Weak Antilocalization of 3DTI Surface States in the Presence of Spin-orbit Impurities P. Adroguer1, W.E. Liu2, D. Culcer2, and E. M. Hankiewicz1 1 Institute
for theoretical physics, Universität Würzburg 2 University of New South Wales DPG Frühjahrstagung 2015, Berlin
Outline • •
•
Introduction to transport in 3D topological insulators
• •
3DTI surface states and transport Regime of coherent transport (weak localization)
Effects of spin-orbit impurities in 3DTI
• • •
Elastic scattering time Diffusion constant Quantum correction to conductivity
Perspectives
ky (
0.0
0.0 –0.1
–0.1 3.1. 2 dimensional topological insulators : HgTe
3D Topological insulators surface states a
0.1
d
b
0.0
19 eV
0
0.0
0
31 eV
-0.2 -0.1 0.0
31 28 25
0.0
22 21 20
–0.1
c
¬0.4
LETTERS
¬0.6
¬0.2 –0.2
¬0.2
¬0.4
¬0.4
–0.4 LETTERS ¬0.6
¬0.2
0
0.2
ky (Ŭ1)
¬0.6 ¬0.2
a
hv (eV)
EB (eV)
¬0.2
EB (eV) EB (eV)
equation 3DTI : insulator with odd number5.2.ofBoltzmann topologically protected surface states (Bi2Te3 , Bi2Se3, strained HgTe... ) EB (eV)
•
21 eV
0
–0.1
High c
Low
NATURE PHYSICS –0.2
Low
0
0.2
ky (Ŭ1)
–0.10 0.0 ¬0.2 0.2
0.1
19 18 17 16 15
–1 ky (Åk¬1x) (Å ) b a High Low
M 3.2
kz (Ŭ1)
LETTERS
2.8
–0.2
2.4
–0.3 ¬0.4 0 EB (eV)
31 eV
22 21 19 15
–0.2 –0.1 0.0 0
¬0.
kx (Å–1) e
Spin polarization
High b 0.2 0.2 Bi2Tedata TunedofBiDirac Figure 2 | Transverse-momentum kz dependence bands �. a, TheBi energy dispersion along the �–MBi cut, mea 2–!Ca !Senear 3 Tuned 3 2–!Ca!Se3 2Te3 Figure 1 | Detection of0.1 spin-momentum locking of spin-he energy of 21 eV (corresponding to 0.3 length along �–Z � kz ), 19 eV (�) and 31 eV (−0.4 k-space length along �–Z0.2 of th 0.1k-space 0.1 0.1
ky (Å–1)
ky (Å–1)
BZ) are shown. Although the bands below −0.4 eV bindingin energy showand strongBikz dependence, the linearly dispersive Dirac-like electrons Bi2Se 3 2Te3 using spin-resolved ARPES broad feature show weaker kz dispersion. The Dirac point is observed to lie inside the bulk bandgap. A careful look at the individu 0.0 the surface offrom tuned stoichiome map0.0 atenergy EF ofdistribution 0.0 (seeintensity 0.0 (111) dependence of the U-shaped continuum b for details). b, The curves obtained0.0 the normal-emission (a;bulk seebands text) and−0.3 of eV Bi(blue (b). lines). Red This arrows denote the 2Te3dotted 15–31 eV photon energies reveal two dispersive below is in addition to the twodirec nonDirac-cone bands inside the gap. –0.1 The Dirac band intensity isaround strongly modulated by thesurface. photon energy changes due to the matrix c , d , ARPES dispersio projection the Fermi –0.1 –0.1 –0.2 –0.1 also observed in BiSb; ref. 5). c, A k-space map of locations in the bulk three-dimensional BZ scanned bykthecut. detector at differen Ca Se ( c ) and Bi Te ( d ) along the The dotte Bi 22d d 3 2 3 x theta (θ ) range of ±30◦ . This map (kz , ky , Ephoton ) was used to explore the kz dependence of the observed bands.
Intensity (a.u.)
EB (eV)
EB (eV)
c and d are our pr guides to the eye. The–0.2 shaded regions in–0.2 0.0 -0.1 –0.1 -0.2 -0.1 0.0 -0.2 0.2 0.0 0.0 –0.1 0.1 0.1 0.1 Te3, isrespectively, onto Ins th bulk bandspoints of pure Bi2SeV-shaped at particular high-symmetry points—the Kramers on the singly degenerate. 3 and Bi2band d c d g c ! surface BZ. In our calculations, the SSs (red dotted lines) are doubly electron-pocket-like U-shaped continuum e, Measured y component of spin-polarization along the Ci Hsieh et al., 2009 degenerate only at � (Fig. 1f). This is generally true forresults all known near This filled U-shaped Figure 3.7: ARPES experiments for Bicuts SeFermi andlevel. Bithe Tesurface top b 2the 3through 2 3 . The 220 meV, only states. EB 5D. 0.0Biwhich 0.0 such spin–orbit-coupled material surfaces as gold25,26 or correspondence to the bottom part of the 1−x Sbx shows the dispersion relation ofdirection. Bi2 Secross one direction ; the brightes 4 0.01f). f,0.0 Measured x (red(Fig. triangles) and z (bl 3 along (ref. 5). In Bi2 Se3 , the SSs emerge fromof thethe bulkcut continuum, band continuum Considering the ! ! components of spin-polarization along the C { M direction each other at �,the pass bulk throughcontribution. the Fermi level (EF )We and can eventually occurring Bi2 Se3 and of by corres being checknaturally the linear dispersion the
merge with the bulk conduction-band continuum, ensuring that calculation, we assign the broad feature to –0.1grey –0.2apply –0.1 Top right : the traverses same results for The ares corr –0.2 2 Teof 3 .the 3 at leaststate. one continuous band-thread the bulk bandgap theBi bottom conduction band. 2 3 locked due to Z topology. This is most clearly see between a pair of Kramers points. Our calculated result shows that To systematically investigate the nature to bulk states[56]. Bottom : the winding of the spin projection arou 2 ",# imaged in our data,–0.2 no surface band crosses the Fermi level if SOC is not included in we have carried out a d –0.2 resolved spectra (I Fig. 1g), which are calculated fr Fermi and surface, left for doped Bi Se , right for Bi Te [56]. y ; dependence –0.4 2 3 2 3 the calculation, only with the inclusion of the realistic values study, of which selected dat 2 –0.4of the topological " # Figure 5.2: Left : ARPES data showingofthe linear dispersion 5 Itot(1 1 PyFig. )/22a,b. andA modulation Iy 5–0.3 Itot(1 Py)/2, w to Iyspectrum SOC (based on atomic Bi) does theing calculated show of 2 incident photo singlyand degenerate gapless surface bands that are guaranteed to cross probe the k dependence of the bands –0.3 z insulator surface states labeled SSB, BVB BCB being the bulk bands [58]. spin-averaged intensity. To 0.1 extract the spin polariza –0.1 0.0 –0.2 –0.1 0.0 0.1 –0.1 0.0 –0.12c), 0.1 SOC study 0.1 0.2 the Fermi level. The calculated band topology with realistic for a way to distin –0.2(Fig. 0.0allowing –1 ) and backward (2k ) moving the forward (1k kx (Å–1) kcontributions Right : ARPES data showing the Fermileads surface ofexperiment, topological surface x photoemissio to a single ring-like surfaceainsulator FS,conducting which is singly degenerate particular x (Å STM is xused to ) scan a–1a)surface, when av kx (Åto kx (Å–1) tip long as the for chemical potential is inside the bulk bandgap. This energy study did notfit indicate a strong kz d performed a standard numerical (Supplemen state, with projection of the spin in redso arrows every direction [93]. bias is applied between the tipFu–Kane–Mele and the surface to characterize electro topology is consistent with the Z2 = −1 class in21the lying energy bands on the ‘U’,:although . The fit results yield 100(615)% polarized (Figth tion) 7 Figure 1 | Detection of spin-momentum locking of spin-heli classification scheme . have some dispersion (Fig. 2). Some variat tunnelFigure from1 the tip topoint thespin-momentum surface. The tunneling probability is a func | Detection of of near spin-helical Dirac along the (k 3locking z)intensity direction, is consistent A global agreement between the experimental band structure Espin-resolved is, however, observed ow ARPES. electrons in Bi Fwhich 2Se3 and Bi 2Te3 using 14,21 the and distance ofin the tip from the surface, applied voltage and ofIn th Bi21f) Te spin-resolved ARPES. a, b, ARPES electrons Bi (Fig. 1a–c) our theoretical calculation (Fig. is obtained by the the electron–photon matrix element. lig 2Se 3 and 3 using
Bi Te Alpichshev et al., 2010
• •
Strong spin-orbit coupling : Spin-momentum locking Dirac fermions Hamiltonian : H = �vF (�k × �σ )z
Transport in mesoscopic physics • • • •
Mesoscopic physics = weak disorder, coherent transport λF � le � L, Lφ Scattering of the electrons on impurities
Each trajectory has a given probability amplitude ai � a∗i aj Conductivity σ ∝ i,j
a∗i
aj
Transport in mesoscopic physics • • • •
Mesoscopic physics = weak disorder, coherent transport λF � le � L, Lφ Scattering of the electrons on impurities
Each trajectory has a given probability amplitude ai � a∗i aj Conductivity σ ∝ i,j
a∗i
ai
aj −→ 0
(kF �e � 1)
σcl ∝
� i
a∗i
a∗i ai
Quantum corrections to conductivity • In case of time-reversal symmetry : added contribution 2.1. GENERAL IDEA
Diffuson
•
Cooperon
and the variance �(g − �g�V )2 �V (B) as a function of the magnetic field. The results are shown on figure 2.2. T
show that beyond fields of order2 4.10−2 T, both the average and variance of conductance over disorder realizat
Constructive interference magnetic field Weakor the magnetic fi become magnetic fieldsuppressed independent, indicatingby that the statistics over the disorder:realization, are equivalent. localization (Altshuler et al., 1980)
Figure 2.2: (A): mean conductance obtained from figure 2.1 and the weak localization fit [67]. (B): the varia
Weak anti-localization • In presence of 3D spin orbit impurities (Hikami et al., 1980) • •
V (�k, �k � ) = U (Id + iλ(�k × �k � ).�σ )
Elastic scattering time modification 1 = 2πρ(EF )nI U 2 (1 + λ2 kF4 ) τe αe2 Quantum correction to conductivity : σ = σcl − 2 ln L π � 2 spin 1/2 : 4 cooperon modes 16
G. Bergmann, Weak localization in thin films
54.0
0.1
Au
~
Mg
14.0
- 3 triplet (+1/2, killed) - 1 singlet (-1/2, preserved) α : 1 → −1/2
Weak anti-localization!
16%Au 8%Au
°°
-1.0
4%Au
7.1
2%Au
~
0
0
3.8
1%Au R~91Q T’4.6K
0.5
-0.1
1.0 0.5 -0.8
-0.6
-0.4
0%Au -0.2
0 H(T)
0.2
0.4
06
0.8
WL to WAL correction induced by SOC from
Fig. 2.10. The magneto-resistance of a thin Mg-film at 4.5 K for different coverages with Au. The Au thickness is given in % of an atomic layer o the right side of the curves. The superposition with Au increases the spin-orbit scattering. The points are measured. The full curves are obtaine with the theory by Hikami et al. The ratio T 1/T5, on the left side gives the strength of the adjusted spin-orbit scattering. It is essentially proportion to the Au-thickness.
impurities for electrons with parabolic dispersion
Coherent transport of Dirac fermions Dirac fermions + scalar disorder : weak anti-localization th : Tkachov and Hankiewicz, PRB 84 (2011) Adroguer et al., NJP 14 (2012) 3
Kong et al. , 2010 5
5 1.75
/h)
Bi2Te3 thin film
1.5K 2K 3K 4K 5K 7K 10K 15K
0.6
0.4
pi , po
2
δσ xx (e /πh)
perimentally plotted in Fig. 3 are derived by inverting the measured resistivity tensor (ρxx , ρxy ) and expressing the data in units of e2 /h [see a flash animation in the supplementary-material-c]. Fits to the two fluid model D D σxx + σ0 and σxy , shown as dotted lines, are found to be quite accurate for all gate voltages. The residual differences between the data and fits is attributed below to the quantum corrections to the conductivity (antilocalization shown in Fig. 4). A qualitative discussion of the data sheds some light on the origins of the parallel conduction σ0 . At Vg ≈ 1.75V (≈ 1V above Vg0 ), the maximum of the Hall magnetoconductance occurs at B=0.6 T. In Eq. 2 this maximum is at B∗ = 2eD/µ(Vg ) where the Hall conductance equals kF #e /2. Since B∗ and kF #e fully specify the Dirac magnetoconductance, the value of σ0 , the parallel conductance, is found to be of order one in units of e2 /h. The value of σ0 varies little above Vg0 but increases linearly in the hole region up-to ≈ 4.5e2 /h@ Vg = −2 V. In this region, σ0 measures the gradual population of the bulk heavy
2
•
0.2
0.0
V g=6V -0.4
-0.2
Strained HgTe
0.0
0.2
0.4
B (T )
FIG. 4: The quantum correction Bouvier to the conductivity are obet al. , 2011 tained by subtracting the two-fluid fit to the measured longitudinal conductivity. The difference are plotted as a function of magnetic field for different temperatures. The curves are fitted to the expected digamma dependence as a function of field. The characteristic field is Bi = 40 mT at T=1.5 K and
Summary of coherent transport • Dirac fermions + scalar disorder : weak anti-localization (dot) • Electrons w/ parabolic dispersion + 3D spin-orbit impurities : crossover from weak localization to weak anti-localization (line) 0.5
δσ(B = 0) (arb . units) 0.2
0.4
0.6
0.8
1.
1.2
1.4
1.6
1.8
2.
nSOI
(arb. units) �0.5
•
�1
What is the effect of the spin-orbit impurities on the Dirac surface states physics?
Outline • •
•
Introduction to transport in 3D topological insulators
• •
3DTI surface states and transport Regime of coherent transport (weak localization)
Effects of spin-orbit impurities in 3DTI
• • •
Elastic scattering time Diffusion constant Quantum correction to conductivity
Perspectives
Elastic scattering time • Model : Dirac fermions + weak scalar disorder + SOC from impurities H = �vF (�k × �σ )z + V (�k, �k � ) V (�k, �k � ) = U (Id + iλ(�k × �k � ).�σ )
• • •
Elastic scattering time via Fermi golden rule� Self energy calculation
1 = πρ(EF )nI U 2 1 + λkF2 + τe �
1 = πρ(EF )nI U 2 1 + λkF2 + τe
λ2 kF4 2 λ2 kF4 2
New : Linear dependance in λ of the elastic scattering time!
� �
2τe
=
Diffusion constant B.
1+α+
2
2
2
σ = e ρ(EF )D
Classical conductivity, solving the Boltzmann equation
Boltzmann equation reads :
•
Solving the kinetic equation � ∇ � �f = −eE. k
�
dk�� �� |V |�k�|2 δ(E(k�� ) − E(�k))(f (k�� ) − f (�k)) 2π|� k (2π)2 2
2
�
2
5λ2 kF4
2
�
� is the electric σclfield = eapplied, ρ(EF )vand λkperturbed + o(λof) states. In the follo his equation, E the density f τe f 1is− F + 4 to assume the electric field is along the x axis, and look for an ansatz of the form f (�k) = nF (�(�k)) Standard diagrammatic technique (ladder diagram) . Consequently, after the integration over |�k| we need to solve the simpler equation :R G (k) � � � ∂� dθ� � 2 ¯ � ¯ −eEx ∗ = 2πρ(EDF ) |�θ |V |θ�| f (θ ) − f (θ)D ai x ∂k 2π + = ai king for a solution of the form f¯(θ) ∝ evF τe Ex cos θ gives an equation to solve for the proportion A nd in the end that : � � G (k−q)
•
•
2 4 5λ kF 2 2 2 σcl = e ρ(EF )vf τe 1 − λkF + + o(λ2 ) Diagrammatic representation equation obeyed by the diffuson structure fac 4 1 + αof+the α2Bethe-Salpeter /2
Figure 7: explicited the spin structure. f¯(θ) = 2
New : Dependence of
evF τe Ex cos θ 2 /4 1 + 2α + 5α the diffusion constant on λ!
each mode, the Bethe-Salpeter can now compute theFor current in the x direction :equation reduces to :
Quantum correction to conductivity •
Cooperon structure factor R
G (k) C
+
=
C
A
•
G (Q−k)
1 singlet mode 3 triplets : one single diffusive Figure 4:and Diagrammatic representation of the Bethe-Salpeter equation obeyed by the cooperon structure factor. (gapless) mode In this equation, T r means a trace over all quantum numbers (momenta and spins), whereas tr is a trace only theCspins, the integration1 over the momenta being calculated in P . We can note that the current renormalizat � we calculated both the ”bubble diagram” and the ”ladder diagram”, this is why the second curr |S��S| Γ (Q)takes =into account s.s. 2 operator is not renormalized. DQ Using the relation between the elastic scattering time and the strength of the disorder �/τ = πρ(E )γ, we C the�conductivity tensor1as a function of the scattering time and finds for the longitudinal conductivity : express Γt.s. (Q) = |Ti ��Ti | 2 � δ DQ + m �σ i � = e v 2 = e ρ(E )v τ δ . ( D
e
Dr αβ
•
2π
2 2 F
αβ
γ
2
F
F
2 F e αβ
Using the Einstein relation σ = e2 ρD, this gives a diffusion constant D = vF2 τe as previously calculated. T
time τ in diffusive motion is defined as D = where d is the dimensionnality of the motion. In t Always transport weak anti-localization case we observe the expected doubling of the transport time compared to the elastic scattering time expected tr
2 vF τtr d
Dirac fermions. This doubling can be interpreted as the fact that because of the suppression of the backscatter
What we learnt in coherent transport? 0.5
Dirac fermions
δσ(B = 0)
Hikami-Larkin-Nagaoka 3D
(arb . units) 0.2
0.4
0.6
0.8
1.
1.2
1.4
1.6
1.8
2.
nSOI
(arb. units)
Hikami-Larkin-Nagaoka 2D �0.5
• •
�1
Dirac fermions : always weak antilocalization Hikami-Larkin-Nagaoka : dependance on the dimension (in 2D, SU(2) symmetry not totally broken, 1 triplet remains)
Conclusions and perspectives •
Linear dependence in λ of the elastic scattering time
• •
Diffusion constant dependence in λ
•
Hikami-Larkin-Nagaoka formula do not give WAL for surface states
•
Derivation of the quantum correction to conductivity in presence of magnetic field
Weak anti-localization preserved
•
Characteristic mag. field
Thanks for your attention! Work supported by DFG grant HA 5893/4-1 within SPP 1666
