Gyrotropic effects in trigonal tellurium studied from first principles
GGyrotropic effects in trigonal tellurium studied from first principles
Stepan S. Tsirkin,
1, 2
Pablo Aguado Puente,
1, 2 and Ivo Souza
1, 3 Centro de F´ısica de Materiales, Universidad del Pa´ıs Vasco, E-20018 San Sebasti´an, Spain Donostia International Physics Centre, E-20018 San Sebasti´an, Spain Ikerbasque Foundation, 48013 Bilbao, Spain (Dated: January 23, 2018)We present a combined ab initio study of several gyrotropic effects in p -doped trigonal tellurium(effects that reverse direction with the handedness of the spiral chains in the atomic structure). Thekey ingredients in our study are the k -space Berry curvature and intrinsic orbital magnetic momentimparted on the Bloch states by the chirality of the crystal structure. We show that the observed signreversal with temperature of the circular photogalvanic effect can be explained by the presence ofWeyl points near the bottom of the conduction band acting as sources and sinks of Berry curvature.The passage of a current along the trigonal axis induces a rather small parallel magnetization, whichcan nevertheless be detected by optical means (Faraday rotation of transmitted light) due to thehigh transparency of the sample. In agreement with experiment, we find that when infrared lightpropagates antiparallel to the current at low doping the current-induced optical rotation enhancesthe natural optical rotation. According to our calculations the plane of polarization rotates in theopposite sense to the bonded atoms in the spiral chains, in agreement with a recent experiment thatcontradicts earlier reports. I. INTRODUCTION
The spontaneous magnetization of ferromagnetic met-als gives rise to Hall and Faraday effects at B = 0. Theseeffects are termed anomalous , in opposition to the ordi-nary (linear in B ) Hall and Faraday effects in metalslacking magnetic order. The scattering-free or intrinsic contribution to the anomalous Hall conductivity (AHC)is given by [1, 2] σ A ab = − e (cid:126) (cid:90) [ d k ] (cid:88) n f ( E k n , µ, T ) (cid:15) abc Ω c k n , (1a) Ω k n = ∇ k × A k n = − Im (cid:104) ∂ k u k n | × | ∂ k u k n (cid:105) , (1b)where A k n = i (cid:104) u k n | ∂ k u k n (cid:105) is the Berry connection, Ω k n is the Berry curvature, E k n is the band energy, f is theequilibrium occupation factor, and the integral is overthe Brillouin zone with [ d k ] ≡ d k/ (2 π ) .The possibility of inducing similar effects in nonmag-netic conductors by purely electrical means was raisedby Baranova et al. [3], who predicted the existence ofan electrical analog of the Faraday effect in chiral con-ducting liquids: a change in rotatory power caused bythe passage of an electrical current. In the following,we shall refer to this phenomenom as “kinetic Faradayeffect” (kFE). In the kFE the induced rotatory powerreverses sign with the applied electric field E , in much thesame way that in the ordinary Faraday effect it reversessign with B . Althought it has not been observed so far Although this is a nonstandard designation, we find it prefer-able to current-induced optical activity [4, 5] since the effect iscloser to Faraday rotation than to natural optical activity. Thename adopted here is also consistent with that of a closely-relatedphenomenom to be discussed shortly, the kinetic magnetoelectriceffect . in liquids, the kFE was measured in a chiral conduct-ing crystal, p -doped trigonal Te [4, 5], following a the-oretical prediction [6]. The effect is symmetry allowedin the 18 (out of 21) acentric crystal classes known as gyrotropic [7], including those for which natural opticalrotation is disallowed.Gyrotropic crystals also display a nonlinear optical ef-fect closely related to the kFE: the circular photogalvaniceffect (CPGE). It consists in the generation of a pho-tocurrent that reverses sign with the helicity of light [6–11], and occurs when light is absorbed via interband orintraband scattering processes, with the latter involvingvirtual transitions to other bands [11].When impurity scattering is treated in the constantrelaxation-time approximation, it becomes possible toidentify a contribution to the intraband CPGE associ-ated with the Berry curvature of the free carriers [12–14].This “intrinsic” contribution, proportional to the relax-ation time τ , is conveniently described in terms of thefollowing dimensionless tensor D ab = (cid:90) [ d k ] (cid:88) n ∂E k n ∂k a Ω b k n (cid:18) − ∂f ∂E (cid:19) E = E k n , (2)where the index k has been dropped for brevity. D trans-forms like the gyration tensor g , but unlike g it is alwaystraceless. This means that D can only be nonzero in 16of the 18 gyrotropic crystal classes; the excluded classesare O and T, for which g is isotropic (its form is tabulatedin Ref. [10] for all the gyrotropic crystal classes). After integrating Eq. (2) by parts, the trace of the tensor D can be expressed as a Brillouin-zone integral of the divergence ofthe Berry curvature, weighted by the occupation factor. The factthat the Berry curvature is divergence-free except at isolated chi-ral band crossings (Weyl points) implies that D is traceless [12]. a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n In addition to the CPGE, the tensor D also describesa nonlinear anomalous Hall effect (AHE) [14] that canbe viewed as the low-frequency limit of the kFE. Indeed,the kFE is governed by a tensor (cid:101) D ( ω ) [Eq. (12) below]that reduces to D at ω = 0.The flow of electrical current that gives rise to the kFEgenerates a net magnetization in the gyrotropic medium,a phenomenom known as kinetic magnetoelectric effect (kME) [15]. It was first proposed for bulk chiral conduc-tors [6, 15] and later for two-dimensional (2D) inversionlayers [16, 17], where it has been studied intensively [18].A microscopic theory of the intrinsic kME effect in bulkcrystals was recently developed [19, 20]. The response,proportional to τ , is described by K ab = (cid:90) [ d k ] (cid:88) n ∂E k n ∂k a m b k n (cid:18) − ∂f ∂E (cid:19) E = E k n , (3)which has the same form as Eq. (2) but with the Berrycurvature replaced by the intrinsic magnetic moment m k n of the Bloch electrons. In addition to the spin mo-ment, m k n has an orbital component given by [2] m orb k n = e (cid:126) Im (cid:104) ∂ k u k n | × ( H k − E k n ) | ∂ k u k n (cid:105) , (4)where we chose e >
0. The tensor K (with units ofamperes) is symmetry allowed in all 18 gyrotropic crys-tal classes, and its symmetric part gives an intrabandcontribution to natural optical rotation at low frequen-cies [20, 21].In this work, we evaluate from first principles in p -doped tellurium ( p -Te) the CPGE and nonlinear AHEdescribed by the tensor D , the kFE described by (cid:101) D ( ω ),and the kME and intraband natural optical activity de-scribed by K , as well as the interband natural optical ac-tivity. We study them as a function of temperature andacceptor concentration, compare with the available ex-perimental data, and establish correlations between themon the basis of a unified microscopic picture.The manuscript is organized as follows. In Sec. II wedescribe the crystal structure of trigonal Te, the energybands, and the form of the gyrotropic response tensors.In the subsequent sections we present and analyze ourfirst-principles results for the various gyrotropic effects.The circular photogalvanic effect is treated in Sec. III,the nonlinear anomalous Hall effect in Sec. IV, the kineticFaraday effect in Sec. V, the kinetic magnetoelectric ef-fect in Sec. VI, and natural optical activity in Sec. VII.In each section, only the essential theory needed to un-derstand the results under discussion is given; all deriva-tions and additional technical details are left to the ap-pendixes. II. CRYSTAL STRUCTURE, ENERGY BANDS,AND SYMMETRY CONSIDERATIONS
Elemental Te is a nonmagnetic semiconductor thatcrystalizes in two enantiomorphic structures with space groups P3
21 and P3
21 (crystal class 32). The unit cellcontains three atoms disposed along a spiral chain that isright-handed for P3
21 and left-handed for P3
21, withthe chains arranged on a hexagonal net. In addition tothe screw symmetry along the trigonal axis, there arethree twofold axes lying on the perpendicular plane.The calculations reported in this work were carried outfor the right-handed Te structure described in Ref. [22].For the left-handed enantiomorph, the tensors D and K flip sign. These two tensors assume the forms D = D (cid:107) − − (5)(note that the trace vanishes ) and K = K ⊥ K ⊥
00 0 K (cid:107) , (6)where (cid:107) and ⊥ denote the directions parallel and perpen-dicular to the trigonal axis, respectively.The fully relativistic density-functional theory calcula-tions were done using the HSE06 hybrid functional [23].Figure 1 shows the calculated energy bands. The energygap of 0.312 eV at the H point is in good agreement withthe value of 0.314 eV obtained with the GW method [24],and with the experimental value of 0.323 eV [25]. Thecharacteristic “camel-back” shape of the upper valenceband around H can be seen in panel (c). The band struc-ture in Fig. 1 is in good agreement with other fully rel-ativistic calculations [24, 26]. It was calculated in thesame way as in Ref. [27], and we refer the reader to thatwork for further details.Below room temperature, the transport and low-frequency optical properties of weakly p -doped Te aregoverned by the upper valence band together with thelower conduction subbands. The conduction subbandshave an anisotropic Rashba-type spin-orbit splittingaround H, visible in Fig. 1(b); their spin textures (notshown) are consistent with those reported in Ref. [24].The three band degeneracies visible in Fig. 1(b) areWeyl points [24]. Of particular interest to the presentstudy is the one at H between the conduction subbands.It has positive chirality in the right-handed structure,which means that it acts as a source (sink) of Berrycurvature in the lower (upper) subband. Time-reversalsymmetry maps the Weyl point at H onto a Weyl pointof the same chirality at H’. More generally, it sends( E k n , Ω k n , m k n ) to ( E − k ,n , − Ω − k ,n , − m − k ,n ), so that k and − k contribute equally to Eqs. (2) and (3). III. CIRCULAR PHOTOGALVANIC EFFECT
A detailed study of the CPGE in Te due to free-carrier absorption was reported in Ref. [9]. The mea-surements were done at room temperature and below A A KM H K LH ′′ (d) (a) (b) (c) FIG. 1. Fully-relativistic band structure of trigonal Te, with energies measured from the valence-band maximum (VBM). (b) Ablow-up of the region demarcated by a dashed rectangle in (a), and (c) shows the top of the upper valence band around H,along the HK line. The Brillouin zone and its high-symmetry points are displayed in (d).
100 200 300 T (K) − j C P G E k ( µ A / c m ) Exp., N a = 4 · cm − Calc., N a = 4 · cm − Calc., N a = 3 · cm − Calc., N a = 7 · cm − FIG. 2. (Solid lines) Temperature dependence, for differentacceptor concentrations, of the intraband photocurrent den-sity induced in right-handed Te by circularly-polarized lightof positive helicity and intensity I = 10 W / cm propagatingalong the trigonal axis in the positive direction. According toEq. (7), the photocurrent is proportional to D (cid:107) . (Dashed line)Open-circuit photovoltage measured in Ref. [9], converted toa current density as described in the main text. on samples with a residual acceptor concentration N a ≈ · cm − , using a CO laser source with frequency (cid:126) ω = 0 .
117 eV. Under these conditions the relaxationtime exceeds 10 − s [26] so that ωτ (cid:29)
1, and Eq. (A7)for the intrinsic contribution to the intraband photocur-rent becomes j CPGE (cid:107) ( N a , T ) = sgn( q (cid:107) ) (cid:0) π a P circ D (cid:107) (cid:1) eI (cid:126) ω . (7)The quantity D (cid:107) ( µ ( N a , T ) , T ) is given by Eqs. (2)and (5), a ≈ /
137 is the fine-structure constant, and I is the intensity of incident light with wavevector q (cid:107) anddegree of circular polarization P circ propagating along thetrigonal axis.The photocurrent density calculated from Eq. (7) withsgn( q (cid:107) ) > P circ = +1 is plotted versus temper-ature in Fig. 2 for several acceptor concentrations, as-suming a laser intensity of 10 W / cm (see below). Thephotocurrent starts out positive at low temperature, andbecomes negative at around room temperature (except atthe highest doping level). Such a sign reversal was indeedobserved experimentally [9]. For a more detailed com-parison, we have converted the open-circuit photovolt-age and longitudinal conductivity measured in Ref. [9]into a current density, shown as a dashed curve after anoverall sign change (the handedness of the sample wasnot determined in Ref. [9]). Since the laser intensity wasalso not reported, we fixed the value of I in Eq. (7) bymatching the experimental values at low temperature. Atthe experimental doping level the calculated photocur-rent changes sign at around 220 K, in good agreementwith experiment.In order to understand the temperature dependence, FIG. 3. Microscopic mechanism of the intraband circular photogalvanic effect in right-handed Te. (a) The quantity D (cid:107) inEq. (8b) versus ε measured from the VBM, calculated using a Fermi smearing of 23 K with (heavy black solid line) and without(dashed gray line) spin-orbit coupling. The light colored lines show the function − ∂f ( E, µ ( N a , T ) , T ) /∂E | E = ε plotted versus ε at fixed N a and two different temperatures, as detailed in the inset. (b) (Solid lines) Fully relativistic band structure in thevicinity of the H point. q z denotes k z measured from the H point along HK, and the arrows denote the z component of theBerry curvature on each band. (Dashed lines) Scalar-relativistic band structure; for comparison purposes, the band edges havebeen aligned with those of the fully relativistic calculation. it is convenient to express the quantity D (cid:107) in Eq. (7) as D (cid:107) ( µ, T ) = (cid:90) + ∞−∞ dε D (cid:107) ( ε ) (cid:18) − ∂f ( E, µ, T ) ∂E (cid:19) E = ε , (8a) D (cid:107) ( ε ) = 1(2 π ) (cid:88) n (cid:90) E k n = ε dS ˆ v z k n Ω z k n , (8b)where D (cid:107) ( ε ) ≡ D (cid:107) ( ε, T ≈ v k n is the unit vectoralong the band velocity. Figure 3(a) shows that D (cid:107) has opposite signs at thetwo band edges, increasing slowly into the valence bandand rapidly into the conduction band, where it peaks. Atthe experimental doping level, − ∂f /∂E at 150 K is non-negligible in the valence band only, resulting in a positive Equation (8) is also convenient for numerical work. Once D (cid:107) ( ε )has been calculated from Eq. (8b), Eq. (8a) can be used to eval-uate D (cid:107) as a function of T and N a at a low computational cost.The temperature dependence of the chemical potential is calcu-lated assuming that at the temperatures of interest all dopantlevels are activated. The same approach will be used in subse-quent sections to evaluate the tensors ˜ D ( ω ), K , and C [Eq. (B6)]. D (cid:107) . At 250 K the chemical potential µ approaches thecenter of the gap, and − ∂f /∂E reaches the conductionband. D (cid:107) now collects contributions of opposite signsfrom the two band edges; the largest one comes fromthe D (cid:107) peak in the conduction band, which renders D (cid:107) negative. (When N a is increased to 7 · cm − , µ staysclose to the valence-band edge even at room temperature.The photocurrent is then dominated by hole-like carriers,and it remains positive over the entire temperature rangeof Fig. 2.)The behavior of D (cid:107) ( ε ) at the two band edges can be un-derstood by inspecting the energy bands and their Berrycurvatures along the HK line [Fig. 3(b)]. Because oftwofold symmetry about ΓK, v z k n and Ω z k n are both oddin q z = k z − k H , z , so that q z and − q z contribute equallyto Eq. (8b). Regarding the Ω z k n profiles, note that theBerry curvature of a band arises from its coupling toother bands [see Eq. (C20)], and that this coupling be-comes resonantly enhanced at (near) degeneracies [1, 2].At the nondegenerate valence-band edge this couplinghas no singularities and as a result Ω z k n varies smoothlywith q z , vanishing at q z = 0. Apart from a small re-gion between the “camel humps” that gives a negligiblecontribution, ˆ v z k n and Ω z k n have the same sign, which ex- ← q ⊥ q k → D k → E n e r g y ( a r b . un i t s ) FIG. 4. (a) Energy bands for the anisotropic 3D Rashbamodel of Eq. (9) with m (cid:107) /m ⊥ = 1, and v (cid:107) /v ⊥ = 0 . v (cid:107) /v ⊥ = 1 . D (cid:107) in Eq. (8b) eval-uated for the same choices of parameters with sgn( v (cid:107) ) > plains the steady increase in D (cid:107) towards positive valuesas ε enters the valence band.At the edge of the conduction band the Berry curva-ture is dominated by the strong intersubband couplingnear the Weyl point, which acts as a monopole of Berrycurvature leading to Ω z k n ∝ ± q − z for small | q z | [2]. When ε is slightly above the crossing energy, the two subbandsgive competing contributions to Eq. (8b): | Ω z k n | is largeron the inner branch, but the outer branch has a larger en-ergy isosurface. For an isotropic three-dimensional (3D)Rashba model these two contributions would cancel out, but the anisotropy of the Rashba splitting in Te is suchthat the inner branch dominates the integral in Eq. (8b),producing a negative peak in D (cid:107) near the Weyl-pointenergy.A minimal model for the conduction-band edge is [28] H R ( q ) = (cid:126) q (cid:107) m (cid:107) + (cid:126) q ⊥ m ⊥ + (cid:126) v (cid:107) q (cid:107) σ z + (cid:126) v ⊥ ( q x σ x + q y σ y ) , (9)where q = k − k H . We have evaluated Eq. (8b) numeri-cally for this two-band model, starting from the analyticexpression for the Berry curvature [29]. As expected D (cid:107) vanishes in the isotropic limit, and when either m (cid:107) (cid:54) = m ⊥ or v (cid:107) (cid:54) = v ⊥ a peak develops around the Weyl crossing. Fora given chirality, the peak can change sign depending onthe ratios m (cid:107) /m ⊥ and v (cid:107) /v ⊥ , as illustrated in Fig. 4.While spin-orbit coupling is not needed to generateWeyl points and Berry curvatures in the bands of Te(in contrast to centrosymmetric collinear ferromagnets,where it is essential), the intraband CPGE would be verydifferent in its absence. The spin-orbit-free D (cid:107) and en-ergy bands are shown as dashed gray lines in Fig. 3. The D (cid:107) peak in the conduction band has been suppressed,and a new peak has appeared in the valence band, againassociated with a Weyl crossing at H.In conclusion, the intrinsic CPGE of p -Te is strongly af-fected by the presence of spin-orbit-induced Weyl points
100 200 300T (K)00 . . σ A x y ( S / c m ) N a = 4 · cm − N a = 4 · cm − N a = 1 · cm − FIG. 5. Anomalous Hall conductivity induced in right-handedTe by a current density j (cid:107) = 1000 A/cm [Eq. (10)], plottedversus temperature at different acceptor concentrations. at H and H’ near the bottom of the conduction band. Thelarge Berry curvature around those chiral band crossingscauses a sign reversal of the photocurrent upon cool-ing a weakly p -doped sample, in agreement with experi-ment [9].We emphasize that the Berry-curvature mechanism forthe intraband CPGE is different from the one discussedin Ref. [9]. It involves elastic scattering from impuritiesrather than inelastic phonon scattering, and it relies onthe spin-orbit splitting of the conduction subbands thatwas neglected in that work. IV. NONLINEAR ANOMALOUS HALL EFFECT
In tellurium, the nonlinear AHE takes the form of anin-plane linear AHE proportional to the current densityflowing along the trigonal axis. Taking j (cid:107) = 1000 A / cm as a reference value [4, 5], the current-induced AHC isgiven by [see Eq. (B7)] σ A xy ( j (cid:107) = 1000 A / cm ) ≈ . D (cid:107) C (cid:107) (A / cm) S/cm . (10)The AHC calculated from Eq. (10) is plotted versustemperature in Fig. 5 at three different doping levels.At high doping it decreases monotonically with temper-ature, while at low doping it drops to negative valuesabove 220 K (due to the sign change in D (cid:107) discussed inthe previous section) and then approaches zero from be-low. Between 50 and 170 K, the AHC is only weaklydependent on N a over a wide doping range. This is dueto a near cancellation between the strong dependenciesof D (cid:107) and C (cid:107) on N a (see Fig. 6).The current-induced AHC displayed in Fig. 5 does notexceed 5 · − S/cm , which is probably too small tobe detected (it is five orders of magnitude smaller thanthe spontaneous AHC of bcc Fe [1]). Nevertheless, the D k e D k (¯ hω = 117 meV)10 − − . D k , e D k N a = 4 · cm − N a = 4 · cm − N a = 1 · cm −
100 200 300T (K)10 − . C k ( A / c m ) (a)(b) FIG. 6. (a) The quantities D (cid:107) [Eq. (2)] and (cid:101) D (cid:107) [Eq. (12)] inright-handed Te, plotted versus temperature on a semiloga-rithmic scale for different acceptor concentrations. The strongdip in log D (cid:107) around 220 K at N a = 4 · cm − signalsthe sign change in D (cid:107) seen in Fig. 2. (b) The quantity C (cid:107) [Eq. (B6)]. associated Faraday rotation has been observed in the in-frared [4, 5]. The analysis of that effect will occupy us inthe next section. V. KINETIC FARADAY EFFECT
So far, p -Te is the only material for which the kFEhas been measured. The first observation was reportedin Ref. [4], and new measurements were taken in Ref. [5].These works established that the current-induced changein rotatory power (∆ ρ ) is linear in j (cid:107) up to at least ± , and that ∆ ρ has the opposite (same) signas the natural rotatory power ρ when light travels par-allel (antiparallel) to the current.We have calculated ∆ ρ from the following expression,derived in Appendix C 3,∆ ρ ( ω, j (cid:107) ) = sgn( q (cid:107) ) a (cid:101) D (cid:107) ( ω ) j (cid:107) n ⊥ ( ω ) C (cid:107) (11)(our sign convention for optical rotation is specified in TABLE I. Natural rotatory power (in units of rad/cm), andcurrent-induced change in rotatory power divided by the cur-rent density (in units of 10 − rad · cm / A) at (cid:126) ω = 0 .
117 eVand T = 77 K for two different doping concentrations. Thesign of ∆ ρ/j (cid:107) corresponds to light propagating in the positivedirection along the trigonal axis [sgn( q (cid:107) ) > ρ ∆ ρ/j (cid:107) HandednessExpt. 1 . ± . a − . ± . , b − c UnknownTheory -0.86 4.5, 4 Right-handed a Ref. [31], undoped samples. b Ref. [5], p -doped samples with N a = 4 · cm − . c Ref. [4], p -doped samples with N a = 1 . · cm − . Appendix C 1). Here q (cid:107) is the wavevector of light, and n ⊥ is the index of refraction; we used the value n ⊥ = 5 . q (cid:107) ) = sgn( j (cid:107) )],∆ ρ has the same sign as the quantity (cid:101) D (cid:107) ( ω ) defined by (cid:101) D ab ( ω ) = (cid:90) [ d k ] (cid:88) n ∂E k n ∂k a (cid:101) Ω b k n ( ω ) (cid:18) − ∂f ∂E (cid:19) E = E k n , (12)a finite-frequency generalization of Eq. (2) obtained byreplacing Ω k n therein with (cid:101) Ω k n ( ω ) given by Eq. (C20).In addition to ∆ ρ , we have calculated the rotatorypower ρ caused by the natural optical activity of Teat j (cid:107) = 0. We used the formalism described in Ap-pendix C 2 a to evaluate ρ ignoring the influence of dop-ing (the effect of doping on ρ will be analyzed in Sec. VII,where it is shown to be negligible at the doping levels usedin the kFE measurements [4, 5]).Table I shows the calculated values of ρ and ∆ ρ/j (cid:107) alongside the experimental ones, measured on samplesof unknown handedness. In agreement with experiment,we find that ∆ ρ has the opposite sign from ρ when lighttravels parallel to the current (we defer the discussion ofabsolute signs to Sec. V C). The calculated | ρ | and | ∆ ρ | are smaller by roughly a factor of two compared to themeasured values, which can be considered a fair level ofagreement. The calculated | ∆ ρ | decreases only slightly as N a is increased from 4 · to 1 . · cm − . The largerdecrease seen in the experimental values was attributedin Ref. [5] to technical differences relative to Ref. [4].At j (cid:107) = 1000 A/cm , ∆ ρ is about five orders of mag-nitude smaller than the spontaneous Faraday rotatorypower of bcc Fe [32]. This is the same difference in or-ders of magnitude that was found in the previous sectionfor the AHC. However, the smallness of the kFE is com-pensated by the high transparency of Te in the infrared,which allows one to measure the optical rotation acrossa cm-sized sample [4, 5], compared to ∼ − cm-thickiron films [32].
100 200 300T (K)00 . . . ∆ ρ ( r a d / c m ) N a = 4 · cm − N a = 4 · cm − N a = 1 · cm − ∆ ρ exp / FIG. 7. Temperature dependence of the change in the rota-tory power of right-handed Te induced by a current densityof 1000 A/cm . The optical frequency is (cid:126) ω = 0 .
117 eV, N a is the doping level, and the sign of ∆ ρ corresponds to lightpropagating parallel to the current along the trigonal axis.The open circles denote experimental data [33, 34] taken at N a = 3 . · cm − , which has been rescaled by a factor of1 / A. Doping and temperature dependence
Figure 7 shows a weak doping dependence of ∆ ρ atlow doping between 50 and 170 K, in good agreementwith the experimental data in Ref. [34] (p. 27), anda monotonic decrease with temperature. The decreaseis by a factor of three to four between 77 and 300 K,in agreement with an earlier theoretical estimate [35].Apart from the previously mentioned overall factor oftwo which at present we cannot account for, the calcu-lated ∆ ρ agrees rather well with the experimental datareported in Refs. [33] and [34] (p. 35), as indicated by theopen circles in Fig. 7.Even at the lowest doping, ∆ ρ shows no sign change(only a dip) around 220 K. This behavior, which is incontrast to the CPGE and the nonlinear AHE, can beunderstood from Fig. 6(a) where at N a = 4 · cm − the quantity (cid:101) D (cid:107) maintains its sign as T goes above 220 K,whereas D (cid:107) changes sign.How close (cid:101) D (cid:107) ( ω ) is to D (cid:107) at a given temperature anddoping level depends on how close (cid:101) Ω k n ( ω ) is to Ω k n inthe relevant energy bands, which in turn depends on how ω compares with ω mn for the dominant transitions inEq. (C20). We proceed as in Sec. III, expressing (cid:101) D (cid:107) ( ω )in terms of (cid:101) D (cid:107) ( ε,ω ) according to Eq. (8). The band- edge behavior of (cid:101) D (cid:107) ( ε, ω ) and (cid:101) Ω z k n ( ω ) at (cid:126) ω = 0 .
117 eVis depicted in Fig. 8, to be compared with Fig. 3. Inthe valence band (cid:101) Ω z k n ≈ Ω z k n and (cid:101) D (cid:107) ≈ D (cid:107) , becausethe dominant coupling is with the conduction bands thatare separated by more than 0.3 eV (the coupling to thevalence band below, which is closer in energy, is sup-pressed by selection rules [5]). In contrast | (cid:101) Ω z k n | (cid:28) | Ω z k n | in the conduction bands, because 0 .
117 eV is a large en-ergy compared to the Rashba splitting of the coupledsubbands. The peak in (cid:101) D (cid:107) is therefore strongly reducedcompared to the peak in D (cid:107) , and this is the reason for (cid:101) D (cid:107) not changing sign with temperature at low doping inFig. 6(a).In conclusion, at the CO laser frequency the kFE isdominated by contributions that to a good approxima-tion can be expressed in terms of the Berry curvatureat the top of the valence band. Since this is the samequantity that governs the intrinsic CPGE at low tem-peratures (Sec. III), one can correlate the sign of ∆ ρ with that of the photocurrent measured on the same sam-ple. When linearly-polarized light travels parallel to thecurrent [sgn( q (cid:107) ) j (cid:107) > ρ has the samesign as the photocurrent induced at low temperatures bylight of positive helicity traveling in the positive direction[sgn( q (cid:107) ) P circ > B. Frequency dependence
The spectral dependence of the kFE was investigatedin Ref. [5] by taking additional measurements with a COlaser, which generates radiation of higher frequency thanthe CO laser. These measurements were again taken at77 K on samples with N a ≈ · cm − .Between (cid:126) ω = 0 .
117 eV and (cid:126) ω = 0 .
23 eV, ∆ ρ wasfound to increase by a factor of 1 .
7. This is significantlyless than the increase by a factor of 4.7 in ρ [31], con-firming that current-induced optical rotation and naturaloptical activity are separate physical effects [5]. Our cal-culated ∆ ρ and ρ increased by factors of 1.4 and 5.6respectively over the same spectral range, in reasonableagreement with the observed trends.The calculated ∆ ρ ( ω ) is plotted in Fig. 9 at differentdoping levels and temperatures. At N a = 4 · cm − the spectral dependence is smooth, becoming weaker asthe temperature increases. The reason is that at thisrelatively high doping (cid:101) D (cid:107) is mostly determined by (cid:101) Ω k n at the valence-band edge, which depends only weakly onfrequency over the subgap spectral range of Fig. 9.Reducing N a to 4 · cm − has practically no effecton the spectral dependence of ∆ ρ in Fig. 9 at low tem-peratures, since (cid:101) D (cid:107) still originates mostly from the topof the valence band. At 300 K, the contribution fromthe conduction bands has become significant at this lowdoping. At frequencies higher than 0.05 eV this leads toa reduction in ∆ ρ , due to the opposite signs of (cid:101) D (cid:107) at the FIG. 8. Microscopic mechanism of the kinetic Faraday effect in right-handed Te at (cid:126) ω = 0 .
117 eV. The figure is similar toFig. 3, but with D (cid:107) replaced by (cid:101) D ab ( ω ) [the low-temperature limit of Eq. (12)], Ω z k n by (cid:101) Ω z k n ( ω ) [Eq. (C20)], and a differentdoping level when plotting − ∂f /∂E in (a). The dotted line in (a) represents D (cid:107) , and is identical to the heavy solid line inFig. 3(a). two band edges [Fig. 8(a)]. Below that frequency, thephoton energy becomes comparable to the Rasha split-ting near the bottom of the conduction band. As a result,∆ ρ exhibits a strong dispersion caused by the couplingin Eq. (C20) between the two conduction subbands. C. Absolute sign of the optical rotation
All gyrotropic effects have equal magnitudes and oppo-site signs for two otherwise identical samples of oppositehandedness. Unfortunately the experimental determina-tion of the handedness is particularly difficult for elemen-tal crystals [36], and there are conflicting claims in theliterature as to which enantiomorph of trigonal Te rotatesthe plane of polarization of light in which sense.We are aware of three studies that tried to establishthe handedness of a Te sample, correlating it with thesign of the rotatory power ρ . The first work used etch-ing techniques [37], the second polarized neutron diffrac-tion [38], and the third resonant x-ray diffraction [36]. InRefs. [37, 38] it was concluded that the plane of polar-ization of light rotates in the same sense as the bondedatoms in the spiral chains (with our sign convention,that means ρ > opposite conclusion, see Er- ratum [41]: right-handed Te has a negative ρ , in agree-ment with our calculations.Let us conclude with a comment on the sign of ∆ ρ calculated in Refs. [4, 5] using a k · p model for theband-edge states. It was found in those works that∆ ρ < q z at the top of the uppermost valence band are domi-nated by atomic states with total angular momentum j z = − / j z = +3 / ρ > ab initio results leads to the opposite conclusion. For example,the lower panel of Fig. 11(c) shows that in right-handedTe the spin magnetic moment of states near the top ofthe upper valence band is negative for q z >
0. In anatomic picture, this corresponds to states with total an-gular j z = +3 / q z . In con-clusion, once the k · p model is matched to our ab initio wavefunctions it yields ρ < ρ > The k · p model of Refs. [4, 5] includes spin-orbit coupling inthe valence bands only. This is an acceptable approximation,given that the spin-orbit induced Weyl points at the edge of theconduction band do not give a large contribution to the kFE atthe CO laser frequency. Recall from Sec. III that this was not T = 300 K T = 150 K T = 77 K N a = 4 · N a = 4 · . . hω (eV)00 . . . ∆ ρ ( r a d / c m ) FIG. 9. Frequency dependence of the change in rotatorypower induced in right-handed Te by a current density j (cid:107) =1000 A/cm , at different temperatures and doping levels. Inorder to avoid singularities in Eq. (C20) at ω = ω k mn , ∆ ρ iscalculated at complex frequencies using Im[ (cid:126) ω ] = 1 meV. VI. KINETIC MAGNETOELECTRIC EFFECT
Along with the Faraday rotation of transmitted light,the flow of a dc current through a gyrotropic crystal pro-duces a macroscopic magnetization. So far, the intrinsiccontribution to this effect has only been calculated formodel tight-binding systems [19, 42]. Our goal in thissection is to make quantitative estimates for p -Te, and toprovide a microscopic picture for the effect.A current flowing along the trigonal axis induces a par-allel magnetization given by (Appendix D) M (cid:107) = − K (cid:107) j (cid:107) πC (cid:107) . (13)The temperature and doping dependence of M (cid:107) calcu-lated at j (cid:107) = 1000 A/cm is shown in Fig. 10. In contrastto 2D inversion layers where the current-induced magne-tization is purely spinlike [16–18], in p -Te it has both or-bital and spin components, shown separately in Fig. 10.They have opposite signs and comparable magnitudes,with the orbital effect being somewhat larger. Their the case for the CPGE: Without spin-orbit coupling in the con-duction bands, the intrinsic part of the intraband CPGE wouldnot change sign with temperature. Without spin-orbit coupling the bulk kME would be purely or- M o r b k ( − µ B / a t o m )
100 200 300 T (K)-1.5-1.0-0.50.0 M s p i n k ( − µ B / a t o m ) N a = 4 · cm − N a = 4 · cm − N a = 1 · cm − (a)(b) FIG. 10. Temperature dependence of the orbital and spinmagnetization induced in right-handed Te, at different accep-tor concentrations, by a current density j (cid:107) = 1000 A/cm . magnitudes are ∼ − µ B /atom, six orders smaller thanthe spontaneous orbital magnetization in bcc Fe [43] (re-call that comparable differences in orders of magnituderelative to bcc Fe were found earlier for the nonlinearAHE and for the kFE).The current-induced spin density at 77 K, N a =4 · cm − , and j (cid:107) = 1400 A/cm was estimated inRef. [5] to be ∼
560 spins/ µ m . Under the same con-ditions our calculation yields 561 spins/ µ m , in a sur-prisingly perfect agreement. While it may be difficult todirectly measure such a small magnetization, indirect ev-idence for the kME in p -Te has already been gathered.In addition to the kFE [4, 5], a current-induced split-ting of nuclear magnetic resonance peaks was recentlydetected [44]. bital [19, 20], and we attribute the presence of a comparable spincontribution to the kME to the strong spin-orbit coupling in Te. FIG. 11. Microscopic mechanism of the kinetic magnetoelectric effect in right-handed Te. (a) Similar to the lower panel ofFig. 8(a), with (cid:101) D (cid:107) replaced by K (cid:107) [the low-temperature limit of Eq. (3)]. The total K (cid:107) (heavy solid line) is decomposed intoorbital (dashed line) and spin (dotted line) parts. (b) and (c) Like the lower panels of Fig. 8(b), with (cid:101) Ω z k n replaced by m orb k n,z [Eq. (4)] in (b), and by m spin k n,z = − g s µ B (cid:104) ψ k n | σ z | ψ k n (cid:105) in (c) — the orbital and spin parts of the intrinsic magnetic moment ofa Bloch electron. In (b), the gray arrows denote orbital moments calculated according to Eq. (14). The dominance of the orbital contribution to M (cid:107) inFig. 10 implies that it remains positive over the entiretemperature range. The signs of M orb (cid:107) and M spin (cid:107) can beunderstood from Fig. 11. Panel (a) shows the quantity K (cid:107) ( ε ) [defined in terms of K (cid:107) in the manner of Eq. (8)]at the top of the valence band, and the signs of its or-bital and spin contributions follow from panels (b) and(c), where it can be seen that the z component of the or-bital (spin) moments of the band states are antiparallel(parallel) to ∂E k n /∂k z .The fact that the spin and orbital moments are an-tiparallel for states in the upper valence band is some-what surprising. Those states can be approximated asa linear combination of atomic states with total angularmomenta j z = ± / m atomic k n,z = − µ B (cid:88) ilm m |(cid:104) u k n | ilm (cid:105)| , (14)where the | ilm (cid:105) are projectors onto spherical-harmonicstates localized on the i th atom in the unit cell, and µ B = e (cid:126) / (2 m e ) is the Bohr magneton. As seen in Fig. 11(b),the moments calculated from Eqs. (4) and (14) differ inboth sign and magnitude. This signals a breakdown ofthe atomic picture of orbital magnetism for states at thetop of the valence band, highlighting the need to use therigorous definition (4) of m orb k n so as to include itinerantcontributions related to the Berry curvature.In fact, the signs of m orb k n,z and Ω z k n are correlated forstates in the upper valence band, as can be seen by com-paring the spectral decomposition of Eq. (4), m orb k n = e (cid:126) (cid:88) m ( E k m − E k n ) Im ( A k nm A k mn ) , (15) with that of Eq. (1b) [given by Eq. (C20) at ω = 0],and recalling from Sec. V A that the upper valence bandcouples most strongly to the lower conduction subbands,for which E k m − E k n >
0. This analysis suggests that m orb k n,z and Ω z k n should be antiparallel, which is indeedthe case: Compare Fig. 11(b) with Fig. 3(b). VII. NATURAL OPTICAL ACTIVITY OFDOPED TELLURIUM
The theoretical value of ρ in Table I was calculatedfor undoped Te, and here we analyze how it changes un-der doping. We consider two effects: the doping depen-dence of the interband contribution, and the appearancein doped samples of an intraband contribution, whosemechanism is closely related to that of the kME [20, 21].We calculate both effects at 77 K, for the CO laser fre-quency.We begin with the doping dependence of the interbandrotatory power, which can be taken into account by re-placing (cid:80) o,en,l with (cid:80) n,l f k n (1 − f k l ) in Eq. (C12) (seeRef. [40]). As shown by the dashed line in Fig. 12, ρ inter0 remain negative over the entire doping range. At first itsmagnitude decreases slightly with increasing N a , due to adepopulation of the upper valence band that blocks someof the interband transitions [40]. It reaches a minimumat N a ≈ . · cm − , and then increases rapidly inmagnitude. The rapid increase is caused by transitionsbetween the two upper valence bands, which become pos-sible at high doping [40]. Although the matrix elementsfor such transitions are small [5, 40], along the HA linethe band separation is close to the CO laser frequencyof (cid:126) ω = 0 .
117 eV [see Fig. 1(b)], producing a resonantenhancement.We now turn to the intraband rotatory power, shown1 N a (cm − ) − − ρ ( r a d / c m ) totalinterbandintraband FIG. 12. Doping dependence of the natural rotatory powerof right-handed Te at 77 K and (cid:126) ω = 0 .
117 eV, decomposedinto interband and intraband contributions. as the dotted line in Fig. 12. In Appendix C 2 b we ob-tained, following Refs. [20, 21], ρ intra0 ( ω ) = ω τ ω τ ρ clean0 , (16a) ρ clean0 = − π a ec K ⊥ , (16b)with K ⊥ given by Eqs. (3) and (6). Using the val-ues of τ ( N a , T ) from Ref. [26], we conclude that up to N a = 10 cm − the “clean-limit” condition ωτ (cid:29) laser frequency and room tempera-ture (and below). Thus, ρ intra0 ≈ ρ clean0 over the entirerange of Fig. 12. ρ intra0 has the opposite sign comparedto ρ inter0 , and a negligible magnitude at low doping. Butwhile | ρ inter0 | initially decreases as the doping level in-creases, | ρ intra0 | increases (more or less linearly) with N a .Interestingly, between 8 · and 7 . · cm − thecompetition between the two contributions results in asign reversal of ρ . VIII. SUMMARY
In summary, we have carried out a combined ab ini-tio study of several gyrotropic effects in p -doped Te. The computer code developed for this project was written asa module of the wannier90 package [45, 46], and will be madepublicly available in a forthcoming release.
The motivation was provided by recent theoretical devel-opments that recognized the central role played by theBerry curvature and by the intrinsic orbital moment inthe description of such effects in the semiclassical regimeof low frequencies compared to the band splittings. Thisprompted us to revisit the pioneering infrared measure-ments of the CPGE [8, 9] and kFE [4, 5] in bulk Te.We found that the intrinsic mechanism for the intra-band CPGE [12–14] accounts for the observed sign re-versal of the CPGE with temperature, and that the signreversal is caused by the presence of Berry-curvaturemonopoles (Weyl points) at the bottom of the conductionband. This provides an interesting example of the wayin which Weyl points can influence physical observablesin semiconductors.Regarding the natural and current-induced optical ro-tation (kFE), our calculations give rotatory powers whosemagnitudes are within a factor of two of the measuredones. In agreement with experiment [5], we find that ∆ ρ and ρ have opposite signs when light propagates in thesame direction as the current.As for the absolute sign of ρ , we find that in un-doped samples the plane of polarization rotates in theopposite sense to the bonded atoms in the spiral chains.This contradicts the result of early attempts to determinethe handedness of a Te sample [37, 38], but agrees withthe most recent experimental determination [36, 41]. Wealso predict a sign reversal of ρ over a significant dop-ing range, due to the competition between interband andintraband contributions to the natural optical activity.In order to compare our fully quantum-mechanical cal-culation of ∆ ρ with the semiclassical limit, the result wasexpressed in terms of a quantity (cid:101) Ω k n ( ω ) that reduces tothe Berry curvature at ω = 0. We found that at the CO laser frequency, (cid:101) Ω k n ( ω ) at the top of the valence bandis very close to Ω k n . Hence, the low-temperature kFEis well described by the same Berry-curvature parameter D (cid:107) that governs the intrinsic CPGE. This leads to a def-inite sign relation between the two effects, which couldbe tested by measuring both on the same sample.We have also provided estimates for the magnitudes ofother gyrotropic effects that have not yet been observed,such as the nonlinear AHE and the kME. Our estimatesindicate that those effects are rather small in p -Te. How-ever, a recent study predicted a sizable nonlinear AHEin Weyl semimetals [47].In closing, we hope that the present work will stimu-late further experimental and theoretical work exploringthe role of the k space Berry curvature, intrinsic orbitalmoment, and Weyl points in connection with gyrotropiceffects in bulk crystals. Note added:
After this work was submitted, a com-plementary theoretical study of the kME in p -Te ap-peared [48]. The authors used a k · p model to investigateextrinsic as well as intrinsic contributions to the kFE. Fordoping concentrations up to a few 10 cm − , they findthat the latter are dominant, with the same magnitudeand sign as reported here.2 ACKNOWLEDGMENTS
We acknowledge support from Grant No. FIS2016-77188-P from the Spanish Ministerio de Econom´ıa yCompetitividad, Grant No. CIG-303602 from the Eu-ropean Commission, and from Elkartek Grant No. KK-2016/00025. We would like to thank V. A. Shalygin foruseful discussions.
APPENDICES
In Appendices A to D we review the theory of the var-ious gyrotropic effects considered in the main text. Thephotogalvanic effect is treated in Appendix A, the nonlin-ear AHE in Appendix B, optical rotation in Appendix C,and the kinetic magnetoelectric effect in Appendix D.Concerning the microscopic theory of these effects, ouraim is to present a coherent picture based on a small num-ber of basic ingredients (Berry connections, curvatures,and intrinsic magnetic moments). We only consider the“intrinsic” contributions that can be calculated from theelectronic structure of the pristine crystal supplementedby a phenomenological relaxation time τ . We thereforeneglect extrinsic effects due to skew-scattering and side-jump processes at impurities [12, 49]. Finally, AppendixE describes some technical details of our Wannier-basednumerical scheme. Appendix A: Photogalvanic effect1. Phenomenology
Consider an oscillating electric field E ( r , t ) = Re (cid:104) E ( ω ) e i ( q · r − ωt ) (cid:105) . (A1)The current density induced at second order in the fieldamplitude can be written as [14] j a ( t ) = Re (cid:0) j a + j ωa e − i ωt (cid:1) , (A2a) j a = 12 σ abc ( ω ) E b ( ω ) E ∗ c ( ω ) , (A2b) j ωa = 12 σ abc ( ω ) E b ( ω ) E c ( ω ) . (A2c)Equations (A2b) and (A2c) describe a dc photocurrentand a second-harmonic current, respectively.Writing σ abc = λ abc + γ abc and E b E ∗ c = Re ( E b E ∗ c ) + i Im ( E b E ∗ c ), where the first and second terms in these ex-pressions are, respectively, symmetric and antisymmetricunder b ↔ c , Eq. (A2b) becomes j a = 12 { λ abc Re ( E b E ∗ c ) − γ ab Im ( E × E ∗ ) b } , (A3)where γ ab = − i(cid:15) bcd γ acd / − i(cid:15) bcd σ acd /2. The first (sec-ond) term describes the linear (circular) photogalvanic effects. λ abc transforms like the piezoelectric tensor, and γ ab like the gyration tensor [10, 11].
2. Berry-curvature (“intrinsic”) contributions
The intrinsic intraband contribution to the nonlinearconductivity of a nonmagnetic crystal can be expressedin terms of the tensor D in Eq. (2) as [14] σ abc = − e τ ω (cid:126) (cid:15) adc D bd , (A4)where τ ω = τ − iωτ . (A5)Combining Eqs. (A2)–(A4) one findsRe (cid:0) j a (cid:1) = j LPGE a + j CPGE a , (A6a) j LPGE a = − e (cid:126) Re( τ ω ) (cid:15) adc D bd Re ( E b E ∗ c ) , (A6b) j CPGE a = − e (cid:126) Im( τ ω ) D ab Im ( E × E ∗ ) b , (A6c)where LPGE stands for “linear photogalvanic effect,”and Tr( D ) = 0 was used to eliminate one term fromEq. (A6c) [12].Consider the CPGE in trigonal Te with light propagat-ing along the trigonal axis. Writing q = q (cid:107) ˆ z , E = | E | ˆ e ,and − Im ( e × e ∗ ) = P circ ˆ q where P circ is the degree ofcircular polarization, and defining the intensity of inci-dent light as I = c(cid:15) | E | /
2, Eq. (A6c) becomes j CPGE (cid:107) = sgn( q (cid:107) ) (cid:0) π a P circ D (cid:107) (cid:1) Im( τ ω ) eI (cid:126) , (A7)where a = e / (4 π(cid:15) (cid:126) c ) is the fine-structure constant. Forpositive helicity ( P circ > D (cid:107) > D (cid:107) < Appendix B: Nonlinear anomalous Hall effect
In the ω → j a = − e τ (cid:126) (cid:15) adc D bd E b E c , (B1)with equal parts coming from the second-harmonic andLPGE currents (the CPGE vanishes at ω → E · j = 0, Eq. (B1) describes a nonlinear anomalous Hallcurrent [14].It is instructive to obtain Eq. (B1) by replacing f inEq. (1) with the change in the distribution function atlinear order in an applied static field,∆ f ≡ f − f = − eτ E · v k n (cid:18) − ∂f ∂E (cid:19) E = E k n . (B2)3Doing so yields ∆ σ A ab = e τ (cid:126) (cid:15) abd D cd E c (B3)for the field-induced AHC, in agreement with Eq. (B1).Inserting Eq. (5) for the tensor D in Te into Eq. (B1)for the current, we obtain j x = 3 e τ (cid:126) D (cid:107) E z E y , (B4a) j y = − e τ (cid:126) D (cid:107) E z E x , (B4b) j z = 0 . (B4c)The nonlinear current flows in the plane perpendicular tothe trigonal axis, and the effect can be viewed as an in-plane linear AHE induced by the out-of-plane field com-ponent E (cid:107) ≡ E z . The effective field-induced AHC is σ A xy ( E (cid:107) ) = 3 e τ (cid:126) D (cid:107) E (cid:107) = 3 e (cid:126) D (cid:107) σ (cid:107) /τ j (cid:107) , (B5)where in the second equality we inverted Ohm’s law toexpress the result in terms of j (cid:107) (the nonzero componentsof the Ohmic conductivity are σ ⊥ ≡ σ xx = σ yy and σ (cid:107) ≡ σ zz ). In the constant relaxation-time approximation wehave σ (cid:107) /τ = (2 πe/ (cid:126) ) C (cid:107) , with C (cid:107) = eh (cid:90) [ d k ] (cid:88) n (cid:18) ∂E k n ∂k z (cid:19) (cid:18) − ∂f ∂E (cid:19) E = E k n (B6)a positive quantity with units of surface current density.With this notation, the current-induced AHC reads σ A xy ( j (cid:107) ) = ( e /h )(3 D (cid:107) / j (cid:107) /C (cid:107) ) , (B7)where e /h is the quantum of conductance, D (cid:107) is dimen-sionless, and j (cid:107) /C (cid:107) has units of inverse length. Appendix C: Optical rotation1. Phenomenology
The dielectric tensor of trigonal Te has the form [5] ε ( ω, q (cid:107) , j (cid:107) ) = ε ⊥ ε A xy ( ω, q (cid:107) , j (cid:107) ) 0 − ε A xy ( ω, q (cid:107) , j (cid:107) ) ε ⊥
00 0 ε (cid:107) . (C1)In equilibrium, the antisymmetric part ε A xy responsiblefor optical rotation is linear in the wavevector q (cid:107) of lightpropagating inside the crystal along the trigonal axis.Under a steady current flow, ε A xy acquires a new contribu-tion closely related to the nonlinear AHC of Appendix B.It is linear in j (cid:107) and zeroth-order in q (cid:107) , giving rise to thekFE. (As for the diagonal elements ε ⊥ and ε (cid:107) , they areindependent of j (cid:107) and q (cid:107) to linear order.) Before proceeding further, let us specify our sign con-vention for optical rotation. We say that the rotatorypower ρ is positive when the sense of rotation of the elec-tric field vector is counterclockwise as seen by an ob-server looking toward the light source. With this choicewe have ρ = πλ Re ( n − − n + ) = ω c Re ( n − − n + ) , (C2)where λ is the wavelength in vacuum, and n + and n − arethe complex indices of refraction for circularly polarizedwaves of positive and negative helicity, respectively, withpolarization vectors given by e ± = ˆ x ± i sgn( q (cid:107) ) ˆ y √ . (C3)Assuming a sufficiently small current density such that | ε Axy /ε ⊥ | (cid:28)
1, one finds [5, 52] n − − n + ≈ − sgn( q (cid:107) ) i ε A xy /ε n ⊥ , (C4)where n ⊥ ≡ (cid:112) ε ⊥ /ε . Converting to conductivities using ε ab ( ω ) = ε (cid:20) δ ab + iωε σ ab ( ω ) (cid:21) , (C5)we obtain ρ ( ω, j (cid:107) ) = sgn( q (cid:107) ) Re σ A xy ( ω, j (cid:107) )2 cε n ⊥ ( ω ) (C6)at nonabsorbing frequencies, with n ⊥ ( ω ) = (cid:20) − ωε Im σ ⊥ ( ω ) (cid:21) / . (C7)In the following, we expand the rotatory power as [5] ρ ( ω, j (cid:107) ) = ρ ( ω ) + ∆ ρ ( ω, j (cid:107) ) + O (cid:16) j (cid:107) (cid:17) . (C8) ρ is the natural rotatory power at j (cid:107) = 0, and ∆ ρ ( j (cid:107) ) isthe change in rotatory power at linear order in j (cid:107) .
2. Natural optical rotation
Natural optical rotation is described by σ A xy ( q , ω ) atfirst order in q z , which is conventionally written as [53] σ A xy ( ω, q ) = ωε γ xyz q z = sgn( q (cid:107) ) ωε γ xyz | q (cid:107) | , (C9) Compare with Eq. (2) in Ch. XIV of Ref. [50], where the oppo-site sign convention for ρ was adopted. Therein, “left-circularpolarization” refers to our positive helicity (see also Ref. [51]). γ xyz has units of length. Using | q (cid:107) | / Re n ⊥ = ω/c ,Eq. (C6) becomes [28] ρ ( ω ) = ω c Re γ xyz ( ω ) . (C10)Note that the natural rotatory power does not reversesign with q (cid:107) . Thus, if a linearly-polarized ray travels backand forth inside the material the plane of polarization isunchanged when it returns to the initial point [52, 53].We now turn to the microscopic theory. The natu-ral optical activity of nonconducting crystals is governedby virtual interband transitions [28, 54, 55], and the ro-tatory power decreases as ω at frequencies well belowthose of interband transitions. Instead, conducting crys-tals remain optically active at such low frequencies dueto intraband processes [20, 21]. Thus, the rotatory powerof a conducting crystal is given by ρ ( ω ) = ρ inter0 ( ω ) + ρ intra0 ( ω ) . (C11)In the following, both contributions are calculated. a. Interband natural optical rotation Following Ref. [55] we write, with ∂ c ≡ ∂/∂k c ,Re γ inter abc ( ω ) = e ε (cid:126) (cid:90) [ d k ] o,e (cid:88) n,l (cid:104) ω ln − ω Re (cid:0) A bln B acnl − A aln B bcnl (cid:1) − ω ln − ω ( ω ln − ω ) ∂ c ( E l + E n )Im (cid:0) A anl A bln (cid:1)(cid:105) . (C12)The summations over n and l span the occupied ( o ) andempty ( e ) states respectively, ω ln = ( E l − E n ) / (cid:126) , and weomit the k subscript for brevity. Here A aln = i (cid:104) u l | ∂ a u n (cid:105) (C13)is the matrix generalization of the Berry connection ap-pearing in Eq. (1b). Finally, the matrix B acnl has bothorbital and spin contributions given by B ac (orb) nl = (cid:104) u n | ( ∂ a H ) | ∂ c u l (cid:105) − (cid:104) ∂ c u n | ( ∂ a H ) | u l (cid:105) (C14)and B ac (spin) nl = − i (cid:126) m e (cid:15) abc (cid:104) u n | σ b | u l (cid:105) . (C15)In Te the spin matrix elements contribute less than 0.5%of the total ρ inter0 , and can be safely ignored. Writing H = (cid:80) m | u m (cid:105) E m (cid:104) u m | , the orbital matrix elements become B ac (orb) nl = − i∂ a ( E n + E l ) A cnl + (cid:88) m (cid:110) ( E n − E m ) A anm A cml − ( E l − E m ) A cnm A aml (cid:111) . (C16) This reduces the calculation of B (orb) to the evaluationof band gradients and off-diagonal elements of the Berryconnection matrix, and both operations can be carriedout efficiently in a Wannier-function basis [56].In our implementation, the summation in Eq. (C16)is restricted to the s and p bands included in the Wan-nierization procedure (see Appendix E). To check howquickly the calculated ρ inter0 converges with the numberof bands, we redid the calculation keeping only the fourbands (two valence and two conduction) closest to thegap, and found that the value changed by only 10% com-pared to a calculation including all s and p states. Thisis consistent with the conclusion of Ref. [40] that thenatural optical activity of Te is contributed mainly bytransitions between states near the energy gap. b. Intraband natural optical rotation Here we calculate ρ intra0 following Refs. [20, 21]. Com-bining Eqs. (5a) and (S61) in Ref. [20] and noting thatin our notation the tensor α GME defined therein is givenby − iω ( e/ (cid:126) ) τ ω K , we findRe γ intra abc ( ω ) = e Im τ ω ωε (cid:126) ( (cid:15) acd K bd − (cid:15) bcd K ad ) . (C17)Using Eq. (6) for the tensor K in Te leads toRe γ intra xyz ( ω ) = − e Im τ ω ωε (cid:126) K ⊥ . (C18)The intraband rotatory power of Eq. (16) is obtained byinserting this expression in Eq. (C10).
3. Current-induced optical rotation
Let us now obtain a microscopic expression for ∆ ρ inEq. (C8), by expanding Eq. (C6) to first order in j (cid:107) .For that purpose, it is sufficient to expand the tensorRe σ A xy ( ω ) in the numerator. At j (cid:107) = 0 it is given by thefollowing finite-frequency generalization of Eq. (1),Re σ A ab ( ω ) = − e (cid:126) (cid:90) [ d k ] (cid:88) n f ( E k n ) (cid:15) abc (cid:101) Ω c k n ( ω ) , (C19)where the quantity (cid:101) Ω k n ( ω ) = − (cid:88) m ω k mn ω k mn − ω Im ( A k nm × A k mn ) (C20)reduces to the Berry curvature at ω = 0. Contrary to the Berry curvature, the divergence of ˜ Ω k n ( ω ) isgenerally nonzero. As a result, ˜ D ( ω ) given by Eq. (12) can havea nonzero trace at finite frequencies, i.e., ˜ D (cid:107) (cid:54) = − D ⊥ in Te. j (cid:107) canbe obtained by replacing f therein with ∆ f given byEq. (B2). Following Appendix B we obtainRe σ A xy ( ω, j (cid:107) ) = ( e /h ) (cid:101) D (cid:107) ( ω )( j (cid:107) /C (cid:107) ) (C21)with (cid:101) D (cid:107) ( ω ) given by Eq. (12), and inserting this expres-sion in Eq. (C6) we arrive at Eq. (11) for ∆ ρ . Note that∆ ρ reverses sign with q (cid:107) , contrary to ρ : Like the conven-tional Faraday effect [52, 53], the kFE is nonreciprocal.The final step is to determine the refraction index n ⊥ appearing in Eq. (11). For that purpose, we evaluate thequantity Im σ ⊥ ( ω ) in Eq. (C7) usingIm σ ⊥ ( ω ) = − e (cid:126) (cid:90) [ d k ] (cid:88) (cid:48) nm f ( E k n ) [1 − f ( E k m )] × ω k mn ω k mn − ω (cid:0) | A x k nm | + | A y k nm | (cid:1) , (C22)where the prime on the summation indicates that theterm m = n is excluded. This expression gives the inter-band contribution to Im σ ⊥ ( ω ). Since at the CO laserfrequency we have ωτ (cid:29) Appendix D: Kinetic magnetoelectric effect
The kME effect in a conducting gyrotropic crystal isdescribed phenomenologically by [20] j B a ( ω ) = iωα ab ( ω ) B b ( ω ) , (D1a) M a ( ω ) = α ba ( ω ) E b ( ω ) . (D1b)In the limit ωτ (cid:28) α ab ( ω ) becomes real we have j B a ( t ) = − α ab (0) ˙ B b ( t ) , (D2a) M a ( t ) = α ba (0) E b ( t ) , (D2b)which for an isotropic gyrotropic medium ( α ab = αδ ab )reduces to Eqs. (1) and (3) of Ref. [15].It is convenient to introduce a reduced (dimensionless)magnetoelectric tensor α r ab ( ω ) = cµ α ab ( ω ) , (D3)in direct analogy with the standard description of magne-toelectric couplings in insulators [57]. The intrinsic part is given in terms of Eqs. (2) and (A5) by α r ab ( ω ) = − π a τ ω e K ab . (D4)It can be verified that at ω = 0 this expression agreeswith that obtained in Ref. [19] for the magnetization in-duced by a static E field. Specializing to Te and followingAppendix B to recast the result in terms of j (cid:107) , we obtain M (cid:107) = ( e/ π a )( α r (cid:107) (0) /τ )( j (cid:107) /C (cid:107) ) , (D5)which combined with Eq. (D4) becomes Eq. (13). Appendix E: Wannier interpolation
In order to interpolate in k space the energy bandsand other quantities (see below), we use the formalismof maximally-localized Wannier functions [58, 59], as im-plemented in the Wannier90 code package [45, 46]. Weconstruct four disentangled Wannier functions per tel-lurium atom and per spin channel, for a total of 24 Wan-nier functions per cell. The 5 s and 5 p bands of trigonalTe are well separated from the lower d states, and theycross with higher-lying sates only in a small region of theBrillouin zone. Thus we set the outer energy window forthe disentanglement procedure [59] from -20 to +5 eVrelative to the valence-band maximum, so as to cover all s and p bands. The inner frozen window spans the rangefrom -20 to +2.5 eV, and we choose atom-centered sp -type trial orbitals for the initial projections. This choiceof Wannier functions differs from that of Ref. [24, 27],where only 5 p states were included in the wannierization.The Wannier basis is also used to evaluate the k spacequantities entering the expressions for the response ten-sors, namely: the band gradient ∇ k E k n , the Berrycurvature Ω k n [Eq. (1b)], the intrinsic orbital moment[Eq. (4)], and the off-diagonal elements of the Berry con-nection matrix A k nm [Eq. (C13)]. The Wannier interpo-lation of these quantities is described in Refs. [43, 56, 60].When evaluating the response tensors, the integrationsover the Brillouin zone are performed using a uniformgrid of 200 × × k points. In the case of responsesthat can be expressed in the form of Eq. (8), when ε is close to the band gap (no further than 100 meV fromthe band edges), only k points in the vicinity of H and H’contribute, due to the factor ( − ∂f /∂E ) in that equation.In such cases, we use a grid of 200 × × k pointswithin a small box centered at H that amounts to lessthan 0.2% of the entire Brillouin zone, and then multiplythe result by two in order to account for H’. This allowsus to increase the numerical accuracy for ε near the bandgap, which is the energy range that contributes most theresponse. [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, “Anomalous Hall effect,” Rev. Mod. Phys. , 1539 (2010).[2] D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. , 1959–2007(2010).[3] N. B. Baranova, Yu. V. Boddanov, and B. Ya.Zel’Dovich, “Electrical analog of the Faraday effect andother new optical effects in liquids,” Optics Commun. ,243 (1977).[4] E. L. Vorob’ev, E. L. Ivchenko, G. E. Pikus, I. I. Farb-shtein, V. A. Shalygin, and A. V. Shturbin, “Opticalactivity in tellurium induced by a current,” JETP Lett. , 441 (1979).[5] V. A. Shalygin, A. N. Sofronov, E. L. Vorob’ev, and I. I.Farbshtein, “Current-Induced Spin Polarization of Holesin Tellurium,” Phys. Solid State , 2362 (2012).[6] E. L. Ivchenko and G. E. Pikus, “New photogalvanic ef-fect in gyrotropic crystals,” JETP Lett. , 604 (1978).[7] V. I. Belinicher and B. I. Sturman, “The photogalvaniceffect in media lacking a center of inversion,” Sov. Phys.Usp. , 199 (1980).[8] V. M. Asnin, A. A. Bakun, A. M. Danishevskii, E. L.Ivchenko, G. E. Pikus, and A. A. Rogachev, “Observa-tion of a photo-emf that depends on the sign of the circu-lar polarization of the light,” JETP Lett. , 74 (1978).[9] V. M. Asnin, A. A. Bakun, A. M. Danishevskii, G. E.Pikus, and A. A. Rogachev, ““Circular” photogalvaniceffect in optically active crystals,” Solid State Commun. , 565 (1979).[10] B. I. Sturman and V. M. Fridkin, The Photovoltaic andPhotorefractive Effects in Noncentrosymmetric Materials (Gordon and Breach, Philadelphia, 1992).[11] E. L. Ivchenko and G. E. Pikus,
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