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Transport in two-dimensional materials

Vassilios VargiamidisWCPM, University of Warwick, UK

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OutlineOverview of graphene

• Crystal structure, band structure, Dirac cones, etc.• Density of states

Transport in silicene

• Tuning the charge conductance and transport gap• Spin-resolved, valley-resolved, and charge conductances• Near perfect spin and valley polarizations• Topological phase transitions

Possible applications

Summary

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Overview of graphene

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Andrey Geim Kostya Novoselov

Philip Kim

Graphene pioneers (Nobel prize 2010)

New 2D electron system (Manchester 2004)

Nanoscale electron system with tunable properties;Field-effect enabled by gating: tunable carrier density

1. very high mobilities in its suspended form: μ ~ 200.000cm2/(V.s)2. ballistic transport over sub-micron distances:3. high thermal conductivity: κ=5000W/(m.K),4. chemically stable, very stiff, . . .

ml P1~

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Semimetal (zero bandgap): electrons and holes coexist

• Hexagonal BZ: 2 inequivalent points and where carriers mimic relativisticmassless Dirac particles

• Linear energy dispersion at and : massless Dirac fermion model in 2D.c.

. Slow, but ultrareletivisticDirac Fermions!

• Electronic energy dispersion: Dirac points

..c

.c.

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Graphene allotropes

3D Graphite

1D Carbon Nanotube (1991)(rolled-up cylinder of graphene)

0D Fullerenes (“buckyball”)

2D Graphene:presumed not to exist in the free state

(1985)

(2004)

Graphene’s honeycomb lattice1st Brillouin zone

Unit cell

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Tight-binding model for electrons on thehoneycomb lattice

Energy bands:Semi-metal

Dirac points at K and K’

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Dirac points

The Dirac points are situated at the points where

Time-reversal symmetry:

Dirac points occur in pairs:twofold valley degeneracy

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Low-energy regime of the TB model: 2D Dirac equation

Valley pseudospin for K and K’ points

Pauli matrices of the sublattice pseudospin

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Transport in silicene

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Crystal structure of silicene

46.02 " ǺBuckled structure

• A potential difference arises between silicon atoms at A sites and B siteswhen an electric field is applied

zE"2vzE

• Energy gap 1.55 meV

• Silicene is compatible with silicon-based technology

meVso 9.3|O

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Low-energy Hamiltonian

� � zzzsozyyxxF skkH WWO[[WWX[ '��� !

1r [ valley index smF /105 5u|X Fermi velocity

iW Pauli matrices of sublattice pseudospin

zE electric field,zz Ee" '

zs electron spin operator normal to silicene plane ( )1r zs

spin-orbit interaction

rkiisk e

eSr

kz

**

** �

¸̧¹

·¨̈©

§

��

<[I[O TOO

TOcos1

cos121)(,,,

Eigenfunctions Eigenvalues

electron (hole) states)1(1 �� O

meVso 9.3|O

2222,,, )( sozzFsk skE

zO[XO[O �'� !

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Band structuresozz OG /' soF k OXE /! nmVEzz /17.01 � G

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Density of states

� � � �> @sozsozF

EEE

ED OOXS

�'�4��'�4 22)(!

The DOS reflects the two gaps that open in thesystem

Evolution of the gap with the electric field:G

zzso s[GOG � 2

The gap closes when 1r zz s[G

This yields withczz EsE [

17/ "socE O meV/Å

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Spin- and valley-polarized transport acrossferromagnetic (FM) silicene junctions

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Low-energy Hamiltonian

� � MsUppH zzsyyxxF z��'�� W[WWX [[

Barrier potential(gate voltage)

Exchange field

sozzs sz

O[[ �' 'with

Single FM junctionU, M

)()( ykxki

N

iFrki

N

iF

Iyxe

Eec

rAeEec

Ar ���

�¸̧¹

·¨̈©

§��¸̧

¹

·¨̈©

§ <

[T[T***

)()( ykxki

F

iFrki

F

iF

IIyxe

ecbe

ecar �c�

��c

¸̧¹

·¨̈©

§ c��¸̧

¹

·¨̈©

§ c <

HH

[I[I***

rki

N

iF

III eEec

tAr*** �

¸̧¹

·¨̈©

§ <

[T

)(

Eigenfunctions

NF EEA 21 ,

Transmission amplitude

zszFFsozFNFFF MsUEsEEkc [HO[X '��� � ,,!

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Transmission and conductance through a FM junction

4/)sinsin2)((sincoscos)(coscoscos)( 212222

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ITDDITITT[ ��c�c

�dkdkT

xxsz

Incidence at an angle :T

FNFF kEk c HD T222 sinFFx kkk �c c

F

soFF

Ek

XO

!

22 �

F

szFF

zMsUE

kX

[

!

22)( '��� c

with and

Fermi wave vectors:

1)( T[ zsT Sndkx cfor as in graphene

Normal incidence :0 T4/))((sin1

1)0( 212 ��c�

DD[ dkT

xsz

(in contrast to grapheneit depends on barrierheight)

zzz sss gGdTGG [

S

S[[ TTT 0

2/

2/0 cos)( ³�

Conductance: hWkeG F S220 ,

W sample width

Spin-resolved conductance

0/,0 Fz EMmG

Resonances for high barrier or deep well with amplitude between 2/3 and 1

2)()(

)(pncpn

pn

� KK gg

g

Transport at the Dirac point (DP) is due to evanescent modes

Transport via evanescent modes is suppressed with increasingand the dip develops to transport gap

zG

nmdmeVEF 110,40

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Tuning the conductance

pn � gggc

T222 sinFFx kkk �c c

becomes imaginary when becomes imaginary

xk c Fk c

MsUEs zFsozcz ��r ' O[

meVMEU F 5.2,2/

For the Fermi level lies in the gapczz '!'

9.11/ ' socz

cz OG

In graphene 0 'zs[

FFF UEk X!� c�

No analog in graphene

It cannot be made imaginary The junction acts as an electric switch

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Band structure sozz OG /' soF k OXE /!

Current is entirely carried by spin-down electrons at the ( oscillatory)K cpg

Only evanescent modes contribute to (monotonically decreasing behavior) ng

0zM

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Conductance and transport gap

0 m

Near the DP is imaginaryFk c

As increases the evanescent modes are gradually damped out transport gap

zG�

socFs E OH /)2,1()2,1(

cFF EE �Transport gap when

sozcFE O�'

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We aim now to explore possible spin and valley polarizations in FM silicene

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Spin-resolved and charge conductances

058.0 m

058.0 m

FEMm /

Relative sift of and in the presence of Mng pgTransport gaps for both

is periodic function of U/EFcg

OGXS dET FFu /! 1!!FEUfor

=0.235

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Valley-resolved and charge conductances

� �� � � �

2pcnc

c

� KKKK

KK

ggg

Kg c

Splitting of the peaks of gc with m

Splitting of the peaks of due to the brokenvalley symmetry in the presence of m

6 zG

6 zG

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Spin polarization

pn

pn

��

gggg

ps

22.0 m5 zG

For large δz or m:

• only a single spin band contributes to the current• the evanescent modes are suppressed• the change of sign is due to the relative shift of and ng pg

Spin polarization can be inverted by changing U

The range of increases with δz or m1|sp

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Spin and valley polarization

pn

pn

��

gggg

ps

KK

KKv gg

ggpc

c

��

8 zG

8 zG

and can be inverted by changing the direction of Msp vp

This is directly related to the relative shift of and ng pg

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Transition from a topological insulator to band insulator regime

Spin-Hall conductivitybecomes zero for highEz

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The results for the polarized transport through a FM silicene junction can be used to realize silicene-based, high-efficiency spin- and valley-filter.

Other possible applications:

•In spintronics (due to longer spin-diffusion time and spin-coherence length)•In quantum computing•Silicene can overcome difficulties associated with potential applications of graphene in nanoelectronics (lack of a controllable gap)•Optoelectronics•Energy harvesting

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Summary• The buckled structure of silicene can facilitate the control of its band gap by the

application of an electric field Ez.

• Above a critical Ez the charge conductance gc through a silicene FM junctionchanges from an oscillatory to a monotonically decreasing function of thejunction width d. A gap develops near the DP with increasing Ez.

• These features can be used for the realization of electric-field controlledswitching.

• The spin ps and valley pv polarizations near the DP increase with Ez or M,and become nearly perfect ( ) above certain Ez and M values.%100|

• ps and pv can be inverted by reversing the direction of M or U: near perfectspin and valley filtering

Most of the results* have no analog in graphene

*APL 105, 223105 (2014), JAP 117, 094305 (2015)