Download Electron transport, interaction and spin in graphene and graphene nanoribbons Artsem Shylau

Linköping Studies in Science and Technology
Dissertation No. 1469
Electron transport, interaction and spin in
graphene and graphene nanoribbons
Artsem Shylau
Department of Science and Technology
Linköping University, SE-601 74 Norrköping, Sweden
Norrköping 2012
Cover illustration: zigzag graphene nanoribbon attached to leads
Electron transport, interaction and spin in
graphene and graphene nanoribbons
© 2012 Artsem Shylau
Department of Science and Technology
Linköping University, SE-601 74 Norrköping, Sweden
ISBN 978-91-7519-816-3
ISSN 0345-7524
Printed by LiU-Tryck, Linköping 2012
To my parents, Valentina and Alexander.
Abstract
Since the isolation of graphene in 2004, this novel material has become
the major object of modern condensed matter physics. Despite of enormous research activity in this field, there are still a number of fundamental
phenomena that remain unexplained and challenge researchers for further
investigations. Moreover, due to its unique electronic properties, graphene
is considered as a promising candidate for future nanoelectronics. Besides
experimental and technological issues, utilizing graphene as a fundamental
block of electronic devices requires development of new theoretical methods for going deep into understanding of current propagation in graphene
constrictions.
This thesis is devoted to the investigation of the effects of electronelectron interactions, spin and different types of disorder on electronic and
transport properties of graphene and graphene nanoribbons.
In paper I we develop an analytical theory for the gate electrostatics
of graphene nanoribbons (GNRs). We calculate the classical and quantum capacitance of the GNRs and compare the results with the exact selfconsistent numerical model which is based on the tight-binding p-orbital
Hamiltonian within the Hartree approximation. It is shown that electronelectron interaction leads to significant modification of the band structure
and accumulation of charges near the boundaries of the GNRs.
It’s well known that in two-dimensional (2D) bilayer graphene a band
gap can be opened by applying a potential difference to its layers. Calculations based on the one-electron model with the Dirac Hamiltonian predict
a linear dependence of the energy gap on the potential difference. In paper
II we calculate the energy gap in the gated bilayer graphene nanoribbons
(bGNRs) taking into account the effect of electron-electron interaction. In
contrast to the 2D bilayer systems the energy gap in the bGNRs depends
non-linearly on the applied gate voltage. Moreover, at some intermediate
gate voltages the energy gap can collapse which is explained by the strong
modification of energy spectrum caused by the electron-electron interactions.
Paper III reports on conductance quantization in grapehene nanoribbons subjected to a perpendicular magnetic field. We adopt the recursive
Green’s function technique to calculate the transmission coefficient which
iv
is then used to compute the conductance according to the Landauer approach. We find that the conductance quantization is suppressed in the
magnetic field. This unexpected behavior results from the interactioninduced modification of the band structure which leads to formation of
the compressible strips in the middle of GNRs. We show the existence of
the counter-propagating states at the same half of the GNRs. The overlap between these states is significant and can lead to the enhancement of
backscattering in realistic (i.e. disordered) GNRs.
Magnetotransport in GNRs in the presence of different types of disorder
is studied in paper IV. In the regime of the lowest Landau level there
are spin polarized states at the Fermi level which propagate in different
directions at the same edge. We show that electron interaction leads to
the pinning of the Fermi level to the lowest Landau level and subsequent
formation of the compressible strips in the middle of the nanoribbon. The
states which populate the compressible strips are not spatially localized
in contrast to the edge states. They are manifested through the increase
of the conductance in the case of the ideal GNRs. However due to their
spatial extension these states are very sensitive to different types of disorder
and do not significantly contribute to conductance of realistic samples with
disorder. In contrast, the edges states are found to be very robust to the
disorder. Our calculations show that the edge states can not be easily
suppressed and survive even in the case of strong spin-flip scattering.
In paper V we study the effect of spatially correlated distribution of impurities on conductivity in 2D graphene sheets. Both short- and long-range
impurities are considered. The bulk conductivity is calculated making use
of the time-dependent real-space Kubo-Greenwood formalism which allows
us to deal with systems consisting of several millions of carbon atoms. Our
findings show that correlations in impurities distribution do not significantly influence the conductivity in contrast to the predictions based on
the Boltzman equation within the first Born approximation.
In paper VI we investigate spin-splitting in graphene in the presence of
charged impurities in the substrate and calculate the effective g-factor. We
perform self-consistent Thomas-Fermi calculations where the spin effects
are included within the Hubbard approximation and show that the effective
g-factor in graphene is enhanced in comparison to its one-electron (noninteracting) value. Our findings are in agreement to the recent experimental
observations.
Populärvetenskaplig
sammanfattning
Ända sedan isoleringen av grafen år 2004, har detta nya material blivit
den viktigaste föremålet för den moderna kondenserade materiens fysik.
Trots enorm forskning inom detta område finns det fortfarande ett antal grundläggande fenomen som förblir oförklarade och utmanar forskare
för vidare undersökningar. Dessutom, på grund av dess unika elektroniska egenskaper, anses grafen vara en lovande kandidat för framtida nanoelektronik. Förutom experimentella och teknologiska frågor, kan grafen
användas som ett grundläggande block av elektroniska komponenter som
kräver utveckling av nya teoretiska metoder för att fördjupa förståelsen av
ström utbredning i nanostrukturer av grafen.
Denna avhandling tillägnar åt utredningen av effekterna av elektronelektron växelverkan, spin och olika typer av oordning inom elektroniska
och transport egenskaper hos grafen och grafen nanoremsor.
vi
Acknowledgments
Despite of my primary background in applied sciences I always wished to
be a theoretical physicist, because I was always impressed by the fact that
human mind is able to unravel Nature secrets just with the use of a piece
of paper and a pencil (or a computer nowadays). This dream got fulfilled
in Sweden where I spent four years as a PhD student at ITN LiU doing my
research in theoretical physics. Tack, Sverige!
I would like to thank a lot of people who surrounded me during this
period.
First of all, I would like to thank Prof. Igor Zozoulenko for his great
supervision, significant contribution to my scientific development and for
motivation when it was really needed.
I am thankful to the research administrator Ann-Christin Norén and
Elisabeth Andersson for their administrative help during my research work.
I had a fruitful collaboration with my colleagues from Germany, Dr.
Hengyi Xu and Prof. Thomas Heinzel, and my polish colleague Dr. Jaroslaw
Klos.
Besides the science there were a lot of other things happening in my life.
I met a lot of nice people here, some of them eventually became my friends.
First of all, I am grateful to Olga Bubnova for our inspiring philosophical
discussions, pleasant lunch time and just for being a good friend. I am
also thankful to Loı̈g, Anton, Brice, Sergei, Julia and especially Taras for
a good company and funny talks.
All this time I kept in touch with my belarusian friends, Sergei and
Alex. We had a good time together whenever I went home, to my lovely
Minsk.
Finally, I would like to thank my wife Marina for her love, understanding
and patience; my parents and my sister’s family for their love and support
which I feel every day no matter where I am.
Artsem Shylau
Norrköping, August 2012
viii
List of publications
Publications included in the thesis
1. A. A. Shylau, J. W. Klos, and I. V. Zozoulenko, Capacitance of
graphene nanoribbons, Phys. Rev. B 80, 205402 (2009).
Author’s contribution: Implementation of the self-consistent numerical model and all numerical calculations, partial contribution to
development of the analytical model. Preparation of all the figures.
Initial draft of the paper.
2. Hengyi Xu, T. Heinzel, A. A. Shylau, I. V. Zozoulenko, Interactions
and screening in gated bilayer graphene nanoribbons, Phys. Rev. B
82, 115311 (2010); selected as ”Editor’s Suggestion”.
Author’s contribution: Development of the analytical model. Discussion and analysis of all the obtained results.
3. A. A. Shylau, I. V. Zozoulenko, Hengyi Xu, T. Heinzel, Generic
suppression of conductance quantization of interacting electrons in
graphene nanoribbons in a perpendicular magnetic field, Phys. Rev.
B 82, 121410(R) (2010).
Author’s contribution: All numerical calculations and preparation
of all the figures. Discussion of the results and writing an initial draft.
4. A. A. Shylau and I. V. Zozoulenko, Interacting electrons in graphene
nanoribbons in the lowest Landau level, Phys. Rev. B 84, 075407
(2011).
Author’s contribution: All numerical calculations and preparation
of all the figures, discussion of the results and writing an initial draft.
5. T. M. Radchenko, A. A. Shylau, and I. V. Zozoulenko, Influence
of correlated impurities on conductivity of graphene sheets: Timedependent real-space Kubo approach, Phys. Rev. B 86, 035418 (2012).
Author’s contribution: Contribution to implementation of the numerical model, performing initial numerical calculations, discussion
of all results.
x
6. A. V. Volkov, A. A. Shylau, and I. V. Zozoulenko, Interactioninduced enhancement of g-factor in graphene, arXiv:1208.0522v1 [condmat.mes-hall], submitted to PRB.
Author’s contribution: Contribution to development of the model,
discussion of all results, writing a part of the manuscript, supervision
of the numerical calculations.
Relevant publications not included in the thesis
1. J. W. Klos, A. A. Shylau, I. V. Zozoulenko, Hengyi Xu, T. Heinzel,
Transition from ballistic to diffusive behavior of graphene ribbons in
the presence of warping and charged impurities, Phys. Rev. B 80,
245432 (2009).
2. T. Andrijauskas, A. A. Shylau, I. V. Zozoulenko, Thomas-Fermi
and Poisson modeling of the gate electrostatics in graphene nanoribbon, Lith. J. of Phys. 52, 63 (2012).
Contents
1 Introduction
2 Quantum transport
2.1 Kubo formalism . . . . . .
2.2 Kubo-Greenwood formula
2.3 Landauer approach . . . .
2.4 S-matrix technique . . . .
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3 Electron-electron interactions
3.1 The many body problem . . . . . .
3.2 Hartree-Fock approximation . . . .
3.3 Density-functional theory . . . . . .
3.4 Kohn-Sham equations . . . . . . .
3.5 Thomas-Fermi-Dirac approximation
3.6 Hubbard model . . . . . . . . . . .
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graphene
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5 Modeling
5.1 Tigh-binding Hamiltonian and Green’s function
5.2 Recursive Green’s function technique . . . . . .
5.2.1 Dyson equation . . . . . . . . . . . . . .
5.2.2 Bloch states . . . . . . . . . . . . . . . .
5.2.3 Calculation of the Bloch states velocity .
5.2.4 Surface Green’s function . . . . . . . . .
5.2.5 Transmission and reflection . . . . . . .
5.3 Real-space Kubo method . . . . . . . . . . . . .
5.3.1 Diffusion coefficient . . . . . . . . . . . .
5.3.2 Transport regimes . . . . . . . . . . . . .
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4 Electronic structure and transport
4.1 Basic electronic properties . . . .
4.2 Graphene nanoribbons . . . . . .
4.3 Warping . . . . . . . . . . . . . .
4.4 Bilayer graphene . . . . . . . . .
4.5 Dirac fermions in a magnetic field
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xii
Contents
5.3.3
5.3.4
5.3.5
5.3.6
5.3.7
6 Summary
6.1 Paper
6.2 Paper
6.3 Paper
6.4 Paper
6.5 Paper
6.6 Paper
Time evolution . . . . . . . . . . . .
Chebyshev method . . . . . . . . . .
Continued fraction technique . . . . .
Tridiagonalization of the Hamiltonian
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of the papers
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II . . . . . . .
III . . . . . . .
IV . . . . . . .
V . . . . . . .
VI . . . . . . .
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Chapter 1
Introduction
Inside every pencil, there is a neutron star waiting to get out.
To release it, just draw a line.
(New Scientist, 2006)
In 2004 the researchers from Manchester University, Kostya Novoselov,
Andre Geim and collaborators, reported on experimental isolation of graphene
[1], a pure 2D crystal consisting of carbon atoms arranged in a honey-comb
lattice. For a long time before graphene had been considered only by theoreticians as a basic block used to build theory for graphite [2] and carbon
nanotubes [3]. Its existence was doubt since the theory predicted that
perfect 2D crystals are not thermodynamically stable [4]. The discovery
of graphene triggered a great scientific interest in this field, as a result
graphene is one of the most extensively studied object in a modern condensed matter physics [5].
Due to its specific lattice structure graphene posses a number of unique
electronic properties which make this material interesting for both theoreticians, experimentalists and engineers. One of the most important feature
of graphene is its linear energy spectrum. This kind of spectrum is known
from high-energy physics where it corresponds to massless particles like
neutrino. Relativistic-like dispersion relation is responsible for such effects
as Klein tunneling - unimpeded penetration of particle through the infinitely large potential barrier [6]. The experimental discovery of graphene
had led to emergence of a new paradigm of ’relativistic’ condensed-matter
physics and provided a way to probe quantum electrodynamics phenomena
[4]. That is why graphene is sometimes called ”CERN on a desk”.
Being massless fermions electrons in graphene propagate at extremely
high velocities, only 300 times smaller than the velocity of light. This
makes graphene the best known conductor with mobility up to 200,000
cm2 V−1 s−1 at room temperatures. Graphene subjected to a perpendicular
magnetic field exhibits the anomalous quantum Hall effect [7]. This is a result of unusual spectrum quantization and the presence of the 0’th Landau
2
Introduction
level, which is equally shared between electrons and holes. Also fractional
quantum Hall effect has been experimentally observed in graphene [8].
The range of possible applications of graphene is very broad. With its
high mobility graphene is considered as the main candidate for a future
post-silicon electronics [9]. It is particularly interesting to use graphene in
transistors operating at ultrahigh radio frequencies [10]. Due to its high
optical transmittance (≈ 97.7%) graphene is proposed to be used as a flexible transparent electrode in touchscreen devices [11]. Also graphene posses
a broad spectral bandwidth and fast responce times, which makes this
material attractive for optoelectronics and, in particular, phototransistors
[12].
Chapter 2
Quantum transport
Electrical transport is a non-equilibrium statistical problem. In principle,
one can solve the time-dependent Schrödinger equation
i~
∂|Ψ(~r, t)i
= Ĥ|Ψ(~r, t)i
∂t
(2.1)
to find a many-body state of the system |Ψ(~r, t)i at any time and then
calculate the expectation value of the current operator
Iˆ =
Z
S
~
ĵ(~r, t) · dS.
(2.2)
In practice, however, the calculation of the many-body state is an unfeasible task. Moreover, the wave function representing the state provides a
detailed information about the system which is often redundant for determination of transport properties. Hence, one needs to introduce a number
of approximations in order to simplify the above problem. Prior doing this
it is useful to formulate viewpoints underlying quantum transport theories
[13]:
Viewpoint 1: The electrical current is a consequence of an applied
electric field: the field is the cause, the current is the response to this field.
Viewpoint 2: The electrical current is determined by the boundary
conditions at the surface of the sample. Charge carriers incident to the sample boundaries generate self-consistently an inhomogeneous electric field
across the sample. Thus the field is a consequence of the current.
In this chapter two approaches for calculation of transport properties
are discussed, namely, the Kubo formalism and the Landauer approach.
The first one belongs to the viewpoint 1, while the latter belongs to the
viewpoint 2.
4
2.1
Quantum transport
Kubo formalism
Kubo formula relates via linear response the conductivity to the equilibrium
properties of the system. First we define the expectation value of the
current density operator
h i
h~ji = Tr ~jρ ,
(2.3)
where ρ is a statistical operator
− k HT
ρ = Z −1 e
B
h H i
−
, Z = Tr e kB T .
(2.4)
The Hamiltonian, H, describing the system can be split into two parts
H = H0 + δH,
ρ = ρ0 + δρ,
(2.5)
where H0 corresponds to the system in a global cannonical equilibrium
described by the statistical operator ρ0 . External electric field introduces
perturbation in the system which is described by the terms δH and δρ.
Here we focus on the case of uniform electric field [14]. It’s assumed that
the field is applied at time t = −∞ and reaches adiabatically its steady
value at t = 0
h
i
~ re(−iωt+αt) .
δH = lim eE~
(2.6)
α→0
Substituting ρ from Eq.(2.5) into Eq.(2.3), we have
h
i
h i
h~ji = Tr ~j(ρ0 + δρ) = Tr ~jδρ ,
(2.7)
h i
where we took into account that Tr ~jρ0 = 0, since there is no current
in the system before applying the external field. If we substitute ρ given
by Eq.(2.5) into the Liouville equation, i~ρ̇ = [H, ρ], which describes the
dynamics of a closed quantum systems, then the change of δρ in time can
be found as
i~δ ρ̇ = [H0 , δρ] + [δH, ρ0 ],
(2.8)
where we used that i~ρ̇0 = [H0 , ρ0 ] and neglected the term [δH, δρ]. Let
us now pass to the interaction picture representation of time-dependence
of an operator
iH0 t
iH0 t
δρ = e− ~ ∆ρe ~ .
(2.9)
Substituting Eq.(2.9) into the left part of Eq.(2.8), we arrive at
iH0 t
~
iH0 t
− ~
[δH,
h ρ0 ]e
i
iH0 t
iH0 t
~ .
= limα→0 e(−iωt+αt) e ~ [e~r, ρ0 ]e− ~ E
i~∆ρ = e
(2.10)
Kubo formalism
5
Both δρ and ∆ρ satisfy the following conditions: have the same value at
time t = 0, δρ(0) = ∆ρ(0), and equal to zero at t = −∞, δρ(−∞) =
∆ρ(−∞) = 0. Thus, integrating Eq.(2.10), we get
Z 0
iH0 t
iH0 t
1
~
δρ(t = 0) =
lim
dte(−iωt+αt) e ~ [e~r, ρ0 ]e− ~ E.
(2.11)
i~ α→0 −∞
Substituting Eq.(2.11) into the expression (2.7) for the expectation value
of the current density, we obtain
Z 0
h i
i
h iH0 t
iH0 t
1
~ .
h~ji = Tr ~jδρ =
lim
dte(−iωt+αt) Tr ~je ~ [e~r, ρ0 ]e− ~ E
i~ α→0 −∞
(2.12)
The conductivity tensor is defined as
jx
σxx σxy
Ex
=
,
(2.13)
jy
σyx σyy
Ey
which allows to deduce from Eq.(2.12) the components of the tensor
Z 0
σµν = lim
dte(−iωt+αt) Kµν ,
(2.14)
α→0
−∞
where
i
iH0 t
1 h ~ iH0 t
Tr jµ e ~ [erν , ρ0 ]e− ~ .
(2.15)
i~
Equation (2.15) can be written in another form if we use the following
relation [14]:
Kµν =
[rν , ρ0 ] = ρ0
Z
1/kB T
dλeλH0 [H0 , rν ]e−λH0 .
(2.16)
0
Taking into account that [H0 , rν ] = −i~ṙν and −eṙν = jν , we get
[erν , ρ0 ] = i~ρ0
Z
1/kB T
dλeλH0 jν e−λH0 .
(2.17)
0
Finally, using Heisenberg representation for time-dependence of the current
~j(t) = e
iH0 t
~
~je−
iH0 t
~
,
(2.18)
we obtain the Kubo formula for conductivity which is formulated in terms
of the current-current response function
Z 1/kB T
Kµν =
dλhjµ (0)jν (t − i~λ)i.
(2.19)
0
6
2.2
Quantum transport
Kubo-Greenwood formula
Equations (2.14) and (2.19), which constitute the Kubo formula for conductivity are very important, since they reflect the underlying physics of
the linear responce theory. However, these equations are not suitable for
a particle calculations and one needs to work out another form of the conductivity formula. As in the previous section we start with splitting the
Hamiltonian, H, into two parts H = H0 + δH corresponding to the system in the equilibrium and the perturbation respectively. Hence, if the full
Hamiltonian is defined as
1
~ 2 + eφ
H=
(~p + eA)
(2.20)
2m
~=A
~0 + A
~ ext ,
and the perturbation is incorporated in the vector potential, A
we get for δH:
~ ext · ~v .
δH = eA
(2.21)
~
If we assume the time dependence of the external electric field to be E(t)
=
~ −iωt , then using the relation E
~ = − dA~ , we arrive at
Ee
dt
e ~
δH = E
· ~v.
iω
(2.22)
h i
The expectation value of the current operator, hji = Tr ρĵ , can be written
as
Z Z
X
′
′
2
hji =
hk|δρ|k ihk |ĵ|ki = V
dEdE ′ g(E)g(E ′)hE|δρ|E ′ihE ′ |ĵ|Ei,
kk ′
(2.23)
where g(E) is a density of states and we used Tr[ρ0 ĵ] = 0. If we assume
δρ(t) = δρ · eiωt and use Eq.(2.8), then we get
fF D (E ′ ) − fF D (E) ′
hE |δH|Ei,
(2.24)
E ′ − E − ~ω − i~α
where α is a small constant and we used that ρ0 |Ei = fF D (E)|Ei [14]. Substituting Eq.(2.24) into Eq.(2.23) and recalling that ~j = − Ve ~v , we obtain
the expectation value of the current operator
Z Z
e e −
hji = V 2
dEdE ′ g(E)g(E ′)
iω
V
fF D (E ′ ) − fF D (E) ′ ~
×
hE |E · ~v |EihE|~v|E ′ i.
(2.25)
E ′ − E − ~ω − i~α
This equation allows us to derive the components of the conductivity tensor
2 Z Z
e
dEdE ′ g(E)g(E ′)
σij (ω) = Re − V
iω
fF D (E ′ ) − fF D (E) ′
′
×
hE |~vi |EihE|~vj |E i ,
(2.26)
E ′ − E − ~ω − i~α
hE ′ |δρ|Ei =
Landauer approach
7
which can be further simplified using the relation
1
1
Re lim
= πδ(E ′ − E − ~ω)
α→0 i E ′ − E − ~ω − i~α
(2.27)
and performing integration over E ′
Z
e2 πV
σij (ω) = −
g(E)g(E + ~ω)hE + ~ω|~vi |EihE|~vj |E + ~ωi
ω
× [fF D (E + ~ω) − fF D (E)]dE.
(2.28)
If one is interested in DC conductivity, Eq.(2.28) should be considered in
D (E)
F D (E)
a limit ω → 0, limω→0 fF D (E+~ω)−f
= ∂fF∂E
,
~ω
σij (ω) = e2 πV ~
Z
∂fF D (E)
|g(E)|2hE|~vi |EihE|~vj |Ei −
dE.
∂E
(2.29)
Finally, if we consider the case of a very low temperature T → 0, the derivative of the Fermi-Dirac function can be substituted by the delta function
∂fF D (E)
= δ(E − EF ), which simplifies the integration over E. Thus, re∂E
calling that g(E) = Tr[E − H], we obtain the relation for the conductivity
tensor
e2 π~
Tr [vi δ(EF − H)vj δ(EF − H)] .
(2.30)
σij =
V
This equation is a starting point of the numerical real-space time-dependent
Kubo method which will be described in details in the Chapter 5.
2.3
Landauer approach
Landauer approach belongs to the viewpoint 2, i.e. a constant current
is forced to flow through a scattering system and the asking question is
what the resulting potential distribution will be due to the spatially inhomogeneous distribution of scatters [15]. Calculation of the current in the
Landauer approach requires to divide formally the system into three parts,
namely, the perfect leads and the scattering region, as depicted on Fig.(2.1).
The leads, in turn, are connected to the infinite reservoirs which represent
the infinity and contain many electrons in a local equilibrium characterized by the Fermi-Dirac distribution function. The basic idea behind this
approach is that the electron has a certain probability to transmit through
the scattering region [16]. Hence, the current carrying by an electron in a
state with a wave-vector k is
Jk = ev(k)T (k)
(2.31)
8
Quantum transport
a)
Reservoir
Sample
(scattering region)
Left
lead
Right
lead
Reservoir
b)
0
L
Figure 2.1: a) Schematic illustration of the system which consists of the
scattering region connected to perfect leads. The leads itself are coupled to
microscopic reservoirs. b) Potential profile. The left and right leads have a
constant potential µL and µR , respectively, equal to the chemical potential
in the reservoirs.
with T (k) being a transmission probability. Full current supplied by the
left lead is a sum over all states
X
IL = 2
Jk fF D (E(k), µL ),
(2.32)
k
where the factor 2 is due to spin-degeneracy, µL is a chemical potential in
the left lead and
fF D (E(k), µ) =
1 + exp
1
E(k)−µ
kB T
(2.33)
is the Fermi-Dirac distribution function. Hence, we have
"
#
Z
X
X
1
IL = 2e
v(k)T (k)fF D (E(k), µL ) =
→
dk (2.34)
2π
k
k
Z
2e ∞
=
v(k)T (k)fF D (E(k), µL )dk.
(2.35)
2π 0
dk
Changing integration variables by dk = dE
dE and using the expression for
1 dE
the group velocity v = ~ dk , we get
Z
2e ∞
IL =
T (E)fF D (E, µL )dE.
(2.36)
h UL
Similarly, the current supplied by the right lead is
Z
2e ∞
IR = −
T (E)fF D (E, µR )dE.
h UR
(2.37)
S-matrix technique
9
Sum of both contributions gives the net current
Z
2e ∞
I = IL + IR =
T (E) [fF D (E, µL ) − fF D (E, µR )] dE.
h UL
(2.38)
The potential drop between the reservoirs is
eV = µL − µR .
(2.39)
In the case of very low bias, the Fermi-Dirac functions can be expanded in
the Taylor series
fF D (E, µL ) − fF D (E, µR ) ≈ −eV
∂fF D (E, µ)
,
∂E
(2.40)
which results in
I=
2e
h
Z
∞
UL
∂fF D (E, µ)
T (E) −eV
dE.
∂E
This equation allows to calculate conductance of the system
Z
I
2e2 ∞
∂fF D (E, µ)
G=
=
T (E) −
dE.
V
h UL
∂E
(2.41)
(2.42)
At a very low temperature the derivative of the Fermi-Dirac distribution
function can be replaced by the Dirac delta function δ(E −µ) which reduces
Eq.(2.42) to
2e2
T (µ).
(2.43)
G=
h
This equation shows that the conductance of the perfect conductor (i.e.
T = 1) is finite and thus the resistance (G−1 ) is non-zero. The following
explanation can be used [16]: in the contacts (reservoirs) the current is
carried by infinitely many transverse modes, however inside the conductor
only few modes supply the current. It leads to redistribution of the current
among current-carrying modes which results in the interface resistance.
2.4
S-matrix technique
As it was shown in the previous section, the current (or conductance) can be
formulated in terms of the transmission function T . The powerful method
to calculate T is the scattering matrix technique. Scattering matrix (or Smatrix) relates the outgoing amplitudes b = (b1 , b2 , ..., bn ) to the incident
amplitudes a = (a1 , a2 , ..., an ), see Fig.(2.1),
′
r(E) t (E)
.
(2.44)
b = S(E)a, S(E) =
′
t(E) r (E)
10
Quantum transport
S matrix has a 2N · 2N dimension, where N is a number of transmission
channels. The transmission probability now equals to
tm←n (E) = |Smn |2
(2.45)
Prior to calculation of the S-matrix, one can determine its general properties. S-matrix must be unitary, that
P is a2 consequence of current conservation:
the
incoming
electron
flux
n |a| must be equal to the outgoing
P
flux n |b|2
b+ b = a+ a,
a (1 − S + S)a = 0,
S + S = I.
+
(2.46)
Moreover S-matrix is also a symmetric matrix, S = S T . This fact reflects
the time-reversal symmetry of the Schrödinger equation, H = H ∗ . A nonzero magnetic field breaks the time-reversal symmetry. In this case, we
have SB~ = S−T B~ .
S-matrix and Green’s function S-matrix can be expressed in terms of
Green’s function. Outside the scattering region solution of the Schrödinger
equation has the form of plane waves ψn (~r) ≈ eikn z , where we assume that
the system with a cross-section area A is uniform in x and y directions, ~r =
(~ρ, z). Each plane wave corresponds to a scattering channel n characterized
by a transverse momenta ~qn and longitudinal momenta kn with the energy
E = (1/2m)(kn2 + qn2 ). If we define the Green’s function G(E) = (E + iη −
H)−1 with matrix elements between scattering channels m and n as
Z
Z
Gmn (z, z ′ , E) = A−1 d~ρ dρ~′ exp (−i~qm ρ~) exp (−i~qn ρ~′ )h~r|G(E)|r~′i.
(2.47)
Then, the transmission coefficient, following Fisher and Lee [17], can be
calculated as
√
tmn = −i~ vm vn Gmn (z, z ′ , E) exp [−i(km z − kn z ′ )],
(2.48)
where z and z ′ are taken outside the scattering region, i.e. z > L and
z ′ < 0, see Fig.(2.1), and vn = kn /m is the velocity in channel n. This
relation is very important, since it shows the connection between different
transport formalisms.
Chapter 3
Electron-electron interactions
With advent of quantum mechanics, the physical laws which govern particles motion and interactions between particles became known. However
the exact analytical solution is possible only for a system consisting of two
particles. A typical piece of solid consists of approximately 1023 particles.
Even if it would be possible to write down all the differential equations
required to describe this system, the solution of these equation is an unfeasible task in principle. The problem of finding the solution arises from
electron-electron interaction which makes the motion of particles correlated
and couples corresponding differential equations. Therefore it is of great
importance to develop approximated methods which provide a simplified
form of the electron-electron interaction and reduces the number of equations needed to be solved.
3.1
The many body problem
The Hamiltonian of a many-body system of interacting particle is written
as
Ĥ = T̂e + T̂n + V̂e−e + V̂e−n + V̂n−n
X ~2
X ~2
1 X e2
∇2i −
∇2I +
= −
2me
2MI
4πεε0 i>j |~ri − ~rj |
i
I
+
1 X ZI ZJ e2
1 X ZI e2
−
,
~I − R
~ J | 4πεε0
~I|
4πεε0 I>J |R
ri − R
i,I |~
(3.1)
where the first two terms describe the kinetic energy of electrons and nuclei.
The last three terms result from the Coulomb interaction between electrons,
electron-nuclei and nuclei-nuclei respectively. The Hamiltonian acts on the
~ I }) which depends on the position of all electrons
wave-function Ψ({~ri }, {R
and nuclei in the system
~ I }) = EΨ({~ri }, {R
~ I }).
ĤΨ({~ri }, {R
(3.2)
12
Electron-electron interactions
An essential simplification of Eq.(3.2) can be done with the use of the BornOppenheimer approximation which neglects the coupling between the nuclei and electronic motion. In thermodynamic equilibrium electrons move
much faster than nuclei, since M ≫ me , which allows to treat nuclei as
stationary particles and neglect the kinetic term T̂n in the Hamiltonian.
Hence one can deal only with the electronic part, Ĥe , of the full Hamiltonian which corresponds to the system of interacting electrons moving in
the effective potential produced by nuclei
Ĥe = T̂e + V̂e−e + V̂e−n + V̂n−n ,
Ĥe Φ({~ri }) = EΦ({~ri }).
(3.3)
(3.4)
Even though electronic wave-function Φ({~ri }) still depends on the positions
~ I are just parameters of Eq.(3.3) and the number of differential
of nuclei, R
equations needed to be solved is greatly reduced.
3.2
Hartree-Fock approximation
The Born-Oppenheimer approximation significantly simplifies the problem
of interacting particles by eliminating the coupling between electrons and
nuclear motion. However determination of the exact solution of the manyparticle electronic wave-function is still not feasible. The basic idea of
the Hartree-Fock approximation is to substitute the system of interacting
electrons by the motion of single electrons in the average self-consistent
field generated by all the other electrons in the system.
The Hamiltonian of the many-particle system is given by
X p2 X
X
X
e2 X
1
Ĥ =
+
V (~rk )+
=
Ĥk +
Ĥkk′ , (3.5)
2me
8πεε0 ′ |~rk − ~rk′ |
′
k
k
k
kk
kk
P
~ I ) describes interaction between kwhere the term V (~rk ) = I V (~rk − R
~ I }. The operator Ĥk is a
electron with all nuclei in the system located at {R
′
one-particle operator, while Ĥkk depends on the position of two particles.
The simplest way to construct the many-particle wave-function is to
write down it in the form of a product of single-particle wave-functions,
φk (~r), which have to be determined,
Y
Φ({~rk }) =
φ(~rk ), hφi (~r)|φj (~r)i = δij .
(3.6)
k
Let us calculate an expectation value of the energy [14]
X
1 e2 X
E = hΦ|Ĥ|Φi =
hφk |Ĥk |φk i +
φk φk ′ φk φk ′ .
8πεε0 ′
~rk − ~rk′ k
kk
(3.7)
Hartree-Fock approximation
13
According to the variational principle the closer values of φk to the exact
solution the smaller value of the energy, thus
!
X
δ E−
Ek (hφk |φk i − 1) = 0,
(3.8)
k
where Ek are Lagrange parameters. Changing φi → φi + δφi and keeping
terms linear in respect to δφi , we arrive at the Hartree equation
"
#
Z
e2 X |φk (~rk )|2
~2
−
∆ + V (~r) +
d~rk φi (~r) = Ei φi (~r). (3.9)
2me
4πεε0 k6=i
|~rk − ~ri |
The third term in the brackets
P has a simple interpretation. If we define the
charge density as n(~r) = e i |φi (~r)|2 , then the term
Z
e
n(r~′ ) ~′
UH (~r) =
dr ,
(3.10)
~
4πεε0
|r ′ − ~r|
called the Hartree term, describes the Coulomb interaction between the
i-th electron located at ~r with all the other electrons in the system.
Since electrons are fermions, the many-particle wave-function must change
the sign under the interchange of the coordinates of any two particles. The
wave-function given by Eq.(3.6) does not satisfy this condition. In order to
construct an antisymmetric wave-function one can use Slater determinant
φ1 (~q1 ) . . . φN (~q1 ) 1 ..
..
..
Φ({~qk }) = √ (3.11)
,
.
.
.
N! φ1 (~qN ) . . . φN (~qN )
where q~i = {~ri , σi } denotes both position and spin of the electron and the
factor √1N ! is used for normalization. Following the same way as before, i.e.
applying the variational principle to the expectation value of the energy,
the new form of the wave-function results in the equation [14]
Z
~2
e2 X |φk (r~′)|2 ~′
−
∆ + V (~r) φi (~r) +
dr φi (~r)
2me
4πεε0
|r~′ − ~r|
k6=i
Z
e2 X φ∗k (r~′ )φi (r~′ ) ~′
−
dr · φk (~r) = Ei φi (~r),
(3.12)
4πεε0 k6=i
|r~′ − ~r|
called the Hartree-Fock equation. The additional term in Eq.(3.12) is
known as the exchange interaction. It does not have classical analog and
results from the Pauli exclusion principle. The exchange interaction term
which arises in the Hartree-Fock approximation has a non-local form in
contrast to the Coulomb interaction. This makes calculations more complicated. In the density-functional theory (discussed in Sec.(3.3)) a number
of approximations are used to deduce a local form of the exchange interaction.
14
3.3
Electron-electron interactions
Density-functional theory
Density-functional theory (DFT) is one of the most widely used modeling method applied in physics and chemistry for calculation of electronic
properties of complex systems. The basic idea behind DFT is to describe
the system in terms of the electronic density instead of operating with a
many-body wave function [18].
Hohenberg-Kohn theorems
In 1964 Hohenberg and Kohn proved two theorems which made the DFT
possible [19]. They state that a knowledge of the ground-state density
can, in principle, determine all the ground-state properties of a many-body
system [18].
Theorem 1: An external potential Vext (~r) uniquely determines the electronic density for any system of interacting particles.
Proof: Assume that the same electron density n(~r) results from two po1
2
tentials Vext
(~r) and Vext
(~r) differing by more than constant. Obviously,
1
2
Vext (~r) and Vext (~r) belong to distinct Hamiltonians Ĥ 1 (~r) and Ĥ 2 (~r) which
produce different wave-functions Ψ1 (~r) and Ψ2 (~r). The ground-state state
energy associated with the Hamiltonian Ĥ 1 (~r) is
E 1 = hΨ1 |Ĥ 1 |Ψ1 i.
(3.13)
According to the variational principle no other wave-function can give lower
energy, i.e.
E 1 = hΨ1 |Ĥ 1|Ψ1 i < hΨ2 |Ĥ 1 |Ψ2 i
(3.14)
Since the Hamiltonians differs by the external potentials only, we can write
1
2
Ĥ 1 = Ĥ 2 + Vext
− Vext
, which gives us for the expectation value
Z
1
2
hΨ2 |Ĥ 1 |Ψ2 i = hΨ2 |Ĥ 2|Ψ2 i +
Vext − Vext
n(~r)d~r.
(3.15)
Substituting it into Eq.(3.14) and recalling that E 2 = hΨ2 |Ĥ 2|Ψ2 i, we
obtain
Z
1
2
E1 < E2 +
Vext − Vext
n(~r)d~r.
(3.16)
Interchanging labels (1) and (2), we find in the same way that
Z
2
1
E2 < E1 +
Vext − Vext
n(~r)d~r.
(3.17)
Addition of Eq.(3.16) and Eq.(3.17) leads to contradiction
E1 + E2 < E1 + E2.
Hence the theorem is proved by reductio ad absurdum.
(3.18)
Kohn-Sham equations
15
Theorem 2: The exact ground-state density n(~r) is the global minimum
of the universal functional F [n].
Proof: Since the electron density n(~r) uniquely determines wave-function
Ψ, the universal functional F [n] can be defined as
F [n] = hΨ|T̂ + Ûe−e |Ψi.
(3.19)
For a given external potential Vext (~r), the energy functional is written as
Z
E[n] = F [n] + Vext (~r)n(~r)d~r.
(3.20)
According to the variational principle, it has a minimum only for the
ground-state wave-function Ψ. For any other wave-function Ψ′ which produces density n′ (~r), we get
Z
Z
E[Ψ] = F [n]+ Vext (~r)n(~r)d~r < F [n′ ]+ Vext (~r)n′ (~r)d~r = E[Ψ′ ]. (3.21)
3.4
Kohn-Sham equations
In the Kohn-Sham method [20] one considers a system of non-interacting
electrons moving in some effective potential vef f (~r) (which will be defined
later)
~2 2
−
∇ + vef f (~r) φi (~r) = ǫi φi (~r).
(3.22)
2m
An obtained set of the Kohn-Sham orbitals φi determines the electron
density
X
n(~r) =
|φi (~r)|2 .
(3.23)
i
The energy functional equals to
E[n] = TS [n] + Vef f [n] = TS [n] +
Z
n(~r)vext (~r)d~r + VH [n] + Exc [n], (3.24)
where the first term,
Ts [n] =
N Z
X
i=1
~2 2
φ∗i (~r) −
∇ φi (~r)d~r,
2m
(3.25)
is a single-electron kinetic energy functional. The second term describes the
potential energy acquired by the charged particles in an external electric
field. The Hartree term is given by
Z Z
e2
n(~r)n(r~′ )
VH =
d~rdr~′ .
(3.26)
8πεε0
|~r − r~′ |
16
Electron-electron interactions
The last term, Exc [n], arises from the exchange-correlation interaction. It
can be shown [21] that the functional given by Eq.(3.24) corresponds to
the the effective potential in the form
Z
e
n(r~′ ) ~′
vef f (~r) = vext (~r) +
dr + vxc (~r).
(3.27)
4πεε0
|~r − r~′ |
Equations (3.22), (3.23) and (3.27) constitute the basis of the Kohn-Sham
method, and are solved self-consistently for the ground state density and
the effective potential.
The explicit form of the exchange-correlation potential vxc can be determined using other approximations. The most widely used are the local
spin density approximation [21] and the generalized gradient approximation
[22].
3.5
Thomas-Fermi-Dirac approximation
According to the Hohenberg-Kohn theorems discussed in Sec.(3.3), the
total energy of the system may be written as
Z
Z
E[n(~r)] = T [n(~r)]d~r + V (~r)n(~r)d~r,
(3.28)
where T [n(~r)] is a kinetic energy functional of the electron density n(~r) and
V (~r) is an external potential. The Thomas-Fermi approximation assumes
that the kinetic-energy functional is a local function of the density. This
assumption allows to rewrite Eq.(3.28) in the form
µ = T [n(~r)] + V (~r).
(3.29)
Let us now derive the relation between the potential and charge density
in graphene. Taking into account dispersion relation for graphene, E =
R d~k
6
±~vF |~k|, and using n = gv gs (2π)
m/s and gv = gs = 2, see
2 (vF = 10
Chapter 4 for details), we get [28]
p
(3.30)
sgn[n(~r)]~vF πn(~r) + V (~r) = µ.
An alternative way is to rewrite Eq.(3.28) in the form which directly relates
electron density to the external potential [23, 24]
Z
n(~r) = dEρ(E − V (~r))fF D (E, µ),
(3.31)
where ρ(E) is a single-electron density of states, calculated in the presence
of homogeneous potential. Figure (3.1) illustrates application of Eq.(3.31).
Hubbard model
17
Figure 3.1: Schematic illustration of Thomas-Fermi model.
Locally the dispersion relation corresponding to the ideal system (with
homogeneous external potential) is preserved. Filling up the states lying
between the charge neutrality point and the Fermi energy level one obtains
local electron density.
Even though the Thomas-Fermi model misses quantum mechanical effects (e.g. quantization), it produces quantitatively similar results in comparison to more rigorous models and widely used in graphene physics
[25, 26, 27, 28].
3.6
Hubbard model
The Hubbard model, originally proposed by John Hubbard in 1963 [29], is
the simplest model of interacting particles in a lattice. The interaction is
assumed to take place only between particles located at the same site (or
atom). Despite of its simplicity rigorous analytical solution is found only
for a one-dimensional problem [30].
The Hubbard Hamiltonian in the tight-binding approximation consists
of two terms, namely the kinetic energy term and the on-site potential
X
X
Ĥ = −t
(a+
ni↑ ni↓ ,
(3.32)
i,σ aj,σ + h.c.) + U
hi,ji,σ
i
where ni,σ = a+
i,σ ai,σ is the occupation number operator. The parameter
U = const describes the strength of on-site Coloumb interaction and can
be determined using ab-initio calculations or extracted from experimental
data.
Equation (3.32) can be rewritten within the mean-field approach using
substitutions
ni↑ = hni↑ i + (ni↑ − hni↑ i),
ni↓ = hni↓ i + (ni↓ − hni↓ i),
(3.33)
(3.34)
18
Electron-electron interactions
where hniσ i denotes average occupation of spin σ at site i. Hence, we have
ni↑ ni↓ = ni↑ hni↓ i + ni↓ hni↑ i − hni↓ ihni↑ i + (ni↑ − hni↑ i)(ni↓ − hni↓ i), (3.35)
|
{z
}
≈0
where the product of two deviations from the average values is assumed
to be small. Substituting the result of Eq.(3.35) into Eq.(3.32) we derive
Hubbard Hamiltonian in the mean-field approximation
X
X
Ĥ M F = −t
(a+
(ni↑ hni↓ i + ni↓ hni↑ i − hni↓ ihni↑ i) .
i,σ aj,σ + h.c.) + U
hi,ji,σ
i
(3.36)
Despite of its simplicity the Hubbard model was applied to investigate the
properties of different materials. It reproduces a variety of phenomena
observed in solid state physics, such as ferromagnetism, metal-insulator
transition and superconductivity.
Chapter 4
Electronic structure and
transport in graphene
4.1
Basic electronic properties
y
B
a)
b)
A
x
Figure 4.1: a) Graphene lattice consisting of two interpenetrating triangular sublattices A (red circles) and B (blue circles) with unit vectors ~a1 , ~a2
and nearest-neighbours vectors ~δ1 , ~δ2 , ~δ3 . The yellow parallelogram marks
unit cell containing two atoms. b) The structure of reciprocal lattice defined by unit vectors ~b1 and ~b2 . The grey hexagon is the first Brillouin
zone.
Real space and reciprocal lattices Carbon atoms in graphene are
arranged in a honeycomb lattice shown on Fig.(4.1). This structure can be
described [31, 32] as a triangular lattice with unit vectors
√
acc √
acc
~a1 =
(3, 3),~a2 =
(3, − 3),
(4.1)
2
2
20
Electronic structure and transport in graphene
where acc = 0.142 nm is a carbon-carbon
distance. Unit
√
√ cell contains two
atoms and has an area Scell = a 4 3 , where a = acc 3 = 0.246 nm is a
lattice constant. Each point of the sublattice A is connected to its nearestneighbors by the vectors
√ !
√ !
1
3
1
3
δ~1 = acc
,
, δ~2 = acc
,−
, δ~3 = acc (−1, 0) .
(4.2)
2 2
2
2
For a given lattice one can easily build a reciprocal lattice with a unit
vectors ~bi defined by the relation ~bi~aj = 2πδij , i.e.
√
√
~b1 = 2π (1, 3), ~b2 = 2π (1, − 3).
3acc
3acc
(4.3)
The first Brillouin zone, which is the Wigner-Seitz primitive cell of the
reciprocal lattice [33], is shown on Fig.(4.1)(b). There are two inequivalent
points which are of special interest in graphene physics
~ = 2π 1, √1 , K
~ ′ = 2π 1, − √1 .
K
(4.4)
3acc
3acc
3
3
Dispersion relation Graphene lattice can be considered as two interpenetrating triangular sublattices A and B which are defined by vectors
A
B
~ p,q
~ p,q
R
= p~a1 + q~a2 , R
= ~δ1 + p~a1 + q~a2 ,
(4.5)
where p, q are integer numbers.
In a single-electron approximation the tight-binding Hamiltonian for
electrons in graphene is given by
X
+
+
Hˆtb = −t
a+
(4.6)
p,q bp,q + ap,q bp−1,q + ap,q bp−1,q+1 + h.c.,
p,q
+
where a+
p,q (ap,q ) and bp,q (bp,q ) create (annihilate) an electron on sublattices
A and R~B respectively and t = 2.77 eV is a nearestA and B at site R~p,q
p,q
neighbor hopping integral.
The wave-function for the lattice can be written in the form
X
A +
B +
|Ψi =
ζp,q
ap,q + ζp,q
bp,q |0i,
(4.7)
p,q
~
A(B)
A(B)
where ζp,q is a probability amplitude to find the electron at site Rp,q
ˆ
Substituting Eqs.(4.6),(4.7) into the Schrödinger equation, Htb |Ψi = E|Ψi,
ˆ +
and calculating the matrix elements h0|ap,q Hˆtb a+
p,q |0i and h0|bp,q Htb bp,q |0i
Basic electronic properties
21
with use of the commutation relation, one arrives to the system of difference
equations
B
B
B
A
−t ζp,q
+ ζp−1,q
+ ζp−1,q+1
= Eζp,q
,
(4.8)
A
A
A
B
−t ζp,q + ζp+1,q + ζp+1,q+1 = Eζp,q .
A
B
The states ζp,q
and ζp,q
can be written in the Bloch form
~ ~A
A
A ik Rp,q
A
A
A
ζp,q
= ψp,q
e
, ψp,q
= ψp+1,q
= ψp+1,q−1
,
i~k R~B
p,q
B
B
ζp,q
= ψp,q
e
(4.9)
B
B
B
, ψp,q
= ψp−1,q
= ψp−1,q+1
.
Substituting Eq.(4.9) and Eq.(4.2) into Eq.(4.8) and omitting indexes
(p, q), one gets
−tφ(~k)ψ B = Eψ A ,
−tφ∗ (~k)ψ A = Eψ B ,
where
(4.10)
~~
~~
~~
φ(~k) ≡ eikδ1 + eikδ2 + eikδ3 ,
or in a matrix form
A A ψ
ψ
Ĥ
=
E
, Ĥ ≡
ψB
ψB
(4.11)
0
−tφ(~k)
∗ ~
−tφ (k)
0
!
.
(4.12)
In order to obtain dispersion relation, one needs to determine the eigenvalˆ =
ues of the matrix Ĥ, which are calculated using the relation det |Ĥ − IE|
0,
E(~k)2 = t2 |φ(~k)|2 ,
v
u
u
~
E(k) = ±tt1 + 4 cos2
√
3
acc ky
2
!
+ 4 cos
√
3
acc ky
2
!
cos
3
acc kx .
2
(4.13)
Dirac equation The spectrum of graphene, given by Eq.(4.13), is symmetric in respect to energy E = 0. If the Fermi energy coincides with
this point (EF = 0), i.e the states are occupied only up to zero energy,
it corresponds to the case of electrically neutral graphene. One is usually
interested in electronic properties close to a charge-neutrality point. There
are six points in the k-space where the energy equals to zero. These points
are at the corners of the first Brillouin zone, see Fig.(4.1). Only two of
~ and K
~ ′ , are inequivalent.
them, K
22
Electronic structure and transport in graphene
a)
b)
3
0
Figure 4.2: a) Dispersion relation calculated using Eq.(4.13). The energy
is given in units of the hopping integral. b) Close to the charge neutrality
point dispersion relation is linear and has a form of a cone determined by
the Dirac equation (see Eq.(4.23) below).
Let’s expand the function φ(~k) in the Hamiltonian of Eq.(4.12) near the
~
K-point
~k = K
~ + ~q,
(4.14)
~ vector having origin at K.
~ Substituting
where ~q is some small (|~q| < |K|)
~
Eq.(4.14) in φ(k), one gets
~ + ~q) =
φ(K
3
X
~~
~
2π
~
~
2π
~
eiK δi ei~qδi = ei 3 ei~qδ1 + ei0 ei~qδ2 + e−i 3 ei~qδ3 .
(4.15)
i=1
Considering the continuum (low energy) limit (acc → 0) [32], one can expand exponents in Taylor series
~
lim ei~qδi ≃ 1 + i~q~δi .
(4.16)
acc →0
After performing some straightforward algebra, we arrive at
"
!
√
√ ! #
3acc
3
1
1
3
~
φ(k) =
−
−i
qx −
+i
qy .
2
2
2
2
2
If we rotate the system of coordinates on angle θ =
R=
cos θ − sin θ
sin θ cos θ
=
√
3
2
1
2
π
6
1
−
√2
3
2
(4.17)
by operator
!
,
(4.18)
Basic electronic properties
23
Eq.(4.15) is finally reduced to
φ(~q) = −
3acc
(qx + iqy ).
2
(4.19)
Substituting Eq.(4.19) into Eq.(4.12), one gets
A A 3
0
qx + iqy
ψ
ψ
acc t
=E
.
B
B
q
−
iq
0
ψ
ψ
2
x
y
(4.20)
With the use of the Pauli matrices, ~σ = (σx , σy ), where
σx =
0 1
1 0
, σy =
0 −i
i 0
,
(4.21)
the Hamiltonian can be written in a vector form
ĤK = ~vF ~σ~q,
(4.22)
t
≃ 106 m/s is a Fermi velocity. Equation (4.22) is alwhere vF = 23 acc
~
gebraically identical to a two-dimensional relativistic Dirac equation with
vanishing rest mass known as Weyl’s equation for a neutrino, where the
two-component wave function (or spinor) represents pseudo-spin which results from the presence of two sublattices [34, 35]. The eigenenergies of ĤK
are
E = ±~vF q.
(4.23)
In order to find eigenfunctions Eq.(4.20) can be rewritten in the following
way
A A 0
eiθ(~q)
ψ
ψ
~vF q
=E
,
(4.24)
ψB
ψB
e−iθ(~q)
0
where θ(~q) = arctan(qy /qx ). Taking into account Eq.(4.23) for eigenenergies and using normalization condition |ψ A |2 + |ψ B |2 = 1, one gets
iθ(~q)/2 1
e
√
,
(4.25)
|ΨK,s
q )i =
~ (~
−iθ(~
q )/2
2 se
where the sign s = ± corresponds to the eigenenergies ±~vF q.
Besides spin-degeneracy each level is double-degenerated due to valley.
One has to operate by a full wave function which includes the contribution
from both valleys. The same procedure can be repeated to obtain the
effective Hamiltonian and wave function near K ′ -point
−iθ(~q)/2 1
e
∗
ĤK ′ = ~vF ~σ ~q, |ΨK~ ′,s (~q)i = √
.
(4.26)
iθ(~
q )/2
2 se
24
Electronic structure and transport in graphene
If the wave-vector ~q rotates once around the Dirac point, i.e. θ →
θ + 2π, the wave-function acquires an additional phase equals to π, hence
ΨK,s
~ (θ ± 2π) = −ΨK,s
~ (θ) which is a characteristics of fermions.
Electron’s wave function in graphene has a chiral nature. Helicity can
be interpreted as a projection of pseudospin vector on direction of motion
and defined by the operator ĥ = ~σ~q/|~q|. It can be easily shown using
Eq.(4.22), that eigenvalues of the helicity operator equal to h = ±1. Since
ĥ commutes with the Hamiltonian, helicity is a conserved quantity and
responsible for such effects as the Klein tunneling [36].
4.2
Graphene nanoribbons
Electronic properties of graphene nanoribbons (GNR) depend on the type
of a edges. One can distinguish two types, namely, zig-zag and armchair
GNR’s.
a)
3
b)
2
L
E/t 0
1
0
-1
-2
-3
-π
ka
x
π
Figure 4.3: a) Structure of an armchair graphene nanoribbon lattice. Each
edge of the ribbon is terminated by both A and B atoms. b) Dispersion
relation of the ribbon with 10 atoms in transverse direction calculated using
the tight-binding Hamiltonian.
Armchair graphene nanoribbons Dispersion relation of armchair GNR
can be derived solving the Schrödinger equation in the following form [37]





0
kx + iky
0
0
ψA
ψA
 kx − iky
  ψB 


0
0
0

 = E  ψB′  ,
~vF 

 ψA 
0
0
0
−kx + iky   ψA′ 
0
0
−kx − iky
0
ψB′
ψB′
(4.27)
Graphene nanoribbons
25
′
where ψA(B) and ψA(B)
are the probabilty amplitudes on the sublattice
~ and K
~ ′ points, respectively. The total wave
A(B) for the state near the K
function has the form
~
~′
Ψ = eiK~r ΨK~ + eiK ~r Ψ′K~ .
(4.28)
Let us first find
a solutionnear the K point. Substituting in Eq.(4.22)
∂
∂
~
wave vector k = −i ∂x
, −i ∂y
, one gets
A ∂
∂ 0
−i ∂x
+ ∂y
ψA
ψ
=ǫ
,
(4.29)
∂
∂
ψB
ψB
−i ∂x
− ∂y
0
where ǫ = ~vEF . Due to translational invariance in ~x-direction the wave
function can be written in the form
A
φ (y)
ΨK~ (x, y) = eikx x
,
(4.30)
φB (y)
which allows us to reduce the problem to a system of two differential equations
(
(
A (y)
B (y)
φB (y) = kǫx φA (y) − 1ǫ ∂φ∂y
kx φB (y) + ∂φ∂y
= ǫφA (y)
⇒
2
∂ φA (y)
A (y)
kx φA (y) − ∂φ∂y
= ǫφB (y)
+ z 2 φA (y) = 0
∂y 2
(4.31)
where z 2 = ǫ2 − kx2 . The general solution of the system of equations is a
sum of plane waves
φA (y) = Aeizy + Be−izy
(4.32)
Aeizy + kx +iz
Be−izy
φB (y) = kx −iz
ǫ
ǫ
Similar derivation can be done for the wave functions describing the states
near K ′ -point, which gives
′
φA (y) = Ceizy + De−izy
(4.33)
φ′B (y) = −kxǫ−iz Ceizy + −kxǫ+iz De−izy
In order to find the unknown coefficients A and B, one can utilize the
boundary conditions. The armchair nanoribbon edge consist of atoms belonging to both sublattices, see Fig.(4.3)(a), therefore one can expect that
both ΨA and ΨB should vanish at the edges
ΨA (0) = ΨB (0) = ΨA (L) = ΨB (L) = 0,
(4.34)
where ΨA(B) is an A(B) component of the total wave function (4.28).
Hence, we have
φA (0) + φ′A (0)
φB (0) + φ′B (0)
eiKy L φA (L) + e−iKy L φ′A (L)
eiKy L φB (L) + e−iKy L φ′B (L)
=
=
=
=
0
0
0
0
(4.35)
26
Electronic structure and transport in graphene
Substituting Eq.(4.32) and Eq.(4.33) into Eq.(4.35) and solving the system
of four unknowns, we arrive at
e2izL = ei∆Ky L ,
where ∆Ky =
4π
.
3a
(4.36)
Therefore, the allowed values of z are
z=
πn 2π
+
.
L
3a
(4.37)
Finally, the dispersion relation is given by
s
2
πn 2π
E = ±~vF kx2 +
+
.
L
3a
(4.38)
Note that if the ribbon consists of 3N + 1 atoms, Eq.(4.38) allows zero
energy solutions when kx → 0. This kind of ribbons are called metallic.
Otherwise the dispersion relation posses an energy gap, i.e. the ribbons are
semiconducting.
a)
3
b)
2
E/t 0
1
L
0
-1
-2
-3
-π
ka
x
π
Figure 4.4: a) Structure of a zig-zag graphene nanoribbon lattice. Each
edge of the ribbon is terminated by either A or B atoms. b) Dispersion
relation of the ribbon with 10 atoms in transverse direction calculated using
the tight-binding Hamiltonian.
Zig-zag graphene nanoribbons The Hamiltonian describing zig-zag
GNR can be derived from the Hamiltonian used for armchair GNR by
rotation of the system on angle π2 , i.e. changing kx → ky and ky → −kx
∂
i ∂y
0
−
∂
∂x
∂
i ∂y
+
0
∂
∂x
ψA
ψB
=ǫ
ψA
ψB
.
(4.39)
Warping
27
Substituting the wave function in the form given by Eq.(4.30) and following
the same procedure as in the case of the armchair GNR, we get
φA (y) = Aeizy + Be−izy
(4.40)
φB (y) = −ikǫx −z Aeizy + −ikǫx +z Be−izy
For zig-zag nanoribbons, as illustrated in Fig.(4.4), the boundary conditions
are
φA (L) = φB (0) = 0.
(4.41)
Solving the system of linear equation, we derive the relation between kx
and z
ikx + z
e−i2zL =
(4.42)
ikx − z
In contrast to armchair nanoribbons, the transverse and longitudinal components of the wave vector are coupled in zGNR. Another interesting feature of zGNR is an existence of surface states. Besides the solutions with
real z describing propagating states the transcendental equation (4.42) supports also solutions with imaginary values of z. It corresponds to the socalled edge (or surface) states, i.e the states spatially localized near the
edges.
4.3
Warping
c
Figure 4.5: a)Experimental 3D constant current STM image of single
layer graphene adopted from Ref.[40] b) Generated surface of a corrugated
graphene sheet [39]. Black line corresponds to a characteristic wave-length
of the ripples. c) Magnified area marked in a) by black rectangular. M and
S stands for maximum (minimum) and saddle points, respectively.
Graphene is a pure two-dimensional crystal. According to the MerminWagner theorem, the long-range order of 2D crystals should be destroyed
28
Electronic structure and transport in graphene
by long-wavelength fluctuations and therefore 2D membranes have a tendency to get crumpled being in a 3D space [38]. In the case of graphene
minimization of energy results in appearance of ripples on graphene sheet.
The existence of the ripples was confirmed by a number of experiments
[40, 41]. Warping can effect electronic properties. For example, bending
the graphene plane changes overlap between p orbitals and in turn hopping
integrals in the tight-binding model. Also warping can be related to the
formation of electron-hole puddels which cause the spatial modulation of
charge density [42]. Another example is a magnetotransport. Electrons
propagating through a warped graphene subjected to a magnetic field are
influenced by spatially correlated effective magnetic field.
The corrugated surface of graphene can be modeled [see Fig.(4.5)] by a
superposition of plane waves
X
h(~r) = C
Cqi sin(~qi~ri + δi ),
(4.43)
i
where h(~r) is a out-of-plane displacement at point defined by in-plane position vector ~r = (x, y). The directions ϕi of wave vectors ~qi = |qi |(cos ϕi , sin ϕi )
and the phases δi were chosen randomly. The length of the wave vectors
qi covers equidistantly the range 2π/L < qi < 2π/(3acc), where L is a
leading linear size of the rectangular area and acc denotes the C-C bond
length (we assume that L ≫ λ∗ ). The
of the mode was given
q amplitude
by the harmonic approximation Cq = 2 h2q for the wave length λ < λ∗ ,
otherwise it was kept constant and equal to Cq∗ , where q ∗ = 2π/λ∗ . We
introduced the normalization constant p
C to keep the averaged amplitude
of the out-of-plane displacement h̄ = 2 hh2 i equal to the experimental
values h̄ ≈ 1 nm for typical sizes of samples.
4.4
Bilayer graphene
In a tight-binding approximation bilayer grpahene can be modeled by the
following Hamiltonian [31]
X
Ĥ = − t
(a+
l,i bl,i+∆ + h.c.)
i;l=1,2
− γ1
− γ3
X
(a+
1,i a2,i + h.c.)
i
X
(b+
1,i b2,i+∆ + h.c.),
(4.44)
i
+
where a+
l,i (al,i ) and bl,i (bl,i ) are the creation (annihilation) operators for
~ i , where i = (p, q). The
sublattice A (B), in the layer l = 1, 2, at site R
Bilayer graphene
29
B1 A1
γ3
t
γ1
γ4
B2
A2
Figure 4.6: Structure of bilayer graphene with the illustration of the hopping integrals used in tight-binding model: t is the intralayer nearestneighbor coupling energy, γ1 is the coupling energy between sublattice A1
and A2 in different graphene layer, and γ3 the hopping energy between
sublattice B1 and B2 in the upper and lower layers, respectively.
meaning of hopping integrals is illustrated in Fig. (4.6): t is the intralayer
nearest-neighbor coupling energy, γ1 = 0.39 eV is the coupling energy
between sublattice A1 and A2 in different graphene layers, and γ3 = 0.315
eV the hopping energy between sublattice B1 and B2 in the upper and
lower layers, respectively. The other coupling energy between the nearestneighboring layers, γ4 ≈ 0.04 eV, is very small compared with γ0 and
ignored below. Since the unit cell of bilayer graphene consist of four atoms,
the wave function can be written in the form
 A1   A 
ψp,q
c 1
B1 
 ψp,q
 cB1  i~kR~
p,q
 

ψp,q = 
.
(4.45)
A2  =  A2  e
 ψp,q
c
B2
B2
c
ψp,q
Using this Bloch form of the wave function,
problem


0
tφ(~k)
γ1
0
cA 1
 ∗~
  B1
~
0
0
γ3 g(k)   c
 tφ (k)


 γ1
0
0
tφ∗ (~k)   cA2
cB2
0
γ3 g ∗(~k) tφ(~k)
0
we arrive at the eigenvalue



cA 1

 B1 
 = −E  cA  , (4.46)

 c 2 
cB2
~
where g(~k) = e−k(~a1 +~a1 ) φ(~k) and φ(~k) is same as in the single layer graphene
and defined by Eq.(4.11).
A minimal low-energy model Electronic properties close to the chargeneutrality point can be obtained using a low-energy effective bilayer Hamiltonian [43, 44]. In this approximation the interaction between layers is described by the interplane hopping term γ1 only (i.e. γ3 is neglected). Also,
~ point, one can make
since we are interested in the properties close to K
30
Electronic structure and transport in graphene
the expansion −tφ(~k) ≈ ~vF k, where k = kx + iky .
these approximations, one gets

0
~vF k γ1
0
 ~vF k ∗
0
0
0

Ĥ = 
γ1
0
0
~vF k ∗
0
0
~vF k
0
Taking into account


.

Solution of det(Ĥ − EI) = 0 gives a dispersion relation
!
r
γ1
γ12
2 2
2
~
+ ~ vF k ,
E(k) = s1 s2 +
2
4
(4.47)
(4.48)
where s1 , s2 = ±1. One can consider this equation in two limits (below
only the conduction band is discussed):
a) low energies, ~vF k < γ1
r
γ1
γ1 γ1 ~2 vF2 k 2
γ12
~
E(k) = s2 +
+ ~2 vF2 k 2 ≈ s2 +
+
,
2
2
2
γ12
| 4 {z
}
γ1
2
1+
E(k) ≈
(4.49)
2~2 v 2 k2
F
γ2
1
(
2 k2
~2 vF
,
γ1
2 k2
~2 vF
γ1 + γ1 ,
s2 = −1
s2 = 1
(4.50)
Equations (4.50) show that the conduction band of bilayer graphene consists of two parabolic bands separated by the energy interval γ1 . In the
vicinity of the Dirac point the dispersion relation of the lowest subband
can be rewritten in a form which is usually used in a conventional 2DEG
systems, i.e.
~2 k 2
E=
,
(4.51)
2m∗
γ
where m∗ ≡ 2v12 is an effective mass.
F
b) high energies, ~vF k > γ1
r
γ
γ1
γ12
1
E(~k) = s2 +
+ ~2 vF2 k 2 ≈ s2 + ~2 vF2 k 2 ,
(4.52)
2
4
2
| {z
}
~vF k 1+
E(k) ≈
γ2
1
8~2 v 2 k2
F
~vF k,
s2 = −1
~vF k + γ1 , s2 = 1
(4.53)
i.e. the dispersion relation consists of two linear bands (as in the case of
the the single layer graphene) separated by the energy interval γ1 .
Dirac fermions in a magnetic field
31
Biased bilayer graphene Electronic properties of bilayer graphene can
be strongly modified by applying external electric field, which causes a
potential difference between layers. In this case the Hamiltonian (4.47) is
modified


U1
~vF k γ1
0
 ~vF k ∗ U1
0
0 
,
Ĥ = 
(4.54)
 γ1
0
U2 ~vF k ∗ 
0
0
~vF k
U2
where U1 and U2 are on-site energies or electrostatic potentials of the 1st and 2-nd layer respectively. The dispersion relation of biased bilayer
graphene is
E±,s (~k) =
r
±
∆U 2
4
U1 +U2
2
+ ~2 vF2 k 2 +
γ2
2
+ (−1)s
q
(4.55)
(∆U 2 + γ 2 )~2 vF2 k 2 +
γ4
,
4
where ∆U = U2 − U1 and s = ±1.
~
Energy gap, ∆Eg , in the vicinity of K-point
is determined by
∆Eg = E+,1 (0) − E−,1 (0) = U2 − U1 = ∆U.
(4.56)
This equation shows that a band-gap can be tuned by application of an
external voltage. This makes bilayer graphene a promising candidate for
future electronics.
4.5
Dirac fermions in a magnetic field
In a high enough perpendicular magnetic field a continues spectrum of any
2DEG systems is usually modified into a series of Landau levels (LLs).
Since the behavior of electrons in graphene described by Dirac rather than
Schrödinger equation it should be reflected in the spectrum as well. We
start the calculation of the dispersion relation from the Hamiltonian given
by Eq.(4.22)
Ĥ = vF ~σ ~p.
(4.57)
~
In a magnetic field the canonical momentum is changed to p~ → ~~k + eA,
~
where A is a vector potential of the magnetic field. Using the Landau gauge
~ = (−By, 0, 0), one gets
in the form A
∂
∂
0 1
0 −i
Ĥ = vF
−i~
− eBy
+ −i~
.
1 0
i 0
∂x
∂y
(4.58)
32
Electronic structure and transport in graphene
Substituting it in the Schrödinger equation, Ĥ|Ψi = E|Ψi, one arrives at
the eigenvalue problem
A ∂ ∂
− eBy − ~ ∂y
0
−i~ ∂x
ψA
ψ
vF
=E
.
∂
∂
ψB
ψB
−i~ ∂x
− eBy + ~ ∂y
0
(4.59)
Due to translational invariance in ~x direction, the solution of the Schrödinger
equation with the Hamiltonian (4.58) can be written in the Bloch form
A A
φ (y)
ψ
ikx
,
(4.60)
|Ψ(x, y)i =
=e
ψB
φB (y)
which gives
0
−~vF
∂
− ∂y
+ lyB − lB k
∂
∂y
+
y
lB
0
− lB k
φA (y)
φB (y)
= lB E
φA (y)
,
φB (y)
(4.61)
q
~
is a magnetic length. Introducing dimensionless length
where lB = eB
y
scale ζ = lB − lB k, one finally gets
0
∂
+ζ
− ∂ζ
∂
∂ζ
+ζ
0
φA (ζ)
φB (ζ)
=ǫ
φA (ζ)
φB (ζ)
,
(4.62)
where ǫ = −lB E/~vF . This system of the first order differential equations
can be reduced to a second order differential equation
∂ 2 φA (ζ) 2
+ (ǫ − 1) − ζ 2 φA (ζ) = 0.
2
∂ζ
(4.63)
This equation
phas a form of the harmonic oscillator equation with eigenenergies ǫ = ± 2(n + 1), where n = 0, 1, 2, .. and eigenfunctions
ζ2
Φn (ζ) = e− 2 Hn (ζ),
(4.64)
where Hn is a n-order Hermitian polynomial. Similarly, √
one can solve an
equation for φB (ζ), which gives the eigenenergies ǫ = ± 2n. Hence, the
final solution is
n−1 y
Φ ( lB − lB k)
|Ψ(x, y)i = eikx
,
(4.65)
Φn ( lyB − lB k)
where we define Φ−1 ≡ 0. Hence, in a magnetic field perpendicular to
graphene layer the spectrum is modified into a series of Landau levels with
the energies
√
E = ±~ωc n,
(4.66)
Dirac fermions in a magnetic field
33
p
where ωc = vF 2eB/~ is a cyclotron frequency. There are to distinctive
features of graphene spectrum in a magnetic field in comparison to the
spectrum of ordinary 2DEG systems. Firstly, the energy of the ground
state (i.e. 0’th LL) of graphene equals to zero and does not depend on a
magnetic field value. Secondly, the difference between to successive levels
is not constant and decreases with increase of energy. An experimental
manifestation of these unusual series is the anomalous quantum Hall effect
2
with the Hall conductivity given by σxy = 4eh (n + 1/2) [45, 46].
34
Electronic structure and transport in graphene
Chapter 5
Modeling
In this chapter we describe two widely used techniques for electronic and
transport properties calculations, namely, the recursive Green’s function
technique (RGFT) and the real-space Kubo method. Both techniques used
in the present thesis have their advantages and disadvantages. The recursive Green’s function technique naturally captures all transport regimes:
ballistic, diffusive and localization. It allows to take into account any arbitrary shape of a considered structure. Also it is easy to include in the model
different types of disorder and a magnetic field. One of the main drawback
of the RGFT is that the computational expenses scales as O(N 3 ). In contrast, the real-space Kubo method scales as O(N) allowing to investigate
structures consisting of up to tens of millions of carbon atoms. However,
this method can not be properly used to study edge physics. Moreover,
even though the real-space Kubo method can be in principle applied to
study any transport regime, straightforward application of the method to
calculation of conductivity is done only for the diffusive regime. Thus,
the recursive Green’s function technique is well suited to study relatively
small structures, where edges play a major role; while the real-space Kubo
method can be used to investigate bulk properties of large systems.
5.1
Tigh-binding Hamiltonian and Green’s
function
In order to model electronic structure and transport properties of graphene
we use standard p-orbital tight-binding Hamiltonian [31] (see also Eq.(4.6)),
X
X
Ĥ =
V (r)a+
tr,r+∆ a+
(5.1)
r ar −
r ar+∆ ,
r
r,r+∆
where the summation runs over all sites in graphene lattice and ∆ includes
the nearest-neighbor only. The effect of an external potential as well as
36
Modeling
the interaction of an electron with all other particles in the system is incorporated through the change of the on-site energy V (r). In the absence
of a magnetic field the hopping integral in Hamiltonian is constant and
equals to tr,r+∆ = t0 = 2.77 eV [31]. The operators a+
r /ar are standard
creation/annihilation operators obeying the following anticommutation relations [47],
{ar , a+
r′ } = δrr′ ,
+
{ar , ar′ } = {a+
r , ar′ } = 0.
(5.2)
The Hamiltonian (5.1) acts on the wave-function which is also expressed
in terms of the second-quantization operators
|Ψi =
X
r
ψr a+
r |0i ,
|0i = |0, . . . , 0i
(5.3)
with |0i representing a vacuum state and ψr = h0| ar |Ψi is a probability
amplitude to find a particle at the site r.
a)
4
3
3
b)
4
2
2
1
1
Figure 5.1: Depending on the C-C bond orientation the hopping integral
acquires different phase in a magnetic field which is determined by Eq.(5.4).
Pierels substitution If the system is subjected to a magnetic field the
hopping integral acquires phase,
φr,r+∆
tr,r+∆ = t0 exp i2π
,
φ0
φr,r+∆ =
Z
r+∆
r
A · dl,
(5.4)
where φ0 = h/e is the magnetic-flux quantum and A is the vector potential.
In the case of uniform perpendicular magnetic field, the convenient choice
for the vector potential is the Landau gauge in the form
A = (−By, 0, 0).
(5.5)
Using Eqs.(5.4)-(5.5), one can evaluate the hopping integral for geometries
depicted on Fig.(5.1). Thus, for zigzag geometry, Fig.(5.1)(a), the phases
Tigh-binding Hamiltonian and Green’s function
are
√
acc 3
1
B y1 + acc ;
2
4
√
acc 3
1
=−
B y3 + acc ;
2
4
φ23 = 0;
φ12 =
φ34
37
√
acc 3
1
B y2 − acc ;
2
4
√
acc 3
1
=
B y4 − acc ;
2
4
= 0;
(5.6)
φ21 = −
φ43
φ32
and for armchair geometry, Fig.(5.1)(b), we get
!
!
√
√
acc
3
acc
3
φ12 =
B y1 +
acc ;
φ21 = − B y2 −
acc ;
2
4
2
4
!
!
√
√
3
acc
3
acc
acc ;
φ32 =
B y3 −
acc ;
φ23 = − B y2 +
2
4
2
4
φ34 = −By3 acc ;
φ43 = By4 acc ;
(5.7)
The Green’s function A standard way to define the Green’s function
is [16]
[E − H ± iη]G± = 1̂ ⇒ G± = [E − H ± iη]−1 .
(5.8)
where η → 0 is an infinitesimal constant. The sign ± distinguishes between
retarded (G+ ) and advanced (G− ) Green’s functions. In this chapter only
retarded Green’s functions are used, therefore we omit the sign thereafter.
The Green’s function can also be expressed in terms of eigenvalues and
eigenfunctions of the Hamiltonian, Ĥ |Ψα i = ǫα |Ψα i,
G(r, r′ ; E) =
X
α
ψrα ψrα∗
′
.
E − ǫα + iη
(5.9)
This equation allows to connect the Green’s function to a local density of
states (LDOS) which is defined as
X
ρ(r, E) =
|ψrα |2 δ(E − ǫα ).
(5.10)
α
Using the relation
1
ℑ lim
= −πδ(x),
η→0 x + iη
we finally arrive at the required equation for LDOS
1
ρ(r, E) = − ℑ lim G(r, r; E) .
η→0
π
(5.11)
(5.12)
Computation of the Green’s function by direct matrix inversion is an
extremely time-consuming routine. If the structure consist of M-slices with
38
Modeling
N-sites on each slice, the real-space Hamiltonian matrix has a dimension
NM × NM, which makes matrix inversion inefficient for structures with a
size relevant for experiments. One way to attack the problem is to switch
to an energy (or momentum) representation using the Fourier transformation. However this method introduce approximation in the original exact
problem. In the next section we describe the recursive Green’s function
technique. This method does not require any simplifications in comparison
with the original problem and allows to avoid the full Hamiltonian matrix
inversion.
5.2
Recursive Green’s function technique:
application to graphene nanoribbons
In this section we use the technique described in Refs. [48, 49] and applied previously to study properties of graphene nanoribbons and photonic
crystals.
5.2.1
Dyson equation
a)
b)
0 1 2
i
N N+1
Figure 5.2: Schematic illustration showing the application of a) Dyson
equation and b) the recursive Green’s function technique.
Imagine that we have two isolated subsystems described by the Hamiltonians H10 and H20 . Then, we put them together and let interact via
potential V [see Fig.(5.2) (a)]. Thus, the total Hamiltonian reads
H = H10 + H20 +V = H 0 + V.
| {z }
(5.13)
0
0 −1
G−1 = E
| −
{zH } −V = (G ) − V.
(5.14)
H0
The Green’s function corresponding to this Hamiltonian equals to
(G0 )−1
Recursive Green’s function technique
39
Multiplying the above equation on the left by G0 and on the right by G
(or vice versa) and rearranging variables, we derive the Dyson equation
G = G0 + G0 V G,
(G = G0 + GV G0 ).
(5.15)
The Dyson equation is an extremely useful relation and is a basis of the
recursive Green’s function technique. If we define the matrix element as
Gmn = hm|G|ni, the Dyson equation reads
Gmn = G0mn +
X
G0mi Vij Gjn ,
(5.16)
i,j
(Gmn = G0mn +
X
Gmi Vij G0jn ).
i,j
Hence, attaching two slices with the use of the Dyson equation, the Green’s
function for the combined system is
G11
G12
G21
G22
=
=
=
=
(I − G011 V12 G022 V21 )−1 G011 ,
G011 V12 (I − G022 V21 G011 V12 )−1 G022 ,
G022 V21 (I − G011 V12 G022 V21 )−1 G011 ,
(I − G022 V21 G011 V12 )−1 G022 .
(5.17)
Let us now consider a typical structure for transport experiments illustrated in Fig. (5.2)(b). It consists of ideal (non scattering) leads and
device (scattering) region. The first step is to split the device region into
slices {i} describing by relatively simple Hamiltonians Hi0
H=
N
X
Hi0 + U.
(5.18)
i
By ’simple’ we imply the Hamiltonian of a such size, for which the corresponding Green’s function can be found by direct matrix inversion G0ii =
[E − Hi0 ]−1 . The leads are assumed to be uniform, which often allows to
find the surface Green’s function (Γ) analytically or by using the technique
described in Sec.(5.2.4). Thus, having the surface Green’s functions ΓL and
ΓR and using Dyson Eqs.(5.17), one can add recursively slice by slice to find
the Green’s function of each slice Gii as a part of the whole system, which
provide information about LDOS, Eq.(5.12); and the Green’s function of
the device GN +1,0 , which allows to calculate transmission coefficient (see
Eq.(5.45) in Sec.(5.2.5)).
40
Modeling
5.2.2
Bloch states
Let us consider an infinitely long ideal graphene nanoribbon consisting of N
sites in the transverse j-direction, as illustrated in Fig.(5.3). The structure
has a translational symmetry in i-direction with unit cell consisting of
M = 2 slices for zigzag GNRs and M = 4 in the case of armchair GNRs.
c) zGNR unit cell
a)
d)
aGNR unit cell
6 6
zigzag graphene nanoribbon
6
5
4
6
6
6
5 5
5
5
5
4 4
4
4
4
3 3
3
b) armchair graphene nanoribbon
2
1
0
1
3
3
3
2 2
2
2
1
2
2
1 1
1
3
0
1
1 2
3 4
5
Figure 5.3: a) zigzag and b) armchair graphene nanoribbons and corresponding to them unit cells c), d).
The Hamiltonian of the structure can be split into three parts
H = Hcell + Hout + U,
(5.19)
where the operators Hcell describes the unit cell spanning slices 1 ≤ i ≤ M,
while Hout describes the region including all other slices −∞ < i ≤ 0
and M + 1 ≤ i < ∞. The coupling between the cell and slices i = 0 and
i = M + 1 is described by the hopping operator U. The total wave function
consists of two parts,
|ψi = |ψcell i + |ψout i,
(5.20)
corresponding to the unit cell and outside region respectively. Substituting
Eqs. (5.20), (5.19) into the Schrödinger equation Ĥ|ψi = E|ψi and making
use of definition of the Green’s function (5.8), we obtain
|ψcell i = Gcell U|ψout i,
(5.21)
where Gcell is the Green’s function of the operator Hcell . This equation
allows us to relate the wave-function in the cell (slice i = 1 and i = M)
with the wave-function in the outside region (slice i = 0 and i = M + 1).
Recursive Green’s function technique
41
Hence calculating the matrix element ψi,j = h0|ai,j |ψi, we get
1,M +
ψ1 = G1,1
cell U1,0 ψ0 + Gcell U1,0 ψM +1 ,
M,M +
ψM = GM,1
cell U1,0 ψ0 + Gcell U1,0 ψM +1 ,
(5.22)
where we used that UM,M +1 = U0,1 due to the periodicity of the ribbon and
+
U0,1 = U1,0
. The vector ψi corresponds to the wave-function on a slice i


ψi,1


ψi =  ...  ,
(5.23)
ψi,N
and the Green’s function on the cell and the coupling matrices are defined
by
(U1,0 )jj ′ = h0|a1,j Ua+
0,j ′ |0i
′
Gi,i
= h0|ai,j Gcell a+
cell
i′ ,j ′ |0i.
(5.24)
jj ′
Equation (5.22) can be rewritten in a compact form
ψM +1
ψ1
T1
= T2
,
ψM
ψ0
where
T1 =
+
−G1,M
cell U1,0
M,M +
−Gcell U1,0
0
,
I
−I
T2 =
0
1,1
Gcell
U1,0
M,1
Gcell U1,0
(5.25)
(5.26)
and I being the unitary matrix. Since we consider the ideal nanoribbon,
the wave-function has the Bloch form with a periodicity equals to the periodicity of the structure,
ψm+M = eikM Iψm .
(5.27)
This allows to rewrite Eq.(5.25) in the form of an eigenvalue problem
ψ1
ψ1
T1−1 T2
= eikM
.
(5.28)
ψ0
ψ0
For a fixed energy E, solution of (5.28) gives a set of eigenvalues {kα } and
eigenfunctions {ψ0α }, {ψ1α }, 1 ≤ α ≤ 2N. Among 2N states there are Nprop propagating states and Nevan -evanescent states, which can be separated by
the value of imaginary part of kα (i.e. |Im[kα ]| > 0 for evanescent states).
The propagating states, in turn, can be separated into two equal parts,
right- and left-propagating states, by calculating their group velocities.
This is described in the next section.
42
5.2.3
Modeling
Calculation of the Bloch states velocity
As it is shown in Sec. (5.2.5), in order to calculate the transmission and reflection coefficients, one needs to know the velocities of propagating states.
Moreover, the sing of velocity of Bloch state is used to separated the states
by its direction of propagation, which is required to construct the surface
Green’s functions (see Sec. (5.2.4)). In order to calculate the velocity we
use the standard formula for the group velocity
1 ∂E
.
(5.29)
~ ∂k
One can estimate the derivative by direct numerical differentiation (E(k2α )−
E(k1α ))/(k2α − k1α ). However it’s not efficient since it requires to solve eigenvalue problem (5.28) twice. Moreover, each time the eigensolver is called,
the eigenvalues are given in different order.
In this section we derive the explicit form of the derivative. The wavefunction of the α-th Bloch state within the unit cell is given by
v=
|ψ α i =
M
X
i=1
|ψiα i,
|ψiα i = eikα xi |ϕαi i.
(5.30)
(Thereafter we omit α to simplify notation.) Making use of the relation
hψi |H|ψi = Ehψi |ψi = E|ϕi |2 , we obtain
M
1 ∂E
1 1 X ∂ hψi |H|ψi
v=
=
,
(5.31)
~ ∂k
~ M i=1 ∂k
|ϕi |2
where the elements of the vector


ϕi,1


ϕi =  ... 
ϕi,N
(5.32)
equal to ϕi,j = h0|ai,j |ϕi. Finally, calculating the matrix element hψi |H|ψi
with Hamiltonian in the form (5.18), the velocity reads
M
h
1 i X ϕ∗T
i
(xi − xi−1 )Ui,i−1 ϕi−1 e−ik(xi −xi−1 )
2
~ M i=1 |ϕi |
i
−(xi+1 − xi )Ui,i+1 ϕi+1 eik(xi+1 −xi ) .
v = −
(5.33)
In the case of zero magnetic field, this equation can be simplified for zigzag
nanoribbons
√
M
=2
X
ϕ∗T
acc 3
i−1
v=
sin k
Ui−1,i ϕi ,
(5.34)
2~
|ϕ
|2
i−1
i=1
where we used that ϕi = ϕi+M due to periodicity.
Recursive Green’s function technique
5.2.4
43
Surface Green’s function
Applying the Dyson equation, one can construct the Green’s function of
the scattering region of an arbitrary shape. However in calculation of transport properties of the system, one usually assumes that the system is open
and attached to a perfect (non-scattering) semi-infinite leads. Hence, one
is interested in finding the surface Green’s function of the leads as a starting point for Dyson equation. Most of the methods for calculation of the
Green’s function rely on searching for a self-consistent solution for ΓR and
ΓL which makes these calculation very time consuming [50]. The method
[48, 49] presented in this section does not require self-consistent calculations, and the surface Greens function is expressed in terms of the Bloch
states of the graphene lattice.
Let us consider a semi-infinite periodic ideal graphene ribbon extended
to the right in the region −M ≤ i < ∞. Suppose that an excitation |si
is applied to its surface slice i = −M. Introducing the Green’s function
of the semi-infinite ribbon, Grib , one can write down the response to the
excitation |si in a standard form [16]
|ψi = Grib |si.
(5.35)
The unit cell of a graphene lattice spans slices 1 ≤ i ≤ M, (M = 2 and
4 for the zigzag and armchair lattices, see Fig. 5.3). Applying the Dyson
equation between the slices 0 and 1, we get
G1,−M
= ΓR U1,0 G0,−M
,
rib
rib
(5.36)
1,1
Grib
.
where we defined the right surface Green’s function as ΓR ≡
Evaluating the matrix elements h0|a1,j |ψi of Eq. (5.35) and making use of Eq.
(5.36), we obtain for each Bloch state α
ψ1α = ΓR U1,0 ψ0α .
(5.37)
This equations can be used to construct ΓR
ΓR U1,0 = Ψ1 Ψ−1
0 ,
(5.38)
where Ψ1 and Ψ0 are the square matrices composed of the matrix-columns
ψ1α and ψ0α , (1 ≤ α ≤ N), Eq. (5.23), i.e.
Ψ1 = (ψ11 , ..., ψ1N );
Ψ0 = (ψ01 , ..., ψ0N ).
(5.39)
The expression for the left surface Greens function ΓL can be derived
in a similar fashion
+
ΓL U1,0
= ΨM Ψ−1
(5.40)
M +1 ,
where the matrices ΨM and ΨM +1 are defined in a similar way as Ψ1 and
Ψ0 above. Note that matrices ΨM and ΨM +1 can be easily obtained from
Ψ1 and Ψ0 using the relation (5.25). Note also that in the case of the zero
magnetic field, the right and left surface Greens functions are identical,
ΓL = ΓR .
44
5.2.5
Modeling
Transmission and reflection
According to the Landauer formula (2.42), conductance can be expressed in
terms of the transmission function. In order to calculate the transmission
coefficient T (E), the system is divided into three regions, namely, two
ideal semi-infinite leads of the width N extending in the regions i ≤ 0
and i ≥ L respectively, and the central scattering region (device), see
Figs.(5.2),(5.3). The left and right leads are assumed to be identical. The
incoming, transmitted and reflected states in the leads, |ψαi i, |ψαt i and |ψαr i,
have the Bloch form,
N
X
|ψαi i =
X
|ψαt i =
XX
tβα eikβ (xi −xL )
|ψαr i =
XX
rβα eikβ xi
+
eikα xi
j=1
i≤0
i≥L
i≤0
φαi,j a+
i,j |0i
+
j=1
β
β
N
X
−
N
X
j=1
(5.41)
φβi,j a+
i,j |0i
φβi,j a+
i,j |0i,
(5.42)
(5.43)
where tβα (rβα ) are the transmission (reflection) amplitude from the incoming Bloch state α to the transmitted (reflected) Bloch state β (α → β),
and we choose x0 = 0. The transmission and reflection coefficients can be
expressed through the corresponding amplitudes and the Bloch velocities
[16]
X vβ
X vβ
|tβα |2 ; R =
|rβα |2 .
(5.44)
T =
v
v
α
α
α,β
α,β
The summation includes only propagating states. For the transmission and
reflection amplitudes we use the equations given in Ref. [49],
Φ1 T = −GL,0 (U0,1 Φ1 K − ΓL −1 Φ0 ),
Φ0 R = −G0,0 (U0,1 Φ1 K − ΓL −1 Φ0 ) − Φ0 ,
(5.45)
(5.46)
where the matrices T and R of the dimension N × Nprop and have the
following meaning, (T )βα = tβα , (R)βα = rβα ; (with Nprop being the number
of propagating modes in the leads); GL,0 and G0,0 are the Green’s function
matrices and ΓL is the left surface Green’s function, Eq. (5.40); U0,1 couples
the left lead and the scattering region; K is the diagonal matrix with the
matrix elements Kα,β = exp(ikα+ x1 )δα,β . The square matrices Φ1 and Φ0
are composed of the Bloch states φα0 and φα1 , Eq.(5.30), on the slices 0 and
1
N
1 of the ribbon unit cell, i.e. Φ1 = (φ11 , ..., φN
1 ) and Φ0 = (φ0 , ..., φ0 ).
Real-space Kubo method
5.3
45
Real-space Kubo method
The real-space Kubo method is a powerful tool to study transport properties of a system consisting of several millions of atoms. It was developed by
S. Roche and D. Mayou [51, 52] and then applied extensively to investigate
properties of carbon nanotubes [53, 54] and graphene [55, 56, 57]. The
method is based on numerical solution of the time-dependent Schrödinger
equation. Having calculated wave-function at different times, one can compute the mean square spreading of a wave packet. This quantity is proportional to the diffusion coefficient, which, in turn, can be related to the
conductivity. The derivations provided in this section are primarily based
on Refs. [58, 59].
5.3.1
Diffusion coefficient
The starting point of the model is the Kubo-Greenwood formula for conductivity (see Chapter 2 for details)
σij (E) =
2e2 π~
Tr [v̂x δ(E − H)v̂x δ(E − H)] ,
V
(5.47)
where V is the system volume, v̂x is a x-component of the velocity operator
and H is the Hamiltonian of the system. The last delta function can be
written as a Fourier transform
Z +∞
1
δ(E − H) =
dtei(E−H)t/~ .
(5.48)
2π~ −∞
Substituting it in Eq.(5.47), we get
Z +∞
2
σ(E) = e
dtTr v̂x δ(E − H)eiEt/~ v̂x e−iHt/~ .
(5.49)
−∞
Using the relation eiEt/~ = eiHt/~ and recalling the Heisenberg form of
operators, v̂x (t) = eiHt/~ v̂x e−iHt/~ , we arrive
Z +∞
σ(E) = e2
dtTr [v̂x (0)δ(E − H)v̂x (t)] .
(5.50)
−∞
Introducing the velocity autocorrelation function
hv̂x (t)v̂x (0)iE =
Tr [v̂x (0)δ(E − H)v̂x (t)]
,
Tr [δ(E − H)]
the equation for conductivity becomes
Z +∞
σ(E) = e2
dtTr [δ(E − H)] hv̂x (t)v̂x (0)iE .
−∞
(5.51)
(5.52)
46
Modeling
It is possible to show that the velocity autocorrelation function hv̂x (t)v̂x (0)i
can be related to the mean value of the spreading in the x-direction,
χ2 (E, t) = h(x̂(t) − x̂(0))2 iE as
d2 2
χ (E, t) = hv̂x (t)v̂x (0)iE ,
(5.53)
dt2
where x̂(t) = eiHt/~ x̂e−iHt/~ is the x-component of the position operator in
the Heisenberg picture. Inserting Eq.(5.53) into Eq.(5.52) and performing
integration, we get
d
(5.54)
σ(E) = e2 Tr [δ(E − H)] lim χ2 (E, t).
t→∞ dt
Using the definition of χ2 (E, t), the conductivity reads as
d Tr [δ(E − H)(x̂(t) − x̂(0))2 ]
σ(E) = e2 Tr [δ(E − H)] lim
t→∞ dt
Tr [δ(E − H)]
d
2
= e Tr [δ(E − H)] lim (tD(E, t)),
(5.55)
t→∞ dt
where we have introduce a new quantity called diffusion coefficient and
defined it as
χ2 (E, t)
1 Tr [δ(E − H)(x̂(t) − x̂(0))2 ]
1 nx (E, t)
=
=
, (5.56)
D(E, t) =
t
t
Tr [δ(E − H)]
t n(E)
where n(E) is a density of states (DOS). Details of calculation of nx (E, t)
are given in Sec.(5.3.3).
5.3.2
Transport regimes
The diffusion coefficient, D(E, t), does depend on time. Figure (5.4) illustrates typical time-evolution of the diffusion coefficient. Analyzing its
behavior one can distinguish three transport regimes, namely
1. ballistic
2. diffusive
3. localization
Ballistic regime For times t < τball , when the number of scattering
events is small, the wave-packet propagates ballistically. This is manifested
by linear growth of the diffusion coefficient as a function of time such that
p
χ(E, t)2 = v(E)t
(5.57)
D(E, t) = v(E)2 t.
(5.58)
If the system is large enough or the impurities concentration ni is high, the
diffusion coefficient saturates and becomes constant at time τball ≈ d/v(E),
√
where d = ni is an average distance between impurities.
Real-space Kubo method
47
Figure 5.4: Diffusion coefficient as a function of time. Three transport
regimes (ballistic, diffusion, localization) are separated by dashed lines.
The slope of D is proportional to the square of velocity v 2 . Dotted line
denotes maximal value of D which corresponds to a semiclassical limit.
Diffusive regime In this regime the diffusion coefficient is independent
of time, D(E, t) = D(E). Hence, the conductivity according to Eq.(5.55)
becomes
σ(E) = e2 ρ(E)D(E),
(5.59)
where ρ(E) = n(E)/S is a density of states per unit area per spin and S
is an area of the system. This equation is also known as Einstein relation
for conductivity [16].
In some cases the plateau on the diffusivity plot is small and linear
growth of D is immediately changed by decay. This is in contrast to the
classical regime where the diffusion coefficient does not change when t →
∞. Thus, in order to calculate conductivity in the diffusive regime, Eq.
(5.59) is modified such that the maximum value of the diffusion coefficient,
D(E, t) → Dmax (E), is used [55]
σsc (E) = e2 ρ(E)Dmax (E).
(5.60)
This corresponds to a semi-classical conductivity within the Boltzmann
approach where quantum effects leading to localization are not taken into
account
v2
σBoltz = e2 ρ(E)τ F ,
(5.61)
2
with τ being a scattering time. Comparing two equations, one can relate
the mean free path to the computed diffusion coefficient
le = vF τ =
2Dmax
.
vF
(5.62)
Localization regime The effect of localization can be viewed as constructing interference between forward and backward electron trajectories.
48
Modeling
When these effects become dominant, the diffusion coefficient decreases
[55, 56]. According to the theory of localization, conductivity decreases
with the increase of the system
length L. In the Kubo approach we can
p
relate L to time by L = χ(E, t)2 . In this case conductivity depends on
the size of the system
L(t)
,
(5.63)
σ(E) ∼ exp −
ξ
with ξ being localization length.
Figure 5.5: Diffusion coefficient as a function of time for different concentration of strong short-range impurities [59]. The graphene sample consists
of 6.8 millions of carbon atoms.
Figure (5.5) [59] illustrates the normalized diffusion coefficient calculated for graphene sample consisting of 6.8 millions of carbon atoms. Depending on the concentration of impurities the system appears to be in
different transport regimes within plotted time interval. If the impurity
concentration is zero, electrons propagate ballistically and the diffusion coefficient (blue line) grows linearly. For concentration n = 2% the diffusion
coefficient undergoes crossover from ballistic to diffusive regime (red line)
at time t = 20 fs. At n = 5% localization effects becomes dominant already
at t = 10 fs which is manifested by the decrease of D (green line) with time.
5.3.3
Time evolution
According to Eq.(5.56), in order to calculate the diffusion coefficient, we
need to compute the quantity
nx (E, t) = T r δ(E − H)(x̂(t) − x̂(0))2 .
(5.64)
Time-dependence of the position operator can be rewritten in the Heisenberg picture x̂(t) = eiĤt x̂e−iĤt , which yields
h
i
nx (E, t) = T r (eiĤt x̂e−iĤt − x̂)δ(E − H)(eiĤt x̂e−iĤt − x̂) .
(5.65)
Real-space Kubo method
49
Using the commutation relation [Ĥ, e−iĤt ] = 0, we get
h
i
nx (E, t) = T r (eiĤt x̂ − x̂eiĤt )δ(E − H)(x̂e−iĤt − e−iĤt x̂)
h
i
= T r [x̂, Û(t)]+ δ(E − H)[x̂, Û (t)] ,
(5.66)
where Û(t) = e−iĤt is a time evolution operator. In order to estimate the
trace we need to know how the operator [x̂, Û(t)] acts on the wave-function
|ψi i, i.e
[x̂, Û (t)]|ψi i = x̂ Û(t)|ψi i −Û(t) (x̂|ψi i) = x̂|ψi (t)i−Û (t) (x̂|ψi i) . (5.67)
If we denote |ψix i = x̂|ψi i, then
X
nx (E, t) =
hΨi (t)|δ(E − Ĥ)|Ψi (t)i,
(5.68)
i
where |Ψi (t)i = |ψi (t)i − |ψix (t)i. Mathematically Eq.(5.68) is similar to
equation for finding of the local density of states. The details of LDOS
calculation are given in Sec.(5.3.7).
5.3.4
Exact solution of the time-dependent Schrödinger
equation: Chebyshev method
As it is seen from Eq.(5.56) and subsequent discussion in Sec. (5.3.2), the
diffusion coefficient is a function of time. In order to estimate D(E, t) at
different times t one needs to know the wave-function ψ(t) in accordance
to Eq.(5.68). In this section we present an efficient method for solution
of the time-dependent Schrödinger equation based on the expansion of the
time evolution operator in an orthogonal set of Chebyshev polynomials
[51, 58, 57].
The starting point is the time-dependent Schrödinger equation
Ĥ |ψ(t)i = i~
∂
|ψ(t)i ,
∂t
(5.69)
with Ĥ being the tight-binding time-independent Hamiltonian. If the initial
wave-function at time t0 , |ψ(t = 0)i = |ψ0 i, is known, the formal solution of
Eq.(5.69) can be expressed via the time-evolution operator (or propagator)
Û (t),
i
|ψ(t)i = Û (t) |ψ0 i , Û(t) = e− ~ Ĥ(t) .
(5.70)
In order to expand Û(t) in a set of the Chebyshev polynomials Tn (x) (which
are defined in the interval x ∈ [−1; 1]), we first renormalize the Hamiltonian
such that its spectrum lies in the above interval,
Ĥnorm =
2Ĥ − (Emax + Emin )Iˆ
,
Emax − Emin
(5.71)
50
Modeling
where Emax and Emin are the largest and the smallest eigenvalues of the
original Hamiltonian Eq. (4.6). (In order to calculate Emax and Emin we
use a computational routine that estimates the largest/smallest eigenvalues
of the operator without calculation of all the eigenvalues).
Expanding Û(t) in Chebyshev polynomials in Eq. (5.70) we obtain for
the wave function,
|ψ(t)i =
∞
X
n=0
cn (t) |Φn i ,
(5.72)
where the functions |Φn i = Tn (Ĥnorm ) |ψ0 i are calculated using the recurrence relations for the Chebyshev polynomials,
|Φ0 i = T0 (Ĥnorm ) |ψ0 i = |ψ0 i
|Φ1 i = T1 (Ĥnorm ) |ψ0 i = Ĥnorm |ψ0 i
|Φn+1 i = Tn+1 (Ĥnorm ) |ψ0 i = 2Ĥnorm |Φn i − |Φn−1 i .
(5.73)
(5.74)
(5.75)
The recursive routine is repeated until the expansion (5.72) converges which
is defined by the condition
N
X
cn (t) |Φn i k − 1 < ǫ,
k
(5.76)
n=0
where ǫ is some predefined computational tolerance. The expansion coefficients cn (t) are calculated making use of the orthogonality relation for the
Chebyshev polynomials
−i
cn (t) = 2e
(Emax +Emin )t
2~
n
(−i) Jn
Emax − Emin
t .
2~
(5.77)
For large t the expansion coefficients cn (t) become exponentially small.
This leads to the fast convergence of the expansion series Eq. (5.72), and
makes the Chebyshev method very efficient for calculation of the temporal
dynamics.
Figure (5.6) illustrates temporal dynamics of a wave-packet in a pure
(undoped) graphene. The wave-paket, originally (at t = 0) localized at
a single site in the middle of the structure, distributes uniformly in all
directions. Less than 1000 iterations is required to achieve convergence for
80 fs of elapsed time.
Real-space Kubo method
51
Figure 5.6: Temporal dynamics of the wave-packet in 2D graphene. At
initial time the wave-packe was localized at a single site in the middle of
the structure.
5.3.5
Continued fraction technique
Consider a Hamiltonian matrix given in a tridiagonal form,


α1 β1 0 · · · · · · · · ·
···
 β1 α2 β2 0 · · · · · ·
··· 


 0 β2 α3 β3 0
···
··· 


 ..

..
 . 0 β3 . . . . . .
.
··· 
Ĥtri = 
.
..
..
..
..
.. 
 ..
.
.
.
. 
. 0
 .
 .

..
.. . .
..
..
 ..
.
.
. βN −1 
.
.


..
..
..
..
..
. βN −1 αN
.
.
.
.
(5.78)
Our aim is to calculate the first diagonal element G11 of the Greens function
Ĝ = (E Iˆ − Ĥtri )−1 without a computing the whole Greens function and all
the eigenvalues/eigenfunctions of the Hamiltonian.
Let us denote λi = G1i . From the definition of the Greens function we
obtain [60],
(E − αi )λi − βi−1 λi−1 − βi λi+1 = 0, 2 ≤ i ≤ N − 1
(5.79)
with (E − αN )λN − βN −1 λN −1 = 0 and (E − α1 )λ1 − β1 λ2 = 1. Expressing
sequentially λN by λN −1 , λN −1 via λN −2 and λN −3 we express G11 as a
continued fraction [58, 59, 60, 61]:
1
G11 =
E − α1 −
.
β12
E − α2 −
(5.80)
β22
β2
E − α3 − . 3
..
Even for a huge matrices calculation of G11 does not require a lot of
computational time. However if the Hamiltonian matrix is given in a general form, i.e. consist of arbitrary number of diagonals, it must be first
52
Modeling
Figure 5.7: LDOS calculated with the use of the continued fraction technique. M is the number of terms included in Eq.(5.81). Energy is in units
of the hopping integral.
transformed into the tridiagonal form [see Sec.(5.3.6)]. Tridiagonalization
procedure is a very time-consuming routine. Therefore it is of great importance to use as less number of terms in Eq.(5.80) as possible. It can be
done by truncation of the continues fraction
1
G11 =
E − α1 −
β12
E − α2 −
.
(5.81)
β22
..
.
2
E − αM − βM
Σ(E)
In the above equation we truncated the order-N continued fraction at the
fraction M < N by introducing the self-energy Σ(E),
1
Σ(E) =
E − αM −
=
2
βM
E − αM −
2
βM
..
.
1
,
2
E − αM − βM
Σ(E)
(5.82)
that includes all the remaining terms M + 1 ≤ i ≤ N. Solving Eq. (5.82)
one easily obtains
p
2
E − αM − i 4βM
− (E − αM )2
Σ(E) =
.
(5.83)
2
2βM
Real-space Kubo method
53
The number of terms M included in the summation in Eq. (5.81) is determined from the condition for the convergence of G11 . As it is seen from
Fig.(5.7), the higher energy the more elements should be included in the
summation. For this particular case (N = 10000), half of the elements is
enough to achieve high accuracy of calculations in a broad energy interval.
5.3.6
Tridiagonalization of the Hamiltonian matrix
In the real-space representation the tridiagonal Hamiltonian, which is given
by Eq.(5.78), usually corresponds to a one-dimensional chain with nearestneighbor interaction. In practice, however, Hamiltonian has more compicated form. For instance, for 2D graphene it can consist of up to 9
diagonals if the periodic boundary conditions are used. In order to utilize
the continued fraction technique to calculate G11 the Hamiltonian should
be first transformed to the tridiagonalized form H → Htri . This is done
by constructing a new orthogonal basis as described below. We start by
selecting the first basis vector |1}. Curled brackets |i} are used to denote
the new basis vectors while straight brackets |ii denote the old ones. If
the tridiagolization is performed in order to find the local density if states
on the i-th site of the system at hand, the first basis vector is selected as
|1} = |ii, where |ii = c†i |0i.
We require that the Hamiltonian in the new basis be of the tridiagonal
form (5.78). By operating Ĥ|i} we arrive to the following equations [58,
59, 60],
Ĥ|1} = α1 |1} + β1 |2},
Ĥ|i} = βi−1 |i − 1} + αi |i} + βi |i + 1}, 2 ≤ i ≤ N − 2,
Ĥ|N} = βN −1 |N − 1} + αN |N}.
(5.84)
(5.85)
(5.86)
Using Eq. (5.84) and the orthogonality relation {1|2} = 0 we obtain the
second basis vector and the matrix elements α1 and β1 ,
1 |2} = √
Ĥ|1} − α1 |1} ,
(5.87)
C2
α1 = {1|Ĥ|1}, β1 = {2|Ĥ|1},
where the normalization coefficient C2 (as well as all other normalization
coefficients Ci , 2 ≤ i ≤ N) are obtained from the normalization requirement
{i|i} = 1.
We then proceed to Eq. (5.85) and recursively calculate the basis vectors |i}, 2 ≤ i ≤ N − 2, and corresponding matrix elements αi and βi ,
1 |i + 1} = √
Ĥ|i} − βi−1 |i − 1} − αi |i} ,
(5.88)
Ci+1
αi = {i|Ĥ|i}, βi+1 = {i + 1|Ĥ|1}, 2 ≤ i ≤ N − 2.
54
Modeling
Finally, from Eq. (5.86) we obtain
|N} = √
1 Ĥ|N} − αN |N} , αN = {N|Ĥ|N},
CN
(5.89)
which concludes the tridiagonalization procedure.
5.3.7
Local density of states
The techniques describe in the previous sections, namely tridiagolization
and continued fraction technique, provides an efficient way to calculate the
local density of states (LDOS). If the system is described by the real-space
Hamiltonian H given in the matrix form, LDOS at site i can be calculated
by (see Eq.(5.12) in Sec.(5.1))
1
(5.90)
ρi (E) = − ℑ lim Gii (E + iη) , G(E) = (E Iˆ − H)−1 .
η→0
π
In order to calculate Gii , the Hamiltonian must be first tridiagonalized. The
first basis vector is chosen in a such way that all elements except i-th are
equal to zero, hence |1} = |ii. Then, then the continued fraction technique
is applied to calculate G11 (E), which in the new basis corresponds to the
required Gii (E).
If one needs to compute DOS, it can be done by calculating LDOS
on each site and then averaged over them. However this is not an efficient
way, since the Hamiltonian must be tridiagonalized for each site separately.
Instead we generate a random state, which extends over M-sites in the
middle of the structure,
1 X 2iπαi
|ψran i = √
e
|ii,
M i
(5.91)
and choose the first basis vector as |1} = |ψran i. Then the Hamiltonian is
tridiagonalized as described in Sec.(5.3.6) and LDOS computed by ρi (E) =
− π1 ℑ [limη→0 Gii (E + iη)] using the continues fraction technique. In this
new basis one ’site’ corresponds to the chosen domain consisting of M real
sites. Therefore calculating LDOS we obtain DOS.
Chapter 6
Summary of the papers
6.1
Paper I
In this paper we consider a system consisting of the graphene nanoribbon
located on an insulating substrate with a metallic back used to tune charge
density. We develop an analytical theory for the gate electrostatics of the
GNRs and calculate the capacitance. The total capacitance of the system is
a sum of two contributions: the classical capacitance which is determined by
the geometry of the system and a dielectric permittivity and the quantum
capacitance which arises from a finite density of state of the GNR. To
complement our study we also perform numerical calculations which are
based on the tight-binding Hamiltonian. The effect of electron-electron
interaction is taken into account within the Hartree approximation. For
a chosen gate voltage the charge density is calculated self-consistently by
integrating LDOS which is computed using the recursive Green’s function
technique.
We show that the distribution of charge density and potential is not
uniform. Due to the electrostatic repulsion electrons tend to accumulate
near the boundaries. It is also demonstrated that electron-electron interaction leads to significant modification of the band structure. Our exact
numerical calculations show that the density distribution and the potential
profile in the GNRs are qualitatively different from those in conventional
split-gate quantum wires with a smooth electrostatic confinement where
the potential is rather flat and the electron density is constant throughout
the wire. At the same time, the electron distribution and the potential
profile in the GNR are very similar to those in the cleaved-edge overgrown
quantum wires (CEOQW) exhibiting triangular-shaped quantum wells in
the vicinity of the wire boundaries. This similarity reflects the fact that
both the CEOQWs and the GNRs correspond to the case of the hard-wall
confinement at the edges of the structure.
56
6.2
Summary of the papers
Paper II
We study interaction and screening effects in a gated bilayer graphene
nanoribbon. We employ the numerical model similar to the model described in paper I in order to study electron distribution and the energy
gap in bGNRs. We also derive an analytical expression for a dependence of
a potential difference between graphene layers and the gate voltage. The
analytical estimations are shown to be in a good quantitative agreement
with the numerical calculations. We demonstrate, see Fig.(6.1), that in
(a)
100
6
Eg (meV)
80
5
60
2
40
d=15 nm
d=50 nm
d=100 nm
1
20
3
0
0
4
5
10
15
EF
E (γ 0 )
0.62
1
30
2
0.95
0.54
0.91
0.5
0.87
-0.08
-0.04
0
0.04
-0.08
0.08
2.12
EF
0.04
0
-0.04
0.08
EF
2.08
3
1.44
E (γ 0 )
25
EF
0.99
0.58
1.48
4
2.04
1.4
2
1.36
1.96
1.32
-0.1
4.5
E (γ 0 )
20
V g (V)
(b)
-0.05
0
0.05
0.1
1.92
-0.15 -0.1 -0.05
8.7
EF
0
0.05
5
8.5
4.3
-0.2
6
8.6
4.4
0.1 0.15
EF
-0.1
0
k (1/a)
0.1
0.2
-0.2
-0.1
0
0.1
0.2
k (1/a)
Figure 6.1: (a) The energy gap as a function of the applied gate voltage
for various dielectric thicknesses d. (b) Representative band structures
corresponding to the Fermi energies shown in (a). The dashed lines indicate
the positions of Fermi energy.
contrast to 2D graphene sheets the energy gap of the bGNRs dependence
nonlinearly on the applied gate voltage. Also the energy gap can collapse
at some intermediate gate voltages which is explained by the strong modification of the energy spectrum caused by the electron-electron interactions.
Paper III
6.3
57
Paper III
Conductance quantization is a hallmark of mesoscopic physics. The first
experimental observation of the conductance plateau was done more than
20 years ago [62]. In magnetic field conductance plateau in a quantum point
contact becomes more clearly defined, which is attributed to formation of
the robust edge states. In this paper we investigate the conductance of
gated graphene nanoribbons in a perpendicular magnetic field. We adopt
the recursive Green’s function technique to calculate the transmission coefficient which is then used to compute the conductance according to the
Landauer approach.
10
20 K
Hartree
one-electron
G (2e2/h)
8
50 K
Hartree
one-electron
(d)
(b)
6
(c)
(a)
4
_ ~ 11
w
l B~
B
2
w
d
n
ribbo
nano
dielectric
gate
0
0
2
4
6
8
10
12
14
filling factor
Figure 6.2: Conductance of the GNR as a function of filling factor for
interacting and noninteracting electrons at temperatures T = 20 K (red
thick lines) and 50 K (blue thin lines) in a magnetic field B = 30 T. Inset:
sketch of the sample geometry.
Figure (6.2) shows the conductance as a function of filling factor at a
fixed magnetic field. We compare both the case of interacting particles calculated within the Hartree approximation (solid lines) and the one-electron
case (dashed lines). In the one-electron case the conductance is quantized
as it is expected for ideal (without disorder) GNRs. However in the interacting case the conductance quantization is destroyed. This surprising
behavior is related to the modification of the band structure of the GNR
due to the electron interaction leading, in particular, to the formation of
compressible strips in the middle of the ribbon and existence of counterpropagating states in the same half of the GNR.
58
Summary of the papers
6.4
Paper IV
ν = 0.0
ν = 0.1
ν = 0.52
0.04
0.04
0.04
b)
c)
0.02
0.02
0.00
0.00
0.00
-0.02
-0.02
-0.02
-2
n (nm )
a)
0.02
-0.04
-10
0.0
y (nm)
10
-0.04
-10
-0.04
10
-10
0.0
y (nm)
0.0
y (nm)
10
0.010
d)
e)
0.105
f)
0.505
0.005
E(k)/t 0
0.100
0.500
0.000
0.095
0.495
-0.005
0.090
0.490
-0.010
-0.2 -0.1
0.0
kx (1/a)
0.1
0.2
-0.2 -0.1
0.0
k x (1/a)
0.1
0.2
-0.2 -0.1
0.0
0.1
0.2
kx (1/a)
Figure 6.3: (a)-(c) Charge density distributions, and (d)-(f) band structure
of a graphene nanoribbon in a perpendicular magnetic field B = 150T.
(a),(d) one-electron approximation, eVg /t0 = 0; (b),(e) self-consistent calculations for eVg /t0 = 0.1 and (c),(f) eVg /t0 = 0.5. Red and blue curves
marked by ↑ and ↓ correspond to the charge densities of spin-up and spindown electrons respectively,n↑ (y), n↓(y). Charge densities are averaged over
three successive sites. Green curves correspond to the total density distribution n(y) = n↑ (y)+n↓ (y). Dashed lines define the Fermi energy position.
Yellow fields correspond to the energy interval [−2πkB T, 2πkB T ]; temperature T = 4.2 K.
One of the interesting peculiarity of graphene is the existence of the
0’th Landau level in a magnetic field [63]. This level is located at the Dirac
point and equally shared between electrons and holes. In a high enough
magnetic fields experiments exhibit the formation of an insulating state at
ν = 0 which is manifested by a peak on ρxx plots [64]. The nature of this
state is still under debate. It was suggested that backscattering of spin
polarized edge states at ν = 0 can be responsible for the increase of ρxx
[65].
Figure (6.3) shows the self-consistent distribution of the charge density
and the band-structure of the GNR for different values of the filling factor.
Our self-consistent calculations demonstrate that, in comparison to the one-
Paper V
59
electron picture, electron-electron interaction leads to the drastic changes
in the dispersion relation and structure of the propagating states in the
regime of the lowest LL such as a formation of the compressible strip and
opening of additional conductive channels in the middle of the ribbon. Also
we study the effect of different types of disorder (short-range impurities,
edge disorder, warping and spin-flipping) on GNRs conductance, focusing
on the robustness of, respectively, edge and bulk state transmissions. The
latter are shown to be very sensitive to the disorder and get scattered even
if the concentration of the disorder is moderate. In contrast, the edge states
are very robust and cannot be suppressed even in the presence of the strong
spin-flipping.
6.5
Paper V
Recent experiments address the effect of correlation in the spatial distribution of disorder on the conductivity of graphene sheet by doping it with
potassium atoms. It has been found that the conductivity of the system at
hand increases as the temperature rises, and argued that this was caused
by the enhancement of correlation between the potassium ions due to the
Coulomb repulsion [66]. In paper V using the efficient time-dependent
real-space Kubo formalism, we performed numerical studies of conductivity
of large graphene sheets with random and correlated distribution of disorder. In order to describe realistic disorder, we used models of the shortrange scattering potential (appropriate for adatoms covalently bound to
graphene) and the long-range Gaussian potential (appropriate for screened
charged impurities on graphene and/or dielectric surface). The calculations for the uncorrelated potentials are compared to the corresponding
predictions based on the semiclassical Boltzmann approach and to exact
numerical calculations performed by different methods. We find that for
the most important experimentally relevant cases of disorder, namely, the
strong short-range potential and the long-range Gaussian potential, the
correlation in the distribution of disorder does not affect the conductivity
of the graphene sheets as compared to the case when disorder is distributed
randomly. Our results strongly indicate that the temperature enhancement
of the conductivity reported in the recent study [66] and attributed to the
effect of dopant correlations was most likely caused by other factors not
related to the correlations in the scattering potential.
6.6
Paper VI
The value of a spin-splitting caused by the Zeeman effect is proportional
to the g-factor. For graphene the g-factor equals to its free-electron value
60
Summary of the papers
g = 2. If the electron-electron interaction is significant, the spin-splitting
is increased. In this case the effective g-factor is introduced to incorporate
this effect. Recent experiments shows that effective g-factor in graphene
is enhanced in comparison to the free-electron value g ∗ = 2.7 ± 0.2 [67],
which indicates that electron-electron interaction effects play an important
role and should be taken into account for explanation of the enhanced
spin-splitting.
Figure 6.4: The effective g-factor as a function of the filling factor ν for different concentrations of charged impurities, ni = 0%, 0.02%, 0.08%, 0.2%,
at the constant perpendicular magnetic field B = 35T. Inset: the dependence of g ∗ on the Hubbard constant U for the fixed ν = 2.5. All the
calculations are done at the temperature T = 4 K.
In this work we employed the Thomas-Fermi approximation in order
to study the effective g-factor in graphene in the presence of a perpendicular magnetic field taking into account the effect of charged impurities
in the substrate. We found that electron-electron interaction leads to the
enhancement of the spin splitting, see Fig. (6.4), which is characterized
by the increase of the effective g-factor. We showed that for a low impurity concentration g ∗ oscillates as a function of the filling factor ν in the
∗
∗
range from gmin
= 2 to gmax
≈ 4 reaching maxima at even filling factors
and minima at odd ones. Also, we outlined the influence of impurities on
the spin-splitting and demonstrated that the increase of the impurity concentration leads to the suppression of the oscillation amplitude and to a
saturation of the the effective g-factor around a value of g ∗ ≈ 2.3.
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