Symmetry and quantum transportin disordered graphene

K Kechedzhi, E McCannJ Robinson, H Schomerus

T Ando (Tokyo IT) B Altshuler (Columbia U - NY)

V Falko

A.Geim and K.NovoselovNature Mat. 6, 183 (2007)

A.Geim and K.NovoselovNature Mat. 6, 183 (2007)

List of properties of electrons in graphene.

Weak localisation vs anti-localisation: qualitative discussion.

Formal WL analysis: effect of different types of disorder, specifically - adatoms.

Universal conductance fluctuationsand correlation function thermometry.

Symmetry and quantum transportin disordered graphene

σrr⋅= pvH

conduction band

valence band

1=⋅nrrσ

1−=⋅nrrσ

pr

pr

)sin,cos( ϑϑ ppp =r

‘chiral’ electrons: sublattice ‘isospin’is linked to the direction of the electron momentum

⎟⎟⎠

⎞⎜⎜⎝

⎛=

B

A

ϕϕ

ψ

σr

⎟⎟⎠

⎞⎜⎜⎝

⎛±

= − ϑψ ip e1

21r

Fε

)( pn rrBerry phase

ψψψ πσσπ3

322 i

i

ee =→ψϑ

ψϑππ

dddi +∫=

2

0

vp−=ε

‘Chirality’ of electrons observed using ARPES of graphene

pKGk rrrr+±=||

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅22

sin~ 2 ϑBARkrr

ARPES of heavily doped graphenesynthesized on silicon carbide

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

2|~| BAARPESI ϕϕ +

⎟⎟⎠

⎞⎜⎜⎝

⎛=

B

A

ϕϕ

ψ

Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann, VF, Boswick, Rotenberg - PRB 77, 195403 (2008)

Nomura and MacDonald - PRL 96, 256602 (2006)Cheianov and VF - PRL 97, 226801 (2006)

Nomura and MacDonald - PRL 98, 076602 (2007)Hwang, Adam, Das Sarma - PRL 98, 186806 (2007)

)(4 22

ve

i

ecl F

nn

he

h=σ

Role of scattering from remote charges for graphene conduction in GraFETs

due to screening of potential of charges by electrons in

graphene

DoS

gatecarriers Vn ∝

holes electrons

Metallic (high-density) regimepFl >>1 and δne<< ne

Electronic properties of graphene.

Weak localisation vs anti-localisation: qualitative discussion.

Formal WL analysis: effect of different types of disorder.

Universal conductance fluctuations.

Interference correction: weak localisation effect…][||||||~ **222

⊃⊂⊃⊂⊃⊂⊃⊂ +++=+ AAAAAAAAw

0|| 2* >= ⊂⊃⊂ AAA

WL = enhanced backscattering in time-reversal-symmetric systems

⊃

⊂

AA ⊂⊃ = AA⊂⊃ = ϕϕ ii ee

( )ττπ

σσ ϕ /ln2

2

he

cl −=

Interference correction: weak localisation effect…][||||||~ **222

⊃⊂⊃⊂⊃⊂⊃⊂ +++=+ AAAAAAAAw)0()0(

⊂−

⊃ ≠ AeAe ii δδ

Broken time-reversal symmetry, e.g., due to a magnetic field B

suppresses the weak localisation effect

WL magnetoresistance

( )τττπ

σσ ϕ /],min[ln2

2

Bcl he

−=

B

clσ

⊃

⊂

AA

Random (path-dependent) phase factordue to a magnetic field

... but …

][||||||~ **222⊃⊂⊃⊂⊃⊂⊃⊂ +++=+ AAAAAAAAw

WL = enhanced backscatteringfor non-chiral electrons in

time-reversal-symmetric systems

WAL = suppressed backscattering for Berry phase π electrons

⊃

⊂

AA

inpr

0|||| 22)2/(2* <−== ⊂⊂−

⊃⊂ AAeAA zi σπ

chiral electrons ini

outze ψψ σφ )2/(−=

outpr

chiral electrons ini

outze ψψ σφ )2/(−=

WAL = suppressed backscattering for Berry phase π electrons

... but …

][||||||~ **222⊃⊂⊃⊂⊃⊂⊃⊂ +++=+ AAAAAAAAw

WL = enhanced backscatteringfor non-chiral electrons in

time-reversal-symmetric systems

( )τττπ

σσ ϕ /],min[ln2

2

Bcl he

−=

chiral electrons

0|||| 22)2/(2* <−== ⊂⊂−

⊃⊂ AAeAA zi σπ

ini

outze ψψ σφ )2/(−=

WAL = suppressed backscattering for Berry phase π electrons

Suzuura, Ando - PRL 89, 266603 (2002)B

clσ

+

⊃

⊂

AA

weak trigonal warping leads to a random phase difference, δ for long paths.

( ) )(ˆ2)(ˆ 22 ruIpppppvH yyxxyxrrr

+−−−⋅= σσμσ

( ) ( )[ ] Fjj

jj vlpp hrr∑ −−= εεδ

)0()0(⊂

−⊃ ≠ AeAe ii δδ

... however …

( )τττπ

σσ ϕ /],min[ln2

2

Bcl he

−=

chiral electrons

0|||| 22)2/(2* <−== ⊂⊂−

⊃⊂ AAeAA zi σπin

iout

ze ψψ σφ )2/(−=

+

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)B

clσ

Some types of disorder (e.g., lattice deformation) lead to a similar effect.

)(ˆ rV r+

( )ivBcl he τττπ

σσ ϕ /],min[ln2

2

−=

B

clσ

Inter-valley scattering restores the WL behaviour typical for electrons in

time-inversion symmetric systems

( ) )(ˆ2)(ˆ 22 ruIpppppvH yyxxyxrrr

+−−−⋅±= σσμσ

mKK AA ⊂⊃ =±

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)

... and, finally, …

Fp εε =)( r

m

rr KKpp

tt

→−→

−→

±;

time-inversionsymmetry

)(ˆ rV r+

DoS

gatecarriers Vn ∝

holes electrons

Metallic (high-density) regimepFl >>1 and δne<< ne

Electronic properties of graphene.

Weak localisation vs anti-localisation: qualitative discussion.

Formal analysis of different types of disorder.

Weak localisation and universal conductance fluctuations.

Dirac electrons

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−

⋅=

pvpv

H rr

rr

σσ0

0ˆ

Dirac electrons in disordered graphene.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

',

',

,

,

KA

KB

KB

KA

ϕϕϕϕ

valley index

sublatticeindex

‘isospin’

xpyp xpyp

+K−K

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+''

ˆˆˆˆ

KKiv

ivKK

VVVV

smooth potential;intra-valley scatterers

intervalley-scattering disorder

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

1000

;

0100

;

0010

;

0001

',',

,,

KAKBKBKA

4-dimensional representation of the symmetry group of the honeycomb lattice

Translation

AAT →⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

34

34

34

34

π

π

π

π

i

i

i

i

ee

ee

Generating elements: xAA SCT ,,3π→

3πCRotation

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

32

32

32

32

π

π

π

π

i

i

i

i

ee

ee

'KKBA

⎯→←⎯→←

Mirror reflection

xS

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

11

11

BA ⎯→←

}{ 6 TCG v ⊗

Symmetry operations and transformations of matrices

Generators of the group G{T,C6v} :

UXUXUX ˆˆˆ]ˆ[ˆ +=→The 16D space of matrices can be separated into irreducible representations of the symmetry group G

Χ

4-components wave-functions arrange a 4D irreducible representations of the lattice symmetry group.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=Φ

K'A,K'B,KB,KA,)()(ˆ)(

33rCUrC rr

Φ=Φ ππ

)()(ˆ)( rSUrS xxrr

Φ=Φ

)()(ˆ)( 11 rTUrT rrΦ=Φ

xAA SCT ,,3π→

⎥⎦

⎤⎢⎣

⎡−

=Σx

xx σ

σ0

0⎥⎦

⎤⎢⎣

⎡−

=Σy

yy σ

σ0

0⎥⎦

⎤⎢⎣

⎡=Σ

z

zz σ

σ0

0

3

321

212],[ s

sssss i Σ=ΣΣ ε

SU2 Lie algebra with: sublattice ‘isospin’ matrices:

SU2 Lie algebra with: ⎥⎦

⎤⎢⎣

⎡=Λ

00

z

zx σ

σ⎥⎦

⎤⎢⎣

⎡−

=Λ0

0

00σ

σz⎥

⎦

⎤⎢⎣

⎡ −=Λ

00

z

zy i

iσ

σvalley ‘pseudospin’ matrices:

3

321

212],[ l

lllll i Λ=ΛΛ ε

[ ] 0, =ΛΣ ls

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=Φ

K'A,K'B,KB,KA,

Examples of convenient 4x4 matrices

four 1D-representations

111111111111

3

−Λ−−ΣΛ−Σ

z

zz

z

x

ITsCπ

2

1

2

1

BBAA

one 4D-representationΣ (x,y) Λ (x,y)

3πC xs T

four 2D-representations

1E

2E

Irreducible matrix representation of G{ T,C6v }

Time-reversal

invert signs

invariant

invariantΛ⊗Σ

ΛΣrr

rr,

,I

Time inversion of Σ, Λ matrices:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=Φ

K'A,K'B,KB,KA,

Fp εε =)( r

'; KKpp

tt

→−→

−→rr

time-inversionsymmetry

Λ⊗Σ

ΛΣrr

rr,

,I

16 generators of group U4

tt −→

invert sign

symmetric

symmetric

⎥⎦

⎤⎢⎣

⎡−

=Σx

xx σ

σ0

0⎥⎦

⎤⎢⎣

⎡−

=Σy

yy σ

σ0

0⎥⎦

⎤⎢⎣

⎡=Σ

z

zz σ

σ0

0

3

321

212],[ s

sssss i Σ=ΣΣ ε

SU2 Lie algebra with: sublattice ‘isospin’ matrices:

⎥⎦

⎤⎢⎣

⎡=Λ

00

z

zx σ

σ⎥⎦

⎤⎢⎣

⎡−

=Λ0

0

00σ

σz⎥

⎦

⎤⎢⎣

⎡ −=Λ

00

z

zy i

iσ

σvalley ‘pseudospin’ matrices:

3

321

212],[ l

lllll i Λ=ΛΛ ε

SU2 Lie algebra with:

[ ] 0, =ΛΣ ls

Full basis of symmetry-classified 4x4 matrices

( ) )(ˆ2)(ˆ 22 rVpppppvH yyxxyxrrr

+−−−⋅±= σσμσ

Dirac term warping term

( ) ( ) ( ) ( ) lszyxlsslxxzx ruruIpppvH ΛΣ++Σ⋅ΣΣΛ⋅ΣΣ−⋅Σ= ∑

=

rrrrrrrr

,,,

ˆˆ μ

general form of time-inversion-symmetric disorder

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)

Microscopic origin of various disorder terms

0~ˆ)( αuuIruComes from potential of charged impurities in the substrate, deposits on its surface (water-ice) and doping molecules screened by electrons in graphene.

Unavoidable in graphene-based field-effect transistors (GraFETs)

in Si/SiO2 substrate.

Nomura and MacDonald - PRL 96, 256602 (2006)Cheianov and VF - PRL 97, 226801 (2006)

Nomura and MacDonald - PRL 98, 076602 (2007)Hwang, Adam, Das Sarma - PRL 98, 186806 (2007)

Geim and Novoselov - Nature Materials 6, 183 (2007)Jang, Adam, Chen, Williams, Das Sarma, FuhrerPRL 101, 146805 (2008)

Lattice deformation – bond disorder

BA dddrrr

−=

0rot ≠= ⊥ubfictr

)(

ˆ

⊥

⊥

Λ+⋅Σ=

⋅ΣΛ+⋅Σ=

up

upH

z

zrrr

rrrrFictitious ‘magnetic field’:

Suppresses the interference of electrons in one valley, similarly to the warping effect in the band structure.

zyxssz ruV ΛΣ= ),()(

⊥⋅ΣΛ= uzrr

dlu z

rrr×⊥ ~

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

BA dddrrr

−=

Lattice deformation – bond disorder

⊥⋅ΣΛ= uV zrrˆ

dlu z

rrr×⊥ ~

The phase coherence of two electrons propagating in different valleys is not affected (real time-reversal symmetry is preserved).

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

'KK →

fictfict bb −→

)(ˆ⊥⊥ Λ+⋅Σ=⋅ΣΛ+⋅Σ= upupH zzrrrrrrr

⊥⊥ = γ221 ur

different energy on A and B sites opens a gap and thus suppresses chiraltyof electrons.

BA εε ≠A

B

)(ˆ ruV zzzr

ΣΛ=

intra-valley AB disorder

Intra-valley disorder Λ zΣsus suppresses the interference of electrons in one valley (however, does not affect coherence of a pair of electrons in the opposite valleys)

zzu γ=2

)2(1⊥

− +≡ γγγτ zFz

BAzu εε −~

Inter-valley disorder

pr'prKK'

ky

ky

Deformations similar to those produced by the phonons in the corners of the

Brillouin zone. Can be induced by deposits on graphene sheet, points of mechanical contact with the substrate,

atomic defects, and sample edges.

intervalley scattering rate )24(~1zFiv ββγτ +⊥

−

lssl ru ΛΣ)( lzzl ru ΛΣ)(yxlyxs ,;, == yxl ,=

⊥= β2slu zzlu β=2

alkali atom

hydrogen

oxygen

Symmetry breaking by adatomsMcCann and VF, Phys. Rev. B 71, 085415 (2005)

alkali atom

AB

Adatom in the middle of hexagon: AB symmetric but scatters between valleys

McCann and VF, Phys. Rev. B 71, 085415 (2005)

0,,0,=

≠

⊥⊥ βγγβα

z

z

)(]I[ izlliv r-rvuuV rrδΣΛ+=

v⊃

yxl or=

i

2A3A

1A

1B

2B3B

AB

On-site adatom (‘hydrogen’)

A or B

yxsl or, =

1±=m

∈w

McCann and VF, Phys. Rev. B 71, 085415 (2005)

0,0,,

=≠

⊥

⊥

z

z

βγβγα

)(]I[ isllsiivzziz r-rwumuuV rrδΣΛ+ΣΛ+=

Aleiner, Efetov - PRL 97, 236801 (2006)

Many adatoms: renormalisation of disorder parameters

Fε

a~

Fλ

},,,,{)(....)'()/(

⊥⊥=ΓΓ⇒⇒Γ⇒Γ

ββγγαεε

zz

Fav

Aleiner, Efetov - PRL 97, 236801 (2006)Ostrovsky, Gornyi, Mirlin, PRB 74, 235443 (2006)

Foster, Aleiner, PRB 77, 195413 (2008)

a~

Fλ

Adsorbate - induced disorder in graphene

Robinson, Schomerus, Oroszlany, VF PRL 101, 196803 (2008)

Strong inter-valley scattering

γεγγ 66.0~;2.2~ HH

Robinson, Schomerus, Oroszlany, VF - PRL 101, 196803 (2008)

DoS

gatecarriers Vn ∝

holes electrons

Metallic (high-density) regimepFl >>1 and δne<< ne

Electronic properties of graphene.

Weak localisation vs anti-localisation: qualitative discussion.

Formal WL analysis taking into account different types of disorder.

Universal conductance fluctuations.

( ) ( ) ( ) ( ) lszyxlsslxxzx ruruIpppvH ΛΣ++Σ⋅ΣΣΛ⋅ΣΣ−⋅Σ= ∑

=

rrrrrrrr

,,,

ˆˆ μ

leading terms do not contain valley operators Λ , thus, they remain invariant with respect to valley transformations SU2

Λ: this allows for all four Cooperons.

00000~ CCCC zyx −++δσ

WL correction)',( rrCl

srr

Particle-particle correlation function ‘Cooperon’

1−slτ relaxation rate of the corresponding ‘Cooperon’

( ) )'()',(12 rrrrCiD lsr sl

rrrrr −=+−∇ − δτω

Λ2SU

Σ2SU

( ) ( ) ( ) ( ) lszyxlsslxxzx ruruIpppvH ΛΣ++Σ⋅ΣΣΛ⋅ΣΣ−⋅Σ= ∑

=

rrrrrrrr

,,,

ˆˆ μ

00000~ CCCC zyx −++δσ

All types of symmetry breaking disorder

inter-valleysame valley

inter-valley disorder

βτ ∝−1iv

The only surviving mode1111 ,, −−−− << zivw ττττϕ

Trigonal warping1−

wτ

inter-valley + intra-valley

disorder11 −− + ziv ττ

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler PRL 97, 146805 (2006)

1111 −−−−∗ ++= ivzw ττττ

Heersche et al,Nature 446, 56-59 (2007)

Morozov et al, PRL 97, 016801 (2006)

WL magnetoresistance is sample-dependent, due to a sample-dependent inter-valley scattering strength.

)0()( σσ −B

∗∗ τ

φD

B 0~

ϕτττ <<<∗ iv

Bϕττ >iv

ϕτττ <∗ iv~

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006)

Tikhonenko et alPRL 100, 056802 (2008)( ))()(2)(~

*2

2

ϕϕϕπσ B

BBB

BBB

B FFFh

eiv

−+Δ ++

)(ln)( 121 −++= zzzF ψ

1111

*

−−−−∗

∗

++=

=

ivzw

DL

ττττ

τ

DL ivi τ=

WL used to test ‘what type’ of disorder:

Li >> L* > l(substrate charge seems to

dominate in the resistance of GraFETs on Si/SiO2).

McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006)

Narrow ribbons of graphene feature strong inter-valley scattering due to edges and thus

shows robust WL magnetoresistance.

Symmetry and quantum transport in disordered graphene

QT is strongly effected both by graphene band structure and the type of disorder in the sample.

Measurements of weak localisation can be used as a tool to grade the efficiency of different types of disorder in material.

Universal conductance fluctuations and correlation function thermometry of graphene.

DoS

gatecarriers Vn ∝

holes electrons

pFl >>1 and δne<< ne

( ) )'()',(12 rrrrDiD lsr sl

rrrrr −=+−∇ − δτωDiffusion pole:

inter-valley same valley

2

)(

2

)(2

'2

'2 ~ symmvalleyKKsymmantivalleyKKKKKK DDDDg −−− +++δ

UCF in graphene

1111 −−−−∗ ++= ivzw ττττ 1−

ivτ1111 ,, −−−− << zivw ττττϕ

Tikhonenko et.al. PRL (2008)

Kechedzhi, Kashuba, VFPRB 77, 193403 (2008)

Narrow ribbon, α=12D:Wide (2D) graphene sheet, 1<α<4

)()()( Δ+≡Δ μδμδ ggF

eBc Tk⋅≈Δ 7.2

)0()(

D1

D1

FF Δ

Δ

cΔ

),(),(2)( 2D1 Δ+′′⎟⎠⎞

⎜⎝⎛≈Δ ∫ μεμεεπ ϕ ffd

LL

LDF h

)(μg

eBTk

eBTkμ

Δ+μ

1−ϕτ

Correlation thermometry function of graphene ribbons

e

e

Correlation thermometry function of graphene ribbons

Width at half maximum of the correlation function of UCF

Kechedzhi, Horsell, Tikhonenko, Savchenko, Gorbachev, Lerner, VF - PRL 102, 066801 (2009)

Electron temperature from the correlation

function

bath temperature(in low current measurements)

≈

Correlation function of UCF can be used to measure temperature of electrons in

graphene ribbons

Summary: Quantum transport in disordered graphene

a) QT is strongly effected both by graphene band structure and the type of disorder in the sample.

a) Measurements of weak localisation enable one to grade the efficiency of different types of disorder in material.

________________________________________________________________

c) Universal conductance fluctuations can be used for the correlation function thermometry of graphene.