Topological Insulators - GDR Meso
Topological Insulators - GDR Meso Topological Insulators - GDR Meso
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
- Page 1 and 2: Some Transport properties of Topolo
- Page 3 and 4: Surface States of 3D Topological In
- Page 5 and 6: Transport measurement on TI Bi2Se3
- Page 7 and 8: Transport measurement on TI Bi2Se3
- Page 9 and 10: Transport measurement on TI ‣ Bi2
- Page 11 and 12: 2D Dirac Matter A B H = ~v F ( y .k
- Page 13 and 14: Diffusion of 2D Dirac states • Tr
- Page 15 and 16: Study of diffusion of 2D Dirac stat
- Page 17 and 18: (Weak) Localization • Probability
- Page 19 and 20: (Weak) Localization / Quantum trans
- Page 21 and 22: Weak localization : Universality cl
- Page 23 and 24: Weak localization : Universality cl
- Page 25 and 26: Anderson Univ. classes and Topologi
- Page 27: Anderson Univ. classes and Topologi
- Page 31 and 32: Anisotropic Scattering of Dirac sta
- Page 33: Conclusion ‣ New playground for u
Universality classes and weak localization<br />
Localization Universality Classes (for metals)<br />
‣ Orthogonal / AI Class (T 2 =+1)<br />
- electrons + scalar disorder<br />
- 4 Diffuson + 4 Cooperon<br />
‣ Symplectic / AII Class (T 2 =-1)<br />
- electrons + spin orbit disorder / Dirac + scalar disorder<br />
- 1 Diffuson + 1 Cooperon<br />
‣ Unitary / A Class (T=0)<br />
- electrons + magnetic disorder / Dirac + magnetic disorder<br />
- 1 Diffuson + 0 Cooperon<br />
For 1 flavor of carrier with spin<br />
(In graphene : treat spin x valley degeneracy !)<br />
0.2<br />
0<br />
-0.2<br />
-0.4<br />
-0.6<br />
2.10 - 5<br />
690 mK<br />
conductance<br />
Weak Localization (d=2)<br />
hgi = hgi(B) hgi(0)<br />
"<br />
!#<br />
= ↵ e 2 ~<br />
ln<br />
1 ⇡ h 4Bel 2 2 + ~<br />
4Bel 2<br />
weak (anti-)localization<br />
↵ =1(Orthogonal), ↵ = 1/2 (Symplectic), ↵ =0(Unitary)<br />
↵ =(N C,T N C,S )/2<br />
-0.8<br />
-2000 0 2000<br />
B<br />
Conductance, same sample, different spins config.<br />
12.4<br />
12.2<br />
12<br />
11.8<br />
11.6<br />
11.4<br />
11.2<br />
11<br />
10.8<br />
0 20 40 60 80 100<br />
Flux Through the Sample<br />
Universal Conductance Fluctuations (d=2)<br />
h( g) 2 i = h(g hgi) 2 i<br />
= N ✓<br />
C + N S e<br />
2<br />
15 h<br />
◆ 2<br />
h( g) 2 i = 2 15<br />
h( g) 2 i = 1 15<br />
✓ e<br />
2<br />
h<br />
✓ e<br />
2<br />
h<br />
◆ 2<br />
◆ 2<br />
Symplectic : Dirac + scalar impurities<br />
Unitary : Dirac + magnetic impurities<br />
mardi 3 janvier 12<br />
Symplectic
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