Download Electron Transport in Semiconductor Nanowires and Electrostatic

Electron Transport in Semiconductor Nanowires
and Electrostatic Force Microscopy on Carbon
Nanotubes
Ph.D. Thesis
Thomas Sand Jespersen
February 2007
Niels Bohr Institute
Faculty of Science
University of Copenhagen
Denmark
Thesis Advisor:
Assoc. Prof. Jesper Nygård
Electron Transport in Semiconductor Nanowires and Studies of Carbon Nanotubes using Electrostatic Force Microscopy
Ph.D. Thesis
c
°Thomas
Sand Jespersen 2007
e-mail: [email protected]
Niels Bohr Institute
Nano-Science Center
Faculty of Science
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen O
Denmark
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Abstract
A study is presented of the electronic transport properties of semiconducting Indium Arsenide nanowires. The wires are grown by molecular beam
epitaxy and have diameters ∼ 50 − 70 nm and lengths ∼ 3 − 5 µm. A
scheme is developed for making electrical devices in a field effect transistor geometry where metal contacts to the wires are lithographically defined.
The transport characteristics are measured as a function of bias voltage,
temperature, gate potential, and magnetic field. Measurements have been
performed on devices in different transport regimes enabled by differences
in the transparency of the nanowire-electrode interfaces. Devices with large
barriers exhibit clear Coulomb blockade and the level structure is analyzed,
showing evidence of transport through spin-degenerate levels. The quantum dot structure is highly regular and the sequential addition of ∼ 150
electrons can be followed. The dependence on electrode separation is investigated showing that the confinement of the quantum dot is determined
by the lithographically defined contacts. For larger transparencies, higher
order transport processes are found to contribute significantly, and the phenomena of cotunneling and the Kondo effect are observed. In devices with
open contacts the nanowires act as phase-coherent diffusive conductors and
electron interference gives rise to conductance fluctuations and weak (anti)
localization in the magnetoconductance.
A section is devoted to measurements on nanowire devices contacted with
leads of a Ti/Al/Ti trilayer turning superconducting below the transition
temperature Tc ∼ 750 mK. Results from the three transport regimes described above are presented, with emphasis on the intermediate coupling
regime where the Kondo effect is observed for temperatures above Tc . Due
to the competition of the Kondo effect and superconductivity the zero-bias
Kondo peaks are (in most cases) suppressed once the device is cooled below
Tc . However, the Kondo effect is found to have a pronounced effect on the
sub-gap structure, greatly enhancing the conductance peak at a finite bias
voltage of ∆/e (where ∆ is the superconducting energy gap). By comparing
with results obtained on carbon nanotubes the effect is shown to be a general quantum-dot phenomena and a possible explanation is discussed. The
problem is currently under theoretical consideration.
Also presented are investigations of single wall carbon nanotubes using a
scanning probe technique known as electrostatic force microscopy (EFM).
The superiority of the technique for fast, large scale characterization of nanotube distributions is demonstrated and it is shown to convey also information about defects on a single nanotube level, as well as the interplay of
nanotubes with static charges on the substrate. Finally it is shown that EFM
can be used for mapping the three-dimensional orientation and position of
nanotubes embedded in a polymer matrix.
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Preface
This thesis is submitted to the Faculty of Science at the University of Copenhagen in partial fulfillment of the requirements for the Ph.D. degree in
Physics.
The experimental work on carbon nanotubes and semiconducting nanowires
presented in the thesis has primarily been carried out at the Ørsted Laboratory, Niels Bohr Institute. The aim of this three-year project was to initiate
the studies of semiconductor nanowires in the nano-physics group utilizing
the existing experimental facilities for semiconductor processing and lowtemperature measurement setups for investigations of transport properties.
I am sincerely grateful to my supervisor Jesper Nygård for introducing me
to the world of mesoscopic transport and for invaluable guidance in all aspects of my project, from overall strategies and interpretation of results to
the details of how-to-run-the-cryostat.
Also, I wish to thank Prof. Poul-Erik Lindelof for many inspiring discussions
and physical insights.
I am very grateful to Claus Sørensen and Martin Aagesen. Without your pioneering work on growing nanowires by molecular beam epitaxy this project
would have developed much differently.
For inspiring discussions (on physics-related as well as non-physics related
issues), for always being willing to help, and for our everyday lunch-breaks
I am very grateful to my co-workers Kasper Grove Rasmussen, Jonas Rahlf
Hauptmann, Anders Mathias Lunde, Henrik Ingerslev Jørgensen, Jeppe
Holm, Martin Aagesen, Søren Stobbe, Pawel Utko, Ane Jensen, Magdalena
Utko, and Brian Skov Sørensen - I hope we will be able to keep in touch
and get together once in a while even though we will probably soon find
ourselves scattered around the globe.
Thanks to Claus Sørensen, Nader Payami and Inger Jensen for all your
technical support and for keeping the lab running smoothly, and to Carsten
Hyldebrand Mortensen, at the workshop, for all your skilled assistance in
designing and constructing the CVD lab.
On the theory side, I wish to thank Jens Paaske, Karsten Flensberg and
Brian Møller Andersen for taking an interest in our experiments, and for
working on a theoretical description of the non-equilibrium properties of a
Kondo-dot contacted by superconductors.
For granting me access to their Raman spectroscopy laboratories, I which to
thank Prof. Eleanor Campbell (Gothenburg University, Sweden) and Prof.
Ole Faurskov Nielsen (Institute of Chemistry, University of Copenhagen).
During the course of the project I stayed six very stimulating months in the
group of Prof. Charlie Marcus, Harvard University, and I am very grateful
to Prof. Marcus for granting me this opportunity and for his many insightful
and encouraging thoughts on the nanowire project. Also, I am grateful to
Hans-Andreas Engel and Emmanuel Rashba for taking an interest in our
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experiments. I wish to thank all the people in the Marcus Lab for making
my stay in Cambridge a very memorable one and I especially wish to thank
Jimmy Williams for many enjoyable hours in the lab, and for giving me an
inside view on the American culture.
Thomas Sand Jespersen, Copenhagen, February 2007
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Contents
1 Introductory comments
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2 Electron Transport in Semiconducting Nanowires
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 InAs nanowires . . . . . . . . . . . . . . . . . . . . .
2.3 Device preparation and experimental details . . . . .
2.4 Nanowire devices in the closed dot regime . . . . . .
2.5 Kondo physics in InAs nanowires . . . . . . . . . . .
2.6 Conductance Fluctuations in InAs nanowires . . . .
2.7 Nanowire devices with superconducting leads . . . .
2.8 Conclusion of nanowire studies . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Electrostatic Force Microscopy on Carbon Nanotubes
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Experimental Details . . . . . . . . . . . . . . . . . . . .
3.3 The EFM technique . . . . . . . . . . . . . . . . . . . .
3.4 Identifying defects using EFM . . . . . . . . . . . . . . .
3.5 Studying surface charges using EFM . . . . . . . . . . .
3.6 EFM for assessment of embedded CNT’s . . . . . . . . .
3.7 Conclusion of EFM studies . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A InAs nanowire growth NBI73
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B Details of nanowire device fabrication.
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C Type-1 devices in the closed-dot regime
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D Additional Kondo data
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E EFM parameters
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F Publication list
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1
2
Chapter 1
Introductory comments
The work presented in this thesis has been carried our during the last three
years. It can be categorized into two rather separate areas and a chapter
has been devoted to each; Chapter 2: ”Electron transport in semiconducting nanowires”, and Chapter 3: ”Electrostatic Force Microscopy on single
walled carbon nanotubes”. Each chapter is self-contained and can be read
separately, however, a few references are made from chapter 3 to the material in chapter 2 and thus the chapters are preferentially read in the order
in which they are presented.
The studies of electron transport in InAs nanowires, presented in chapter
2, constitute the subject of the originally proposed project and contains the
main body of material.
The two chapters contain their own introductory sections but before continuing, a comment should be made on the link between the presented material: The studies presented in chapter 3 - investigating the possibilities of
the technique of electrostatic force microscopy (EFM) for carbon nanotube
studies - may seem somewhat unrelated to those of the first chapter on electron transport in InAs nanowires (except for both of them belonging to the
general field of nano-science). The reason for this separation is to be found
in the circumstances and practical problems encountered during the course
of the project. During the start-up phase, no nanowires were available for
the study of transport and a considerable amount of time was spent dealing with practical issues concerning the infrastructure and construction of a
CVD lab for the growth of silicon nanowires. In this period also some efforts
were put into optimizing a nanotube CVD setup, and the EFM technique
was introduced during the search for the best way of characterizing the nanotube growth products. The various studies in chapter 3 emerged while
investigating subtle features of the obtained EFM images.
Once semiconducting nanowires were finally grown in our group (enabled
by the decision to introduce gold into the existing MBE-chamber) all efforts
were directed towards making nanowire devices for electrical transport stud3
ies, and the EFM investigations were paused.
After these preliminary comments we can now begin with the presentation of
the results of experimental investigations of low-temperature electron transport in semiconductor nanowires. Please enjoy!
4
Chapter 2
Electron Transport in
Semiconducting Nanowires
2.1
Introduction
During the last 50 years the technological development in the semiconductor industry has been aimed at making electronic devices smaller. At some
point the size reaches the scale of the coherence length of the electrons in
which case the wave nature of the electron becomes important for the device
behavior, leading to a wealth of novel phenomena. This mesoscopic regime
has been intensively studied since the 1980ies, in particular utilizing the
two-dimensional electron gas of a semiconductor heterostructure (2DEG) as
the experimental realization. This was later supplemented by an intense
research in mesoscopic devices incorporating carbon nanotubes and other
nano structures.
This chapter presents our studies of electron transport in one of the most recent classes of mesoscopic systems: semiconducting nanowires (NW). Nanowires, occasionally also denoted nanorods or nanowhiskers, are high aspectratio needle-like crystals typically measuring 2 − 100 nm in diameters and
5 − 30 µm in lengths. During the last few years the research into the properties and possibilities of nanowires has accelerated, leading to many exciting
discoveries. This introductory section first provides a short survey of the
field of nanowire research followed by a brief introduction to mesoscopic
electron transport being at the heart of the work presented in the chapter.
The introduction ends with an outline of the chapter.
Nanowires
In this section we outline the general field and history of nanowire research,
postponing, to the subsequent section, the treatment of the properties of
the particular type of wires which was used for our measurements: semicon5
Au
As
In
As
In
In
InAs
In
As
Figure 2.1: Top left, schematic illustration of the VLS growth of nanowires
(see text). Bottom left, TEM micrographs of Si nanowires grown by laser
ablation (adapted from Ref. [1]). Right, SEM micrograph of InAs nanowires
grown by CBE from lithographically defined gold particles (adapted from
Ref. [2]).
ducting Indium Arsenide (InAs) nanowires. The section does not (by any
means) cover the complete field which, despite its young age, already encompasses a substantial number of published works. Instead, some general
aspects and applications are described being a part of the author’s motivation for entering this field of research.
Already in the 1960ies the growth of whisker or wire-like semiconductor crystals were demonstrated[3] and the growth mechanism, the so-called vaporliquid-solid (VLS) growth, is still the governing principle in most presentday nanowire growths1 . The VLS-mechanism is sketched in Fig. 2.1(a):
The starting point of the NW growth is a substrate prepared with metal
(usually gold) nanoparticles either by deposition from commercially available colloidal suspensions[6], by direct deposition using aerosol techniques[7]
or made by metal evaporation and subsequent thermal annealing[8]. Subsequently, the substrate is heated and placed in a gas flux containing the
constituents of the desired semiconductor. These are incorporated into the
molten nanoparticle thereby forming an eutectic alloy and at the point of
supersaturation, crystal growth takes place beneath the particle (rightmost
part of the schematic). By examining the phase diagram of the alloying
between the metal nanoparticle and the semiconductor species, the growth
parameters can be rationally chosen making the VLS mechanism a viable
route for the synthesis of a nanowires from wide range of group IV (Si, Ge,
SiGe), group III-V (GaAs, GaP, GaAsP, InAs, InP, InAsP) and II-VI (ZnS,
1
Recent work indicates that nanowires in some cases grow by a vapor-solid-solid
mechanism[4] and also solution-based techniques have been developed[5].
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ZnSe, CdS, CdSe) semiconductors[6, 3, 1].
The diameter of the resulting wire is determined by the size of the metal
catalyst particle and the growth of epitaxial wires with diameters as low as
3 nm has been reported[9, 10]. The length is determined primarily by the
growth time and can be up to many micrometers.
The method used for introducing the semiconductor species varies and nanowire growth has been demonstrated using laser ablation[1], chemical vapor
deposition (CVD), metal-organic vapor phase epitaxy (MOVPE)[11], chemical beam epitaxy (CBE)[7], and molecular beam epitaxy (MBE)[12]. The
latter method has been employed for growing the wires used in this work
(see section 2.2). Figure 2.1 shows examples from the literature of nanowires: The bottom-left images show transmission electron microscope (TEM)
images of a Si NW grown by laser ablation[1] and the rightmost panel shows
an scanning electron microscopy (SEM) image of InAs nanowires grown by
CBE from lithographically defined Au-particles.
The nanowires grown by the VLS method possess many unique and exciting
properties. The increasing interest in recent years is probably partly spurred
by the improvements and better understanding of the growth process enabling rational design of wire geometries and composition. Importantly, in
a particular growth, all nanowires are, to a large extend, identical. This contrasts the other famous one-dimensional system, the carbon nanotubes, for
which a given growth contains nanotubes of a wide range of crystal structures
leading to very different properties (section 3.1). Despite extensive research,
this aspect still remains an obstinate problem for the technological application of carbon nanotubes. The VLS growth process furthermore opens up
for the possibility of changing the nanowire composition during growth by
altering the semiconductor species introduced in the growth chamber. In
this way, epitaxial heterostructures, as are known from the two-dimensional
growth of semiconductors, can be designed along the wires (axially), as well
as radially. In the case of two-dimensional growth the materials which can
be epitaxially combined are limited by their lattice mismatch since, for large
mismatch, the interface becomes defective due to strain. For the nanowires, however, their small cross-sectional area allows the strain to relax to
the surface, enabling the realization of novel material-combinations thereby
adding even further flexibility to the nanowire system. The possibilities
(fabricating radial and axial heterostructures, diameter control etc.) enable
a wide range of functionalities to be introduced in the wires, already at the
growth stage, and p-n junctions[13] and grown barriers[14] etc. have been
reported. Figure 2.2 shows examples from the literature of nanowire heterostructures. Panel (a) shows a TEM image of a Ge/Si core/shell nanowire
with an atomic-resolution zoom shown in (d). In (b),(c) elemental mappings
of the Si and Ge in the same view as panel (a) are shown, clearly emphasizing the core/shell structure. This is further elaborated in panel (e) showing
the elemental mapping in a cross-section through the wire (adapted from
7
f
Figure 2.2: (a) TEM micrograph of a Si/Ge core/shell nanowire with a high
resolution zoom in (d). (b),(c) Si and Ge elemental mapping of the image
from (a) showing clearly the core/chell structure. This structure is further
elaborated on in (e) showing the elemental composition at a cross section
of the wire - blue trace Si, red Ge (adapted from Ref. [15]). (f) An TEM
image of an InAs nanowire containing InP segments. The image is colored
with respect to the lattice spacing (adapted from Ref. [16]).
Ref. [15]). Panel (f) shows an image of an InAs nanowire grown with InP
segments (barriers). The composition changes on the level of monolayers
and the image has been colored according to the lattice spacing (adapted
from Ref. [16]).
Another characteristic which distinguishes nanowires from conventional nanostructures, which are lithographically defined from two-dimensional substrates, is the possibility of transferring the nanowires from the growth substrate to virtually any other substrate. This enables device fabrication on,
for example amorphous SiO2 -wafers suitable for low temperature devices
(section 2.3) or plastic films for realizations of flexible electronics[17].
A feature which has received considerable attention is the use of NW’s for
sensor applications. The electrical conductivity of semiconducting NW’s is
highly sensitive to electrostatic gating and combining this with their very
high surface/bulk ratio makes NW’s extremely sensitive to their immediate
environment. Much research has been directed towards this application and
advanced schemes for single-molecule detection have been realized[18].
As for low temperature investigations, which is the focus of the work presented in the upcoming sections, a number of important results have also
8
been established early on. Resonant tunneling diodes and quantum dots
with barriers defined during NW-growth have been realized[19, 14] followed
by reports on electrostatically defined single and double quantum dots[20].
Finally, the flexibility offered by nanowire devices concerning the contact
material (unlike the strict demands of conventional 2DEG-based mesoscopic devices) has been exploited in Refs. [21, 22, 23] for fabricating nanoscale
devices with superconducting contacts. This has enabled the experimental investigation of a number of novel effects such as a tunable[22] and
reversible[21] supercurrent. The reported low temperature studies will be
further discussed below during the presentation of our own measurements.
Mesoscopic transport
This chapter describes measurements of the electrical resistance (or conductance) of nanowires. For a macroscopic bulk sample, the electrical resistance
is simply proportional to its length and inversely proportional to its cross
sectional area, the coefficient of proportionality being the resistivity. Thus,
a sample of half the width will simply have twice the resistance but brings
no new physical phenomena. This is the Ohmic regime of charge transport.
A furious technological development has for the last decades aimed at downsizing of electronic devices. Obviously, at some point the physical behavior
must change since eventually the atomic scale is reached, a scale, which is
well known to be governed by entirely different principles. New phenomena
are, however, observed long before the atomic scale as the sample size L approaches the scale of the single electron coherence length `ϕ . The coherence
length is normally determined by inelastic scattering e.g., scattering by lattice vibrations, and `ϕ can be long at low temperatures. In this mesoscopic
regime (L ¿ `ϕ ) effects associated with the wave nature of the electrons
become important leading to a wealth of new physical phenomena.
Several approaches exist for experimentally realizing mesoscopic devices. An
important class of systems utilizes the 2DEG which can exist at an epitaxial
grown interface between two semiconductors. Employing lithographic techniques to define electrostatic gates the 2DEG can be shaped into advanced
geometries allowing a large variety of devices to be realized. Another important class of mesoscopic devices are provided by carbon nanotubes which are
less flexible with regard to device design but, on the other hand, are ”born”
as nearly ideal one-dimensional conductors.
The nanowire devices, which are treated in this chapter, also belong to
the mesoscopic regime. There is of course common phenomena associated
with the nanowires and the aforementioned systems but (as discussed in
the previous section) they also provide novel opportunities as well as unique
(experimental) difficulties.
Figure 2.3 shows a 3D rendering of an atomic force microscopy (AFM) measurement of a nanowire device. Electrical contacts are made by evaporating
9
Metal
electrode
Γ2 ee-
Insulating
substrate
Γ1
Metal
electrode
Nanowire
Figure 2.3: A perspective view of an AFM measurement, schematically illustrating the system under consideration: a nanowire with metal electrodes.
The coupling of the wire to the leads is characterized by the tunnel rate
Γ = Γ1 + Γ2 .
metal on the ends of the nanowire2 allowing investigations of the current
flow through the wire while varying external parameters. To describe how
well the wire and the metal leads are coupled we use the electron tunnel
rate Γ as indicated on the figure. Another parameter which is important for
the device behavior is the so-called charging energy EC which denotes the
electrostatic energy required to add an electron to the wire3 .
One of the features of the nanowire system has been the possibility of realizing devices with different barrier transparencies Γ, either electrically by
means of a nearby electrostatically coupled metal electrode (gate) or by
varying the fabrication techniques. This aspect is illustrated in Fig. 2.4
which shows the linear conductance G (inverse resistance) as a function of
the voltage Vg on a gate electrode situated below the insulating top-layer as
shown in the device schematic (inset). Obviously, the behavior of the device
is qualitatively different in the three Vg regimes. The reason is that the coupling Γ depends on the back gate potential, and as Vg is sweeped the device
thus enters different regimes dominated by different physical phenomena.
The measurements will be treated in subsequent sections but the regimes
can be distinguished by the relationship between Γ and EC [24, 25]:
• Γ ¿ EC . In this weak coupling regime corresponding to the lowest
Vg region of Fig. 2.4, an electron entering the device spends a long
time on the nanowire before leaving. The charge on the wire is a
well-defined integer multiple of the elementary charge e, and charging
effects dominate. The transport is thus governed by Coulomb blockade
2
3
The fabrication will be described in section 2.3
See section 2.4
10
Vsd
3
Contact
NW
I
G [e2/h]
SiO2
2
1
0
-9.0
p++ Si
Vg
x20
-8.8
-8.6
-4.4
-4.0
-3.6
6
8
10
Vg [V]
Figure 2.4: The linear conductance G of a nanowire device at T = 300 mK
as a function of the voltage Vg on a capacitively coupled back-gate situated
beneath the device (see schematic, inset). The device transparency increases
with the gate voltage and three transport regimes can be identified; closed
dot (lowest gate region), intermediate coupling (middle gate region) and
open device (highest Vg region), respectively.
physics. This is the subject of section 2.4.
• Γ . EC (middle region of Fig. 2.4). Coulomb blockade is still important in this intermediate coupling regime but higher order tunneling
processes contribute significantly to the transport. Cotunneling and
the so-called Kondo effect are observed (section 2.5).
• Γ À EC (highest Vg region of Fig. 2.4). In this strong-coupling regime
the charge on the nanowire is no-longer well-defined and the transport
is dominated by effects of interference of electron waves (conductance
fluctuations) as treated in section 2.6.
Other characteristic parameters relevant in describing the various regimes
will be discussed in the subsequent sections in relation to the measurements.
Organization of this chapter
The outline of this chapter is as follows: After this introductory description
of nanowires and mesoscopic transport, the particular nanowire material
used in this work is presented. This is followed by a section on the experimental techniques used for device fabrication and electrical measurements at
low temperatures (section 2.3). The subsequent presentation of the results
is given in sections 2.4, 2.5, and 2.6 which are organized according to the
three transport regimes listed above; closed-dot, intermediate coupling, and
open dot, respectively.
11
Further interesting physical phenomena are anticipated for mesoscopic devices with superconducting leads and this is the subject of section 2.7. Initially, the additional experimental efforts required for attaching superconducting Al electrodes to the nanowires are described, followed by a presentation of the results. Again, these are organized according to increasing
barrier transparency with emphasis on the effect of the superconductors in
the regime of intermediate coupling (Kondo effect).
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2.2
InAs nanowires
In the previous section a brief introduction to the general field of nanowire
research was provided illustrating that a large number of semiconductor nanowires can be grown. In this section we focus on the properties of the particular nanowires used in this work. Actually, two types of nanowires have
been investigated: Indium Arsenide (InAs) nanowires grown by MBE at University of Copenhagen by Martin Aagesen and Claus Sørensen (see details
below) and Silicon (Si) nanowires provided by Jiwoong Park at the Rowland Institute, Harvard University. Due to obstinate difficulties in making
electrical contacts to the Si nanowires this work was abandoned/postponed
when the InAs nanowires became available. The InAs wires are easier to
contact as will be discussed below but pose other experimental problems
(section 2.3). In this section the InAs nanowire material is introduced and
some of their relevant properties are discussed.
The nanowire material
As mentioned the InAs nanowires were grown by molecular beam epitaxy
following the vapor-liquid-solid method described in the previous section
(Fig. 2.1). These are the first InAs wires grown by MBE4 and the details
on the growth are provided elsewhere[27, 28]. It follows along these lines: A
GaAs substrate is loaded in the MBE chamber which is held at a background
pressure of ∼ 10−11 torr. The substrate is heated to ∼ 650 ◦ C and ∼ 1 nm
gold is deposited in situ. Due to the elevated substrate temperature the gold
forms nanosize droplets on the surface which will act as growth promoters
for nanowires. During the whole process the sample sits in a flux of As and
following the Au deposition the temperature is lowered to ∼ 480 ◦ C and the
nanowire growth is started by a few nanometers of GaAs by turning on a
Ga source5 . Subsequently the temperature is further lowered to ∼ 400 ◦ C
and by opening an In source the growth is continued as an InAs NW.
It turns out that the nanowires are epitaxially oriented with respect to the
substrate and that they grow in the <111> direction. Figure 2.5(a) shows
4
InAs wires reported in the literature are most often grown by CBE. This technique differs most significantly from the MBE technique in the way the semiconductor compounds
are handled. In CBE, the In and As come in the form of metallo-organic molecules (e.g.
trimethylindium for the In) whereas in MBE they come in pure form. Thus we may expect
that the MBE method introduces less carbon doping ([26]), however, no conclusions can
be drawn on the basis of the present early stage of the investigations.
5
Starting the NW growth by a small section of GaAs is required due to the lattice
mismatch between InAs and the GaAs growth substrate. Once the nanowire growth has
started and the materials are changed to InAs the NW growth continues since the strain
can be relaxed to the surface. This step is not needed for growth of InAs NW’s on the
(more expensive) InAs substrates. The small section of GaAs in the end of the nanowire is
not important for the electrical measurements since the device is patterned in the middle
section of the wire.
13
(a)
(b)
(d)
(c)
Figure 2.5: (a) Picture of a GaAs substrate after MBE growth of InAs
nanowires. The substrate assumes a grey color after the wire growth. (b)
SEM micrograph showing a side view of the nanowire forest close to a cleaved
edge. The substrate is (100) GaAs and as the wires grow in the <111>
directions, they do not stand perpendicular to the substrate as in (c) showing
the result of growth on a (111) substrate. (d) Inset: top view of a growth
on a (111) substrate; the wires have hexagonal cross-sections epitaxially
oriented to the substrate. Upon tilting the sample the SEM image (main
panel) shows directly the faceted nanowire. (SEM images courtesy of Martin
Aagesen).
14
a picture of a wafer after growth6 . the and in (b) a side-view SEM image
shows the nanowire ”forest” close to a cleaved edge of the wafer. In panel (b)
the substrate is (100)-GaAs and thus the nanowires do not grow perpendicularly to the surface but form angles consistent with the <111> direction.
Indeed, panel (c) shows the result of wires grown on a (111) GaAs substrate
where all wires stand in the vertical direction. In the inset to panel (d)
such a growth is viewed from the top and the wires can be seen to have a
hexagonal cross-section and all wires are oriented the same way confirming
that they keep their epitaxial orientation during growth7 . Upon tilting the
sample the facets of the nanowire are clearly observed.
Figure 2.6 shows high-resolution TEM micrographs of MBE grown InAs
nanowires8 . Firstly, from panels (a) and (c) it is noted that the wires are
covered with a few nanometers (. 5 nm) of amorphous (oxide) coating. This
shows that an oxide-removing treatment is needed to achieve electrical contact to a nanowire (section 2.3). Secondly, the wire obviously contains a
number of crystal faults. Consider first the image in (a) and the zoom in
(b). From detailed analysis of the image it is found that this nanowire has
cubic zinc blende (zb) crystal structure like bulk InAs crystals. The nanowire grows in the <111> direction and contains multiple defects which can
be identified as twinning faults, i.e., at certain points the stacking suddenly
changes from ABCABC to CBACBA (mirror symmetry). Twinning has
been studied in details in Ref. [29] for InP nanowires where it is shown to
facilitate micro-faceting of the wire surface. It is uncertain, however, how
the defects will affect electron transport. Figure 2.6(c) shows a TEM micrograph of a different InAs nanowire: Again segments of twin-related zb
structures are observed (labeled a1 , a2 ) but large sections of the crystal (labeled b) are found to have a hexagonal wurtzite (wz ) structure (i.e. ABAB
stacking) which is not found in bulk InAs. The wz structure has been observed in a number of nanowire studies (see e.g. Ref. [30, 31]) but very little
is know about the properties of this phase. Two recent theoretical works
(Ref. [32, 33]) report on band-structure calculations and estimate a ∼ 13%
(55 meV) increase in the band gap of the wz structure, compared to the
band gap of zb InAs Egzb = 0.43 eV (at T = 0 K,[34]). Thus, we may expect
the wz nanowires to behave slightly different and abrupt changes between wz
and zb structures in the same wire, as observed in Fig. 2.6(c), to influence
electron transport. The calculations do not, however, include the finite size
of the NW crystals which may also affect the band structure significantly.
6
The wire forest causes the greyish color of the central area of the wafer, and the darker
and more shiny appearance of the last few millimeter closest to the edge corresponds to
the naked substrate. No wires are grown close to the edge due to the clamping of the
wafter during growth.
7
There are exceptions to this, and in some growths, wires with different directions may
be present. For further discussions see Refs. [27, 28].
8
Imaging and analysis courtesy of Erik Johnson, University of Copenhagen
15
(a)
(b)
<111>
(c)
10 nm
0001h
b
0110h
2110h
a1
b
111c
211c
a2
110c
a1
Figure 2.6: TEM micrographs of InAs nanowires. (a) Low-resolution image
showing a ∼ 40 nm diameter nanowire of cubic zinc blende structure containing multiple twinning defects. The zoom in (b) illustrates the mirror
symmetry in the planes of the twin defects. (c) TEM of a nanowire having
regions of zinc blende structure (labeled a) and regions of hexagonal wurtzite
structure (labeled b). Only the zinc blende structure is found in bulk InAs
crystals. (Images and analysis courtesy of Erik Johnson).
More work is needed to understand these issues.
All measurements presented in this chapter were carried out on devices fabricated from the InAs nanowire growth #NBI73 using a (100) GaAs substrate.
The growth contains wires of various lengths and diameters but those which
were chosen for transport studies have diameters ∼ 60 nm, lengths up to
4 µm and most show virtually no tapering. SEM images from this particular growth are provided in Appendix A.
Electronic properties of InAs nanowires
Due to the lack of knowledge about the properties of the wz structure,
we briefly mention here some of the relevant well-established properties of
bulk zb InAs. As we will see later, the measurements are to a large extend
consistent with these properties (and effects of confinement).
Bulk InAs has a small, direct band gap of Egzb = 0.43 eV and an low effective
mass 0.026 me [34]. An important property of InAs for our purpose is that it
16
makes a Schottky barrier free contact when brought in contact with Au (and
Ti) metals due to surface states which pins the Fermi level of the metal in the
conduction band[35]. Thus, contacts are established by simply evaporating
metal onto the wires, in contrast to e.g. Si NW, for which carefully controlled
thermal alloying is required to avoid the otherwise all-dominating Schottky
barriers at the interfaces.
Furthermore, bulk InAs has a large g-factor |gbulk | = 15 leading to a large
Zeeman splitting of the energy levels in a magnetic field (section 2.5). The
g-factor in nanowires is, however, suppressed with respect to the bulk value
due to the geometric confinement[30]. Finally, the bulk crystal exhibits a
strong spin-orbit coupling being responsible for the presence of weak antilocalization in magneto-transport measurements[36, 37] (section 2.6).
17
2.3
Device preparation and experimental details
In the previous section the synthesis and properties of semiconducting InAs
nanowires were described. As shown in the SEM image of Fig. 2.5 the nanowires are grown as a dense nanowire forest on a substrate. This section
describes the methods that were developed to fabricate devices enabling
electrical measurements on individual nanowires from the forest. The methods overlap considerably with the techniques used for fabrication of carbon
nanotube devices, however, the nanowires do pose their own obstacles which
need special attention. In the final part of the section the methods used for
performing low-temperature measurements are briefly described.
Device fabrication
The goal of this section is to briefly describe the methods used to achieve
macroscopic electrical contacts to nanowires in a simple field-effect-transistor
(FET) geometry shown schematically in Fig. 2.7(c) with two electrical contacts and the possibility of electrostatic gating by a global back-gate electrode. A detailed account of the fabrication scheme, including various parameters, are given in Appendix B. The fabrication follows along these lines:
• The starting-point is substrate wafer of highly doped silicon capped
with an insulating layer of SiO2 . The wafer is prepared with alignment
marks and bonding pads by lithographic techniques.
• Wires are transferred from the growth substrate (Fig. 2.5(b)) by gently
pressing the two wafers together. In this way the longest wires break
of the growth substrate and sticks to the silicon wafer.
• Wires are located with respect to the alignment marks by SEM or optical microscopy, and an appropriate contact pattern is designed. Fig.
2.7(a) shows a typical optical microscopy image of nanowires deposited
in a grid of metal alignment marks.
• Contacts were defined by e-beam lithography. Prior to metal evaporation, the samples were cleaned in an oxygen plasma etch to remove
resist residues from the wire surface. This was followed by a brief
wet etch in buffered hydrofluoric acid (HF) to remove the oxide from
the nanowire surface (cf. TEM image, Fig. 2.6). In order to limit
re-oxidation of the wire, efforts were made to bring down the time
between the end of the HF etch and loading in the evaporation chamber. Usually the load-lock was evacuated within 2 minutes of the HF
etch. Based on the relatively few devices (< 20) which were electrically characterized it seems that omitting the plasma and HF etches
result in devices with large contact barriers (devices in the Coulomb
blockade regime). In contrast, by including the cleaning procedure,
18
(a)
Connection to
bonding pads
Alignment marks
Nanowire
20 µm
(c)
(b)
Ti/Au contact
Nanowire
Au
Ti
Nanowire
SiO2
doped Si
Figure 2.7: (a) Optical microscope image showing a grid of alignment marks
fabricated on a Si wafer. The marks are used as references when designing
contact patters for nanowire devices. Part of the larger structure which
allows connection to macroscopic bond pads is also seen. Dark lines corresponds to InAs nanowires which have been deposited on the sample after
the fabrication of the alignment grid. (b) An SEM micrograph of a typical
nanowire device. Metal Ti/Au contacts have been defined with an electrode
spacing of ∼400 nm. (c) Schematic side-view of the device layout of (b).
19
better contact is achieved and the device transparency can be tuned
by the gate (section 2.1). For the contacts either a Ti/Au bilayer was
used (for normal-metal contacts) or a Ti/Al/Ti trilayer for superconducting contacts (see also section 2.7). Figure 2.7(b) shows an SEM
micrograph of a typical finished device.
• An ultrasonic ball bonder was used for bonding devices, however, special care was needed since the nanowire devices were found to be extremely sensitive to electrical/static shocks. Hundreds of devices were
destroyed in the bonding process and the subsequent handling. A number of precautions enabled, however, a reasonable fraction of devices
to reach the measurement step (see Appendix B for details). Most
importantly it was found that after bonding each device the sample
should be mounted in the cryostat and checked for conduction. In
case the device survived the bonding it was cooled and investigated at
low temperature. Attempting to bond additional devices on the same
chip usually results in the loss of all previously bonded devices. This
procedure greatly limits the throughput, but no other solution has yet
been found.
Electrical measurements
This section briefly describes the setups used for performing electrical measurement on the nanowire devices. Almost all the measurements were carried
out in a voltage-biased setup using lock-in techniques as schematically shown
in the diagram of Fig. 2.8. The measurements were controlled by Lab-View
software running on a PC, equipped with a National Instruments data acquisition (DAQ) card (DA and AD converters). To reduce noise, the sample
was isolated from the mains by battery driven opto-couplers (isolation amplifiers with unity gain) on all lines connecting to the sample. The output
range of the DAQ is ±10 V and one output (DAC1 on Fig. 2.8) controls
directly the gate potential through a large 10 MΩ resistor which protects
the sample in case of a gate-leak. The source-drain bias is supplied through
DAC0 but since we are usually interested in biases in the milli-Volt range,
the output is passed through a 1000:1 voltage divider before it is connected
to the sample. On top of the DC bias a small AC signal is added for lock-in
measurement. The current is measured using a low-noise battery powered
current amplifier (DL Instruments, model 1211) which outputs a voltage
proportional to the current. This output (DC current) is logged by ADC0
and is fed into the lock-in amplifier which measures the AC part of the current (at the same frequency as the applied AC bias). Finally, the output
from the lock-in, corresponding to the differential conductance dI/dV of the
device, is logged by ADC1.
Initially measurements were carried out in a simple dip-stick setup in which
20
Sample
Current amp.
I
DAC0
1000:1
(DC bias Vsd)
10000:1
10MΩ
(AC source)
ADC0
(DC current I)
DAC1
(gate Vg)
Opto
coupler
Ref. in
ADC1
(dI/dV)
Lock-in amp.
Figure 2.8: Schematic diagram of the electrical setup used in the measurements. A computer controls/sweeps the DC bias and gate potential and
measures the DC current as well as the differential conductance which is
extracted using a standard AC lock-in setup.
the sample can be lowered gradually into a transport dewar with liquid helium. At different distances from the He-level the sample reaches equilibrium
at different temperatures, thus enabling measurements from room temperature to 4.2 K. It turned out, however, that the phenomena of interest in this
project required even lower temperatures and thus the samples needed to
be transferred to a more advanced setup (described below). The dip-stick
setup is, however, usually a very fast and convenient method for screening samples for the best devices which is then subsequently cooled to lower
temperatures. For the nanowires, however, once the sample is mounted in
the dip stick and the bonding-wire protection loops removed to enable measurements (see above) it has proven exceedingly problematic to transfer the
sample to another setup without loosing the device9 . Thus, after a while the
otherwise convenient dip-stick pre-screening was abandoned and the samples were directly mounted in an Oxford Instruments Heliox system.
In the Heliox the sample sits in vacuum and the whole inset is submerged
into liquid He. By pumping on liquid 4 He, present in a small can in the
Heliox inset, a temperature of ∼ 1.3 K is reached. This is sufficient for condensation of 3 He of which the Heliox holds a small amount. Due to the
higher vapor pressure of 3 He (for a given temperature) even lower temperatures are reached by subsequently pumping on the condensed 3 He liquid.
The sample temperature in this setup reaches, in this way, a temperature
of ∼300 mK and the hold-time is roughly 24 hours. For further details on
9
This problem is obviously not a fundamental one and a scheme has subsequently been
developed to keep the sample properly grounded at all times while transferring.
21
(a)
(b)
(c)
(d)
1µm
Figure 2.9: (a)-(d) A series of successive AFM images showing the effects
of AFM manipulation. The wires are relatively easily moved around on
the surface and the figure illustrates the possibility of assembling advanced
nanowire devices geometries, e.g., using AFM manipulation.
the principle and operation of the relatively simple Heliox setup, the reader
is referred to the manual and Ref. [38]. The setup is equipped with a 2 T
superconducting magnet and is placed in a shielded room.
Advanced device geometries
We conclude this section by commenting on the possibility of assembling
the InAs nanowires into advanced device geometries. A number of device
architectures incorporating multiple silicon nanowires have been developed
for sensor purposes[18] and for making junctions between wires with different properties in a crossed-wire geometry[39]. In these studies the wires
were assembled either by methods of directed flow or by direct mechanical
micro/nano manipulation in an optical microscope. These methods benefit
from the extended length (> 10 µm) of these silicon wires and may not be
applicable to the shorter MBE-grown wires. Figure 2.9 demonstrates the
use of AFM manipulation for moving individual InAs nanowires which were
deposited on a silicon substrate as described above. The wires are found to
be robust towards pushing by the AFM (in contrast to single-wall carbon
nanotubes, see section 3.5) and they are, with relative ease, assembled into
a crossed-wire geometry in four manipulation steps. Future studies may
benefit from this possibility.
22
2.4
Nanowire devices in the closed dot regime
In this section we begin the presentation of the low temperature electron
transport measurements performed on the devices described in the previous
section. The results are presented in the order of increasing barrier transparency starting with this section on nanowire devices with large barriers. In
this regime, the device acts as a quantum dot and transport is dominated by
the phenomenon called Coulomb blockade (CB). Coulomb blockade physics
has been observed in a large number of mesoscopic devices and the phenomenon is well understood. Thus, the observation of CB in the nanowire
measurements comes as no surprise, however, these measurements are the
first performed on the ”Copenhagen-wires”, which differ from other wires
in that the growth is performed by MBE and, as shown below, that the
barriers are defined by the contacts. Thus, the CB measurements add to
the understanding of these wire’s ability to act as mesoscopic devices and
the results are therefore relevant for this work. Furthermore, a thorough
understanding of the CB in these simple devices is a prerequisite (or at least
an aid) for future development of advanced nanowire devices. Lastly, this
section introduces the terminology and sets the scene for the following sections describing the results in the regimes of stronger coupling.
Devices in the closed dot regime have been realized using two fabrication
schemes differing in the procedure used for ”cleaning” the wire interface
before metal evaporation (section 2.3). As mentioned in section 2.1, exposing the wire to both oxygen plasma etching and a HF etch prior to metal
evaporation, results in devices where the barriers are tunable with the back
gate (type-1). In this case (see Fig. 2.4) the closed-dot regime is relevant for
the lowest gate-voltages when the barriers are large. Alternatively, devices
born with large barriers can be fabricated by omitting one of the cleaning
steps. Here we show results from devices fabricated without the plasma
etching (type-2). The type-1 devices are generally regular and stable in the
intermediate coupling regime but in the closed-dot regime, relevant for this
section, they become noisy and exhibit multiple gate-switches. Therefore,
regular Coulomb blockade was only observed in relatively short gate-ranges.
The type-2 devices, on the other hand, are usually stable and regular in the
closed-dot regime but it has not been possible to tune these devices into the
intermediate and open regimes.
Figure 2.10 shows the linear conductance G as a function of gate-voltage
Vg for various temperatures T between room temperature and 4.2 K. The
leftmost inset shows an SEM micrograph of the device. The contacted wire
has a diameter d ≈ 70 nm and the (Ti/Au) electrodes are separated by
L ≈ 300 nm. Importantly, for all temperatures the conductance increases
upon increasing Vg showing that the current is carried by electrons; i.e.,
the nanowire acts as an n-type semiconductor. The n-type behavior has
been observed for all investigated InAs nanowire devices agreeing with the
23
G [e2/h]
2.0
G [10-3 e2/h]
2.5
295K
1.5
6
4.2K
4
13.0
0.25
1µm
13.3
Gate [V]
13.6
210K
100K
0.00
50K
4.2K
-10
0
10
20
Gate [V]
Figure 2.10: The linear conductance G as a function of gate voltage Vg
measured at various temperature from room temperature to 4.2 K for a type2 nanowire device with a ∼ 70 nm diameter wire and ∼ 300 nm electrode
separation (SEM micrograph, leftmost inset). The rightmost inset shows a
magnification of the T = 4.2K trace in the region marked with a red oval
in the main panel. The oscillatory behavior is due to Coulomb blockade.
These characteristics are typical of all devices.
measurements on CBE grown InAs nanowires of Ref. [26] where the carbon present during the CBE-growth was speculated to be the source of the
donors. In our case, however, there is no carbon present during the (MBE)
growth but the wires may be subsequently doped by the environment or
the n-type behavior may be an effect of the peculiar contact properties of
InAs (section 2.2). Another feature of the measurements in Fig. 2.10 is the
decreasing of conductance observed upon lowering the temperature which
shows the importance of thermal excitation of carriers into the conduction
band of the wire.
The rightmost inset shows a blow-up of the T = 4.2 K trace at the region marked in the main panel with a red oval. The conductance exhibits
pronounced periodic oscillations which are caused by Coulomb blockade as
described below.
Coulomb blockade - model
In the following, the nanowire device is modeled as a conducting island (dot)
coupled by tunnel-contacts to the two metal electrodes. For poorly coupled
devices G < e2 /h the charge on the island is a well-defined integer number
N of the elementary charge e. If each tunnel junction is represented by a
capacitance and a resistance the model can be illustrated by the circuit in
Fig. 2.11, where the back gate electrode is included as a capacitive coupling
Cg . If we assume that the total capacitance C = Cg + Cs + Cd accounts
24
Rd
Nanowire
Rs
N
Cd
Cs
Cg
Vg
Vsd
Figure 2.11: Circuit diagram schematic of the nanowire device. Each
wire/electrode junction is represented by a resistance and a capacitance
and the gate couples capacitively to the central region. For high-resistance
devices the charge on the nanowire-dot is a well-defined integer number N
of electron charges.
for the Coulomb interaction of the electrons on the island, the change in
electrochemical potential associated with the addition of an extra charge on
the dot is the charging energy EC = e2 /C independent of N . Obviously,
this energy is large for small objects and for low temperature measurements
where kB Tk < EC effects associated with single electron transport through
the device become important. Single electron transport has been thoroughly
analyzed in the reviews of Refs. [40, 41] and the details will not be repeated
here. The treatment in this section serves to describe the physical origin
of the oscillatory behavior of G observed in Fig. 2.10, and to introduce the
relevant energy scales.
In the model of Fig. 2.11 the reduced size of the nanowire is not evident.
However, due to the quantum nature of the electrons, the nanoscale confinement of the nanowire is expected to lead to a discrete set of electron energy
levels {En }. These are important for the measured transport characteristics
if the experiment is carried out with kB T < ∆En , where ∆En = En − En−1
are the spacings between the successive single electron levels. In this case
the island is denoted a quantum dot. An alternative view in which these
discrete energy levels are included is provided by the sketch in Fig. 2.12(a)
of the potential landscape through the device: Tunnel barriers separate the
dot (quantum well) in the center from the source and drain electrodes which
are kept at the chemical potentials µs and µd , respectively. A source-drain
bias Vsd = (µs − µd )/e is applied across the dot. The single electron levels
are illustrated by horizontal lines; solid for the occupied and dashed for the
unoccupied. We assume in the following that the level structure is independent of N .
The electrochemical potential of the dot is calculated from the total ground
25
(a)
(b)
(c)
..
...
∆Ei
Eadd
e
µs
Eadd
µd
µs
µd
µs
Eadd
Vsd
µd
..
..
.
..
.
..
Vsd
e
Vsd
Figure 2.12: Schematic potential landscape of the nanowire device. The
electrode/wire junctions are represented by tunnel barriers and a small bias
Vsd = (µs − µd )/e is applied across the device. Due to the nanoscale confinement of the wire (central region) the electrons occupy discrete energy levels
illustrated by horizontal solid(dashed) lines for occupied(unoccupied) levels.
In (a) there are no levels for transport in the bias window and transport is
blocked. The gate voltage displaces the level structure vertically and in (b)
a level has been tuned into the bias window and a current can flow by single
electron tunnel events on (b) and off (c) the dot (see text).
state energy µN = U (N − 1) − U (N ), where U contains a contribution
P
N
Ei due to the discrete energy levels and an electrostatic contribution
2
Q /2C = (e(N − N0 ) + Cg Vg )2 /2C with N0 being the number of electrons
at zero gate voltage. This leads to[40]
µN = EN +
1´
e2 ³
N − N0 −
− eαVg ,
C
2
(2.1)
where the coupling of the gate is described by α = Cg /C. Thus, the addition
energy Eadd depicted in Fig. 2.12(a) is given by Eadd = µN +1 − µN =
e2 /C + ∆EN .
In the situation illustrated in Fig. 2.12(a) there are no levels available for
transport within the bias window. Thus, transport through the device is
blocked and the number of electrons on the dot is fixed; this is known as
Coulomb blockade. As seen from Eq. 2.1, however, the gate potential Vg
continuously changes the energy of the electrons on the dot and thereby
displaces the level structure of the dot-region in Fig. 2.12(a) in the vertical
direction. Thus, Coulomb blockade can be lifted by sweeping Vg until the
situation of Fig. 2.12(b) is reached. Here (say) µN now falls in the bias
window and an electron can tunnel onto the dot leading to the situation of
Fig. 2.12(c) with N electrons on the dot. This N th electron can, however,
leave the dot again by tunneling to the drain thereby returning the system
to the situation of 2.12(b). Obviously, the cycle can be repeated and by
such successive single electron tunnel events, a current can flow through the
device. Upon further changing the gate voltage, the levels are tuned out of
26
(a)
(b)
(c)
∆Ei
∆Ei
µs
µd
µs
Vsd
..
µd
µd
Eadd
Eadd
Eadd
Vsd
∆Ei
e
µs
e
..
Vsd
..
Figure 2.13: Lifting of Coulomb blockade by increasing the bias. (a) No
current flows at low bias due to Coulomb blockade. By increasing Vsd a
level becomes accessible in the bias window (b) and transport occur by a
cycle analogous to Fig. 2.12(b),(c). Upon further increasing Vsd the first
excited state enters the bias window resulting in a further increase of the
current (c).
the bias window leading again to a blocked situation analogous to the one in
Fig. 2.12(a), however, with the electron number now fixed at N + 1. Thus,
from Eq. 2.1 it is seen that upon sweeping Vg , conductance peaks (Coulomb
peaks) will appear spaced by
∆Vg = Eadd /(eα),
(2.2)
and this is the origin of the oscillatory behavior observed in the measurements of Fig. 2.10.
The line shape of the Coulomb peaks depends on the coupling strength and
the temperature. The lifetime broadening hΓ gives the peaks a Lorentzian
line shape with a finite width even at the lowest temperatures and for larger
temperatures the width of the Fermi distribution of the electrons in the
source and drain contacts results in thermal broadening of the CB peaks.
In the quantum limit hΓ ¿ kB T ¿ ∆E, EC the line shape of the peaks are
given (for a peak at zero gate) by
G(Vg ) ∝
³ αeV ´
1
g
cosh−2
,
kB T
2kB T
(2.3)
which has a full-width-at-half-max of eα∆VgF W HM = 3.5kB T and a peak
height which scales as T −1 . In the classical regime ∆E < kB T < EC the
peak shape is similar, but the width scales as ∼ 4.4kB T /(eα) and the peak
height becomes temperature independent [42]. The temperature broadening
is the cause of the large width of the conductance peaks in the measurement
of Fig. 2.10 as well as the non-zero conductance in the valleys (see below).
In the above treatment only the linear response limit (Vsd ¿ ∆E, EC ) was
considered, and CB peaks appeared whenever a level was tuned by the gate
27
Vsd
(c)
(d)
(e)
(f)
(h)
(g)
(f)
(b)
(c)
(d)
(e)
(h)
(g)
(f)
(e)
(a)
(b)
(c)
(d)
N
(g)
(f)
(e)
(d)
(a)
(b)
(c)
(f)
(e)
(d)
(c)
(a)
(b)
(c)
(d)
∆Ecd
Eadd
∆Ecd
N-1
(f)
(e)
(d)
(c)
N
N+1
Eadd
(e)
(f)
(g)
(h)
.
eαVg
Vsd
(f)
(g)
(h)
(e)
(d)
(c)
(b)
(e)
(f)
(g)
(h)
(d)
(c)
(b)
(a)
(d)
(e)
(f)
(g)
(c)
(b)
(a)
.
(c)
(d)
(e)
(f)
Figure 2.14: (Left) Schematic illustration of Coulomb diamonds and the
excited state structure in the differential conductance of a quantum dot as a
function of bias and normalized gate. The states are labeled according to the
schematic on the right which shows the configuration for a point (Vsd , αeVg )
in the center of the middle diamond. The small arrows indicate the level
spacing ∆Ecd also indicated in the schematic on the right.
into the (small) bias window. The sketches in Fig. 2.12 show that this
situation can also be reached upon increasing the bias. This is illustrated
in Fig. 2.13: in (a) transport is prohibited due to CB but in (b) it is seen
how current can flow at higher bias. In the sketch in (b) the positions of
the levels with respect to the drain are the same as in (a). This requires
a compensation by the gate, since the levels in the dot are dragged along,
when increasing the bias due to the capacitive coupling of the contacts (Fig.
2.11). It is clear in (b) that by moving the levels slightly up with the gate
potential the device would re-enter the CB situation and a slightly larger bias
would be needed to reestablish transport. Thus, in a plot of the current, or
dI/dVsd as a function of Vg and Vsd , the pairs of (Vg , Vsd ), where transport
is blocked, form diamond shaped regions. Outside the diamonds features
are expected each time an excited state becomes accessible in the transport
window as illustrated in panel (c). Such a plot is called a stability diagram
or a bias spectroscopy plot, and is schematically illustrated in Fig. 2.14,
where different levels can be tracked. Below, these considerations will be
used for analyzing the nanowire data.
InAs nanowire quantum dots
After these general considerations of the transport properties of quantum
dots, we now return to the measurements on the InAs nanowire devices.
In Fig. 2.10 the overall temperature dependence of the linear conductance
28
was shown and at 4.2 K an oscillatory behavior was observed. From the
above discussion, these oscillations are interpreted at temperature broadened Coulomb oscillations. Figure 2.15(a) shows an analogous measurement
at a lower temperature T = 300 mK. The measurement is performed in a
different cool-down and the transport onset has been shifted to a somewhat
lower gate voltage probably due to rearrangements of the charges in the
substrate10 . At low Vg no current flows but above ∼ 3.5 V a series of pronounced peaks separated by regions of zero conduction are observed, being
clear indications of Coulomb blockade. The peak spacings are regular over
the entire gate region spanning ∼ 150 peaks. Superimposed on the fluctuating peak amplitudes an overall gate dependent modulation is observed and
they are almost entirely suppressed in the middle region (see below). At the
highest gate voltages this modulation causes a stronger coupling to the wire
and the conductance does not fall to zero in the CB valleys.
The present section focuses on the regular, weakly coupled regions. Panel
(b) shows a blow-up of 6 CB peaks measured at 4 different temperatures.
As expected from the previous section the width of the CB peaks increases
with temperature and in panel (c) the FWHM, ∆VgF W HM , is shown as a
function of temperature for the 6 peaks. The dashed line shows the linear
dependence of Eq. 2.3 allowing an estimate of the gate coupling coefficient
α = Cg /C ≈ 0.086. This shows that the capacitance of the dot is primarily
to the contacts which is intuitively reasonably given the geometry of the
device (if the device acts as a classical dot, cf. the previous section, the
fit corresponds to α = 0.068). As for the peak heights, panel (d) shows
the temperature dependence for the peaks in (b): the heights increase as
expected for a quantum dot and scales in reasonable agreement with the
expected T −1 dependence.
Figure 2.15(e) shows a bias spectroscopy plot showing 58 regular low conductance Coulomb diamonds measured in the low Vg region of panel (a). The
overall modulation of the contact is observed as regions of increased conductances appear in the vicinity of Vg ∼ 3.9 V and Vg ∼ 5.1 V corresponding
to the larger peak amplitudes in panel (a) at these gate voltages11 . For a
dot in the quantum regime, the Coulomb peak amplitudes are expected to
fluctuate randomly between peaks due to transport through differently coupled levels. In Fig. 2.15(a) an obvious correlation is observed between the
amplitudes of adjacent peaks but as seen from Fig. 2.15(e) this appears to
be related to the overall modulation of the contact. The origin of this modulating behavior is not clear but it has been observed for all devices, however,
10
Such large differences in the threshold potential between successive cool-downs are
frequently observed. Moreover, often a hysteretic behavior is observed upon reversing the
sweep-direction of Vg , showing the dependence on the particular charge configuration in
the substrate.
11
A small gate shift has occurred in the time between the measurements (days).
29
(a)
G [10-2 e2/h]
1.5
1.0
0.5
0.0
4
6
10
8
Gate [V]
0.30K
0.65K
1.00K
1.60K
0.4
0.2
0.0
4.1
4.0
2.0
0.0
0.0 0.5 1.0 1.5
4.2
Vsd [mV]
(d)
10-3
Gate [V]
3.0
10-2
6.0 (c)
G0 [e2/h]
(b)
FWHM [mV]
G [10-2 e2/h]
0.6
0.3
1.0
T [K]
T [K]
(e)
0.0
-3.0
3.5
4.0
4.5
5.0
5.5
6.0
Gate [V]
Figure 2.15: Measurements of Coulomb blockade physics in InAs nanowires. (a) The linear conductance G as a function of gate volgate measured at 300 mK. At low gate voltages no conduction occurs but above
Vg ∼ 3.5 V CB peaks appear. (b) The measurement of 6 CB peaks for
temperatures T = 1.6 K, 1.0 K, 0.65 K, 0.30 K; upon lowering the temperature the peaks narrow and the peak values G0 increases as emphasized
in panels (c) and (d), respectively. In (c) the dashed lines corresponds
to ∆VgF W HM = 3.5kB T /(eα) (Eq. 2.3) with a gate-coupling coefficient
α = 0.086. In (d) the solid line corresponds to G0 ∝ T −1 . (e) Measurement at T = 300 mK of dI/dVsd as a function of Vg and Vsd in a gate range
corresponding to the left part of panel (a), (darker = less conductive). Coulomb diamonds and large scale conductance modulations are observed (see
text).
30
with large differences between devices12 . We speculate that it may be related to charging effects in the electrode-nanowire interface and/or effects
of defects in the wire itself close to the interface, cf. section 2.2.
Non-linear transport
Figure 2.16 focuses on the details of a smaller region of Fig. 2.15(e). Panel (a)
shows the linear conductance (left axis) as a function of gate voltage, revealing the regularly spaced CB peaks. The extracted peak spacings (right axis)
are grouped around ∼ 40 mV but show a systematic alternating (even/odd)
pattern over the entire region. Within the constant interaction model discussed in the previous section, the spacing is given by ∆Vg = Eadd /(eα)
(Eq. 2.2) where Eadd = e2 /C + ∆E, and, thus, an alternating behavior is
expected in the case of two-fold degenerate levels. Here, the eα∆Vg -sequence
would be: (EC , EC + ∆Ei , EC , EC + ∆Ei+1 , . . .). The origin of the degeneracy is not revealed in the present measurement but an obvious candidate
is spin-degeneracy which is consistent with measurements discussed below
(Fig. 2.17 and Fig. 2.22) of the magnetic field dependence showing that electrons enter the dot in order of alternating spins (↑, ↓, ↑, . . .). In the case of
spin-degeneracy being at the root of the alternating ∆Vg the larger spacings
in Fig. 2.16(a) correspond to the number of electrons on the dot N being
even, while the smaller ones, to N being odd.
Thus, from the constant interaction model the smaller spacings should be
identically EC /(eα). This is in good agreement with the measurements giving an average charging energy of ECavg /(eα) = 39 ± 1 mV (below, a better
estimate of α will be found). The deviations observed on the figure are
due to the uncertainty in determining the peak positions or from a slightly
varying charging energy. The alternating sequence in principle allows a determination of the subsequent level spacings, however, due to the small level
spacings compared to the charging energy and the mentioned uncertainty
of the peak positions such a mapping is not meaningful in the present case.
The average level spacing ∆Eavg /(eα) = 3.1 ± 1.9 mV, however, gives an
idea of the associated scale.
Panel (b) shows a stability diagram measured in the gate region marked in
(a) with a horizontal dashed line. Five Coulomb diamonds can be clearly
seen along with a complicated pattern of closely spaced excited states (cf.
Fig. 2.14). The diamonds appear slightly skewed due to an asymmetry in the
capacitances of the two barriers13 . From the height of the diamonds we can
directly determine the addition energy Eadd /e which should then be linearly
related to the peak separations from (a), with a slope corresponding to the
gate coupling coefficient. This is confirmed in the inset to panel (a) giving
12
Note that only type 2 devices are considered, cf. the introduction.
The asymmetry of the biasing, cf. Fig. 2.8, is not expected to be important since the
capacitance to the gate is much smaller than the capacitance to the contacts (α ≈ 0.06).
13
31
(a) 2.0
∆V [mV]
1.0
1
0
0
0.0
3.8
30
2
20
∆Vg [mV]
50
10
0
4.0
4.2
Gate [V]
4.4
(b) 5.0
0.01
dI/dV [e2/h]
Vsd [mV]
∆Vg [mV]
G [10-2 e2/h]
40
3
∆E/e
0.0
0
-5.0
4.30
4.40
4.35
4.45
Gate [V]
Figure 2.16: (a) Linear conductance G (left axis) as a function of Vg showing a series of CB peaks. The peak spacings ∆Vg = Eadd /(αe) (right axis)
exhibits an alternating pattern due to a two-fold degeneracy of the levels
(see text). (b) Stability diagram for the gate region marked in (a) with a
horizontal dashed line (darker = more conductive). Regular coulomb diamonds are observed along with a densely spaced pattern of excited states.
The vertical bar illustrates the average level spacing ∆Eavg ≈ 0.18 meV deduced from the even-odd pattern of Eadd in (a) - see text. The inset to (a)
shows the diamond width ∆Vg vs. the diamond height in ∆V from (b). The
linear fit gives an estimate of the gate coupling coefficient α = 0.06. All
measurements were performed at T = 300 mK.
32
α = 0.06 in reasonable agreement with the estimate above based on the temperature dependencies. With this value for α, the average charging energy
and level spacing becomes ECavg = 2.3 meV and ∆Eavg = 0.18 ± 0.10 meV,
respectively.
By comparing with ∆Eavg (vertical bar) it appears that the observed excited states in the stability diagram of Fig. 2.16 do indeed correspond to
individual separated levels despite the close spacing of the excited levels and
the fact that some appear very faintly.
We note that the double degeneracy of the levels identified above should in
principle lead to diamonds having pairwise identical excited state spectra.
However, it has not been possible to firmly identify such behavior in the
stability diagrams in Fig. 2.16(b), due to the large number of closely and
evenly spaced levels.
Figure 2.17(a) shows a stability diagram measured in the high gate region
of Fig. 2.15(a). Again a very regular series of Coulomb blockade diamonds
are observed corresponding to addition of 12 electrons. The charging energy
EC ≈ 2.3 meV is identical to the previous measurements for lower Vg , however, no even/odd behavior can be identified in Eadd in this region. Except
for one strongly coupled level, all lines corresponding to excited states appear with a negative slope. This is in contrast to Fig. 2.16(b) which displays
no difference between the two sets of lines (positive/negative slope) and it
shows that the dot in the present case has developed asymmetric tunnel couplings to the two leads14 (cf. the above discussion of the overall modulation
of the barriers). As in Fig. 2.16(b), the excited states appear closely spaced
and identifying the individual level spacings in each diamond is not possible.
In contrast to Fig. 2.16(b), however, some of the excited states are much
more strongly coupled to the electrodes and the position of these levels can
be tracked between neighboring diamonds. The symbols around diamond
#3 mark five such states and in panel (b) the threshold biases for each of
these have been mapped as a function of diamond number n. As the levels
are filled, the positions of the strongly coupled levels shift according to the
schematic in Fig. 2.14 and thus, in principle, each set of points in (b) maps
the excited state spectrum. As before, however, the relative uncertainty in
determining the level positions hinders such detailed analysis. The linear
fits to the level positions (dashed lines) are approximately identical (except
for a sign) as expected since the same level spacings are responsible for all
the position shifts. The average slope yields a good estimate for the average
level spacing ∆Eavg = 0.10 meV in reasonable agreement with the (less accurate) value deduced from the even-odd pattern in 2.16(b). We note that
the absence of the two-fold (spin) degeneracy in this region is also reflected
14
In the case of asymmetric couplings the transport will be dominated by the thickest
barrier and at a finite bias only for those excited states which are accessible for the thick
barrier will a line appear in the stability diagram. This can be realized by considering
schematics like those of Fig. 2.13.
33
Vsd [mV]
5.0
(a)
2.5
0.0
1
2
4
5
∆V7
6
8
9
10
11
12
-2.5
-5.0
6.8
6.9
7.0
Gate [V]
7.1
7.2
2.5
(b)
(c)
6.95
2.0
57 (d)
g=2
∆V6
∆V6
1.0
∆V7
∆Vg [mV]
7.00
1.5
Gate [V]
Vexc [mV]
51
∆V8
7.05
∆V7
45
39
∆V8
∆V9
0.5
33
7.10
∆V9
0.0
0
4
8
12
0.0
1.0
B [T]
n
2.0
0.0
1.0
B [T]
2.0
Figure 2.17: Grey scale representation of dI/dVsd vs. Vg and Vsd at the
high-Vg end of Fig. 2.15(a) showing regular Coulomb diamonds numbered
according to increasing electron number n (darker = more conductive). The
symbols in diamond #3 label five particularly strongly coupled excited states
whose position can be tracked for different n (see Fig. 2.14) and panel (b)
shows the threshold bias for each of these states as a function of n. The
dashed lines are least square linear fits and the average slope gives an estimate of the average level spacing ∆Eavg = 0.10 meV. Panel (c) shows a grey
scale plot of dI/dVsd vs. Vg and perpendicular magnetic field B measured at
zero bias for the gate region of diamonds #5-10 (darker = more conductive).
The positions of the CB peaks are nearly constant but small wriggles can be
observed as emphasized in (d) showing the CB peak separations ∆Vn (three
top curves, each offset by 6 mV). For low fields the peaks shift systematically by an amount consistent with a g-factor of ∼ 2 (dashed line, see text).
All measurements were performed at T = 300 mK.
34
in level positions of panel (b) which, in the case of spin degeneracy, would
only change for every second n.
Nanowire dots in magnetic fields
Additional information about the nanowire quantum dots can be obtained
by applying an external magnetic field B. In the case of spin degeneracy, the
degeneracy is lifted due to the Zeeman splitting ∆EZ = 2× 12 gµB B where µB
is the Bohr magneton and g the effective g-factor. Thus the level spacings
change proportionally to ∆EZ resulting in a shift in the addition energies
which can be directly measured by mapping the Coulomb peak separations
as in Fig. 2.16. Such measurements were analyzed in details in Ref. [30] for
InAs nanowire dots of various sizes showing a variation of the g-factor from
the bulk value gbulk = 15 in large dots to ∼ 2 for very small dots.
A similar effect is expected also in the absence of spin degeneracy as the
mutual distances between the levels shift according to the Zeeman splitting. Figure 2.17(c) shows a grey scale representation of the linear conductance measured as a function of gate and perpendicular magnetic field
B for the gate region of diamonds #5-10 in the stability diagram. The
Coulomb peaks follow nearly parallel paths but small wriggles can be seen.
The relative shifts are small since EC À ∆E and no shifts larger than
∼ ∆Eavg are expected as level crossings then occur. Systematic shifts can,
however, be identified in panel (d) where peak the separations, ∆Vg , have
been extracted. For low fields (B . 0.3 T) the addition energies alternately
shrink and grow consistent with an alternating spin filling. The dashed line
shows the expected shift magnitude for g = 2 (with the above value for the
gate coupling coefficient α) in good agreement with the measurement. By
comparing with Ref. [30], this value of the g-factor corresponds to a very
small dot size15 which, as discussed below, is not consistent with a simple
capacitance estimate based on the charging energy. This discrepancy may
result from coupling between the levels causing a suppression of the shifts.
Another explanation may be that the dot geometry is not expected to be
a well-defined segment of the wire, as is the case in the dots of Ref. [30],
which are confined by growth-defined barriers. For larger fields a complicated level crossing behavior is observed in Fig. 2.17(d) which will not be
discussed further.
Nanowire quantum dot geometry
The results presented above were all obtained from measurements on one
device which was particularly thoroughly investigated, however, measure15
In Ref. [30] g ≈ 2 was found for dots defined in a wire segment of length 8 nm in a
wire of diameter 70 nm.
35
L ~700nm
G [a.u.]
15
Cg [aF]
10
5.7
5.8
Vg [V]
5.9
L
5
0
0
250
500
750
L [nm]
Figure 2.18: The capacitance Cg of the back gate to the quantum dot as a
function of the electrode separation L. The linear relation shows that the dot
size is determined by the electrode spacing rather than the random defects
along the wire. The linear fit extrapolates to zero at a finite L indicating
a substantial contact induced depletion of the wires. The rightmost inset
shows an SEM micrograph of a large L ∼ 700 nm device (scale bar, 1 µm) and
the leftmost inset shows its regular Coulomb peaks in the linear conduction
G as a function of gate voltage at T = 300 mK.
ments on other devices were found to exhibit similar characteristics.
An important feature is the striking similarity in the dot characteristics
in the measurements of Fig. 2.16 and Fig. 2.17 despite the fact that they
were performed at significantly different gate voltages (separate ends of Fig.
2.15(a) corresponding to addition of ∼ 150 electrons). This shows that the
dot confinement is relatively insensitive to the electrostatic environment and
that the wire behaves as a single dot over the whole range. Such behavior
is generally observed in these devices, however, in a few cases an additional
period in the Coulomb peaks was found at the highest gate voltages characteristic of two dots in series.
Since no special efforts were made to tailor the confinement of the dot, such
as growth defined barriers or electrostatic gates, it is difficult to conclude
about the geometry from the above measurements. Some information is,
however, conveyed when using the CB peak spacings ∆Vg ∼ 40 mV to calculate the gate capacitance which, for ∆E ¿ EC , is given by (eq. 2.2)
Cg ≈ e/∆Vg ∼ 4aF.
(2.4)
This can be repeated for all the devices and Fig. 2.18 shows Cg as a function
of the electrode spacing L. The rightmost inset shows an SEM image of the
36
device having the largest electrode spacing L ∼ 700 nm and the leftmost
inset shows G vs. Vg for this device at 300 mK, showing the regularity of the
Coulomb peaks. As seen, the capacitance scales linearly with L for these
devices, showing that the size of the dot is determined by the electrode
spacing rather than random defects or potential fluctuations along the wire,
which would result in random dot sizes. The linear fit extrapolates to zero
at a non-zero electrode spacing ∼ 120 nm and we speculate that the dot is
defined in a middle segment of the wire with barriers induced by depletion
of the wire near the contacts (i.e., the 120 nm corresponds to the depleted
segment). Assuming that the dot extends the entire cross section of the
wire, we can compare the capacitance with a model of a cylinder with radius
rnw ≈ 35 nm and length lnw situated h ≈ 535 nm (center to plane) above a
conducting back-plane. For this geometry the capacitance is
C0 =
2π²0 ²r lnw
= 0.06 aF/nm × lnw ,
ln(2h/rnw )
(2.5)
where ² = 3.9 is the dielectric constant of the silicon dioxide substrate. For
Cg = 4aF this gives lnw ≈ 65 nm for the effective length of that segment
of the wire which comprises the quantum dot. This is substantially shorter
than the electrode separation L = 300 nm in qualitative agreement with the
above picture.
In Ref. [10] similar contact-defined barriers were found for quantum dots
based on silicon nanowire devices and it is often the case for single-wall
carbon nanotube devices[43]16 , but due to the unusual contact properties of
InAs such regular behavior comes as a surprise. More work (adding data
points to Fig. 2.18 from additional device batches) is needed to substantiate
whether this behavior is indeed a general property of the MBE grown InAs
nanowires.
Tunable nanowire devices in the closed dot regime
The results discussed above were all measured for the devices which were
fabricated to have large barriers (type-2, cf. the introduction). The type-1
devices exhibit tunnel barriers which are tunable with the back-gate and
below, the measurements in the intermediate coupling regime are presented.
In the closed dot regime, these devices are unstable and exhibit multiple
gate-switches. However, for completeness a stability diagram measured for a
type-1 device, in the close-dot regime, is shown in Appendix C for a Vg range
covering a few Coulomb diamonds. The behavior is qualitatively similar to
the results presented in this section, however, the charging energies and
level spacings are generally larger, EC ≈ 6 meV and 0.3 . ∆E . 1 meV,
respectively.
16
In these cases the dot size fits the electrode separations, i.e., the depletion is negligible.
37
Summary: InAs nanowire devices in the closed dot regime
In summary, we have, in this section, treated nanowire devices in the closed
dot regime exhibiting quantum dot physics. Initially the constant interaction model was discussed providing the terminology for the presentation of
the wire data. Subsequently, low temperature measurements (T = 300 mK)
of a device with L = 300 nm electrode separation were treated in detail. The
results show that the wire acts as a stable and regular quantum dot over
the entire gate voltage range corresponding to the addition of ∼ 150 electrons. The characteristic energies were found to be EC ∼ 2.3 meV for the
charging energy, and ∆Eavg ∼ 0.1 meV for the average level spacing. In one
Vg -region an alternating pattern of addition energies were found, caused by
the presence of two-fold (most likely spin) degenerate levels. In a different
Vg -region the magnetic field dependence of the addition energies was investigated showing behavior consistent with an alternating spin for the ground
states at zero field.
In the final paragraph the geometry of the quantum dot was discussed. It
was shown that the back gate capacitance Cg , being a measure of the dot
size, scales linearly with electrode separation and the results indicated a
substantial depletion of the wire at the contacts. Furthermore, by comparing with a cylinder-plane model for the capacitance, the data was found to
be consistent with the dot being confined to a wire segment considerably
shorter than the electrode separation. More work is needed to support this
picture.
The results in this section were all for devices in the regime of weak coupling to the electrodes where transport occurs by sequential tunneling events,
i.e., an electron must tunnel off the dot before the next can tunnel on. In
the upcoming section we treat devices in the intermediate-coupling regime
where also higher-order processes contribute significantly, leading to additional characteristic features in the transport measurements.
38
2.5
Kondo physics in InAs nanowires
This section treats measurements on InAs nanowire devices where the coupling to the leads is stronger than was the case for the results treated in
the previous section. The barriers are still sufficiently opaque that Coulomb
blockade plays a substantial role, however, higher order processes contribute
significantly to the transport in this regime, leading to the observation of
features associated with cotunneling and the Kondo effect. The present
results are the first to demonstrate the presence of the Kondo effect in nanowire systems and the main contents of this section has been presented in
Ref. [44]. Before discussing the measurements, however, we first consider
the physical origin of these phenomena.
Co-tunneling
In the closed dot regime of the previous section, the number of electrons
residing on the dot is a well-defined integer number N , and at zero bias
transport only occurs at discrete degeneracy points. Measuring G vs. Vg
results in narrow peaks separated by stretches of zero conductance corresponding to Coulomb blockade. A schematic of the blockaded situation is
repeated in Fig. 2.19(a). Classically, transport is blocked as it requires a
finite energy for the electron on the dot to tunnel off (or for an additional
electron to tunnel on). Quantum mechanically, however, such an event, leading to the intermediate virtual state depicted in panel (b), can occur as long
as the system only exists in the virtual state for a time ∆t sufficiently short
not to violate the Heisenberg uncertainty relation ∆t . ~/EC (the charging
energy EC being the energy associated with the charge fluctuations). Thus,
as long as another electron tunnels onto the dot again within ∆t, thereby
returning the dot to its original state, a current can flow by the cotunneling
sequence (a)-(c), leading to a non-zero conductance in the Coulomb valleys
(diamonds) for stronger coupling[45]. An example of such behavior in a nanowire device was already given in the introduction (Fig. 2.4).
Since the above sequence occurs at zero bias and leaves the dot in the ground
state the process is called elastic cotunneling. For larger bias voltages, inelastic cotunneling processes, which leave the dot in an excited, state become
accessible as illustrated in the schematic in panel (d). The onset occurs at
eVsd = ∆E (the excitation energy of the dot) and in a bias spectroscopy
measurement, such processes result in features horizontally truncating the
Coulomb diamonds as illustrated in panel (e).
The Kondo effect
The transport properties of bulk alloys containing magnetic impurities (e.g.,
Fe in Au) have been studied for almost a century. In the 1930’ies it was found
39
(a)
(b)
Virtual
..
.
..
(c)
..
.
..
(d)
..
.
..
(e)
N-1
N
N+1
Vsd
Vsd
..
.
..
Vg
Figure 2.19: Schematic illustration of cotunnel processes. In the sequence
(a)-(c) an electron is transferred from source to drain electrode through
an intermediate virtual (classically forbidden) state. The process is elastic
leaving the dot in the ground state. (d) Example of an inelastic cotunneling
process leaving the dot in an excited state. Such processes become accessible
when the bias provides the required energy Vsd = ∆E. The consequence for
the stability diagram is schematically illustrated in (e). In the middle of
the diamond only the elastic processes are possible (light gray) and above
Vsd = ∆E also inelastic cotunneling contribute (dark gray).
that the temperature dependence of the electric resistivity was described by
the relation ρ(T ) = ρ0 + aT 5 − b log(T ). The first two terms were well
understood to arise from scattering off static impurities (T -independent)
and of phonons (which ”freeze” out at low temperatures), respectively. A
satisfactory explanation for the logarithmic term, which causes an increase
in the resistivity at low temperature, was not given until the 1960’ies. At
that point Jun Kondo showed that the measurements could be explained
by considering multiple spin-flip scatting of the conduction electrons on the
impurity spin[46]. In this way the conduction electrons screen the impurity
spin leading to the formation of a many-body Kondo state (Kondo cloud)
accompanied by a resonance at the Fermi level. This increased scattering
results in the observed resistivity increase below a temperature TK being
the characteristic binding energy of the Kondo state17 .
This original observation of the Kondo effect is concerned with bulk alloys
containing many magnetic impurities. Consider now a quantum dot hosting
an odd number of electrons. In the simplest case, pairs of electrons with
anti-aligned spins are formed, leaving a single unpaired spin-1/2 in the dot,
17
For further material on the Kondo effect in bulk metals see Refs. [47, 48, 25]
40
(b)
(a)
(d)
Virtual
(c)
(e)
DOS
TK
N
Even
Vsd
N+2
N+1
Vg
Γ
Figure 2.20: Schematic illustration of the processes leading to the Kondo
effect in odd-N quantum dots. In (a)-(c) an electron is transported through
the dot while flipping the dot spin. The sum of all such processes lead to
a many-body Kondo state appearing as a sharp resonance in the density of
states at the Fermi level leading to an enhanced conductance at zero bias. At
higher biases this enhancement is lost and the Kondo appears in a stability
diagram measurement as illustrated in (e).
as illustrated in Fig. 2.20(a). In this case the dot and the metallic electrodes constitute a system in many ways analogous to a single magnetic
impurity embedded in a metallic host, and indeed the two systems share
the intriguing phenomenon of the Kondo effect. As for the bulk alloys, the
dot spin is screened by the conduction electrons through a large number
of spin-flip cotunnel events. An example of such a process is schematically illustrated by the sequence (a)-(c) of Fig. 2.20. This again leads to a
correlated many-body state extending into the leads, however, contrary to
the bulk case, the increased scattering contributes to the current and thus
leads to an enhanced conductance. Thus, the Kondo effect will appear as
a conductance increase in the odd-N Coulomb valleys upon lowering the
temperature. This is opposite to the behavior expected for ordinary Coulomb blockade which dominates in the even-N valleys where the net spin (in
the simplest case) is zero and no Kondo effect is expected. The Kondo state
leads to a peak of width ∼ TK in the density of states locked to the chemical
potentials of the leads as illustrated in Fig. 2.20(d). In the non-equilibrium
situation at finite bias, the Kondo enhanced conductance is lost and in a
stability diagram measurement, the signature of the Kondo effect is therefore a zero-bias conductance enhancement through the odd-N diamonds as
41
illustrated in panel (e). Another distinct feature of the Kondo effect is the
splitting of the Kondo-ridge in a magnetic field B into two peaks at finite
bias eVsd = ±gµB B where g is the g-factor and µB = 58µ eV/T, the Bohr
magneton[49, 50].
Investigating the Kondo effect in a quantum dot allows much extended flexibility compared to bulk systems. First of all, such measurements probe
a single impurity spin rather than an ensemble as in the bulk diluted alloys. Furthermore, it allows individual tuning of various parameters such
as the coupling to the leads Γ, the distance from the Fermi level to the impurity level ²0 (controlled by the gate), and the possibility of investigating
non-equilibrium effects by applying a finite bias across the impurity. For example, midway along the Kondo ridge (Fig. 2.20(e)) the Kondo temperature
is given in terms of these parameters as[47]
1
1
kB Tk = (ΓEC ) 2 e−πEC /2Γ
(2.6)
2
For these reasons, the Kondo effect has received renewed attention since
it was first discovered in quantum dots in 1998 by Goldhaber-Gordon et.
al. [51]. This pioneering work was based on dots defined by electrostatic
gating of the 2DEG in a GaAs/GaAlAs heterostructure. Subsequently the
effect has been observed in devices based on single and multi-wall carbon
nanotubes[52, 53], single molecules[54, 55], fullerene peapods[56], and the
present chapter adds the InAs nanowire system to this list[44]. For further
details on the Kondo effect in quantum dots see Refs. [25, 47, 49, 57, 58].
Kondo physics in InAs Nanowire quantum dots
This section considers the experimental evidence for the existence of the
Kondo effect in the InAs nanowire quantum dots. The Kondo effect is only
visible for temperatures lower than TK and as seen from Eq. 2.6 making TK
large requires a strong coupling to the leads. The results considered in the
previous section were measured on devices which had large contact barriers
due to the fabrication scheme (type-2). In this section we consider the (type1) devices where the wire surfaces received both oxygen plasma etching and
a HF wet-etch (immediately) prior to evaporation of the contacts (cf. section
2.3). It turns out that the contact transparency in these devices varies with
gate potential rendering it possible to investigate the intermediate coupling
regime relevant for the Kondo effect.
Figure 2.21 shows the temperature dependence of the G vs. Vg trace for
a typical type-1 device. At high temperatures, the conductance varies
monotonically from 0.9 e2 /h at Vg = −10 V to 2.8 e2 /h at Vg = 10 V thereby
identifying, as for the type-2 devices, the carriers as n-type. At lower temperatures, the slope of the G(Vg )-trace increases but the conductance at
Vg = 10 V remains effectively unchanged. This is in contrast to the behavior of the type-2 devices (Fig. 2.10) and indicates a low barrier at the
42
5
0
G[e2/h]
-8.9
2
-8.8
-8.7
V [V]
g
G [e2/h*10-2]
3
-8.6
270K
130K
15K
0.3K
1
0
-10
-5
0
Gate [V]
5
10
Figure 2.21: The linear conduction G vs. Vg for temperatures T =
0.3 K, 15 K, 130 K, 270 K for a type-1 device showing gate-dependent barriers:
At 0.3 K and Vg > 2 V the contacts are open and reproducible conductance
fluctuations are observed. For −6 V < Vg < −1 V the Kondo effect is observed and for Vg < −7 V the device is dominated by CB as expanded in
the inset (see text).
contacts at Vg = 10 V increasing with smaller gate-voltages. At 0.3 K, the
behavior is as follows: For Vg . −7.5 V (inset) the device behaves as a quantum dot in the CB regime with large tunnel barriers between the leads and
the dot with transport exhibiting sharp peaks separated by regions of zero
conductance as discussed in the previous section. For Vg & 2 V large-scale
reproducible conductance oscillations are observed. There are no indications
of charge quantization (Coulomb blockade), supporting the view of low barriers between the leads and the wire in this gate range. This regime will
be treated in the upcoming section, where the fluctuations are interpreted
by electron interference effects. The focus of this section is on the intermediate gate region where broad CB peaks appear (see also Fig. 2.4). The
qualitative behavior of Fig. 2.21 is typical for the type-1 devices, however,
the gate-voltage scale may change between devices and between subsequent
cool-downs of the same device due to sensitivity to the charge configuration
in the substrate. Also variations are found in the size of the gate interval in
which the device is in the intermediate coupling and closed regimes, respectively.
In Fig. 2.22(a) the G vs. Vg trace for a short gate range has been enhanced. Solid and dashed traces show measurements for T = 300 mK and
T ≈ 800 mK, respectively. Broadened CB peaks are observed and as in
the previous section each valley corresponds to a fixed number of electrons
(N ) on the dot. The non-zero valley conductance Gv indicates a significant
contribution to the conductance from elastic co-tunneling processes and Gv
exhibits an alternating pattern where high-Gv valleys are followed by valleys
43
G [e2/h]
1.0
75
0 Valley # 4
8
(c)
∆Vg [mV]
100
∆Vg [mV]
(a)
0.5
#0
100
90
80
70
0.0
-4.50
-4.25
0
-4.00
0.25
B [T]
Vg [V]
0.5
(b)
B [T]
1.5
1.0
0.5
0.0
-4.50
-4.25
Vg [V]
-4.00
(d) 2.0
Vsd [mV]
E
O
E
O
E
O
E
O
1.0
0.0
-1.0
(c)
-2.0
-4.50
-4.25
Vg [V]
-4.00
Figure 2.22: (a) Measurement of the linear conduction as a function of gate
voltage in the intermediate regime in Fig. 2.21 for T = 300 mK (solid) and
T = 800 mK (dashed) showing 8 broad Coulomb peaks. The arrows indicate
the temperature dependence of the valley conductance for increasing T .
Inset shows the peak separation identifying a clear even-odd pattern due to
spin degeneracy. (b) The linear conductance at T = 300 mK as a function
of gate voltage and perpendicular magnetic field B showing the evolution
of the CB peaks (darker = more conductive). In (c) the low-field peak
separations have been extracted establishing the odd-N electron occupation
for the narrow CB valleys. (d) Corresponding stability diagram at T =
300 mK showing 8 CB diamonds numbered E/O corresponding to even/odd
N , respectively (darker = more conductive). In each odd-N diamond a high
conductance ridge is observed at zero bias (black arrow) due to the Kondo
effect. The white arrows point to examples of features attributed to inelastic
cotunneling.
44
of lower Gv and vice versa. This even-odd pattern repeats in the peak separations ∆Vg shown in the inset extending 9 Coulomb diamonds. As for the
even-odd effect discussed in connection with Fig. 2.16, the corresponding
even-odd pattern of the addition energies suggests a twofold degeneracy of
the dot levels. The electron spin is identified as the origin of this degeneracy by measuring the evolution of the peaks in a magnetic field. Figure
2.22(b) shows the linear conductance measured as a function of Vg and perpendicular magnetic field B analogous to the measurement for closed dots
shown in Fig. 2.17(c). Valleys corresponding to odd(even) N are expected
to widen(shrink) by the Zeeman splitting gµB B[59] and at low fields this
pairing behavior is readily observed in panel (c) where the peak separations
have been extracted from (b)18 . The complicated pattern observed at higher
fields in (b) is due to level-crossing/anti-crossing and will not be discussed
further. Thus, in the valleys in Fig. 2.22(a) corresponding to the short ∆Vg ,
the dot holds an odd number of electrons N , and the Kondo effect can be
anticipated. As seen, these valleys indeed show an increased conductance
with respect to their neighboring valleys. By considering the T = 800 mK
trace (dashed line) the conductance in the even-N valleys is seen to increase
upon heating the sample (arrows) as expected for normal Coulomb blockade.
Contrary, in the odd-N valleys Gv exhibits the opposite behavior showing
that transport occurs by different mechanisms in the even and odd-N valleys.
These observations agree with the presence of the Kondo effect and in the
stability diagram for this region, shown in Fig. 2.22(d), there are indeed
pronounced high-conductance ridges appearing at zero bias in every odd-N
diamond (one is indicated by a black arrow).
Temperature dependence of the Kondo ridge
The temperature dependence of a Kondo peak is described by the interpolation function
³ T 02 ´s
Gv (T ) = G0 2 K 02
(2.7)
T + TK
0 = T /(21/s −1)1/2 and s = 0.22 expected for a spin-half system[61,
Here TK
K
49]. The temperature dependence thus allows for an estimate of the Kondo
temperature. As seen in Fig. 2.23(a), however, only a small increase in
valley conductance is observed upon lowering the temperature from 800 mK
to 300 mK and it has not been possible to make a meaningful fit to the above
formula for these data. A crude estimate of the coupling Γ ≈ 0.7 meV from
18
We note, that an estimate of the g factor from the peak separations is not straight
forward since the Kondo effect (which will be seen below to be present for the odd-N
diamonds) modifies the peak separations[60] and as a magnetic field affects the Kondo
ridge, it contributes to the behavior in Fig. 2.22(c). Below it will be shown that the
splitting of the Kondo ridge in a magnetic field allows for determination of g.
45
T [K]
(a) 1.5
0.3
3.0
(b) 2
1.0
Vsd [mV]
dI/dVsd [e2/h]
0
0.5
x1/2
0.0
-2
ΓK
-2
2
0.3K
0
-2
2
0.9K
0
1.6K
-2
0
Vsd [mV]
2
-3.9
-3.7
Vg [V]
-3.5
Figure 2.23: (a) Top curve: temperature dependence of the conductance
for the strong Kondo ridge. The solid line is a fit to the formula given in
the text with a Kondo temperature of 2.1 K. Lower curve shows dI/dVsd
vs. Vsd through the middle of the ridge (scaled by a factor 0.5). The peak
width corresponds to TK = 2.2 K. (b) Stability diagrams measured for T =
0.3 K, 0.9 K, 1.6 K (top, middle, bottom) showing a particular strong Kondo
ridge at 300 mK (intersected a by dashed line). At higher temperatures the
Kondo ridge disappears but the Coulomb diamond pattern prevails.
the width of the Coulomb peaks gives an estimate of TK ≈ 0.8 K from Eq.
2.6.
The top panel of Fig. 2.23(b) shows a stability diagram measured for T =
300 mK in a different cool-down of the same device. A particularly strong
Kondo resonance is observed, and Fig. 2.23(a) shows dI/dVsd vs. Vsd through
the middle of the ridge at 300 mK (lower curve) showing a pronounced Kondo
peak at zero bias. The temperature dependence of the peak conductance is
shown in the upper curve (circles) and the solid line shows a fit to equation
2.7. The agreement is excellent and the fit yields TK = 2.1 K and s = 0.22
supporting the Kondo nature of the ridge. The Kondo temperature can also
be estimated from the full width at half maximum ΓK ≈ 2kB Tk /e of the
Kondo peak and for the data in Fig. 2.23(a) we find ΓK = 340 mV and thus
TK = ΓK e/2kB = 2.2 K in good agreement with the estimate above. The
lower and middle panels of Fig. 2.23(b) show the stability diagram measured
at higher temperatures T = 900 mK and T = 1.6 K, respectively. The Kondo
ridge diminishes with increasing T and at 1.6 K only the Coulomb diamond
pattern persists.
In the case of equal coupling to the leads the conductance in the Kondo valley
is expected to saturate for T ¿ TK at the ideal value G0 = 2e2 /h (unitary
46
(b)
(a)
0.2
0.5T
-2.2 -2.5
G [e2/h]
-3 0T
-2.5
Vg [V]
-2.2
Vg [V]
0.1
0T
0.0
δΚ
gFIT
1.5
0
∆V [mV]
Vsd [mV]
3
∆0 (mV)
∆ Κ (g )
7.9 +- 0.1
0.2 +- 0.01
δΚ (c)
7.7 +- 0.1
-0.02 +- 0.01
∆c (o)
8.5 +- 0.7
-0.03 +- 0.03
1.0
Vsd=Vc
0.5
0.9T
∆c
Vsd=0
0.0
0.0
∆Κ
-1
0
1
0.5
1.0
1.5
B [T]
Vsd [mV]
Figure 2.24: (a) Stability diagrams at B = 0 T (leftmost inset) and B =
0.5 T (rightmost inset) of an odd-N diamond with a Kondo resonance. Main
panel shows traces through the middle of the diamond in perpendicular
magnetic fields of 0 T, 0.1 T, . . . , 0.9 T (each off-set by −0.0075 e2 /h). The
peak at Vsd ≈ 1.25 mV indicates the on-set of inelastic cotunneling. (b) The
separation of the peaks in magnetic fields as indicated in (a). The upper
inset shows the g-factor and extrapolated splitting ∆0 at B = 0 T for the
linear fits. Rightmost inset: Schematic diagram of cotunneling processes for
zero source-drain bias (left) giving the usual Kondo effect, and at finite bias
(right) yielding Kondo enhanced cotunneling peaks out of equilibrium. All
states have total spin S = 1/2. All results are for T = 300 mK.
limit). We have not for any of the investigated Kondo ridges observed a
saturation of Gv within the accessible temperature range T & 300 mK. For
the Kondo ridge in Fig. 2.23, Gv seems to approach a value . 1.5e2 /h which
is smaller than the ideal value probably due to asymmetric coupling to the
leads in which case G0 is modified by a factor ∝ ΓL ΓR /(ΓL + ΓR )2 .
Magnetic field dependence of the Kondo ridge
In a magnetic field the spin-degenerate states of the quantum dot are split
by the Zeeman splitting gµB B, which for Kondo resonances leads to peaks
at Vsd = ±gµB B/e, independent of gate voltage[49]. The stability diagrams
inset to Fig. 2.24(a), show a Kondo ridge through an odd-N diamond measured at zero field (leftmost inset) and at B = 0.5 T (right). Indeed, a gate
independent splitting of the ridge is clearly observed. The main panel shows
dI/dVsd vs. Vsd for Vg corresponding to the middle of the diamond and for
different magnetic fields B = 0−0.9 T. As expected the peak splitting is
proportional to B. Different methods for determining g from the splitting
of the Kondo peak have been suggested in the literature. Early works used
the distance δK between the peaks in dI/dV [50, 60], however, subsequent
theoretical work has suggested the use of the distance ∆K between peaks in
47
d2 I/dV 2 (steepest points in dI/dVsd )[62] or a combination of the two[63].
Figure 2.24(b) shows ∆K (open squares) and δK (solid squares) extracted
from the data of panel (a) and measurements at higher fields (not shown).
Both ∆K and δK show a clear linear dependence and the parameters of the
linear fits are shown in the inset with g calculated from the slope and ∆0
being the extrapolated splitting at B = 0 T. Both methods gives g ≈ 8 considerably smaller than the g-factor of bulk InAs (gbulk = 15) but larger than
the value found in the previous section. The value is, however, as discussed
earlier expected to be strongly dependent on dimensionality and size of the
system[64, 30].
As discussed above, inelastic cotunneling processes, which leave the dot in
an excited state, set in when Vsd reaches the excitation energy of the dot.
This gives rise to horizontal ridges at finite bias within the Coulomb diamonds as seen in Fig. 2.22(d) and Fig. 2.24(a). At zero field, Fig. 2.24(a)
shows a sharp peak appearing at this onset (Vc ≈ 1.25 mV), in addition to
the zero bias Kondo peak (odd N ). In a magnetic field, the doublet excited
state splits by ∆c = gµB B, i.e., half that of the Kondo peak, allowing for an
independent estimate of g. The splitting is readily observed, and the linear
dependence of ∆c vs. B extracted in panel (b) (circles) yields a g-factor of
8.5 ± 0.7 consistent with that measured from the Kondo peak.
Figure 2.25 shows the splitting of the 6 Kondo ridges from two samples that
have been investigated in a magnetic field. For all the ridges the splitting is
close to that of Fig. 2.24 (also included in Fig. 2.25). The small variation in
g is interpreted as a result of slightly different dot geometries/sizes.
Finally it is noted that non-equilibrium population of the excited states at
finite bias[62] may cause a broad cusp at the cotunneling onset. However,
the peak in Fig. 2.24(a), being considerably narrower than the threshold
bias, is inconsistent with this mechanism. We speculate that the finite bias
peak is a signature of a Kondo resonance existing out of equilibrium. For
carbon nanotubes, a detailed quantitative analysis recently showed that for
an even-N quantum dot, inelastic cotunneling processes can result in a nonequilibrium singlet-triplet Kondo effect, accompanying transitions between
the singlet ground state (S = 0) and an excited triplet state (S = 1)[65].
The Kondo correlations are in this case indicated by peaks at the cotunneling onset being narrower than the threshold bias. We see similar, sharp
peaks in our devices in even-N diamonds, however, for the odd-N example in
Fig. 2.24(a), the peak is intensified by (spin-flipping) transitions promoting
the dot into excited states with the same total spin S = 1/2 as the ground
state19 . More experimental and theoretical work is needed to substantiate
these findings.
19
For a dot with nearly equidistantly spaced levels such as the region analyzed in Fig.
2.22(a) (insert), two different excited orbital states can contribute to this resonance, cf.
the schematic in Fig. 2.24(b)
48
2.0
0T 0.3
G [e2/h]
0.0
1.5
Vg [V]
1.0
1.1
2
1.0
-0.5
0
Vsd [mV]
0.5
0
Vsd [mV]
∆V [mV]
-0.3
1T
0.5
0T
-2
2
0
-2
0.0
0.0
0.5
1.0
B [T]
1.5
0.25T
2.0
Figure 2.25: The splitting of 6 Kondo ridges in a perpendicular magnetic
field. Rightmost insets show a stability diagram measured with B = 0 T
(top) and B = 0.25 T (bottom). A zero bias Kondo ridge is seen in the top
inset, and a gate-independent splitting is seen in the bottom inset. Leftmost
inset shows dI/dVsd vs. Vsd through the middle of the diamond in the stability diagrams for B = 0 − 1 T (curves offset by −0.0075(e2 /h) ∗ (B/T)).
A linear splitting is clearly observed as shown in the main panel with leftpointing triangles (C) along with the splittings of all the Kondo ridges that
were investigated in a magnetic field. The data of Fig. 2.24 is shown with
squares. The points fall almost along the same line showing that the g-factor
g ∼ 8 only varies slightly in these samples.
Summary: Kondo effect in InAs-nanowire devices
In summary we have in this section treated nanowire devices in the intermediate coupling regime. Evidence of the higher order transport processes of
inelastic cotunneling and the Kondo effect was provided, and the latter was
treated in detail. From the splitting of the Kondo ridge in a perpendicular
magnetic field a g-factor of ∼ 8 was inferred for all the Kondo ridges that
were studied in a magnetic field (6 ridges from 2 devices).
49
2.6
Conductance Fluctuations in InAs nanowires
In the previous sections the phenomena associated with nanowire devices
with low and intermediate coupling to the leads were investigated. Continuing along this line we treat, in this section, the regime of vanishing barriers
at the contacts. At low temperature the device in this regime acts as a
diffusive (or quasi-ballistic), phase coherent conductor. Due to electron interference the conductance fluctuates as a function of gate potential and
magnetic field as these change the wavelength and phase of the electrons.
Below, we first investigate the fluctuation as a function of Vg and then study
the effect of a magnetic field on the average conductance, where an average
over impurity configurations is achieved by averaging over Vg . In this way
it has been possible to study the phenomena of weak (anti) localization in
single wires.
Conductance fluctuations
In the discussion of Fig. 2.21 it was already mentioned that the open regime
is relevant for type-1 devices at the highest gate-voltages. Figure 2.26(a)
shows the linear conductance as a function of gate voltage and temperature for such a device with L ≈ 300 nm electrode separation measured on
the SEM micrograph in the inset to panel (b)20 . The conductance is high
∼ 2e2 /h and at low temperatures there is no trace of Coulomb blockade oscillations. The conductance exhibits large-scale fluctuations as a function of
gate which are reproducible in the sense that the exact same trace is found
upon repeating the measurement21 . Indeed, the corresponding stability diagram shown in panel (b) shows no Coulomb blockade diamonds but instead
an irregular pattern of low-conductance regions appear. The irregularities
contrasts the regular Fabry-Perot interference pattern found in ballistic nanotube wave guides[66] where reflection only occur at the contacts, but our
results bear great resemblance to results found for high conductance diffusive or quasi ballistic nanotube samples[53, 67]. The results of Fig. 2.26 are
interpreted as an effect of quantum interference due to scattering off random impurities along the wire - the pattern arises since, changing the Fermi
wavelength by means of Vg , has the same effect as reconfiguring the impurity distribution and thus changes the transmission probability[42] (similar
changes are expected upon applying a magnetic field). Considering a conductor of length L and mean free path `e as a sum of L/`e uncorrelated
p
fluctuating segments the root-mean-square
fluctuations δGrms = Var(G)
p
of the conductance is a factor `e /L smaller than the average conductance
hGi and thus for L À `e no significant fluctuations would be expected.
20
The measurements of the Kondo effect in Fig. 2.23 are from this device at a lower
gate-voltage.
21
Except for occasional hysteretic effects
50
(c)
273K
68K
35K
2
18K
1
11K
7K
4K
0. 3K
0 (a)
2
4
6
Gate [V]
8
10
δGrms [e2/h]
G [e2/h]
3
0.1
1µm
0.01
0.1
1
T [K]
2
0
0.3
dI/dVsd [e2/h]
2.7
(b)
Vsd [mV]
10
-2
2
4
3
5
Gate [V]
Figure 2.26: (a) The linear conductance as a function of gate voltage for
different temperatures for a type-1 nanowire device with vanishing contact
barriers and L = 300 nm electrode separation. Upon lowering the temperature large reproducible conductance fluctuations develops due to quantum
interference. (b) Differential conductance measured at T = 300 mK as a
function of gate and source-drain bias. An irregular pattern of low and
high G regions is observed. (c) The root-mean-square fluctuations δGrms of
G(Vg ) as a function of temperature after subtracting a second order polynomial (see text). δGrms saturates at low temperature to ∼ 0.5 e2 /h - dashed
line is a guide to the eye. Inset to (c) shows an SEM micrograph of the
device.
Quantum correlations, however, extends over the phase coherence length `ϕ
which at low temperatures can be much longer than `e and for `ϕ À L and
zero temperature, it is a remarkable result that fluctuations on the order of
e2 /h are expected regardless of the degree of disorder or sample size. For this
reason these fluctuations are often called universal conductance fluctuation
(UCF) (see Ref. [42] and references herein). For `ϕ < L the sample can be
considered as consisting of L/`ϕ uncorrelated segments and the fluctuations
are suppressed by a factor (`φ /L)3/2 [42].
As seen in Fig. 2.26 the G(Vg ) trace is monotonic at higher temperatures
with a slightly positive slope due to the changing carrier concentration and
the fluctuations develop upon lowering the temperature (and increasing `φ ).
Panel (c) shows δGrms calculated from the traces in (a) after subtracting a
second order polynomial to account for the monotonic background. Indeed
δGrms increases rapidly for decreasing T and saturates around T . 1 K at a
51
value ∼ 0.5 e2 /h relatively close to the expected value ∼ 0.73 e2 /h for a conductor which is 1D with respect to `φ and wire length much larger than the
diameter[42]. Thus the results indicate that `φ & L = 300 nm for T . 1 K
and the appearance of the conductance fluctuations shows that the wires
are diffusive (or quasi-ballistic) conductors (`e < L).
Magneto conductance
In this section we briefly consider the effect of applying a magnetic field
on the transport in nanowire devices in the open regime. In view of the
above considerations, random conductance fluctuations are again expected
and, in addition, another effect of quantum interference, known as weak
localization (WL), may be anticipated. The weak localization-correction
to the conductivity is caused by the constructive interference of coherently
backscattered time-reversed paths (leading to an enhanced probability for
the electron to stay where it is). This enhanced backscattering probability reduces the conductance. Upon applying a magnetic field, the electrons
along such two paths pick up different phases and the constructive interference is lost, leading to a positive magnetoconductance. In the presence of
strong spin-orbit scattering the backscattering interference can be destructive leading to the opposite behavior in a magnetic field, an effect known as
weak anti-localization (WAL)[68].
The low temperature magnetoconductance of InAs nanowires were studied
in details by Hansen et. al. in Ref. [36]. A crossover from WL to WAL
was found upon increasing Vg due to a decreasing spin relaxation length `so
(relative to `φ ). The presence of the random conductance fluctuation impedes the analysis of WL and WAL in the magnetoconductance traces, and
in Ref. [36] this obstacle was overcome by measuring 40 identical nanowire
devices in parallel thus averaging out the UCF and leaving only the weak
localization correction.
Figure 2.27(a) shows the conductance as a function of Vg for an open nanowire device measured at 4.2 K showing conductance fluctuations. Panel (b)
shows G as a function of perpendicular magnetic field −0.5 T ≤ B ≤ 1.5 T
for four different gate voltages. As expected, G(B) is symmetric in B and
exhibits large scale reproducible conductance fluctuations. A peak is observed at zero field for all Vg , but the peak width and height is modified by
the conductance fluctuations. Instead of averaging out the fluctuations by
measuring multiple devices in parallel, G was measured as a function of B
and Vg 22 and subsequently averaged over Vg ∈ [−10, 10 V] yielding hGiVg as
illustrated in panel (c). Note, that the measurement was only performed for
one direction of B, but from panel (b), G is known to be symmetric in B,
and in panel (c) the measurements have been repeated for both directions for
22
Measurement performed with Vg on the inner loop.
52
Vg = 10V
(b)
(a)
4.2 K
-5V
2.0
-10V
1.0
4.2 K
B = 0T
-5
0
Vg [V]
5
10
<G> [e2/h]
G [e2/h]
0V
-10
(c)
1.65
3.0
1.60
4.2 K
0.0
0.5 1.0
B [T]
1.5
(-)0.5
0.0
B [T]
0.5
Figure 2.27: (a) Measurement of G vs. Vg at T = 4.2 K showing large reproducible fluctuations due to quantum interference. (b) Measurement of
G as a function of perpendicular magnetic field B at four different gate
voltages (marked in panel (a)). The conductance is symmetric in B and
exhibits fluctuations due to quantum interference. (c) The average conductance hGiVg ∈[−10,10V ] as a function of B. Since G is symmetric (as confirmed
in (b)) only the part B > 0 T is measured and the data is repeated for
B < 0 T to show clearer the features attributed to WAL (negative magnetoconductance at low fields) and WL (positive magnetoconductance at higher
B). The solid line is a fit to the theory (see text).
clarity. A clear negative magnetoconductance is observed at low fields interpreted as an effect of weak anti-localization, thus showing that a significant
spin-scattering takes place in these wires. At higher fields the interfering
paths are shorter and the spin does not scatter significantly and the positive
magnetoconductance of (normal) weak localization is recovered. Since the
conductance changes significantly over the averaging interval, showing that
the carrier density and possibly also the scattering lengths changes, a quantitative analysis of these data should not be taken too seriously. However, to
get a rough estimate for the parameters which would result in a WAL curve
like the one in Fig. 2.27(c) we fit the data to the model which was found
in Ref. [36] to best describe the InAs nanowire results23 . In this model the
magnetoconductance with respect to the value at zero field is given by
4
1 ´− 12 1 ³ 1
1 ´− 12
2e2 n 3 ³ 1
+
+
−
+
hL 2 `2ϕ 3`2so DτB
2 `2ϕ DτB
3³ 1
1
4
1 ´− 12 1 ³ 1
1
1 ´− 21 o
+
,
+
+
+
+
+
2 `2ϕ `2e
3`2so DτB
2 `2ϕ `2e
DτB
∆G(B) = −
−
23
Equation 3 of Ref. [36]: ’pure’ model assuming `e greater than wire diameter and `φ ¿
L which is expected to hold for the present data because of the relative high temperature
T = 4.2 K, i.e., above the saturation temperature of Fig. 2.26(c)
53
where D is the diffusion constant and DτB = C1 `4m `e /W 3 + C2 `2mp
`2e /W 2 ,
with W being the wire diameter and `m the magnetic length `m = ~/eB.
This yields
• `φ = 220 ± 20 nm,
• `so = 195 ± 20 nm, and
• `e = 100 ± 15 nm,
The fit is shown on the figure with a solid line. If we assume that the
resistance of the device in the open regime is dominated by the resistance
of the wire, rather than contact resistances24 , the change in G observed in
Fig. 2.27(a) is attributed to a changing carrier density. Relating the elastic
mean free path to the conductivity in a simple Drude model we find that
the conductance-range 1 − 3e2 /h in Fig. 2.27(a) corresponds to a carrier
density of ∼ 1016 − 1017 cm−3 . These values are close to what was found in
Ref. [36] for CBE grown InAs nanowires25 but again we stress that these
values should only be taken as a rough guide and more experimental work
is needed to make a detailed comparison based on the magnetoconductance.
The measurement show, however, the importance of spin-scattering and the
possibility of eliminating the conductance fluctuations by averaging over Vg
thus enabling measurements of weak localization on individual wires. The
averaging interval should, however, be chosen in a region where G does not
change significantly (e.g. as in Fig. 2.26(a))26 .
24
Without a four-terminal measurement it is not possible to directly evaluate the contact
resistances and therefore we do not know if this is actually the case.
25
`φ ∼ 250 nm, `so ∼ 150 nm, `e = 60 nm - for a gate corresponding to a conductance
per wire similar to the present case.
26
We note, that repeating the measurement of Hansen et. al. with many wires measured
in parallel, giving the scattering lengths as a function of Vg , is not applicable for our
devices since this requires good reproducibility in device performance and at present, a
considerable difference is observed between devices (threshold etc.). Further control of the
fabrication may enable such measurements.
54
2.7
Nanowire devices with superconducting leads
In the previous sections we have investigated electrical transport in nanowire
devices contacted by normal metals (N). As discussed in Section 2.3 one of
the advantages of new mesoscopic systems, such as nanotubes and nanowires, is the wider range of possible contact materials as opposed to traditional
mesoscopic devices based on a two dimensional electron gas in a semiconductor heterostructure. For instance, applying ferromagnetic contacts may
enable studies of spin polarized transport in mesoscopic devices[69, 70] and
other exciting phenomena are anticipated if, instead of the normal metals, the wires/tubes are contacted by superconductors (S). In this case
the nanowire provides a tunable link between the two superconductors and
the system naturally lends itself for studies of the effects of size quantization, charge quantization, and electron interactions in transport between
superconductors[71, 72, 73, 74, 75, 76, 77, 78, 21, 22, 23]. This section
presents results on S-Nanowire-S junctions in the three regimes that were
treated in the past sections for the case of normal metal leads (Coulomb
Blockade, Kondo, UCF). Due to time limitations while performing the measurements the main body of results have been obtained in the Kondo regime,
allowing only for a brief discussion of the closed-dot and open regimes. Future experiments are planned in order to investigate these regimes further.
SNS junctions
Superconductor-Normal metal-Superconductor (SNS) junctions (weak links)
have been studied extensively in the past decades[79] and the most wellknown property of such junctions is probably that, in the absence of an applied bias, a dissipationless supercurrent Is = Ic sin(∆ϕ) can flow through
the junction; an effect known as the dc Josephson effect[80, 81]. Here ∆ϕ is
the phase difference between the two superconductors and Ic is the critical
current the junction can sustain. Applying a voltage V across the junction
results in a high frequency alternating current since the phase difference
evolves according to the ac Josephson relation d(∆ϕ)/dt = 2eV /~. Thus, a
current biased setup is required in order to measure the supercurrent in a
weak link, and this has recently been achieved for both nanowire and nanotube devices[77, 78, 22, 21]. The supercurrent has not been the focus of the
studies presented in this section and the measurements have been carried
out in a voltage biased setup like those of the preceding sections enabling the
investigation of another interesting feature of superconducting weak links,
the subharmonic gap structure (SGS).
The SGS is observed at finite bias and is caused by the process of Andreev
reflections. An electron in the normal region impinging on the superconductor interface with an energy below the superconducting gap, cannot directly
enter the superconductor since there are no available states in the gap. It
55
(a)
∆
(b)
2∆/3
Figure 2.28: Schematic illustration of the process of Andreev reflections
(AR) in an SNS junction. For each superconductor the density of states is
shown and solid/open circles represents electrons and holes, respectively. (a)
Illustration of the process which opens at Vsd /e = ∆ involving a single AR.
(b) Process involving two subsequent AR’s becoming possible at Vsd /e =
2∆/3. Only processes starting with an electron are shown.
can, however, be transmitted as a Cooper pair if, at the same time, a hole is
reflected. This process is known as an Andreev reflection (AR) and due to
energy conservation, the reflection occurs symmetrically around the Fermi
level as schematically illustrated in Fig. 2.28(a). In an SNS junction the
reflected hole can subsequently be Andreev reflected as an electron at the
other NS-interface as illustrated in Fig. 2.28(b). Such processes are known
as multiple Andreev reflections (MAR) and from simple schematics as those
in Fig. 2.28 it can be realized that upon increasing the bias of the junction,
MAR processes involving n − 1 AR’s becomes accessible at bias voltages
Vsd = 2∆/ne, n = 1, 2, ... while at the same threshold MAR’s processes involving n+1 AR’s are closed and therefore the IV-characteristics (or dI/dV )
will contain structure at these voltages - the SGS[80]. As we will discuss
below, further considerations are needed if the normal region of the junction constitute a quantum dot with conduction through discrete levels and
possible interactions of the AR with the electrons on the dot[73, 76, 77].
Sample preparation
The samples for these studies were prepared in very much the same way
as described in section 2.3 for the devices with normal contacts. The main
difference being the contact material which, instead of the Ti/Au bilayer
in section 2.3, was chosen to be a Ti/Al/Ti trilayer (14/65/8 nm) inspired
by the measurements of in Refs. [22, 74]. In bulk form, aluminum has a
superconducting transition temperature of ∼ 1.2 K, and due the proximity
effect the lower Ti layer, which makes the actual contact to the nanowire,
get an enhanced Tc from its bulk value (400 mK). From the measurements
presented below it is found that the contact trilayer has a transition tem56
Closed QD
Kondo Physics
Conductance Fluctuations
15
G [e2/h]
10
2
1
0
-34
-33
-22
-21
-3
0
3
Gate [V]
Figure 2.29: Linear conductance G as a function of gate volgate Vg measured
at 300 mK for a nanowire device with superconducting Ti/Al/Ti leads. The
inset shows an SEM micrograph of the device.
perature Tc ≈ 750 mK. The top Ti-layer is added to protect the aluminum
from oxidation. An SEM image of a typical device can be seen in the inset
to Fig. 2.29.
Since, as in the previous section, Ti makes the actual contact to the nanowire we expect the contact characteristics to be similar. This is indeed
the case, as illustrated in Fig. 2.29 which shows the linear conductance as a
function of gate voltage measured at 300 mK. The results are qualitatively
similar to those presented in section 2.3 and the three regimes treated in
the previous sections can again be identified. Below, the influence of the
superconductor on the transport characteristics is treated separately for the
three regimes.
S-Nanowire-S junction in the closed-dot regime
In this section we briefly consider the closed-dot regime observed for the
lowest gate voltages in Fig. 2.29 where the coupling to the dot is Γ ≈
150 − 200 µV as determined from the width of the Coulomb peaks, i.e.,
EC À ∆ ≈ Γ27 . This regime has been addressed experimentally in Refs.
[57, 72, 76] for nanotube weak links and in Ref. [71] for metal particles coupled to superconducting leads. Figure 2.30(a) shows a bias spectroscopy plot
measured for this region. The familiar pattern of Coulomb diamonds is observed, however, the diamond degeneracy points do not touch at Vsd = 0 V
as was the case for normal metals (Fig. 2.16) but are instead separated by
an amount ∼ 0.48 mV as indicated by the arrows. Such behavior has been
observed before for nanotube weak links[57] and can be qualitatively understood by considering the condition for the onset of quasi particle (QP)
tunneling through the resonant level as illustrated in the schematics in Fig.
27
Note, that this estimate is performed while the leads are in the superconducting state
since no magnetic field was available to suppress the superconductivity.
57
(a) 3.0
Vsd [mV]
1.5
0.0
-1.5
-3.0
-26.75
-26.50
-26.25
-26.0
Vg [V]
(b)
(c)
(d)
(e)
∆
∆
∆
∆
eVsd
∆
Figure 2.30: (a) Bias spectroscopy plot of a nanowire device contacted with
superconducting leads in the closed dot regime. The Coulomb diamonds do
not come together as indicated by the arrows. This is due to the gap in the
superconducting contacts leading to a finite bias threshold Vsd = ±2∆/e
for QP tunneling as sketched in (b)-(d); see text. (e) Schematic illustration
of the origin of the gate-dependent structure between the diamond tips observed in (a). The sketch illustrates that the process involving one Andreev
reflection (n = 1) is only possible when |Vsd | ≥ ∆/e and then only occurs
through the resonant level if the gate is adjusted such that αVg = Vsd /2.
2.30(b)-(c). For voltages |Vsd | < 2∆/e no direct quasi-particle tunneling
occurs since there are no available states to tunnel into (Fig. 2.30(b)). The
threshold is at Vsd = 2∆/e as illustrated in Fig. 2.30(c) and if the position
of the level is changed by the gate a larger bias is required (Fig. 2.30(d)).
Except for this ±2∆ offset, the condition for QP tunneling is the same
as that which determines the diamond edge for a QD with normal leads
(section 2.4) and thus the Coulomb diamond pattern appears unaltered except for a ±2∆ gap. Note, that for Tc = 750 mK the value of the gap is
4∆ ∼ 4 ∗ 1.75kB Tc ∼ 460 µ eV in good agreement with the observed value.
As seen in Fig. 2.30(a) there is, however, a pronounced highly gate-dependent
structure observed for |Vsd | < 2∆/e below the diamond tips, i.e., when the
resonant level is in the gap. This structure is interpreted as a signature of
Andreev reflections. For gate voltages away from the diamond tips (multiple) AR’s are highly suppressed due to Coulomb blockade and no SGS is
observed. Contrary, between the opposing diamond tips, sub-gap peaks are
observed when MAR processes can occur through the resonant level. The
dependence on the position of the level has been analyzed theoretically in
58
Refs. [73, 82] in the case of negligible Coulomb interaction, and experimentally in Ref. [76] for nanotube quantum dots with superconducting leads.
Since our emphasis will be on the more open regime discussed below we will
not go into the details of these models. The gate-dependence can, however,
be qualitatively understood by considering the condition for the processes
involving one AR (n = 2) to occur through the resonant level. In Fig.
2.28(b) the process was illustrated in the non-resonant case occurring at a
threshold bias Vsd = ∆/e. The corresponding situation in the case of a
resonant junction is illustrated in Fig. 2.30(e). It is seen, that for the level
positioned symmetrically between the source and drain, the n = 2 process
does not connect through the level and therefore does not contribute significantly to the current. Aligning the level to the Fermi-level of the drain by
changing the gate potential αVg = ∆/2e reestablishes the resonance (here α
is the scaling parameter for the coupling of the gate to the dot - cf., section
2.4). This constitutes the threshold for the n = 2 process in the resonant
junction and for increasing bias the gate must be further adjusted to maintain the level in resonance. Similarly, the threshold of the higher order MAR
processes depends on the position of the level resulting in a gate-dependent
structure as observed in Fig. 2.30(a).
For the rightmost diamond of Fig. 2.30(a) the coupling to the leads is
stronger, and cotunneling contributes to the current leading to a faint subgap structure even in middle of the diamond. Below, we discuss the device
for the middle region of Fig. 2.29 where the coupling is even stronger and
the cotunneling contributes significantly.
59
Influence of Kondo correlations on the subharmonic gap structure of quantum dots
Since the discovery of the Kondo effect in quantum dots (QD)[51] this system has received extensive theoretical and experimental attention. As discussed in section 2.5 the effect emerges for QD’s strongly coupled to the
leads when the total spin of the electrons on the QD is non-zero e.g., if it
hosts an odd number of electrons N . The conduction electrons in the leads
then screen the spin through multiple spin-flipping cotunnel events resulting
in a correlated many-electron Kondo-state which is experimentally observable as an increased linear conduction through the dot. In the present case
where the leads are superconductors the conduction electrons form spinsinglet Cooper-pairs incapable of flipping the dot spin and thus, the Kondo
effect and superconductivity constitute competing many-body effects. In
the pioneering work by Buitelaar et. al., this competition was experimentally confirmed in investigations of the linear conductance of Kondo-dots in
multi-wall carbon nanotubes connected to superconductors. It was indeed
found that the Kondo-state persists when the energy needed for breaking
the Cooper-pairs are compensated for by the energy gained by forming the
Kondo state. Otherwise the Kondo state is suppressed and the Kondoinduced increase in the linear conductance disappears.
As discussed above, electron transport for finite bias, smaller than the gap
of the superconductors, occurs through the process of Andreev reflections
where electrons impinging on a superconductor interface are transferred as
Cooper pairs by the reflection of holes. In the previous section these features
were suppressed due to Coulomb blockade. In this section we treat nanowire
devices in the intermediate regime where the Kondo effect is observed for
normal-state leads. No experimental study of the influence of Kondo correlations on the SGS has yet been reported and this is the focus of the present
section.
We first characterize the device with the contacts in the normal state.
Because of the relatively high critical field and the very high g-factor of
InAs[30, 44] g ≈ 8 driving the contacts normal with a magnetic field will
significantly perturb the Kondo state, and instead we study the device
characteristics for temperatures above Tc . Figure 2.31(a) shows the linear conductance G as a function of backgate voltage Vg for temperatures
T = 750 − 950 mK when the contacts are normal (solid lines). A series of
overlapping Coulomb peaks are observed and the temperature dependencies
of the valley conductances Gv (for T > Tc ) are indicated by the arrows. In
four valleys the conductance decreases upon lowering the temperature as expected for normal Coulomb blockade, however, for the three valleys labeled
κ1 − κ3 the reverse behavior is observed indicating Kondo physics. The left
inset shows the stability diagram measured at T = 800 mk showing the familiar pattern of Coulomb diamonds and confirming the presence of a high
60
G [e2/h]
2
0
κ1
-3
(b)
3
800 mk
Vsd [mV]
3
Vsd [mV]
3
κ3
b3
0
0.0
-3
Vg
1.5
300 mk
κ1
Vg
G [e2/h]
(a)
κ2
b2
0.6
0.0
b2
1
b1
0
-21.5
0.6
κ1
b1
κ1
κ2
-21.25
b3
-21.0
Vg [V]
κ3
0.0
-20.75
-0.6
0.0
0.6
Vsd [mV]
Figure 2.31: (a) Linear conductance in the Kondo-regime for temperatures
T = 750 mK − T = 950 mK (black-red) and at 300 mK (dashed). The
qualitative temperature dependence of the valley conductances for temperatures above Tc are indicated by arrows. The three high-conductance valleys
κ1 − κ3 exhibit the temperature dependence expected for the Kondo effect.
Leftmost inset shows the stability diagram for the region of κ1 at 800 mK
clearly showing the zero-bias Kondo ridge (darker = less conductive). At
300 mK, i.e. below Tc , the Kondo ridge has been suppressed (rightmost inset). Instead, a double peak structure is observed as indicated by the arrows
(see text). (b) Show dI/dVsd vs. Vsd through the center of all the diamonds
for T = 800 mK confirming the presence Kondo peaks in the κ-valleys and
the absence of such in the remaining diamonds.
conductance Kondo ridge at Vsd = 0 V through the diamond of κ1. Panel
(b) confirms the presence of Kondo peaks above Tc in the κ-diamonds and
the absence of such in the even-N diamonds. The black dashed line in (a)
shows G vs. Vg for T = 300 mK, i.e., below the superconducting transition
temperature. Instead of continuing its increase as expected for the Kondo effect (without superconductivity) the valley conductances of κ1−κ3 decrease
below their values at 950 mK. This result is consistent with the findings of
Ref. [75] and shows that the binding energy ∼ kB TK of the Kondo states
κ1 − κ3 are less than the binding energy of the superconducting Cooper
pairs ∆ ≈ 1.75kB Tc . Here kB is Boltzmanns constant and TK is the Kondo
temperature and thus Tκ1 , Tκ2 , Tκ2 < ∆/kB ≈ 1.3 K (The temperature dependencies are investigated in detail below). The suppression of the Kondo
state is confirmed by the disappearance of the Kondo ridge in the stability
diagram measured at 300 mK shown for κ1 in the rightmost inset to Fig.
2.31(a). In consistency with Ref. [75] we have also observed Kondo ridges
that survive the transition to superconducting contacts and are further enhanced by the superconductor as expected for TK > ∆/kB . Such behavior
61
is demonstrated in Appendix D, but for the present discussion we focus on
the case of ”suppressed” Kondo diamonds.
The finite bias peaks indicated by the arrows in Fig. 2.31(a) are observed
throughout the stability diagram and appear symmetrically around Vsd =
0 V. These are manifestations of the MAR processes as discussed in the
previous section, however, in the present case the dot is more strongly coupled and the peaks can be observed also in the middle of the diamonds.
The outermost peak are at Vsd = 2∆/e when the peaks in the density of
states at the gap edges line up as illustrated in Fig. 2.30(c)28 . The transport for energies below the superconducting energy gap (|Vsd | < 2∆/e)) is
mediated by multiple Andreev reflections (MAR) and peaks in dI/dVsd are
expected each time a new Andreev process becomes accessible. The stability diagram in Fig. 2.32(a) shows a detailed measurement of these low-bias
features for the Coulomb diamonds of κ1 and κ2. As discussed above, a
complicated peak structure is observed close to the degeneracy points of the
Coulomb diamonds since, in this region, the MAR occur resonantly through
the gate-voltage dependent dot level[76, 73]. We restrict our discussion to
the middle region of the Coulomb diamonds where the peak positions are
gate-independent and transport occurs by cotunneling between the two superconductors. In this case, the MAR peaks at |Vsd | = 2∆/ne, n = 1, 2, . . .
are expected to appear with decreasing intensities ∝ tn (t being the effective transparency of the device) due to the strong Coulomb repulsion[83].
The black trace in Fig. 2.32(b) shows the differential conductance along
the black dashed line in Fig. 2.32(a) through the middle of an even-N diamond. Indeed, peaks are observed at |Vsd | = 2∆/e with fainter shoulders at
|Vsd | = ∆/e.
As clearly seen in Fig. 2.32(a) the sub gap structure in the odd-N diamonds
of the suppressed Kondo ridges, κ1 and κ2, is clearly modified. Unexpectedly, the peaks at |Vsd | = ∆/e are more than 5 times more intense than
the peaks at |Vsd | = 2∆/e. This is emphasized by the red trace in Fig.
2.32(b) which shows a trace through the middle of the κ1-diamond. This
contrasts the expectations for simple tunneling between the two superconductors and the presence of the Kondo effect in the normal state points
towards electron-electron correlations as the origin of the modified sub gap
structure29 . Further support of the connection between the Kondo effect
and the enhanced ∆/e-peak is provided in Fig. 2.32(c): The lower panel
shows the linear conductance (right axis) for a gate-range of 5 odd-N and
6 even-N Coulomb valleys. In the three odd-N valleys κ1 − κ3 from Fig.
2.31(b) and an additional one, κ0, the Kondo effect was observed in the
normal state, and as seen in the upper panel, the enhanced ∆/e-peak is ob28
Ignoring the level for the present discussion.
We note, that a similar enhanced ∆/e-peak in the SGS of a carbon nanotube QD
was observed in Ref. [75] and its possible connection to the Kondo effect was suggested,
however, no conclusions were drawn
29
62
Vsd [mV]
0.3
b1
b2
κ1
b3
κ2
0.0
dI/dVsd (e2/h)
(a)
2
1
-0.3
0
-21.5
-21.0
(c)
(b)
dI/dVsd
b2
0.0
Vsd [mV]
0.4
2
0.2
1
ο1
0.0 κ0
0.30
κ1
κ2
-22.0
κ3
G [e2/h]
Bias [mV]
κ1
e2
h
-0.30
Vg [V]
-21.25
0
-21.0
Gate [V]
Figure 2.32: (a) Bias spectroscopy for the Vg -region of κ1 and κ2 and
−0.35 mV ≤ Vsd ≤ 0.35 mV emphasizing the structure around the superconducing gap (darker = less conductive). (b) dI/dVsd vs. Vsd through the
middle of two neighboring diamonds (lines in (a)) showing the enhanced
peaks at Vsd ≈ ∆ for the Kondo-diamond (curves offset for clarity). (c) Bias
spectroscopy for a larger Vg -region showing the enhanced ∆-peak for four
Kondo diamonds. Main panel shows the linear conductance (right axis) and
the distance between ∆ and 2∆-peaks (left axis). Dashed lines indicate the
position of 2∆ and 4∆.
63
served in the SGS of each Coulomb diamond. In the remaining diamonds,
including o1 with odd-N (which did not show the Kondo effect in the normal state), the conventional SGS is observed. Thus, the effect seems not
to be caused alone by N being odd. In the lower panel of Fig. 2.32(c) the
separations of the SGS peaks are extracted for each Coulomb valley (left
axis). The separations of the 2∆/e-peaks depend slightly on N and are
increased in the κ-valleys with respect to the even-N values30 , however, the
enhanced ∆/e-peaks in the κ-valleys always appear at half the separation
of the 2∆/e-peak in the same valley.
The physical mechanism responsible for the peak at Vsd = ∆/e was schematically illustrated in Fig. 2.28(a): An electron co-tunnels trough the dot and
gets reflected at the superconductor interface as a hole with the creation of a
Cooper-pair in the superconductor. In the discussion of the MAR structure
in the closed dot regime it was noted, in connection with Fig. 2.30(e), that
the ∆-peak was only present when the gate was adjusted to tune the level
in alignment with the chemical potential of one of the leads. This effect has
been experimentally and theoretically established[73, 76]. In the present
case the level is far away from the gap but, by a similar reasoning, we note
that the enhancement of the ∆-peaks in the κ-diamonds is consistent with
the presence of a narrow Kondo-related peak in the density of states at the
chemical potentials of the superconductors. In the case of normal leads, this
is indeed what is expected for the Kondo effect in the presence of an applied
bias; the Kondo peak splits into two components pinned to the chemical
potentials of the leads. There are no theoretical works on the finite-bias
analog for the case of superconducting leads but a few studies have focused
on the equilibrium situation[84, 85, 86] where it is found that correlations
do indeed result in a peak within the gap in the spectral function, also for
TK < ∆. The position of the peak, however, is found to depend on TK
(also for TK > ∆) and some calculations predict the presence of two peaks
symmetrically situated in the gap. This contrasts the present measurements
which would be consistent with a single peak being situated at the center for
all TK . The situation is still under theoretical consideration and the finite
bias case remains to be treated.
Temperature dependence of the ∆-peak
One of the most distinct features of the conventional Kondo effect is provided
by the temperature dependence of the valley conductance, and to investigate
further the origin of the ∆/e-peak we study the temperature dependence of
the sub-gap structure. For T > Tc the increasing Gv for lower temperatures
is qualitatively indicated by the arrows in Fig. 2.31(b) for κ1 − κ3. Figure
2.33(b) shows Gv (T ) (solid symbols) for κ1 exhibiting the increase for T &
30
Similar fluctuations were observed in Ref. [75] for nanotube based quantum dots.
64
2
(a)
(b)
dI/dVsd
310 mK
dI/dVsd [e2/h]
e
2h
1.15
1.00
0.85
1700 mK
-0.5
0.5
0.5
1.0
T [K]
2.0
(c)
(d)
1.6
dI/dVsd
dI/dVsd [e2/h]
2
e
2h
0.0
Vsd [mV]
1.4
1.2
1.0
-0.2
0.0
Vsd [mV]
0.2
0.3
0.4
0.5
T [K]
0.6
0.7 Tc
Figure 2.33: (a), dI/dVsd vs. Vsd through κ1 for different temperatures
310 mK - 1700 mK (offset for clarity). The formation of the ∆/e-peaks can
be followed. Symbols correspond to traces in (b) showing the suppression of
the valley conductance (dashed line guides the eye) and the increase of the
average ∆-peak heights upon cooling below Tc (solid line shows ∆bcs (T ) see text). (c) Measurements as in (a) from another device. The ∆ and 2∆
peaks can be clearly resolved and the average peak heights are extracted in
(d) - the 2∆-points offset for clarity. The average 2∆-peak height is well
described by the temperature dependence of the gap (solid lines) but the
behavior of the 2∆-peak is not captured by ∆bcs (T ) (solid lines) - see text.
1 K followed by a saturation around Tc and a suppression of the conductance
when the superconducting gap develops at lower temperatures.
To investigate the temperature dependence of the finite-bias peaks in the
sub-gap structure, traces of dI/dVsd vs. Vsd were measured through the
middle of the Coulomb diamonds for different temperatures. Figure 2.33(a)
shows the result for κ1 for temperatures 1700 mK − 310 mK. Upon lowering
the temperature, the initial formation of the Kondo peak for T > Tc is
observed followed by the formation of the sub-gap peaks at |Vsd | = ∆/e
with shoulders at |Vsd | = 2∆/e for T < Tc . As shown on the figure (red
dashed lines), the temperature dependence of the peak positions follow the
expected temperature dependence of the BCS gap ∆BCS (T ). To allow for
a better comparison of the evolution of the ∆ and 2∆ peaks we include in
65
Vsd [mV]
*
*
0.4
**
*
0
-.4
-5
-4
-3
Gate [V]
Figure 2.34: Low bias stability diagram of a nanowire quantum dot with
superconducting leads. The bare arrows points to diamonds exhibiting an
enhancement of the ∆-peak, starred arrows points to diamonds with a surviving zero-bias Kondo ridge and the double starred arrow indicated diamond of the measurement presented in Fig. 2.33(c) and (d). The slight bias
drift evident in (b) is due to a changing optocoupler-offset during the course
of the measurement.
panel (c) a similar measurement performed for a suppressed Kondo ridge
in a different device where the four peaks can be clearly distinguished. In
panels (b) and (d) the temperature dependence of the peak heights and
the valley conductances have been extracted. In both cases the ∆-peaks
increases monotonically whereas the 2∆-peaks of the measurement in panel
(c) saturates below ∼ 0.5 K. Since the temperature dependence is governed
by the temperature dependence of the gap we expect the measured peak
heights31 (in the absence of correlations) to be proportional to ∆(T )[87, 88].
Such a fit is made in panel (d) and indeed describes the behavior of the 2∆peak very well. For the ∆-peak, however, the continuing strengthening for
lower temperatures is not well reproduced by ∆(T ), thereby adding further
evidence to the importance of correlations for the origin of this peak.
Finally, we note that the observed phenomenon is not limited to the four
odd-N diamonds presented here. Figure 2.34 shows a low-bias stability
diagram for another device exhibiting the same enhancement of the ∆-peak
in 11 diamonds. Altogether we have observed the increased ∆/e peak for
∼ 20 suppressed Kondo ridges in two different nanowire devices and the
same phenomenon has been observed in single-wall carbon nanotube-based
devices fabricated and measured by K. Grove-Rasmussen and H.I. Jørgensen.
In appendix D an example from such a device is presented. This shows that
the phenomenon is general for Kondo-dots connected to superconducting
leads. More work is needed, however, to fully understand its origin.
31
The measured conductance peaks are proportional to the magnitude of the currentjumps at eVsd = 2∆/n in the IV-curves.
66
S-Nanowire-S junctions in the open regime
We now briefly turn to the S-Nanowire-S junction in the rightmost gate range
of Fig. 2.29 displaying the characteristics of a diffusive conductor as treated
in section 2.6 for the case of normal metal leads. Measurements on devices
in the open regime have been reported in Ref. [89] for diffusive multi-wall
carbon nanotube devices, in Refs. [77, 78] for single-wall carbon nanotube
devices in the Fabry-Perot regime, and in Refs. [22, 23] for semiconducting
InAs and Si/Ge nanowires, respectively. As mentioned earlier our investigation of the superconducting devices has been focused on the Kondo regime
discussed in the previous section and only a few measurements have been
performed in this regime. This section is included, however, for the sake of
completeness.
Figure 2.35(a) shows a bias spectroscopy plot of the device in the open
regime. For |Vsd | & 0.3 mV the pattern interpreted in section 2.6 as caused
by quantum interference (UCF) is again recognized. As expected. in view
of the preceding sections, a pronounced structure is observed at low bias
due to the superconductors. Panel (b) shows a detailed bias-spectroscopy
measurement of this sub-gap structure32 . Since in this gate range, Γ À Uc
there is no suppression of MAR due to the Coulomb interaction and the
peaks due to quasi particle tunneling are clearly observed in the whole gate
range for Vsd ≈ ±190 µV. Higher order structures can be identified as well,
as indicated by the arrows in panel (c). The lower value of 2∆ in these measurements as compared to those discussed in the preceding section might
be due to local heating which is expected to be more important for these
measurements due to the larger conductance (i.e. higher current). It is clear
from the extracted sections in panel (c) that the relative strength of the
different MAR processes depends on the transmission of the device, ranging
from enhancement to suppression of the higher order processes. A similar
behavior has been observed in carbon nanotube based devices[89, 57]. A
more systematic investigation might be pursued in a future nanowire study,
but the present data does not allow for such an analysis.
An interesting feature evident in the measurements of Fig. 2.35(b) and (c)
is the narrow (FWHM ∼ 10 − 14 µV) zero-bias peak observed for all gate
voltages. Recently, a zero bias dissipationless (super) current was observed
in nanowire devices in the open regime [22, 23]. This would correspond to a
divergent conductance peak at Vsd = 0 mV. The supercurrent is, however,
very sensitive to noise and in Refs. [22, 23] the cryostats were equipped with
special (cold) copper-powder filters in order to observe the supercurrent. In
the absence of such precautions (as is the case for the present measurements)
32
The data of Fig. 2.35(a) and (b) were measured with a few days and a few 3 He
condensation-cycles in between, and despite the overlapping gate interval there is not
complete correspondence between the two probably due to a reconfiguration of the charges
in the substrate in the meantime.
67
the supercurrent will be smeared and thus, it is tempting to attribute the
zero-bias peak in our measurements to a noise-smeared super-current peak.
However, direct observation of the dissipationless current is required to prove
this.
Vsd [mV]
(a) 2.5
0.0
-2.5
6
7
8
Vg [V]
10
(c) 7.0
dI/dV [e2/h]
Vsd [mV]
(b) 0.3
9
0.0
7
(ii)
(i)
3.0
iii
ii i
-0.3
(iii)
5.0
8
9
10
Vg [V]
-3
0
Vsd [mV]
3
Figure 2.35: (a) Bias spectroscopy plot (darker = less conductive) of a
nanowire device contacted with superconducting leads in the open regime
exhibiting quantum interference. At |Vsd | = 2∆/e an enhanced conductivity
is observed due to the enhancement of the quasiparticle transport by the
peaks in the density of states. Below this bias threshold transport occur
by the process of Andreev reflections giving rise to the sub-gap structure
emphasized by the detailed low-bias spectroscopy plot in (b) and the sections
extracted in (c). The peak at Vsd = 0 mV observed in (b) and (c) for all Vg
is interpreted as a smeared super-current peak. The discontinuities in (a) at
Vg ∼ 9.1 V and Vg ∼ 9.4 V are due to gate-switches and the slight bias drift
evident in (b) is due to a changing optocoupler-offset during the course of
the measurement.
68
2.8
Conclusion of nanowire studies
In conclusion, we have in this chapter investigated the electrical transport
characteristics of devices fabricated from semiconducting InAs nanowires. A
fabrication scheme has been developed which has allowed a reasonable fraction of devices to reach the state of low-temperature measurements (despite
extensive and obstinate handling problems due to extreme device sensitivity to static electricity). Data has been obtained on nanowires contacted
by normal-metal (Ti/Au) leads and the transport characteristics have been
investigated in regimes of different barrier transparencies. These investigations were enabled by a tunability of the barriers with the backgate potential. Three transport regimes were treated separately; the Coulomb blockade regime, the Kondo regime, and the open regime. In the CB regime
un-tunable devices ”born” with large contact barriers, due to a slight variation in the fabrication scheme, were most extensively studied. A regular
CB structure was found, persisting at least over the gate range corresponding to the addition of ∼ 150 electrons. Results of non-linear transport
measurements were presented and the level structure was analyzed in details giving charging energies EC ∼ 2.5 meV and average level spacings
∆Eavg ≈ 0.2 meV. By investigating the influence of electrode separation
on the back-gate capacitance the confinement of the dot was shown to be
defined by the electrodes rather than random defects along the wire. For
device with L = 300 nm electrode spacing, the capacitance of the dot corresponded to an effective length of ∼ 65 nm for the wire segment comprising
the dot.
For larger barrier transparencies, evidences of the higher order transport
processes of cotunneling was found. For Coulomb diamonds corresponding to the dot hosting an odd number of electrons, the Kondo effect was
observed. Being the first observation of the Kondo effect in nanowire systems its characteristics were studied in detail and found to agree with the
conventional theory for quantum dots. The splitting of the Kondo ridge in
an external perpendicular magnetic field was found to be a convenient way
of determining the Lande g-factor of the devices and g ≈ 8 was found for
all Kondo ridges and devices. The value is considerably down-shifted with
respect to the bulk value but agrees with reported studies of the g-factor
in confined InAs structures. The observation of the excited state structure
and the Kondo effect in the nanowire quantum dots show unambiguously
that these relatively simple devices constitute well behaved and coherent
quantum devices at low temperatures.
For even higher transparencies the devices acted as coherent diffusive conductors and electron interference leads to the phenomena of (universal) conductance fluctuations when varying the external parameters of gate, magnetic field and bias. In the magneto conductance, weak (anti) localization
was found and by averaging over gate potential while varying the magnetic
69
field the weak localization correction to the conduction could be separated
from the random fluctuations enabling an estimate of the elastic scattering
length le ≈ 100 nm, the spin-orbit scattering length lso ≈ 200 nm and the
phase-coherence length lϕ ≈ 220 nm (at 4.2 K).
Nanowire devices contacted by superconducing leads were realized following
the same fabrication scheme but choosing a (Ti/Al/Ti) trilayer for contacting which turns superconducting below Tc ≈ 750 mK. The effects of the
superconducting leads were studied in the three transport regimes, however,
with emphasis on the intermediate regime where the conventional Kondo
effect was observed for temperatures above Tc . Upon cooling the devices
below Tc , the fate of the Kondo state depends on its binding energy with
respect to the binding energy of the Cooper-pairs of the superconductor
and in most cases the Kondo ridge was suppressed. Nevertheless, Kondocorrelations were found to have a pronounced influence on the subharmonic
gap structure, greatly enhancing the ∆-peak in the differential conductance.
The same effect has also been observed by K. Grove-Rasmussen and H.I.
Jørgensen in our group, for quantum dots in carbon nanotubes contacted
by superconductors, confirming the general character of the phenomenon.
The measurements are consistent with a Kondo-related peak situated at each
chemical potential, as is the case for the finite-bias situation for Kondo-dots
with normal leads, however, no theoretical model has yet been solved to substantiate this further. These issues remain under theoretical consideration.
In the closed-dot regime the main effect of the superconductor was to open
a gap in the stability-diagram and in the open regime the effects of multiple
Andreev reflections were observed and indications of a supercurrent were
found but the details remain for future studies.
All the results presented in this chapter have been observed in more than
one device.
Outlook
As discussed in the introduction, the nanowire system in general, provides
a large range of possibilities for nanoscale research and applications. For
electrical device applications alone, countless combinations exist of possible
nanowire crystals (with various properties), the choice of contact materials
and the possibilities for tailored heterostructures (both radial and axial).
In this respect the nanowire system is unique and is still in its infancy and
therefore holds plenty of room for significant scientific discoveries.
There are a number of obvious extensions of the experiments presented in
this thesis. Useful information for quantifying possible differences between
the MBE-grown wires of this project and those synthesized by ”conventional” techniques could be extracted from a systematic study of the magnetoconductance measurements of weak localization (cf. Ref. [36]). Such
experiments should be carried out in the near future. Fabrication wise, fol70
lowing the recent reports in Refs. [20, 22, 90] on local electrostatic control
of the quantum dot barriers, would allow extended flexibility and the possibility of defining multiple dots. Such systems are interesting for quantum
information and recent theoretical proposals for electrical control over the
quantum dot spin via the spin-orbit interaction in InAs nanowires provide
an exciting perspective[91]. Also in extension of the measurements on nanowires contacted by superconductors, the development of local gate control
would allow exciting investigations of the possibility of forcing the individual
electrons of a Cooper-pair through two spatially separated dots while still
preserving their spin entanglement[92]. The possibilities and perspectives
are great and it will be exciting to follow the future developments.
71
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78
Chapter 3
EFM on Carbon Nanotubes
3.1
Introduction
Figure 3.1(a) shows a schematic illustration of the crystal structure of a single wall carbon nanotube (SWNT). SWNT’s can be viewed as the structure
which emerges when rolling a 2D graphene layer into a seamless cylinder.
Due to the properties of the underlying graphene structure and the quantization of the electronic wave function around the circumferential direction
of the tube, SWNT’s acquire remarkable mechanical, optical, and electrical properties[1] and ever since their discovery in 1991[2] they have received
enormous attention from almost every corner of the nano science community.
Electrically, SWNT’s can be either metallic or semiconducting depending on
the exact crystal structure and they constitute a near-ideal realization of a
one-dimensional system. Therefore, many groups have focused on making
electrical measurements on carbon nanotubes (CNT) for fundamental as
well as applied science.
Almost all CNT-based electrical devices share the same starting-point geometry as the nanowire devices discussed in the previous chapter. This geometry is schematically illustrated in Fig. 3.1(b). CNT’s are supported on a thin
layer of insulating oxide (e.g., silicon dioxide) which caps a conducting substrate (e.g., highly doped silicon) that can act as a global electrode for electrostatic gating. The CNT’s are usually deposited on the substrate from a
suspension in an organic solvent or directly grown on the substrate by chemical vapor deposition (CVD). The electrical contact to the CNT’s is realized
by metal electrodes which are usually fabricated by standard top-down lithographic techniques. For many purposes it is crucial to be able to image the
CNT distribution prior to electrode fabrication. For instance, the exact positions of the CNT’s on the substrate are often required for single-nanotube
devices, and for devices made by evaporation of random contacts it is useful to know the density of CNT’s. Two standard techniques are capable of
imaging CNT’s in this geometry. Scanning electron microscopy (SEM) and
79
(a)
(b)
CNT
SiO2
doped Si
Figure 3.1: (a) Side-view schematic of the atomic configuration of a section of
a carbon nanotube. The crystal structure of this particular tube is denoted
(7, 7). (b) Side-view schematic of the typical starting point geometry for
CNT-based electrical devices: The CNT is supported on a conducting wafer
(e.g. doped Si) which is capped with ∼ 500 nm of insulator (e.g. SiO2 ). The
CNT’s studied in this chapter are all in this geometry.
tapping-mode atomic force microscopy (AFM). SEM has a number of major
drawbacks. Due to the very small size of CNT’s as well as charge build-up
in the insulating substrate it is often difficult to achieve sufficient resolution to image CNT’s by SEM. Furthermore, carbon contaminants are often
deposited by the scanning beam (thus questioning subsequent use of the imaged tubes) and, more importantly, it has been shown that even low energy
SEM induces defects in CNT’s greatly affecting device performance[3, 4, 5].
Tapping mode AFM is (currently believed to be) a non-invasive technique
capable of imaging surfaces with very high resolution and it has been widely
employed for imaging CNT samples. The technique is, however, relatively
slow and if SWNT’s having diameters of ∼ 1 − 3 nm are to be clearly resolved, the image-area is limited to ∼ 20 × 20 µm, making assessment of the
larger scale distribution of SWNT’s exceedingly time consuming.
In this section our explorations of a relatively new technique for assessing CNT’s in the aforementioned geometry are described. The technique:
Electrostatic force microscopy (EFM) employs an atomic force microscope
equipped with a conducting cantilever for measuring the electrostatic interaction between the tip and the sample. Since SWNT’s are highly conducting,
compared to their insulating support, they are easily resolved with the EFM
technique. As EFM makes use of the Coulomb interaction, of longer range
than the van der Waals interaction of standard AFM, much faster and larger
scans can be reliably carried out without loosing important information. The
technique has previously been employed for the study of charging effects in
carbon nanotubes[6], for comparison of the conductivity of low-dimensional
objects (thereby establishing the insulating nature of λ-DNA)[7, 8] and even
for assessment of the density of states of individual CNT’s [9].
The following sections initially provides a brief introduction to the properties
of SWNT’s followed by a description of the principle of EFM, the experimental details and the details of the sample preparation. The technique is
subsequently employed for
80
• characterization of the large scale distribution of SWNT’s,
• assessing the quality of SWNT’s by investigating defects on a single
nanotube level,
• demonstrating the interplay between SWNT’s and surface charges, and
• mapping SWNT’s suspended in a polymer matrix (a highly relevant
geometry as will be described below)
Most of these results are presented in Refs. [10, 11, 12, 13].
Briefly about carbon nanotubes
In this section we briefly review the properties and terminology of singlewall carbon nanotubes relevant for understanding the EFM measurements
in the following sections. Already Refs. [1, 14] provide excellent general reviews of SWNT-properties, and detailed accounts of electronic properties
and electron transport can (among other places) be found in Refs. [15, 16].
For details on the mechanical and vibrational properties Refs. [17, 14] are
good sources. Since detailed knowledge is not needed for our purpose and
since most readers will, anyway, most likely be familiar with the basic theory of SWNT’s I simply list below the relevant properties and refer to the
literature for the details
• Geometric properties. As noted above, single wall carbon nanotubes can be imaginatively formed by rolling a strip of graphene into
a seamless cylinder. Obviously there are many different ways of choosing an appropriate strip - leading to tubes of different diameters and
symmetries. Figure 3.2(a) shows four examples of different SWNT’s
having approximately the same diameters dt ∼ 1.4 nm but different
symmetries. The crystal structure of a SWNT can be uniquely identified by the chiral vector C which describes the vector in the underlying
2D graphene which needs to be rolled end-to-end to form the particular tube. Thus, as indicated on the figure, a tube can be designated
by the pair of integers (n, m) used in writing C = na1 + ma2 as a
linear combination of the graphene unit vectors a1,2 . The special case
of tubes with indices (n, n) or (n, 0) have chiral vectors coinciding
with a symmetry axis of the graphene and they attain special symmetries themselves - they are non-chiral and are designated armchair
and zigzag, respectively.
Typically, SWNT’s come with lengths up to a few micrometers, however, occasionally tubes with lengths of hundreds of micrometers are
observed1 (section 3.2). The diameters of SWNT’s produced in the
1
The current record is ∼ 4 cm [19].
81
(a)
(12,7)
(13,7)
(10,10)
(b)
(15,0)
(d)
(e)
(a)
(b)
(c)
(c)
Figure 3.2: (a) 3D illustrations of the structures of non chiral (10, 10),(15, 0)
and chiral (12, 7),(13, 7) SWNT’s. The (12, 7) and (13, 7) are almost identical to the naked eye but have different electronic properties. (b),(c),(d)
High-resolution TEM micrographs showing individual single and multi wall
carbon nanotubes and a section of a bundle of SWNT’s respectively (from
[18]). A bundle of (10, 10) SWNT’s is schematically illustrated in (e); in
reality the SWNT’s comprising are not identical.
82
lab usually fall in the range 0.7 nm . dt . 3 nm; thinner SWNT’s
are energetically unfavorable due to the large strain and only form by
use of special growth methods[20] and thicker SWNT’s collapse since
graphene is rather soft towards out-of-plane deformations2 . Figure
3.2(b),(c) show high-resolution TEM micrographs of individual single
and multi-wall carbon nanotubes respectively.
Often, SWNT’s come in the form of thin bundles/ropes comprising a
few tens of individual SWNT’s. Figure 3.2(d) shows a TEM micrograph of a section of such a rope and panel (e) shows schematically
the tube arrangement in a rope of (10, 10) tubes.
• Electronic properties The wave-function of an electron in a SWNT
is obviously exposed to periodic boundary conditions in the circumferential direction, and thus the momentum will we be quantized in
this direction. For an infinitely long tube the parallel momentum is
continues and the band structure can be obtained from the electronic
structure of graphene by slicing the graphene dispersion along a set of
lines parallel to the tube axis. Graphene is a semi-metal and the valence band touches the conduction band at the discrete corner points
(K-points) of the hexagonal Brillouin zone. Thus, a particular (n, m)nanotube will be metallic only if the allowed lines in k-space happen
to slice through a K-point. Otherwise, it will be semiconducting. The
number and orientation of the lines on the underlying graphene dispersion depend on the tube indices (n, m) and it can be shown that it
will be metallic if n−m is divisible by 3 [1, 14]. Therefore, even though
they appear almost identical to the eye, the (12, 7) tube of Fig. 3.2(a)
is semiconducting whereas the (13, 7) tube is metallic. There is much
more to be said about the electronic structure of SWNT’s especially
if considering (low-temperature) electronic transport, however, for the
purpose of the EFM studies, this is all that is required, and the reader
is referred to the literature for further details.
These points summarize what is needed to know about nanotubes for understanding the EFM measurements in the upcoming sections. Before presenting the results, however, the methods used for preparing the samples
and carrying out the experiments are described.
2
Thicker carbon nanotubes are, however, available in the form of multi-wall carbon
nanotubes (MWNT’s) where the thick shells are stabilized by thinner tubes formed in
their interior.
83
Gas
Sources
Tube
Furnace
Control
unit
Pressure
Gauges
Mass-flow
controllers
Figure 3.3: (a) The CVD oven. (b) picture of ceramic boat with wafers
3.2
Experimental Details
This section is devoted to describing the preparation of the SWNT samples
in suitable form for EFM measurements, i.e., growth of SWNT’s and processing of metal alignment marks to enable easy navigation. Subsequently, the
details of the EFM measurements are presented.
Sample preparation
As described in section 3.1, the geometry with SWNT’s supported on a conducting substrate capped with an insulating layer is the geometry-of-choice
for electrical measurements and, therefore, a number of fabrication techniques have been developed to realize it. First of all, the method depends
on the form of the tube material, which in turn depends on the method
of synthesis. SWNT’s grown from the carbon-vapor produced either in an
electrical arc-discharge between two graphitic electrodes or in the vapor produced by laser-ablating a graphitic target, usually come in bulk quantities
and are commercially available from various sources. The tubes grown in this
way, however, come in the form of thick bundles (Fig. 3.2(d)) and separating them into individual tubes is a difficult task which involves suspending
them in an organic solvent (usually di-chloro-ethane) by ultra-sonication.
Subsequently, the tubes are deposited on a suitable substrate by allowing
a small amount of the suspension to evaporate on the substrate. The first
electrical measurements [21, 22, 23, 24] were made in this way with relatively
high quality laser-ablation material (Smalley, Rice University). The sonication may, however, damage the tube structure and nowadays most research
groups, including ours, primarily use tubes grown directly on the substrate
using chemical vapor deposition (CVD). It should be noted, though, that the
various EFM-characterizations demonstrated in this chapter apply equally
well to samples prepared from bulk SWNT material.
For CVD-growth of SWNT’s the substrates are initially prepared with catalyst particles. A wafer of highly doped silicon capped with 500 nm of SiO2
84
is cut into handleable sizes (∼ 7 × 10 mm, Fig. 3.3(b)) and catalyst particles
of iron nitrate (Fe(NO3 )3 ) are deposited following Ref. [25]: Iron nitrate
is dispersed in IPA (2-propanol) (∼ 1 mg in 20 ml IPA - solution appears
faintly yellow) and each wafer is dipped in the solution for a few seconds
followed by a 10 sec. dip in n-hexane and left to dry in air. The dippingprocedure can be repeated for a denser catalyst deposition. Figure 3.3(a)
shows the CVD-setup used for growing SWNT’s. A commercial tube furnace3 is equipped with three separate gas lines - one for Argon (Ar), which
acts as an inert carrier/diluter of Methane (CH4 ), acting as the carbon feedstock and Hydrogen (H2 ) which reduces the catalyst Fe particles. The flow
rate for each gas is controlled by mass-flow controllers4 and after loading the
prepared substrates in the oven, a typical growth-run involves: 1) heating
of the furnace to 950 ◦ C under Argon, 2) 10 min. of H2 -flow (0.25 l/min), 3)
5 min CH4 (0.75 l/min) and 4) cool down under Argon. Usually the cooldown time was shortened by external air cooling. It has been difficult to get
complete reproducibility in the growth but it seems that the growth is selfterminating; i.e. extending the growth-time (step 3) does not increase the
density of the growth product. We speculate that this is due to oxidation of
the catalyst particles stoping the growth and, indeed, hydrogen reactivates
the growth (removes the oxide); i.e., repeating the steps 2) and 3) usually
results in denser growths5
After the growth, the samples can be processed using the standard techniques described in section 2.3, and for many of the EFM measurements
described in this chapter, a grid of metal alignment marks were fabricated
by e-beam lithography, metal evaporation (usually 10 nm Cr, 40 nm Au) and
subsequent liftoff. For measurement using the scanning probe microscopy
such alignment marks are essential for navigating on the sample and for
relocating areas of interest after changing the cantilever.
The scanning probe microscope
Figure 3.4(a) shows an image of the scanning probe microscope6 (SPM)
used in all the measurements presented in this chapter. In addition to the
standard features of an atomic force microscope (AFM), the instrument is
equipped with an automatic xy-sample stage, camera and for vibrational
damping the microscope is supported on a heavy table floating on pressurized pistons and everything is acoustically shielded during operation. The
3
Carbolite MTF tube furnace
Mass-flow controller:Brooks Digital Thermal Mass Flow Meter & controllers 5850S.
Electronics: Brooks Gas and Liquid Mass Flow Secondary Electronics, Model 0154
5
Thus, mixing the methane with hydrogen during growth may result in longer tubes,
however, it has not been possible to reach any conclusions. At present, repeating the steps
2) and 3) seems like the best way of increasing the tube density.
6
Digital Instruments/Veeco Metrology Group, Dimension 3100 Scanning Probe Microscope
4
85
Control
electronics
(a)
(b)
225 µm
Acoustic
Shield
AFM
10-15 µm
(b)
Vibrationally
damped support
Figure 3.4: (a) Image of the microscope and controllers. (b) SEM image of
the conducting cantilevers used in the EFM studies (SCM-PIT, images from
www.veecoprobes.com).
AFM is one of the fundamental experimental tools of nano-physics and the
following assumes a basic knowledge about its operation. For EFM measurements, a conducting cantilever is required and Fig. 3.4(b) shows an SEM
image of those used in these studies7 . For high-resolution topographic measurements, standard tapping mode cantilevers8 tend to give better results
(i.e. less noisy) and these have occasionally been used as well. The procedure for tuning the resonant frequency of the cantilever and achieving good
performance of the microscope in usual tapping mode follows the standard
operation of an AFM and will not be described here; typical parameters
can be found in appendix E. For acquiring EFM data, the microscope software’s9 interleave-mode is activated and the desired lift height is entered
in the associated field. This enables the dual-scan mode as discussed in
below and the desired parameters for the interleave scan can be chosen
(tip/substrate voltages, lift-height, scan sizes and velocities etc.). In order
to study the EFM response of the same CNT while varying the parameters,
the slow scan axis is disabled; i.e., the same line is scanned continuously. To
study, for example, the dependence on the tip bias Vt , usually a sequence
of 10 scans for each value of Vt followed by a few scans with Vt = 0 V was
used. The Vt = 0 V scans make it easy, in further analysis, to distinguish
the data-blocks belonging to different values of Vt . A typical image resulting
from this procedure is shown in Fig. 3.5(b). The microscope control software
7
SCM-PIT cantilevers from Veeco Instruments, www.veeco.com. Spring constant, k ∼
2.8N/m, Resonant frequency ω0 ∼ 70 kHz.
8
RTESP, MPP-11100-10, k = 40N/m, ω0 ∼ 270 kHz.
9
Nanoscope V6.11r1, Digital Instrument/Veeco 2003
86
(a)
(b)
Vt = 8V
Vt = 0V
Vt = 8V
Vt = 6V
Vt = 0V
Vt = 4V
Vt = 0V
1 µm
Φ [deg.]
Φ [deg.]
-4
0
...
0
-4
Figure 3.5: Illustration of the method used when investigating the dependence of the EFM signal on various parameters. (a) Typical EFM image of
CVD grown carbon nanotubes (see section 3.3). (b) Measuring along the
dashed line in (a) with the slow scan axis disabled. The tip voltage Vt is
changed allowing for ∼ 10 scan lines for each voltage and leaving ∼ 2 scan
lines with Vt = 0V to separate the blocks of data.
as well as other SPM software packages10 is designed to display the data as
images and provides a number of image-processing tools. However, for statistical data analysis of the individual scan lines, as called for by the data in
Fig. 3.5, the tools provided by these programs are very limited. For further
analysis the data were usually exported to software capable of handling large
matrices.
3.3
The EFM technique
In this section we consider the principle of the EFM technique. Most of the
results derived here can also be found in Refs. [8, 7]. Figure 3.6(a) shows
a schematic illustration of the EFM operation of an atomic force microscope. For each scan line, the topography is first attained using standard
tapping-mode AFM. The line is then retraced while using the topographic
data to keep a fixed tip-surface distance h. During the second scan the
tip is oscillated at its free resonance frequency ω0 and the EFM signal is
then the measured phase difference between the driving force and the actual oscillation of the cantilever. Letting γ denote the damping coefficient
of the oscillation it is a well known result of the damped-driven harmonic
oscillator[26] that this phase difference is given by tan(φ) = γω/(ω02 − ω 2 ) as
shown in Fig. 3.6(b) (solid curve). At resonance, φ = π/2 and it is standard
convention in scanning probe measurements to measure phase-shifts Φ with
respect to this value. In the presence of a force gradient F 0 , the resonance
frequency changes and as illustrated in Fig. 3.6(b) (dashed curves) a phase
10
The free program WSxM from Nanotec Electronica (www.nanotec.es) has often been
used for imaging processing.
87
(a)
(b)
ω0
π
Vt
h
Φ
φ
SiO2
Φ
t
0
CNT
p++ Si
ω0
ω
Figure 3.6: (a) Side-view schematic of the EFM operation of an atomic force
microscope (see text). The black shape illustrates the AFM cantilever and
the sample geometry is that of Fig. 3.1(b).
difference develops. It is straightforward to show that this phase difference
(in radians) is given by
Q
(3.1)
Φ ≈ F 0,
k
where k and Q = ω0 /γ are the spring constant and quality factor of the
cantilever, respectively. For the EFM operation, the SPM is equipped with
a conducting tip and letting C denote the tip-sample capacitance, and Vt
an applied tip-sample bias, we get F 0 = 21 C 00 Vt2 and thus the EFM signal
becomes
Q 00 2
C Vt
(3.2)
Φ=
2k
This is an important relation for the analysis of EFM data as it distinguishes
between capacitive forces (∝ Vt2 as above) and the force on the tip due to
static charges on the substrate (∝ Vt , see section 3.5). The conventions used
in SPM for the sign of the phase shift is such that for the attractive force
between the cantilever and a conducting sample the phase shift appears negative[8].
Topographic imaging of carbon nanotubes requires high resolution and relatively slow scanning since the typical SWNT feature size is on the order
of the diameter ∼ 1 nm close to the typical noise level (usually a few Å)
and the roughness of typical substrates used for nanotube devices (SiO2 ).
Electrically, however, their conductivities are much larger than that of the
insulating substrate (see below) and the tip-sample capacitance is greatly
affected by the presence of a nanotube on the surface. Thus, through Eq.
3.2 this translates into contrast for the EFM image. This is apparent in
Fig. 3.7(a) and (b) showing a topographic and corresponding EFM image
of a sample of CVD grown SWNT’s. The observed branching and the apparent height dt ∼ 1.5 nm (Fig. 3.7(c)) show that the object is probably
a thin bundle containing a few tubes. As expected, the bundle appears in
the EFM image with a pronounced contrast and due to the longer range of
88
(b)
5
0
z [nm]
Φ [deg.]
(a)
1 µm
(c)
(d)
Φ [deg.]
0
-2
3
-3
9
-4
6
0
Vt [V]
8
x [µm]
5
10
W
Φ0
-1
(f)
FWHM [µm]
(e) 3
7
z [nm]
Φ [deg.]
0
|Φ0| [deg.]
1 µm
-4
0
2
1
6
7
0
Vt [V]
8
x [µm]
9
5
10
100
200
h [nm]
300
0.6
0.4
0.2
0
0
100 200
h [nm]
0
300
Figure 3.7: (a) Topographic and (b) EFM of SWNT’s grown on SiO2 by
chemical vapor deposition. (c) Lower curve shows the hight profile (right
axis) along the black line in (a) across a ∼ 1.5 nm SWNT bundle. Upper
curve shows corresponding EFM data (dashed line in (b)) for Vt = 6 V: The
red dots show data from 10 scans for h = 30 nm and the black curves show
the average for h = 30, 40, 50, 75, 100, 125, 150, 200, 250, 300 nm (bottomtop). (d) The average EFM signal for h = 75 nm, Vt = 6 V (black) and
a Lorentzian fit (red line). The definition of the EFM amplitude Φ0 and
FWHM W of the CNT-signal is indicated. (e) The fit amplitude W and (f)
width Φ0 as a function of lift height for Vt = 6 V (2) and of Vt for h = 100 nm
(◦). In (e) solid lines show quadratic fits to Φ0 (Vt ) and Φ0 (h) ∝ h−1 and in
(f) the solid lines show linear fits to the signal widths.
89
the Coulomb interactions the bundle appears with a large apparent width
W ∼ 400 nm. As a direct consequence, EFM images can be obtained much
faster and require lower resolution making the EFM technique advantageous
for larger scale sample characterization (see below).
The line shape of the EFM response depends on the exact geometry of the
tip[27, 28], however, the data agree well with the Lorentzian fit in Fig. 3.7(e)
and this is used for analyzing the results. Figure 3.7(d) shows the amplitude Φ0 of the CNT EFM signal as a function of cantilever voltage Vt and
the quadratic dependence of Eq. 3.2 is readily observed. Obviously, the
parameters of the fit provide information about the capacitance, but since
this section is only meant for introducing basic concepts we postpone such
detailed analysis for section 3.5 where the result is needed for analyzing additional data. The height dependence of the EFM amplitude enters in Eq.
3.2 through ∂ 2 C(h)/∂h2 and again depends on the exact geometry of the
sample and cantilever. In Refs. [28, 27] the relation Φ0 ∝ h−1 was used to fit
the height dependence of Φ0 , however, we have found a considerably better
agreement using Φ0 = A(h + B)−3 which is seen in Fig. 3.7(e) to fit the data
very well in the range considered here (30 nm < h < 300 nm).
As seen in Fig. 3.7(f), the width of the EFM signal increases linearly with
lift height and is approximately independent of Vt . Thus, in cases where
large lift heights are required; e.g., for the purpose of characterization of
CNT’s embedded in thick dielectrics (see section 3.6), the weaker signal can
be compensated for by increasing Vt effectively without the loss of resolution.
After having investigated the lift height and tip voltage dependence of the
EFM signal of a particular nanotube we now consider how the signal depends
on the nanotube length l and one dimensional conductivity G0 . Consider the
schematic in Fig. 3.8(a). If we let C0 denote the tube-backlane capacitance
per unit length we have Ctb = lC0 and considering the capacitive voltage
divider we get
lC0
Vt
Vtt =
lC0 + Ctt
for the tip-tube voltage Vtt . Thus the EFM signal associated with a SWNT
of length l becomes
Q 00 ³ lC0 ´2 2
Φ(l) =
C
Vt
2k tt lC0 + Ctt
and putting a = (Vt2 Ctt00 Q/2k)−1/2 we get
³
Ctt −1 ´
Φ−1/2 = a 1 +
l
,
C0
(3.3)
i.e. Φ−1/2 depends linearly on l−1 . This relation is verified experimentally in
Fig. 3.3(b) showing measurements from 30 well-isolated CNT’s. The noiselevel is indicated on the figure and by extrapolation, the shortest observable
90
(a)
Vt
(b) 3
~ 250nm
Ctt
-1/2
1
-1/2
[deg.
Ctb
]
2
Φ0
1µm
0
2
4
6
[µm-1]
Figure 3.8: (a) Illustration of the terminology used for analyzing the length
dependence of the EFM signal. (b) The dependence of Φ0 on nanotube
−1/2
length l for 30 well isolated CNT’s (shown as Φ0
vs. l−1 , see text).
◦
The grey area illustrates the noise level (0.15 ) in the measurement and the
shortest detectable tube is estimated to be ∼ 250 nm consistent with the
fact that no EFM signal could be distinguished for a l ≈ 160 nm tube. The
inset shows a typical EFM measurement.
tube for the given experimental conditions is lmin ∼ 250 nm (see section 3.4).
In section 3.4 we study how defects in individual SWNT’s can effectively
separate long tubes into shorter sections thus causing deviations from Eq.
3.3.
In the above considerations, it was assumed that the SWNT’s could be safely
regarded as metallic objects. We now show (along the lines of Ref. [7]) why
this is a good approximation. If the EFM signal Φ of a SWNT is to achieve
its full value, the time, τscan , spend by the scanning cantilever in the vicinity
of the tube must be larger than the time it takes for the tube to be fully
polarized which is the characteristic RC time τRC of the tube. If R0 is the
one-dimensional resistivity of the tube τRC = R0 C0 l2 and thus
τscan > R0 C0 l2
(3.4)
for a fully observable CNT. As an order of magnitude estimate we treat
the CNT’s as infinite conducting wires of radius r ∼ 1 nm situated a distance t = 500 nm above a conducting plane. Then C0 ∼ 2π²0 / log(2t/²r r) ∼
1 × 10−17 F/µm where ²r = 3.9 is the dielectrical constant of SiO2 . Taking
vscan ∼ 25 µm/s as a typical scan velocity and 250 nm as a typical length
scale for the tip-tube interaction we find τscan = 1 × 10−2 s. Using these
values, and working with the one-dimensional conductivity G0 rather than
R0 , Fig. 3.9(b) shows the limits set by Eq. 3.4 as well as lmin of Fig. 3.9(a).
The shaded area corresponds to combinations of (G0 , l) which are fully ob91
10
Observable
CNT's
4
Metallic
l [µm]
6
Semiconducting
8
2
0
10-22
10-19
10-16
10-13
G0 [cm/Ω]
10-10
10-7
Figure 3.9: Illustration of the limit on G0 = 1/R0 for a CNT of length l set
by eq. 3.4 and lmin of panel Fig. 3.8. The shaded area corresponds to pairs
of (G0 , l) for fully observable CNT’s. Also indicated is typical conductivities
of (defect-free) metallic and semiconducting CNT’s.
servable with the EFM technique. The typical conductivities of metallic
(10−9 cm/Ω) and semiconducting (10−11 cm/Ω) SWNT’s[29] are also indicated and the figure clearly shows that both types are expected to be fully
observable in an EFM measurement.
As mentioned above one of the major advantages of EFM for research involving carbon nanotubes is the possibility of imaging and characterizing CNT
samples much faster than with conventional AFM, and in the following sections we investigate what information the EFM technique can provide. We
conclude this section by showing in Fig. 3.10 a ∼ 250µm × 330µm EFM image of a CVD grown CNT-sample. The image was automatically acquired
in about ∼ 12 hours and consists of 12 separate 90µ×90µm images subsequently stitched together. Individual CNT’s can clearly be resolved and
the image provide information about the density, length, orientation of the
CNT’s as well as the homogeneity of the distribution - information which
would be exceedingly time consuming to obtain with other techniques.
92
10 µm
Figure 3.10: Large EFM image (∼ 250µm×330µm) of a CVD-grown sample
of carbon nanotubes. The image was acquired in ∼ 12 hours and demonstrates the ability of EFM to investigate the nanotube distribution on a
large scale.
93
3.4
Identifying defects using EFM
Due to the unique one-dimensional nature of carbon nanotubes the performances of CNT-based electrical devices are dramatically affected by the
presence of structural defects. For instance it has been shown that geometric
defects such as kinks can make an otherwise metallic tube show semiconducting behavior[30] and defects induced by an electric discharge from an
atomic force microscope can completely destroy the electrical conduction
of a tube device[31]. Moreover, defects can make nanotubes increasingly
sensitive to their electrostatic environment which can be exploited for better performance of nanotube based chemical sensors[32] and it has been
shown that defects improve the field emission properties of multi-wall CNT’s
(MWNT)[33, 34, 35]. It is, therefore, important to study defects in CNT’s
and for many applications ways of assessing defect-related properties on a
single nanotube level prior to device fabrication would be beneficial. Most
characterization tools, however, either require electrical contact (such as
scanning tunneling microscopy and direct transport measurements), are difficult to interpret (micro Raman spectroscopy), or require freely suspended
CNT’s (high resolution transmission electron microscopy11 ) which is an inappropriate geometry for standard device applications which usually involves
CNT’s supported on insulating substrates as discussed in the previous section.
As was also demonstrated in the previous section, EFM provides a noninvasive technique for assessment of electrostatic properties of individual
carbon nanotubes in this geometry, and in this section we demonstrate that
EFM can provide useful information about their defects. This ability is
rooted in the dependence of the EFM signal on the length (Fig. 3.8) and the
conductivity (Fig. 3.9) of the nanotubes. A nanotube containing a defect,
say halfway along its length, which effectively separate the tube into two
conducting segments, will exhibit an EFM amplitude which does not correspond to the CNT’s full length, and from Fig. 3.8 it is possible to assess
the actual length of the conducting segments. If the defect is not situated
halfway along the tube (as is usually the case) its position will be directly
evident in the EFM image as a discrete jump will occur in the EFM phase
Φ upon passing the defect.
Below we initially discuss measurements where a mild oxygen plasma etch
is used for intentionally inducing defects in nanotubes and we show that
these are easily detectable by mapping individual nanotubes using EFM.
Subsequently, it is demonstrated that the general principle can be used to
assess the quality in as-grown samples where defects are not intentionally
induced.
11
Note, that TEM will most likely also damage the tubes in the process of imaging.
94
(a)
(b)
2µm
(c)
Intensity [A.U.]
(d)
Pristine
Etched
G
D
1600
1400
1200
Frequency [cm-1]
Figure 3.11: EFM images of CVD grown CNT’s (a) before and (b) after
a brief oxygen plasma etch. The double-square in the center is a metal
alignment mark used for navigation on the sample. Most of the CNT’s
loose their EFM signal by the ash (dashed lines). No change is found in
the topographic data (c). The dark spots seen in the topographic image are
minor (5-10 nm) pits in the oxide substrate and are unimportant for this
work. (d) Normalized Raman spectra using an excitation energy of 1.17 eV
of bulk CNT’s before (red) and after (black) plasma etching. The enhanced
D-band of the etched tubes indicates an increase in the defect density (see
text).
Plasma Induced Defects
The technique of oxygen plasma etching12 is widely used for removing organic material and it completely removes the CNT’s from the CVD samples
within ∼ 60 − 90 sec. For much shorter times, however, such an etch is expected to leave the bulk of the CNT’s intact and merely introduce occasional
defects.
Figure 3.11(d) shows Raman spectra measured at four different areas of a
CNT13 sample before (red traces) and after (black) a brief (∼ 1 sec.) oxy12
A commercial plasma system Plasma-Preen II-862 from Plasmatic Systems Inc. operating at a O2 pressure of 1-5 torr and a power of ∼ 600W was used for this study.
13
For Raman measurements samples were prepared by deposition of HiPCO CNT’s[36]
(rather than CVD-growth) in order to achieve higher nanotube densities and thereby
increase the intensity of the Raman signal. Substrates for the Raman measurement were
95
gen plasma etch14 . The spectra have been normalized with respect to their
G-band intensity and the (near) indistinguishability of the individual spectra in each group (emphasized in the inset) shows the homogeneity of the
distribution of tubes probed by the measurement. The double-peak structure of the G-band shows that the sample consists of single-wall CNT’s and
since the processes of the D-band are mediated by scattering events involving defects[14] the large increase in D-band intensity after the etch directly
confirms that this treatment increases the density of defects.
Turning to the effect on the EFM-signal, metal alignment marks were fabricated on a sample to allow easy navigation and Fig. 3.11(a) and 3.11(b)
show EFM images before and after the etch, respectively. The sample has a
relative high density of tubes as is clearly seen in 3.11(a) and the observed
branching indicates that they come in form of thin bundles. There is no
apparent effect of the etch observed in the topographic data (Fig. 3.11(c)),
however, in panel 3.11(b) it is clearly seen that it causes the EFM-signals of
most of the nanotubes to drop below the noise level. There is no change upon
reversing the tip polarity, and repeating the scan after one month yields the
same result. This rules out charging effects as the cause of the loss of EFM
signal[6] and leaves defect-induced changes in the CNT properties as a natural explanation. The signal remains only for the thickest bundles probably
because the outermost tubes of the bundle partly protect the inner ones
from the oxygen plasma. Notice that in some cases the signal disappears
abruptly, and no tubes with homogeneous but faint signals are observed.
This indicates that for the EFM signal, the most significant effect of the
etch is to introduce a number of discrete defects which effectively separate
the tubes into conducting sections shorter than lmin ∼ 250 nm rather than
inducing a homogeneous defect distribution which would increase the scattering and suppress the overall conductivity below Gmin . This is consistent
with the study of Ref. [31] where it was shown that defects not observable
in topographic data could, nevertheless, reduce the conductance of a CNT
to zero. These results demonstrate that EFM, in some situations, can reveal
defects in CNT’s which would pass unnoticed in standard topographic measurements. We note that our findings of the etch making the tubes highly
insulating, may add to the explanation of the improved field emission properties of plasma etched MWNT’s[33, 34, 35]: Etching MWNT’s will affect
the outer walls first thus leaving a thinner (and thereby more strongly emitting) conducting core to act as the emission source.
covered by an evaporated layer of Au to prevent an interfering signal form the Si and
amorphous SiO2 layer. HiPCO tubes were ultrasonically suspended in di-chloro-ethane
and a small amout left to dry on the substrate. The tubes precipitate as small grains on
the surface which can be seen in the Raman microscope.
14
The plasma is turned off immediately after it is lit making the effective etching time
as short at possible (. 1 sec.)
96
(a)
(b)
4
4
5
6
5
6
3
3
1
2
1
2
1µm
Φ0 [deg.]
(c)
4
4
2
2
5
3
1
0
1
x [µm]
4
Figure 3.12: Topography (a) and corresponding EFM image (b) of an asgrown CNT sample with defect-related effects indicated by arrows (see text).
(c) EFM signal Φ0 along the CNT indicated by arrows 1-5 in (a): The section
2-3 has no EFM response and at 4 a defect causes an abrupt change in Φ0 .
Defects in as-grown samples
In Fig. 3.12 we explore applications of EFM in situations where defects are
not intentionally induced. Figures 3.12(a) and (b) show topography and
EFM image, respectively, from an as-grown sample of CNT’s and by carefully correlating the two images several defect related features are observed.
Consider first the CNT indicated by arrows 1-5. The section 2-3 is significantly thinner than the rest and as seen in 3.12(b) it is completely absent
in the EFM signal showing that this section is probably highly defective or
poorly graphitized. Without the EFM data, however, one might wrongly
attribute the structure observed in the topography to a rope which along
2-3 just contains fewer tubes. The CNT section 3-5 shows no obvious defectrelated features in the topographic image. In the EFM image, however, an
abrupt intensity increase occurs at 4, as is emphasized in panel (c) showing
Φ0 (x) along 1-5. Thus, the EFM data show that at the point 4 the tube
contains a defect which separates the segment 3 − 4 from the rest and since
this segment is shorter (l3−4 ≈ 1.4 µm) than the remaining part of the tube
it appears with a weaker EFM signal. The combination of length and EFM
signal of the segment 3 − 4 agrees with the linear relation in Fig. 3.8(b)
showing that the defect at 4 is highly resistive (i.e. transport across the
defect at 4 occurs with conductivity ≤ Gmin ) and that the section 3 − 4
is by itself not separated into even shorter sections. For more examples of
defects in as-grown samples and the effects on the EFM signal see Ref. [12].
We note, that the techniques described here could be used for assessing the
97
frequency of defects in nanotube samples grown under various conditions information which would aid in optimizing the synthesis of SWNT’s.
98
3.5
Charge trapping in Carbon Nanotube Loops
demonstrated by EFM
The electronic interaction between carbon nanotubes and local charges plays
a vital role for applications of CNT’s in molecular electronics. Carbon nanotube field effect transistors (FET’s) exhibit hysteric behavior upon sweeping
the back gate due to rearrangements of charges or ions in water which locally gate the semiconducting nanotube[37, 38, 39]. This hysteresis can be
exploited in CNT-based memory elements[37, 38] but must be eliminated
for reproducible operation of the CNT FET’s in electronic circuitry. Also
applications of carbon nanotubes as the active element in nanoscale sensors for chemical compounds[40, 41] exploit the high sensitivity of carbon
nanotube FET’s to local charges in the environment[42]. In the previously
reported studies of these effects, the role of the charges has been elucidated
by monitoring the change in conductance of CNT FET’s upon changing the
nanotube environment. In this section we demonstrate the use of EFM to investigate the interaction of surface charges on SiO2 with carbon nanotubes.
We find that tubes strongly influence the dynamics of surface charges and
that charges can be trapped on regions of the substrate fenced by nanotubes.
Figure 3.13(a),(b),(c) show EFM-images of SWNT’s grown by CVD as described in section 3.2: For this particular growth, the CNT’s are extremely
long (up to ∼ 100 µm) and form loops and coils, as also reported in Ref.
[43] for long CVD grown CNT’s. An example of a nanotube coil formed at
the end of a CNT is shown in Fig. 3.13(b).
As seen in Figure 3.13(a),(b), a pronounced EFM signal Φl is observed in the
interior of some of the loops. This reflects that the tip experiences a force
gradient over the substrate enclosed by a nanotube loop different from when
it is outside a loop. This force cannot be attributed to a mere capacitive
coupling to the nanotubes since it is often observed that in two loops made
from the same tube and of similar geometries only one exhibits the contrast in the interior. Such cases are marked by the arrows in Figure 3.13(a).
Figure 3.14(a) shows the phase shift along a line through a nanotube coil
with radius r ∼ 2µm (Figure 3.14(b)). Sharp peaks ΦN T are observed when
the tip passes by a nanotube, and the enhanced phase difference Φl in the
interior of the coil is clearly observed. Figure 3.14(c) shows how Φl and ΦN T
depend on the potential difference Vs between tip and substrate15 . The relation ΦN T ∝ Vs2 derived in section 3.1 (equation 3.2) and expected for a
capacitive coupling to the carbon nanotubes is clearly observed, but for the
interior of the loop we find Φl ∝ Vs , indicating different physical origins for
the two forces. We ascribe Φl to static charges trapped in the interior of
15
Note that for the measurements in this section the voltage was applied to the substrate
in interleave mode while keeping the tip grounded rather than the reverse as indicated
in Fig. 3.6. This has no practical significance, however, in this section we work with Vs
rather than Vt .
99
(a)
50
z [nm]
100 150
(d)
200
Phase shift [deg.]
0
-5
-10
-15
-20
-25
0
20 40 60 80 100
2
2
Vs [V ]
(c)
(b)
Figure 3.13: (a) EFM image of long CVD-grown nanotubes forming loops.
Enhanced EFM phase-shift Φl is observed from the interior of some of the
loops. Arrows point to pairs of similar loops formed on the same CNT of
which only one exhibit enhanced Φl . (b) EFM-image of CNT loop and coil
from a different spot on the sample. (c) EFM image showing many loops.
(e) Phase difference over the bare substrate as a function of Vs2 for z = 60nm
(¤) and lift height z for Vs = −10V (•). The data has been fitted to theory,
see text. Scale bars for all images are 10µm, intermediate lift heights are
z = 60nm, and Vs = −6V .
the nanotube loops. If a charge Qs rests on the substrate, image charges
qt = AQs and qb = (1 − A)Qs will be induced in the tip and back gate,
respectively. The parameter A ∈ [0, 1] describes the division of the image
charges between tip and substrate. With a capacitance Ctb between tip and
backgate the total charge on the tip is Qt = qt + Ctb Vs and the force on the
tip can be approximated by[44, 45]
F (z) ≈
−Qs (AQs + Ctb Vs ) 1 0 2
− Ctb Vs .
4π²0 z 2
2
(3.5)
Here we have neglected the van der Waals forces which are negligible for
z > 10nm [46]. The force gradient entering equation 3.1 is then
·
¸
Qs Vs Ctb 1 ∂Ctb
AQ2s
1 00 2
0
F =
−
+
− Ctb
Vs .
(3.6)
2
3
2π²0 z
z
2 ∂z
2π²0 z
2
The last term is canceled by the offset of our measurement. Thus, for the
static charge contribution, F 0 depends linearly on Vs as we measure inside
the loop.
To estimate the charge density in the loop we must determine the capaci100
(a)
(b)
(e)
Phase shift [deg.]
0.2
Φl
0,0
-0.2
ΦNT
-0.4
-0.6
(f)
-0.8
0
2
4
6
8
10 12 14 16 18
x [µm]
0.0
(c)
(d)
Φl
0.0
Phase shift [deg.]
Phase shift [deg.]
0.5
-0.5
-1.0
ΦNT
-0.5
-1.5
-6
-4
-2
0
2
4
6
8
-1.0
10
Sample bias [V]
40
80
120
160
200
Lift height [nm]
Figure 3.14: (a) EFM phase shift with Vs = −5 V through a nanotube loop
along the line in (b) for Vs = −5 V. x = 0 µm corresponds to the top of the
line. The two edges of the loop is seen as sharp peaks with phase shift ΦN T
and the trapped charges give rise to shift Φl from the interior of the loop.
(c) The phase shifts Φl and ΦN T from (a) as a function of sample bias. The
measured points are averages from ∼ 15 scans along the line shown in (b). A
linear fit to Φl and a quadratic fit to ΦN T are shown as solid lines. (d) Phase
shifts Φl for the loop shown in (e) and (f) [AFM and EFM, respectively] as
a function of lift height for Vs = −5V (¥) and VS = −10V (¤). The solid
lines are fits to powers az −3 + b. Scale bars in (b) and (e) are 5 µm and (b),
(f) are measured with Vs = −5 V and z = 60 nm.
tances called for in equation 3.6. Figure 3.13(c) shows Φ vs. Vs measured
over the naked substrate. Again, the Vs2 dependence is found and from
the slope of the linear fit (2.8 × 10−3 rad./V2 ) and equation 3.2 we find
7.0 × 10−5 F/m2 for the second derivative. This, however, corresponds to
the entire cantilever-substrate capacitance, but at these heights only about
one third of this is due to the tip of the cantilever which is the part sensitive
to the charge[47]. Thus, approximating the tip as a circular disc of radius
R and the tip-substrate system as a parallel plate capacitor with z = 60 nm
air and t = 400 nm SiO2 with dielectric constant ²s = 3.9²0 , we estimate[8]
00
Ctb
=
2πR2 ²0
F
= 2.3 × 10−5 2 ,
3
(z + t/²s )
m
(3.7)
0 = − 1 C 00 (z + t/² ) = −1.9 × 10−12 F/m and
and from this we find Ctb
s
2 tb
0 (z + t/² ) = 3.1 × 10−19 F. In this approximation, the tip acts as
Ctb = −Ctb
s
101
a circular disc with effective radius r ≈ 40 nm in reasonable agreement with
the observation that CNT’s appear in EFM-measurements with widths of
about 200 nm which is close to 2R (Fig. 3.14(a)). In Fig. 3.13(c) we also
plot the z-dependence, and fitting the data to equation 3.7 (solid line) we
find again effectively R ≈ 40nm identical to the estimate from the voltage
dependence.
Using these capacitances in equation 3.6 and relating the measured phase
shifts to F 0 by equation 3.1 we use the slope of the line in Fig. 3.14(c) to
calculate Qs ≈ −7.7 × 10−19 C. The sign of the charge is inferred by noting
that conducting objects (e.g. CNT’s) which attract the tip appear with a
negative phase shift, and negative Φl is measured with Vs negative, i.e. with
a positive charge on the tip. Thus, the loop charge Qs must be negative.
This charge is situated within an area πR2 which gives a charge density of
2.2 × 10−8 C/cm2 , i.e., the coil on Fig. 3.14 contains a charge of about 17500
e− . With a similar analysis it is found that the r ∼ 1µm loops in Fig. 3.13(a)
contain between 1500 and 3500 e− each, corresponding to densities around
0.8 − 1.8 × 10−8 C/cm2 . Studies of charges induced on sapphire substrates
by corona discharge from an SPM-tip[48] yield densities of 2 × 10−8 C/cm2
close to our measured value.
The third term of equation 3.6 gives the Vs -independent phase shift due to
the charges on the surface. Using our calculated Qs , we find a value of 0.1×A
deg. which is consistent with the fact that we do not observe a phase shift
for Vs = 0 V if 0 < A < 0.1. Figure 3.14(d) shows the height dependence of
Φl for Vs = −5 V and Vs = −10 V for the loop shown in Figure 3.14(e),(f).
Assuming the Coulomb interaction dominates the h-dependence we have
fitted the data to the function ah−3 + b, i.e. the first term16 of equation 3.6.
The fit gives charge densities of 9.9 × 10−10 C/cm2 and 1.8 × 10−9 C/cm2 for
the Vs = −5 V and Vs = −10 V respectively. Using the Vs -dependence as
above we find a charge density in the loop of ∼ 2.4 × 10−8 C/cm2 , i.e., an
order of magnitude larger. We expect the estimate from the h-dependence
to be the less accurate since the relative scales of the tip-sample-substrate
system changes when altering h and this will influence the interaction of the
charges with the tip.
Further evidence that Φl is indeed due to charges on the substrate is provided
by AFM manipulation. Figure 3.15(c) shows an AFM image of two loops
formed on the same nanotube which initially each hold ∼ 6500 e− trapped
in their interior as seen on the EFM-image (panel a). By touching the
substrate in the interior of the lower loop with the grounded AFM-tip the
loop is discharged as observed on the subsequent EFM-image (panel (b)).
It was not possible to recharge the loop by applying voltages of ± 20V to
the tip while touching the substrate. The disappearance of the loop force
unambiguously shows that Φl is not due to a geometric effect. Furthermore,
16
The fit remains the same if the entire expression of equation 3.6 is used.
102
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 3.15: (a) EFM image of two loops formed on the same nanotube,
each trapping charges as seen by the phase difference measured in their
interior. (b) EFM-image of same area as (a) after touching the interior of
the lower loop by the grounded AFM-tip. (c) Topographic image of the
nanotube showed in (a) and (b). (d)-(f) AFM and EFM images of nanotube
coil before (d),(f) and after (e),(g) the coil is broken by AFM manipulation.
All measurements are performed with z = 60nm and Vs = −6V . Scale bars
are 2µm.
since the entire loop is discharged by just touching the middle of the loop
and the observation that the charges are uniformly distributed within the
loops suggests that the charges are relatively mobile on the surface and not
trapped deeply in the substrate. In panels (d)-(g) a similar series of images
is shown where the charges (∼ 3000 e− ) are removed by cutting a nanotube
coil using the AFM-tip in contact mode to ensure that the surface was indeed
touched when removing the charges.
Only a fraction of all loops have trapped charges as seen from Fig. 3.13(d)
which shows 22 loops on a different sample, none of which show trapped
charges, and Fig. 3.13(a) which shows loops both with and without charges.
We have not observed significant trapped charges in regions fenced by CNT’s
in other geometrical configurations than closed loops or coils. We speculate
that the charges stem from ions present during the CVD growth of the
nanotubes. The CNT may deplete neighboring regions of the substrate from
free charges, effectively making the region in the interior an isolated island
where charges can reside. Alternatively, the CNT loops could themselves be
103
negatively charged and thus trap charges through the Coulomb interaction.
In the latter case, however, it should be possible to observe by EFM the static
charges on the CNT which has not yet been possible. It it also possible that
the hydrophobic CNT encloses a thin layer of adsorbed water molecules on
a surface which can contain ions [39]. Tribological effects where charges
are induced on the substrate by the scanning cantilever can be ruled out
since scanning does not recharge loops which were once discharged as in
Fig. 3.15(a). In the lower loop of Fig. 3.15(a) charges did not reappear
after 12 hours. Further work where charge is deliberately injected during
the experiments could aid in resolving the issue.
In conclusion, we have used EFM to quantitatively address the interaction
between surface charges and carbon nanotubes. We find that surface charges
can be effectively trapped on the substrate in regions enclosed by CNT’s.
Static charges are responsible for the hysteretic behavior of CNT FET’s
which play a key role in CNT memories, sensors and logic circuits. This
suggests that the hysteretic behavior of CNT-loop FET’s might be different
from that of ordinary CNT FET’s. Furthermore, we have shown that EFM
provides a powerful tool for rapid characterization of large area CNT samples
and the present work shows how to identify contributions from static charges
in such data.
104
3.6
EFM for assessment of embedded CNT’s
Individual single wall carbon nanotubes posses extraordinary mechanical,
electrical and thermal properties and great effort has been put into transferring these properties to bulk systems by incorporating SWNT’s in polymer
matrixes[49, 50]. Due to difficulties in solubilization and chemical functionalization of SWNT’s, as well as strong tendency for bundle formation, it
has proven difficult to produce high quality suspensions of SWNT’s in polymers. Furthermore, it is difficult to evaluate, on a single nanotube level, the
structure of such polymer suspensions since the polymer matrix makes the
use of conventional characterization techniques questionable. Atomic force
microscopy (AFM) and scanning electron microscopy (SEM) are capable of
imaging only tubes at, or very close to the surface[51, 52, 53] and transmission electron microscopy (TEM) requires delicate sample preparation[49].
As is apparent from the first section of this chapter, EFM is capable of assessing nanotubes imbedded in an insulating matrix since the technique relies on
the electrostatic interaction. In this section we demonstrate the use of EFM
as a simple, non-invasive tool for mapping the three dimensional orientation
and position of individual SWNT’s in thin films of SWNT/polymer composites. The technique is used for studying spin-cast films of the composite
formed by ultrasonically suspending SWNT’s in poly-(methyl methacrylate)
(PMMA)[54, 53]. It is demonstrated that EFM technique reliably provides
statistics on the density as well as the length and orientation distribution
of the suspended nanotubes. Towards the end we discuss the limitations of
the technique and it is shown that for long tubes & 3 µm composite samples
with thicknesses up to ∼ 1 µm can be imaged.
Basic characterization of composites
The samples studied here differ from the samples of the previous sections
and were prepared in the following way. A small amount (< 1 mg) of commercially available SWNT’s17 grown by the electrical arc technique was suspended in a 2% solution of poly-(methyl methacrylate) (PMMA) in anisole
by low power ultrasonication for 30 minutes. The resulting homogeneous
black suspension was immediately spin-casted on wafers of highly doped
silicon capped with 500 nm of insulating SiO2 by spinning at 3000 rpm for
45 sec. and subsequent evaporation of the anisole on a hotplate (185 ◦ C) for
90 sec., This results in a 50 − 60 nm film of the SWNT/PMMA composite on
the substrate as schematically shown in Fig. 3.16(a). Figure 3.16(b) shows
an AFM topography image of the resulting surface. 10 − 20 nm particles are
observed probably due to the amorphous carbon ash present in the nanotube material, however, no nanotubes can be identified on the surface. Due
17
As-grown material from Carbon Solutions
105
15
180
#counts
(d)
20
#counts
0
15
0
(c)
(a)
Vt
90 θ [deg.]
h
PMMA
10
Si/SiO2
0
1
2 3
L [µm]
4
0
10
-4
z [nm]
(b)
Φ [deg.]
0
0
5 µm
5 µm
Figure 3.16: (a) Schematic illustration of the spin-cast polymer/SWNT sample and EFM operation of a SPM (see text). (b) AFM image of the surface
topography of a ∼ 60 nm thick film of PMMA/SWNT composite. Due to
the polymer the tubes cannot be observed, however, in the corresponding
EFM image (h = 35 nm, V = 7 V) in panel (c) individual embedded SWNT’s
are clearly seen as dark lines. (d) The distribution of the lengths and orientation (inset) of 95 embedded SWNT’s measured in a 90 µm × 90 µm area of
the sample (see text). The angle is measured with respect to the horizontal
direction in (c).
to the much higher conductances of SWNT’s as compared to the PMMA
matrix the tip-sample capacitance is significantly altered in the presence of
a SWNT sandwiched between the tip and the substrate. The embedded
SWNT’s are therefore clearly revealed in Fig. 3.16(c) showing the EFM image corresponding to the area in (b). Since the individual nanotubes can be
identified, important parameters characterizing the nanotube distribution
can be reliably obtained. In Fig. 3.16(d) we show the distribution of lengths
and in-plane orientations for 95 individual tubes measured in a 90 × 90 µm
area of the sample. The length distribution peaks around ∼ 750 nm and the
orientations are almost random, however, with a slight tendency for aligning
along the ∼ 50 ◦ directions which is most likely an effect of the spin-casting.
We note that as the nanotubes are much longer than the thickness of the
composite film the projected lengths measured in the EFM image are very
close to the actual lengths (see below). One of the obstacles for preparing
nanotube suspensions is the strong tendency of the tubes to aggregate and
to form bundles. It is not possible in the EFM image to distinguish a single
SWNT from a thin bundle but it is possible to evaluate to which degree the
tubes are freely suspended or form larger aggregates. In the image in Fig.
3.16(c) most tubes are isolated but occasionally two or more are touching
106
suggesting that additional sonication may yield a better suspension18 . Detailed investigations and optimizations of the parameters of the composite
preparation scheme are, however, not the focus of this study.
3D mapping of SWNT’s in composites
For fixed Vt the amplitude of the EFM signal from two identical SWNT
buried at different depths in the polymer film depends on the backplanetube-tip distances. Due to the large SiO2 thickness (500 nm) the relative difference in backplane-nanotube distances is small and thus the change in the
EFM signal is effectively caused by the relatively larger change in tip-tube
distance. In the following we neglect the effects of differences in backplanenanotube distances and use the EFM image to map the three-dimensional
position of the SWNT’s in the polymer matrix. For this demonstration
a sample consisting of a composite-PMMA-composite (60/50/60 nm) trilayer was fabricated by repetition of the spin-casting procedure described
above. Figure 3.17(a) shows an EFM image from this sample measured
with Vt = 7 V and a tip/sample separation h = 35 nm. Two SWNT’s, T1
and T2 with lengths LT 1 ≈ 1.5 µm and LT 2 ≈ 1.6 µm are clearly seen. T2
clearly has a lower EFM amplitude ΦT0 2 than T1 . In addition to the distance
of the nanotubes from the sample surface the EFM signal depends on their
lengths as discussed in relation to Fig. 3.8, and since we are interested in
the position of the nanotube in the polymer matrix we first compensate for
the length dependence as described below.
The inset to Fig. 3.17(b) shows Φ0 vs. L for 30 isolated SWNT’s grown on a
SiO2 substrate without polymer19 (i.e., h is constant). The solid line shows
−1/2
a fit to the theoretical prediction Φ0
∝ L−1 derived in section 3.1 and
the figure shows that the length differences between T1 and T2 can (as a
first approximation) be compensated for by a ∼ 10% decrease in ΦT0 2 . Figure 3.17(b) shows the resulting values of Φ0 for T1 and T2 measured as a
function of lift height at the positions indicated in Fig. 3.17(a). For each
lift height ∼ 10 scans were measured and the average nanotube signal fitted
to a Lorentzian line shape Φ(x) = Φ0 /(1 + 4((x − xc )/W )2 ) with amplitude Φ0 and full-width-at-half-max (FWHM) W . The results for T2 are
obviously horizontally off-set by an amount δh, i.e., for T2 the tip-surface
distance does not correspond to the tip-tube distance. To estimate δh the
data is fit to the relation Φ0 = A/(h + B)3 . The solid lines show fits using
A = 0.124 deg./µm for both, and we get δh = BT 2 − BT 1 = 32 nm as the
18
In other samples having received less ultrasonication most tubes are found as large
aggregates containing hundreds of tubes.
19
The measurement was performed with h = 60 nm and Vt = 5 V and to compensate
for the different Vt the signal has been normalized to coincide with ΦT0 1 measured for
h = 60 nm
107
3
T2
Φ0 [deg.]
1 µm
T1
Φ0 [deg.]
(a)
T1
2
4
2
0
0
T2
1
2
L [µm]
1
y
(b)
x
0
-2
-4
δh
0
Φ [deg.]
50
150
h [nm]
250
(c)
T1
z [nm]
0
6
5
4
3
2
-40
-80
y [µm] 1
-120
1
2
3
4
X [µm]
5
6
Figure 3.17: (a) EFM image showing two SWNT’s embedded in a ∼ 170 nm
thick film of SWNT/PMMA composite. (b) The lift-height dependence of
the length-corrected EFM signal of the two tubes in (a). The measurements
were performed at the points indicated in (a) - see text. The measured
amplitudes have been fit to Φ0 = A/(h + h0 )3 . The inset shows Φ0 vs.
nanotube length L for 30 isolated SWNT’s measured with known tip-tube
separation h = 60 nm. The red line shows a fit to the theoretical prediction
−1/2
Φ0
∝ l−1 . (c) A projection view of the three dimensional map of the
two nanotubes as inferred from the data in (a) and (b). The blue region
illustrates the PMMA matrix (see text).
108
effective distance between T2 and T1 . Since T2 is embedded in PMMA with
the dielectric constant ²P = 3.2 this corresponds to T2 being situated a distance δz = ²P δh ≈ 100 nm below T1 .
The xy-coordinates along each nanotube is found by tracing the minima in
Fig. 3.17(a) and for each minimum the relative z-coordinate is found from
the particular Φ0 and the fits in Fig. 3.17(b) (and adding δz for T2 ). A
projection view of the resulting map is illustrated in Fig. 3.17(c). Since T1
emerges from a catalyst particle which can be seen in the topographic data
as in Fig. 3.16(b) we expect T1 to be situated close to the surface of the
sample and the blue region in Fig. 3.17(c) illustrates the PMMA matrix.
The z-extension of the tubes agrees with the fabrication of the sample from
which we expect nanotubes in the lower and upper ∼ 60 nm of the composite but a nanotube-free middle section of ∼ 50 nm. In Refs. [28, 27] the
relation Φ0 ∝ h−1 was found to describe the height dependence of Φ0 . For
the data presented here as well as data obtained without PMMA (Fig. 3.7),
i.e., where h is known, we find Φ0 = A(h + B)−3 to provide the best fit.
However, using ΦT0 1 = A/h to estimate A and using this value in fitting
ΦT0 2 = A/(h + δh) yields δh = 32 nm in agreement with the above analysis.
We note that once the length and h dependencies are known for a given
sample and setup, a single image provides sufficient information for a complete three-dimensional mapping of all the SWNT’s, i.e., the above analysis
does not need to be repeated for each SWNT in finding its z-position.
Limitations of the technique
We now discuss the limitations of the technique for mapping SWNT’s in
polymer composites. Firstly, the decreasing EFM amplitude for larger tiptube distances limits the thickness of the composite films which can be
studied with the technique. In Fig. 3.17(b) the SWNT signal can be clearly
observed up to h = 225 nm, however, we have routinely imaged SWNT’s
with h up to 300 nm and taking the dielectric constant of the polymer into
account we estimate the limiting thickness to be ∼ 1 µm. We have made no
attempts to reduce the noise in the commercial AFM setup but we expect
that such efforts will increase the limiting thickness further. Furthermore,
since the amplitude also decreases for shorter SWNT’s the length Lmin of
the shortest observable SWNT increases with the tip-tube distance. Using
the parameters of the fits to the L and h dependencies in Fig. 3.17(b) we
show by the shaded area in Fig. 3.18(a) the combinations of L and h which
are observable by the EFM technique with the boundary given by the noise
level in the measurement. Thus, close to the surface the technique detects
all SWNT’s longer than ∼ 250 nm, however, from the most distant areas of
thicker samples only tubes with L & 3µm will be observed. Secondly, due
to the long range of the electrostatic interaction, the spacial resolution of
109
300
0.6
(a)
(b)
W [µm]
h [nm]
250
200
150
0.4
100
50
0.5
1.0
1.5
L [µm]
2.0
2.5
0.2
0
100
h [nm]
200
Figure 3.18: (a) Graph showing the combinations of CNT length L and
effective tip-tube separation h which are observable with the EFM technique
(shaded area). The graph is calculated using the parameters of the fits
in Fig. 3.17(b) and hatched region illustrates the uncertainty (due to the
uncertainty in the values of the fits). (b) The FWHM of the Lorentzian fit
to the EFM signal of T1 as a function of the lift height. The solid line is a
linear fit.
the technique is expected to decrease for SWNT’s further away from the
sample surface. This is confirmed in Fig. 3.18(b) showing a clear linear increase of the FWHM of the EFM signal from T1 as a function of tip-tube
distance[28, 27]. We note that this dependence can be exploited for reaching
a better estimate of the z-position of the SWNT’s20 . Thirdly, the technique
as presented here is only suited for samples where the length as measured
with EFM is close to the actual tube length, i.e., the SWNT’s must be approximately aligned with the substrate plane. This is, however, naturally
achieved by spin casting the composite and is therefore not a major constraint. In addition, we suspect that by including also the width dependence
shown in Fig. 3.18(b) and supporting the results with model calculations[55]
including the effect of larger tube-substrate angles, this constraint may be
further relaxed. Finally, at high filling factors of SWNT’s in the polymer
matrix the EFM 3D-mapping method as described here becomes unsuitable
since it requires the nanotubes to have well-defined lengths, i.e., each nanotube needs to be electrically isolated. The technique may, however, still
provide a valuable tool for imaging the suspended tubes closest to surface.
To summarize, we have in this section showed the use of electrostatic force
microscopy as a non-invasive technique for the three dimensional mapping of
individual SWNT’s in polymer/SWNT composites. The technique is shown
to provide important information about the homogeneity of the incorporated
SWNT’s and since individual nanotubes can be identified the distribution
of lengths and orientation can be measured (information otherwise inac20
The measurement of ΦT0 2 (h) in Fig. 2(b) was not performed with a scan direction
exactly perpendicular to the tube axis and thus the width cannot be directly compared
with that of T1 in Fig. 3.18(a). Extracting the perpendicular FWHM of T2 from Fig.
3.17(a) gives a width of ∼ 330 nm and with h = 35 nm this corresponds to a horizontal
offset in Fig. 3.18(a) of ∼ 45 nm in good agreement with δh deduced from Φ0 (h).
110
cessible). Such 3D mapping has not previously been shown for nanotube
composites and we expect that the technique could become a valuable tool
for the future studies and developments of these materials.
111
3.7
Conclusion of EFM studies
In the preceding sections we have seen the use of electrostatic force microscopy for different studies of single wall carbon nanotubes. The work
with the EFM technique originated from an attempt to find the fastest and
best way of characterizing our CVD-grown nanotube samples. It was quickly
realized that EFM was the easiest way of measuring basic parameters such
as densities, lengths, orientation, homogeneity etc. and it was used to build
a catalog of the various samples grown. The different studies presented
in this section appeared when trying to understand various features in the
measured images. It was found that the technique also convey information
about the quality of the tubes and their interplay with static charges. Moreover, it was shown that EFM, as the only known technique, can be used for
the three-dimensional mapping of individual nanotubes in SWNT/polymer
composites. I believe that the technique will find many uses as a powerful tool in the nanotube research and will become recognized as a standard
characterization tool of nano science.
Considering the differences between the subject of this chapter and that
of the first chapter on electron transport in InAs nanowires (at least) two
questions would be reasonable to raise; how does EFM apply as a characterization tool for the studies of nanowires and what are the differences
in the transport characteristics of the two systems? Concerning EFM as a
tool for nanowire characterizations it has not found many uses - EFM may
be a superior technique for characterizing nanotubes (as advocated for in
this chapter). However, since nanowires are considerably larger (∼ 50 times
larger diameter) they can be imaged using standard optical microscopy or
by SEM21 , thereby rendering scanning probe techniques obsolete. Also, for
EFM studies of nanowires, the large diameter of the wires causes the tipbackgate distances to be significantly altered when scanning a wire and this
effect makes interpretation of the EFM images more involved than for the
nanotubes, where the tip-backgate distance can effectively be considered
constant. We note, however, that the EFM technique (or its close analog Kelvin probe microscopy) may provide important information about the
potential landscape if imaging contacted nanowire devices[56] and such experiments are planned for the near future.
Regarding electric transport in carbon nanotubes, this has been the subject of numerous studies. Carbon nanotubes constitute a truly fascinating
system and many remarkable results have been found. Their properties resemble in many ways those of the nanowires as presented in previous chapter; well-defined quantum dots can nowadays be routinely made in carbon
nanotubes exhibiting the physical phenomena of Coulomb blockade, Kondo
21
Also, due to the larger scale of the nanowires SEM imaging is not expected to significantly damage the wires as is possible the case for nanotubes (section 3.4).
112
SWNT loop
contacts
1 µm
Figure 3.19: AFM image of a looping nanotube with lithographically defined
metal contacts for future investigations of electron transport.
physics and open quantum dots (Fabry Perot interference). In this respect,
the subject of transport in nanowires is still in its infancy. The original aim of
this project was to study transport in semiconducting nanowires and transport in nanotubes was not pursued systematically. However, it would be
interesting to investigate electron transport in the nanotube loop-structures
of section 3.5 for the possible interference of the electrons going around the
loop and the influence of a magnetic flux through the loop. Devices were
fabricated for these investigations as shown in Fig. 3.19, however, due to
limited time (and technical issues) their transport properties have not been
measured yet but is scheduled for the near future.
113
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Appendix A
InAs nanowire growth
NBI73
The InAs wires which have been studied in this thesis was grown by MBE
on a (100) GaAs substrate. A thin Au film was deposited with a substrate temperature of 760 ◦ C(Au source at 1300 ◦ C) , followed by 15 min of
GaAs growth (substrate temp. 580 ◦ C, As source 200 ◦ C, Ga source 910 ◦ C)
and 30 min of InAs growth (substrate temp. 470 ◦ C, As source 200 ◦ C, In
source 887 ◦ C). The growth was developed and performed by Martin Aagesen (manuscript in preparation). Figure A.1 shows additional images of the
growth product (see section 2.2).
119
Figure A.1: SEM images of the nanowire growth NBI73 used for the devices
in chapter 2. For some of the wires gold nano particles can be observed in
the ends.
120
Appendix B
Details of nanowire device
fabrication.
This appendix provides the details for the fabrication of nanowire devices
as well as a discussion of the precautions needed to allow a reasonable fraction of (extremely sensitive) devices to reach the stage of low-temperature
measurements. The fabrication follows along these lines:
• The starting point is a substrate wafer of highly doped silicon (which
will stay conducting to dilution fridge temperatures) capped with
500 nm of insulating SiO2 . The final device will be supported on this
substrate and the conducting silicon will act as a global electrostatic
(back) gate. The substrate is cut into manageable sizes of ∼7×10mm
and prepared with a grid of metal alignment marks fabricated by standard e-beam lithography (for parameters, see below) and subsequent
metal evaporation (Cr/Au, 10/40 nm). The grid is needed for finding and contacting nanowires as described below, and a grid-size of
∼10 − 20 µm was found suitable. Furthermore, large metal bonding
pads were fabricated around each area of alignment marks, using standard UV-lithography and (Cr/Au, 20/180 nm) metal evaporation.
• To transfer the nanowires from the high density growth-wafer to the
device substrate a very simple mechanical approach was employed. A
small (4×4 mm) piece of the growth substrate was placed face-down on
the device substrate and moved gently back and forth a few times while
applying as little downwards pressure as possible. In this way some
wires break off the growth substrate and stick to the sample. The
result can easily be checked in a high resolution optical microscope
and the process can be repeated until the desired density of wires is
achieved. Figure 2.7(a) shows a typical optical microscopy image of
InAs nanowires deposited in an alignment grid.
• Using either the optical images or SEM images a contact pattern is
121
now designed using the alignmentmarks as reference. SEM images
have the advantage of higher resolution and are therefore required for
advanced device geometries, however, it is faster to obtain the optical
images and furthermore, they can be obtained after spinning on the
e-beam resist. This is important since the wires, in some cases, tend to
move when spin coating the sample with resist, leading to subsequent
misalignment. This is avoided by obtaining the optical images for
alignment after spinning on the resist and since we were only interested
in making simple two-terminal devices the limited resolution was not
a concern and the optical method was adopted in most cases.
• The contacts are defined by standard e-beam lithography: The chip is
coated with resist layers of 6% copolymer and 4% PMMA by spinning
at 4000 rpm for 45 s and baked on a hotplate at 185 ◦ C for 90 s. The
lithography is performed using a JEOL JSM-6320F scanning electron
microscopy running Elphy lithography software using an exposure voltage of 30 keV and a current of ∼20 pA. The patterns were developed
for 60 s in MIBK:IPA (1:3) and flushed in IPA.
• Prior to metal evaporation the samples were ashed for 5 − 10 s in an
oxygen plasma etch to remove resist residues from the wire surface.
This was followed by a brief ∼5 s wet etch in buffered hydrofluoric
acid to remove the oxide from the nanowire surface (cf. TEM image,
Fig. 2.6). The sample was then flushed with deionized water and
blow-dried with N2 and immediately loaded in the load-lock of the
metal evaporation chamber. In order to prevent (as far as possible)
re-oxidation of the wire, efforts were made to bring down the time
between the end of the HF etch and loading in the evaporation chamber. Usually the load-lock was evacuated within 2 minutes of the HF
etch. Based on the relatively few devices (< 20) which were electrically
characterized it seems that omitting the plasma and HF etch result in
devices with large contact barriers (devices in the Coulomb blockade
regime). In contrast, by including the cleaning procedure, better contact is achieved and the device transparency can be tuned by the gate
(section 2.1). For the contacts either a Ti/Au (∼10/50 nm) bilayer
was used (for normal-metal contacts) or a Ti/Al/Ti (∼10/60/10 nm)
trilayer for superconducting contacts (see section 2.7). Lift-off was
performed in acetone and Fig. 2.7(b) shows an SEM micrograph of a
typical finished device.
• To achieve good contact to the substrate (back gate) the back of the
sample was metallized (Cr/Au, 10/40 nm) and the sample glued to
a chip-carrier using a conducting silver-paste. The remaining and final step is to wire-bond the sample to the chip-carrier. An ultrasonic
ball bonder (Kliche & Soffa 4124) was used for this purpose, how122
ever, special care was needed since the nanowire devices were found to
be extremely sensitive to electrical/static shocks (much more so than
nanotube based devices). Hundreds of devices were destroyed in the
bonding process and the subsequent handling, but taking the following precautions enabled a reasonable number of devices to reach the
measurement step: *) Always ensuring extensive grounding of bonding
machine (and user). *) Using an ionizing fan to limit static buildup.
*) Before bonding from the chip-carrier to the sample, all pads on the
chip-carrier were connected by bonding wire-loops large enough to be
subsequently removed without the use of microscope (after mounting
the sample in the cryostat). *) Using the bonder in ”manual” mode,
i.e., preventing automatic electrical cutting of the bonding wire. *) After bonding the two leads of a device, the SEM was used to assess if the
wire was still intact. In this case the sample was mounted in the cryostat - bonding additional devices was sometimes/often found to destroy
all previously bonded devices. *) Finally, the sample was mounted in
the grounded cryostat and the bonding-loops removed with tweezers
(user grounded). Note the absence of a room-temperature probingstep; it has been found to be nearly impossible to use a probe-station
to characterize the devices before bonding: in such attempts nearly all
devices were lost.
123
Appendix C
Type-1 devices in the
closed-dot regime
5.0
Vsd [mV]
2.5
0.0
-2.5
-5.0
-7.5
-7.0
Vg [V]
-6.5
Figure C.1: Stability diagram measured in the closed-dot regime of a type-1
device measured at T = 300 mK (darker = more conductive). The arrows
indicate the level spacings which are generally larger than for the type-2
devices treated in section 2.4. The measurement has been corrected for a
number of gate-switches. See discussion in the main text.
124
Appendix D
Additional Kondo data
Figure D.1 provides additional data measured on the same device as discussed in connection to Fig. 2.31 (section 2.7). The data illustrates the
possibility of a large TK Kondo peak (TK & ∆) surviving the transition to
superconducting leads (see figure caption).
Figure D.2 shows a stability diagram measured at T = 50 mK for a device with a single-wall carbon nanotube contacted by superconducting leads
in much the same way as the nanowire devices presented in this thesis.
The device fabrication and the measurement were carried out at the Niels
Bohr Institute by K. Grove-Rasmussen and H.I. Jørgensen - for details see
K. Grove-Rasmussen et. al. ”Interplay between supercurrent and Kondo effect in single wall carbon nanotube Josephson junctions” cond-mat/0601371.
The pronounced ∆-peaks are observed along a suppressed Kondo ridge which
shows that the observation discussed in section 2.7 is not related to the nanowire system, but is rather a general phenomenon of Kondo-dots connected
to superconducting leads.
125
3
3
800 mK
Vsd [mV]
Vsd [mV]
κ
0 b
300 mK
0
(a)
(c)
-3
-3
Vg
Vg
9.0
5.0
G [e2/h]
G [e2/h]
κ
b
0.0 (b)
-0.6
7.0
5.0
(d)
0.0
Vsd [mV]
0.6
0.5
1.0
T [K]
2.0
2∆
∆
0.2
0.2
Vsd [mV]
Vsd [mV]
Figure D.1: (a) Stability diagram for T = 800 mK, i.e., above the superconducting transition temperature. A clear Kondo ridge is seen as also
emphasized in (b) showing line traces through the center of the Kondo ridge
and the adjacent CB valley (cf. 2.31(b)). (c) Corresponding stability diagram for T = 300 mK. The 2∆ peaks are clearly seen and the Kondo ridge
remains. (d) Shows the temperature dependence of the valley conductance
exhibiting an increase through Tc rather than a decrease as for Kondo valleys with lower TK . The reason for the very high conductance (even in the
normal state) is not understood.
0.0
0.0
-∆
-2∆
-0.2
-0.2
dI/dV
[e2/h]
-5.0
-4.8
Vg [V]
-4.9
-5.0
-4.8
Vg [V]
-4.9
1.4
0.0
Figure D.2: Measurement from a CNT-based quantum dots contacted by
superconducting leads (see text). For normal leads (left) a Kondo resonance
is observed which is suppressed for superconducting leads (right). Along
the suppressed ridge the enhanced ∆-peaks are clearly observed. Colorscale
identical for the two plots.
126
Appendix E
EFM parameters
The various parameters used when scanning in EFM mode can readily be
found in the screen-shot from the SPM controller software shown in Fig.
E.1.
127
Figure E.1: Screen-shot of a typical EFM run showing the typical parameters
and settings used. For greater magnification, view the electronic version of
the thesis.
128
Appendix F
Publication list
During the course of the project the following papers have been published
(or are in preparation):
• Williams J.R., Sand-Jespersen T., Marcus C.M., et. al., ”Weak localization
and conductance fluctuations in mesoscopic graphene”, In preparation (2007)
• Sand-Jespersen T., Grove-Rasmussen, K., Ingerslev, H., Aagesen, M., Sørensen,
C., Lindelof, P.E., Paaske, J., Andersen, B.M., Flensberg, K., Nygård J. ”Effect of Kondo Correlations on the Subharmonic Gap Structure of a Quantum
Dot Coupled to Superconductors”, In preparation (2007) - section 2.7.
• Sand-Jespersen T., Nygård J. ”3D-Mapping of Individual Carbon Nanotubes
in Polymer/Nanotube Composites using Electrostatic Force Microscopy”,
Submittet to Appl. Phys. Lett. (2007) - section 3.6.
• Sand-Jespersen T., Nygård J. ”Exposure of Carbon Nanotubes in Dielectric
Layers using Electrostatic Force Microscopy”, Submittet to Appl. Phys.
Lett. (2006) - section 3.1 partly.
• Sand-Jespersen T., Nygård J. ”Probing Induced Defects in Individual Single
Wall Carbon Nanotubes using Electrostatic Force Microscopy”, Accepted,
Appl. Phys. A. (2007) - section 3.4.
• Sand-Jespersen T., Aagesen, M., Sørensen, C., Lindelof, P.E., Nygård J.
”Kondo effect in tunable semiconductor nanowire quantum dots”, Phys. Rev.
B. 74, 233304 (2006) - section 2.5.
• Sand-Jespersen T., Nygård J. ”Charge Trapping in Carbon Nanotube Loops
Demonstrated by Electrostatic Force Microscopy”, Nano Lett. 9, 1838 (2005)
Cover Story - section 3.5.
• Sand-Jespersen T., Lindelof P.E., Nygård J. ”Characterization of Carbon
Nanotubes on Insulating Substrates using Electrostatic Force Microscopy”,
AIP Conference Proceedings,786, 135 (2005) - section 3.1 partly.
129