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Transport in Graphene

Klaus Richter

Universitat Regensburg

Transport in Graphene Strasbourg, July 2012

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Graphene layers

Graphite

3d layered material withhexagonal 2d layers

Bilayer graphene

Monolayer graphene

2-dimensional material;zero-gap semiconductor;Dirac spectrum of electrons

figures: Courtesy of E McCannTransport in Graphene Strasbourg, July 2012

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Outline:

1 monolayer graphene: electronic properties and tight-binding model physics near K-points: Dirac equation

2

quantum transport (Landauer formalism)

3 transport in bulk graphene:Chiral tunneling and Klein paradox

4 transport through graphene nanoribbons:edge magnetism and spin currents


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Tight binding model of monolayer graphene

+ lecture notes by Ed McCannTransport in Graphene Strasbourg, July 2012

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Bond formation of graphene

Carbon: 6 electrons 2 core electrons 4 valence electrons: one 2s and three 2p orbitals

sp2 hybridization: 2s and two 2p orbitals form three -bonds in plane

stability of graphene as real 2D-system


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Bond formation of graphene

Carbon: 6 electrons 2 core electrons

4 valence electrons: one 2s and three 2p orbitals

sp2 hybridization: remaining 2pz orbital: orbital perpendicular to plane

at energies of interest for transport measurements:

only -orbital relevant keep only this orbital per site in a corresponding tight-binding model


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Graphene lattice

honeycomb lattice two different ways of bond orientation two different types of atomic sites (chemical identical)


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Graphene lattice

two different atomic sitestwo interpenetrating triangular sub-lattices

triangular lattice with a basis of two atoms (A and B) per unit cell

2 sublattice degrees of freedom:


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Reciprocal lattice

triangular reciprocal latticehexagonal Brillouin zone withtwo unique corner points:

K-points or K-valleys,

2 valley degrees of freedom:


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Tight binding model

Schrodinger equation:

H(k, r) =

p2

2m+ V(r)

(k, r) = E(k)(k, r)

one -orbital per site

two orbitals per unit cell

tight binding approximation: expansion in Bloch functions(label j = 1 (A sites) and j = 2 (B sites)):

j(k, r) =1N

Rj

eikRj j (rRj)

with

PRj

: sum over all type j atomic sites

j : atomic wavefunction


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Tight binding model

Eigenfunction j(k, r) (for j = 1, 2)written as a linear combination of Bloch functions:

j(k, r) =2

j=1

Cjj (k) j(k, r) = CjA(k) A(k, r) + CjB (k) B(k, r)

Eigenvalues Ej (for j = 1, 2) written as:

Ej(k) =j |H|jj |j

in terms of Bloch functions:

Ej(k) =i,l CjiCjli|H|l

i,l CjiCjli|l

= i,l CjiCjlHili,l C

jiCjlSil

with transfer matrix elements Hil = i|H|l

and overlap matrix elements Sil = i|l


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Tight binding model

calculation of transfer and overlap integrals:

j(k, r) =1

N Rj eikRj j (r

Rj)

Hij = i|H|j , Sij = i|j

diagonal matrix elements:

HAA = HBB = 0 ; SAA = SBB = 1

A and B sites are chemically identical;every A site has three neighbours

off-diagonal matrix elements:(nearest-neighbour transfer integrals)

HAB =

A

|H

|B

=

tf(k) ; SAB = sf(k) ; f(k) =

3

j=1 eikj


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Ti h bi di d l b d

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Tight binding model: band structure

six corners of the Brillouin zone (K-points),but only two are non-equivalent


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Graphene effective hamiltonian

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take into account both, K and K-points:

(AK, BK, AK , BK)t

effective 4 4-Hamiltonian:

H = vF

0 px ipy 0 0px + ipy 0 0 0

0 0 0 px ipy0 0 px + ipy 0

inter-valley coupling often of relevance



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Hamiltonian for one K-point, e.g. = 1:

H = vF 0 px ipypx + ipy 0 = vF (xpx + ypy) = vF p = vFp nwith

x =

0 11 0

; y =

0 ii 0

helical electrons: chiral operator n projects pseudospin onto quantization axis n eigenstates of H are also eigenstates of n with eigenvalues 1 pseudospin direction is linked to momentum direction

conduction band (electrons):

n = 1valence band (holes):

n = -1

absence of backscattering in ideal graphene !


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Quantum transport:

Landauer formalism


From macroscopic to microscopic conductors

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From macroscopic to microscopic conductors

classical (Ohms law) quantum


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Green function formalism for transport

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Green function formalism for transport

m n

Conductance:

G = (e2/h)T with

T =N

n=1N

m=1|tnm|2 = Tr(lGrlGa)

retarded Green function: Gr = (EHscat r)1

self-energies: r =

leads

rl

coupling to lead l: l = i(r

l a

l )

use recursive Green function techniques

(matrix reordering strategies; graph-theoretical approaches)M. Wimmer and KR, J. Comp. Phys (2009), arXiv:0806.2739


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Quantum transport through graphene

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Quantum transport through graphene

Landauer approach + tight-binding approximation


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Klein backscattering phase shift

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g p

consider a phase-coherent npn junction

prediction of a half-period phase shift in the associated Fabry-Perotoscillation:

= 2WKB + 1 + 2

A.V. Shytov, M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 101, 156804 (2008).


A key experiment on Klein backscattering

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y p g

A.F. Young and P. Kim, Nature Phys. 5, 222 (2009)


Observation of Klein backscattering phase shift

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A.F. Young and P. Kim, Nature Phys. 5, 222 (2009)


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Transport in graphene nanoribbons:

edge magnetism and spin currents


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Edge states in zigzag nanoribbons

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scanning tunnelingexperiment

DFT calculation oflocal DoS

X. Zhang et al., Crommie lab. (2012)


Edge magnetism in zigzag nanoribbons

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Zigzag graphene nanoribbons: flat-band high density of states magnetism of the edge state

Fujita et al., JPSJ (1996) (mean field Hubbard model)

Y.-W. Son et al., Nature (2007) (DFT)

H. Feldner et al. (2011)

edge magnetization within each sublattice A and B

ground state: opposite magnetization of A and B edge states

staggered magnetization


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Edge magnetism in zigzag nanoribbons

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Zigzag graphene nanoribbons: flat-band high density of states magnetism of the edge state

Fujita et al., JPSJ (1996) (mean field Hubbard model)

Y.-W. Son et al., Nature (2007) (DFT)

H. Feldner et al. (2011)

Xk

-3

-2

-1

0

1

2

3

E-EF

[eV]

edge magnetization within each sublattice A and B

ground state: opposite magnetization of A and B edge states

staggered magnetization


Modelling edge state magnetism

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staggered magnetization of edge state:

Hmag =M s on sublattice AM s on sublattice B

fit magnetization to DFT calculations:


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Spin injection in graphene

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net spin conductance !

M. Wimmer, I. Adagideli, S. Berber, D. Tomanek, KR, Phys. Rev. Lett. (2009)


Ribbons with rough edges

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more realistic in todays experiments: rough edges:

1

20 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1

R

R

Spin conductance: Gs = G GIf R1 = R2, Gs = 0.However: mesoscopic conductance fluctuations!

Spin conductance fluctuations:

Var Gs = Var G + Var G = Var Gtot

rms Gs = Var Gs = rms GtotTransport in Graphene Strasbourg, July 2012

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Spin conductance fluctuations

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Disordered graphene nanoribbon: single ribbon


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