Quantum transport in graphene L1 Disordered graphene (G) graphene - - PowerPoint PPT Presentation

Quantum transport in graphene

L1 Disordered graphene (G)

graphene 101 QHE in G and quantum resistance standard weak localisation regimes in graphene

L2 Ballistic electrons in graphene L3 Moiré superlattice effects in G/hBN heterostructures

• bonds

hybridisation forms strong directed covalent bonds between carbons (at 120o) which determine the honeycomb lattice structure

2

sp

C

• *
• 2

sp

empty full pz-bands

2

sp

Graphenes

Exfoliated from bulk graphite onto a substrate, or hanged suspended (highest quality G/hBN in L2, L3) Grown using chemical vapor deposition (CVD) on metals (Cu, Ni), or insulators: polycrystalline and strained Epitaxial graphene sublimated

• n Si-terminated surface of SiC:

wafer-scale single-crystalline carpet

Wallace, Phys. Rev. 71, 622 (1947) Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

eV 3 ~

Wallace, Phys. Rev. 71, 622 (1947) Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

G

• '

G

• Valley

Valley FirstBrillouin zone

i

e

3 / 2 i

e

3 / 2 i

e

p

• )

( ) ( ,

3 2 2 3 2 3 3 2 2 3 2 y a x a y a y a x a

p p i i p i p p i i K AB

e e e e e H

• )

(

2 3 , y x K BA

ip p a H

• )

(

2 3 y x

ip p a

• sec

8 2 3

10 ~

cm

a v

• p

v ip p ip p v H

y x y x

• ˆ

Bloch function amplitudes (e.g., in the valley K)

• n the AB sites (‘isospin’) mimic spin components
• f a massless relativistic particle.
• B

A

• McClure, PR 104, 666 (1956)

p v H

• Bloch function amplitudes on the A/B sites

' K A B

• Wave function.

sublattice composition is linked to the axis determined by the electron momentum. for conduction band electrons, valence band (‘holes’)

1

• n
• 1
• n
• p
• p
• )

sin , cos (

• p

p p

• i

p

e 1

2 1

• vp

c

• x

p

y

p

vp

v

• B

A K

Angle-resolved photo-emission spectroscopy (ARPES)

• f heavily doped graphene synthesized on silicon carbide
• A. Bostwick et al –— Nature Physics 3, 36 (2007)

high-energy photon ~100eV Simultaneous detection

• f the energy, E and

propagation angle of photo-electrons enables

• ne

to restore completely the band structure.

E A p mE p

• )

( cos 2

|| ||

• work function

Electronic states in graphene photographed using ARPES

vp

• Electronic states in graphene photographed using ARPES

p K G k

• ||
• 2

2 sin ~

2

• BA

R k

• ARPES of heavily doped graphene

synthesized on silicon carbide

Bostwick et al - Nature Physics 3, 36 (2007)

2

| ~|

B A ARPES

I

• B

A

• Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann,

Fal’‚ko, Boswick, Rotenberg - PRB 77, 195403 (2008)

conduction band valence band

vp n

• ,

1

• vp

n

• ,

1

• p
• p
• sublattice ‘isospin’ is

linked to the direction of the electron momentum

• i

p

e 1

2 1

• p
• 3

3

2 2 i i

e e

• d

d d i

2

vp

• vp
• x

p

y

p

p v v H

• i

y x i y x

pe ip p pe ip p p p p

• )

sin , cos (

) ( 1 ˆ x U p v H

• Simple A-B symmetric potential

(smooth at the scale of lattice constant cannot scatter Berry phase electrons in exactly backward direction.

p

• 2
• i

i p p

w

• i

i

• p

i a b p i b a

z z

Ae Ae

• 2

2

• a

b a b i b a

z

e

• ]

[

2 ) , ( 2 ) , ( 2

• b

a b a a b b a i i p p

w

• ‘Unstoppable’ Berry phase electrons

DoS

gate carriers

V n

• holes

electrons

Graphene-based field-effect transistor: GraFET (bipolar)

Geim and Novoselov, Nature Mat. 6, 183 (2007) Wallace, Phys. Rev. 71, 622 (1947)

Graphene: gapless semiconductor

Quantum transport in graphene

L1 Disordered graphene (G)

graphene 101 QHE in G and quantum resistance standard weak localisation regimes in graphene

L2 Ballistic electrons in graphene L3 Moiré superlattice effects in G/hBN heterostructures

filling factor

eB c r s cm v

c B

• )

( 8

/ 10 ~

• ‘relativistic’ Landau levels
• 2
• 4

4

ne (1012 cm-2)

2 2 4 6

xx (k) xy (4e2/h)

1 2

• 1
• 2
• 4
• 3

4 3

eB hcn n

e LL e

• v

for valley K on A sublattice and valley K’ on B sublattice

McClure, PR 104, 666 (1956)

y x y x

ip p ip p A e i p

• B

v c n

v n

• 2

/

with 4-fold degenerate Landau level

McClure Phys.Rev.104,666(1956)

B

v

• 2

2

• the largest gaps in

the LL spectrum

2

• good for the quantum Hall effect in graphene

with Rxy=h/2e2

• )

( A p v H

c e

2

• 2
• 5

1 0 1 5 2 0 2 5 3 0 3 5 4 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0

E

B , T g ra p h e n e G a A s 3 0 0 K 1 6 2 7 K

Novoselovetal.,Science315,1379(2007).

Lauffer, Emtsev, Graupner, Seyller (Erlangen), Ley PRB 77, 155426 (2008) Gaskill et al, (HRL Malibu) ECS Trans. 19, 117 (2009)

Epitaxial G/SiC (Si face)

Dead layer with a large unit cell carries defects (missing C, Si substitutions of C, interstitial Si) in a large variety of positions, therefore, provides a broad band of surface donor/acceptor states which transfer charge to graphene

donors bulk G donors surface G

A A A A A A

• ~

surfacedonorsDoS bulkdonorsdensity Schottky barrier

• /

2

2 2 l

e U n v

F

• classical

capacitance ‘’quantum capacitance’‚

Kopylov, Tzalenchuk, Kubatkin, Fal’ko - Appl. Phys. Lett. 97, 112109 (2010)

‘Quantum capacitance’ and charge transfer in G/SiC

n v

F

• G/SiC:

filling factor pinning

Due to the filling factor pinning, the largest QHE breakdown current is not at a nominal B(=2), but appears at a higher field.

2 4 2 4 , 2

• N

eB nh N v N

B F

• Janssen, Tzalenchuk, Yakimova, Kubatkin,

Lara-Avila, Kopylov, Fal’ko - PRB 83, 233402 (2011)

Graphene-based resistance standard

•‣ 500Aat14Tand300mK •‣ 87pptrillion(ppt)

20 40 60 80 100 120

• 1.2
• 0.8
• 0.4

0.0 0.4 0.8 1.2

V1 source source source

(RGaAs/AlGaAs-RGraphene )/(h/2e2) (ppb) Source-drain current (A)

Graphene SiC

source drain V1 V2 V3 V4 V5 V6

20 m

Janssen,Tzalenchuk,LaraAvila,Kubatkin,Fal’‚ko Rep.Prog.Phys.76,104501(2013) Tzalenchuk,LaraAvila,Kalaboukhov,Paolillo,Syväjärvi,Yakimova,Kazakova,Janssen,Fal’‚ko,Kubatkin NatureNanotechnology5,186(2010)

Resistance metrology

Wire resistor: a unique artefact which drifts in time Quantum Hall effect: universal and accurate

25 812.807 557 I- I+ V+ V-

XIXXXcenturies XXIcentury

•‣ 87pptrillion(ppt)

density

• f

states

E

localised states in the 2D bulk, extended states which become edge states near the sample edge kT

• hot spots at

the current injection points

• Hall current is carried by electrons in the edge states

extended along the edges and equipotential near metallic contacts, terminated at the current injection points

• Hot spots at the current injection contacts limit applicable

current and therefore practical accuracy of quantisation

Edge states in graphene

B=0 QHE regime

Akhmerov & Beenakker, PRB 77, 085423 (2008) Slizovskiy & Fal'ko, arXiv:1705.02866

Slizovskiy & Fal'ko, arXiv:1705.02866 Nam, Hwang, Lee, Phys. Rev. Lett. 110, 226801 (2013)

Current injection hot spot, chiral heat transport, and edge states cooling by phonons in G in vdW structures

Electrostatics of edge states in G/SiC

shorter T-decay length due to slower edge modes

Slizovskiy & Fal'ko, 2017

B=5T

Rozhko, Tzalenchuk Janssen (2017)

Low-field QHE in G/SiC

• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6

• 15
• 10
• 5

5 10 15 20 25 30 35

Rxx, Rxy (kOhm)

T=3.5K

n=1.6x10

11cm

• 2
• 6
• 4
• 2

2 4 6 10 15 20 25 30 35 40 45 50 55 60 Breakdown Current (A) Magnetic Field (T)

Photochemical gating

UV

• 248

nm

UV dose Carrier density

Lara-Avila, et al Adv Mat 23, 878 (2011)

Commercial application of QHE: push-button QRS calibration tool

Oxford Instruments cryo-free system NPL Cryogenic Current Comparator

• ptimal QRS device design (NGI)

•‣ Quantumstandard

1 ppb

•‣ Secondarystandard

10ppb

•‣ Calibrationlaboratory

100ppb

•‣ Company‘’master’‚item

1ppm

•‣ Companyproduction equipment

10ppm

•‣ Product

100ppm

3.2 K @ 5 T

Quantum transport in graphene

L1 Disordered graphene (G)

graphene 101 QHE in G and quantum resistance standard weak localisation regimes in graphene

L2 Ballistic electrons in graphene L3 Moiré superlattice effects in G/hBN heterostructures

] [ | | | | | | ~

* * 2 2 2

• A

A A A A A A A w

WL = enhanced backscattering for non-chiral electrons in time-reversal-symmetric systems

• /

] , min[ ln 2

2 B cl

h e

• A

A

Interference correction to conductivity: Weak Localisation.

de-coherence suppresses interference contribution time reversal symmetry breaking suppresses interference correction, leading to negative magnetoresistance

] [ | | | | | | ~

* * 2 2 2

• A

A A A A A A A w

WL = enhanced backscattering for non-chiral electrons in time-reversal-symmetric systems

• /

] , min[ ln 2

2 B cl

h e

• chiral electrons

| | | |

2 2 ) 2 / ( 2 *

• A

A e A A

z

i

in i

• ut

z

e

• )

2 / (

• WAL = suppressed backscattering

for Berry phase electrons in MLG

• A

A

Interference correction to conductivity: Weak Localisation.

p i p i

z z

e A e A

• 2

2

~

• '
• '

'

• '

' '

• 3

2 3 2

• i

i

e e

3 2 3 2

' ' ' ' ' '

• y

x i i

i e e

• Strained graphene
• ]

[ ˆ

def def

v p v p v H

• z

def eff

r v B )] ( [

• shiftoftheDiracpointinthemomentumspace,

likesomevectorpotential:oppositeinK/K’‚valleys.

Iordanskii, Koshelev, JETP Lett 41, 574 (1985) Ando - J. Phys. Soc. Jpn. 75, 124701 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006)

pseudomagneticfield,asiftime inversionisliftedforelectronsin eachvalley(=±1 forK/K’‚valleys)

Levy, Burke, Meaker, Panlasigui, Zettl, Guinea, Castro Neto, Crommie - Science 329, 544 (2010)

Strain-induced ‘’100Tesla’‚ pseudo–—magnetic fields in nanobubbles

PseudoLandaulevelsdueto pseudomagneticfieldinthe nanobubbles

• def

r U I p v H

• )

( ˆ ˆ

K K

A A

• /

] , min[ ln 2

2 B cl

h e

• chiral electrons

| | | |

2 2 ) 2 / ( 2 *

• A

A e A A

z

i

in i

• ut

z

e

• )

2 / (

• B

cl

• Inhomogeneous strain

Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006)

) ( *

*

B

*

• Relaxation time

• iv

B cl

h e

• /

] , min[ ln 2

2

• B

cl

• and the inter-valley

scattering restores the WL behaviour typical for electrons in time-inversion symmetric systems.

• K

K

A A

• ... but strain has the opposite effect on electrons in K and K’ valleys, so

that the true time-reversal symmetry is preserved,

F

p

• )

(

• K

K p p t t

• ;

time-inversion symmetry

) ( iv

• iv
• Intervalley time

Heersche et al, Nature 446, 56-59 (2007) Morozov et al, PRL 97, 016801 (2006) ) (B

• iv

McCann, Kechedzhi, Fal’ko, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006) Tikhonenko et al PRL 100, 056802 (2008)

• )

( ) ( ) ( 2 ~

2 2

*

• B

B B B B B B B B

F F F h e

iv iv

• )

( ln ) (

1 2 1

• z

z z F

• Ki et al,

PR B 78, 125409 (2008)

B

cl

• iv

iv iv

D e L e B

,*, ,*, ,*,

4 / 4 /

• 2

2 1

2 ln / / ) ( e h e h T T

Couto, Costanzo, Engels, Ki, Watanabe, Taniguchi, Stampfer, Guinea, Morpurgo - PRX 4, 041019 (2014)

is an indication for that random strain fluctuations are the dominant source of disorder

data for graphene on SiO2, SrTiO3, hBN

• ~

*

Kozikov,Horsell,McCann,Falko Phys.Rev.B86,045436(2012) LaraAvila,Tzalenchuk,Kubatkin,Yakimova,Janssen, Cedergren,Bergsten,Falko–— PRL107,166602(2011)

G/SiC

Saturationof decoherence time atlow temperatureshints thatthereare spinflipprocesses

Engels,Terres,Epping,Khodkov,Watanabe, Taniguchi,Beschoten,StampferPRL113,126801 (2014)

WL in epitaxial graphene on SiC

defect polarisation by in-plane magnetic field should restore a longer phase coherence time

Lara-Avila, Kubatkin, Kashuba, Folk, Luscher, Yakimova, Janssen, Tzalenchuk, Fal'ko. PRL 115, 106602 (2015)

) ( ˆ

i i i

r r S s J H

• no-spin-flip:

this does not cause decoherence, but scattering amplitude/phase depend on the mutual orientation

• f defect’s and arriving electron’s spins

spin-flip in scattering, suppresses interference correction, in addition to spin relaxation

Influenceofspinflipscattering andscatter’‚sspindynamicsonWL

) 1 (

2 / 1

• S

S J e n

kT B g i s

i B

• 1

1 1

• s
• Hikami, Larkin, Nagaoka - Prog Theor Phys 63, 707 (1980)

i e

• t

t

• t

t

i e

• ||

B

• t

Difference of scattering conditions between clockwise and anti-clockwise trajectories, at leads to faster de-coherence:

| | /

1 || e i B s

g g B

• 1

) ( | |

• i

e

t t

• Influence of scatterer’s spin dynamics on WL

Kashuba, Glazman, Fal’ko - PRB 93, 045206 (2016)

For difference of scattering conditions between clockwise and anti- clockwise trajectories leads to a faster decoherence for

i B e i B s

g kT B g g h

• ||

1

| |

Influence of scatterer’s spin dynamics on WL

1 2

• i

e

g g

Si substitutions of C in the dead carbon layer on SiC (Si has stronger SO coupling than carbon)

e i

g g

• Kashuba, Glazman, Fal’ko - PRB 93, 045206 (2016)

Lara-Avila, Kubatkin, Kashuba, Folk, Luscher, Yakimova, Janssen, Tzalenchuk, Fal'ko - PRL 115, 106602 (2015)

l y x l z y x l z z y x z

r u r u r u I p v H

• ,

, , , ,

ˆ ˆ ) (

z BR z z

l s s

• l

s z y x l y x s sl l z z y x l lz

s r a s r a

• ,

, , , ,

z-z asymmetric, relaxes all spin components z-z symmetric: conserves sz but breaks time-inversion for the

• rbital motion of

spin-up/down electrons.

McCann,Fal’‚ko PRL108,166606(2012)

SOcouplingandWAL/WLcrossoveringraphene

inter-valley scattering strain and sublattice asymmetry random potential

MoS2 MoSe2 WSe2 WS2 2 1 2 , 2 , 1 1

) / ( 2 | | ~

• BR

F all s l s l so iv

a

• McCann,Fal’‚ko PRL108,166606(2012);Wang,Ki,Khoo,Mauro,Berger,Levitov,Morpurgo PRX6,041020(2016)

2 1

) / ( 2 ~

• BR

asy

• WALduetoproximityinducedSOcouplingingrapheneon

transitionmetaldichalcogenides

Dyakonov-Perel relaxation Bychkov-Rashba type SO coupling (G does not become not topological insulator)

QHE in G and quantum resistance standard weak localisation regimes in disordered graphene

Ed McCann (Lancaster) Sergey Kopylov (kopylov.com ltd) Sergey Slizovskiy (NGI) Oleksiy Kashuba (Wurzburg) Leonid Glazman (Yale) Boris Altshuler (Columbia) Alexander Tzalenchuk (NPL) JT Janssen (NPL) Sergey Kubatkin (Chalmers) Joshua Folk (Vancouver) Rositsa Yakimova (Linkoping) Ziad Melhem (Oxford Instruments)

Quantum transport in graphene

L1 Disordered graphene (G)

graphene 101 QHE in G and quantum resistance standard weak localisation regimes in graphene

L2 Ballistic electrons in graphene L3 Moiré superlattice effects in G/hBN heterostructures

Bilayer inclusions in a monolayer matrix formed

• n the step edges

T.Yager etal.,NanoLett. 13,42174223(2013)

Influence of bilayer inclusions

Interlayer asymmetry gap

• pened by the transverse

electric field Substrate Gate Metallic bilayer

Chua,Connolly,Lartsev,Yager,LaraAvila,Kubatkin,Kopylov,Fal’‚ko, Yakimova,Pearce,Janssen,Tzalenchuk,Smith NanoLetters,14,3369(2014)

Influence of bilayer inclusions

Bilayer inclusions act as metallic shunts

Influence of bilayer inclusions

Chua,Connolly,Lartsev,Yager,LaraAvila,Kubatkin,Kopylov,Fal’‚ko, Yakimova,Pearce,Janssen,Tzalenchuk,Smith NanoLetters,14,3369(2014)

Perfect quantisation

Influence of bilayer inclusions

Chua,Connolly,Lartsev,Yager,LaraAvila,Kubatkin,Kopylov,Fal’‚ko, Yakimova,Pearce,Janssen,Tzalenchuk,Smith NanoLetters,14,3369(2014)