Diffusion at the surface of Topological Insulators P. Adroguer, D.Carpentier, J. Cayssol, E.Orignac Laboratoire de Physique, Ecole Normale Supérieure de Lyon, France
http://arxiv.org/abs/1205.5209
Outline •
Topological Insulators surface states
•
Classical conductance
/(b=0)
2
b
N.P
b4
•
Coherent transport
Exp.
!∆Σ "B#$ !∆Σ "B $ 0#$ 1.0
0.8
Bi 2 Te3
0.6
0.4 E=0.1 eV 0.2 E=0.2 eV E=0.3 eV !1.0
!0.5
0.0
EF (eV)
0.5
B 1.0 B0
Topological insulators •
Insulating bulk with robust conducting surface states : Quantum Hall effect ?
B
•
Paradigm : 2D + Time Reversal symmetry breaking
Topological insulators • Paradigm : 2D + Time reversal symmetry breaking • 2D + Time-reversal symmetry : Spin orbit coupling Vs. magnetic eld Kane, Mele 2005 ; Bernevig, Zhang, 2006
Pict. : Murakami
•
3D + TRS : topological insulators ; realized with Bi1-xSbx , Bi2Te3 , Bi2Se3 , Strained HgTe , etc
Topological insulators surface states Quantum Hall Effect
•
Robust edge states
3D TI
•
Robust surface states (odd number)
•
Responsible of electronic transport Topological Insulator
!1.5
•
Responsible of electronic transport Buttiker, 1982
•
!1.0
!0.5
0.0
0.5
1.0
1.5
EF
Linear dispersion + momentum-spin locking : Dirac fermions
Transport of these surface states
ARPES data for the surface states • •
Dirac fermions : linear dispersion Magnetic spin in the plane, winding around vertical axis
S.Y. Xu et al. (2011)
•
Richer structure, hexagonal shape of the Fermi surface in Bi2Te3 and Bi2Se3
Y.L. Chen et al. , 2009
Transport experiments • •
Thin lms : improve surface/bulk ratio, gating both surfaces Eto et al. , 2010
Strained HgTe : no bulk conductance
Magneto-transport : weak anti-localization 1.5K 2K 3K 4K 5K 7K 10K 15K
0.6
0.4
pi , po
2
•
Residual bulk conductance
δσ xx (e /πh)
•
Bi2Te3 thin lm
Bulk HgTe
0.2
0.0
V g=6V -0.4
Kong et al. , 2010
-0.2
Bouvier et al. , 2011
0.0
B (T )
0.2
0.4
Dirac fermions system • • • •
Graphene : sublattice 2 x 2 cones TRS : no constraint Trigonal warping at high energies
H = !vf (!σ × !k).ˆ z
• • • •
TI surface state : magnetic spin 1 cone (odd) TRS : constraint Hexagonal warping at high energies
Departure from Dirac fermions • •
Dirac point burried in bulk valence band
•
Hexagonal warping : effect on transport
High energy regime natural
1.5K 2K 3K 4K 5K 7K 10K 15K
0.6
2
δσ xx (e /πh)
0.4
pi , po
0.2
0.0
V g=6V -0.4
-0.2
0.0
B (T )
0.2
0.4
Outline •
Topological Insulators surface states
•
Classical conductance
/(b=0)
2
b
N.P
b4
•
Coherent transport
Exp.
!∆Σ "B#$ !∆Σ "B $ 0#$ 1.0
0.8
Bi 2 Te3
0.6
0.4 E=0.1 eV 0.2 E=0.2 eV E=0.3 eV !1.0
!0.5
0.0
EF (eV)
0.5
B 1.0 B0
Model • Fermi surface deformation
Different energies Fermi surfaces
kmax
kmin S.Y. Xu et al. (2011)
•
Warping hamiltonian λ 3 3 ! H = !vF (!σ × k).ˆ z + (k+ + k− )σ z 2 ( L. Fu, 2009 )
λEF2 w(w + wmax )2 b= = 2(!vF )3 2(wmax − w)3
;
Experimentally : 0 ! b " 0.6
w = wmax
kmax − kmin , kmax + kmin
Regime of diffusive transport • • • •
Experimental regime : far from the Dirac point (good metal) λ 3 3 ! Hamiltonian : H = !vF (!σ × k).ˆ z + (k+ + k− )σ z + V (!r) 2 $V (!r)% = 0 $V (!r)V (r!" )% = γδ(!r − r!" ) Sample length # mean free path $e (weak disorder)
Semi classical approach, kf $e # 1 (perturbative approach) - Boltzmann equation - Diagrammatics
Boltzmann approach • Density of states : f (!k) ! |V |!k%| |$ k Scattering probability : • "
k!"
|$k!" |V |!k%| = g!k (θ) 2
e σ= h
γ
θ !k
!k
|$k!" |V |!k%|2 = g!k (θ)
θ k!"
Perturbative result 2!2 vF2
= g!k (θ) spinor overlap
θ
8
2
2
/(b=0) 2
b
6 4
(1 + 8b2 − 58b4 + o(b4 ))
2 0.0
b4 0.2
0.4
0.6
0.8
1.0
b
Diagrammatic approach •
Kubo formula : σαβ ∝ +
ΓD
!
j α GR j β G A !k ΓD (!q )
+
=
ΓD (!q )
!k − !q
•
Dyson equation :
"V (!r)# = 0 "V (!r)V (r!" )# = γδ(!r − r!" )
GR
=
+
+
+
=
+
+
$GR/A % = (E ± i!/2τe − H0 )−1
+ ...
+
+
+ ...
Diagrammatic approach •
Kubo formula : σαβ ∝
! +
•
j α GR j β G A ΓD
Disorder induced coupling %GR GA &
−→ 0 (kF !e # 1)
Diagrammatic approach •
σαβ ∝
Diffuson mode %GR GA &
!
j α GR j β G A ΓD
+
Anisotropy +
vF2 τtr vF2 τe −→ D= d 2
+...
•
b=0 +
ΓD
=
+
+
+
+...
4 modes 1 - 1 diffusive singlet D(b)q 2 τ − iωτ /! e e - 3 triplet, non diffusive
= −2evF σ x
tr = 2 e • b '= 0
jx = −evF (σ x + 2bk 3 cos 2θσ z ) ΓD
Non perturbative results • • •
Non perturbative in warping Correction to Dirac physics Possible to probe experimentally
/(b=0)
λEF2 b= 2(!vF )3
/(b=0) 2
2
b
b
N.P
N.P
b
b4
4
E (eV) F
b Exp.
Exp.
Non perturbative results • • • 1.0
2 Einstein relation σ = e ρD
Opposite effects Strong effect on diffusion constant w.r.t. Dirac physics (b)/(b=0)
D(b)/D(b=0) N.P
b2
0.8 0.6 0.4
0.0
0.2
0.4
0.6
0.8
1.0
b
b4
b Exp.
Outline •
Topological Insulators surface states
•
Classical conductance
/(b=0)
2
b
N.P
b4
•
Coherent transport
Exp.
!∆Σ "B#$ !∆Σ "B $ 0#$ 1.0
0.8
Bi 2 Te3
0.6
0.4 E=0.1 eV 0.2 E=0.2 eV E=0.3 eV !1.0
!0.5
0.0
EF (eV)
0.5
B 1.0 B0
Coherent transport •
Phonons : nite coherence time Mesoscopic physics : low T (
), small samples
Quantum interferences Diffuson Cooperon
•
Universal values : weak (anti)localization / UCF
Coherent transport : diagrammatics • Interferences effects : 2 diffusive modes •
Γ(d)
= × + ××
Γ(c)
=
×××
+
××××
+···
+
×
+
×
+···
Weak anti-localization −k$"
k$"
H
•
×
+
$k
$k
k$"
$k
+
=
C
$k
$k
$k
$k
−k$"
k$"
−k$"
$k
+ $k
−$k
k$"
−k$"
$k
$k
e2 %δσ& = π!
Conductance uctuations $ − $k, EF − Ω Q
$k
! ΓC (Ω, Q)
$ −K $ ", Q EF − Ω
HC k$"
$ − k$" , EF − Ω Q
$k − $ q , EF − Ω
$k
! ΓC (−Ω, Q)
ΓD (Ω, !q)
$ − K, $ Q EF − Ω
$ K
$" − $ q, K EF − Ω
HD k$"
$" K
ΓD
ΓD
HC
ΓD
ΓD
ΓC
ΓC
ΓC
ΓC
$" K
HD q , EF − Ω k$" − $
$ −$ K q, EF − Ω
ΓD (−Ω, −!q)
$ K
%δσ 2 & = 12
#
2
e h
$2
" 1 V
! Q
1 Q2 "
q !
1 q4
Same results as non-relativistic electrons with random spin-orbit coupling !
Anderson problem • Coherent metal + weak disorder : Anderson problem • Universality classes for transition (strong disorder) : universal metallic properties (weak disorder) • •
Time Reversal Symmetry
λ 3 3 $ (k+ + k− H = !v ($ σ × k).ˆ z + )σ z + V ($r) , F 2
T2 = -1
Symmetry T
P
C
0
A
0
0
0
AI
1
0
AII
−1
0
d−1 1
2
0
Z
0
Unitary
0
0
0
0
Orthogonal
0
0
Z2
Z2
Symplectic
Wigner - Dyson Classes
•
Symplectic class/AII crossover to Unitary/A (mag. eld) 2 / 1 diffusive modes Diffuson
Cooperon
Symplectic and unitary classes results •
Diffuson
• •
Unitary class
Symplectic class, TRS Diffusive modes
Diffuson
Cooperon
Weak Anti Localization (WAL) " 1 e2 %δσ& = π! Q ! Q2 Conductance uctuations # 2 $2 " 1 1 e 2 %δσ & = 12 h V q! q 4
%δσ& = 0 2
%δσ & = 6
#
2
e h
$2
1 V
"
q !
1 q4
Universal results : specicity of Dirac in the crossovers
WAL crossover • •
Phase coherence length : Lφ = Result for Lφ ) L :
Lφ e2 ln( ) %δσ& = π! 'e
•
%
D(b)τφ
Function of b (or EF) !∆Σ "B#$ !∆Σ "B $ 0#$
Crossover :
1.0
! ! Bφ (b) = 4eD(b)τe 4eD(b)τφ & # $ # $' 2 1 Be 1 Bφ e + + %δσ(B)& = Ψ −Ψ 4π 2 ! 2 B 2 B Be (b) =
Function of b (or EF)
0.8
Bi 2 Te3
0.6
0.4 E=0.1 eV 0.2 E=0.2 eV E=0.3 eV
!1.0
!0.5
0.0
0.5
B 1.0 B0
In plane Zeeman magnetic eld • •
$ σ Extra term in the hamiltonian : gµB B.$
Effect in absence of hexagonal warping
3
2
1
0
!1
!2
!3
• •
!3
!2
!1
0
1
2
3
No change of the scattering probability Cooperon stays massless
1 D(b)Q2 τe − iωτe /!
No change in conductivity
In plane Zeeman magnetic eld Brillouin Different zones mag.fields 3
•
Effect in absence/presence of hexagonal warping
2
1
0
!1
!2
•
!3 !3
TRS broken, crossover to unitary class. Cooperon is no longer massless L %l
!2
!1
0
1
2
3
B e
1 D(b)Q2 τe − iωτe /! + m(b, B)
B=1T
2500 2000 1500
Magnetic length ( LB = D(b)τe /m(b, B)
Bi2Se3
1000
Bi2Te3
500
0.10
0.15
0.20
0.25
E " 0.30 F
Conclusions • Hexagonal warping taken into account non-perturbatively
!∆Σ "B#$ !∆Σ "B $ 0#$ 1.0
/(b=0)
0.8
Bi 2 Te3
2
b
0.6
N.P
0.4 E=0.1 eV 0.2
b4
•
E F (eV)
E=0.2 eV E=0.3 eV !1.0
!0.5
0.0
0.5
Exp.
In-plane Zeeman magnetic eld effect Brillouin Different zones mag.fields
3
2
1
0
!1
!2
http://arxiv.org/abs/1205.5209
!3 !3
!2
!1
0
1
2
3
B 1.0 B0
