MS Proposal

8/6/2019 MS Proposal

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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,


46/47

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


47/47

(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

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