ISSN 0021-3640, JETP Letters, 2017, Vol. 105, No. 12, pp. 782–785. © Pleiades Publishing, Inc., 2017. Original Russian Text © L.E. Golub, E.L. Ivchenko, B.Z. Spivak, 2017, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 105, No. 12, pp. 744–747.
CONDENSED MATTER
Photocurrent in Gyrotropic Weyl Semimetals a Ioffe b
L. E. Goluba, *, E. L. Ivchenkoa, and B. Z. Spivakb
Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia Department of Physics, University of Washington, Seattle, WA 98195, USA *e-mail:
[email protected] Received May 12, 2017
Photocurrents in the Weyl semimetals belonging to the gyrotropic symmetry classes have been theoretically studied. It has been shown that the circular photocurrent transverse to the direction of light incidence appears in weakly gyrotropic crystals with the Cnv (n = 3, 4, 6) symmetry only when spin-dependent terms both linear and quadratic or cubic in the quasimomentum, as well as a spin-independent term resulting in the tilt of the cone dispersion, are taken into account in the electron effective Hamiltonian. A polarization-independent magnetic-field-induced photocurrent, which is allowed only in gyrotropic systems, has been predicted. For crystals with the C2v symmetry, a microscopic mechanism of the photocurrent in a quantized magnetic field, which is generated in direct optical transitions between the ground and first excited magnetic subbands, has been considered. It has been shown that this photocurrent becomes nonzero in the presence of the anisotropic tilt of dispersion cones. DOI: 10.1134/S0021364017120062
1. INTRODUCTION A distinctive property of the symmetry of a gyrotropic crystal is the presence of components of the polar (R) and axial (L) vectors, which are transformed according to equivalent representations of the crystal point group. This property allows effects where the physical quantities Rα and Lβ are related to each other by a second-rank pseudotensor Cαβ or an invariant physical quantity I is linearly related to the products Rα Lβ. The most famous of such effects is natural optical activity or the rotation of the polarization plane of a light wave propagating in a gyrotropic medium. It is described by linear terms in the expansion of the permittivity tensor ελν(ω, q) in powers of the wave vector q of light: a gyrotropic correction to the electric displacement vector δD can be represented as the vector product i(E × g), where E is the electric field of the light wave and g is the gyration vector linearly related to the vector q [1]. Another gyrotropic effect is the appearance of an electric photocurrent j proportional to the vector product i(E × E*). It was called the circular photogalvanic effect (CPGE), predicted independently in [2, 3] and observed for the first time on a bulk tellurium crystal in [4] and on a GaAs/AlGaAs quantum-well structure in [5, 6]. In theoretical works [7–9], the CPGE was studied in Weyl semimetals. It was established that the contri-
bution of each Weyl node to the circular photocurrent has the universal form [7]
j = # Γ 0τ pi(E × E ∗),
(1)
where Γ0 Planck constant, # = ± 1 is the chirality (or topological charge) of the node, and τ p is the electron momentum relaxation time. The universality of Eq. (1) means that this formula includes the numerical factor π/ 3, the world constants e and h, and the only nonuniversal factor τp. The authors of [8] considered a pair of Weyl nodes with opposite chiralities. They showed that their contributions to the circular photocurrent do not cancel each other if the effective electron Hamiltonian contains not only the terms Aαβ σα kβ, but also the tilt term a · k with the vector a different at different nodes (σα are the Pauli spin matrices, and k is the electron wave vector measured from the kW node). However, the equation for the photocurrent loses its universality in this case. The authors of [10] have recently observed a circular photocurrent in a TaAs crystal excited by CO2 laser radiation. In this work, we analyze how the presence of a reflection plane in the point-symmetry group of a gyrotropic crystal affects the CPGE and discuss the influence of a magnetic field on the photocurrents in gyrotropic Weyl semimetals. Particular attention is paid to the crystals with C4v and C2v symmetries because it has been established recently that these symmetries are characteristic of TaAs, NbP, and
782
= πe3 /3h2, e is the elementary charge, h is the
PHOTOCURRENT IN GYROTROPIC WEYL SEMIMETALS
NbAs monopnictides [11–13] and WTe2 tungsten telluride [14, 15], which are Weyl semimetals. We show that the CPGE in a TaAs-type Weyl semimetal can be described with the effective electron Hamiltonian including terms not only linear but also cubic in electron wave vector. 2. GYROTROPIC CRYSTALS IN THE ABSENCE OF A MAGNETIC FIELD We consider a crystal with the effective electron Hamiltonian near the point kW in the form
* = d(k) ⋅ σ + d 0(k)σ 0,
(2)
where σ0 is the identity matrix of dimension 2 × 2 and dl (k) (l = 0, x, y, z) are functions whose expansions in powers of k contain no terms of the zeroth order. The eigenvalues of this Hamiltonian are E±, k = d0(k) ± d(k), where d (k) = |d(k)|. Here and below, the energy is measured from the electron energy at the Weyl point. Usually, only linear terms are taken into account in d0(k) for simplicity, i.e., d0(k) = a · k, where a is a vector. This term describes the tilt of the Weyl cone. Direct optical transitions near the point kW are accompanied by the generation of the photocurrent given by the formula
j=e
∑τ k
p
2 ∂ d(k)W (k), +− ! ∂k
(3)
where the rate of optical transitions per unit volume per unit time is given by the expression 2 (4) W +− = 2π M +− F (k)δ(2d − ! ω). ! Here, M+− is the matrix element of the optical transition that does not depend on d0(k) and
F (k) = f ( E −,k ) − f (E +,k ),
(5)
where f (E) is the equilibrium Fermi–Dirac distribution function. A factor of 2 in Eq. (3) takes into account the contributions to the current from the photoelectrons and photoholes. In much the same way as in [7], one can show that, under the circularly polarized excitation, the polarization-dependent contribution to the square of the absolute value of the matrix element is proportional to the Berry curvature 2 2 2 (6) M +− circ = 2e d 2 | E| 2 κ ⋅ Ω, (! ω) where κ = i(e × e*), e is the unit polarization vector, and Ω is the Berry curvature related to the vector d (k) as
⎛ ⎞ Ω j = d 3 ⎜ ∂d × ∂d − ∂d × ∂d ⎟ , 4d ⎝ ∂ k j +1 ∂ k j + 2 ∂ k j + 2 ∂ k j +1 ⎠ JETP LETTERS
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and the cyclic permutation of the indices j is assumed. In view of the energy conservation law, the argument E±, k of the distribution function can be replaced by d0(k) ± ħω/2. Therefore, for a fixed frequency ω, the difference of occupation numbers given by Eq. (5) is a function of the scalar d0(k). We introduce the pseudotensor ˆγ , which describes the CPGE in accordance with
jα = γ αβ κ β | E| 2.
(8)
The structure of the second-rank pseudotensor ˆγ for all 18 gyrotropic classes is known (see, e.g., [16]). The purpose of our analysis is to find the simplest form of the Hamiltonian given by Eq. (2) which satisfies two conditions. First, it leads to a nonzero contribution of the kW node to the γαβ component allowed by the crystal symmetry group. Second, this contribution does not disappear after the summation over the star of the vector kW. We first consider gyrotropic classes that do not contain reflection planes and take into account only terms linear in k in the Hamiltonian (2): dα = Aαβ kβ, with the chirality of the node kW equal to sgn{Det(Â)}. In this case, all the Weyl nodes obtained by the symmetry transformations are characterized by the same chirality. In the absence of a tilt, d0(k) = 0, the tensor ˆγ is isotropic: the off-diagonal components are absent, while the diagonal components coincide and are equal to the contribution of a single node (1) multiplied by the number of vectors n in the star of the vector kW if this star contains the vector −kW and by 2n if kW and −kW belong to different stars. The doubling is due to the symmetry with respect to time reversal, which transforms the point kW to −kW, conserving the chirality. The difference in the diagonal components allowed by the symmetry is obtained taking account the tilt. To calculate the off-diagonal components γαβ in crystals with the C1 and C2 symmetries, one needs to take into account the tilt with nonzero coefficients aα and aβ. In the Hamiltonian with terms linear in k, the off-diagonal components γxy = −γyx in the C3, C4, and C6 classes do not appear even with allowance for the tilt. As will be shown below, it is necessary in this case to include the higher powers of k in the Hamiltonian. The gyrotropic classes contain reflection planes; only terms linear in k are taken into account in the Hamiltonian. The off-diagonal components of γαβ in the groups Cs, C2v, and S4 and the diagonal components γxx = −γyy in the groups S4 and D2d arise in the calculation with allowance for the tilt. Nonzero off-diagonal components in the groups C3v, C4v, and C6v do not appear in the linear Hamiltonian model. Thus, the six gyrotropic classes Cn, Cnv (n = 3, 4, 6) stand apart from the rest: the components γxy = −γyx
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for them can be obtained by adding terms of the second or third order in k to the spin-dependent part of * . In the simplest case, this condition is satisfied by a Hamiltonian of the mixed form 2
2
d x = β k x + Dk y k ⊥ , d y = β k y + Dk x k ⊥ , d z = β k z , d0 = ax k x + ayk y,
k ⊥2
k x2
(9)
k y2 .
where In this case, the Berry curvature = + is given by the expressions
β ⎡⎛ 2 2 4⎞ 2 2 ⎤ ⎢⎜ β − D k ⊥ ⎠⎟ k x + 2β D(k x − k y )k y ⎦⎥ , 2d 3 ⎣⎝ β Ω y = 3 ⎡⎛⎢⎣⎝⎜ β 2 − D 2k ⊥4 ⎞⎠⎟ k y − 2β D(k x2 − k y2 )k x ⎤⎦⎥ , 2d βk Ω z = 3z ⎛⎜⎝ β 2 − 3D 2k ⊥4 + 4β Dk x k y ⎞⎟⎠ . 2d In the k · p method, the cubic terms in Eq. (9) arise from the contribution of remote bands in the third order of perturbation theory and, therefore, can be considered as small compared to the linear terms. The components γxy = −γyx become nonzero in the first Ωx =
order in D with allowance for the tilt at a x2 ≠ a y2 . 2.1. Circular Photocurrent in Crystals of the C4v Symmetry We assign the number 1 to one of the Weyl nodes kW1 lying in the region of the Brillouin zone with the positive components kW1,α > 0. In view of eight elements of spatial symmetry and the time reversal symmetry, there are 16 equivalent nodes. To analyze the total photocurrent, it is sufficient to consider two additional nodes kW2 and kW3 obtained from the node kW1 by the reflection σy in the plane perpendicular to the y axis and the rotation C4 around the fourth-order axis, respectively. Upon passage from node 1 to node 2, in the effective Hamiltonian given by Eqs. (2) and (9), the coefficient β changes its sign, the coefficient D does not change, and the function F(kx, ky, kz) defined by Eq. (5) is transformed to F(kx, −ky, kz). When passing to node 3, the coefficient β is unchanged, the coefficient D changes the sign, and the function F(kx, ky, kz) is transformed into F(ky, −kx, kz). In the presence of the tilt but for D = 0, the contributions of nodes 1 and 3 cancel each other and no electric photocurrent is generated. The inclusion of cubic terms is sufficient for the sum over 16 nodes to be nonzero. In particular, the inclusion of this term in Ωx leads to the current
jy ∝ κ x
3
2
e | E| 2 βD (" ω) 2
2 y
∑k (k
2 x
∂d(k)/∂(ħkα). According to Eq. (10), the measurement of the circular photocurrent allows us to determine the sign of the coefficient D rather than the sign of the coefficient β, which determines the chirality of the Weyl node at D = 0.
2 − k y )F! (k)δ(2d − " ω), (10)
k
where F! (k) = F (k x , k y , k z ) − F (k y , −k x , k z ). A comparable contribution comes from cubic terms in the argument of the δ function and the group velocity
3. PHOTOCURRENTS IN THE PRESENCE OF A MAGNETIC FIELD An additional effect specific to gyrotropic media is a magnetic-field-induced photocurrent independent of the light polarization. In this effect, in the approximation linear in the magnetic field, the photocurrent density, which is a polar vector, is related to the magnetic field, which is an axial vector. For example, the following currents can be generated in crystals of the C2v symmetry: 2
j x = (S xx B x + S xy B y )| E| , 2
j y = −(S xx B y + S xy B x )| E| ,
(11)
where the Cartesian coordinate system is chosen with the z axis || C2 and the x axis making the angle of 45° with the reflection planes σv. In the quantized magnetic field, the current flows only in the field direction. Therefore, the transverse effect of generation of the current jx in the field B || y or the current jy in the field B || x is suppressed. Taking into account the photocurrent depending on the circular polarization of light, we obtain the macroscopic equations j x = ⎡⎣S xx (| B x |)B x + γ xx (| B x |)κ x ⎤⎦ | E| 2, j y = − ⎡⎣S xx (| B y |)B y + γ xx (| B y |)κ y ⎤⎦ | E| 2,
(12)
where the coefficients Sxx and γxx are even functions of the magnetic field strength. The detailed calculation of these coefficients and frequency dependence of the magnetic-field-induced photocurrent will be reported elsewhere. Here, we briefly consider the photocurrent generated in direct optical transitions between the ground (chiral) and first excited magnetic subbands under unpolarized excitation. In the quantizing magnetic field B || y, the dispersion relation in these subbands has the form [17, 18]
E 0 = [a y − sgn(# B y )! v" 0 ]k y ,
(13)
! 2c , E1 = a y k y + " (v! 0k y ) 2 + ω
(14)
where v! 0 = v 0 / γ , v 0 = β/ ! is the Weyl velocity in the absence of the magnetic field,
! c = v! 0 ω
2| eB y | , "c
1
(15) , 2 1 − (a x / v 0 ) ax, ay are the coefficients of the first-order term in the k expansion of the tilt term in Eq. (2) (|ax, y| < v0), and the coefficient az is taken to be zero for simplicity. We JETP LETTERS
γ=
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consider here one of the possible mechanisms of magnetic-field-induced photocurrent in the Weyl semimetal of the C2v symmetry, where the points kW lie in the kz = 0 plane, two of them are characterized by the chirality # and the pairs of the coefficients (ax, ay) and (−ax, −ay), and two others are characterized by the chirality − # and the pairs of coefficients (ay, ax) and (−ay, −ax). In this mechanism, the optical transitions E0 → E1 occur predominantly at the Weyl nodes with the chirality 1 or –1, depending on the sign of the field component By. The photoelectrons return to the subband E0 within the energy relaxation time τ ε , and only a minor part of them are scattered into a valley of the Weyl node with the opposite chirality. At zero temperature, at the chemical potential µ > 0 lying below the bot! c ), and for frequentom of the excited subband (µ < " ω
! 2c ± µ/ " , cies ω − < ω < ω + , where ω ± = (µ/ ")2 + ω the photocurrent density is given by Eq. (12), where | E| 2 = | E x | 2 + | E z | 2 ,
ττ 2 2 S xx = # e 2 ε [Φ(a x ) − Φ(a y )]. 8π! τ1 3
(16)
Here, 2
Φ(a x ) =
2 ⎡ ⎛ a x ω ⎞2 ⎤ !c⎞ η ⎛ω exp ⎢−⎜ ⎥, ⎜ ⎟ ! c ⎟⎠ ⎥⎦ |B y| ⎝ ω ⎠ ⎢⎣ ⎝ "v 0ω
(17)
where η = 1 + γ + γ 2 − γ 3 , and τ and τ1 are the times of elastic scattering between the monopoles for carriers in the ground and excited subbands, respectively ( τ ε ! τ, τ1 ). The photocurrent remains nonzero after summation over the monopoles if a x2 ≠ a y2 . The work of L.E.G. and E.L.I. was supported by the Russian Science Foundation (project no. 17-1201265). REFERENCES 1. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Pergamon, New York, 1984).
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Translated by R. Tyapaev
