, which is a realistic size considering the dimensions
of our structure. Also the stability diagram (Figure 6.1) shows that there is
no direct coupling between the donor, since this would result in hybridization
of the orbitals of both donors, reflected by a shift in the stability diagram.
Therefore, we conclude that the inter-donor distance must be ≳ 20 nm [17].
76
Coherent transport through a . . .
(b)
12
9
9
10
7
8
6
5
6
4
4
3
2
2
1
8
Magnetic field [T]
Magnetic field [T]
8
0
510
(c)
10
Drain Current [nA]
G [µS]
7
6
5
4
3
2
1
514
518
Gate voltage [mV]
0
0
−0.5
0
0.5
arg(q)/π [rad]
12
B=1T
10 arg(q) < 0
8
6
4
500
(d)
Drain Current [nA]
10
6.5 T
(a)
5
510
520
Gate Voltage [mV]
B=5T
4 arg(q) > 0
3
2
500
510
520
Gate Voltage [mV]
Figure 6.3: a) We plot differential conductance traces as function of gate voltage and
magnetic field. This reveals, by the shift to higher gate voltages, that the Fano resonance
carries spin down electrons which is consistent with the state being a charged donor state D−
[4]. We fit these traces using a phenomenological formula and obtained the complex Fano
parameter q (see main text). Furthermore, we fit a linear function to the peak positions
(black dots) and convert this to energy, using the gate coupling α, to find the shift of the
peak as a function of the field, we find 0.12 ± 0.02 meV/T consistent with a shift dominated
by the Zeeman energy [10]. b) We plot the argument of q (arg(q)) to quantify the magnetic
field dependence, in particular, the symmetry transition of the peak. The period of this
symmetry transition is found to be 6.5 T. (c, d) The Fano formula fits well, R2 ∼ 0.9, and
the peak shows a symmetry transition as a function of the magnetic field.
Supported by the found projected loop size, we argue that this is also consistent
with the coherent transfer of electron between two independent donors.
Furthermore, we observe that the background as well as the Fano resonance
decreases with magnetic field (Figure 6.3b). Well away from the resonance,
e.g. at Vg ∼ 475mV, the background is due to the Kondo effect, and is thus
quenched by the magnetic field [18]. Therefore, we speculate that the Fano
resonance is the results of interference between a Kondo transport channel and
a direct transport processes.
6.4
Summary & conclusions
In summary, we demonstrate phase coherent exchange of electrons between
two donors at low temperature. This is a key ingredient to single-donor quantum device applications. The observation of a Fano resonance, due to the
interference between two conduction paths induced by the two donors, is a
proof of phase coherence in our device. We speculate that this interference
effect originates from the interplay between a Kondo and a direct transport
channel. The phase difference between the two conduction paths can be tuned
by means of a magnetic field, analogous to the AB-effect. This analysis in-
References
77
dicates that the distance between these donors is on the order of the device
dimensions. Consistent with this, the transport measurements show no signs
of direct interaction between the two donors. Thus, we conclude that the
donors are physically separated and only coherently coupled in transport.
Acknowledgment
Financial support was obtained from the European Community’s seventh framework under the grant agreement nr: 214989-AFSiD and the Dutch Fundamenteel Onderzoek der Materie (FOM).
References
[1] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[2] Hollenberg, L. et al. Charge-based quantum computing using single
donors in semiconductors. Phys. Rev. B 69, 113301 (2004).
[3] Eriksson, M. et al. Spin-based quantum dot quantum computing in silicon.
Quantum Information Processing 3, 133–146 (2004).
[4] Sellier, H., Lansbergen, G. P., Caro, J. & Rogge, S. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett.
97, 206805 (2006).
[5] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[6] Rahman, R. et al. Orbital stark effect and quantum confinement transition of donors in silicon. Phys. Rev. B 80, 165314 (2009).
[7] Sellier, H. et al. Subthreshold channels at the edges of nanoscale triplegate silicon transistors. Appl. Phys. Lett. 90, 073502 (2007).
[8] Lansbergen, G. P. et al. Tunable kondo effect in a single donor atom.
Nano Lett. 10, 455–460 (2010).
[9] Beenakker, C. Theory of coulomb-blockade oscillations in the conductance
of a quantum dot. Phys. Rev. B 44, 1646–1656 (1991).
[10] Weis, J., Haug, R., Klitzing, K. & Ploog, K. Competing channels in
single-electron tunneling through a quantum dot. Phys. Rev. Lett. 71,
4019–4022 (1993).
[11] Fano, U. On the absorption spectrum of noble gases at the arc spectrum
limit. Nuovo Cimento 12, 154–161 (1935).
78
Coherent transport through a . . .
[12] Miroshnichenko, A., Flach, S. & Kivshar, Y. Fano resonances in nanoscale
structures. Rev. Mod. Phys. 82, 2257–2298 (2010).
[13] Göres, J. et al. Fano resonances in electronic transport through a singleelectron transistor. Phys. Rev. B 62, 2188–2194 (2000).
[14] Yacoby, A., Heiblum, H., Mahalu, D. & Shtrikman, H. Coherence and
phase sensitive measurements in a quantum dot. Phys. Rev. Lett. 74,
4047–4050 (1995).
[15] Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in
the quantum theory. Physical Review 115, 485–491 (1959).
[16] Clerk, A., Waintal, X. & Brouwer, P. Fano resonances as a probe of phase
coherence in quantum dots. Phys. Rev. Lett. 86, 4636–4639 (2001).
[17] Koiller, B., Hu, X. & Sarma, S. D. Electric-field driven donor-based charge
qubits in semiconductors. Phys. Rev. B 73, 45319 (2006).
[18] Meir, Y., Wingreen, N. & Lee, P. Low-temperature transport through a
quantum dot: The anderson model out of equilibrium. Phys. Rev. Lett.
70, 2601–2604 (1993).
7
Non-local coupling of two
donor-bound electrons in silicon
J. Verduijn, R. R. Agundez, M. Blaauboer, S. Rogge
We report the results of an experiment investigating coherence and correlation
effects in a system of coupled donors. Two donors are strongly coupled to two
leads in a parallel configuration within a nano-wire field effect transistor. By
applying a magnetic field we observe interference between two donor-induced
Kondo channels, which depends on the Aharonov-Bohm phase picked up by
electrons traversing the structure. This results in non-monotonic conductance
as a function of magnetic field and clearly demonstrates that donors can be
coupled through a many-body state in a coherent manner. We present a model
which shows good qualitative agreement with our data. The presented results
add to the general understanding of interference effects in a donor-based correlated system which may allow to create artificial lattices that exhibit exotic
many-body excitations.
This chapter has been submitted for publication.
A preprint is available under arXiv:1209.4726v1
79
80
7.1
Non-local coupling of two donor- . . .
M
Introduction
7.2
Single donor transport
any physical effects in modern solid state physics are a manifestation
of quantum interference and, therefore, rely on the preservation of coherence of quantum states. Quantum dots embedded in an Aharonov-Bohm
ring are ideal for studying coherent effects in solid state nano-structures [1–
3]. Shallow dopants in silicon have recently gained much interest because of
their extremely long spin coherence times [4] and the reproducible confining
potential [5, 6]. For the same reasons that make them interesting for quantum
device applications, single dopants are ideal to act as a flexible model system
to investigate fundamental open problems in correlated systems in the solid
state [7, 8]. Here, we study a system of two dopants which are coherently coupled in an Aharonov-Bohm ring configuration using transport spectroscopy.
We observe a peculiar type of Kondo effect mediating the interactions in the
Coulomb-blockade regime of this system.
The system studied here consists of two arsenic donors in a field effect
transistor that are coupled to leads in a parallel configuration. This geometry
allows for the tuning of the phase acquired by electrons as they traverse the
structure. Changing the net magnetic flux threading the loop enclosed by the
two current paths changes the acquired phase through the Aharonov-Bohm
(AB) effect. In a previous publication we have shown that this device exhibits phase-coherent transport in the sequential tunneling regime, evidenced
by the presence of a Fano resonance [9]. Here, new data of the same device in
the Kondo regime is presented and it is shown that the Kondo effect can be
coherently modulated.
By studying the magnetic-field dependence of the Kondo transport, we
link the phase modulation to the AB ring arrangement of the donors. In a
broader context, this system allows to study the rich behavior of occurrences
of universal physical phenomena such as the Kondo effect [10], the AharonovBohm effect [11] and the Fano effect [12] in a mesoscopic system.
Single dopant transport spectroscopy has proven to be a very powerful tool
to study properties of dopants in nano-structures, e.g. [13–17]. Here we use
three-dimensional field effect transistors (FinFETs) with dopants embedded in
the channel of the device as a platform for our experiments [13, 15, 18]. The
device investigated here consists of two coupled dopants that dominate the
sub-threshold transport. All presented data has been obtained from a device
which has a channel height of 60 nm, a gate length of 60 nm and channel width
of 60 nm. Even though the channel is nominally only doped with boron, we
occasionally find arsenic donors that have diffused into the p-type channel, see
Figure 7.1. This allows us to perform transport spectroscopy on few-donor
systems at temperatures ≲ 4 K [13].
We measure the DC differential conductance G = dI/dVsd as a function
7.3. Aharonov-Bohm effect in the . . .
b
c
Id
d
drain
e
sourc
B
D1
20
15
D2
gate
gate
25
Conductance [μS]
a
81
s
Vg
Vs
10
5
0
B=5T
450 500 550
Gate voltage [mV]
Figure 7.1: To study single donor transport a three-dimensional FinFET device is used.
a) Colored blue, the gate, wrapped around the channel (green), is shown. The device is
fabricated on top of silicon dioxide (orange). b) In the channel region, below the gate, two
arsenic donors (labeled D1 and D2 ) are coupled to the source (s) and drain (d) contacts
in an Aharonov-Bohm ring arrangement, c) resulting in a Fano resonance in the sequential
tunneling regime, denoted by the arrow. A magnetic field, B, is applied which induces a
magnetic flux piercing through the loop enclosed by the current paths. Voltages are applied
to source, Vs , and gate, Vg , while the drain current, Id , is measured.
of magnetic field, B, gate voltage Vg and source/drain bias, Vsd , see Figure
7.1b. The hallmarks of single donor transport are a large charging energy and
odd/even spin filling of the donors [13]; both have been observed in the device
presented here. We perform all measurements at 0.3 K and Vsd = 50 µV except
when explicitly mentioned otherwise.
7.3
Aharonov-Bohm effect in the Kondo regime
In the Coulomb-blockade region, where both donors are occupied by a single
electron, we observe considerable zero-bias conductance. This is unlikely due
to thermal effects, since the charging energy of the donors is large compared to
the thermal energy kB T at the experimental temperature, i.e. U /2kB T ≈ 500.
Zero-bias conductance is one of the characteristics of the spin-1/2 Kondo effect
in single donors [7] and quantum dots [10, 19]. To confirm this we investigate
the temperature and magnetic field dependence of the conductance at a gate
voltage of Vg = 485 mV and Vg = 490 mV respectively, see Figure 7.2b and c. For
the Kondo effect in quantum dots the conductance increases logarithmically as
the temperature is ≲ TK and saturates at a zero-temperature maximum as the
temperature ≪ TK , where TK is the Kondo temperature. Figure 7.2b shows
a fit to the temperature dependent data of a phenomenological formula [20]
which describes this behavior. The Kondo temperature resulting from the fit is
TK =(12 ± 5) K, a value that justifies that we use an effective zero-temperature
model to analyze the conductance later on. Figure 7.2c shows a plot of the
measured differential conductance as a function of source/drain bias voltage
and magnetic field. Since the transport processes in the Kondo regime involve
spin-flip tunneling transitions, a gap ∼ 2gµB B wide (with g = 2 for silicon),
82
Non-local coupling of two donor- . . .
centered at zero bias, is expected to open [10, 21]. This is indeed what we
observe in the data, see Figure 7.2c. At various gate voltages, we plot the
Kondo conductance as a function of magnetic field and show that the Kondo
effect is quenched as the magnetic field increases, see the corresponding trace
in Figure 7.2a. This has been recognized as one of the hallmarks of a Kondo
effect in similar systems [10].
3.2
2.4
0
2
4
6
Magnetic field [T]
8
10
2
TK = 12 K
10
Magnetic field [T]
c
1
fit
data
2.8
8
1
10
Temperature [K]
G [μS]
3.6
2gμBB
6
4
2
0
-2
Vg= 490 mV
Vg= 480 mV
Vg= 490 mV
Vg= 500 mV
Vg= 510 mV
Conductance [μS]
b
10
Conductance [μS]
a
-1
0
1
2
Source/drain bias [mV]
0.36
Figure 7.2: The Kondo conductance is measured in the Coulomb-blockade region at several
gate voltages as a function of magnetic field, temperature and source/drain bias voltage. a)
At ∼ 6 T a clear upturn of the conductance, as the magnetic field increases, is visible. This is
attributed to constructive interference of Kondo channels induced through both dopants, see
also the discussion in Section 4. b) The temperature dependence of the Kondo conductance
at Vg = 485 mV. A fit of a semi-empirical spin-1/2 Kondo model for T < 5 K has been
performed, with the scaling parameter left free (red line) [20]. This fit results in a Kondo
temperature of 12 K and scaling parameter 0.23 ± 0.1, consistent with a spin-1/2 Kondo
effect. c) The Kondo conductance as a function of magnetic field and source/drain bias
voltage at Vg = 490 mV. A zero-bias gap, characteristic for the Kondo effect, opens when a
magnetic field is applied.
There is, however, a discrepancy between our data and a simple spin-Kondo
effect, which we believe is due to the magnetic field dependence of the phase,
which is carried by electrons and tuned through the Aharonov-Bohm effect.
Figure 7.2a shows that the zero-bias conductance unexpectedly increases with
increasing field and subsequently decreases again, exhibiting a maximum at
B ∼ 7.7 T. We find that this effect persists throughout the Coulomb-blockade
region and the position of the maximum is almost independent of gate voltage.
Because the donors are coupled to the contacts in a parallel configuration, this
upturn may be explained by an interference effect that depends on the acquired
AB phase during tunneling. To further investigate this idea we first present
7.4. Results and discussion
83
an analysis of the current in the sequential tunneling regime. We then develop
a simple model which describes the dependence of Kondo conductance on the
magnetic field.
7.4
Results and discussion
Since the two donors present in the channel of our device are coupled in an
AB-ring configuration, see Figure 7.1b, the transport phase acquired by electrons traversing the structure contains, in addition to the geometric phase,
a flux-dependent part [1, 22]. This mechanism allows to tune the symmetry
of the Fano resonance that is a result of the interference between sequential
transport channels induced by the two donors in the device. To extract the
tunnel coupling as well as the detuning of the donor levels we fit our data to
an extended version of the model that was used in the analysis presented in
[9]. The improved fitting formula is derived using a non-interacting scattering
approximation, see Figure 7.3b.
7.4.1 Sequential transport & the Kondo effect
This model treats both donor levels in a symmetric way and allows the resulting
resonances to be either symmetric (Lorentzian) or asymmetric (Fano) [23].
Figure 7.3b shows that the resonance around Vg ≈ 530 mV stays symmetric
over the entire magnetic field range up to 10 Tesla. The narrower resonance
at Vg ≈ 513 mV, on the other hand, changes significantly. This behavior is
consistent with an AB phase that is tuned by the magnetic flux piercing the
loop enclosing the two donor-induced current paths. We cannot obtain the
AB phase directly from the fit, because the shape of the Fano resonance is
influenced by the effect of the thermal occupation of the leads at our finite
experimental temperature [22]. Instead, we use the periodic change of the
Fano line shape to determine the period of the AB oscillations. This results in
an AB period of ∼6.5 T. In addition, we extract the tunnel coupling of both
donors as 0.5 ± 0.1 mV and 5.2 ± 0.5 mV.
Next, we turn back to the data in the Coulomb-blockade region shown in
Figure 7.2a. The data show a change in the conductance as a function of magnetic field, very similar to conventional AB oscillations in mesoscopic systems
[1, 24], the main difference being that the magnitude decreases with magnetic
field. In the conventional AB effect, the interference alternates between constructive and destructive as the field is swept, depending on the magnetic field
dependent phase picked up as electrons traverse the circumference of the loop.
In our experiment, however, the continuum conduction paths are formed by
spin Kondo channels in both arms. This explains why the amplitude of the oscillation decreases with magnetic field since the spin Kondo effect is quenched
[10]. Another mechanism that can potentially enhance the conductance in the
Kondo regime at finite field, is the crossing of spin singlet and triplet states
at finite magnetic field [25]. The latter possibility can be excluded since this
84
Non-local coupling of two donor- . . .
a
3
60
Conductance [μS]
Conductance [2e2/h]
1
b
AB phase [π]
1
2
0
9.8 T
40
0.5
6.5 T
4.9 T
20
0
4.9 T
8.2 T
3.3 T
1.6 T
0T
0
2
4
6
Magnetic field [T]
8
10
0
480
500
520
Gate Voltage [mV]
540
Figure 7.3: The conductance in the Kondo and sequential tunneling data has been modeled, see main text. a) A plot of the calculated Kondo conductance (Eq. 7.1) as a function
of magnetic field shows the non-monotonic behavior, very similar to the data in Figure 7.2a.
On the top axis the AB phase using a period of 6.5 T is given. b) By fitting a non-interacting
formula for the sequential conductance to the gate voltage traces at each magnetic field [23],
we extract the coupling of the donors to the source and drain contacts and the period of the
AB oscillations. The inset shows an enlargement of the Fano resonance at Vg ≈ 513 mV.
would also be visible as a kink in the shift of the Coulomb-blockade peaks [26],
which we do not observe for the magnetic field range covered in our experiment.
7.4.2 Interfering Kondo channels
Taking a phenomenological approach, we obtain an analytical expression for
the Kondo conductance using a scattering formalism. Assuming that both
donors are fully in the Kondo regime, the Kondo temperature does not change
over the magnetic field range probed and the inter-donor Coulomb interaction
is negligible, an expression for the magnetic-field dependent conductance can
be derived. The assumption of vanishing inter-donor Coulomb interaction is
supported by the analysis of low-temperature transport measurements that
are not presented here. Therefore, the zero-field Kondo effect can be modeled
as a single resonance at the Fermi energy at zero temperature, i.e. T ≪ TK as
can be seen in Figure 7.2b. The latter assumption is confirmed by slave boson
mean field calculations for parameters similar to the experimental conditions
[27, 28]. The effect of the magnetic field is incorporated through the AB
phase picked up during tunneling and a shift of the resonance away from the
Fermi energy by the Zeeman energy [21, 27]. The magnetic field dependent
conductance is then given by:
G(B) = GB=0
Γ̃21 + Γ̃22 + 2Γ̃1 Γ̃2 cos(φ)
Γ̃21 + Γ̃22 + ∆2Z + 2Γ̃1 Γ̃2 cos2 (φ/2) + Γ̃21 Γ̃22 sin4 (φ/2)/∆2Z
(7.1)
with ∆Z = gµB B, Γ̃1,2 the tunnel coupling of the donors to the leads and
φ = 2πB/BAB the AB phase where BAB ≈ 6.5 T is the AB period. The tunnel
7.5. Conclusions
85
coupling Γ̃1,2 represents an effective tunnel coupling for the coherent higher
order tunneling processes associated with the Kondo effect. They are different
from the “bare” tunnel coupling in that they take the interaction effects into
account, as is customary in the framework of slave boson mean field approaches
[27, 28]. Figure 7.3a shows a plot of the calculated zero-bias conductance
(Eq. 7.1) as a function of magnetic field. In the limit where Γ̃1 = Γ̃2 and
∆Z → ˜, with ˜ the detuning of the level with respect to the Fermi energy and
zero Zeeman shift, this expression is identical to the expression obtained by
Lopéz et al. [29]. Using parameters similar to the ones in the experiment [30],
we obtain a curve that is qualitatively the same as the Kondo data in Figure
7.2a. At low magnetic field the conductance is less strongly quenched in the
experimental data as compare to the theory. This can be understood by the
fact that the shift of the Kondo resonances away from zero bias is reduced
by interactions, especially at small fields [27]. Therefore, we conclude that
the conductance enhancement at ∼ 7.7 T can be explained as the constructive
interference of Kondo channels mediated by two donors.
7.4.3 Phase coherence
The modulation of the Kondo conductance by the magnetic field through the
Aharonov-Bohm effect is a clear experimental proof that the phase is preserved
in this system. Our experiment shows that the Aharonov-Bohm effect persists,
even though the phase is carried by a many-body Kondo state in each arm.
The role of the Kondo effect is to provide coherent transport channels, even
though sequential transport is completely blocked due to Coulomb blockade.
Since the presence of the AB oscillations are a consequence of the many-body
Kondo states being delocalized over both arms into the contacts, there could be
a spin-spin interaction present in that region and induce correlations between
localized spins [31–33]. An interesting open question is whether these kind of
interactions are useful in the context of the development of a scalable quantum
computer architecture.
Recently demonstrated placement of single donor atoms with atomic precision should allow to fabricated lattices of dopants [16, 34]. Using a gate, each
donor atom can be tuned such that it holds a single electron at low temperature. Such a structure can for example be used to study the formation of exotic
collective spin states at low temperature [35, 36]. This work demonstrates the
tunable properties of coupled donors and suggests that they can be used to
build a testbed system to study correlation effects.
7.5
Conclusions
In conclusion, we present experimental data of an Aharonov-Bohm effect for
two donors in a ring-shaped geometry in the regime where transport is mediated by a donor-induced Kondo effect. The Aharonov-Bohm phase is carried
by the many-body Kondo states in both arms of the interferometer. This observation is consistent with the phase modulation observed in the sequential
86
Non-local coupling of two donor- . . .
transport regime of the same device. A phenomenological model for the Kondo
conductance versus magnetic field confirms this by reproducing qualitatively
the same trend of the conductance versus magnetic field as the data. These
results contribute to the understanding of a system in which the advantageous
properties of single donors can be used utilized to create a testbed system to
study correlation effects in artificial lattices.
Acknowledgments
We are grateful to S. Biesemans and N. Collaert for providing us with the devices. This research was partly conducted by the Australian Research Council
Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027). S.R. acknowledges an ARC Future
Fellowship. This work is part of the research program of the Foundation for
Fundamental Research on Matter (FOM), which is part of the Netherlands
Organization for Scientific Research (NWO).
References
[1] Yacoby, A., Heiblum, H., Mahalu, D. & Shtrikman, H. Coherence and
phase sensitive measurements in a quantum dot. Phys. Rev. Lett. 74,
4047–4050 (1995).
[2] Zaffalon, M., Bid, A., Heiblum, M., Mahalu, D. & Umansky, V. Transmission phase of a singly occupied quantum dot in the kondo regime. Phys.
Rev. Lett. 100, 226601 (2008).
[3] Hatano, T. et al. Aharonov-bohm oscillations changed by indirect interdot
tunneling via electrodes in parallel-coupled vertical double quantum dots.
Phys. Rev. Lett. 106, 076801 (2011).
[4] Steger, M. et al. Quantum information storage for over 180s using donor
spins in a 28 Si “semiconductor vacuum”. Science 336, 1280–1283 (2012).
[5] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[6] Morton, J., McCamey, D., Eriksson, M. & Lyon, S. Embracing the quantum limit in silicon computing. Nature 479, 345–353 (2011).
[7] Lansbergen, G. P. et al. Tunable kondo effect in a single donor atom.
Nano Lett. 10, 455–460 (2010).
[8] Tettamanzi, G. et al. Magnetic-field probing of an SU(4) kondo resonance
in a single-atom transistor. Phys. Rev. Lett. 108, 046803 (2012).
[9] Verduijn, J. et al. Coherent transport through a double donor system in
silicon. Appl. Phys. Lett. 96, 072110 (2010).
References
87
[10] Goldhaber-Gordon, D. et al. Kondo effect in a single-electron transistor.
Nature 391, 156–159 (1998).
[11] Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in
the quantum theory. Physical Review 115, 485–491 (1959).
[12] Miroshnichenko, A., Flach, S. & Kivshar, Y. Fano resonances in nanoscale
structures. Rev. Mod. Phys. 82, 2257–2298 (2010).
[13] Sellier, H., Lansbergen, G. P., Caro, J. & Rogge, S. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett.
97, 206805 (2006).
[14] Tan, K. Y. et al. Transport spectroscopy of single phosphorus donors in
a silicon nanoscale transistor. Nano Lett. 10, 11–15 (2010).
[15] Pierre, M. et al. Single-donor ionization energies in a nanoscale CMOS
channel. Nature Nanotechnology 5, 133–137 (2010).
[16] Fuechsle, M. et al. A single-atom transistor. Nature Nanotechnology 7,
242 (2012).
[17] Roche, B. et al. Detection of a large valley-orbit splitting in silicon with
two-donor spectroscopy. Phys. Rev. Lett. 108, 206812 (2012).
[18] Lansbergen, G. P. et al. Gate-induced quantum-confinement transition
of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661
(2008).
[19] Cronenwett, S., Oosterkamp, T. & Kouwenhoven, L. A tunable kondo
effect in quantum dots. Science 281, 540–544 (1998).
[20] Goldhaber-Gordon, D. et al. From the kondo regime to the mixed-valence
regime in a single-electron transistor. Phys. Rev. Lett. 81, 5225–5228
(1998).
[21] Meir, Y., Wingreen, N. & Lee, P. Low-temperature transport through a
quantum dot: The anderson model out of equilibrium. Phys. Rev. Lett.
70, 2601–2604 (1993).
[22] Kobayashi, K., Aikawa, H., Katsumoto, S. & Iye, Y. Tuning of the Fano
effect through a quantum dot in an Aharonov-Bohm interferometer. Phys.
Rev. Lett. 88, 256806 (2002).
[23] For the fit we used: G() = G0 + {[Z1 ( − 1 )Γ1 + Z2 ( − 1 )Γ2 ]2 cos2 (φ/2) +
(Z1 ( − 2 )Γ1 − Z2 ( − 1 )Γ2 )2 sin2 (φ/2)}/{[( − 1 )( − 2 ) − Γ1 Γ2 (1 +
cos2 (φ/2))]2 + [Γ1 ( − 2 ) + Γ2 ( − 1 )]2 }, where Γ1,2 is the tunnel coupling to the donors (1,2), taking asymmetry into account, ( − 1,2 ) is the
88
Non-local coupling of two donor- . . .
detuning of donor level (1,2), φ the Aharonov-Bohm phase and the gate
voltage up to a conversion factor from voltage to energy. .
[24] der Wiel, W. V. et al. The kondo effect in the unitary limit. Science 289,
2105–2108 (2000).
[25] der Wiel, W. V. et al. Two-stage kondo effect in a quantum dot at a high
magnetic field. Phys. Rev. Lett. 88, 126803 (2002).
[26] Weis, J., Haug, R., Klitzing, K. & Ploog, K. Competing channels in
single-electron tunneling through a quantum dot. Phys. Rev. Lett. 71,
4019–4022 (1993).
[27] Dong, B. & Lei, X. Kondo-type transport through a quantum dot under
magnetic fields. Phys. Rev. B 63, 235306 (2001).
[28] Details of these calculations will be published elsewhere: R. R. Agundez,
J. Verduijn, S. Rogge and M. Blaauboer, in preparation.
[29] López, R. et al. Probing spin and orbital Kondo effects with a mesoscopic
interferometer. Phys. Rev. B 71, 115312 (2005).
[30] For the plot in Figure 3a the parameters are chosen as Γ1 = 0.3, Γ2 = 0.12
and BAB = 6.5.
[31] Craig, N. J. Tunable nonlocal spin control in a coupled-quantum dot
system. Science 304, 565–567 (2004).
[32] Vavilov, M. & Glazman, L. Transport spectroscopy of Kondo quantum
dots coupled by RKKY interaction. Phys. Rev. Lett. 94, 086805 (2005).
[33] Heersche, H. et al. Kondo effect in the presence of magnetic impurities.
Phys. Rev. Lett. 96, 017205 (2006).
[34] Schofield, S. et al. Atomically precise placement of single dopants in Si.
Phys. Rev. Lett. 91, 136104 (2003).
[35] Aynajian, P. et al. Visualizing heavy fermions emerging in a quantum
critical kondo lattice. Nature 486, 201 (2012).
[36] Park, W. et al. Observation of the hybridization gap and fano resonance
in the kondo lattice URu2 Si2 . Phys. Rev. Lett. 108, 246403 (2012).
A
Appendix A: The valley-orbit
correction
The energy spectra of the confined phosphorus donor in Figure 3.5 of chapter
3 are calculated within the effective mass framework described in that chapter. For simplicity, only the single valley ground-states were considered in
that discussion. Depending in the problem, it may however be necessary to
include excited states of the single valley Hamiltonian to accurately describe
the ground state and the lowest few excited states. For the bulk donor, the
contribution of the excited states to the lowest six perturbed states is negligible, since the valley orbit correction does not change the nature of the wave
function much. For a more strongly perturbed system, such as a donor subject
to a strong electric field, this may be not true anymore. In these cases, the
unperturbed wave functions can be very different from the perturbed ones.
Here, single valley wave functions are obtained by solving the Hamiltonian
on a grid. One advantage of this approach is that the wave functions that
form the basis for the perturbation are already close to the perturbed wave
function for an arbitrary potential, as opposed to for example a set of Gaussian
functions centered at the donor site. Therefore, it is expected that irrespective
of the shape of the potential, only a relatively small number of excited states
need to be included to obtain an accurate description of the perturbed ground
state. For both graphs in Figure 3.5a and b of chapter 3, 10 single valley
excited states have been used for the calculation. The method presented here
differs from the methods published by Debernardi et al. [1] and Baena et al.
[2]. Debernardi et al. use a large Gaussian basis (1800 functions). Baena et
al. attempt to guess a small basis in a variational way that is expected to
describe the wave function over the entire field range studied.
In general, a wave function can be written as a linear combination of states
in the six conduction band valleys:
6
N
ψ(r) = ∑ ∑ cn,µ ϕn,µ (r)
(A.1)
µ=1 n=1
where the conduction band valleys µ are labeled for convenience as {1, .., 6},
ϕµ,n (r) is the wave function n in the conduction band minimum, n labels the
N excited states within a valley and the coefficients are referred to as valley
2
coefficients, with the condition ∑6µ=1 ∑N
n=1 ∣cµ,n ∣ = 1. The valley state ϕµ,n (r)
89
90
Appendix A: The valley-orbit . . .
can be written as:
ϕn,µ (r) = Fn,µ (r)uµ (r)e−ir⋅kµ .
(A.2)
Here, the factor uµ (r)e
is the Bloch wave function at kµ , which is rapidly
varying in real space. Fn,µ (r) is a slowly varying envelope function which
satisfies the effective mass Schrödinger equation given by:
−ir⋅kµ
HFn,µ = En,µ Fn,µ
(A.3)
where En,µ is the energy corresponding to the state Fn,µ . The Hamiltonian,
H, in cartesian coordinates is given by:
̵ 2 1 ∂2
h
1 ∂2
1 ∂2
e2
+ eF ⋅ r.
(A.4)
H =− [
+
+
]−
2
2
2
2 mx ∂x
my ∂y
mz ∂z
4π∣r − r0 ∣
Here, mx,y,z = m∗x,y,z me is the effective mass of the electron in the lattice, with
me the free electron mass, = r 0 is the effective dielectric constant with 0
the dielectric constant of vacuum and F = (Fx , Fy , Fz ) is an applied electric
field. The Coulomb potential is centered at the position of the donor, r0 =
(x0 , y0 , z0 ). The silicon conduction band minima have an anisotropic effective
mass, a large mass in the lateral direction and a smaller mass in the transversal
one. For example, at the ±x minima m∗y = m∗z = 0.191 and m∗x = 0.916. The
Schödinger equation is solved on a grid using a finite difference method. By
discretizing the Schrödinger equation up to second order on a square grid, the
problem is turned into finding the eigenvalues/vectors of a matrix. This is
done by filling the elements of a sparse matrix in Matlab® and solving for
the eigenvalues using the “eigs()” function provided with Matlabs’ ‘parallel
computing’ toolbox , which uses an efficient parallel Lanczos algorithm. The
resulting eigenvectors (wave functions) are used as a basis for a perturbative
calculation as explained below.
For a bulk donor wave function the coefficients, cn,µ , are given by group
theory [3]. The crystal field splits the six-fold degenerate ground state into a
singlet ground state, a doublet and a triplet state with tetrahedral symmetry:
c(1,..,6)
=
1
√ (1, 1, 1, 1, 1, 1)
6
1
(1, 1, −1, −1, 0, 0)
2
1
√ (−1, −1, −1, −1, 2, 2)
12
1
√ (1, −1, 0, 0, 0, 0)
2
1
√ (0, 0, 1, −1, 0, 0)
2
1
√ (0, 0, 0, 0, 1, −1).
2
91
The splittings can be calculated in a perturbative way after Kohn and Luttinger [3] with a core potential as derived by Pantelides and Sah [4]. The idea
is basically that the behavior of the rapidly varying part of the wave function,
reflecting the atomic scale nature, and slowly varying part, dictated by the
long-range donor potential and possibly additional confinement, can be considered separately. The slowly varying envelope function satisfies an effective
mass equation and the effect of the rapid modulation is included afterwards
using with a perturbation to these wave functions. The perturbed Hamiltonian
of the system is given by:
H = H0 + H ′
(A.5)
Where the envelope functions Fn,µ (r) are eigenfunctions of H0 and H ′ serves
to include the overlap between different valley states. Physically, the binding energy increases because the confining potential deviates from a simple
Coulomb potential for ∣r − r0 ∣ ≲ aB , where aB is the Bohr radius and r0 the
position of the donor nucleus. This correction can be included by:
−e2
−e2
→
+ W (r)
4π∣r − r0 ∣
4π∣r − r0 ∣
where W (r) is a correction in terms of a varying dielectric constant obtained
from first principles [4]. The purely hydrogenic part in included in H0 and the
correction in H ′ . The function W (r) for phosphorus in silicon is given as:
W (r) = −Ae−α∣r−r0 ∣ − (1 − A)e−β∣r−r0 ∣ + e−γ∣r−r0 ∣
(A.6)
with A = 1.175 and α, β and γ equal 0.2524 nm−1 , 0.1041 nm−1 and 0.6813
nm−1 respectively [4]. This function has been derived from first principles
using the potential due to the core electrons of the phosphorus donor and the
high frequency dielectric response of silicon. Now the energies are given by:
⟨ψ∣H∣ψ⟩ = ⟨ψ∣H0 ∣ψ⟩ + ⟨ψ∣H ′ ∣ψ⟩
N
=
6
0
∑ ∑ cn,µ En,µ +
(A.7)
(A.8)
n=1 µ=1
N
∑
6
′
∑ cn,m,µ,ν En,m,µ,ν
n,m=1 µ,ν=1
where (n, m) label the single valley (excited) states and (µ, ν) the valleys.
Instead of using the local approximation of the nuclear wave function, W (r),
to calculate the correction to the energy directly, it is used to compute the
overlap between the states Fn,µ and Fm,ν induced by the donor nucleus. The
eigenenergies of the perturbed system can be found by inverting the above
equation and determining the eigenvalues of the 6N × 6N matrix [Hn,m,µ,ν ].
The matrix elements of the Hamiltonian H are given by:
C
Hn,m,µ,ν = βn,m,µ,ν
∆vo
µ,ν
(A.9)
92
Appendix A: The valley-orbit . . .
where ∆vo
µ,ν the bulk inter-valley overlap chosen to reproduce the correct splitC
ting of the A1 , T2 and E states. βn,m,µ,ν
is the interaction of Fn,µ and Fm,ν ,
normalized by the interaction for a bulk donor, and is given by:
C
βn,m,µ,ν
=
∗
∫ Fn,µ W (r)Fm,ν dr
.
∗ W (r)F
m,ν dr∣bulk
∫ Fn,µ
(A.10)
The value of the inter-valley overlap is the same for all lateral valleys ∆vo
l =
∆vo
µ,−µ . Also, all perpendicular valleys have the same overlap. The elements
C
on the diagonal Hn,n,µ,µ = βn,n,µ,µ
∆vo
µ,µ are equal to the single valley energy
En,µ , corresponding to the wave function Fn,µ (r). All other matrix element
vo
describe a transversal overlap ∆vo
t . So the matrix [∆µ,ν ] where the diagonal
elements are subtracted, is given by:
⎡
⎢
⎢
⎢
⎢
⎢
vo
∆ =⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
∆l
∆t
∆t
∆t
∆t
∆l
0
∆t
∆t
∆t
∆t
∆t
∆t
0
∆l
∆t
∆t
∆t
∆t
∆l
0
∆t
∆t
∆t
∆t
∆t
∆t
0
∆l
∆t
∆t
∆t
∆t
∆l
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎦
(A.11)
For for a phosphorus donor ∆l = −1.51 meV and ∆t = −2.16 meV results in
the correct splittings [5]. The binding energy is still somewhat off, 42.0 meV
instead of 45.6 meV, but the splitting is exactly match for all states.
The electric field induced valley splitting due to the mismatch of the wave
function at the interface interface can be included in a similar way and simply added to the matrix [∆vo
In case only a field in the z-direction,
µ,ν ].
F = (0, 0, Fz ), is considered, the inter-valley overlap matrix is given by [6]:
⎡
⎢
⎢
⎢
⎢
⎢
∆vs = ⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∣Vvs ∣eiθ
0
0
0
0
∣Vvs ∣e−iθ
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎦
(A.12)
Here, ∣Vvs ∣ is the magnitude of the inter-valley coupling which is a linear function of the field [7, 8] and θ is the phase of the coupling at the interface. This
phase, which depends on the details of the interface, is particularly important
if the tunnel coupling between a donor-like and interface-like state is considered. The valley splitting in this case is given by 2∣Vvs ∣. For Figure 3.5a in
chapter 3 the valley splitting was made dependent on the wave function density at the interface in a somewhat artificial way by integrating the overlap of
the wave function with an exponent that decays with a characteristic length of
References
93
1 nm from the interface. The absolute value of the valley splitting was tuned
to match the value of 0.087 meV⋅nm/mv as calculated by Rahman et al. [9].
Summary of the procedure:
○ Solve a single valley Hamiltonian H∣µ∣ to find the lowest N single valley
envelope wave functions and the corresponding energy. Note that the
envelope wave functions are the same for parallel valleys µ and −µ.
○ Compute the interaction integrals of the states in the µ and ν valleys
induced by the core potential for the bulk case. For phosphorus it was
found the a box of 133×133×133 with a grid spacing of 0.3 nm (40×40×40
nm) suffices.
C
○ These states are used to compute the interaction βn,m,µ,ν
for all N single
valley eigenstates in all six valleys. And, when considering a problem
with a non-zero electric field, compute the wave function density of the
wave function Fn,µ at the interface.
○ Then the Hamiltonian matrix [Hn,m,µ,ν ] is constructed by adding all
the perturbations and taking the diagonal elements as the energies of
the unperturbed single valley states.
○ Diagonalizing [Hn,m,µ,ν ] results in the corrected energy and valley coefficients cµ . From the valley coefficients the valley population can be
computed as P∣µ∣ = ∣cn,µ ∣2 + ∣cn,−µ ∣2 . Also the wave function can be constructed as a linear combination of single valley wave functions. It should
be noted, however, that depending on the problem and the level of accuracy required, a much larger basis set may be needed to get and accurate
description of the wave function than generally is needed to obtain the
energy spectrum with a sufficient accuracy.
References
[1] Debernardi, A., Baldereschi, A. & Fanciulli, M. Computation of the Stark
effect in P impurity states in silicon. Phys. Rev. B 74, 035202 (2006).
[2] Baena, A., Saraiva, A., Koiller, B. & Calderón, M. Impact of the valley
degree of freedom on the control of donor electrons near a Si/SiO2 interface.
Phys. Rev. B 86, 035317 (2012).
[3] Kohn, W. & Luttinger, J. Theory of donor states in silicon. Physical
Review 98, 915–922 (1955).
[4] Pantelides, S. & Sah, C. Theory of localized states in semiconductors. I.
New results using an old method. Phys. Rev. B 10, 621–637 (1974).
94
Appendix A: The valley-orbit . . .
[5] Koiller, B., Hu, X. & Sarma, S. D. Strain effects on silicon donor exchange:
Quantum computer architecture considerations. Phys. Rev. B 66, 115201
(2002).
[6] Saraiva, A., Calderón, M., Hu, X., Sarma, S. D. & Koiller, B. Physical
mechanisms of interface-mediated intervalley coupling in Si. Phys. Rev. B
80, 081305 (2009).
[7] Sham, L. & Nakayama, M. Effective-mass approximation in the presence
of an interface. Phys. Rev. B 20, 734–747 (1979).
[8] Saraiva, A. et al. Intervalley coupling for interface-bound electrons in
silicon: An effective mass study. Phys. Rev. B 84, 155320 (2011).
[9] Rahman, R. et al. Orbital stark effect and quantum confinement transition
of donors in silicon. Phys. Rev. B 80, 165314 (2009).
B
Appendix B: Supporting information
to chapter 5
Methods
In chapter 5 some well established methods were used. This section gives a
summary of these methods.
The low temperature transport measurements were carried out at 4.2 K
with the device submerged in liquid helium. A home-build low noise battery
operated measurement setup was used to measure the drain current and apply
voltages. Since the substrate wafer is not very well conducting at the experimental temperature, the current traces as a function of topgate voltage are
acquired taking some wait time into account after setting the backgate voltage.
In most cases five minutes was found to be sufficient.
To extract the position and magnitude of the resonance induced by the
donor in a consistent way, a thermally broadened Coulomb blockade resonance [1] was fit to the data Id = Γ′ e2 /4kB T cosh−2 (αtg Vtg /2kB T ), with e the
electron charge, αtg the capacitive coupling to the topgate and Γ′ the tunnel
coupling, up to a factor accounting for a possible asymmetry in the tunnel
coupling to source and drain. Near the anti-crossing
shown in Figure 7.3d a
√
dispersion relation of a two-level system, E = ± 2 + ∆2 , is fit to extract the
hybridization gap ∆, where E is the resonance position and the detuning of
the uncoupled donor and interface-well states. To convert the fit from voltages
to energies, the conversion factors are determined from the transport data. By
measuring a charge stability diagram as a function of source drain voltage and
topgate voltage at various backgate voltages, it is found that the coupling to
the topgate is approximately a linear function of the backgate voltage and
varies between 0.44 meV/mV at Vbg = 0 V and 0.18 meV/mV at Vbg = 12.5 V.
A linear fit results in αtg = −0.020 × Vtg + 0.45 in meV/mV, with an estimated
uncertainty of 10%. Using the data in Figure 7.2a the backgate coupling αbg
with respect to the topgate is obtained as αbg /αtg ≈ 32 mV/V over the same
voltage range.
Comparison to data of doped and undoped device
The subject of chapter 5 is a device with a sub-threshold resonance associated
with a single phosphorus donor at low temperature. Here, these data are
compared to a similar undoped device. The undoped device has a channel
95
96
Appendix B: Supporting information . . .
length of 25 nm a width of 20 nm and a channel height of 12 nm for the doped
device this is 40 nm, 60 nm and 20 nm respectively.
a
b
Back gate voltage [V]
20
10
10
0
−10
0
Id [nA]
8
0
undoped
doped
−10
−400 −200 0
200 400
−600 −400 −200
0
200
Top gate voltage [mV]
Top gate voltage [mV]
Figure B.1: Low temperature (4.2 K) transport data of similar undoped a) and doped b)
devices. For both devices the color scale is the same and the data has been acquired at 0.2
mV source/drain bias voltage.
Figure B.1 shows data of two devices, one undoped (Figure B.1a and one
doped Figure B.1b, the latter data is the same to Figure 5.2a of chapter 5).
Except for the doping, these devices are very similar. The change in slope of
the onset of the current at slightly positive backgate voltage can be understood
as charge moving in the channel and will be discussed in more detail in the
next section. In contrast to the undoped device, the doped device shows more
disorder, which can be understood as induced by the 3-7 phosphorus donor
present in the channel. One of these dopants results in the Coulomb blockade
resonance present well below the threshold conduction band. The binding
energy associated with this donor is estimated from the maximum distance
of the resonance to the intrinsic device threshold. Using the gate coupling
extracted with the method described in the “Methods” section above, a value
of ∼ 35 meV is found. This significant reduction with respect to the bulk value
(45.6 meV) may be due to electrostatic screening by image charges of in the
non-uniform dielectric environment of the device [2] or by the electric fields
subject to the donor [3].
Capacitive coupling
In principle, the electrostatic coupling of the donor, as extracted by sweeping
the voltages on the gates, present in the device can be used to estimate its
position in the channel. This requires, however, a model describing the electrostatics of the device, which is not trivial for the used nano-wire FET. As
many microscopic details of the device structure are unknown, the position of
the donor will only be discussed on a qualitative level.
97
Figure B.2: The derivative of the donor threshold voltage and the intrinsic device threshold
compared. Both insets show the location of localized states in the device channel with respect
to the topgate “tg” and backgate “bg”. The solid vertical line denotes the flatband voltage of
2.5 V as from the fit of the tunnel current, i.e. the minimum in the donors’ resonant tunnel
current.
From the data shown in Figure B.1b, the position of the donor resonance
and the device threshold, indicated by the green dashed line in Figure B.1b, the
derivative dVtg,th /dVbg was calculated, with Vtg,th the topgate threshold and
Vbg the backgate voltage. This derivative can be interpreted as a the capacitive
coupling of the localized state to the topgate with respect to the backgate.
Figure B.2 shows the resulting graph. Apart from the region between ∼ 2.5 V
and ∼ 3.5 V, the coupling is constant. The change in capacitive coupling of the
intrinsic device threshold (the dashed line) can be understood as the charge
density moving from the top side at low backgate voltage to the bottom side
of the channel at high backgate voltage, see insets Figure B.2. For this charge
the linear dielectric screening of the silicon in the 20 nm thick channel results
in a different capacitive coupling for these two cases. For the donor resonance
this is not expected to play a role, as, over the shown backgate voltage range,
the charge is localized on the donor (see chapter 5) and therefore no shift is
expected. A possible explanation could be that interface states, that do not
carry any measurable current, are responsible for this screening effect. The
reason these transitions take place at a positive backgate voltage instead of ∼ 0
V is most likely a work function difference between the polycrystalline silicon
gate and the channel.
Since the slope is consistently smaller below flatband and larger above
flatband, the donor is must be located in the bulk of the channel, at some
distance from that top and back side of the channel, as the capacitive coupling
is always somewhat weaker than for a state at interface (dashed line). Linear
screening of the silicon between the donor and the bottom side of the channel
reduces the coupling to the gates. Further, interface states not visible in the
transport may be screening the donor.
Modeling of the tunnel coupling
As discussed in chapter 5, the tunnel current is proportional to the wave
function overlap between the states in the source and drain contacts and the
localized state. An approximate value of the tunnel coupling can be calculated
by integrating over the electron wave function at some distance away from the
donor nucleus, similar to Bardeen’s approximation. Implicitly, the part of the
wave function penetrating into the barrier region from the source or drain
contact to the donor, is assumed to be exponentially decaying.
To obtain the donor wave at a certain distance from the interface subject
98
Appendix B: Supporting information . . .
on an electric field the following Hamiltonian is used:
̵ 2 ∂2
̵ 2 ∂2
̵ 2 ∂2
h
h
h
e2
H =−
−
−
−
+ eF z,
(B.1)
2
2
2
2mt ∂x
2mt ∂y
2ml ∂z
4π0 Si ρ
√
were ρ = x2 + y 2 + (z − d)2 and F the electric field. We take mt and ml as
0.191 and 0.916 times the free electron mass respectively and Si as 11.9.
0.7
Tunnelcoupling [−]
0.6
0.5
0.4
0.3
0.2
0.1
0
−15
−10
−5
0
5
10
Electric field [mV/nm]
15
Figure B.3: Tunnel coupling as a function of electric field for the z-valley for a donor in
the center of a Lx × Ly × Lz = 40 × 40 × 20 nm box, so at 10 nm from the interface. This
curve has been fit to the data in the first fitting step as described in the next section.
By solving the Schrödinger equation Hψ = Eψ on a grid using a finite
differences method, the ground state wave function is computed for different
values of the electric field F . For all calculations presented, a grid of 133 ×
133 × 76 is used with a spacing between the grid points of 0.3 nm (the box is
40 × 40 × 20 nm). For the zero field situation the result is close to a bulk donor
value and Bohr radii of 2.61 nm for the transversal direction and 1.46 nm for
the lateral direction are found. Integrating these wave functions over a plane
at 18 nm from the donor nucleus perpendicular to the transport direction
(an yz-plane), a field dependent tunnel coupling as shown in Figure B.3 is
obtained. Because many details of the system, such as the density of states
in the contacts, are unknown, an absolute value for the tunnel current cannot
be calculated. Instead, a pre-factor and offset free as well as a linear scaling
factor for the electric field conversion to backgate voltage is left free in the
fitting procedure explained in the next section.
Fit procedure to obtain the valley composition
The valley population of the electron wave function as discussed in the main
text can only be determined by making a couple of assumptions. Here, the
99
assumptions, the used model and the fitting procedure are discussed in detail.
Inspired by the work of Baena et al. [4], the following smooth function is
used to qualitatively describe the valley composition of the x and y valleys as
a function of backgate voltage:
Vbg − V1
Vbg − V2
1
1 1
1
Px,y (Vbg ) = [ arctan (
) + ] [ arctan (
)+ ]
π
w1
2 π
w2
2
(B.2)
where V1,2 are the points at which the valley compositions decays to zero over
a gate voltage range w1,2 . For the fit constraints are chosen such that V1 < 0
and V2 > 0. Since at Vbg ≈ 2.5, the flatband voltage, the donor is assumed
to be bulk-like, the Px,y,z (Vbg = 2.5) = 1/3. Since the x and y valleys are
affected in the same way by an electric field in the z-direction (they both
have a light mass in the z-direction) and the valley population is normalized,
Pz (Vbg ) = 1 − 2Px,y (Vbg ).
1
10
data
fit 1
fit 2
ii
Peak drain Current [nA]
0
10
−1
10
i
−2
10
−3
10
i:
ii :
fit 2 :
−5
0
5
Backgate Voltage [V]
10
Figure B.4: A fit is performed in two steps: First the curves i and ii are fit to the data are
indicated. Then Eq. B.2 combined with these fit results are to all data points in the central
region of the graph to extract the valley composition of the ground state.
The fit is performed in two steps:
○ Step 1: Fitting the tunnel coupling as obtained from a single valley
Hamiltonian to the data. Once for small backgate voltage (or weak
electric field) and once for larger backgate voltage (stronger field).
○ Step 2: Using these two curves as obtained from the fit, and the backgate
voltage dependent valley composition, a second fit is performed to fix V1,2
and w1,2 .
100
Appendix B: Supporting information . . .
Figure B.4 shows the data with colored data points to indicate how the fitting
was performed. Curve i was fit on the green and blue data points and curve
ii on the green and read data points. The fit results are denoted to as fi (Vbg )
and fii (Vbg ) respectively. Then a second fit was performed using the function:
1
1
g(Vbg ) = Pz (Vbg )[a1 fii (Vbg ) + b1 ] + [1 − Pz (Vbg )][a2 fi (Vbg ) + b2 ]. (B.3)
2
2
Here, it was assumed that half of the current was carried by the y-valley and
the other half by the z-valley since they have initially (at Vbg = 2.5 V) the same
spatial extent in the transport direction. The x-valley is assumed to be not
significantly contributing to the current. At high field the contribution of the
z-valley will increase, shifting the valley population from the x- and y-valley
into the z-valley resulting in the curves shown in Figure 5.3b in chapter 5.
Finally, a remark should be made regarding the quantitate value of this
result. The details of the function Pz (Vbg ) are highly dependent on how the
data points are chosen for the fit as shown in Figure B.4. The the overall trend,
however, is consistently an increase of the z-valley population. Therefore, the
graph in Figure 5.3b should not be considered a quantitate result, but rather
a qualitative one.
References
[1] Beenakker, C. Theory of coulomb-blockade oscillations in the conductance
of a quantum dot. Phys. Rev. B 44, 1646–1656 (1991).
[2] Calderon, M. J. et al. Heterointerface effects on the charging energy of the
shallow D− ground state in silicon: Role of dielectric mismatch. Phys. Rev.
B 82, 075317 (2010).
[3] Rahman, R. et al. Electric field reduced charging energies and two-electron
bound excited states of single donors in silicon. Phys. Rev. B 84, 115428
(2011).
[4] Baena, A., Saraiva, A., Koiller, B. & Calderón, M. Impact of the valley
degree of freedom on the control of donor electrons near a Si/SiO2 interface.
Phys. Rev. B 86, 035317 (2012).
Conclusions & Outlook
Building on the experimental results and new insights presented in this thesis,
some possible future directions for the research are given. For comprehensiveness first a very brief summary of the results is given. Quite a conservative
approach to indicate possible future directions is chosen: mostly established
and proven techniques are utilized.
Summary of main results:
○ Understanding mechanisms that determine the ground-state/excited-state
splitting is important in the context of silicon quantum device applications. The splitting between the ground state and first excited state
in nano-structures with donors and without any donors present is investigated. By comparing these splittings to a large-scale tight binding
model, the intimate relation between valley-splitting in quantum dots
and quantum wells and the valley-orbit interaction in the presence of
donors is studied. The results indicate that it is possible to tune the
ground-state/excited-state gap from 12 or 21 meV, for bulk phosphorus
and arsenic respectively, down to values < 1 meV, by properly designing
the nano-structure and positioning of the donor.
○ Electrical control over the wave function control of a single donor is
demonstrated. Even though many proposed donor-based quantum computer schemes rely on the possibility to manipulate the wave function
of an electron bound to a single donor, there have been very few experiments that address this aspect, so far. By probing the single donor
wave function using transport spectroscopy in a gated nano-structure, it
is shown that by tuning the electric field in the device, the wave function
can be deformed. Furthermore, a non-trivial change in the conduction
band valley population of the ground state was observed.
○ Coherent coupling of single donors is another key ingredient for quantum device applications. The observation of a Fano resonance and the
modulation of sequential transport in a magnetic field, shows that two
donors in a field effect transistor are coherently coupled. This coupling
is mediated by delocalized electrons in a metallic region that are tunnel
coupled to both donors.
○ As an example of a system where coherence is carried by a many-body
state, i.e. two donors coupled by a Kondo interaction, are investigated.
The magnetic field dependence shows that this results in constructive
101
102
Conclusions & Outlook
and destructive interference. This demonstrates that coherent coupling
of donors in nano-structures can be carried by a many-body interaction.
It also suggests that coupled dopants could serve as a model system to
study correlated physics.
Next, building on the knowledge presented in this thesis, some new experiments
are proposed. Some of them require only modifications to the experimental
setup and some require modifications to the device as well.
Implementing fast gating capabilities on the experimental setup will allow
to control the dynamics of the donor-interface-well system [1]. Combining this
with local charge sensing capabilities [2], it becomes possible to investigate the
dynamics of a one-donor building block of the donor-based quantum computer
[3] as well as to investigate the implementation of a charge-based qubit [4]. Recently, it has been shown that for sufficiently high doping levels of the channel
region, it is possible to investigate tunnel coupled donors [5]. With the capability of applying a modulated magnetic field, a true two-donor Kane quantum
computer can be demonstrated. It should be noted however, that a critical
requirement here is the ability to tune the exchange coupling. Calculations
that are published show that this should be possible [6, 7], but no experiments
have confirmed this yet. Further, a system like this is not easily scalable and
will therefore probably not be useful as a quantum computer. Nevertheless,
this structure will surely provide fundamental new insights in how this goal
can be achieved. A detailed understanding of properties, such as the role of
valley-orbit interaction in the dynamic properties of this system, are crucial
to any donor-based device application.
Drawing from the knowledge obtained about systems of indirectly coupled donors [8], it is interesting to investigate the role of spin interactions in
these systems. It has been shown that charge coherence can be preserved and
Kondo effects are often observed for donors that are strongly tunnel coupled
to the source and drain contacts in the studied FinFETs [9]. So far, the work
on donors in transport has been focussed mainly on single donors or simple
coupled systems. Accepting that this is a disordered system with (a few) randomly placed donors that may be coupled, the effect of spin-spin interactions
can be studied. In addition, these interaction effects may be used to sense
the spin state of localized electrons or even the nuclear spin of donors [10]. A
combination of a microwave field and a static magnetic field should allow for
manipulation of these spins as well.
Another approach to take advantage of the precision placement of single
donors using hydrogen lithography techniques with a scanning tunneling microscope (STM) that have been recently developed [11]. This would allow to
create arrays of dopants, thereby creating a superlattice. Combining this with
local gating, it should be possible to study collective excitations. This can be
done either in a transport device or in situ using the tip of the STM.
References
103
References
[1] Dupont-Ferrier, E. et al. Coupling and coherent electrical control of two
dopants in a silicon nanowire. arXiv cond-mat.mes-hall (2012). 1207.
1884v1.
[2] Morello, A. et al. Single-shot readout of an electron spin in silicon. Nature
467, 687 (2010).
[3] Kane, B. A silicon-based nuclear spin quantum computer. Nature 393,
133–138 (1998).
[4] Koiller, B., Hu, X. & Sarma, S. D. Electric-field driven donor-based charge
qubits in semiconductors. Phys. Rev. B 73, 45319 (2006).
[5] Roche, B. et al. Detection of a large valley-orbit splitting in silicon with
two-donor spectroscopy. Phys. Rev. Lett. 108, 206812 (2012).
[6] Fang, A., Chang, Y. & Tucker, J. Effects of J-gate potential and uniform
electric field on a coupled donor pair in Si for quantum computing. Phys.
Rev. B 66, 155331 (2002).
[7] Wellard, C. et al. Electron exchange coupling for single-donor solid-state
spin qubits. Phys. Rev. B 68, 195209 (2003).
[8] Verduijn, J. et al. Coherent transport through a double donor system in
silicon. Appl. Phys. Lett. 96, 072110 (2010).
[9] Lansbergen, G. P. et al. Tunable kondo effect in a single donor atom.
Nano Lett. 10, 455–460 (2010).
[10] Vincent, R., Klyatskaya, S., Ruben, M., Wernsdorfer, W. & Balestro,
F. Electronic read-out of a single nuclear spin using a molecular spin
transistor. Nature 488, 357–360 (2012).
[11] Fuechsle, M. et al. A single-atom transistor. Nature Nanotechnology 7,
242 (2012).
Summary: silicon quantum
electronics
Quantum mechanical effects can be exploited for fundamentally different device operation principles. Integrated circuits consisting of transistors have been
improved over time in an exponential way, following Moore’s Law. In order to
keep improving pcs, laptops, smartphones, etc. this way, device designs that
are radically different from the conventional planar transistor geometry are
introduced. Besides looking for novel device designs, researchers are searching
for different device operation principles. A design that is currently being developed is the Tunnel FET. For this device controllable tunneling of electrons
is used to switch it on and off.
The most common way to perform a computation is by using Boolean logic
bits (0s and 1s). There are, however, theoretical proposals which promise that
computations for certain classes of problems can be performed much faster on
a quantum computer. A quantum computer uses quantum bits, or qubits, to
perform calculations. Instead of the two states of a classical bit, “0” and “1”,
the qubit can be partly in a state ∣0⟩ and partly in ∣1⟩ at the same time. The
power of the quantum computer lies in the fact that it can be used to perform
computations in parallel. A physical implementation of a quantum computer
has proven to be difficult to realize, mainly because of the loss of coherence
(information) in the states of the qubits.
Besides the investigation of quantum mechanical systems for quantum computation, they also can provide new insights in open problems in fundamental
physics. Dopants embedded in nano-scale devices can be used as a testbed
system for this. A device that is particularly suitable for this is a threedimensional field effect transistor (FET) with a fin-shaped channel, the FinFET. The unique properties of dopants combined with the flexible device design of the FinFET, allows to perform experiments that can not easily be done
in a different way.
The devices have been fabricated in a foundry cleanroom on a 200 mm
wafer. Part of the used fabrication processes are standard CMOS (Complementary Metal Oxide Semiconductor) processes, and some steps, such as the
device patterning, are customized. Despite this, mass production of these devices should in principle be possible with current industrial technologies. By
measuring currents as a function of voltages on the gates at low temperature,
localized electron states of the device are probed. Because of the Coulomb
repulsion, electrons tunnel one-by-one through a single donor in the FinFET.
This enables the investigation of the states of single donors. Further, the effect
105
106
Summary
of inference of electrons and correlations, closely associated with the transport,
can be investigated. These physical principles, together with the reproducible
fabrication of the devices, provides a powerful system to study single dopants.
Donors in silicon, such as arsenic and phosphorus atoms, are promising
candidates to serve as the basic building blocks for a quantum computer (i.e.
qubits). The main reason for this is that, in contrast to, for example, quantum
dots, the potential confining electrons is naturally formed by the donor nucleus
which substitutes a silicon atom in the lattice. Therefore, all donors are virtually identical. This is a good starting point to build a large-scale network
of coupled donor-based devices. Another favorable property is the long spin
coherence time. The spins of either the donor nucleus or the electron bound
to the donor can preserve the coherence of a quantum state for a very long
time. A disadvantage however is that manipulation of these spins is relatively
slow, allowing a limited number of computational operations to be performed
within the coherence time. Using the electron charge to encode a qubit state
instead of the spin is much faster and may be a feasible alternative as recent
experiments indicate.
Manipulation of single donors can be achieved by embedding them in nanostructures. In a nano-structure the donor-bound electrons are likely to be
subject to external perturbations which modify the hydrogenic donor potential
of the nucleus. The energy spectrum differs from a simple scaled hydrogen
spectrum due to the indirect bandgap nature of the conduction band in silicon.
This makes the splitting between the ground state and the first excited state
very sensitive to external perturbations. An electric field can change this
splitting from tens of milli-electronvolt in the bulk case to close to zero.
Because of the sensitivity of the level spectrum of donors to external perturbation by an electric field, gate control is possible. By performing transport
spectroscopy measurements the excited state spectrum of single gated donor
in nano-scale FinFETs was determined. Combining this with large scale tight
binding calculation the distance of the donor from the channel/gate interface
as well as the local electric field was determined. Also devices without any
donors present were investigated and the ground state/excited state splitting
was determined by temperature dependent measurements. These measurements on different devices provide a proof of principle result for the tunability
of the gap between the ground state and first excited state.
Combining the top gate of a FinFET and the ability to use the substrate as
a back gate, the local electric field subject to a donor was tuned, independent of
the chemical potential. This allowed to deform the electron wave function, and
thereby tune its energy. From the observed capacitive coupling it was deduced
that the electron state can be tuned from bulk-like, with the electron tightly
bound to the donor, to a state localized in an electric field induced interface
well state. Furthermore, the amplitude of the tunnel current mediated by the
electron state was used to probe the character of the wave function. It was
Summary
107
shown that the population of the six conduction band minima (valleys) changes
from an approximately equal distribution across the six valleys in the bulk-like
case, to only populating two valleys parallel to the field. This demonstrates the
manipulation of a single donor electron, as is required for the implementation
of single donor-based quantum device applications.
In order to build a scalable structure of coupled donors, coherent coupling
is necessary. Coherent coupling of two donors was demonstrated in transport
in a FinFET. The coupling resulted in a resonance with an asymmetric Fano
line shape as a function of gate voltage in the sequential electron tunneling
current. As a function of magnetic field the symmetry of the Fano resonance
changed. This observation was used to show that the donors are not directly
coupled, but spatially separated and coupled to each other via the metallic
leads in a ring configuration.
At low temperature, electron-electron correlation effects start to play a
role. Donors in FinFETs are ideal systems to study these effect because they
provide a robust tunable system. In particular, two coupled donors in a FinFET channel showed evidence of coherent coupling mediated by a many-body
Kondo state. By studying the magnetic field dependence, it was shown that
Aharonov-Bohm oscillations persists even though the coherence is carried by
a correlated system. These results suggest that these type of interactions may
be useful in the context of creating a scalable donor-based quantum computer. Combined with the recently developed atomic precision dopant placement techniques, it may be possible to create a model system in which exotic
correlated states can be studied.
Donors embedded in FinFETs provide a robust and flexible platform to
study a range of physical effects. Transport spectroscopy is a powerful tool to
probe the electronic structure of single and coupled donors. Furthermore, a
donor is a naturally occurring system and, therefore, all donors are virtually
identical. This makes them attractive building blocks for quantum device
applications.
Jan Verduijn
September 2012
Samenvatting: silicium
kwantumelektronica
Kwantummechanische effecten kunnen worden gebruikt voor fundamenteel
nieuwe werkingsprincipes van componenten in geïntegreerde circuits. Conventionele geïntegreerde circuits zijn opgebouwd uit transistors. Transistors
zijn in de loop van de tijd verbeterd op een exponentiële manier volgens de Wet
van Moore. Om pc’s, laptops, smartphones, etc. te kunnen blijven verbeteren
met deze snelheid, is het noodzakelijk om nieuwe, onconventionele ontwerpen
te introduceren. Deze ontwerpen zijn vaak radicaal anders dan het ontwerp
van de conventionele vlakke transistor. Daarnaast werken wetenschappers en
ingenieurs aan nieuwe werkingprincipes voor de bouwstenen van circuits. Een
ontwerp dat op dit moment veelbelovend lijkt is de tunnel veld effect transistor. Bij dit ontwerp wordt gebruik gemaakt van het gecontroleerd tunnelen
van elektronen om de transistor te schakelen.
Veruit de meest gebruikelijke manier om berekeningen uit te voeren is met
behulp van Booleaanse logische bits (nullen en enen). Maar er bestaan theoretische voorstellen die voorspellen dat een zogenaamde kwantum computer een
bepaald soort problemen veel sneller op kan lossen. Een kwantum computer
maakt gebruik van kwantum bits om berekeningen uit te voeren. In plaats van
de twee toestanden van een klassieke bit, kan een kwantum bit op hetzelfde
moment gedeeltelijk in een toestand ∣0⟩ en gedeeltelijk in ∣1⟩ kan zijn. De
kracht van een kwantum computer vindt zijn oorsprong in het feit dat berekeningen op een parallelle manier uitgevoerd kunnen worden. Het blijkt in de
praktijk moeilijk te zijn om een kwantum computer te bouwen. Dit komt voor
een belangrijk deel doordat de toestand van een kwantum bit zeer gevoelig is
voor decoherentie, ofwel verlies van informatie.
Naast het onderzoeken van kwantum systemen voor toepassingen als de
kwantum computer, kan het onderzoeken van zulke systemen ook leiden tot
belangrijke inzichten in fundamentele fysische vragen. Doteringsatomen, ingebed in elektronische componenten met nanometer afmetingen, kunnen worden
gebruikt als een modelsysteem om dit te doen. Een ontwerp dat hiervoor bij
uitstek geschikt is, is de veldeffect transistor met een vin-vormig kanaal, ook
wel een FinFET genoemd (Engels: Fin Field Effect Transistor). De unieke
eigenschappen van doteringsatomen, gecombineerd met het flexibele ontwerp
van de FinFET, maken experimenten mogelijk die niet gemakkelijk op een
andere manier uitgevoerd kunnen worden.
De FinFETs worden gefabriceerd in een foundry cleanroom op 200 mm
wafers. Een gedeelte van gebruikte fabricageprocessen zijn standaard CMOS
109
110
Samenvatting
processen (Engels: Complementary Metal Oxide Semiconductor). En andere
processen, zoals het schrijven van de structuren, zijn op maat ontwikkeld. Desondanks is met de huidige beschikbare technologieën de massaproductie van
deze structuren in principe mogelijk. Door bij lage temperatuur stromen te
meten als functie van de voltages op de FinFET kan de toestand van opgesloten elektronen worden waargenomen. Door de Coulomb afstoting tunnelen
de elektronen een-voor-een door een geïsoleerd doteringsatoom in de FinFET.
Dit maakt het bestuderen van de eigentoestanden van één geïsoleerd doteringsatoom mogelijk. Daarnaast kunnen ook interferentie-effecten van elektronen
en het effect van interacties op het transport van de elektronen worden onderzocht. Deze fysische principes, samen met het feit dat de gebruikte structuren
op een reproduceerbare manier gemaakt worden, maken dit een krachtig systeem om geïsoleerde doteringsatomen te onderzoeken.
Doteringsatomen in silicium, zoals arseen en fosfor atomen, zijn veelbelovende kandidaten om als bouwstenen voor een kwantum computer (kwantum
bits) te dienen. De belangrijkste reden hiervoor is dat, in tegenstelling tot
bijvoorbeeld kwantum dots, het potentiaal wat de elektronen opsluit op een
natuurlijke manier wordt gevormd door de nucleus van het doteringsatoom.
Daarom zijn alle doteringsatomen nagenoeg identiek. Dit is een goede eigenschap om geïsoleerde doteringsatomen te kunnen gebruiken voor kwantum
toepassingen. Een andere gunstige eigenschap is de dat de coherentie van de
spin toestanden relatief lang behouden blijft. Maar tegelijkertijd is deze spin
niet heel erg snel te manipuleren. Hierdoor kunnen maar een beperkt aantal
bewerkingen binnen de coherentietijd uitgevoerd worden. Een alternatieve,
veel snellere manier om doteringsatomen te gebruiken is door de lading te gebruiken om de kwantuminformatie in op te slaan. Recente experimenten geven
aan dat dit misschien in de toekomst tot de mogelijkheden kan behoren.
Een elektron gebonden aan een geïsoleerd doteringsatoom kan worden beïnvloed door het in te bedden in een nano-structuur. In deze nano-structuur
zijn de doteringsatomen vaak onderhevig aan elektrische velden en voelen opsluiting naast het potentiaal van de nucleus. Doordat silicium een materiaal
is met een indirecte bandkloof, zijn de kwantummechanische toestanden niet
simpelweg hetzelfde als die van een geschaald waterstof atoom. Dit heeft als
gevolg dat de afstand tussen de grondtoestand en eerste aangeslagen toestand
heel gevoelig is voor externe verstoringen. Bijvoorbeeld onder invloed van
een extern elektrisch veld kan deze afstand variëren van een tientallen millielektronvolts in de onverstoorde toestand tot bijna nul.
Deze gevoeligheid voor elektrische velden maakt het mogelijk de toestanden
met behulp van elektrische potentialen nauwkeurig in te stellen. Door middel
van elektrontransportspectroscopie kan het energiespectrum van toestanden
van de geïsoleerde doteringsatomen worden bepaald. Voor data van een aantal
FinFETs wordt dit vervolgens vergeleken met een berekening op basis van een
methode die gebruik maakt van miljoenen atomaire orbitalen (Engels: large-
Samenvatting
111
scale tight binding). Hieruit volgt de afstand van het doteringsatoom tot het
oppervlak van het kanaal en de grote van het lokale elektrisch veld. Ook
de afstand van de grondtoestand tot de eerste aangeslagen toestand is voor
structuren zonder doteringsatomen met behulp van temperatuurafhankelijke
metingen onderzocht. Door al deze informatie te combineren wordt duidelijk
te het in principe mogelijk moet zijn om de afstand tussen de grondtoestand
en de eerste aangeslagen toestand in te stellen.
Het elektrisch veld in het kanaal van een FinFET kan worden ingesteld door
de stuurspanning op de bovenste elektrode van de FinFET te combineren met
de juiste spanning op het substraat. Deze aanpak maakt het mogelijk om het
elektrisch veld onafhankelijk van de potentiaal in het kanaal te variëren. Hiervan is gebruik gemaakt om voor het eerst aan te tonen dat de golffunctie van
een elektron gebonden aan een doteringsatoom op een gecontroleerde manier
kan worden vervormd. Door middel van de capacitieve koppeling kon worden
aangetoond dat de golffunctie een transitie ondergaat van onverstoord, waarbij
het elektron strak aan de nucleus is gebonden, naar een oppervlaktetoestand.
Daarnaast leverde de amplitude van de tunnelstroom door de toestand informatie op over de aard van de golffunctie. Er kon op deze manier worden
aangetoond dat de bezetting van de geleidingsbandminima varieert. In een onverstoorde toestand is deze bezetting gelijk voor alle zes minima. Maar onder
invloed van een elektrisch veld gaat de bezetting naar twee van de zes minima
parallel aan het veld. Dit experiment laat zien dat de controle over een enkel
elektron gebonden aan een doteringsatoom inderdaad mogelijk is. Hoewel dit
vaak voor gegeven wordt aangenomen als men praat over kwantumtoepassingen van doteringsatomen was dit nooit eerder experimenteel bevestigd voor
een enkel doteringsatoom.
Om een netwerk van enkele gekoppelde doteringsatomen te bouwen is het
noodzakelijk om deze op een coherente manier te koppelen. Experimenten
aan twee doteringsatomen in een FinFET hebben laten zien dat dit inderdaad
mogelijk is. De koppeling die hierbij een rol speelde resulteerde in een asymmetrische Fano resonantie in de tunnelstroom als functie van de stuurspanning
op de FinFET. Wanneer een magnetisch veld werd aangelegd bleek de resonantie van vorm te veranderen. Met dit laatste geven kon worden aangetoond
dat het in dit experiment gaat om doteringsatomen die zich ruimtelijk vrij ver
uit elkaar bevinden. De koppeling tussen de doteringsatomen vindt op een
indirecte manier via de metallische contacten van de FinFET plaats.
Bij lage temperaturen gaan elektron-elektron interacties een rol spelen.
Vanwege hun eigenschappen vormen doteringsatomen in FinFETs een krachtig
systeem om deze effecten te bestuderen. In een experiment aan twee doteringsatomen in een FinFET kon worden aangetoond dat ze aan elkaar gekoppeld
zijn via een gecorreleerde meer-deeltjes Kondo interactie. Deze interactie bleek
als functie van magnetisch veld te resulteren in Aharonov-Bohm oscillaties wat
ook weer aantoont deze koppeling coherente is. Dit is misschien verassend om-
112
Samenvatting
dat de coherentie in dit geval word gedragen door een meer-deeltjes toestand.
Daarom suggereert dit resultaat dat dit type interacties mogelijk bruikbaar
zijn voor het bouwen van een kwantum computer gebaseerd op doteringsatomen. Gecombineerd met de recent ontwikkelde atomaire fabricage principes
is het misschien ook mogelijk om deze structuren te gebruiken om exotische
gecorreleerde toestanden te bestuderen.
Concluderend kunnen we zeggen dat doteringsatomen in FinFETs een flexibel system vormen om een groot aantal fysische effecten mee te onderzoeken.
Elektrontransportspectroscopie kan hierbij worden gebruikt als een manier om
toegang te verkrijgen tot de elektrische structuur van enkele en gekoppelde doteringsatomen. Dit maakt doteringsatomen in FinFETs een krachtig system
om fundamentele fysica mee te bestuderen. Daarnaast zijn alle doteringsatomen nagenoeg gelijk aan elkaar. Dit maakt ze aantrekkelijk voor het gebruikt
als bouwstenen voor toepassingen in kwantum circuits.
Jan Verduijn
September 2012
Curriculum Vitae
Jan Verduijn
Born in Rotterdam, The Netherlands, on 21 November 1981
1986 – 1994
Basisschool
School met de bijbel Koningin Beatrix in Ouddorp
1994 – 1998
VBO Metaal (vocational education)
Technische School Middelharnis
1998 – 2002
Laboratory Instrument Maker, MBO
ROC Zadkine, dept. Christiaan Huygens Rotterdam
2002 – 2006
B.Eng. in Physics (Cum Laude)
University of Technology TH-Rijswijk
subject: Implementation of a large parabolic reflector antenna for TARA
2006 – 2008
M.Sc. Applied Physics
Delft University of Technology
subject: Single dopant electronic transport in nano-scale transistors
2008 – 2012
Ph.D. research
Delft University of Technology &
University of New South Wales, Sydney
subject: Silicon quantum electronics
113
List of Publications
Peer reviewed publications
19. Fano-Kondo effect in a parallel double quantum dot system
R.R. Agundez, J. Verduijn, S. Rogge, M. Blaauboer
To be submitted for publication
18. Wave function control and mapping of a single donor atom
J. Verduijn, G.C. Tettamanzi, S. Rogge
Submitted for publication
17. Non-local coupling of two donor-bound electrons in silicon
J. Verduijn, R.R. Agundez, M. Blaauboer, S. Rogge
Submitted for publication, preprint available under arXiv:1209.4726
16. Few electron limit of n-type metal oxide semiconductor single electron transistors
Enrico Prati, Marco De Michielis, Matteo Belli, Simone Cocco, Marco Fanciulli, Dharmraj Kotekar-Patil, Matthias Ruoff, Dieter P Kern, David A Wharam,
Jan Verduijn, Giuseppe C Tettamanzi, Sven Rogge, Benoit Roche, Romain
Wacquez, Xavier Jehl, Maud Vinet and Marc Sanquer
Nanotechnology 23, 215204, March 2012
15. Magnetic-Field Probing of an SU(4) Kondo Resonance in a Single-Atom Transistor
G.C. Tettamanzi, J. Verduijn, G.P. Lansbergen, M. Blaauboer, M.J. Calderón,
R. Aguado and S. Rogge
Physical Review Letters 108, 046803, Jan 2012
14. Balanced ternary addition using a gated silicon nanowire
J.A. Mol, J. van der Heijden, J. Verduijn, M. Klein, F. Remacle and S. Rogge
Applied Physics Letters 99, 263109, Nov 2011
13. Integrated logic circuits using single-atom transistors
J.A. Mol, J. Verduijn, R.D. Levine, F. Remacle, and S. Rogge
Proceedings of the National Academy of Sciences 108, 13969, Jan 2011
12. Lifetime-enhanced transport in silicon due to spin and valley blockade
G.P. Lansbergen, R. Rahman, J. Verduijn, G.C. Tettamanzi, N. Collaert,
S. Biesemans, G. Klimeck, L.C.L. Hollenberg, and S. Rogge
Physical Review Letters 107, 136602, Sep 2011
115
116
List of Publications
11. Electric field reduced charging energies and two-electron bound excited states
of single donors in silicon
R. Rahman, G.P. Lansbergen, J. Verduijn, G. Tettamanzi, S. Park, N. Collaert,
S. Biesemans, G. Klimeck, L.C.L. Hollenberg, and S. Rogge
Physical Review B 84, 115428, Sep 2011
10. Engineered valley-orbit splittings in quantum-confined nanostructures in silicon
R. Rahman, J. Verduijn, N. Kharche, G.P. Lansbergen, G. Klimeck, L.C.L. Hollenberg, and S. Rogge
Physical Review B 83, 195323, May 2011
9. Electrically addressing a molecule-like donor pair in silicon: An atomic scale
cyclable full adder logic
Yonghong Yan, J.A. Mol, J. Verduijn, S. Rogge, R.D. Levine, and F. Remacle
Journal of Physical Chemistry C 114, 20380, Dec 2010
8. Drain current modulation in a nanoscale field-effect-transistor channel by single dopant implantation
B.C. Johnson, G.C. Tettamanzi, A.D.C. Alves, S. Thompson, C. Yang, J.
Verduijn, J.A. Mol, R. Wacquez, M. Vinet, M. Sanquer, S. Rogge and D.N.
Jamieson
Applied Physics Letters 96, 264102, June 2010
7. A novel Kondo effect in single atom transistors
G.C. Tettamanzi, G.P. Lansbergen, J. Verduijn, N. Collaert, S. Biesemans, M.
Blaauboer, and S. Rogge
International Conference on Nanoscience and Nanotechnology (ICONN), 2010,
319, Feb 2010
6. Heterointerface effects on the charging energy of the shallow D− ground state
in silicon: Role of dielectric mismatch
M.J. Calderon, J. Verduijn, G.P. Lansbergen, G.C. Tettamanzi, S. Rogge, and
Belita Koiller
Physical Review B 82, 075317, Jan 2010
5. Ternary logic implemented on a single dopant atom field effect silicon transistor
M. Klein, J.A. Mol, J. Verduijn, G.P. Lansbergen, S. Rogge, R.D. Levine, and
F. Remacle
Applied Physics Letters 96, 043107, Jan 2010
4. Tunable kondo effect in a single donor atom
G.P. Lansbergen, G.C. Tettamanzi, J. Verduijn, N. Collaert, S. Biesemans, M.
Blaauboer, and S. Rogge
Nano Letters 10, 455, Jan 2010
3. Coherent transport through a double donor system in silicon
J. Verduijn, G.C. Tettamanzi, G.P. Lansbergen, N Collaert, S Biesemans, and
List of Publications
117
S. Rogge
Applied Physics Letters 96, 072110, Jan 2010
2. Thermionic emission as a tool to study transport in undoped nfinfets
Giuseppe C. Tettamanzi, Abhijeet Paul, Gabriel P. Lansbergen, Jan Verduijn,
Sunhee Lee, Nadine Collaert, Serge Biesemans, Gerhard Klimeck, and Sven
Rogge
IEEE Electron Device Letters 31, 150, Jan 2010
1. Orbital stark effect and quantum confinement transition of donors in silicon
Rajib Rahman, G.P. Lansbergen, Seung H. Park, J. Verduijn, Gerhard Klimeck,
S. Rogge, and Lloyd C. Hollenberg
Physical Review B 80, 165314, Oct 2009
Other publications
5. Single dopant transport
J. Verduijn, G.C. Tettamanzi and S. Rogge
As a chapter in “Single-Atom Nanoelectronics”, edited by E. Prati and T.
Shinada, To be published 31st March 2013 by Pan Stanford Publishing, ISBN:
978-981-4316-31-6
4. Mapping of single donors in nano-scale MOSFETs at low temperature
J. Verduijn, G.C. Tettamanzi, R. Wacquez, B. Roche, B. Voisin, X. Jehl, M.
Sanquer, S. Rogge
IEEE Silicon Nanoelectronics Workshop (SNW), 10-11 June 2012
3. Mass Production of Silicon MOS-SETs: Can We Live with Nano-Devices’
Variability?
X. Jehl, B. Roche, M. Sanquer, B. Voisin, R. Wacquez, V. Deshpande, B. Previtali, M. Vinet, J. Verduijn, G.C. Tettamanzi, S. Rogge, D. Kotekar-Patil, M.
Ruoff, D. Kern, D.A. Wharam, M. Belli, E. Prati and M. Fanciulli
Proceedings of the 2nd European Future Technologies Conference and Exhibition 2011 (FET 11) in Procedia Computer Science 7, 266, 2011
2. Single dopant impact on electrical characteristics of SOI nmosfets with effective
length down to 10nm
R. Wacquez, M. Vinet, M. Pierre, B. Roche, X. Jehl, O. Cueto, J. Verduijn,
G.C. Tettamanzi, S. Rogge, V. Deshpande, B. Previtali, C. Vizioz, S. PauliacVaujour, C. Comboroure, N. Bove, O. Faynot, and M. Sanquer
VLSI Technology, 2010 Symposium, 194, 2010
1. Sample variability and time stability in scaled silicon nanowires
M. Pierre, X. Jehl, R. Wacquez, M. Vinet, M. Sanquer, M. Belli, E. Prati,
M. Fanciulli, J. Verduijn, G. C. Tettamanzi, G. P. Lansbergen, S. Rogge, M.
Ruoff, M. Fleischer, D. Wharam, D. Kern
Proceedings of the 10th International conference on ULtimate Integration of
Silicon, 2009
Afterword
After doing a masters project in the Atomic Scale Electronics group of Sven
Rogge in Delft, I started my PhD research in the same group. Before I came
to Delft to do an applied physics master, I had gone through pretty much
the longest educational trajectory possible in The Netherlands. Starting out,
after primary school, on a very practical engineering school, I went on, as one
of the few of my year, to ‘study’ in Rotterdam, also with a strong focus on
manual skills. Then, encouraged by my teachers, I decided to try my luck and
go for a Bachelors degree in engineering physics in Rijswijk. Because I felt
that there were still more interesting things to learn for me, I followed this
bachelor up by a masters program in applied physics in Delft. It was during
the master research in his group, that Sven asked me if I wanted to start a
PhD project on silicon quantum electronics. Even though I also had a job offer
and thought I wanted to go into industries to start a ‘real job’, it was not so
difficult to persuade me. With Sven in the Photronic Devices group, lead by
Huub Salemink, I was given the opportunity and freedom to dive into things
that I found endlessly exciting and meet and work with numerous interesting
people from all around the world.
The period during which I did my PhD research was pretty much divided into era’s, before and after ‘The Great Move’. After arriving in Sydney, Richard Newbury made me feel welcome in the well-organized chaos that
was going to be the cryogenic measurement lab and the emptiness of the new
CQC2 T offices. Even though you would expect major disruptions in the experiments when you move all setups from Delft on a boat to Sydney and install
everything in a brand-new lab, Sven managed to minimize this to the ultimate
minimum possible under the circumstances. I admire Sven’s determination
and organizational skills that made the group into what it is now. The freedom and trust Sven gives to the people he works with, is already starting to
pay off in terms of results and will surely result in loads of great publications
in the near future.
Science is a collaborative effort by definition, and all the concepts and
ideas presented in this thesis are the result of the combined work of numerous
people, but my special thanks go to my peers within the group. Throughout
my PhD, a constant factor and someone I could count on, has been Giuseppe
Tettamanzi. Thank you for always supporting and trusting me! Then there is
Garbi Lansbergen, who supervised me during my masters project. No one I
know is more optimistic and cheerful than you. Thank you for the enjoyable
and productive time we had together.
119
120
Afterword
While in Delft, I had the pleasure to be part of the European Union FP7
project AFSiD. As a consequence, I was traveling to different places in Europe
for project meetings and reviews 2-3 times a year. During these meetings I met
too many, clever, interesting and sociable people to all list here, but I would
like to thank everyone involved in the AFSiD project. My special thanks go
to Marc Sanquer at CEA/Leti in Grenoble, the project leader of AFSiD, who
led the fabrication of hundreds, perhaps thousands, silicon nano-devices of
unprecedented quality that I used during my research.
Further, I had to opportunity to collaborate on modeling with Rajib Rahman and Gerhard Klimeck at Purdue University, and Lloyd Hollenberg at
Melbourne University. Our numerous Skype and email conversations were a
great help for me. And inspired me to try to understand everything there is to
understand about silicon nanostructures and donors in silicon. I would like to
thank Gavin Morley and Gabriel Aeppli for giving me the opportunity to visit
LCN for a week and work together on the extremely high frequency characterization of single donor in nanodevices. Many thanks to Miriam Blaauboer and
Rodrigo Mojarro for their enthusiasm and inspiring openness in the fruitful
collaborations we had, and still have. I admire the scientific style of Maria
Calderón, Belita Koiller and Francoise Remacle with whom I had the pleasure
to publish work together. Finally, I would like to thank Michelle Simmons for
the opportunity to work with Suddho Mahapatra and Sam Hile on single spin
resonance experiments on STM fabricated donor devices in the first year after
I arrived in Sydney.
The technical electronics support in the Nanoscience department in Delft
was sublime. What would I have done without Ron Hoogerheide? Your help
with the setups and our usual morning tea formed a very welcome distraction.
Raymond Schouten, Ruud Ooijik and Jack Maat always provided me with
selfless help with the world-class measurement electronics that I used in my
research. In Sydney Dave Barber thought me how to use a dilution refrigerator
and I enjoyed his company during the time I spent in the NML.
The social aspect of my PhD is very important to me. Therefore, I feel very
lucky that I was surrounded by many supportive and sociable people. First of
all, I sometimes wonder what would my PhD would have looked like without
my friend and college (dr.) Jan Mol. We endlessly discussed the nitty gritty
details of our work. I also enjoyed our numerous beers in the Roundhouse, the
Gaslight Inn and many other places in Delft and Sydney and the weekly dinners
in Randwick and Surry Hills together with Ingrid and Elfi. Also invaluable to
me were the numerous parties and coffee breaks with the members of Photronic
Devices in Delft. Gabri, Delphine, Felipe, Dmytry, Oleg, Huang (we share the
same age and birthday), Stefan, Thijs, Ahmet and Jeroen. And in Sydney I
had an equal amount of fun in the new group in Sydney with Joost, Ward,
Floor, Chunming, Joe, Gabriele, Rodrigo, Ludo, Tim, Max, Andrea, Karsten,
Steefan and Jay and outside the group with Huw, Craig and Sarah.
Afterword
121
Then, my special thanks go to my family. My mother and late father always
supported me when I decided to take on another study, I am very grateful for
that. During the difficult times we experienced in the past few years, I realized
what it meant to be part of a family that stands by each other. Miranda, Eelco,
Christa, Marja, Piet, Micky and Fay, thank you for your loving support!
Last, but certainly not least, I thank Elfi for her unselfish and loving support. Thank you for all you encouragement and good advice!
Sydney, November 2012