Transport properties of mesoscopic graphene Björn Trauzettel Journées du graphène Laboratoire de...
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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