Diffusion of Dirac fermions P. Adroguer

Laboratoire de Physique, Ecole Normale Supérieure de Lyon, France

Outline

• • •

Diffusion, regime of weak disorder Diffusion of Dirac fermions

• •

3D Strong topological insulators Graphene

Diffusion of semi-Dirac excitations

Outline

• • •

Diffusion, regime of weak disorder Diffusion of Dirac fermions

• •

3D Strong topological insulators Graphene

Diffusion of semi-Dirac excitations

Diffusion

• • • • •

Continuity limit of a random walk (mean free path le ) Semi classical approach λF � le

Experimental regime, high Fermi energy : good metal Sample length � mean free path le Weak disorder regime

- Boltzmann equation - Diagrammatics

Boltzmann equation

5.3. Standard diagrammatic techn

�k) : •atIntegro-differential a given wavevector f (equation � ∂f = � −eE. ∂�k

�

dk�� (2π)2

�

�

2π|�k�� |V |�k�|2 δ(E(k�� ) − EF ) f (k�� ) − f (�k) � . (5.10)

•

solve it, we that the local density of statesand f (�k)density responds linearly 2 To quantities ofassume interest : scattering probability toof thestates electric field, and look to the first order terms in the electric field. We use ∂nF ¯ � �

the ansatz f (k) = nF (E(k))+ ∂E f (θ) where θ accounts for the direction of the wavevector �k. Using the value of the spinor overlap |�θ� |θ�|2 = (1+cos(θ� −θ))/2 we solve the equation in f¯ assuming the electric field is along the x-axis and find : f¯(θ) = 2evF τe cos θEx . (5.11)

•

Linear response : Drude’s conductivity (no quantum We can derive electric current along the x-axis caused by the electric corrections, nothe UCF) � d�k 1 dE ¯ �k) e f (θ)δ(E( 2 (2π) � dkx

field jx = − EF ) and dividing by the electric field we find the classical Drude conductivity : 2 2

Standard diagrammatic technique : Kubo formula • Linear response � � � σαβ =

• •

2πΩ

� T r j α GR j β GA

Quantum mechanics

σ∝

�

a∗i aj

i,j

Classical part (diffuson) = Drude’s conductivity

Bi2Te3

•

Quantum interferences (cooperon) : WL, UCF Kong et al. , 2010

Outline

• • •

Diffusion, regime of weak disorder Diffusion of Dirac fermions

• •

3D Strong topological insulators Graphene

Diffusion of semi-Dirac excitations

ky (Å–1)

EB

0.1

M

–0.1

0.6

0.0

3D Strong topological insulators LETTERS • Insulators with topologically • Surface experiments (ARPES, STM) protected surface states

0.8

1

–0.1 3.1. 2 dimensional topolo

0.0

0.4

0.2

0.6

0.8

–kx (Å–1)

Γ

M

1.0

EB (eV)

0.0

Low

a

–0.2 –0.1 0.0 21 EeV B (eV) 0 0

31 eV

EB (eV)

EB (eV)

topological Hall states Bi2Te3 Alpichshev et al. ,2010 ¬0.4

0

–0.2 ¬0.2 –0.2

EB (eV)

EB (eV)

8

0.0

–0.1

0.0

19 eV

Figure 3.5: ARPES experiment showing the presence of 5 surface states in b c Bi1−x Sbx . The white stripes correspond to the bulk bands. Dirac ¬0.2 insulator with ¬0.2 x=0 x = 0.1

High c –0.1

¬0.4

¬0.4

Energy (eV)

0.10

0

200

100

0.4

6

0.3

300

0.2

T (K)

4 2

0 ¬2

Sb2Te 3

Semimetal

Bi

0.1 0

y

¬1

y

EB (eV)

EB (arbitrary units)

b

¬1

BZ) are shown. Although the bands below −0.4 eV bindingin energy showand strongBikz depen electrons Bi2Se 3 2Te3 broad feature show weaker kz dispersion. The Dirac point is observed to lie inside the bu the map0.0 atenergy EF ofdistribution 0.0 T(seeintensity Insulator8of the U-shaped continuum 0.0 (111) dependence b for details). b, The cu –0.08 ( a ; see text) and of Bi (b). l 2Te3dotted 15–31 eV photon energies reveal two dispersive bulk bands below −0.3 eV (blue 6 Dirac-cone bands inside the gap. –0.1 The Dirac band intensity isaround strongly modulated by thesu p projection the Fermi –0.1 –0.1 4 also observed BiSb; ref. 5). c, A k-space map of locations in the bulk three-dimensiona 0.6 0.8 Bi0.9inSb Bi22dCadSe3 (c) and Bi2Te3 (d) x = 0.1 2 of ±30◦ . This map (kz , ky , Ephoton ) was used to explore the kz dependenc theta (θ ) range guides to the eye. The–0.2 shaded –kx (Å–r 0 0.0 -0 –0.1 0.0 –0.1 0.1pure Bi Se-0.2 bulk bandspoints of at particular high-symmetry points—the Kramers on the2 V-shape 3 and

c

d electron surface BZ.¬2 In our calculations,cthe SSs (red dotted lines)yare doubly e, Measured component of sp ¬4 degenerate only at � (Fig. 1f). This is generally true for all known near Figure 3.7: ARPES experiments results for Bi Se ¬6 2the Figure 3 | The topological gapless surface function of x E(based on band calculation meV, only cut ¬0.2 states in bulk insulating B 5 220 25,26 0.0Biwhich 0.0 spin–orbit-coupled material surfaces such as gold or Sb correspo 1−x x ¬0.15 ¬8 ¬6 shows thedirect-gap dispersion relation ofdirection. Bi2 Seinsulator one f,0.0 Measure ofthethe 3 along second-derivative image Bi0.9KSb0.1. a, TheΓ surface-band-dispersion bulk Dirac point M Γ K M5). In Bi2 Se3 , the (ref.of SSs emerge from bulkcut continuum, cross bandwe co ! ! components of spin-polarizatio each other at �, pass through the Fermi level (E ) and eventually naturall being the bulk contribution. We can check the l F Sb along C { M . The shaded white area shows the projection of the Bi0.4 and its surface forms a ‘topological met d 0.9 0.1 8 8 0.2 merge with the bulk conduction-band continuum, ensuring that calculati Bi2Se3 Bi2Te 3 –0.2 –0.1 Top right : the traverses samestates results apply for bulk dimensional edge inbandgap quantum spi2 –0.2 0.3 bands based on ARPES data, 6as well as a rigid shift of the tight at binding leaststate. one6 continuous band-thread the bulk theBi bott 0.1 locked towinding Zof2thatE topology. between of Kramers points. Our calculated result shows sy integrated within 610 meV bands to sketch the unoccupied bands above the Fermi level. A non-intrinsic toa pair bulk states[56]. Bottom :due the ofTo th 0.2 4 F origin 4 ",# no surface band crosses the Fermi level if SOC is not included in imaged –0.2 0 resolved spectra ; Fig. 1 by analysis of the second-derivative crossings. image was plotted stac flat0.1 band of intensity near EF generated 2 Fermi surface, left for doped Se right for B y by 2–0.4 3 , (I 2 the calculation, and only with the The inclusion of theBi –0.4 "realistic values depende 5 Itot(1 1 the PyFig. )/2 a to Iyspectrum 0 0 of SOC (based on atomic Bi) does theing calculated show image was rejected to isolate the intrinsic dispersion. The Fermi crossings of a series of scans taken parallel to k2a,b ¬0.1 y-d 0 singly degenerate gapless surface bands that are guaranteed to cross probe –0.3 0.0 ¬0.1 surface state are denoted by yellow spin-averaged intensity. Tomt ¬2 –0.1 surface state ARPES intensity near E the circles, with the band near F ¬2 –0.1 0.1 the Fermi level. The calculated band topology with realistic SOC study (F– –0.2 0.0 ¬0.2 21 –1)b ˚ ¬0.2 < 0.5 A ¬4 to double degeneracy. The red lines are ) and the forward (1k (Å k x counted twice owing f , Surface band scan directions ‘1’ and ‘2’. 2k leads to a single ring-like surface FS, which is singly degenerate contribu x x STM ¬4 experiment, a conducting kx (Å–1) tip is used to s ¬0.3 ¬0.3 so long the as the chemical potential is28 inside the bulkthe bandgap. This energy ¬6 performed acorresponding standard nus guides to the eye. An in-plane rotation of the sample by 60u produced with hn 5 eV, and Γ M K Γ K M bias is applied between the tipFu–Kane–Mele and the surface topology is consistent with the Z2 = −1 class in21the lying en ¬0.05

¬4

¬0.1

•

EB (eV)

c

EB (eV)

Numerical simulations

Energy (eV)

N

0

0

8

¬0.10

Energy (eV)

•

0.05

Sb2Se3

¬1

ky (Å–1)

0.15

y

ky (Å–1)

a

2

Energy (eV)

Resistivity (mΩ cm)

0.6 0.8 LETTERS –0.4 LETTERS 0.04 possibly a larger gap. It was shown[11] that a family of compounds (Bi 2 Se3 , ¬0.6 ¬0.6 ¬0.6 2 –0.1 0.0 Ls Moreover, the fact that Bi2 Te3 and Sb62 Te3 ) presents a unique surface state. ¬0.2 0 0.2 La ¬0.2 0 0.2 ¬0.2 0 0.2 Hsieh al.a ,High 2009bk (Åkx) (Å–1) k (Å )et 0.00 they are stoichiometric compounds, instead of an alloy in the case of Bi1−x Sbxk (Å ) aD.Low Low 0.2 Ls 4 with higher purity. Simulations of the expected dispersions in allows a growth ×80 Tuned Bi Ca Se Figure 2 | Transverse-momentum kz dependence of Dirac �. a, TheBi energy 2–! bands ! near 3 Tuned 2–!Ca Figure 1 | Detection of spin-mo energy of 21 eV (corresponding to 0.3 k-space length along �–Z � k ), 19 eV (�) and 31 eV z these materials are plotted in Fig. 3.6. –0.04 0.1 La NATURE PHYSICS DOI: 10.1038/NPHYS1270 0.1 0.1 ARTICLES T

Surface states : Odd number of Dirac cones

Figure 5.2: Left : ARPES data showing the linear disper insulator surface states labeled SSB, BVB and BCB bein

!

!

7

tion) . The fit results yield 1

3DTI Transport experiments ETO et al.

• •

Thin films : improve surface/bulk ratio, gating both surfaces

Bi2Se3 Eto et al. , 2010

Strained HgTe : no bulk conductance

Magneto-transport : weak anti-localization

FIG. 1. "Color online# Temperature dependence of !xx in3 0 T. Upper inset shows the x-ray diffraction pattern of the Bi2Se3 single perimentally plotted in Fig. 3 are derived by inverting crystal used for transport measurements. Lower inset shows !yx for the measured resistivity tensor (ρxx , ρxy ) and expressing 1.5K $ C measured at 1.5 K. B The slope of !yx"B#, shown by the thin 2 0.6 the data in units of e /h [see a flash animation in the 3 2K po solid line, suggests that the electrons whose den3K main carrierspi ,are supplementary-material-c]. Fits to the two fluid model 18 −3 4K D D σxx + σ0 and σxy , shown as dotted lines, are found tosity be is 3.4# 10 cm . 5K

quite accurate for all gate voltages. The residual differ7K 10K ences between the data and fits is attributed below to the III. RESULTS AND DISCUSSIONS 0.4 15K quantum corrections to the conductivity (antilocalization shown in Fig. 4). A. Resistivity and SdH oscillations A qualitative discussion of the data sheds some light on the origins of the parallel conduction σ0 . At Vg ≈ 1.75VFigure 1 shows the temperature dependence of !xx of the Bi2Se3 single0.2crystal studied in this work. It shows a metallic (≈ 1V above Vg0 ), the maximum of the Hall magnetoconductance occurs at B=0.6 T. In Eq. 2 this maximum behavior d! / dT " 0 down to !30 K, and saturate at lower is at B∗ = 2eD/µ(Vg ) where the Hall conductance equals temperature "there is actually a weak minimum near 30 K, as kF #e /2. Since B∗ and kF #e fully specify the Dirac magneis usually observed28,30 in low-carrier-density Bi2Se3#. The toconductance, the value of σ0 , the parallel conductance, single-crystal nature of the sample is evident from the x-ray is found to be of order one in units of e2 /h. The value 0.0 data shown in the upper inset of Fig. 1. The lower of σ0 varies little above Vg0 but increases linearly in diffraction the V g=6V !yx measured at 1.5 inset of Fig. 1 shows the Hall resistivity hole region up-to ≈ 4.5e2 /h@ Vg = −2 V. In this region, σ0 measures the gradual population of the bulk heavy K for the field-0.4direction along the C axis, suggests -0.2 0.0 0.2 which 0.4 3 B (Tand ) that theBouvier main carriers electrons the carrier density ne et al. , are 2011 Kong et al. , 2010 2

•

Residual bulk conductance

δσ xx (e /πh)

•

Bi2Te3 thin film

Bulk HgTe

is 3.4# 1018 cm−3 "in a one-band model#. From the values of

Regime of diffusive transport • • • • •

Semi classical approach, λF � le (perturbative approach) Experimental regime : far from the Dirac point (good metal) � � Hamiltonian : H = �vF �k × �σ .ˆ z + V (�r)

�V (�r)� = 0

�V (�r)V (r�� )� = γδ(�r − r�� )

Sample length � mean free path le Weak disorder regime

Boltzmann equation •

Scattering anisotropy (spinor overlap)

θ

• •

Absence of backscattering (TRS) Doubling of the transport time

e2 σ = ρ(E)vF2 τe h

2 v 2 F τtr D = v F τe = 2

As we will see in the section 6.4, we expect the presence of a single diffusive mode instead of the four that could be present. This reduction of the number of diffusive mode is due to the symmetries of the problem, in this case, because of the spin 1/2 nature of the surface states, the time-reversal operation Θ squares to the opposite of the identity Θ2 = −Id. This corresponds to a given universality class in the Anderson problem (either unitary or symplectic, in Diffuson spinorial this case symplectic) where thestructure number of diffusive mode is fixed to 1.

Diagrammatic approach •

D

Γ long (� q )distance, = fS (� q )is|S��S| + f1to (� q )keep |T1only ��T1 | To describe the diffusion at it thus necessary the singlet mode so we approximate the+diffuson limit��T | f2 (� q ) structure |T2 ��T2 |factor + fby qits) |T 3 (� 3 3 �q → 0 :

•

One single diffusive mode :

ΓD (�q ) = γ

•

1 1 x x y y z z [Id ⊗ Id + σ ⊗ σ − σ ⊗ σ + σ ⊗ σ ] . Dq 2 τe 4

(5.31)

Current operator renormalization

5.5. Diffuson

However, the importance of the diffuson in the diffusion process does not anisotropy) come (scattering only from the diffusive mode, but also from a renormalization of the D + massive modes. current operator, and this renormalization is due= to one of the Jα = 2jα 86

Figure 5.10: Diagrammatic representation of the current operator renormal2 ization.

vF τtr e2 2 2 D = v F τe = σ = ρ(E)vF τe 2 h5.5.2 Renormalization of the current operator

By definition, the current operator is obtained by introducing electric potential Doubling of transport�time from anisotropy via the substitution p� = p� − eA, and deriving the hamiltonian with respect to the potential, jα =

δH δA

. In the case of Dirac fermions we obtain that the

Coherent transport 2.1. GENERAL IDEA

33

and the variance �(g − �g�V )2 �V (B) as a function of the magnetic field. The results are shown on figure 2.2. They

show that beyond fields of order2 4.10−2 T, both the average and variance of conductance over disorder realization

•

•

become magnetic field independent, indicating that the statistics over the disorder realization, or the magnetic field

Universal values : weak (anti)localization / UCF

CHAPTER 2. PROBING MATTER VIA DIFFUSIVE ELECTRONIC WAVES

are equivalent.

Figure 2.2: (A): mean conductance obtained from figure 2.1 and the weak localization fit [67]. (B): the variance

Inelastic scattering : finite coherence time τφ Mesoscopic physics : low T (τφ ), small samples 2.1.3 Probing matter by coherent electronic waves scattering Quantum interferences

disorder configurations (at fixed magnetic field) as a function of the magnetic field. Extracted from [65]. re 2.1: Reproducible magneto conductance curves at T = 45mK of the same Si doped GaAsover afterthe 46 46 annealing

esses. Extracted from [65].

de formula for the conductivity σ0 = ρ(EF )e2 τe /m depends only on the mean free time τe between elastic

usions, i.e. on the density of impurities and not their exact positions. Being of quantum origin, these fluctuations

We will come back in much more details to the description of these fluctuations in the next chapter. Nevertheless,

ct the interference effects between the contribution of different diffusive paths, or sequence of scattering on

we already have all the ingredients to understand the core idea of the MesoGlass project : we consider a metallic

urities, on the conductivity. Along such a given path, the phase of an electronic state |uk (r)� of momentum

wire of µm size with enough magnetic impurities to develop a spin glass phase in the temperature range where the

increased by δφL = kL where L is the length of the path. For electrons at the Fermi level, k � kF , and

transport is coherent. The main goal is to gain access to the overlap between two spin configurations, phase δφL � 2πL/λF appears extremely sensitive on the length L, the Fermi wavelength λelectronic F being of atomic

these configurations correspond to two different quench below the spin glass critical temperature Tg , or two e. After a annealing procedure the positions of these impurities are modified, hence all whether the path lengths L

times tw , tw + t at a fixed temperature below Tg (see the discussion in section 2.1.1). We want to extract modified by at least λF , and correspondingly the phases δφL are redistributed randomly.different The conductivity

the information on the two frozen spin configurations by measuring the two corresponding magneto-conductance g a non self-averaging quantity, its value is then different from the initial one. A different procedure allows to

stribute these phases along the diffusive path : the application of a transverse magnetic field. The at presence traces such aoflow temperature that the spin dynamics is quenched. By this procedure, we thus obtain two � a field can be accounted for by the Peierls substitution which amount to add a extra dephasing e P A.dl snapshots ofalong the spin configurations, similarly to the case of figure 2.1 although we now change the orientations of

Diffuson

Cooperon

each path P, A being the vector potential. As the shape of these paths P is random, the associated the frozenmagnetic spins of the impurities, and not their positions. By considering the correlation between the two magnetoes are random : similarly to a change of impurity positions, the magnetic field redistribute the phases associated

conductance traces, we expect to access the correlation between the corresponding spin configurations, the overlap

each path in a sample and changes accordingly the value of the conductivity. Whenever a new quantum of

we look after.

Cooperon :

� = γ 1 |S��S| ΓC (Q) (B.3) The calculation of the cooperon structure factor ΓC is slightly more τe DQ2 tle xbecause we zneedz to take into account the direction of propagation of γ 1 1 x y y = [1 ⊗ 1 − σ ⊗ σ − σ ⊗ σ − σ ⊗ σ ], (B.4) τe DQ2 4 Green’s function in the recursive equation. Since it must be done "backwar

Coherent transport : diagrammatics • Interferences effects : 2 diffusive modes where D is the diffusion constant, D = vF2 τe in thecorrespond absence of warping. this to take

D

+

=

+

+ HC

C

•

+

=

Γ (Q)

+...

+

Weak anti-localization

� =γ ΓC (Q)

1 1 [Id ⊗ Id − σ x ⊗ σ x − σ y ⊗ σ y − σ z ⊗ σ z ] . 2 DQ τe 4

Figure B1. Diagrammatic representation of the quantum correction to conductivity

−k��

k��

�

�k

�k

�k

We observe that the cooperon structure is a traditional singlet Γ [Id ⊗ Id − σ x ⊗ σ x − σ y ⊗ σ y − σ z ⊗ σ z ] as opposed to the diffuson struc � Indeed, the diff D ∝ [Id ⊗ k�Id−+ k� σ x ⊗ σ x − σ y ⊗ σ y + σ z2⊗ σ z ]. −k� factor k� Γ e 1excitation a �k �k +structure + is� associated with the diffusion of a particle-like �δσ� = −k k� � � k cooperon is associated hole-like with the �k π� Q2 of two � diffusion −�kexcitationk ; the Q tations allowing for a traditional expression of the sin H1Cof the same nature, H2C �

�

�

H Conductance fluctuations 23

C 0

Diffusion at the Surface of Topological Insulators � ΓC (Ω, Q) � − �k, E − Ω � Q k

� −K � �, Q EF − Ω

F

H

24

C

� ΓC (Ω, Q)

� − k�� , EF − Ω Q

k

23 Surface of Topological Insulators Diffusion at the

H Figure B2. Diagrammatic representation of the dressing of the Hikami box

C

� − �k, EF − Ω Q

�k ��

�� K

�

�

�

�k

k Diffusion at the Surface of Topological Insulators

�k

=

HC

•

the transpose of the matrices. The composition o minus sign from σ ˜ and the transpose corresponds to add a minus sign t z and σ matrix of the advanced Green’s function branch, but the minus does not affect the σ y matrix since −(σ y )T = σ y . In the end, from Eq. (5 +... the cooperon structure factor can be approximated when Q −→ 0 : C �

�Q �K �−−K, � �, Q EE ΩΩ FF−−

� ΓC (−Ω, Q)

6.2 ΓD

�� K � K

Quantum correction to conductivity � � ΓD

�

The weak anti-localization correction is obtained by the contraction of a Cooperon 2 2 D D Γ Γon Fig. B1. This Hikami 2 propagator and a Hikami box, as represented diagrammatically The quantum correction to conductivity (or weak anti-localization) is box is the sum of three different contributions represented in ΓFig. B2. We express the C 4 Figure C1. Diagram for the conductance fluctuations with Cooperons ΓC similarly to the diffuson correction by calculating (cf Fig.6.4) q � : first of these contributions as � � � � C � R (Q � − �k)Σ �� −A�k) , �ΓCx GA (Q �δσ0 � = Tr GA (�k)ΣxΓGR (�k)ΓC (Q)G (B.5) R � C � R �� A �� � � � � 2π �δσαβ � = T r G (k)Jα G (k)Γ (Q = k + k )G (k )Jβ G (k ) Figure C3. Diagrams for the second contribution to conductance fluctuations 2π where we the notations introduced in the article Tr for the trace over all the Figure C2. Diagram for the used conductance fluctuations with Diffusons. D. Quantum correction a warped Fermi surfacethe spin indices, and quantum numbers (spin Appendix and momenta), tr forforthe trace over Diffuson (resp. Cooperon) structure factor between two H (resp. H ). Summing � these two diagrams (Fig. C2 and Fig. C1) we obtain : � B and Appendix show explicitly the derivation weakdevoted antilocalizationto the order for trace over the Appendix momentum k. CSpecial care has oftothebe � � the �k Diagram Figure C2. for �the conductance fluctuations with Diffusons. H

C

H

C

� − K, � Q E −Ω

� � − k�� , EF − Ω Q Figure k�C1. Diagram for the conductanceF fluctuationsK� with Cooperons � ΓC (−Ω, Q)

D

Γ (Ω, �q)

�k − �q , EF − Ω

�k

� � − �q , K EF − Ω

HD

��

HD

−Ω �k −k���q− , E�qF, E−F Ω

�kk

H

� − �q , K EK �F �−−Ω�q ,

ΓD (Ω, �q)

H

k�� − �q , EF − Ω

� − �q , K EF − Ω

ΓD (−Ω, −�q)

e2 h

2

1 V

1 . 4 q� q

1 q

D

� K

D

C

Same results as non-relativistic electrons 92 with random spin-orbit coupling !

correction(C.2) and the conductance fluctuations for Dirac fermions. As expected, the =8 corresponding results display no dependence in the only relevant parameter to D C Diffuson (resp. part Cooperon) structure factor between two H the(resp. H ). Summing The second of the conductance fluctuations come from diagrams represented characterize diffusion : the diffusion constant. These results are naturally expected these C2not andyet Fig. C1) we obtain in two Fig.diagrams C3 that (Fig. we have considered. They :require the determination of two to hold when taking into account the hexagonal warping term. To explicitly show this � 2(one �2 for�Diffusons and one for Cooperons) : additional Hikami boxes e � 1� 1 independence, 3 �δσ12 � = 8 . (C.2)we determine the value of the quantum correction to conductivity in the 4 ˜ D = ρ(EhF ) 2τVe q� qπ [1 ⊗ 1 + σ x ⊗ σ x ] H (C.3) general case where the Fermi surface possesses the hexagonal deformation. �δσ12 �

1 V

�� �K K

EF − Ω

ΓD (−Ω, −�q)

D

k��

�� K

�δσ � = 12

e h

obta

.

Anderson problem • Coherent metal + weak disorder : Anderson problem • Universality classes for transition (strong disorder) : universal metallic properties (weak disorder)

4.2. COHERENT DIFFUSION OF DIRAC SURFACE STATES OF D = 3 TOPOLOGICAL INSULATORS 89

• •

� � z + V (�r) Time Reversal Symmetry , H = �vF �k × �σ .ˆ

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

•

Chiral classes AIII

Symplectic class/AII crossover toBDIUnitary/A 1 (mag. 1 1 field) Z 0 2 / 1 diffusive modes Diffuson

CII

0

0

1

Z

0

Z 0

−1

−1

1

Z

0

Z2

C

0

0

0

Z

0

CI

1

−1

1

0

0

Z

BD

0

−1 1

0

Z2

Z

0

DIII

−1

1

1

Z2

Z2

Z

Bogoliubov-de Gennes classes

Cooperon

Table 4.1: Table of topological insulators and superconductors in dimension d=1,2,3. The 10 symmetry classes of

Anderson localization are labeled using the notation of Altland and Zirnbauer [89]. They are defined by the presence

Symplectic and unitary classes results •

Symplectic class, TRS Diffusive modes

Diffuson

• •

Unitary class

Cooperon

Weak Anti Localization (WAL) � e2 1 �δσ� = π� Q � Q2 Conductance fluctuations � 2 �2 � e 1 1 2 �δσ � = 12 h V q� q 4

Diffuson

�δσ� = 0 2

�δσ � = 6

�

2

e h

�2

1 V

�

q �

Universal results : specificity of Dirac in the crossovers

1 q4

Hexagonal warping • Fermi surface deformation

Different energies Fermi surfaces

kmax

kmin S.Y. Xu et al. (2011)

•

Warping hamiltonian 1: Warped state and buried (inaccessible) Diracofpoint of3 . Bi Bi2 Te highly surf war ped surface surface state and buried (inaccessible) Dirac point Bi2 Te Bi22Te Te33. has a highly 3 has awarped (non-ideal Dirac cone) with Dirac nodeunder buriedtrivial undersurface trivial states surfaceinstates which the topological al Dirac cone) with Dirac node buried whichinthe topological transporttransport regime λ 3 3b, ARPES z b, ARPES e realized. a, = Experimental geometry of spin-resolved ARPES. measurement of 3D Dirac surfacecone Di ed. a, Experimental geometry of spin-resolved ARPES. measurement of 3D surface � H �v (� σ × k).ˆ z + (k + k )σ F + measured − e3 . represent Arrows represent the in-plane component spin texture. c, state Surface state warping wa ws the in-plane component of the measured spin texture. c, Surface warping factor w factor as a funct 2of the √ √¯ ( L. Fu, 2009 )¯ ¯ ¯ ¯ ¯ ¯ ¯ kF F (Γ− MK) )−kF2+ (Γ−K) (Γ− 2+√3 F (Γ−M )−k √3¯. × rface carrier (n). density The warping defined as kkw(n) = . w = 0 fully implies fu rier density The(n). warping factor is factor definedis as w(n) = × w = 0 implies isotro ¯ ¯ ¯ ¯ ¯ ¯ ¯ kF F (Γ− MK) )+kF2− (Γ−K) (Γ− 2− 3 3 F (Γ−M )+k w = 1 perfect implies perfect hexagon w > 1 snowflake-shaped implies snowflake-shaped FS. d, High-resolution wcircle); = 1 implies FS; w >FS; 12implies ARPES ARPES mapping λEF2 hexagon w(w +w kmax − FS. kmind, High-resolution max ) b with = binding =binding ; wrepresent = win-plane , of the measured i evolution surface evolution with Arrows component of the measured spin teT energy. energy. Arrows 3represent the component spin texture. maxthe in-plane 3 −2 2(�v ) 2(w − w) k + k −2 −2 −2 F×10 max ˚ ce carrier (in density (in unit of˚ A warping ) and warping factorare values aremin indicated at left the and top left and right bottom r r density unit of A ×10 )max and factor values indicated at the top bottom corn chsurfaces. Fermi surfaces. Experimentally : 0 � b � 0.6

Boltzmann approach • Density of states : f (�k) • Scattering probability : |�k� |V |�k�| �

k�� |�k�� |V |�k�|2 = g�k (θ)

2

= g�k (θ) spinor overlap

θ �k θ

�k

|�k�� |V |�k�|2 = g�k (θ)

θ

k�� 8

Perturbative result 2

e σ= h

2�2 vF2 γ

σ /σ(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 • • •

Result non perturbative in warping term b Correction to Dirac physics Possible to probe experimentally D D Hb = 0L 5

σ /σ(b=0)

λEF2 b= 2(�vF )3 2

2

b

4

b

N.P

N.P

3 2

b

1

4

E (eV) F

Exp.

0.0

b4 0.1

0.2

0.3

0.4

0.5

b 0.6

Exp.

the results are represented as a ratio with σ(λ = 0) which is independent on the ene disorder strength.

Theory of diffusion of 3DTI surface states Indeed, the shape of the typical measurement of t • • •

through the dependance of the conductivity on a solely on the diffusionθconstant. Dirac physics : anisotropysurface of thedepends scattering the Fermi energy, the associated anti-localizatio Symplectic class, universal resultas (WAL energy shown on Fig. ??. While the amplit

correction)

�∆Σ �B�� �∆Σ �B � 0��

Specificity of the hexagonal warping

• • • •

1.0

Departure from pure Dirac physics

0.8

Bi 2 Te3

To be treated non-pertubatively 0.6

Dependance of the crossovers In-plane magneto-transport

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 �1.0

Figure 12. Dependance of the weak lo

Outline

• • •

Diffusion, regime of weak disorder Diffusion of Dirac fermions

• •

3D Strong topological insulators Graphene

Diffusion of semi-Dirac excitations

0

0.1

0

0

-0.1

-0.1

-0.2

-0.2

Dirac fermions system 0.2

-0.2

-0.1

0

0.1

0.2

0.2

-0.1 -0.2 0.2

-0.2

0

0.1

0.2

Graphene

-0.1

• • • • 0

0

σ : sublattice -0.1 -0.2

0.1

0.2

-0.2

-0.1

0

0.1

2 x 2 cones

0.2

urface states in Bi2 Te3 from ARPES and STM studies. The Fermi surface evo-

TRS : no constraint

mical potential (n-type doping) is shown as observed in ARPES measurements.

rom [158].

H = �vf (�σ × �k).ˆ z STI surface state

0.1

0

0.1

-0.1

0.2

0.1

0

-0.2

Trigonal warping at high energies

• • • •

σ : magnetic spin 1 cone (odd) TRS : constraint Hexagonal warping at high energies

Weak localization in graphene Valley degeneracy : possibility of intra- and inter- valley scattering Intravalley scattering only

With Intervalley scattering

•

2 independant Dirac cones

•

2 disorder-coupled Dirac cones

•

Absence of backscattering

•

Possibility of backscattering

•

Weak anti-localization

•

Weak localization

Strong disorder limit : disorder always opens a gap (insulator) as opposed to 3DSTI (always at least one

Outline

• • •

Diffusion, regime of weak disorder Diffusion of Dirac fermions

• •

3D Strong topological insulators Graphene

Diffusion of semi-Dirac excitations

Semi-Dirac excitations • Hexagonal lattice (graphene) qD , m∗ cx = qD /m∗ qD , cx m∗ = qD /cx m∗ , cx qD = m∗ cx p ∗ m , ∆ cx = −2∆/m∗ cx , ∆ qD = −2∆/cx 2 qD , ∆ m∗ = −qD /2∆

4

2 ∆ = −qD /2m∗ ∆ = −cx qD /2 ∆ = −m∗ c2 /2 √ qD = −2m∗ ∆ m∗ = −2∆/c2x cx = −2∆/qD

n to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) parameters may be deduced

III.

PROPERTIES OF THE UNIVERSAL HAMILTONIAN

•

2 Dirac fermions, with topologic number H = �vF �σ .� q

elocity cy , the universal Hamiltonian is described by two independent parameters (left column) rstomay be deduced the velocity cy , the universal Hamiltonian is described by two independent parameters (left column)

Semi-Dirac excitations

parameters may be deduced

PROPERTIES OF THE UNIVERSAL HAMILTONIAN III.

PROPERTIES OF THE UNIVERSAL HAMILTONIAN

•

�2 qx2 x )σ + �vF σ y q y Merging of the Dirac cones H = (∆ + 2m Montambaux et al., 2009

•

�2 q 2 σ + �v σ q Δ = 0 : Semi-Dirac excitation, H = 2m

FIG. FIG. 1: 1: Evolution Evolution of of the the spectrum spectrum when when the thequantity quantity∆∆isisvaried variedand andchanges changesininsign signatatthe thetopolo topo x x y y units). units). The The low-energy low-energy spectrum spectrum stays stayslinear linearininthe theqyqydirection. direction. F

Without Without loss loss of of generality, generality, we we assume assume mm∗∗ >>0.0. When When∆∆varies variesfrom fromnegative negativetotopositi posi transition transition from from aa semi-metallic semi-metallicphase phasewith withtwo twoDirac Diraccones conesand anda aband bandinsulator insulatorwith witha agapp ga the the transition, transition, the the spectrum spectrum isishybrid, hybrid,aareminiscence reminiscenceofofthe thesaddle saddlepoint pointininthe thesemi-metalli semi-meta When When ∆ ∆< < 0, 0, the the spectrum spectrum exhibits exhibitstwo twoDirac Diracpoints pointsthe theposition positionofofwhich whichalong alongthe thex xaxa

Also predited in VO2/TiO2 √√ heterostructures ∗ q = −2m∗ ∆

qDD = −2m ∆ the spectrum when the quantity ∆ is varied and changes in sign at the topological Pardo andtransition Pickett, 2009 (arbitrary when the is qvaried and spectrum changes in sign these at theDirac topological (arbitrary and the linear around points by and linear spectrum around these Dirac pointsistransition ischaracterized characterized bythe thevelocity velocitycxcxalong along yum spectrum staysquantity linear in∆the direction. y the m stays linear in the qy direction. !!

Boltzmann equation • Spinorial nature : Anisotropy of the scattering Pure Dirac fermions

|ψ(�k)� =

θ

•

�

1 eiθ�k

�

1 + cos θ 2 � � � |�ψ(k)|ψ(k )�| = 2

Anisotropy of the density of states

2

High density

�2 qx2 x H= σ + �vF σ y q y 2m

1

�1.0

�0.5

0.5

�1

Low density

�2

1.0

Boltzmann equation •

Stronger anisotropy of the scattering for semi-Dirac excitations compared to Dirac fermions |�ψ(k�y )|ψ(k�� )�|2 θ |�ψ(k�x )|ψ(k�� )�|2 y

0.5

0.6

θ 0.2

0.4

0.6

0.8

0.4

θ 1.0

0.2

x

�1.0

�0.5

0.5 �0.2 �0.4 �0.6

�0.5

•

Combination of these two anisotropies : Anisotropy of the diffusion Dx �= Dy

θ

1.0

Diagrammatics •

Direction dependant elastic mean free time 1 cos θ x −�Σ = Id + ∗ σ τe τe

• • σxx

2 diffusive modes, 1 diffuson and 1 cooperon Drude conductivity tensor

e2 2E�2 2e = h mγ a ˜

•

σyy

e2 �2 vF2 2c = h γ a ˜

σxy = σyx = 0

Weak anti-localization (Quantum interferences)

,∆

D ∗

m =

x 2 −qD /2∆

x

cx = −2∆/qD

Topological phase transition

universal Hamiltonian is described by two independent parameters (left column) ced

TIES OF THE UNIVERSAL HAMILTONIAN

•

σxx

Dependance in Δ of the conductances

e2 2E�2 2e 2 the2 topological 2 = FIG. 1: Evolution ofe�ã FIG. 1: Evolution of the the quantity spectrum∆when the quantity ∆ is varied changes in signtransition at e � v 2c transitio the spectrum when is varied and changes in signand at the topological (arbitrary F h mγ a ˜ units). The low-energy linear in the qy direction. σyy = c�ã units). The low-energy spectrum stays linearspectrum in the qystays direction.

h

1.2

γ

a ˜

∗ Without we lossassume of generality, we When assume∆mvaries > 0.from When ∆ varies from negative values, a Without loss 1.5 of generality, m∗ > 0. negative to positive values,toa positive topological 1.0 transition from a semi-metallic phase with two Dirac insulator cones andwith a band insulator with a occurs. gapped At spectrum transition from a semi-metallic phase with two Dirac cones and a band a gapped spectrum transition, the spectrum is hybrid, a reminiscence point in phase, the semi-metallic phase, see fi the transition, the the spectrum is hybrid, a reminiscence of the saddle pointofinthe thesaddle semi-metallic see figure (1). 0.8 When ∆ < 0, the two spectrum twoposition Dirac points the along position x axis is given by When ∆ < 0, the spectrum exhibits Dirac exhibits points the of which the of x which axis is along given the by ±q D with

1.0

qD

√ = −2m∗ ∆

qD

0.6 √ = −2m∗ ∆

(13)

0.4

0.5 spectrum Dirac points characterized by thethe velocity cx along quantity ∆ is varied inand signthe atlinear the topological transition (arbitrary andand the changes linear spectrum around these Diracaround points these is characterized byisthe velocity cx along x direction : the x directio n the qy direction. !0.2 ! q −2∆cx = D = −2∆ . qD . m∗ (14) cx = ∗ = m∗ ∗ ∗ ��� m m ��� m > 0. When ∆ varies from negative to positive values, a topological �1.0

�0.5

0.5

1.0

�1.0

�0.5

0.5

1.0

∗ The two Dirac points are separated a saddle at position qS is=±|∆|. 0 whose energy The mass m two Dirac conesThe andtwo a band with a gapped spectrum AtqSpoint Dirac insulator points are separated by a saddle point occurs. atbyposition = 0 whose energy The mass is m±|∆|. describes

4 2 ∗ q , m∗ cx =∗ qD /m∗ ∆ 2 = −q ∗ D /2m qD , m∗ D cx = qD /m ∆ = −q /2m D q ,c m∗ = q /c ∆ = −c q /2 qD , cx D∗ m∗x = qD /cx D∗∆x= −cx qD /2 x∗ D2 m ,c q = m cx ∆ = −m c /2 m∗ , cx ∗ qDx = m∗Dcp ∆ = −m∗ c2 /2 x √ p √ ∗ ∗ m , ∆ c −2∆/m q ∗ x = D = ∗ −2m ∆ ∗ m , ∆ cx = −2∆/m qD = −2m ∆ ∗ 2 qD = −2∆/c m 4 = 2x−2∆/cx cx , ∆ cxq,D∆= −2∆/c m∗x= −2∆/c x 2 m∗2 /2∆ = −qD cx = −2∆/qD qD , ∆ qDm, ∗∆= −q c/2∆ x = −2∆/qD D

Conclusion • Significative difference with Dirac or non-relativistic excitations : anisotropic diffusion

∗ 2 ∗ qD , m∗ cx = qD /m ∆I: = D /2m to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) TABLE In−q addition TABLE I: In addition to the velocity cy , the universal Hamiltonian is described by two independent parameters (left column) from which other parameters be deduced qD , cx m∗from = qDwhich /cx two ∆other =two −c x qD /2 parameters may bemay deduced ∗ ∗ ∗ 2 m , cx qD = m cx ∆ = −m c /2 p √ ∗ m , ∆ cx = −2∆/m∗ qD = −2m∗ ∆ III. PROPERTIES OF THE UNIVERSAL HAMILTONIAN III. PROPERTIES OF THE UNIVERSAL HAMILTONIAN cx , ∆ qD = −2∆/cx m∗ = −2∆/c2x 2 qD , ∆ m∗ = −qD /2∆ cx = −2∆/qD

city cy , the universal Hamiltonian is described by two independent parameters (left column) may be deduced

•

Study of the topological phase transition

• •

Weak antilocalization (symplectic class)

PROPERTIES OF THE UNIVERSAL HAMILTONIAN

1: Evolution the spectrum when the quantity ∆ is and varied and changes at the topological transition (arbitrary FIG. 1:FIG. Evolution of the of spectrum when the quantity ∆ is varied changes in sign in at sign the topological transition (arbitrary units). The low-energy spectrum stays linear in the q direction. y units). The low-energy spectrum stays linear in the qy direction.

Details soon on ArXiv

∗ Without of generality, we assume > 0. When ∆ varies from negative to positive a topological Without loss ofloss generality, we assume m∗ > m 0. When ∆ varies from negative to positive values, values, a topological transition a semi-metallic with two cones Dirac and cones and ainsulator band insulator a gapped spectrum At transition from afrom semi-metallic phase phase with two Dirac a band with a with gapped spectrum occurs.occurs. At the transition, the spectrum is hybrid, a reminiscence of the saddle point in the semi-metallic phase, see figure (1). the transition, the spectrum is hybrid, a reminiscence of the saddle point in the semi-metallic phase, see figure (1). < 0, the spectrum exhibits two points Dirac points the position of along which the along the is x axis given ±qD with When When ∆ < 0,∆the spectrum exhibits two Dirac the position of which x axis givenisby ±qDbywith