Topological Insulators - GDR Meso

Some Transport properties of
Topological Insulator Surface States
(...or Some Charge Transport Properties of Dirac Fermions)
P. Adroguer, D. Carpentier, A. Fedorenko, E. Orignac (Lyon)
J. Cayssol (Berkeley)
P. Adroguer, D. Carpentier, J. Cayssol and E. Orignac, unpublished
A. Fedorenko, D. Carpentier, and E. Orignac, unpublished
Posters of P. Adroguer and A. Fedorenko
Aussois, Déc. 2011
mardi 3 janvier 12

Surface States of 3D Topological Insulators
• The Z2 topological order in the bulk
(band inversion induced by strong spin orbit)
• Robust edge (surface) states : Dirac fermions (odd number)
«strong» topological insulator,
ν0 =1, not layered
Z2 Top. Ins.
Empty Top. Band Insulator
E
Filled Band
mardi 3 janvier 12

Surface States of 3D Topological Insulators
• The Z2 topological order in the bulk
(band inversion induced by strong spin orbit)
• Robust edge (surface) states : Dirac fermions (odd number)
• First proposed candidate : Bi1-xSbx
Fu and Kane PRB 76 (2007)
«strong» topological insulator,
ν0 =1, not layered
• Second generation 3D Topological Insulators
Bi2Se3, Bi2Te3, Sb2Te3, ...
• Reference material : Bi2Se3
‣ single Dirac cone at the surface, stoichiometric, large band gap : 0.3 eV
• «Third generation» 3D Topological Insulators
‣ TlBiTe2, Bi2Te3 ·(GeTe)0.5
‣ strained HgTe : Ideal material ?
L. Molenkamp group, PRL (2011)
T. Meunier et L. Lévy : cf talk by C. Bouvier
How to probe experimentally these 3D Top. Insulators ?
‣ existence of surface states (Dirac fermions) :
‣ (spin resolved) ARPES
‣ STM
‣ transport ... problem : get rid of bulk contribution
mardi 3 janvier 12
Checkelsky et al., PRL. 103, 246601 (2009)
Zhang H. et al., Nat. Phys. 5, 438 (2009)
ARPES of topological insulator
First observation by D. Hsieh et al. (Z. Hasan group), Princeton/LBL, 2008.
This is later data on Bi2Se3 from the same group in 2009:
The states shown are in the “energy gap” of the bulk material--in general n

Probing Surface States : ARPES
Difficulty : doping of bulk / edge
Momentum - Spin locking : helical Dirac fermions
Not exactly a perfect cone :
Hexagonal warping
Measure as many properties as possible of the outgoing electron
to deduce the momentum, energy, and spin it had while still in the solid
This is “angle-resolved photoemission spectroscopy”, or ARPES.
Y.L.Chen etal., Science 325,178(2009)
K. Kuroda et al., PRL 105, 076802 (2010)
mardi 3 janvier 12

Transport measurement on TI
Bi2Se3 : good candidate
‣Large band gap : 300 meV
‣Single Dirac surface state
Checkelsky et al. PRL, 103 (2010)
... but μb in the conduction band
➡ chemical bulk doping by Ca (CaxBi2-xSe3)
Pb of residual transport by bulk states ?
Undoped versus doped samples
Gi,Mi : Various Ca doping
bulk sample (2x2x0.05 mm)
mardi 3 janvier 12

Transport measurement on TI
Bi2Se3 : good candidate
‣Large band gap : 300 meV
‣Single Dirac surface state
Checkelsky et al. PRL, 103 (2010)
... but μb in the conduction band
➡ chemical bulk doping by Ca (CaxBi2-xSe3)
Pb of residual transport by bulk states ?
Anti-localization cusp at low field
mardi 3 janvier 12

Transport measurement on TI
Bi2Se3 : good candidate
‣Large band gap : 300 meV
‣Single Dirac surface state
Checkelsky et al. PRL, 103 (2010)
... but μb in the conduction band
➡ chemical bulk doping by Ca (CaxBi2-xSe3)
Pb of residual transport by bulk states ?
2
es of ρ vs. H in Sample G4 at 0.3< T

Transport measurement on TI
‣ Bi2Se3 : good candidate
‣Large band gap : 300 meV
‣Single Dirac surface state
Checkelsky et al. PRL, 103 (2010)
2
... but μb in the conduction band
➡ chemical bulk doping by Ca (CaxBi2-xSe3)
Pb of residual transport by bulk states ?
‣ Thin Films (reduction of bulk)
mardi 3 janvier 12
FIG. 2: Curves of ρ vs. H in Sample G4 at 0.3< T

Transport measurement on TI
‣ Bi2Se3 : good candidate
‣Large band gap : 300 meV
‣Single Dirac surface state
Checkelsky et al. PRL, 103 (2010)
2
... but μb in the conduction band
➡ chemical bulk doping by Ca (CaxBi2-xSe3)
Pb of residual transport by bulk states ?
‣ Thin Films (reduction of bulk)
‣ New Materials : strained HgTe ?
mardi 3 janvier 12
FIG. 2: Curves of ρ vs. H in Sample G4 at 0.3< T

2D Dirac Matter
A
B
H = ~v F ( y .k x x .k y )
Topological Insulators surface states
‣ strong spin-orbit : momentum-spin locking
➡ real spin in Dirac equation (Zeeman effect, spintronic)
‣ no additional degeneracy : a single cone
‣ 1 cone + real spin : strong constraint by T-symmetry
‣ necessity to include hexagonal warping at high
doping
H W = 2
z(k 3 + + k 3 )
k ± = k x ± ik y
Graphene
‣ pseudo spin (AB) in Dirac
‣ 4-fold degeneracy : valley x spin
‣ T-symmetry relates both cones
also : α-(BEDT-TTF)2I3 under pressure
See Talk by M. Monteverde, A. Kobayashi et al., Phys. Rev. B 84,075450 (2011)
mardi 3 janvier 12

2D Dirac Matter
A
B
H = ~v F ( y .k x x .k y )
Topological Insulators surface states
‣ strong spin-orbit : momentum-spin locking
➡ real spin in Dirac equation (Zeeman effect, spintronic)
‣ no additional degeneracy : a single cone
‣ 1 cone + real spin : strong constraint by T-symmetry
‣ necessity to include hexagonal warping at high
doping
H W = 2
z(k 3 + + k 3 )
k ± = k x ± ik y
Graphene
‣ pseudo spin (AB) in Dirac
‣ 4-fold degeneracy : valley x spin
‣ T-symmetry relates both cones
also : α-(BEDT-TTF)2I3 under pressure
A. Kobayashi et al., Phys. Rev. B 84,075450 (2011)
mardi 3 janvier 12

2D Dirac Matter
A
B
H = ~v F ( y .k x x .k y )
Topological Insulators surface states
‣ strong spin-orbit : momentum-spin locking
➡ real spin in Dirac equation (Zeeman effect, spintronic)
‣ no additional degeneracy : a single cone
‣ 1 cone + real spin : strong constraint by T-symmetry
‣ 2 independant cones
(Nielsen-Ninomyia Theorem)
‣ necessity to include hexagonal warping at high
doping
H W = 2
z(k 3 + + k 3 )
k ± = k x ± ik y
Graphene
‣ pseudo spin (AB) in Dirac
‣ 4-fold degeneracy : valley x spin
‣ T-symmetry relates both cones
also : α-(BEDT-TTF)2I3 under pressure
A. Kobayashi et al., Phys. Rev. B 84,075450 (2011)
mardi 3 janvier 12

Diffusion of 2D Dirac states
• Transport in metallic regime :
k F
1/l e
(high doping)
k F
1/l e
F
l e
mardi 3 janvier 12

Diffusion of 2D Dirac states
• Transport in metallic regime :
k F
1/l e
(high doping)
k F
1/l e
F
l e
• Transport in near Dirac point:
k F apple 1/l e
(high doping)
F
k F apple 1/l e
l e
Mirlin et al. PRB, (2009)
Fedorenko et al. (unpublished)
mardi 3 janvier 12

Study of diffusion of 2D Dirac states
• Transport in metallic regime :
k F
1/l e
(high doping)
k F
1/l e
F
l e
weak (anti-)localization
(long wires)
0.2
0
-0.2
-0.4
-0.6
-0.8
2.10 - 5
B
690 mK
-2000 0 2000
conductance
universal conductance fluctuations
(short wire)
Conductance, same sample, different spins config.
12.4
12.2
12
11.8
11.6
11.4
11.2
11
10.8
0 20 40 60 80 100
Flux Through the Sample
mardi 3 janvier 12

(Weak) Localization
• Probability to diffuse from ⇥r to ⇥r
⇥
P (⌅r ⌅r ⇥ ) ⇥
path C ⇥r
• Phase variation along a diffusion path :
• Average interferences vanish,
except if ”C = C ” (reversed ring)
⇥r ⇥
A C
2
= ⇥ |A C | 2 + ⇥ A C A C ⇥
C
C⇤=C ⇥
⌅ C =2⇤ L C
⇥ F
Classical
Interferences
random in a metal
• Phase coherent corrections
‣ loop contributions
➡Effective diffusion of «pseudo particles» Diffuson (same dir.)
O
mardi 3 janvier 12

(Weak) Localization
• Probability to diffuse from ⇥r to ⇥r
⇥
P (⌅r ⌅r ⇥ ) ⇥
path C ⇥r
• Phase variation along a diffusion path :
• Average interferences vanish,
except if ”C = C ” (reversed ring)
⇥r ⇥
A C
2
= ⇥ |A C | 2 + ⇥ A C A C ⇥
C
C⇤=C ⇥
⌅ C =2⇤ L C
⇥ F
Classical
Interferences
random in a metal
• Phase coherent corrections
‣ loop contributions
➡Effective diffusion of «pseudo particles» Diffuson (same dir.) / Cooperon (opp. dir.)
O
mardi 3 janvier 12

(Weak) Localization
• Probability to diffuse from ⇥r to ⇥r
⇥
P (⌅r ⌅r ⇥ ) ⇥
path C ⇥r
• Phase variation along a diffusion path :
• Average interferences vanish,
except if ”C = C ” (reversed ring)
⇥r ⇥
A C
2
= ⇥ |A C | 2 + ⇥ A C A C ⇥
C
C⇤=C ⇥
⌅ C =2⇤ L C
⇥ F
Classical
Interferences
random in a metal
B
• Phase coherent corrections
‣ loop contributions
O ⇤ C,C =2⇥ C,C / 0
➡Effective diffusion of «pseudo particles» Diffuson (same dir.) / Cooperon (opp. dir.)
0.2
0
-0.2
-0.4
-0.6
-0.8
2.10 - 5 690 mK
11.2
11
10.8
0 20 40 60 80 100
-2000 0 2000
Conductance, same sample, different spins config.
12.4
12.2
12
11.8
11.6
11.4
Flux Through the Sample
mardi 3 janvier 12

(Weak) Localization / Quantum transport
• Probability to diffuse from ⇥r to ⇥r
⇥
P (⌅r ⌅r ⇥ ) ⇥
path C ⇥r
• Phase variation along a diffusion path :
• Average interferences vanish,
except if ”C = C ” (reversed ring)
⇥r ⇥
A C
2
= ⇥ |A C | 2 + ⇥ A C A C ⇥
C
C⇤=C ⇥
⌅ C =2⇤ L C
⇥ F
Classical
Interferences
random in a metal
O ⇤ C,C =2⇥ C,C / 0
• Phase coherent corrections
‣ loop contributions
➡Effective diffusion of «pseudo particles» Diffuson (same dir.) / Cooperon (opp. dir.)
‣ Universality properties of (weak) localization
➡ numbers of Cooperon/Diffuson (symmetry)
mardi 3 janvier 12

Weak localization : Universality classes
Diagrammatic techniques
Kubo formula
↵ = ~
2⇡V Re Tr ⇥ j ↵ G R j G A⇤
Coherent Diffusive Regime
Perturbative expension in
1/k F l e
Correlate sequences of scatterers in same / opposite order
➡Diffusons / Cooperon propagators
Average conductance
H C
Γ C ( ⃗ Q)
Fluctuations of conductance
⃗ k
⃗ Q − ⃗ k, EF − Ω
Γ C (Ω, ⃗ Q)
⃗Q − ⃗K ′ ,
EF − Ω
⃗K ′
⃗ k
⃗ k − ⃗q, EF − Ω
Γ D (Ω, ⃗q)
⃗K ′ − ⃗q,
EF − Ω
⃗K ′
H C Γ C (−Ω, ⃗ Q)
H C
H D
H D
⃗k ′
⃗ Q − ⃗ k ′ ,EF − Ω
⃗Q − K, ⃗
EF − Ω
⃗K
⃗k ′
⃗ k ′ − ⃗q, EF − Ω
Γ D (−Ω, −⃗q)
⃗K − ⃗q,
EF − Ω
⃗K
Corrections of Hikami boxes (anisotropic scattering !)
⃗k ′ k ⃗′
⃗ k ⃗ k k ⃗′ k ⃗′
⃗ k
k ⃗′
⃗
⃗ k ⃗ k ⃗ k ⃗ k k ⃗ k
H D = + +
H0 D H1 D H2
D
⃗k ′
⃗ k
⃗k ′
⃗k ′
⃗ k
⃗ k
End Formula depends only on number of
Cooperons / Diffusons : How many Singlet / Triplets ?
Based on microscopic Hamiltonian
mardi 3 janvier 12
See e.g. Book by Akkermans and Montambaux

Weak localization : Universality classes
Diagrammatic techniques
Kubo formula
↵ = ~
2⇡V Re Tr ⇥ j ↵ G R j G A⇤
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 g
Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
Perturbative expension in
1/k F l e
Correlate sequences of scatterers in same / opposite order
➡Diffusons / Cooperon propagators
Average conductance
Fluctuations of conductance
⃗k ′
⃗ k
⃗ Q − ⃗ k, EF − Ω
Γ C (Ω, ⃗ Q)
H C ⃗Q − K, ⃗
Q ⃗ − k ⃗′ ,EF − Ω
EF − Ω
Γ C (−Ω, Q) ⃗
Based on microscopic Hamiltonian
⃗Q − ⃗K ′ ,
EF − Ω
⃗k ′ k ⃗′
⃗ k ⃗ k k ⃗′ k ⃗′
⃗ k
k ⃗′
⃗
⃗ k ⃗ k ⃗ k ⃗ k k ⃗ k
H C
⃗K
⃗K ′
⃗k ′
⃗ k
H C
H D
⃗ k − ⃗q, EF − Ω
⃗ k ′ − ⃗q, EF − Ω
Γ D (Ω, ⃗q)
Γ D (−Ω, −⃗q)
Corrections of Hikami boxes (anisotropic scattering !)
mardi 3 janvier 12
Γ C ( ⃗ Q)
H D = + +
⃗k ′
⃗ k
H0 D H1 D H2
D
End Formula depends only on number of
Cooperons / Diffusons : How many Singlet / Triplets ?
See e.g. Book by Akkermans and Montambaux
⃗K ′ − ⃗q,
EF − Ω
⃗K − ⃗q,
EF − Ω
⃗k ′
H D
⃗k ′
⃗K
⃗K ′
⃗ k
⃗ k
Classification : what is the target manifold
➡ i.e. : how many Cooperon / Diffuson
(encoded in field Q(x) )
Based on symmetry argument (T-reversal / C conjugation)
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
Based on symmetry arguments
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)

Weak localization : Universality classes
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
Classification : what is the target manifold
➡ i.e. : how many Cooperon / Diffuson
(encoded in field Q(x) )
Based on symmetry argument (T-reversal / C conjugation)
2m + iV SO ~ .ˆk ⇥ ˆk 0 ‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffuson / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Based on symmetry arguments
mardi 3 janvier 12
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)

Weak localization : Universality classes
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
Classification : what is the target manifold
➡ i.e. : how many Cooperon / Diffuson
(encoded in field Q(x) )
Based on symmetry argument (T-reversal / C conjugation)
2m + iV SO ~ .ˆk ⇥ ˆk 0
with magnetic disorder : +J~. V ~ m
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffuson / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Based on symmetry arguments
mardi 3 janvier 12
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)

Weak localization : Universality classes
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 g
Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
Dirac Fermions Classification : what is the target manifold
➡ i.e. : how many Cooperon / Diffuson
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
(encoded in field Q(x) )
Electrons + random spin orbit :
1/k F l e
H = (~k)2
Based on symmetry argument (T-reversal / C conjugation)
2m + iV SO ~ .ˆk ⇥ ˆk 0
with magnetic disorder : +J~. V ~ m
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
But wait : what about
topological robustness ???
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Based on symmetry arguments
mardi 3 janvier 12
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)

Anderson Univ. classes and Topological Order
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
2m + iV SO ~ .ˆk ⇥ ˆk 0
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
mardi 3 janvier 12

Anderson Univ. classes and Topological Order
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Not Topologically Protected Topologically Protected
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
2m + iV SO ~ .ˆk ⇥ ˆk 0 ‣ AII Class in d=2 : Topological Term allowed
➡ Prevents Anderson localization
➡ 2 subclasses
- top. term (cf Berry Phase) : Top. Protection
from d+1 Bulk
- no top. term : standard AII class
‣ Classification of all (d+1) topological phases from their d-
surface states properties
S. Ryu et al., NJP 12 (2010)
mardi 3 janvier 12

Anderson Univ. classes and Topological Order
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Not Topologically Protected Topologically Protected
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
2m + iV SO ~ .ˆk ⇥ ˆk 0 ‣ AII Class in d=2 : Topological Term allowed
But : Topological Term does not contribute to the weak
localization of surface states
➡ both AII / Symplectic classes equivalent
➡ known results for electrons + random SO sufficient
➡ Prevents Anderson localization
➡ 2 subclasses
- top. term (cf Berry Phase) : Top. Protection
from d+1 Bulk
- no top. term : standard AII class
‣ Classification of all (d+1) topological phases from their d-
surface states properties
S. Ryu et al., NJP 12 (2010)
mardi 3 janvier 12

Universality classes and weak localization
Localization Universality Classes (for metals)
‣ Orthogonal / AI Class (T 2 =+1)
- electrons + scalar disorder
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1)
- electrons + spin orbit disorder / Dirac + scalar disorder
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0)
- electrons + magnetic disorder / Dirac + magnetic disorder
- 1 Diffuson + 0 Cooperon
For 1 flavor of carrier with spin
(In graphene : treat spin x valley degeneracy !)
0.2
0
-0.2
-0.4
-0.6
2.10 - 5
690 mK
conductance
Weak Localization (d=2)
hgi = hgi(B) hgi(0)
"
!#
= ↵ e 2 ~
ln
1 ⇡ h 4Bel 2 2 + ~
4Bel 2
weak (anti-)localization
↵ =1(Orthogonal), ↵ = 1/2 (Symplectic), ↵ =0(Unitary)
↵ =(N C,T N C,S )/2
-0.8
-2000 0 2000
B
Conductance, same sample, different spins config.
12.4
12.2
12
11.8
11.6
11.4
11.2
11
10.8
0 20 40 60 80 100
Flux Through the Sample
Universal Conductance Fluctuations (d=2)
h( g) 2 i = h(g hgi) 2 i
= N ✓
C + N S e
2
15 h
◆ 2
h( g) 2 i = 2 15
h( g) 2 i = 1 15
✓ e
2
h
✓ e
2
h
◆ 2
◆ 2
Symplectic : Dirac + scalar impurities
Unitary : Dirac + magnetic impurities
mardi 3 janvier 12
Symplectic

Anisotropic Scattering of Dirac states
Consider Dirac surface states with
‣ phase coherence (small sample, low T)
Hexagonal warping :
‣ scalar disorder
‣ semi-classical limit
H W = 2
z k 3 + + k 3 k ± = k x ± ik y
H 0 = ~v F ẑ ⇥ (~ ⇥ ~ k)
+V (~r )
(high doping in the gap)
L. Fu, PRL 103 (2009)
‣ allowed by Time Reversal Symmetry (same univ. class)
‣ experimentally extracted value in TI : bw=0.4-0.6
‣ important at high kF (weak localization regime)
‣ Perturbative parameter (deformation of Fermi surface) bw
➡ depends on EF
k min
kmax
b w =
E2 F
2~ 3 v 3 F
w = w max
1
k min
k max
1+ k min
k max
Z. Alpichshev et al., PRL 104 (2010)
S.Y. Xu et al., (2011)
mardi 3 janvier 12

Anisotropic Scattering of Dirac states
Consider Dirac surface states with
‣ phase coherence (small sample, low T)
‣ scalar disorder
‣ semi-classical limit
H 0 = ~v F ẑ ⇥ (~ ⇥ ~ k)
+V (~r )
(high doping in the gap)
Hexagonal warping :
L. Fu, PRL 103 (2009)
H W = z 2
k+ 3 + k 3 k ± = k x ± ik y
b w = E2 F
‣ allowed by Time Reversal Symmetry (same univ. class)
‣ experimentally extracted value in TI : bw=0.4-0.6
‣ important at high kF (weak localization regime)
‣ Perturbative parameter (deformation of Fermi surface) bw
➡ depends on EF
Anisotropic scattering :
‣ comes from pure (Dirac) Hamiltonian, not disorder
‣ strongly increased by warping term
k min
2~ 3 v 3 F
kmax
Scattering amplitude
bw=0.6 bw=0.0
f(✓, ✓ 0 ✓)=|h ~ k|V | ~ k 0 i|
✓ =0 ✓ = ⇡/6
mardi 3 janvier 12

Anisotropic Scattering of Dirac states
P. Adroguer, D. Carpentier, J. Cayssol and E. Orignac, unpublished
H 0 = ~v F ẑ ⇥ (~ ⇥ ~ k) +V (~r ) H W = 2
z k 3 + + k 3
k ± = k x ± ik y
Diagrammatic
Boltzmann equation
b w =
‣ Double perturbation
E2 F
2~ 3 v 3 F
‣ Anisotropic scattering
‣ warped Fermi surface
e ~ E.@ ~k ñ =
Z
~ k 0
|h ~ k 0 | ~ ki| 2 (E ~k E ~k 0) ñ ~k 0 ñ ~k
L i = p D⌧ i
k min
kmax
H C
Γ C ( ⃗ Q)
Classical Conductivity
‣ Renormalized density of states
‣ Renormalized Transport time
+ Univ. Class (AII) :
‣ Various quantum contributions
‣ All dephasing lengths depends on bw
mardi 3 janvier 12

Warping + (in plane) Zeeman
P. Adroguer, D. Carpentier, J. Cayssol and E. Orignac, unpublished
H 0 = ~v F ẑ ⇥ (~ ⇥ ~ k) +V (~r ) H W = 2
z k 3 + + k 3
k ± = k x ± ik y
Zeeman field :
H Z = gµ B ~ . ~ B
‣ in plane without warping : no effect on transport
‣ + Warping : modifies the scattering amplitude, and the transport
k min
kmax
No Warping
Fermi Surface
Increasing B
Zeeman Field
+ Warping
Zeeman field = Constant Gauge field = Shift of momenta
Transport unchanged
Zeeman field ≠ Constant Gauge field
Transport modified
Modification of UCF, weak loc, etc
mardi 3 janvier 12

Conclusion
‣ New playground for unusual transport
‣ 1 Dirac cone as opposed to graphene
‣ hexagonal warping term
‣ spin degree of freedom ?
P. Adroguer, D. Carpentier, J. Cayssol and E. Orignac, unpublished
Poster by P. Adroguer
‣ Experimental progress
‣ improve existing materials ?
‣ new systems : strained HgTe (Würzburg, Grenoble)
Talk by C. Bouvier
‣ Algebraic correlation of disorder A. Fedorenko, D. Carpentier, and E. Orignac, unpublished
Poster by A. Fedorenko
‣ ripples in graphene
‣ atomic steps (extended scatterers) for TI
mardi 3 janvier 12