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P. Adroguer, D. Carpentier, J. Cayssol and E. Orignac, unpublished 

A. Fedorenko, D. Carpentier, and E. Orignac, unpublished 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









Anti-localization cusp at low field 

mardi 3 janvier 12









2 

es of ρ vs. H in Sample G4 at 0.3< T


‣ Bi2Se3 : good candidate 




2 




‣ Thin Films (reduction of bulk) 


FIG. 2: Curves of ρ vs. H in Sample G4 at 0.3< T


‣ Bi2Se3 : good candidate 




2 




‣ Thin Films (reduction of bulk) 

‣ New Materials : strained HgTe ? 


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


A 

B 

H = ~v F ( y .k x x .k y ) 







doping 

H W = 2 

z(k 3 + + k 3 ) 

k ± = k x ± ik y 

Graphene 





A. Kobayashi et al., Phys. Rev. B 84,075450 (2011) 

mardi 3 janvier 12


A 

B 

H = ~v F ( y .k x x .k y ) 






‣ 2 independant cones 

(Nielsen-Ninomyia Theorem) 


doping 

H W = 2 

z(k 3 + + k 3 ) 

k ± = k x ± ik y 

Graphene 





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 


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 


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



⇥ 

P (⌅r ⌅r ⇥ ) ⇥ 

path C ⇥r 




⇥r ⇥ 

A C 

2 

= ⇥ |A C | 2 + ⇥ A C A C ⇥ 

C 

C⇤=C ⇥ 

⌅ C =2⇤ L C 

⇥ F 

Classical 

Interferences 




➡Effective diffusion of «pseudo particles» Diffuson (same dir.) / Cooperon (opp. dir.) 

O 

mardi 3 janvier 12



⇥ 

P (⌅r ⌅r ⇥ ) ⇥ 

path C ⇥r 




⇥r ⇥ 

A C 

2 

= ⇥ |A C | 2 + ⇥ A C A C ⇥ 

C 

C⇤=C ⇥ 

⌅ C =2⇤ L C 

⇥ F 

Classical 

Interferences 


B 



O ⇤ C,C =2⇥ C,C / 0 


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 


12.4 

12.2 

12 

11.8 

11.6 

11.4 


mardi 3 janvier 12

(Weak) Localization / Quantum transport 


⇥ 

P (⌅r ⌅r ⇥ ) ⇥ 

path C ⇥r 




⇥r ⇥ 

A C 

2 

= ⇥ |A C | 2 + ⇥ A C A C ⇥ 

C 

C⇤=C ⇥ 

⌅ C =2⇤ L C 

⇥ F 

Classical 

Interferences 


O ⇤ C,C =2⇥ C,C / 0 




‣ 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 


See e.g. Book by Akkermans and Montambaux


Diagrammatic techniques 

Kubo formula 

↵ = ~ 

2⇡V Re Tr ⇥ j ↵ G R j G A⇤ 


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 !) 


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 


‣ 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)




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) 








- 1 Diffuson / 0 Cooperon 












S = 1 Z 

d d x Tr [@ µ Q(x)@ µ Q(x)] 

g 





: 

H = ~v F ( y .k x x .k y )+V (x) 


1/k F l e 

H = (~k)2 





2m + iV SO ~ .ˆk ⇥ ˆk 0 

with magnetic disorder : +J~. V ~ m 









- 1 Diffuson / 0 Cooperon 












S = 1 g 

Z 

d d x Tr [@ µ Q(x)@ µ Q(x)] 

Dirac Fermions Classification : what is the target manifold 





: 

H = ~v F ( y .k x x .k y )+V (x) 



1/k F l e 

H = (~k)2 


2m + iV SO ~ .ˆk ⇥ ˆk 0 

with magnetic disorder : +J~. V ~ m 










But wait : what about 

topological robustness ??? 









Anderson Univ. classes and Topological Order 


S = 1 Z 

d d x Tr [@ µ Q(x)@ µ Q(x)] 

g 





: 

H = ~v F ( y .k x x .k y )+V (x) 


1/k F l e 

H = (~k)2 

2m + iV SO ~ .ˆk ⇥ ˆk 0 













mardi 3 janvier 12



S = 1 Z 

d d x Tr [@ µ Q(x)@ µ Q(x)] 

g 

Not Topologically Protected Topologically Protected 





: 

H = ~v F ( y .k x x .k y )+V (x) 


1/k F l e 

H = (~k)2 













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



S = 1 Z 

d d x Tr [@ µ Q(x)@ µ Q(x)] 

g 

Not Topologically Protected Topologically Protected 





: 

H = ~v F ( y .k x x .k y )+V (x) 


1/k F l e 

H = (~k)2 













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 


‣ Symplectic / AII Class (T 2 =-1) 

- electrons + spin orbit disorder / Dirac + scalar disorder 


‣ Unitary / A Class (T=0) 

- electrons + magnetic disorder / Dirac + magnetic disorder 


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 


12.4 

12.2 

12 

11.8 

11.6 

11.4 

11.2 

11 

10.8 

0 20 40 60 80 100 


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 


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


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



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 


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 ? 


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

Topological Insulators - GDR Meso

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