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Quantum Transport in Graphene-
Superconductor (GS) Structures using
Quantum Super-Field Theory (QSFT)
by
ARVEL JOHN M. LOZADA
A Thesis Proposal Submitted to the
Department of Physics
College of Arts and SciencesUniversity of San Carlos
Technological Center
Talamban Campus, Cebu City
As Partial Fulfillment of the Requirements
for the degree of
MASTER OF SCIENCE in PHYSICS
November 2009
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TABLE OF CONTENTS
Contents
I. Introduction ----------------------------------------------------- 1
A. The 2-D Form of Carbon ----------------------------- 2
B. Discovery of Graphene ------------------------------- 3
C. Electronic Structure ----------------------------------- 4
D. Chiral Dirac Electrons -------------------------------- 5
E. Quantum Tunneling of Chiral Particles ----------- 7
F. Klein Paradox ------------------------------------------- 8
G. Absence of Localization ------------------------------ 9
H. Graphene Devices ------------------------------------- 10
II. Basic Physics of Graphene --------------------------------- 11
A. Tight Binding Model ------------------------------------ 12
B. Geometrical Structure --------------------------------- 13C. Lattice of Graphene ----------------------------------- 14
D. Electronic Configuration ------------------------------ 14
E. Bloch Functions ----------------------------------------- 15
F. Secular Equation ---------------------------------------- 15
G. Calculation of Transfer and Overlap Integrals -- 15
H. Calculation of Energy --------------------------------- 15
I. Exactly at the K-Point ---------------------------------- 15J. Linear Expansion --------------------------------------- 15
K. Dirac-like Equation ------------------------------------- 15
L. Absence of Backscattering --------------------------- 15
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III. Quantum Transport ------------------------------------------- 16
A. Dirac-Bogoliubov-deGennes Equation ------------ 17
B. Quantum Super-Field Theory ------------------------ 18
References ----------------------------------------------------------- 19
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Chapter 1
INTRODUCTIONCarbon plays a unique role in nature. The formation of carbon in stars as a result of the
merging of three -particles is a crucial process that leads to the existence of all the relatively
heavy elements in the universe. The capability of carbon atoms to form complicated networks
is fundamental to organic chemistry and the basis for the existence of life, at least in its known
forms. Even elemental carbon demonstrates unusually complicated behavior, forming a
number of very different structures.
As well as diamond and graphite (three-dimensional allotropes of carbon), which have
been known since ancient times, recently discovered fullerenes (zero-dimensional allotropes
of carbon) and nanotubes (one-dimensional allotropes of carbon) are currently a focus of
attention for many physicists and chemists. Thus, the only missing link to the family of carbon
allotropes is its two-dimensional nature, the so called GRAPHENE.
Figure 1: Allotropes of Carbon
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The 2-D Form of Carbon
The elusive form of carbon is named graphene, and ironically, it is probably the best
studied carbon allotrope theoretically. Graphene planar, hexagonal (honeycomb lattice)
arrangements of carbon atoms is the starting point for all the calculations on graphite, carbon
nanotubes, and fullerenes (Figure 1).
At the same time, numerous attempts to synthesize these two-dimensional atomic
crystals have usually failed, ending up with nanometer-size crystallites. These difficulties are
not surprising in light of the common belief that truly two-dimensional crystals cannot exist (in
contrast to the numerous, known quasi-two-dimensional systems).
Moreover, during synthesis, any graphene nucleation sites will have very largeperimeter-to-surface ratios, thus promoting collapse into other carbon allotropes.
Discovery of Graphene
In 2004, a group of physicists from Manchester University, UK, led by Andre Geim and
Kostya Novoselov, used a very different and, at first glance, even naive approach to obtain
graphene and lead a revolution in the field. They started with tree-dimensional graphite and
extracted a single sheet (a monolayer of atoms) using a technique called micromechanical
cleavage (Figure 2).
Graphite is a layered material and can be viewed as a number of two-dimensional
graphene crystals weakly coupled together exactly the property used by the Manchester
team. By using this top-down approach and starting with large, three-dimensional crystals, the
researchers avoided all the issues with the stability of small crystallites.
Furthermore, the same technique has been used by the group to obtain two-
dimensional crystals of other materials, including boron nitride, some dichalcogenides, and
the high-temperature superconductor Bi-Sr-Ca-Cu-O. This astonishing finding sends an
important message, two dimensional crystals do exist and they are stable under ambient
conditions.
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Amazingly, this humble approach allows easy production of large (up to 100-m in
size), high-quality graphene crystallites, and immediately triggered enormous experimental
activity. Moreover, the quality of the samples produces are so good that ballistic transport and
a quantum Hall effect (QHE) can be observed easily.
The former makes this new material a promising candidate for future electronicapplications, such as ballistic field-effect transistors (FETs). However, while this approach
suits all research needs, other techniques that provide a high yield of graphene are required
for industrial production.
Among the promising candidate methods, one should mention exfoliation of
intercalated graphitic compounds and Si sublimation form SiC substrates, demonstrated
recently by Walt de Heer's group at Georgie Institute of Technology.
Figure 2: Micromechanical Cleavage of Graphene
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Electronic Structure
The electronic structure of graphene follows from a simple nearest-neighbor, tight-
binding approximation. Graphene has two atoms per unit cell, which results in two 'conical'
points per Brillouin zone where band crossing occurs, Kand K'. Near these crossing points,the electron energy is linearly dependent on the wave vector. Actually, this behavior follows
from symmetry considerations, and thus is robust with respect to long-range hopping
processes (Figure 3).
What makes graphene so attractive for research is that the spectrum closely resembles
the Dirac spectrum for massless fermions. The Dirac equation describes relativistic quantum
particles with spin , such as electrons. The essential feature of the Dirac spectrum, following
from the basic principles of quantum mechanics and relativity theory, is the existence of
antiparticles.
More specifically, states at positive and negative energies (electrons and positrons) are
intimately linked (conjugated), being described by different components of the same spinmor
wave function. This fundamental property of the Dirac equation is often referred to as the
charge-conjugation symmetry. For Dirac particles with mass m, there is a gap between the
minimal electron energy, E0 = mc2, and the maximal positron energy, -E0 (c is the speed of
light). When the electron energy E >> E0, the energy is linearly dependent on the wave vector
k, E = ck.
Formassless Dirac fermions, the gap is zero and this linear dispersion law holds at any
energy. In this case, there is an intimate relationship between the spin and motion of the
particle that is, spin can only be directed along the propagation direction (say, for particles) or
only opposite to it (for antiparticles).
In contrast, massive spin particles can have two values of spin projected onto any
axis. In a sense, we have a unique situation here that is, charge massless particles.
Although this is a popular textbook example, no such particles have been observed before.
The fact that charge carriers in graphene are described by a Dirac-like spectrum, rather
than the usual Schroedinger equation for nonrelativistic quantum particles, can be seen as a
consequence of graphene's crystal structure. This consists of two equivalent carbon
sublattices A and B (Figure 3). Quantum-mechanical hopping between the sublattices leads to
the formation of two energy bands, and their intersection near the edges of the Brillioun zones
yields the conical energy spectrum.
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As a result, quasiparticles in graphene exhibit a linear dispersion relation E = kvF, as
if they were massless relativistic particles (for example, photons) but the role of speed of light
is played here by the Fermi velocity vF c/300. Because of the linear spectrum, one can
expect that quasiparticles in graphene behave differently from those in conventional metals
and semiconductors, where the energy spectrum can be approximated by a parabolic (free-electron-like) dispersion relation.
Figure 3: Electronic Structure of Graphene
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Chiral Dirac Electrons
Although graphene's linear spectrum is important, it is not the spectrum's only essential
feature. Above zero energy, the current-carrying states in graphene are, as usual, electron-like and negatively charged. At negative energies, if the valence band is not full, unoccupied
electronic states behave as positively charged quasiparticles (holes), which are often viewed
as a condensed matter equivalent of positrons.
Note, however, that electrons and holes in condensed matter physics are normally
described by separate Schroedinger equations, which are not in any way connected (as a
consequence of the so-called Seitz sum rule, the equations should also involve different
effective masses). In contrast, electron and hole states in graphene should be interconnected,
exhibiting properties analogous to the charge-conjugation symmetry in quantumelectrodynamics.
For the case of graphene, the latter symmetry is a consequence of the crystal
symmetry, because graphene's quasiparticles have to be described by two-component wave
functions, which are needed to define the relative contributions of the A and B sublattices in
the quasiparticles' make-up.
The two-component description for graphene is very similar to the spinor
wavefunctions in QED, but the 'spin' index for graphene indicates the sublattice rather thanthe real spin of the electrons and is usually referred to as pseudospin . This allows one to
introduce chirality formally a projection of pseudospin on the direction of motion which is
positive and negative for electrons and holes, respectively.
The description of the electron spectrum of graphene in terms of Dirac massless
fermions is a kind of continuum-medium description applicable for electron wavelenghts much
larger than interatomic distances. However, even at these length scales, there is still some
retention of the structure of the elementary cell, that is, the existence of two sublattices. In
terms of continuum field theory, this can be described only as an internaldegree of freedom
of the charge carriers, which is just the chirality.
This description is based on an oversimplified nearest-neighbor tight-binding model.
However, it has been proven experimentally that charge carriers in graphene do have this
Dirac-like gapless energy spectrum. This was demonstrated in transport experiments (Figure
4) via investigation of the Schubnikov-de Haas effect, i.e. resistivity oscillations at high
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magnetic fields and low temperatures.
Figure 4: Quantum Transport Properties of Graphene
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Quantum Tunneling of Chiral Particles
The chiral nature of electron states in graphene is crucial importance for electron
tunneling through potential barriers, and thus the physics of electronic devices such as'carbon transistors'.
Quantum tunneling is a consequence of very general laws of quantum mechanics,
such as the Heisenberg uncertainty relations. A classical particle cannot propagate through a
region where its potential energy is higher than its total energy (Figure 5). However, because
of the uncertainty principle, it is impossible to know the exact values of a quantum particle's
coordinates and velocity, and thus its kinetic and potential energy, at the same time instant.
Therefore, penetration through 'classically forbidden' region turns out to be possible.This phenomenon is widely used in modern electronics, beginning with the pioneering work of
Esaki.
Figure 5: Tunneling in Graphene (Top) and Conventional Semiconductor (Bottom).
The amplitude of electron wave function (Red) remains constant (Top)
while it decays exponentially (Bottom). The size of the sphere indicates
the amplitude of the incident and transmitted wave functions.
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Klein Paradox
When a potential barrier is smaller than the gap separating electron and hole bands in
semiconductors, the penetration probability decays exponentially with the barrier height andwidth. Otherwise, resonant tunneling is possible when the energy of the propagating electron
coincides with one of the hole energy levels inside the barrier. Surprisingly, in the case of
graphene, the transmission probability for normally incident electrons is always equal to unity,
irrespective of the height and width of the barrier.
In QED, this behavior is related to the Klein paradox. This phenomenon usually refers
to a counterintuitive relativistic process in which an incoming electron starts penetrating
through a potential barrier, if the barrier height exceeds twice the electron's rest energy mc2.
In this case, the trasmission probability T depends only weakly on barrier height,
approaching perfect transparency for very high barriers, in stark contrast to conventional,
nonrelativistic tunneling.
This relativistic effect can be attributed to the fact that a sufficiently strong potential,
being repulsive for electrons, is attractive for positrons (holes), and results in positron states
inside the barrier. Matching between electron and positron (holes) wave functions across the
barrier leads to the high-probability tunneling described by the Klein paradox (Figure 5).
In other words, it reflects an essential difference between nonrelativistic and relativistic
quantum mechanics. In the former case, we can measure accurately either the position of the
electron or its velocity, but not both simultaneously. In relativistic quantum mechanics, we
cannot measure even electron position with arbitrary accuracy since, if we try to do this, we
create electron-positron pairs from the vacuum and we cannot distinguish our original electron
from these newly created electrons.
Graphene opens a way to investigate this counterintuitive behavior in a relatively
simple benchtop experiment, whereas previously the Klein paradox was only connected with
some very exotic phenomena, such as collisions of ultraheavy nuclei or black hole
evaporations.
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Absence of Localization
The tunneling anomalies in graphene systems are expected to play an important role in
their transport properties, especially in the regime of low carrier concentrations wheredisorder induces significant potential barriers and the systems are likely to split into a random
distribution of p-n junctions.
In conventional two-dimensional systems, sufficiently strong disorder results in
electronic states that are separated by barriers with exponentially small transparency. This is
known to lead Anderson localization. In contrast to graphene materials, all potential barriers
are rather transparent, at least for some angles.
This means that charge carriers cannot be confined by potential barriers that smoothon the atomic scale. Therefore, different electron and hole 'puddles' induced by disorder are
not isolated but effectively percolate, thereby suppressing localization. This is important in
understanding the minimal conductivity e2/h observed experimentally in graphene.
Graphene Devices
The unusual electronic properties of this new material make it a promising candidate
for future electronic applications. Mobilities that are easily achieved at the current state of
'graphene technology' are ~20,000 cm2/V.s, which is already an order of magnitude higher
than that of modern Si transistors, and they continue to grow as the quality of sample
improves. This ensures ballistic transport on submicron distances - the holy grail for any
electronic engineer. Probably the best candidates for graphene-based FETs will be devices
based on quantum dots.
Another promising direction for investigation is spin-valve devices. Because ofnegligible spin-orbit coupling, spin polarization in graphene survives over submicron
distances, which has recently allowed observation of spin-injection and a spin-valve effect in
this material.
It has also been shown by Morpurgo and coworkers at Delft University that
superconductivity can be induced in graphene through the proximity effect (Figure 6).
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Moreover, the magnitude of the supercurrent can be controlled by an external gate voltage,
which can be used to create a superconducting FET.
Figure 6: Intrinsic Superconducting Property of Graphene via Proximity Effect
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Chapter 2
BASIC PHYSICS OF GRAPHENEGraphene is a single sheet of atomic thickness with carbon atoms arranged
hexagonally. Though it was an ideal two-dimensional material of theoretical interest and one
of the earliest material on which tight binding band structure calculation was done, it has
triggered recently a lot of interest among people including reinvestigation of many earlier
results since its experimental discovery in 2004.
Tight Binding Model
A large number of physicists have recalculated the tight binding band with nearest
neighbor hopping but without overlap integral correction, some even have calculated the
same by taking account the overlap integral correction, out of these only few calculations are
there which take care of second and third nearest neighbors along with overlap integral
corrections.
It is noticed that the first nearest neighbor hopping integral lies around 2.5eV 3.0eV
when tight binding band is fitted with the first principle calculation or experimental data, near
the Kpoint of the Brillouin zone of graphene but interestingly, when one tries to have a good
matching of the tight binding band over the whole Brillouin zone by including up to third
nearest neighbor hoppings and overlap integrals, the tight binding parameters are considered
as merely fitting parameters, not as physical entities i.e., the values of parameters do not
decrease consistently as one moves towards second and third nearest neighbors.
Here, we present only the first nearest neighbor tight binding approximation of the
energy band structure of graphene using the nearest neighbor tight binding model (NNTBM).
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Geometrical Structure
Since the geometrical structure of a material plays a crucial role in determining the
electronic dispersion of the material, it is important to look at the details of the structure of
graphene before going into the discussion of band structure of graphene. The structure of an
ideal graphene sheet is a regular hexagonal arrangement of carbon atoms in two-dimensions
(Figure). It consists of two nonequavalent (with respect to orientations of bonds) trianguilar
sublattices called A-sublattice and B-sublattice. The unit cell contains one A and one B type of
carbon atoms contributed by respective sublattices. Each carbon atom has three nearest
neighbors coming from the other sublattice, six next nearest neighbors from the same
sublattice and three next to next nearest neighbors from the other sublattice. A0 (1.42 ) is
the nearest neighbor lattice distance. a1 and a2 are the unit vectors with magnitudea=3a
0 i.e., 2.46 .
Lattice of Graphene
2 different ways of orienting bonds means that there are 2 different types of atomic sites [but
chemically the same]
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Electronic Configuration
Carbon has six electrons with the electronic configuration 1s22s22p2. 2s and 2p levels
of carbon atoms can mix up with each other and give rise to various hybridized orbitals
depending on the proportionality of s and p orbitals.
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Graphene has sp2 hybridizations: 2s orbital overlaps with 2px and 2py orbitals and
generates three new inplane sp2 orbitals each having one electron. The 2pz orbital remains
unaltered and becomes singly occupied. Due to overlap of sp 2 orbitals of adjacent carbon
atoms strond (bonding) and * (antibonding) bonds are formed. The bonding bonds, lyingin a plane, make an angle of 120 with each other and is at the root of hexagonal planar
structure of graphene. Pz orbitals being perpendicular to the plane overlap in a sidewise
fashion and give (bonding) and * (antibonding) bonds. sp2 orbitals with a lower binding
enerrgy compared to 1s (core level) are designated as semi-core levels and pz orbitals having
lowest binding energy are the valence levels.
Overlapping of pz energy levels gives the valence band (bonding band) andconduction band (antibonding *band) in graphene. Thus, we see that while the structure of
graphene owes to bonds, band is responsible for the electronic properties of graphene
and hence as far as electronic properties of graphene are concerned, concentration is given
only on bands.
Since the pz orbitals overlap in a sidewise manner, the corresponding coupling is
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weaker compared to that of bonds (where sp2 orbitals overlap face to face). So the pz
orbitals retain their atomic character. Hence, to describe the electronic structure of graphene,
tight binding model could be a good choice.
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At low energies, there are only two bands the bands that arise from the weak
bonding between the 2pz orbitals.
Bloch Functions
We take into account one orbital per site, so there are two orbitals per unit cell.
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where A and B are Bloch functions summing over all type A and B atomic sites in N units
cells respectively, while A and B are the atomic wavefunctions of site A and B.In general, we
will change the subscript A and B with j = 1 [for A sites] and j = 2 [for B sites].
Secular Equation
In this case, the Eigenfunction j (for j = 1 or 2) is written as a linear combination of the
Bloch functions:
Thus, eigenvalue Ej (for j = 1 or 2) can now be written as:
Substituting the expressions in terms of Bloch functions would yield:
( ) ( )AAN
R
Rki
A RreN
rkA
A
= .1,
( ) ( )BBN
R
Rki
B RreN
rkB
B
= .1,
( ) ( )jjN
R
Rki
j RreN
rkj
j
= .1,
( ) ( ) ( )rkkCrk jj
jjj
,, '
2
1'
'=
=
( )jj
jj
j
HkE
=
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where Hil = < i | H | l > is the transfer integral matrix elements while the Sil = < i | l > is
the overlap integral matrix elements.
If the Hil and Sil are known, we can find the energy by minimizing with respect to C*jm:
Explicitly writing out the sums would yield:
( )
=
2
,
*
2
,
*
2
,
*
2
,
*
lijljiil
li
jljiil
lili
jlji
li
lijlji
j
CCS
CCH
CC
HCC
kE
( )
=
2
,
*
2
,
*
li
jljiil
li
jljiil
j
CCS
CCH
kE
22
,
*
22
,
*
2
,
*
2
*
=
li
jljiil
l
jlml
li
jljiil
li
jljiil
l
jlml
jm
j
CCS
CSCCH
CCS
CH
C
E
==
== 2
1
2
1
*0
l
jlmlj
l
jlml
jm
jCSECH
C
E
===
2
1
2
1 l
jlmlj
l
jlml CSECH
( )222121222121
212111212111
2
1
jjjjj
jjjjj
CSCSECHCHm
CSCSECHCHm
+=+=
+=+=
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Rewriting as a matrix equation will give you:
Secular equation gives the eigenvalues:
Calculation of Transfer and Overlap Integrals
The diagonal matrix elements are given by the following:
For same site only:
=
2
1
2221
1211
2
1
2221
1211
j
j
j
j
j
C
C
SS
SSE
C
C
HH
HH
jjj SCEHC =
( ) 0det = ESH
jiijjiij SHH == ;
( ) ( )jjN
R
Rki
j RreN
rkj
j
= .1,
( ) ( ) ( )AjAAiAN
R
N
R
RRki
AAAA RrHRreN
HH Ai Aj
AiAj
== .1
( ) ( )
( ) ( )
0
1
=
=
AiAAiA
N
R AiAAiAAA
RrHRr
RrHRrN
HAi
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Since A and B sites are chemically identical:
The off-diagonal matrix elements are given by:
Since every A site has 3 B nearest neighbors:
( ) ( )
( ) ( )1
1
=
=
AiAAiA
N
R
AiAAiAAA
RrRr
RrRrN
SAi
0== BBAA HH
1
== BBAASS
( ) ( ) ( )BjBAiAN
R
N
R
RRki
BAAB RrHRreN
HH Ai Bj
AiBj
== .1
==
==
==
32,
2
;32
,2
;3
,0
33
2211
aaRR
aaRR
aRR
AiB
AiBAiB
( ) ( ) ( ) ( )BjBAiAki
N
R
BjBAiA
ki
AB RrHRreRrHRreN
H
j
j
Ai j
j
=
=
==
3
1
.3
1
.1
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Parameterize the nearest neighbor transfer integral:
Calculation of Energy
Secular equation gives the eigenvalues:
( ) ( )
( ) ( )=
==
=3
1
.
0
0
;
j
jki
AB
BjBAiA
ekfkfH
RrHRr
( ) ( )
( )kfsS
RrRrs
AB
BjBAiA
=
=
( ) ( )232/3/
3
1
.
cos2 akaikaikki
xyy
j
j eeekf =
+==
( )( )
( )( )
=
= 11
; *0
*
0
00
ksf
ksfSkf
kfH
( ) 0det = ESH
( ) ( )( ) ( )
( ) ( ) ( ) 0
0det
22
0
2
0
0
*
0
00
=+
=
+
+
kfEsE
EkfEs
kfEsE
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Typical parameter values [quoted in Saito et al]:
( )( )kfs
kfE
1
00 =
( ) ( )2
32/3/cos2
akaikaik xyy eekf+=
129.0,033.3,000
=== seV
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Exactly at the K-Point
At the corners of the Brillouin zones (K points), electron states on the A and B
sublattices decouple and have exactly the same energy. (Note: K points are also referred to
as valleys).
6 corners of the Brillouin zones (K points) but only two are nonequivalent.
= 0,3
4
aK
3
2.;
32
,
2
3
2.;
32,
2
0.;3
,0
33
22
11
=
=
=
=
=
=
Kaa
Kaa
Ka
( ) 03/23/203
1
. =++== =
iiKi eeeeKfj
j
b1
b2
K
K
K
K
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We consider two K points with the following wave vectors:
Linear Expansion
=
= 0,3
4';0,
3
4
a
K
a
K
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Consider two nonequivalent K points:
and small momentum near them:
Linear expansion in small momentum:
p
1;0,3
4', =
=
aKK
p
ak +
= 0,3
4
( ) ( ) ( ) 2/2
3
paOipp
akf yx +=
( )
( )
+
= 0
0
0
0
*0
0
yx
yx
ipp
ipp
vkf
kf
H
( )( )
+
=
spa
Oksf
ksfS
10
01
1
1*
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New notation for components on A and B sites:
Dirac-like Equation
For one K point (e.g. = +1), we have a 2 component wave function,
with the following effective Hamiltonian:
where
Bloch function amplitudes on the AB sites ('pseudospin') mimic spin components of a
sma
v /102
3 60 =
=
=B
A
j
j
j C
CC
2
1
=
+
=
B
A
B
A
yx
yx
jjj Eipp
ippvCEHCS
0
01
=
B
A
( ) pvppvvipp
ippvH yyxx
yx
yx .
0
0
0
0
=+=
=
+
=
+
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relativistic Dirac fermion.
To take into account both K points ( = +1 and = -1), we can use a 4-component wave
function,
with the following effective Hamiltonian:
Helical electrons with pseudospin direction is linked to an axis determined by electronic
momentum.
For conduction band electrons,
and valence band holes,
=
'
'
BK
AK
BK
AK
+
+ =
000
000
000000
yx
yx
yx
yx
ipp
ipp
ippipp
vH
1=n
1=n
vpE =
xpyp
p
p
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Absence of Backscattering
angular scattering probability:
Under pseudospin conservation, helicity suppresses backscattering in monolayer graphene.
( )
==
=
=
+
2/
2/
2
1;0
0
0
0
i
i
i
i
e
evpE
e
evpvH
( ) ( ) ( )2/cos0 22
==
( )0=
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Chapter 3
QUANTUM TRANSPORT
Some of the quantum transport techniques widely used:
(1) Boltzmann Approach
(2) Blonder-Tinkham-Klapwijk (BTK)
(3) Matsubara Greens Function (imaginary-time)
What makes Quantum Super-Field Theory (QSFT) a leading technique for transport?
Straightforward procedure for deriving the equations for the super-correlation functions
that enter the theory (a procedure exactly paralleling that of many-body quantum field theory
at zero temperature).
A real time approach and the relevant quantum transport equation comes out to be
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For x > 0 the pair potential vanishes identically, disregarding any intrinsic
superconductivity of graphene. For x < 0 the superconducting electrode on top of the
graphene layer will induce a nonzero pair potential(x) via the proximity effect [similarly to
what happens in a planar junction between a two-dimensional electron gas and a
superconductor. The bulk value 0ei (with the superconducting phase) is reached at a
distance from the interface which becomes negligibly small if the Fermi wavelength 'F inregion S is much smaller than the value F in region N. We therefore adopt the step-function
model
We assume that the electrostatic potential U in regions N and S may be adjusted
independently by a gate voltage or by doping. Since the zero potential is arbitrary, we may
take
For U0 large positive, and EF 0, the Fermi wave vector k'F 2 / 'F = (EF + U0) / vin
S is large compared to the value kF 2 / F = EF / vin N (with vthe energy-independentvelocity in graphene).
The single-particle Hamiltonian in graphene is the two-dimensional Dirac Hamiltonian,
acting on a four-dimensional spinor (A+ , B+ , A- , B- ). The indices A,B label the two
sublattices of the honeycomb lattice of carbon atoms, while the indices label the two valleys
of the band structure. (There is an additional spin degree of freedom, which plays no role
here.) The 2x2 Pauli matrices i act on the sublattice index.
The time-reversal operator interchanges the valleys,
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with C the operator of complex conjugation. In the absence of a magnetic field, the
Hamiltonian is time-reversal invariant, THT-1 = H. Substitution into (Equation) results in two
decoupled sets of four equations each, of the form
Because of the valley degeneracy it suffices to consider one of these two sets, leading to a
four-dimensional electron spinor then has components (u1, u2) = (A+ , B+), while the hole
spinorv Tu has components (v1, v2) (*A ,
*B). Electron excitations in one valley are
therefore coupled by the superconductor to hole excitations in the other valley. (Both valleys
are needed for superconducting pairing because time-reversal symmetry is broken within a
single valley.)
A plane wave (u, V)exp(ikxx + ikyy) is an eigenstate of the DbdG equation in a uniform
system at energy
with |k| - (k2x + k2y)
1/2. The two branches of the excitation spectrum originate from the
conduction band and the valence band. The dispersion relation is shown (Figure) for the
nomal region (where = 0 = U). In the supERconducting region there is a gap in the spectrum
of magnitude || = 0. The mean-field requirement of superconductivity is that 0 0.
Simple inspection of the excitation spectrum shows the essential physical difference
between these two regimes. Since ky and are conserved upon reflection at the interface x=0,
a general scattering state for x > 0 is a superposition of the four k x values that solve
(Equation) at a given ky and . The derivative -1
d/dk is the expectation value vx of thevelocity in the x-direction, so the reflected state contains only the two kx values is an electron
excitation (v = 0), the other a hole excitation (u = 0).
As illustrated (Figure), the reflected hole may be either an empty state in the
conduction band (for < EF) or an empty state in the valence band ( > EF). A conduction-
band hole moves opposite to its wave vector, so vy changes sign as well as vx
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(retroreflection). A valence-band hole, in contrast, moves in the same direction as its wave
vector, so vy remains unchanged and only vx changes sign (specular reflection). For 0 the
retroreflection dominates if EF >> 0, while specular reflection dominates if EF