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‫אוניברסיטת בן‪-‬גוריון בנגב‬
‫הצעת תוכנית מחקר ללימודי דוקטורט‬
‫אפקטים של אינטראקצית ספין‪-‬מסילה במערכות מזוסקופיות‬
‫‪Effects of spin-orbit interaction in mesoscopic systems‬‬
‫שלומי מתתיהו‬
‫אוקטובר ‪3102‬‬
‫חתימת מנחה‪:‬‬
‫חתימת מנחה‪:‬‬
‫חתימת מנחה‪:‬‬
‫חתימת יו"ר ועדת מוסמכים מחלקתי‪:‬‬
‫תקציר‬
‫הפיזיקה של אינטראקצית ספין‪-‬מסילה בננו‪-‬מבנים בממדים נמוכים מעוררת עניין רב כיון שהיא‬
‫מאפשרת לשלוט ולהשפיע על ספין האלקטרון בתחום הספינטרוניקה‪ .‬אינטראקצית הספין‪-‬‬
‫מסילה מצמדת את ספין האלקטרון עם דרגות החופש המסלוליות ובמידה רבה ניתנת לבחירה‬
‫ושליטה במערכות מזוסקופיות וננוסקופיות‪ ,‬למשל על ידי מתח שער חיצוני באמצעות‬
‫אינטראקצית הספין‪-‬מסילה של רשבא‪ .‬לכן חקר התוצאים המגוונים של אינטראקצית הספין‪-‬‬
‫מסילה במערכות מסוג זה הוא בעל חשיבות רבה‪ ,‬הן מבחינה טכנולוגית והן מבחינת מחקר‬
‫בסיסי‪ .‬במחקר המוצע נתמקד באופן תיאורטי במספר היבטים של אינטראקצית ספין‪-‬מסילה‬
‫במערכות מזוסקופיות וננוסקופיות‪ .‬השלב הראשון הוא חקר של סינון ספין כתוצאה מהתאבכות‬
‫באינטרפרומטרים מזוסקופיים וברשתות קוונטיות‪ .‬יצירה יעילה של זרם מקוטב‪-‬ספין באופן‬
‫מושלם היא נדבך חשוב בהתקנים ספינטרוניים ובתורת האינפורמציה הקוונטית‪ .‬התוצאות‬
‫הראשוניות שהתקבלו בנושא זה מוצגות בהצעה הנוכחית‪ .‬בשלב הבא נחקור מספר מודלים‬
‫במטרה לשפוך אור על תופעת ההולכת תלוית הספין של אלקטרונים דרך מולקולות כיראליות‬
‫אורגניות‪ .‬החלק האחרון של המחקר יוקדש להשפעות ההדדיות של אינטראקצית הספין‪-‬מסילה‬
‫ותרמו‪-‬אלקטריות בננו‪-‬מבנים‪ .‬תיבחן האפשרות להגדיל את יעילות ההמרה של חום לחשמל‬
‫(אפקט סיבק) ולהיפך (אפקט פלטייר) בננו‪-‬מבנים עם אינטראקצית ספין‪-‬מסילה חזקה‪.‬‬
Abstract
The effects of the spin-orbit interaction (SOI) on the physics of low-dimensional mesoand nanostructures has drawn much attention due to the ability to control and manipulate the electron’s spin in the field of spintronics. The SOI couples the electron’s spin
with its orbital degrees of freedom and to a large extent can be tailored and tuned in
meso- and nanoscopic systems, e.g. by an external gate voltage through the Rashba SOI.
Studying the various effects of SOI in such systems is therefore of great importance, both
from technological and fundamental points of view. In this proposed research we will
address theoretically several aspects of the SOI in meso- and nanoscopic systems. The
first step is the investigation of interference-induced spin filtering in mesoscopic interferometers and quantum networks. The effective generation of a perfect spin-polarized
current is a major ingredient in spintronic devices and in quantum information theory.
Preliminary results obtained on this subject are presented. Next we will study several
models which may shed light on the phenomenon of spin-dependent electron transport
through organic chiral molecules, observed in recent experiments. The last part of the
research will be devoted to the interplay between SOI and thermoelectricity in nanostructures. The possibility to enhance the efficiency of conversion of heat into electricity
(Seebeck effect) and vice versa (Peltier effect) in nanostructures with strong SOI will be
examined.
i
Contents
1 Introduction
1
2 Research Objectives
4
2.1
Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1
Spin filtering in mesoscopic systems . . . . . . . . . . . . . . . . .
5
2.1.2
Spin-dependent transport through chiral molecules . . . . . . . .
8
2.1.3
The interplay between SOI and thermoelectricity in nanostructures 11
2.2
Significance of the Research . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.3
Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3 Preliminary Results
3.1
14
Spin filtering in a Rashba-Dresselhaus-Aharonov-Bohm double-dot interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Robustness of spin filtering against current leakage in a Rashba-DresselhausAharonov-Bohm interferometer . . . . . . . . . . . . . . . . . . . . . . .
3.3
14
17
Two-legged ladder quantum network with Rashba spin-orbit interaction
and Aharonov-Bohm flux . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 References
18
21
ii
1 Introduction
Spintronics (spin electronics) is a subfield of condensed matter physics devoted to the
active manipulation of the electron’s spin. 1–6 Adding the spin degree of freedom to the
conventional charge-based technology has the potential advantages of multifunctionality,
longer decoherence times and lengths, increased data processing speed, decreased electric
power consumption, and increased integration densities compared with conventional
semiconductor devices. 1,2 Moreover, the control and manipulation of single-spin systems
may serve as a basis for quantum information processing and quantum computation in
the solid state. 7–10 Fundamental studies of spintronics thus include investigations of spin
transport in electronic materials, as well as of spin dynamics and spin relaxation.
Several paradigms of spintronics can be distinguished. Perhaps the most famous
of these paradigms is spintronics using ferromagnetic metals. A prominent example
of this paradigm is the giant magnetoresistance (GMR) observed in structures composed of alternating ferromagnetic and non-magnetic metallic thin films. 11–13 In the last
two decades, this effect has yielded fascinating new devices, e.g. magnetic positioning
sensors, reading heads for magnetic hard discs and magnetic random access memories
(MRAMs). 1
A further paradigm of spintronics is semiconductor spintronics. 14,15 This paradigm
brings the promise of integrating spin-based device schemes with existing semiconductor
technologies. At first glance, it seems obvious that ferromagnetism would be a necessary
and integral component of semiconductor spintronics. For instance, ferromagnetic electrodes may be used to inject and/or collect spin-polarized current into a semiconductor,
as in the seminal proposal of Datta and Das for the spin field-effect transistor (SFET). 16
Such a spin-polarized current is necessary for spintronic devices as well as for quantum
1
computation. However, the connection of ferromagnetic metals to semiconductors is
inefficient, due to a large impedance mismatch between them. 17–20
Such problems have inspired an alternate track to semiconductor spintronics, the socalled ”spintronics without magnetism”. 21 In this subfield of spintronics one manipulates
carrier spins through intrinsic properties of semiconductor nanostructures such as strong
spin-orbit interaction (SOI). In the presence of an electric field, free electrons experience
the well-known SOI in vacuum, 22
HSO = Λσ · [p × ∇V (r)] .
(1.1)
Here, Λ = e~/ (2m0 c)2 (m0 and −e are the mass and electric charge of a free electron and
c is the speed of light in vacuum), p is the electron momentum, V (r) is the electric scalar
potential, and the Pauli matrices σ are related to the electron spin via s = ~σ/2. In a
two-dimensional electron gas (2DEG) formed in mesoscopic structures, made of narrowgap semiconductor heterostructures, the microscopic SOI of Eq. (1.1) modifies the band
structure, and often introduces a spin splitting of bands. 23 The final result can often be
written as an effective SOI Hamiltonian, of the general form HSO = (~kSO /m)(π · σ),
where π is a linear combination of the electron momentum components px and py and
m is the effective mass, usually much smaller than m0 .
Two distinct mechanisms of SOI in two-dimensional systems should be emphasized,
namely the Rashba SOI and the Dresselhaus SOI. The Rashba SOI is a result of a
confining potential well which is asymmetric under space inversion. For an electric field
E = −∇V in the z direction, this SOI has the form
HR =
~kR
~k
σ · (p × ẑ) = R py σx − px σy .
m
m
(1.2)
The coefficient kR depends on the magnitude of E, and can be controlled by a gate
voltage, as shown in several experiments. 27–33 The Dresselhaus SOI is a consequence of
a host crystal which lacks bulk inversion symmetry. For a 2DEG the linear Dresselhaus
SOI is given by
HD =
~kD
px σx − py σy ,
m
2
(1.3)
where kD is a material constant which depends weakly (if at all) on the external electric
field. These SOIs can be interpreted as a Zeeman interaction in a momentum-dependent
effective magnetic field. As the electron propagates in the presence of these SOIs, its
spin precesses around this effective magnetic field. Therefore, it becomes possible to
control and manipulate the electron’s spin in high-purity semiconductor nanostructures.
Motivated by such prospects, there has been an increasing interest in electron transport
in the presence of SOIs in the last two decades. Nevertheless, numerous fundamental
questions are yet to be answered and some of them are the subject of this research.
3
2 Research Objectives
The main goal of the research is to study several aspects of spin-dependent electron
transport in the presence of SOIs in meso- and nanostructures. Specifically, we address
theoretically the following questions:
1. Spin filtering in mesoscopic systems - How to efficiently generate spin-polarized
current of electrons out of an unpolarized source, using small electric and magnetic
fields? We will examine several models for spin filters based on quantum interference
and on SOIs in mesoscopic systems. The properties of such spin filters will be analyzed
and compared.
2. Spin-dependent transport through chiral molecules - Recent experiments have
shown that ordered films of chiral organic molecules on metallic surfaces exhibit a
spin-dependent electron transmission, an effect that was termed as chiral-induced
spin selectivity (CISS). 34 Although some theoretical progress in understanding the
origin of this phenomenon has been made recently, further study of this phenomenon
is necessary to elucidate the role of SOIs in such systems. 34
3. Effects of SOI on thermoelectricity in nanostructures - In the last decade it
has been suggested that meso- and nanoscale systems may be good candidates for
thermoelectric device applications. 35–42 The figure of merit of thermoelectric materials, ZT , which determines their efficiency in a thermoelectric device, remains low for
most conventional bulk materials. Nanoscale thermoelectric materials are promising
for increasing ZT relative to the bulk. In this context, it is important to study the
effects of SOI on the thermoelectric properties of low-dimensional nanostructures.
4
2.1 Related Work
2.1.1 Spin filtering in mesoscopic systems
For many spintronics applications and for a quantum computation, it is necessary to
provide well-defined polarized spins. Hence, a major aim of spintronics is to build mesoscopic spin filters, which polarize the spins going through them along tunable directions.
At first glance, an elementary way to obtain polarized electrons is to inject them from a
ferromagnet. However, the connection of ferromagnetic metals (FM) to semiconductors
(S) is inefficient, due to a large impedance mismatch between them. 17–20 In the diffusivetransport regime, the spin-polarized current across a FM/S junction is dominated by
the ratio RF M /RS where RF M (RS ) is the resistance of the FM (S). 17–20 Since typically
RF M RS , there is a fundamental obstacle hindering the electrical injection of spinpolarized current from an FM into an S. To circumvent this problem, it was suggested
to introduce a spin-dependent tunnel barrier between the FM and the S. 43 Since the
tunnel barrier contact resistance is much larger than the FM and the S resistances, it
will dominate the spin-polarized current, thus providing a method for an efficient spin
injection. 18–20,43,44 Recently, several groups used this method to inject spin-polarized
current from a FM/tunnel barrier contact into an S. 45–48 However, the spin polarization
in the S has not exceeded 30% − 40%. To the best of our knowledge, currently there is
no effective way to inject a ≈ 100% spin-polarized current from an FM into an S. For
this reason, we are interested in spin filters that avoid ferromagnets.
Several proposed filters use quantum dots, in which the filtering is based on either the
Coulomb blockade and the Pauli principle, 49–52 or on the Zeeman energy splitting. 53,54
All the above filters usually generate only a partial spin polarization and require the
confinement of strong magnetic fields into a small region.
Different proposals for spin filters exploit the strong SOIs in 2DEG. For example, the
authors of Refs. [55–59] suggested a spin filter based on SOI in mesoscopic T junctions,
which split the unpolarized electron beam into two polarized ones. These filters are
advantageous since they produce two polarized beams, thus allowing transmission of all
5
the electrons in the original beam, and since they do not use magnetic fields. However,
the outgoing polarization depends on the electrons energy. At finite temperature and/or
finite bias voltage, the average over energies mixes different polarization directions and
eliminates the possibility of obtaining full polarization. It is thus advantageous to have
energy-independent polarizations.
Further suggestions for spin filters take advantage of the interference of electronic
waves in quantum networks which contain closed loops. The phases of these waves
can include the Aharonov-Bohm (AB) phase, 60 which results from a magnetic flux Φ
penetrating the loop, and the Aharonov-Casher (AC) phase, 61 which is caused by the
presence of an electric field that generates the SOIs and affects the spin degree of freedom.
As a consequence, when an electron propagates from site u to site v whose relative
position vector is ruv = rv − ru = Lĝ, its spinor |χi transforms into |χ0 i = U |χi, with
the unitary matrix U = eiϕ eiK·σ . 62–65 Here, ϕ is the AB phase given by
2π
ϕ=−
Φ0
Z
v
A · dr,
(2.1)
u
where Φ0 = hc/e is the flux quantum and
K = αR −gy , gx , 0 + αD −gx , gy , 0 ,
(2.2)
with the dimensionless coefficients αR,D ≡ kR,D L. Many papers proposed a single circular loop interferometer which would be sensitive to this phase and/or to its competition
with the AB phase. 65–72 The loop is connected to two leads, and the destructive interference of the waves in the two paths can sometimes block electrons with one polarization
and fully transmit electrons with the opposite polarization.
Recently, Aharony et al. studied the scattering of electrons through a diamond interferometer made of quantum dots connected by one-dimensional wires, and subject
to both an Aharonov-Bohm flux and (Rashba and Dresselhaus) SOIs. 73 It has been
shown that with some symmetry between the two branches of the diamond, and with
appropriate tuning of the electric and magnetic fields (which control the AB and AC
phases), the interferometer completely blocks electrons with one polarization and allows
only electrons with the opposite polarization to be transmitted. The directions of these
6
polarizations are tunable by these fields, and do not depend on the energy of the scattered electrons. Moreover, when the spin filtering conditions are obeyed, one can tune
the interferometer parameters (site and hopping energies) so that the transmission of
the fully polarized electrons is close to unity. These authors have used a tight-binding
model to describe electron transport through the interferometer.
One aim of the of the proposed research is to study several generalizations of the above
spin filter. For instance, in the diamond interferometer discussed in Ref. [ 73], the spin
filtering conditions require perfect symmetry between the two paths of the interferometer. Although the realization of such a highly symmetric interferometer is possible in
principle (e.g. by tuning gate voltages), it may still be a difficult task in practice. It is
thus desirable to have a perfect spin filtering in an asymmetric interferometer. Probably,
this can be achieved in similar interferometers with a larger number of parameters (see
preliminary results in subsection 3.1).
A further generalization concerns the robustness of spin filtering against current leakage, for example due to quantum tunnelling out of the loop. Furthermore, in a practical
device, with the application of finite source-drain bias voltage, the wires which connect
the four dots are inevitably charged up if they are electrostatically isolated. To avoid
such probably unfavorable effects, one needs to ground them, which leads to current
leakage. In the proposed research we will examine the effect of current leakage in the
context of the diamond interferometer (see preliminary results in subsection 3.2).
Several groups have also considered interference and spin-dependent transport in infinite quantum networks. For instance, SOI-induced interference has been studied in a
quantum network made of a chain of square loops. 74,75 Indeed, interference due to the
Rashba SOI has been measured on a nanolithographically defined square loop array. 76
The effects of Rashba SOI and an AB magnetic flux on the ballistic electron transport
through a chain of quantum rings has been studied in Ref. [77]. However, the possibility to use such networks to achieve spin filtering has not been considered in those
papers. This possibility has been studied in an infinite chain of diamond-shaped rings
by Aharony et al.. 78,79 Such a network was found to give a wide range of electric and
magnetic fields which yield full spin polarization at the output. In the next step we
7
would like to examine the effect of SOI and an AB flux on the band structure and ballistic conductance of a two-legged ladder quantum network and to explore the possibility
for spin filtering in this network (see preliminary results in subsection 3.3).
2.1.2 Spin-dependent transport through chiral molecules
An increasing interest in the spintronics community is focused on ”organic spintronics”, where organic molecules are used within spin-based devices. 80,81 Recently, electron
transmission experiments have shown that ordered films of chiral organic molecules on
metallic surfaces can act as electron spin filters at room temperature, an effect that was
termed as chiral-induced spin selectivity (CISS). 34 The spin polarization in those exper
iments was defined as P = I+ − I− / I+ + I− , where I+ and I− are the intensities of
the signals corresponding to the spin oriented parallel and antiparallel to the electrons’
velocity.
The first indications for the CISS effect were obtained from low-energy photoelectron
transmission (LEPET) spectroscopy studies of thin ordered films comprised of chiral
molecules, deposited on polycrystalline Au substrate. 82 In this experiment, circularly
polarized light was used to eject spin-polarized photoelectrons from the underlying Au
substrate, and the quantum yield and kinetic energy distribution of the photoelectrons
were measured for different molecular assemblies. It was found that the quantum yield
of the photoelectrons depended on the relative polarization of the light and the chirality
of the molecules in the film. For example, right-handed circularly polarized light showed
a higher quantum yield of photoelectrons through a left-handed assembly than that
through a right-handed assembly, even though the film thickness and order were the
same. By studying multilayer films, the chirality of the molecules could be changed
between the layers, and it was found that changing the molecules chirality between
layers decreased the quantum yield of photoelectrons.
Following this first study, several other LEPET studies on self-assembled monolayers (SAMs) of chiral molecules were conducted. 83–85 All of these studies confirmed the
dependence of the electron transmission yield on the chirality of the molecules and the
8
handedness of the light that was used to eject photoelectrons from the underlying Au
substrate. While these experiments reveal a clear dependence of the photoelectron yield
on the light polarization and the chirality of the molecular films, the relationship of
these effects to the spin polarization of the photoelectrons was only inferred. A recent
experiment by Göhler et al. 86 has demonstrated convincingly that molecular assemblies
of chiral molecules act as efficient spin filters for photoelectrons. Göhler et al. performed
a LEPET experiment in which they measured the spin of the photoelectrons with a Mott
detector after they were transmitted through a SAM of double-stranded DNA molecules
adsorbed on a Au surface. The spin polarization of the photoelectrons does not change
significantly with the light polarization, ranging from -35% to -29%. Furthermore, the
experiment shows that the degree of spin polarization increases monotonically with the
length of the DNA molecule.
The CISS effect has been demonstrated also in the tunneling regime. 87 In this study,
Xie et al. measured the conductance of a single double-stranded DNA molecule connected
to a Ni substrate from one side and to a gold nanoparticle from the other side. The I −V
curves reveal that the conductance of the DNA molecule depends on the magnetization
direction of the Ni substrate and on the length of the DNA molecule. The effective barrier
increases with increasing length of the DNA molecule, but the difference between the
barriers for opposite magnetizations of the Ni substrate is constant for all lengths and
is about 1eV . Hence, this experiment shows that the conductance of a single DNA
molecule is spin dependent.
On the theoretical side, several models have been developed to explain the CISS effect
in molecules with helical geometry. In Ref. [88] the authors considered an effective
tight-binding Hamiltonian for an electron that moves along the axis of a helix, which
creates an electric field with helical symmetry. These authors showed that partial spin
filtering occurs for large values of the SOI strength and small values of the hopping
energy. Later, these authors considered a similar model in which the electron moves on
an inner helix. 89 It has been demonstrated that spin filtering occurs provided that each
tight-binding site has at least two energy levels with unequal values of SOI strength and
hopping energy. Although these models demonstrate spin filtering due to SOI, they do
9
not take into account the effect of the transverse potential which confines the electrons to
the one-dimensional curve (a line in Ref. [88] and a helix in Ref. [89]). Such a transverse
potential can have important implications on the effective one-dimensional Hamiltonian,
as proved in the context of spin-dependent transport through quantum rings. 90
In another study, Guo and Sun invoked a tight-binding model which describes electron
transport through a double helix with interchain interactions. 91 In this model a dephasing term is necessary to achieve spin polarization. This study predicts that for a given
dephasing strength, there exists an optimal length of the DNA molecule for which the
spin polarization reaches a maximum. The optimal length is found to be inversely proportional to the dephasing strength. Beyond this optimal length, the spin polarization
slightly declines with increasing length.
In a different approach, the authors of Refs. [ 92,93] considered a spin-dependent
electron scattering off a helical molecular model of six carbon atoms. Their calculations
require multiple scattering for the generation of a spin polarization. The mechanism
for spin filtering in this model involves an interference between the spin-orbit scattering
amplitudes and those from the pure electrostatic terms. For the model system of six
carbon atoms in a single turn, the authors found a spin polarization of about 1%. While
this value is significant in magnitude, it is much smaller than that found in experiments.
The authors suggested that when the density of atoms is higher, as is the case in DNA,
the spin polarization will be higher. Moreover, the increase of the longitudinal spin
polarization with the length of the molecule was demonstrated by constructing successive
turns of the helix and considering incoherent multiple scattering through different turns.
While these studies differ in important ways, they possess several similar conclusions.
In all of these models a chiral potential is necessary for the CISS effect. In addition,
they reveal that the value of the SOI strength for the chiral molecule must be larger
than the atomic SOI strength by 1 − 2 orders of magnitude. However, the origin of such
a strong SOI in chiral molecules is not clear. Moreover, in all of the theoretical models
presented so far, rather strong assumptions are needed to account for the magnitude
of the observed effects. Further theoretical and experimental investigations are thus
necessary to shed a light on the basic mechanisms underlying the CISS effect.
10
2.1.3 The interplay between SOI and thermoelectricity in
nanostructures
Thermoelectric materials can be used to convert heat to electricity (power generation)
through the Seebeck effect, or can be utilized for heating or cooling applications through
the Peltier effect. 94–98 While the basic principles underlying thermoelectric effects are
well known, the use of thermoelectric materials is not very widespread due to their low
efficiency. The efficiency of a thermoelectric material is described by its dimensionless
figure of merit, ZT , defined as ZT ≡ S 2 σT /κ, where S is the Seebeck coefficient (or
thermopower), σ and κ are the electrical and thermal conductivities, respectively, and T
is the temperature. For thermoelectric systems to be competitive with current technologies, ZT must be above 1.5 for power generation and above 2 − 4 for cooling applications. 99,100 While each property of ZT , namely S, σ and κ, can individually be changed
by several orders of magnitude, the interdependence of these properties have made it
extremely difficult to increase ZT above 1 in bulk materials. 101 In the last two decades
several theoretical predictions 102–104 and experimental demonstrations 105–110 have indicated the promise of a higher ZT in thermoelectric nanostructures (for review, see Refs.
[35–41]). Following a suggestion by Mahan and Sofo, 111 attempts to enhance the thermoelectricity of nanostructures are being made (i) by breaking electron-hole symmetry
and making the electronic density of states a strong function of the energy (for example, enhancing it) to increase the thermopower, 112 and (ii) by increasing the presence
of interfaces and surfaces to enhance phonon scattering, thereby reducing the phononic
thermal conductivity. 113,114
Thermoelectricity at the nanoscale has been studied in quantum point contacts, 115–117
quantum dots, 118–123 Carbon nanotubes, 124–126 graphene, 127–129 and more. 130 The effects of electron-electron interactions, 131–133 as well as time-reversal breaking magnetic
field 134–136 and inelastic processes due to coupling to vibrational modes, 42,137,138 on thermoelectric transport were considered.
Recently, several spin-dependent thermoelectric phenomena have been discovered, giving rise to a promising subfield of spintronics, the so-called ”spin caloritronics”. 139 These
11
phenomena can be roughly classified into (i) independent electrons, (ii) collective and
(iii) relativistic (or SOI-induced) effects. The first class is the thermoelectric generalization of effects such as GMR and tunnel magnetoresistance (TMR), and is referred
to as the spin-dependent Seebeck and Peltier effects. 140 The authors of Refs. [141–143]
studied spin-dependent thermoelectric transport in metallic ferromagnets. Following
these studies, thermally driven spin injection from a ferromagnet into a non-magnetic
metal due to the spin-dependent Seebeck effect has been demonstrated. 144 Later, the
same group observed the spin-dependent Peltier effect in a nanostructure consisting of
a non-magnetic metal sandwiched between two ferromagnetic layers. 145
The second class of effects is generated by the collective dynamics of the magnetization in ferromagnets that couple to single spins, as in the recently observed spin Seebeck
effect. In this effect, first observed by Uchida et al. in 2008, 146 a temperature gradient in
a ferromagnet causes a spin injection into an attached non-magnetic metal over a macroscopic scale of several millimeters. This phenomenon surprised the community because
the length scale seen in the experiment was extraordinarily longer than the spin-flip
diffusion length of conduction electrons, suggesting that conduction electrons in the ferromagnet are irrelevant to the phenomenon. Subsequently, the spin Seebeck effect was
observed in various materials ranging from metallic ferromagnets 147 to semiconducting
ferromagnets, 148 as well as in insulating ferromagnets. 149,150 Recent theoretical and experimental efforts have shown that the magnon and phonon degrees of freedom indeed
play crucial roles in the spin Seebeck effect. 151
Finally, the third class of effects includes the thermoelectric generalizations of various
SOI-induced Hall effects. Theoretical work has been carried out, 152–154 and experimental
evidences for Nernst effects (i.e. the Hall voltage induced by a longitudinal heat current)
have been found. 155–158 Further studies of thermoelectric effects in the presence of SOI
include the thermoelectric properties of a 2DEG with Rashba SOI, 159 spin-dependent
thermoelectric properties of an AB interferometer with an embedded quantum dots in
the presence of Rashba SOI 160,161 and spin-dependent thermoelectric properties of quasi
one-dimensional ballistic electron system with Rashba SOI. 162 However, further work
is necessary to better understand the effects of SOI on thermoelectric properties and
12
thermoelectric efficiency of nanostructures.
2.2 Significance of the Research
As outlined in the previous sections, SOI might play a crucial role in spintronics, as
far as manipulating the carriers spin and controlling the spin dynamics is concerned.
Especially the tunability of the Rashba spin-orbit strength by electrostatic means, and
by quantum well design, offer new possibilities that did not exist in the old metal-based
counterpart. Particularly, two emerging subfields of spintronics are promising. The first
subfield is organic spintronics which opens the way to cheap, low-weight and mechanically flexible spinronic devices. The second one is spin caloritronics, which explores
the interaction of spins with heat currents. Combining the spin degree of freedom with
thermoelectric effects in meso- and nanoscopic structures provides new strategies to increase the thermoelectric figure of merit as well as offering new functionalities. On a
more fundamental level, Rashba SOI gives rise to many interesting phenomena. Studying systems with SOIs, and understanding the physics that those experiments reveal,
is therefore of great importance for the applied spintronic community, as well as for
fundamental reasons.
2.3 Research Methodology
Several theoretical methods will be used in our analyses. We will apply scattering theory
in the framework of the tight-binding formalism to study spin-dependent electron transport in single interferometers and in quantum networks with SOIs and AB flux. Similar
techniques will be used to study electron transport in a helical potential. Symmetry
analysis of the scattering matrix will be carried out to better understand the role of
the helical potential in the CISS effect. Thermoelectric coefficients will be calculated in
the presence of various driving fields, assuming linear-response. More details on some
of these techniques are included in the preliminary results below. Other techniques are
described in Refs. [42,136–138,163–165].
13
3 Preliminary Results
3.1 Spin filtering in a
Rashba-Dresselhaus-Aharonov-Bohm double-dot
interferometer
In Ref. [166] we have studied a spin filter made of two elongated quantum dots (QDs),
which are subject to both an Aharonov-Bohm flux and (Rashba and Dresselhaus) spinorbit interactions. The double-dot interferometer is sketched schematically in Fig. 3.1(a).
Recently, the electrical control of SOI was demonstrated in such InAs self-assembled
elongated QDs and nanowires. 33,167–170 Furthermore, in such nanostructures, one can
define several electrodes and control different parts of the nanostructure separately. 167–170
Therefore, we model the interferometer as a square, as shown in Fig. 3.1(b). Each
QD is replaced by a bond connecting two sites [a, b and c, d in Fig. 3.1(b)] with the
corresponding bonds subject to SOI.
In the framework of the nearest neighbors tight-binding model, the Schrödinger equation for the spinor |ψv i at site v is written as
(ε − εv ) |ψv i = −
X
Juv Uuv |ψu i,
(3.1)
u
where εv is the site energy, Juv is a real hopping amplitude and Uuv is a 2 × 2 unitary
matrix which describes the AB and AC phases acquired by an electron moving from
site u to site v. Except for the sites 0, a, b, c, d and 1, the nearest-neighbor hopping
energy along the leads is j, with no SOI, and the site energies εv on the leads are zero.
14
(a)
(b)
Figure 3.1: The double-dot interferometer. (a) Schematic. The interferometer is penetrated by a magnetic flux Φ, and its horizontal edges are subject to spin-orbit
interactions. (b) Tight-binding model. The bonds ab and cd are subject to
spin-orbit interactions.
With a lattice constant a, the states on the leads are combinations of einka , and the
corresponding energy is ε = −2j cos(ka).
To calculate the transmission, one first writes the tight-binding equations for the
spinors at sites 0, a, b, c, d and 1. Eliminating the spinors at sites a, b, c, and d from
these equations in terms of the spinors |ψ0 i and |ψ1 i, one obtains effective tight-binding
equations for hopping between sites 0 and 1. These equations are of the form
(ε − y0 ) |ψ0 i = W† |ψ1 i − j|ψ−1 i,
(ε − y1 ) |ψ1 i = −j|ψ2 i + W|ψ0 i.
(3.2)
Here, W is a matrix which contains all the details of the interferometer and represents
an effective hopping from site 0 to site 1, and y0 , y1 are the effective site energies,
which also depend on the details of the interferometer. It should be noted that these
equations are valid for any two-path interferometer. All the details of the interferometer
are embodied in the effective hopping matrix W and effective site energies y0 and y1 .
The 2 × 2 spin-dependent transmission and reflection amplitude matrices are then found
−1
−1
to be T = 2ij sin (ka) W Y I − W† W
and R = −I−2ij sin (ka) X1 Y I − W† W ,
respectively. Here, I is the 2 × 2 unit matrix and X0,1 = y0,1 + je−ika , Y = X0 X1 .
As discussed in Ref. [ 73], the matrix W has the general form W = γupper Uupper +
γlower Ulower , where γlower , γupper and Ulower , Uupper are the effective hopping energies
15
(which depend on the electron energy ε) and unitary matrices corresponding to transitions through the lower and upper paths of the interferometer, respectively. By analyzing
the properties of the spin-dependent transmission matrix we have shown that the conditions for spin filtering are
γlower = γupper ≡ γ,
cos(φ + ω) = −1,
(3.3)
where φ and ω are the AB and AC phases gained by an electron that goes around the
interferometer loop. The first condition in Eqs. (3.3) can be interpreted as a requirement
for a symmetry relation between the two paths. The second condition in Eqs. (3.3),
namely ω = −φ + π, imposes a relation between the AB flux and the SOI strength.
However, once these conditions are satisfied, the blocked and transmitted spin directions,
−n̂ and n̂0 , are completely determined by the AC phase and are independent of the
electron energy ε.
In Ref. [166] we have shown that by enlarging the number of interferometer parameters
(namely, hopping and site energies), spin filtering can be achieved in an asymmetric
interferometer. This is in contrast to the diamond interferometer discussed in Ref. [
73], in which the first condition of Eqs. (3.3) requires perfect symmetry between the
two paths of the interferometer. However, the transmission of the polarized electrons
gets narrower as the interferometer becomes more asymmetric. Several experimental
procedures to achieve perfect spin filtering in this interferometer have been suggested
and the simplification of the spin filtering conditions in the linear-response regime (i.e.
at low temperature and bias voltage) has been emphasized. In addition, we have derived
analytical expressions for the directions −n̂ and n̂0 of the filtered and the transmitted
electrons in terms of the Rashba and Dresselhaus spin-orbit strengths.
16
3.2 Robustness of spin filtering against current leakage
in a Rashba-Dresselhaus-Aharonov-Bohm
interferometer
In Ref. [171] we have examined the effects of current leakage on spin filtering in the
diamond interferometer discussed in Ref. [ 73] [Fig. 3.2(a)]. To allow for a possible
current leakage, we considered M − 1 intermediate sites with lattice constant a and site
energies ε0 = 0 on each edge uv (uv = Lb, Lc, cR, bR) of the diamond [Fig. 3.2(b)].
Each of these sites is connected to an absorbing channel, modelled as a one-dimensional
tight-binding chain with site energies ε0 = 0 and free of SOI. Electrons can tunnel out
of the interferometer through these absorbing channels. The hopping amplitude on the
first bond on each absorbing channel is Jx,uv , while the other bonds have a hopping
amplitude j.
(a)
(b)
Figure 3.2: The (a) lossless (Ref. [ 73]) and (b) lossy diamond interferometer. The
diamond is penetrated by a magnetic flux Φ, and its edges are subject to spinorbit interaction. In the lossy interferometer electrons can tunnel out of the
interferometer from sites n = 1, . . . , M − 1 on edge uv (uv = Lb, Lc, cR, bR)
through absorbing channels.
Assuming only outgoing waves on the absorbing channels, we have shown that by
eliminating the spinor at the first site of the absorbing channel, 172 one obtains effective
tight-binding equations, similar to Eqs. (3.2). However, the effective hopping and site
17
energies are now complex numbers. Nevertheless, as we have shown in Ref. [171], spin
filtering may still be obtained in the presence of current leakage by appropriately tuning
the gate voltages and the magnetic field. In the presence of current leakage, the spin
filtering conditions read
|γupper | = |γlower | ≡ γ,
cos φe + ω = −1,
(3.4)
where φe = φ + δlower − δupper , with δupper and δlower being the phases of γupper and γlower ,
respectively. Several possibilities to satisfy the conditions (3.4) have been discussed in
Ref. [ 171]. In addition, the transmission of the polarized electrons through the lossy
filter has been compared with that of the lossless one. Many properties of the filter, such
as the dependence of the AC phase ω on the SOI strength and on the geometry of the
diamond (through the opening angle 2β) and the directions −n̂ and n̂0 of the filtered
and the transmitted electrons are the same as in the lossless filter. It is important to
note that although the effects of current leakage have been examined in the context of
the diamond interferometer, the conclusions are quite general and can be applied to an
arbitrary two-path interferometer.
3.3 Two-legged ladder quantum network with Rashba
spin-orbit interaction and Aharonov-Bohm flux
Recently, we have studied the effect of Rashba SOI and an AB flux on the spin-dependent
ballistic electron transport through a two-legged ladder quantum network (Fig. 3.3) with
nearest neighbor (NN) and next nearest neighbor (NNN) bonds. 173 We used the tightbinding Hamiltonian
H=
X
u,σ
εu c†u,σ cu,σ −
X
Juv eiφuv (Vuv )σ,σ0 c†v,σ cu,σ0 + H.C.,
(3.5)
u<v,σ,σ 0
where c†u,σ cu,σ is the creation (destruction) operator of an electron at site u with spin
σ, Juv is a hopping amplitude and εu is the site energy, which we set to be zero. The
18
Figure 3.3: The infinite square two-legged ladder with lattice constant a. Solid and
dashed lines represent nearest and next nearest neighbor bonds, respectively.
The unit cell is shown in a dotted (blue) square.
AB and AC phases are φuv = − Φ2π
0
Rv
u
A · dr and Vuv = e−iθuv (r̂uv ×ẑ)·σ . Here A is the
vector potential, Φ0 = hc/e is the flux quantum, ruv = rv − ru = ruv r̂uv is the vector
connecting the two ends of the bond and θuv = kR ruv is the spin-precession angle. We
˜ respectively, and
denote the hopping amplitudes for NN and NNN bonds as J and J,
√
√
the corresponding spin-precession angles as θ = m∗ ηa/~2 and θ̃ = 2m∗ ηa/~2 = 2θ.
Solving the tight-binding Schrödinger equations with a Bloch-type spinors, we have
analytically analyzed the band structure of the network. The ballistic conductance can
then be inferred from the band structure by considering the number of propagating solutions (which correspond to solutions with real wave vector) for a given energy. For a
given energy, the number of these right-moving (or, equivalently, left-moving) propagating modes, g, completely determines the conductance in the ballistic regime, through the
Landauer formula G = (e2 /h) g. 163,176,177 Such an analytical solution enables us to obtain
several general insights into the spin-orbit dominated physics of quasi one-dimensional
(Q1D) systems. For instance, in accordance with the perturbative and numerical approaches of Refs. [174] and [175], the energy bands and the ballistic conductance turn out
to be quite sensitive to the SOI strength. The SOI produces non-parabolic bands which
in turn causes the ballistic conductance to be strongly energy dependent. Furthermore,
in agreement with Refs. [74] and [75], we have found that the SOI causes localization and
delocalization of the electron states in particular energies, due to quantum interference
effects.
As in the Q1D quantum network of diamond loops, 78,79 we have shown that the
19
combination of both Rashba SOI and an AB flux creates the possibility of spin filtering in
the two-legged ladder quantum network as well. In the absence of an AB flux, one always
has g = 0, 2, or 4. In such a case, the spinor of the propagating electron is a combination
of two or four right-moving modes, thereby carries no definite spin. However, for g = 1
the spinor of the right-moving electron consists of one propagating mode (the other three
modes being evanescent) and the resulting current is spin polarized. As argued in Ref.
[78], this conclusion is valid also for a finite quantum network, provided that the network
is longer than the decay lengths of the evanescent modes. It should be emphasized that
the polarization direction is not unique but changes with energy, SOI strength and AB
phase. For each set of parameters (for which g = 1) there exists a different spin direction.
The possibility for spin filtering in the two-legged ladder quantum network has not been
considered in Refs. [ 74] and [ 75] since these authors averaged the conductance over
the electron energies. This mixes different polarization directions and eliminates the
possibility of obtaining spin-polarized current.
20
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