k kechedzhi, e mccann j robinson, h schomerus t ando...
TRANSCRIPT
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.