Transport properties of mesoscopic graphene

Björn Trauzettel

Journées du graphène

Laboratoire de Physique des Solides

Orsay, 22-23 Mai 2007

Collaborators: Carlo Beenakker, Yaroslav Blanter, Alberto Morpurgo, Adam Rycerz, Misha Titov, Jakub Tworzydlo

Outline

• Brief introduction

• Transport in graphene as scattering problem

• Conductance/conductivity and shot noise

• Photon-assisted transport in graphene

• Summary and outlook

Honeycomb lattice

real lattice(2 atoms per unit cell)

1. Brillouin zone

† †

,i ii j i jR RR R

i j

H t A B AB

Tight binding model

† † 0i i

i i

ik R ik Rk k kR R

i i

BAe e

��������������������������������������� ���

†

†0i

i

k R

ki k R

iik R

B

Ae

�������������� ����������������������������

orpseudospin

structure

Eigenstates:

Solution to Schrödinger equation

( )H E k

conduction and valence band touch each otherat six discrete points: the corner points of the 1.BZ (K points)

Effective Hamiltonian Dirac equation

2

2

0 ( )( )

( ) 0k kx y

Fk kx y

k ik O kv E k

k ik O k

( )k kF

k k

v k E k

�������������������������� ��

Dirac equation in 2D for mass-less particles

( )q qH E q

q K k

with

effective Hamiltonian ( ) | |FE k v k

Low energy expansion:

… similar for the other K-point

Outline

• Brief introduction

• Transport in graphene as scattering problem

• Conductance/conductivity and shot noise

• Photon-assisted transport in graphene

• Summary and outlook

Schematic of strip of graphene

different boundary conditions in y-direction

voltage source drives current through strip

gate electrode changes carrier concentration

Problem I: How to model the leads?

electrostatic potential shifts Dirac points of different regions

large number of propagating modes in leads

zero parameter model for leads for V

Problem II: boundary conditions

(iii) infinite mass confinement

(i) armchair edge

(ii) zigzag edge

(mixes the two valleys;metallic or semi-conducting)

(one valley physics;couples kx and ky)

(one valley physics; smooth on scale of lattice spacing)

Brey, Fertig PRB 73, 235411 (2006)

Berry, Mondragon Proc. R. Soc. Lond. (1987)

see also: Peres, Castro Neto, Guinea PRB 73, 241403 (2006)

Experimental feasibility

Geim, Novoselov Nature Materials 6, 183 (2007)

Underlying wave equation

( ) ( )[ ]2x x y y zvp vp v M y xs s s m e+ + + Y = Y

( ),

,,

1

1n n

n kiq y iq y ikx

n k n nn k

zr a e b e ez

-æ æ ö æ ö ö÷ ÷ ÷ç ç ç÷ ÷ ÷Y = +ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è è ø è ø ø

r

, 2 2n

n kn

k iqz

k q

+=

+

kinetic term

boundary term (infinite mass confinement)

gate voltage term

Ansatz:

2 2lead nv k qe m= + +h

2 2gate nv k qe m= + +%h

in leads:

in graphene:

Scattering state ansatz

Dirac equation (first order differential equation)

continuity of wave function at x=0 and x=L

determines tn and rn

transmission Tn=|tn|2

( )

, ,

, ,

,

( ) ( ) ; 0

( ) ( ) ;0

( ) ;

ikx ikxn k n n k

ikx ikxn nn k n k

ik x Ln n k

y e r y e x

y e y e x L

t y e x L

a b

--

-

-

ì C + C <ïïïïïïY = C + C < <íïïï C >ïïïî

% %% %

Solution of transport problem

( )

( ) ( )

222

2 22 2

n n

n nn k L k L

n n n n

q kT

e q k i e q k ikk k-

- -=

- + + - -

/gateeV vk = h

( )[ ]21

cosh /nT n L Wa p=

+

Transmission coefficient (at Dirac point):

/N W L?

phase depends on boundary conditions

for

propagating modesin leads

In the limit |V| (infinite number of propagating modes in leads):

2 2

2 2

;

;

n n

n

n n

q qk

i q q

k k

k k

ìï - >ïï= íï - <ïïî

Transmission through barrier

( )[ ]21

cosh /nT n L Wa p=

+

( ) ( )2 12 2

2 22 21/ cosh 1 1 sinh

2lead lead

n n nn n

k kT q L q L

q q

-é ùæ ö æ ö÷ ÷ê úç ç= + - -÷ ÷ç ç÷ ÷ê ú÷ ÷ç çç çè ø è øê úë û

• send L ; W ;• keep W/L = const.

transmission remains finite

In contrast: Schrödinger case

transmission Tn 0 for klead n

nq

Wp

=

Outline

• Brief introduction

• Transport in graphene as scattering problem

• Conductance/conductivity and shot noise

• Photon-assisted transport in graphene

• Summary and outlook

Conductivity: influence of b.c.

12

0

4 N

nn

L eT

W hs

-

=

= å

infinite mass confinement

metallic armchair edge

universal limit:W/L 1

12

0

4 N

nn

eG T

h

-

=

= å

Landauer formula:

conductivity:

at Dirac point (in universal regime): conductance proportional to 1/L

Conductivity: Vgate dependence

Tworzydlo, et al. PRL 96, 246802 (2006)

Experiment:

Novoselov, et al. Nature 438, 197 (2005)

gateVmµ

Possible explanations: charged Coulomb impurities Nomura, MacDonald PRL 98, 076602 (2007)strong (unitary) scatterers Ostrovsky, Gornyi, Mirlin PRB 74, 235443 (2006)

Our theory:

Alternative data (Delft group)

gateVmµ

Delft data:

LG

Ws =

Our theory:

conductivity vs. conductance:

-40 -20 0 20 400.0

0.5

1.0

1.5

G (

mS

)

Vg (V)

H. Heersche et al., Nature 446, 56 (2007)

Current noise

( )I j t=

( ) ( ){ }) , 0i tS( dte j t jww+¥

+- ¥

= D Dò

Average current:

Current fluctuations:

We are interested in the zero frequencyand zero temperature limit. shot noise

Shot noise: effect of b.c.

( )1

01

0

1

2

N

n nn

N

nn

T TS

FeI

T

-

=-

=

-= =

å

å

Fano factor:

metallic armchair edge

infinite mass confinement

universal limit:W/L 1

Tworzydlo, Trauzettel, Titov, Rycerz, Beenakker, PRL 96, 246802 (2006)

Maximum Fano factor

sub-Poissonian noise

universal Fano factor 1/3 for W/L 1

same Fano factor as for disordered quantum wireBeenakker, Büttiker, PRB 46, 1889 (1992); Nagaev, Phys. Lett. A 169, 103 (1992)

unaffected by differentboundary conditions &

scaling system sizeto infinity

Sweeping through Dirac point

( )[ ]21

cosh /nT n L Wa p=

+

‘normal’ tunneling (CB CB): Klein tunneling (CB VB):

directly at the Dirac point:transport through evanescent modes resembles diffusive transport

( )21

cosh /nn

TL z

=

How good is the model for leads?

Schomerus, cond-mat/0611209

If graphene sample biased close to Dirac point

difference between GGGand NGN junctions is only

quantitative

GGG

NGN

see also: Blanter, Martin, cond-mat/0612577

Experimental situation IArrhenius plot:

Egap 28meV for ribbon of graphene withlength of 1m and width of 20nm

Chen, Lin, Rooks, Avouris cond-mat/0701599

Similar results: Han, Oezyilmaz, Zhang, Kim cond-mat/0702511

Experimental situation II

Miao, Wijeratne, Coskun, Zhang, Lau cond-mat/0703052

Outline

• Brief introduction

• Transport in graphene as scattering problem

• Conductance/conductivity and shot noise

• Photon-assisted transport in graphene

• Summary and outlook

Motivation: Zitterbewegung

• superposition of positive and negative energy solution

• current operator with interference terms

1( ) 1 1

2ivpt ivpt

p p pp p

t e ep ps s-éæ ö æ ö ù÷ ÷ç çY = + Y + - Yê ú÷ ÷ç ç÷ ÷ç çè ø è øê úë û

r r r r

†p p pj s= Y Yr r

electron-like hole-like

† † † †( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )e e h h e h h ep p p p p p p p pj s s s s= Y Y + Y Y + Y Y + Y Yr r r r r

Zitterbewegung in current operator

Katsnelson EPJB 51, 157 (2006)

Zitterbewegung contribution to current(due to interference of e-like and h-like solutions to Dirac equation)

( ) ( ) ( ) ( )

( )( )

( )( )

( )

†0 1 1

†0 2

† 21 22

ppp

ivptpp

p

j t j t j t j t

p pj t ev

p

p pev ij t p e

pp

sy y

sy s s y

= + +

=

é ùê ú= - + ´ê úë û

å

å

rrr

rrr

r r r r

r r rr

r r rr r r r

Can Zitterbewegung explain the previous shot noise result?

( ) ( ), i te ex t x e e-Y = Y

r r( ) ( ), i t

h hx t x eeY = Yr r

Answer: I don’t think so.

Question: Why not?

In the ballistic transport problem, the wave function is either ofelectron-type or of hole-type, but not a superposition of the two!

no interference term in ballistic transport calculation

How to generate the desired state

( ) ( )cosS acx eV eV tm w= +

Trauzettel, Blanter, Morpurgo, PRB 75, 035305 (2007)

( ) ( ) /,

ac in i m ttr m m m

m

eVt J e e w

w

¥- +

+=- ¥

Y = Yå h h

h

,,

,

/ 2

,,cos

in min m

in m

iiqy ik xinm iinm

ee

e

a

aa

-+

+

æ ö÷ç ÷ç ÷Y = ç ÷ç ÷ç ÷ç ÷è ø

Transport properties

( ) ( ) * ( ')' '

, '0

4eV

ac ac i m m tm m m m

mm

eW eV eVI dq d J J t t e

hwe

p w w-= åò ò h h

( ) ( ) 224 acm m

m

eW eVG dq J t eV

h p w= åò h

The current oscillates

due to applied ac signal andnot due to an intrinsic zitterbewegung frequency.

Differential conductance (in dc limit) can be used

to probe energy dependence of transmission

Summary

• ballistic transport in graphene contains unexpected physics: conductance scales pseudo-diffusive 1/L

• conductivity has minimum at Dirac point

• shot noise has maximum at Dirac point

• universal Fano factor 1/3 if W/L1

• photon-assisted transport in graphene

Aim: spin qubits in graphene quantum

dots

Trauzettel, Bulaev, Loss, Burkard, Nature Phys. 3, 192 (2007)

Why is it difficult to form spin qubits in graphene?

• Problem (i): It is difficult to create a tunable quantum dot in graphene. (Graphene is a gapless semiconductor. Klein paradox)

• Problem (ii): It is difficult to get rid of the valley degeneracy. This is absolutely crucial to do two-qubit operations using Heisenberg exchange coupling.

1 2exchH J S S= ×r r

Solutions to confinement problem

generate a gap by suitable boundary conditionsSilvestrov, Efetov PRL 2007

Trauzettel et al. Nature Phys. 2007

magnetic confinementDe Martino, Dell’Anna, Egger PRL 2007

biased bilayer grapheneNilsson et al. cond-mat/0607343

Illustration of degeneracy problem

1 2exchH J S S= ×r r

One K-point only: Two degenerate K-points:

T SJ E E= -

based on Pauli principle

Solution to both problems

( ) ( ') ( ) ( ')2 / 30 0 0 0/ / / /| | ; | |K K K K

x x x xA B A B A B A Be p±= = = =Y = Y Y = Y

0( )

0

x x y y

x x y yi v eV y

s se

s s

¶ + ¶æ ö÷ç ÷ç- Y + Y = Y÷ç ÷- ¶ + ¶ ÷ççè øh

K point

K’ point

ribbon of graphene with semiconductingarmchair boundary conditions

K-K’ degeneracy is lifted for all modes

Brey, Fertig PRB 2006

Emergence of a gap

bulk graphene with local gates

ribbon of graphene (withsuitable boundaries)

local gating allows us to form true bound states

Calculation of bound states

gate n barriereV vq eVe e- ³ ³ -h

solve transcendental equation for

appropriate energy window

( ) ( ) ( )

( )( ) ( )

2 2

2tan n barrier

barrier gate n

vk vq eVkL

eV eV vqe

e e- -

=- - -

%h h%h

Energy bands for single dot

Energy bands for double dot

Long-distance coupling

ideal system for fault-tolerant quantum computing

low error rate due to weak decoherence high error threshold due to long-range coupling

( ) ( )/ / ln 4 /VB QB gapL W EG G » D