Transport properties of mesoscopic graphene

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Transport properties of mesoscopic graphene
Journées du graphène Laboratoire de Physique des
Solides Orsay, 22-23 Mai 2007

• Björn Trauzettel

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

• Brief introduction
• Transport in graphene as scattering problem
• Conductance/conductivity and shot noise
• Photon-assisted transport in graphene
• Summary and outlook

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Honeycomb lattice
1. Brillouin zone
real lattice (2 atoms per unit cell)
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Tight binding model
Eigenstates
or
pseudospin structure
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Solution to Schrödinger equation
conduction and valence band touch each other at
six discrete points the corner points of the
1.BZ (K points)
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Effective Hamiltonian ? Dirac equation
with
Low energy expansion
effective Hamiltonian
Dirac equation in 2D for mass-less particles
similar for the other K-point
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Outline

• Brief introduction
• Transport in graphene as scattering problem
• Conductance/conductivity and shot noise
• Photon-assisted transport in graphene
• Summary and outlook

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Schematic of strip of graphene

• different boundary conditions in y-direction
• voltage source drives current through strip
• gate electrode changes carrier concentration

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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? ? ?

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Problem II boundary conditions
(i) armchair edge
(mixes the two valleys metallic or
semi-conducting)
Brey, Fertig PRB 73, 235411 (2006)
(ii) zigzag edge
(one valley physics couples kx and ky)
(iii) infinite mass confinement
(one valley physics smooth on scale of lattice
spacing)
Berry, Mondragon Proc. R. Soc. Lond. (1987)
see also Peres, Castro Neto, Guinea PRB 73,
241403 (2006)
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Experimental feasibility
Geim, Novoselov Nature Materials 6, 183 (2007)
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Underlying wave equation
kinetic term
gate voltage term
boundary term (infinite mass confinement)
Ansatz
in leads
in graphene
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Scattering state ansatz

• Dirac equation (first order differential
  equation)
• continuity of wave function at x0 and xL
• determines tn and rn
• ? transmission Tntn2

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Solution of transport problem
In the limit V??? (infinite number of
propagating modes in leads)
Transmission coefficient (at Dirac point)
for
propagating modes in leads
phase ? depends on boundary conditions
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Transmission through barrier

• send L ? ? W ? ?
• keep W/L const.
• ? transmission remains finite

In contrast Schrödinger case
? transmission Tn ? 0 for klead ? ?
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Outline

• Brief introduction
• Transport in graphene as scattering problem
• Conductance/conductivity and shot noise
• Photon-assisted transport in graphene
• Summary and outlook

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Conductivity influence of b.c.
metallic armchair edge
Landauer formula
universal limit W/L ? 1
conductivity
infinite mass confinement
at Dirac point (in universal regime) conductance
proportional to 1/L
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Conductivity Vgate dependence
Experiment
Our theory
Tworzydlo, et al. PRL 96, 246802 (2006)
Novoselov, et al. Nature 438, 197 (2005)
Possible explanations charged Coulomb
impurities Nomura, MacDonald PRL 98, 076602
(2007) strong (unitary) scatterers Ostrovsky,
Gornyi, Mirlin PRB 74, 235443 (2006)
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Alternative data (Delft group)
Delft data
Our theory
H. Heersche et al., Nature 446, 56 (2007)
conductivity vs. conductance
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Current noise
Average current Current fluctuations
We are interested in the zero frequency and zero
temperature limit. ? shot noise
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Shot noise effect of b.c.
infinite mass confinement
Fano factor
universal limit W/L ? 1
metallic armchair edge
Tworzydlo, Trauzettel, Titov, Rycerz, Beenakker,
PRL 96, 246802 (2006)
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Maximum Fano factor
unaffected by different boundary conditions
scaling system size to infinity

• sub-Poissonian noise
• universal Fano factor 1/3 for W/L ? 1

same Fano factor as for disordered quantum
wire Beenakker, Büttiker, PRB 46, 1889 (1992)
Nagaev, Phys. Lett. A 169, 103 (1992)
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Sweeping through Dirac point
normal tunneling (CB ? CB)
Klein tunneling (CB ? VB)
directly at the Dirac point transport through
evanescent modes resembles diffusive transport
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How good is the model for leads?

• If graphene sample biased
• close to Dirac point
• difference between GGG
• and NGN junctions is only
• quantitative

GGG
NGN
Schomerus, cond-mat/0611209
see also Blanter, Martin, cond-mat/0612577
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Experimental situation I
Arrhenius plot
Egap ? 28meV for ribbon of graphene with length
of 1?m and width of 20nm
Chen, Lin, Rooks, Avouris cond-mat/0701599
Similar results Han, Oezyilmaz, Zhang, Kim
cond-mat/0702511
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Experimental situation II
Miao, Wijeratne, Coskun, Zhang, Lau
cond-mat/0703052
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Outline

• Brief introduction
• Transport in graphene as scattering problem
• Conductance/conductivity and shot noise
• Photon-assisted transport in graphene
• Summary and outlook

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Motivation Zitterbewegung

• superposition of positive and negative energy
  solution
• current operator with interference terms

electron-like
hole-like
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Zitterbewegung in current operator
Zitterbewegung contribution to current (due to
interference of e-like and h-like solutions to
Dirac equation)
Katsnelson EPJB 51, 157 (2006)
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Can Zitterbewegung explain the previous shot
noise result?
Answer I dont think so.
Question Why not?
In the ballistic transport problem, the wave
function is either of electron-type or of
hole-type, but not a superposition of the two!
no interference term in ballistic transport
calculation
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How to generate the desired state
Trauzettel, Blanter, Morpurgo, PRB 75, 035305
(2007)
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Transport properties
The current oscillates
due to applied ac signal and not due to an
intrinsic zitterbewegung frequency.
Differential conductance (in dc limit) can be used
to probe energy dependence of transmission
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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/L?1
• photon-assisted transport in graphene

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Aim spin qubits in graphene quantum dots
Trauzettel, Bulaev, Loss, Burkard, Nature Phys.
3, 192 (2007)
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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.

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Solutions to confinement problem
generate a gap by suitable boundary
conditions Silvestrov, Efetov PRL 2007 Trauzettel
et al. Nature Phys. 2007
biased bilayer graphene Nilsson et al.
cond-mat/0607343
magnetic confinement De Martino, DellAnna, Egger
PRL 2007
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Illustration of degeneracy problem
based on Pauli principle
One K-point only
Two degenerate K-points
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Solution to both problems
K point
K point
ribbon of graphene with semiconducting armchair
boundary conditions
Brey, Fertig PRB 2006
? K-K degeneracy is lifted for all modes
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Emergence of a gap
bulk graphene with local gates
ribbon of graphene (with suitable boundaries)
? local gating allows us to form true bound states
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Calculation of bound states
appropriate energy window
solve transcendental equation for ?
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Energy bands for single dot
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Energy bands for double dot
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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