Tilted Dirac cones in 2D and 3D Weyl semimetals implications of pseudo-relativistic covariance

Transcription

1 Tilted Dirac cones in 2D and 3D Weyl semimetals implications of pseudo-relativistic covariance Mark O. Goerbig J. Sári, C. Tőke (Pécs, Budapest); J.-N. Fuchs, G. Montambaux, F. Piéchon ; S. Tchoumakov, M. Civelli Le Studium Workshop Tours, 13-15/06/2016

2 Outline Massless Dirac fermions in 2D materials: from graphene to α-(bedt-ttf) 2 I 3 Tilted Dirac cones theoretical description Pseudo-relativity in α-(bedt-ttf) 2 I 3 in a magnetic fields Effects on magneto-optics Weyl fermions in 3D materials

3 It all started with graphene one-atom thick layer of graphite, isolated in 2004 electronic conductor flexible membrane of exceptional mechanical stability Nobel Prize in Physics, 2010 zoom on atomic scale (x 1000) 1µm Chuan Li, physique mésoscopique, LPS, Orsay Interest for fundamental research: Quantum mechanics meets relativity in condensed matter (electrons behave as 2D massless Dirac fermions)

4 Band structure of graphene Dirac Hamiltonian (two valleys ξ = ± fermion doubling) ( ) Hq ξ 0 ξq v x iq y F ξq x +iq y 0 t ~ 3 ev ~ K

5 α-(bedt-ttf) 2 I 3 : another 2D Dirac material BEDT-TTF =bis(ethylenedithio)tetrathiafulvalene quasi-2d (stacked layers) 4 molecules/unit cell 4 bands electronic filling 3/4 (c) Energy k kx t i mev under pressure: low-energy tilted Dirac cones

6 α-(bedt-ttf) 2 I 3 : electronic band structure schematic view on upper two bands Brillouin zone energy no pressure pressure energy high pressure pressure E F E F wave vector wave vector Katayama, Kobayashi, Suzumura, J. Phys. Soc. Japan 75, (2006)

7 α-(bedt-ttf) 2 I 3 : electronic band structure energy bands under pressure Brillouin zone pressure Katayama, Kobayashi, Suzumura, J. Phys. Soc. Japan 75, (2006)

8 Theoretical description of tilted 2D Dirac cones Most general 2D Hamiltonian (2 2 matrix) with linear dispersion (generalised Weyl Hamiltonian): H = 3 v µ qσ µ (σ 0 = ½, σ 1 = σ x, σ 2 = σ y, σ 3 = σ z ) µ=0 ˆ= (w 0 q ½+w x q x σ x +w y q y σ y ) k y Energy dispersion ( 1, λ = ±): ǫ λ (q) = w 0 q+λ wxq 2 x 2 +wyq 2 y 2 w 0 : tilt velocity k x λ= (VB) tilt axis λ=+ (CB) Energy 5 Graphene: w 0 = 0, w x = w y = v F 4 4

9 How to obtain tilted Dirac cones? In graphene: σ denotes A B sublattice isospin Term proportional to ½: nnn hopping (A A, B B) In continuum limit: H diag = 9 4 t nnn q 2 a 2 ½ i.e. not linear in q, but quadratic Reason: Dirac points [ǫ(q D ) = 0] coincide with K,K (points of high crystallographic symmetry) Drag Dirac points away from K,K!

10 Graphene under strain (I) Distortion: a a = a+δa t t = t+ t a δa t nnn t nnn = t nnn + t nnn a δa a a a t t a a axis of t deformation t nnn t nnn t nnn t nnn t nnn t nnn Dirac points move from K, K to: ξ: valley index q D y = 0, q D x a = ξ 2 3 arccos ( t 2t )

11 Graphene under strain (II) Estimation of tilt: w 0 (w0x w x ) 2 + ( w0y w y ) δa a 0 w 0 < 1: tilt parameter Effect linear in δa/a! MOG, J.-N. Fuchs, F. Piéchon, G. Montambaux, PRB 78, (2008)

12 Criterion for maximal tilt closed Fermi surface (ellipse) open Fermi surface (hyperbolas) MOG, J.-N. Fuchs, G. Montambaux, F. Piéchon, PRB (2008) see also G. Volovik and M. Zubkov, NPB (2014)

13 Criterion for maximal tilt arxiv: Nature (2015) definition of surface for topological characterisation closed Fermi surface open Fermi surface (ellipse) (hyperbolas) MOG, J.-N. Fuchs, G. Montambaux, F. Piéchon, PRB (2008) see also G. Volovik and M. Zubkov, NPB (2014)

14 Tilted cones in a strong magnetic field How is the Landau level spectrum affected by the tilt? H ξ = ξ(w 0 q ½+w x q x σ x +w y q y σ y ) w 0 = (w0x w x ) 2 + ( ) 2 w0y w y Peierls substitution: q q+ea(r) semiclassics: q 2n/l B, (l B = 1/ eb: magnetic length) energy spectrum (as for graphene): ǫ λ,n = λ v F 2n l B 0.2 effective tilt parameter effective Fermi velocity effect of the tilt: renormalisation v F = w x w y (1 w 2 0) 3/4 MOG, J.-N. Fuchs, F. Piéchon, G. Montambaux, PRB 78, (2008)

15 Intermezzo: electrons in crossed B and E fields (I) 2D electrons in a perpendicular magnetic B = A and inplane electric E fields H 0 ( q) H 0 (p+ea(r)) eey (non-relativistic) Schrödinger fermions Galilei transformation to comoving frame of reference v D Landau levels E B Hall drift ǫ n,k = eb m ( n+ 1 ) 2 v D k v = E/B D

16 Intermezzo: electrons in crossed B and E fields (I) 2D electrons in a perpendicular magnetic B = A and inplane electric E fields H 0 ( q) H 0 (p+ea(r)) eey relativistic electrons (graphene) Lorentz transformation to frame of reference v D [Lukose et al., PRL 2007] B B = B 1 (v D /v F ) 2 ǫ ǫ 1/l B B energy in lab frame E B Hall drift v = E/B D ǫ ±n,k = ± v F[1 (v D /v F ) 2 ] 3/4 l B 2n vd k

17 Intermezzo: electrons in crossed B and E fields (II) non relativistic electrons relativistic electrons (graphene) n=2 n=1 n=0 0Energy h ω n=2 n=1 n=0 Energy 0 hv /l F B k=y /l 0 B k=y /l 0 B n= 1 n= 2 no electric field

18 Intermezzo: electrons in crossed B and E fields (II) non relativistic electrons relativistic electrons (graphene) n=2 n=1 n=0 0Energy h ω (=const.) k=y /l 0 B n=2 n=1 n=0 n= 1 n= 2 Energy 0 hv * /l F B k=y /l 0 B in the presence of an electric field

19 Pseudo-covariance in α-(bedt-ttf) 2 I 3 (Weyl) Hamiltonian H 0 (q) = ( 0 (w x q x iw y q y ) (w x q x +iw y q y ) 0 ) H 0 (q)+ w 0 q x ½ tilt term in a magnetic w 0 q x ½ w 0 (p x +ea x (r))½ = w 0 (p x eby)½ same (relativistic) model as before with v D = w 0 = E/B [MOG, Fuchs, Montambaux, Piéchon, EPL 2009] maximal tilt (v D = v F ) related to maximal velocity for Lorentz boost

20 Covariance and wave functions Lorentz boost in x-direction (with w 0 = E eff /B): x µ = Λ µ νx ν (v F t,x ) = (v D t+βx)/ 1 β 2 transformation of wave function (tanhθ = β = v D /v F ): ψ (v F t,x,y) = S(Λ)ψ(v F t,x,y) with S(Λ) = e θσ x/2...needed in matrix element of light-matter coupling v : velocity operator ψ vψ

21 Light-matter coupling Peierls substitution q q+ e [A(r)+A rad(t)] A(r) = B (magnetic field), A rad (t) (radiation field) in Hamiltonian (linear expansion in radiation field) H(q) H B +ev A rad (t) H B Landau levels, velocity operator v = q H/ dipolar selection rules (in comoving frame): λn λ (n+1) for right handed light λn λ (n 1) for left handed light

22 Magnetooptical selection rules selection rules in comoving frame v D (field E = 0) λn λ (n±1) new transitions in lab frame (E 0) matrix elements absorption coefficient β =v D /v =0.2 F

23 Magnetooptical selection rules selection rules in comoving frame v D (field E = 0) λn λ (n±1) new transitions in lab frame (E 0) matrix elements β =v /v D =0.5 F absorption coefficient λ n λ n

24 Magnetooptical selection rules selection rules in comoving frame v D (field E = 0) λn λ (n±1) new transitions in lab frame (E 0) matrix elements absorption coefficient β =v D /v =0.8 F selection rules (absorbed frequencies) depend on frame of reference [Sári, MOG, Tőke, PRB 2015]

25 Tilted Cones in 3D Materials Weyl Semimetals in collaboration with : S. Tchoumakov and M. Civelli arxiv:

26 Theory of Weyl fermions with a tilt 2 2 matrix Hamiltonian with linear dispersion in 3D H = (w 0 q ½+w x q x σ x +w y q y σ y +w z q z σ z ) Energy dispersion ( 1, λ = ±): ǫ λ (q) = w 0 q+λ wxq 2 x 2 +wyq 2 y+w 2 z q z w 0 : tilt velocity ( ) 2 w0 x + ( ) 2 w0 y + ( ) 2 w0 z < 1 type I WSM w x ( ) 2 w0 x + w y ( ) 2 w0 y + w z ( ) 2 w0 z > 1 type II WSM w x w y w z

27 Role of the magnetic field and Landau quantisation tilt parameter (vector) ( w0x t =, w 0z, w ) 0z w x w z w z inplane tilt parameter t = t B B = ( w0x w x, w 0y w y ) Landau level quantisation if B-field close to tilt axis sinα < 1/ t

28 Landau quantisation same recipe as for 2D: Lorentz boost to a frame of reference, where t vanishes 1D Landau bands ǫ λ,n (k z ) = w 0z k z +λ 1 β 2 where β = t w 2 zk 2 z +2 w xw y 1 β 2 l 2 B n

29 Landau bands in the magnetic regime type I WSM type II WSM WSM type confered to 1D bands n=0 n=0 t z = w 0z w z = tcosα 1 t2 sin 2 α n=0 k z n=0 k z (1D tilt parameter)

30 Optical conductivity of a WSM in the magnetic regime Re σ ll (ω) = σ 0 2πl 2 B ω j,j u l v j,j 2 [f(ǫ j ) f(ǫ j )]δ(ω ω j,j ) type I WSM type II WSM again: violation of dipolar selection rules

31 Electric regime in a type-i WSM? add a true electric field to the effective one: w 0 w ξ = w 0 ξ E B B 2 new tilt parameter w ξ (E) = w 2 ξx w 2 x + w2 ξy w 2 y + w2 ξz w 2 z Energy [a.u.] +,3 +,2 +,1 0 0,1,2,3 ξ = + ξ = +,4 +,3 +,2 +,1 0,1,2,3,4 Landau level spectrum depends on valley index ξ : ǫ ξ λ,n;k (E) = λ wx w y l B [ 1 wξ (E) 2] 3/4 2n+ E B k MOG, J.-N. Fuchs, G. Montambaux, F. Piéchon, EPL 85, (2009)

32 Conclusions massless fermions in 2D and 3D tilted cones quasi-2d organic material α-(bedt-ttf) 2 I 3 3D Weyl semimetals (Cd 3 As 2, MoTe 2, WTe 2,...) intimite relation with Lorentz boosts 2D: magnetic regime = type-i (massless) Dirac semimetal electric regime = type-ii (massless) Dirac semimetal signatures expected in magneto-optical meaurements (violation of dipolar selection rules, n n±1)