Observation of three-dimensional massless Kane fermions in a zinc-blende crystal

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1 DOI:.38/NPHYS857 Observation of three-dimensional massless Kane fermions in a zinc-blende crystal M. Orlita D. M. Basko M. Zholudev 3 4 F. Teppe 3 W. Knap 3 V. Gavrilenko 4 N. Mikhailov 5 S. Dvoretskii 5 P. Neugebauer 6 C. Faugeras A.-L. Barra G. Martinez and M. Potemski Laboratoire National des Champs Magnétiques Intenses CNRS-UJF-UPS-INSA Grenoble France Université Grenoble /CNRS LPMMC UMR 5493 B.P Grenoble France 3 Laboratoire Charles Coulomb LC) UMR CNRS 5 GIS-TERALAB Université Montpellier II 3495 Montpellier France 4 Institute for Physics of Microstructures RAS Nizhny Novgorod Russia 5 A.V. Rzhanov Institute of Semiconductor Physics Siberian Branch Russian Academy of Sciences Novosibirsk 639 Russia 6 Institut für Physikalische Chemie Universität Stuttgart Pfaffenwaldring Stuttgart Germany SAMPLE STRUCTURE The studied sample was grown using the standard MBE technique on the 3)-oriented semi-insulating GaAs substrate. The growth sequence started with ZnTe and CdTe transition buffer) regions followed by the MCT epilayer with gradually changing cadmium content x. The profile of cadmium content is shown in Fig.. The prepared MCT layer contains a region with x.7 of thickness d 3.µm. The cadmium profile has been controlled during growth using in situ single wavelength ellipsometry see e.g. Ref.. FIG.. The profile of cadmium content in the studied sample. KANE MODEL AND THE EFFECTIVE BAND HAMILTONIAN In zinc-blende semiconductors the orbital degeneracies of the conduction and valence bands are and 3 respectively. At k = we can choose a real Bloch function u c r) for the conduction band and three real functions u X r) u Y r) u Z r) for the valence band. The function u c r) transforms according to the identical representation Γ of the crystal group T d while u X r) u Y r) u Z r) transform according to the vector Γ 5 representation equivalently to the functions xyz. Out of the three real functions u α r) one can make linear combinations u m r) corresponding to eigenfunctions of the z-projection of the orbital angular momentum l = : u + = i u X + iu Y ) u = iu Z u = i u X iu Y ). S) The spin structure of the wave functions can be accounted for by introducing two spinors χ χ corresponding to the two values of the spin projection on the z axis. Spin-orbit interaction splits the l + )-fold degenerate valence NATURE PHYSICS 4 Macmillan Publishers Limited. All rights reserved.

2 DOI:.38/NPHYS857 band into two subspaces corresponding to the total angular momentum J =/ and J =3/ the latter manifold corresponding to the topmost valence band. Explicitly u 3/;+3/ = u + χ u 3/;+/ = /3 u χ + /3 u + χ u 3/; / = /3 u χ + /3 u χ u 3/; 3/ = u χ u /;+/ = /3 u χ /3 u + χ u /; / = /3 u χ + /3 u χ S) It is convenient to arrange the basis vectors as ) uc χ u 3/+3/ u 3/ / u /+/ u c χ u 3/ 3/ u 3/+/ u / / S3) = ) u c χ u X χ u Y χ u Z χ u c χ u X χ u Y χ u Z χ U S4) / i /6 i /3 i / /6 /3 /3 i /3 i U = S5) /3 i / i /6 i /3 / /6 /3 i /3 i then the time reversal matrix is just σ y the second Pauli matrix acting in the space made of 4 4 blocks. In this basis the electronic Hamiltonian at k = is given by Hk =)= E g E g S6) where the energy is counted from the top of the J =3/ valence band. is the spin-orbit splitting between the J =3/ and the J =/ valence bands. The band gap is parametrized by E g x x c ).9 ev []. Since E g < at x<x c the semimetallic MCT is sometimes called a negative-gap semiconductor. The linear in k terms in the effective band Hamiltonian are obtained in the first order of the k p perturbation theory. The momentum matrix elements between the conduction and the valence band Bloch functions are determined by u α r) u cr) x β d 3 r = Pδ αβ S7) where P is the Kane s matrix element and P /m E P is called Kane s energy m is the free electron mass). NATURE PHYSICS 4 Macmillan Publishers Limited. All rights reserved.

3 DOI:.38/NPHYS857 SUPPLEMENTARY INFORMATION The effective Hamiltonian to Ok) is given by: Hk) =Hk = ) + U P m ik x ik y ik z ik x ik y ik z ) U = E g vk + 3/ vk / vk z / vk z vk / vk 3/ vk + / vk z = vk z / vk / vk z vk + / E g vk 3/ vk+ / vk z / vk + 3/ vk z vk / vk + / vk z / S8) where v 3/ P/m and k ± k x ± ik y. This Hamiltonian obeys the time-reversal symmetry σ y H k)σ y = H k) where σ y is the second Pauli matrix acting in the space made of 4 4 blocks. The eigenvalues of the Hamiltonian S8) can be found from the equation deth E) =E { E 3 + E g )E [E g + 3/)v k ]E v k } =. They do not depend on the direction of k. In the limit of large the Hamiltonian can be easily projected on the subspace orthogonal to the the split-off band. If we are not interested in terms quadratic in k the projection is done by simply eliminating the fourth and the eight row and column of the matrix in Eq. S8): E g vk + 3/ vk / vk z vk 3/ Hk) = vk + / vk z vk z E g vk 3/ vk+ / vk + 3/ vk z vk / This matrix has three doubly-degenerate eigenvalues: E k = E k = E g ± S9). S) E g 4 + v k. S) The eigenvalue E = corresponds to the heavy-hole band which in this approximation is completely flat. Let us see how the existence of the flat band follows from the property U c Hk)U c = Hk) with Hk) given by Eq. S) and U c = diag ). Consider the general situation: an n + m) n + m) matrix A anticommuting with a matrix U c which has m eigenvalues equal to and n eigenvalues equal to and m<n. Let us work in the basis of the eigenvectors of U c which are arranged in such an order that U c = diag......). The condition U c AU c = A implies that in this basis the matrix A has the following block structure: n n A A = n m A m n m m ). S) Consider now the n-dimensional subspace of column vectors x =x x...x n...) T. All these vectors satisfy the first n equations of the linear system Ax =. The remaining m equations leave an n m) dimensional subspace of solutions Ax = which corresponds to the zero eigenvalue of A with multiplicity n m. OPTICAL ABSORPTION AT ZERO MAGNETIC FIELD Let us start from the standard expression for the optical conductivity obtained from the Kubo formula for the response of the current to the monochromatically oscillating vector potential: 6 σ ij ω) = ie d 3 k π) 3 ll = f lk f l k E lk E l k l k v i l k l k v j l k ω E l k + E lk + i +. S3) NATURE PHYSICS Macmillan Publishers Limited. All rights reserved.

4 DOI:.38/NPHYS857 Here l l =...6 label the eigenstates of Hk) which is given by Eq. S) f lk are the occupations of these eigenstates and the velocity matrices are v i = Hk)/ k i = vj i where i j = xyz label the Cartesian components. To calculate the velocity matrix elements we note that the projection of the vector J on an arbitrary direction n = sin ϑ cos φ sin ϑ sin φ cos ϑ) determined by the spherical angles ϑ φ can be related to J z by a rotation J n = J x sin ϑ cos φ + J y sin ϑ sin φ + J z cos ϑ = U φj x sin ϑ + J z cos ϑ)u φ = U φu ϑ J zu ϑ U φ S4) U φ = diag e iφ/ e 3iφ/ e iφ/ e iφ/ e 3iφ/ e iφ/) S5) c s c 3 3cs s 3 3c s U ϑ = 3cs c 3 cs 3c s s 3 c s s c s 3 c cos ϑ s sin ϑ. S6) 3c s c 3 3cs 3c s s 3 +c s 3cs c 3 cs Thus the eigenstates l k for an arbitrary direction of k can be related to those for k along z by l k = U φu ϑ lkz where ϑ φ are the spherical angles of k. By symmetry the tensor structure of the conductivity is trivial σ ij ω) =σω). This can also be shown by the direct calculation whose details we do not give but which is fully analogous to the one given below. We calculate just one component σ zz. Since the energies E lk depend only on k we can integrate over the angles using Eq. S4): J ll = sin ϑ dϑ dφ l k J z l k = 8π 3 lkz J x l k z + 4π 3 lkz J z l k z. S7) The eigenvectors of the Hamiltonian S) for k along the z axis are in the order of decreasing energy) S C S C C S C S where we have denoted C = cos ϕg + π ) ϕg S = sin 4 + π ) E g / ϕ g arcsin. 4 E g/4+v k This gives 4c g s g 3 + s g ) 4s g c g J ll = π s g 4c g 3 s g ) c g 4s g 3 + s g ) 3 s g ) 3 3 s g ) 3 + s g ) 4s g c g 3 s g ) 4c g s g c g 4s g 3 + s g ) s g 4c g c g = cos ϕ g s g = sin ϕ g. S8) Substituting this into Eq. S3) we finally obtain Re σω >) = π e [ ) ξ dξ E g E 3 vω 8π 3 6 δ + g 4 + E ξ g ω +4 + Eg +4ξ [ = e θω E g E g ) E ) g ω 4πv ω ωe g + 6 θω E g ) ) δ Eg +4ξ ω) ] = + E g ω ) ω E g ]. S9) For the gapless case E g = we obtain Re σω > ) = 3 ω e vπ. The imaginary part of the dielectric function εω)=+iσω)/ε ω) then becomes Im εω >) = 3 c α where α is the fine structure constant. v 4 NATURE PHYSICS 4 Macmillan Publishers Limited. All rights reserved.

5 DOI:.38/NPHYS857 SUPPLEMENTARY INFORMATION LANDAU LEVELS In the presence of a magnetic field described by the vector potential in the Landau gauge A x = By A y = A z = we make the standard Peierls substitution p p ea in the Hamiltonian S) and seek the eigenstates in the form ψx y) =e ip xx x Φ n y Φ n z Φ n x Φ n y Φ n 3 z Φ n ) T S) where Φ n =Φ n y + p x l B ) are the harmonic oscillator wave functions and l B is the magnetic length. It can be checked directly that the form S) is preserved upon action on ψx y) by the Hamiltonian. The coefficients satisfy the following linear system we denote ζ p z l B for brevity): E g E 3n n x + v/l B y z ζz = 3n x E y = v/l B n x E z ζx = v/l B ζz + E g E 3n ) n x y + z = v/l B 3n ) x E y = v/l B n ζx + x E z =. v/l B S) Its analysis is especially simple at p z = when the system is split into two decoupled 3 3 blocks for x y z and x y z respectively. It is convenient to shift n n in the block. In each block the Landau levels can be labeled by n =... ζ =. At n = only ζ = ± are allowed while at n = only ζ = exists: E nζ = ζ E g + ζ ψ n> = ψ n>± = ψ n> = ψ n>± = 4n Eg 4 + v lb 4n ± ) S) n Φn 3n Φn E + n /)v/l B ) 4n 3 ψ = ϕ E Φ n 3n/v/l B )Φ n n )/v/lb )Φ n 3n n Φn ψ = ) Φn E + n 3/)v/l B ) Φ E Φ n 3n )/v/lb )Φ n n/v/l B )Φ n The selection rules for the optical absorption at p z = are obtained by calculating the matrix elements of J ± = J x ±ij y :. S3) S4) S5) S6) n ζ σ J + nζσ δ σσ δ n n δ ζ δ ζ ). S7) At p z = the Landau levels can be found directly from the system S): E nζ = ζ E g + ζ Eg 4 + v lb 4n ± ) + v p z. S8) For E g = this expression reduces to Eq. 5) of the main text. NATURE PHYSICS Macmillan Publishers Limited. All rights reserved.

6 DOI:.38/NPHYS857 [] N. N. Mikhailov R. N. Smirnov S. A. Dvoretsky Yu. G. Sidorov V. A. Shvets E. V. Spesivtsev and S. V. Rykhlitski Int. J. Nanotechnology 3 6). [] M. H. Weiler in Semiconductors and Semimetals vol. 6 ed. by R. K. Willardson and A. C. Beer Elsevier 98). 6 NATURE PHYSICS 4 Macmillan Publishers Limited. All rights reserved.