Quantized circular photogalvanic effect in Weyl semimetals

# Quantized circular photogalvanic effect in Weyl semimetals

Fernando de Juan Department of Physics, University of California, Berkeley, CA 94720, USA Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), 28049 Madrid, Spain Rudolf Peierls Centre for Theoretical Physics, Oxford, 1 Keble Road, OX1 3NP, United Kingdom    Adolfo G. Grushin    Takahiro Morimoto Department of Physics, University of California, Berkeley, CA 94720, USA    Joel E. Moore Department of Physics, University of California, Berkeley, CA 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
July 20, 2019
###### Abstract

The circular photogalvanic effect (CPGE) is the part of a photocurrent that switches depending on the sense of circular polarization of the incident light. It has been consistently observed in systems without inversion symmetry and depends on non-universal material details. Here we find that in a class of Weyl semimetals (e.g. SrSi) and three-dimensional Rashba materials (e.g. doped Te) without inversion and mirror symmetries, the injection contribution to the CPGE trace is effectively quantized in terms of the fundamental constants and with no material-dependent parameters. This is so because the CPGE directly measures the topological charge of Weyl points, and non-quantized corrections from disorder and additional bands can be small over a significant range of incident frequencies. Moreover, the magnitude of the CPGE induced by a Weyl node is relatively large, which enables the direct detection of the monopole charge with current techniques.

Introduction

When the Fermi surface of a solid is close to a linear crossing of two bands, the low-energy quasiparticles are relativistic Weyl fermions Murakami (2007); Wan et al. (2011); Burkov and Balents (2011). This linear crossing, known as a Weyl node, is protected from becoming gapped because it carries a monopole source of Berry curvature, which leads to many unique experimental consequences. Materials with this band structure have recently been predicted Weng et al. (2015); Huang et al. (2015a) and discovered Xu et al. (2015a); Huang et al. (2015b); Lv et al. (2015); Yang et al. (2015), primarily through observation in angle-resolved photoemission of an unusual surface state known as a Fermi arc. However, so far it has been challenging to find truly quantized signatures induced by the existence of the monopole, which would be analogous to the quantum Hall effect in two-dimensional systems or the half-integer Hall effect at topological insulator surfaces. The principle that quantized effects can exist in metallic systems is demonstrated by graphene, where for a broad range of frequencies the transmission of incident light is , where is the fine structure constant Nair et al. (2008).

A feature of Weyl fermions examined recently as a potentially quantized linear response is the anomaly induced chiral magnetic effect Fukushima et al. (2008); Aji (2012); Zyuzin et al. (2012); Grushin (2012); Goswami and Tewari (2013): the generation of a current by an applied magnetic field. While it is now clear that there is no equilibrium current Vazifeh and Franz (2013), a finite current is possible in the transport limit with the frequency after the transferred momentum  Chen et al. (2013); Chang and Yang (2015); Goswami et al. (2015). This current has the same origin as natural optical activity Zhong et al. (2016); Ma and Pesin (2015). It is determined by orbital moments rather than the chiral anomaly and its magnitude depends on a non-universal material-dependent property: the energy splitting between Weyl points. Other potential probes of the chiral anomaly are nonlinear responses to both and  Son and Spivak (2013); Parameswaran et al. (2014); Cortijo (2016), which present a characteristic angular dependence measurable in magnetotransport experiments. Current measurements show a strong angular dependence  Xiong et al. (2015), but the direct relation to the chiral anomaly is subtle Burkov and Kim (2016). Other promising scattering proposals could access distinct Weyl node properties Yu et al. (2016); Kourtis (2016).

The main finding of this paper is that in a Weyl semimetal where nodes of opposite chirality lie at different energies, the circular photogalvanic effect (CPGE) becomes a truly quantized response that depends only on fundamental constants and the monopole charge of a Weyl node. The CPGE is the part of a DC photocurrent that switches with the sense of circular polarization. It has been measured in a variety of conventional semiconductors Olbrich et al. (2009); Yuan et al. (2014) and more recently in topological insulators McIver et al. (2012); Okada et al. (2016). The typical magnitude of the CPGE at low frequency corresponds to an observed switchable photocurrent pA for incident intensity of W over a cm-sized sample in quantum wells that have time-reversal symmetry but low spatial symmetry Olbrich et al. (2009). It has been obtained theoretically as a Berry phase effect Moore and Orenstein (2010); Deyo et al. (2009); Sodemann and Fu (2015), possibly the first in nonlinear optics, but there is no quantization: the effect measures the strength of the leading allowed Berry curvature term, which in three-dimensional (3D) materials Sodemann and Fu (2015) can be viewed as the dipole moment of Berry curvature.

In contrast, we find that that the CPGE induced current for a Weyl point is quantized and given by

 12[dj↻dt−dj↺dt]=2πe3h2cϵ0ICi=4παehICi, (1)

where is the fine structure constant defined above, is the integer-valued topological charge of Weyl point and is the applied intensity. In this equation, the currents for left and right circular polarization are perpendicular to the polarization plane, and summed over three mutually orthogonal planes. While the quantization we find is not expected to be exponentially protected as in gapped systems like in the quantum Hall effect, it is robust under small material changes in the sense that it is a direct measurement of the monopole charge in units of fundamental constants, as opposed to the transparency of graphene which enjoys no such interpretation.

Eq. (1) describes a current whose increase in time is proportional to intensity, known as an injection current Sipe and Shkrebtii (2000). It is generated by resonant transitions at frequency between the occupied valence band and the unoccupied conduction band of the Weyl node (see Fig. 1). It contrasts previous finite frequency proposals Moore and Orenstein (2010); Deyo et al. (2009); Sodemann and Fu (2015); Taguchi et al. (2016) that originate in the low frequency response of electronic states near the Fermi level or other high-frequency Taguchi et al. (2016) and interband phenomena Ishizuka et al. (2016) where the CPGE is not quantized. Interestingly, a CPGE was predicted for tilted Weyl nodes that lie at the same energy Chan et al. (2017) but this effect is not quantized.

In a real material, the total Weyl node charge in the Brillouin zone must be zero Nielsen and Ninomiya (1981). Crucially, this does not preclude the observation of a finite CPGE: Weyl nodes of opposite chirality need not be at the same energy in a low-symmetry material and resonant transitions for a given node can be Pauli blocked, rendering it inactive (see Fig. 1). In this case, the response is constant and quantized for a finite range of frequencies. In addition, the key fact for experimental observability is that the prefactor of Eq. (1) is large in comparison to ordinary CPGE magnitudes. For typical relaxation times, the quantized Weyl node contribution will dominate other metallic or insulating contributions Moore and Orenstein (2010); Deyo et al. (2009); Sodemann and Fu (2015) by more than an order of magnitude, suggesting that the total CPGE observed in experiment will indeed reveal the quantization. In what follows we analytically derive the quantized response Eq. (1) for two-band models and then consider corrections including those arising from additional bands. We provide supporting numerical evidence and suggest candidate materials as well as ideas for detection.

Results

The circular photogalvanic effect (CPGE) - In materials with time reversal symmetry, an injection current can only be produced by circularly polarized light. The CPGE injection current is defined as the second order response

 djidt=βij(ω)[E(ω)×E∗(ω)]j, (2)

to an electric field , where latin indices span the cartesian components . The tensor is purely imaginary and only non-zero if inversion is broken and the material belongs to one of the gyrotropic point groups 111The gyrotropic point groups are (ferroelectrics) and and (see  Belinicher and Sturman (1980)). Of them, the subset of enantiomorphic (or chiral) groups and are mirror-free and can support a quantized CPGE. The only other mirror-free group has an improper rotation which constrains the CPGE trace to be zero. The presence of at least one mirror symmetry constrains all the diagonal components to be zero, while the off-diagonal ones can be finite and give a non-quantized CPGE, as in Ref. Chan et al. (2017). Mirror-symmetry constrains the Weyl node at momentum to have the same energy as its partner of opposite chirality at , and hence the effect must vanish. The key for the quantized response to be observed is therefore that inversion and all mirror symmetries are broken, as in enantiomorphic crystals, allowing for the nodes to occur at different energies. In this case the trace of is quantized for a finite range of frequencies as we proceed to show.

The CPGE tensor can be written in general as Sipe and Shkrebtii (2000)

 βij(ω) = πe3ℏVϵjkl∑k,n,mfknmΔik,nmrkk,nmrlk,mnδ(ℏω−Ek,mn),

where is the sample volume, and are the difference between band energies and Fermi-Dirac distributions respectively, is the cross gap Berry connection and .

Exact quantization of the CPGE for two-band Weyl semi-metal models - The position operator matrix elements in Eq. (Quantized circular photogalvanic effect in Weyl semimetals) can be related to Berry curvatures with the general expression Sipe and Shkrebtii (2000)

 Ωck,n=iϵabc∑m≠nrak,nmrbk,mn, (4)

where is the Berry curvature of band . For a model with only two bands, this relation allows us to write

 βij(ω) = iπe3ℏ2V∑k∂kiEk,12Ωjkδ(ℏω−Ek,21), (5)

where 1,2 correspond to valence and conduction bands, is assumed, and . At a given frequency , the delta function selects the surface in k-space where . Since by definition is normal to this surface, the trace of can be written as (see Methods)

 Tr[β(ω)]=ie32h2∮SdS⋅Ω, (6)

where denotes the oriented surface element normal to . Thus the CPGE trace measures the Berry flux penetrating through . In particular, when the surface surrounds a Weyl node (e.g. located at , see Fig. 1), the above formula reduces to the monopole charge of the Weyl node, yielding a quantized CPGE

 (7)

where is the monopole charge of the Weyl node at . In terms of the applied intensity the quantization is given by Eq. (1) as anticipated. For , the second node contributes with opposite sign to and quantization is generically lost. Thus, in the ideal case of two linear Weyl nodes at energy from the chemical potential the quantization holds as long as and . For isotropic Weyl fermions (i.e. linear dispersion with isotropic Fermi velocity and no tilting), measuring only one component of CPGE already suffices since .

To support these findings we have numerically calculated the injection current for a two-band model with a characteristic energy scale (see Methods). Our results are summarized in Fig. 2. Panel a) shows the band structure for representative parameters as a function of the momentum along separating the Weyl nodes (). The dashed lines outline four different chemical potentials for which the injection current is calculated using Eq. (Quantized circular photogalvanic effect in Weyl semimetals) and shown in panel b). Consistent with our discussion, when the chemical potential is chosen such that the CPGE is zero (orange flat-line). When coincides with the right-most node (blue dashed line) and the CPGE is quantized to from . Note that although in the idealized Weyl semimetal model quantization is expected to hold up to , in a lattice model can exceed the band width. This is the case of all non-trivial cases in Fig. 2 and thus the quantization disappears at a frequency . With this caveat, for all generic choices of parameters the CPGE is numerically quantized consistent with our analytics.

Higher band corrections - In practice, corrections from higher bands can lead to a non-universal CPGE since the CPGE can only be written exactly as a Berry curvature flux for two-band models.

To quantify the importance of these corrections consider a three band model with two lower bands forming the Weyl nodes as above that are complemented by a third unoccupied band. Using that it is possible to rewrite Eq. (Quantized circular photogalvanic effect in Weyl semimetals) as for small (see Methods). These corrections become arbitrarily small when for because remains a non-singular function for any pair of bands, while is arbitrarily small for the two bands at resonance forming the Weyl node. Explicitly, the correction scales as

 δβ(ω)∝|ivk,13×vk,13|v2Fω2E213,ω/t≪1, (8)

where is a characteristic Fermi velocity around a Weyl node and is the typical energy difference between the occupied first band and the unoccupied third band. The corrections vanish as and are inversely proportional to the energy separation to higher bands, thus becoming unimportant at low enough frequencies. Note as well that the matrix elements in Eq. (8) are typically small for different orbitals rendering the departure from quantization even less observable in practice.

We have assessed numerically the effect of higher bands on quantization by calculating the CPGE of a generic four-band model Vazifeh and Franz (2013). This model can describe Weyl semimetals with nodes at different energies such as SrSi Huang et al. (2016) (see Fig. 3 top left) relevant for our purposes and, with straightforward modifications, Dirac semimetals. It can also describe materials where the band edge takes the form of a 3D Rashba-like Hamiltonian Anderson et al. (2012) . This is the natural spin-orbit splitting of parabolic bands in the absence of inversion and mirror symmetries. It generates a single Weyl node near the band edge, and the node of opposite chirality naturally appears at significantly different energies (Fig. 3c). Since the contribution of the outer Fermi surface to the CPGE is expected to be zero, a 3D Rashba material can show a quantized CPGE, and concrete examples are discussed below.

In Fig. 3 we show the injection current for two representative band structures: a Weyl semimetal with broken inversion symmetry and a 3D Rashba material. In all plots coincides with a Weyl node; other choices behave qualitatively as described in Fig. 2. Despite the presence of higher bands, the quantization is robust for small frequencies for all studied cases. We find this encouraging for experiments from both our numerical and analytic analysis that predict that the corrections due to higher bands are expected to be small.

Discussion

We now elaborate on the main practical aspects that suggest that the quantized CPGE can be observed with current experiments. In the absence of any scattering mechanism, Eq. (2) predicts a quantized rate of unbounded current growth. In practice, disorder will introduce a finite scattering rate and the linear growth of current can only be observed for times , which in existing Weyl semimetals is 1 ps Parameswaran et al. (2014); Behrends et al. (2016). In the limit of , the current saturates to that can be computed from Fermi’s golden rule or with Floquet theory (see Methods), resulting in

 jsati=τβij(ω)(E(ω)×E∗(ω))j, (9)

with defined as above. The total stationary photocurrent will therefore be , with the universal coefficient . Note this assumes the intensity should remain constant throughout the sample, so light absorption should be small. The attenuation depth of a Weyl semimetal scales as , and we estimate that for THz, (See Methods). Absorption is thus negligible for typical thin film dimensions. Taking a thickness of 10nm, an area of 1mm, and an irradiation time of 1 ps ( ), the induced photocurrent reaches . This is much larger than the reported CPGE current of 10-100 pA for a topological insulator thin film Okada et al. (2016), and thus its measurement is experimentally feasible. The additional Fermi surface contribution that can be described semiclassically Moore and Orenstein (2010) is estimated to be much smaller so that the quantized CPGE contribution is dominant. Surface corrections can also be neglected at normal incidence, since the expected current is normal to the surface, and are in any case small compared to the bulk CPGE we describe.

Experimentally, the rate of current injection can be extracted from an all-optical setup with no free parameters as in Ref. Bas et al., 2016. There, direct time-resolved measurements of photocurrent are possible using short pulses of intense light. The time-dependent photocurrent can be measured as a radiated signal of low frequency set by the envelope of the incident pulses. Alternatively, if only the simpler steady-state measurement is available, the relaxation time can be estimated from other measurements such as the broadening of the drop at in the linear optical conductivity or from the CPGE itself by measuring the width of the jump at . The measured value of could be divided into the photocurrent to get the universal CPGE quantum.

Observing quantization requires a Weyl semimetal where inversion and all mirror symmetries are absent. The recently realized inversion breaking Weyl semimetals in the monopnictide class, such as TaAs Xu et al. (2015a); Huang et al. (2015b), do have a mirror plane in their structure. Shear strain for example can break this symmetry, opening a small window of frequencies to observe the effect. A better candidate is SrSi Huang et al. (2016): all mirror symmetries are broken, the Weyl nodes of opposite chiralities are separated significantly in energy ( eV) and the chemical potential is close to one of the Weyl nodes. Other material candidates for mirror-free Weyl semimetals have been recently predicted in Ref. Chang et al. (2016). As we have shown, the quantized CPGE can also be observed in 3D Rashba materials Anderson et al. (2012) as in the conduction band of trigonal elemental Te Hirayama et al. (2015). Note that BiTeI Ishizaka et al. (2011) does not have a 3D Rashba band structure despite its strong spin-orbit splitting because of its mirror symmetries, which could also be broken by strain. Synthetic 3D Rashba materials can be also engineered in cold atoms Anderson et al. (2012) which can be driven periodically to study the effects presented here.

The quantization of the CPGE is not limited to linear nodes. It can occur for any node with as long as the node is formed only by two bands Volovik and Konyshev (1988); Fang et al. (2012) and Eq. 5 applies, as it happens in SrSi where . Nodal crossings with three or more bands Bradlyn et al. (2016), however, are not expected to display quantization of this type as the corrections from Eq. (8) cannot be made small. We also note that the quantized value of the CPGE response is independent of any tilting of the nodes as long as they remain of type I, but the frequency window to observe it will depend on the tilt parameter Carbotte (2016). If the tilting is strong enough to create a type II node Xu et al. (2015b); Soluyanov et al. (2015), the surface of allowed transitions encompasses only a fraction of the sphere surrounding the node and the quantization is lost at all frequencies. We also note that, unlike optical gyrotropy which is non-quantized and allowed for any metal with broken inversion, the quantized CPGE can occur only in the presence of Weyl nodes.

In conclusion, we have shown that the trace of the circular photogalvanic tensor is quantized for Weyl semimetals and 3D Rashba materials that break inversion and all mirror symmetries. We have identified several candidate materials to observe this effect, which we estimate to be an order of magnitude larger compared to other more conventional contributions.

Methods

Analytical computation of CPGE coefficient - The CPGE tensor for a two band model is given by Eq. (5) in the main text. The trace of this tensor is

 Tr[β(ω)]=iπe3ℏ2V∑k∂kiEk,12Ωikδ(ℏω−Ek,12). (10)

To perform the integral we use that for an isotropic Weyl node where and therefore , where , and that Berry curvature of such Weyl node is given by . We then get

 Tr[β(ω)] =e3πℏ2i∫dΩ(2π)3∫k2dk2vFkik12kik3δ(k−k(ω))2vF =ie3πℏ24π2(2π)3=ie3πh2. (11)

To relate this response coefficient to the applied intensity, we consider circularly polarized light for which with a unit vector normal to the polarization plane. For the plane, for example, we have and ). From Eq. (2) the injection current induced in the direction is given by

 ∂tjz=βzzi|E|2. (12)

To get the trace, we add up the contributions from the three orthogonal directions, defining , and use

 ∂tj↺=e3πh2|E2|=e32πh2cϵ0I=4παehI, (13)

in terms of the fine structure constant . The saturation current density with finite lifetime is simply

 jsat↺=4παehτI=22.17τpsIW/cm2Acm2. (14)

Finally, note this quantity is by construction the one that reverses sign when circular polarization is reversed. In practice other contributions that do not change sign exist in addition to the quantized CPGE. These can be removed by taking or as in the main text.

Absorption and attenuation length - When light is irradiated in any conducting material, the intensity decays exponentially from the surface due to light absorption. The attenuation constant is expressed in terms of the dielectric function as

 α=ω/c√2(−Re[ϵ]+|ϵ|) (15)

which is related to conductivity by . The conductivity of a Weyl semimetal for is given by Ashby and Carbotte (2014), which gives an attenuation length

 δ=λ12π√2(−1+√1+(αc6vF)2)∼0.23λ (16)

where we have used a typical Fermi velocity m/s Behrends et al. (2016). For frequencies below meV ( THz), we have and so absorption is negligible for thin films in the THz range.

Higher band corrections - In this section we discuss how the quantization is modified by the presence of higher bands. We consider the case of three bands: Bands 1 and 2 host the Weyl nodes while we choose band 3 to be higher in energy and unoccupied. The CPGE coefficient is explicitly

 Tr[β]=πe3ℏV∑kϵabc[Δak,12rbk,12rck,21δ(ℏω−Ek,12) +Δak,13rbk,13rck,31δ(ℏω−Ek,13)]. (17)

If we assume that the probing frequencies are always smaller than then will never contribute and can be discarded. For this case

 Tr[β]=πe3ℏV∑kϵabcΔak,12rbk,12rck,21δ(ℏω−Ek,12) (18)

The existence of an extra band now modifies the sum rules to

 Ωck,1=iϵabc(rak,12rb21+rak,13rbk,31), (19) Ωck,2=iϵabc(rak,21rbk,12+rak,23rbk,32), (20)

we may write

 Tr[β]=−πe3ℏV∑kΔak,12(iΩa1+ϵabcrbk,13rck,31)δ(ℏω−Ek,12) (21)

Using that with the quantization will be preserved if for every direction . To make a quantitative estimate we take an isotropic Weyl node with and . The correction to the quantized value can be estimated as the dimensionless ratio of the moduli of the two vectors inside the parenthesis

 δβ∝|ivk,13×vk,13|E2k,13/|k|2. (22)

Assuming zero chemical potential and small probing frequency and using that around the Weyl node we have we obtain

 δβ∝|ivk,13×vk,13|v2Fω2E213. (23)

Therefore at low frequencies, the corrections to the quantization of vanish quadratically, since is a derivative of the Hamiltonian and cannot be singular at the node.

Lattice models - In the main text we have used a two and a four band lattice model which we describe here with more detail. The two band model is defined by with

 dk = −{tsinkx,tsinky,−M+t∑i=x,y,zcoski}, (24) εk = γsin(kz), (25)

where is the identity matrix and the Pauli matrices. For it has a pair of Weyl cones at with at energies . The band structure shown in Fig. 2 corresponds to for and . The chemical potential can be controlled by adding a constant term proportional to . Note that both inversion and time reversal symmetry are broken in this model. By doubling the model one can restore time reversal symmetry while still being inversion odd. Since multiple copies of the model defined by Eqs. (24) and (25) will only result in an additional prefactor in Eq. (1) proportional to the number of optically active Weyl nodes, in the main text we use the model defined by Eq. (24) and Eq. (25).

The second model that we use to investigate the effect of higher bands is a four band model that can originate from an orbital degree of freedom and spin . The Hamiltonian in the basis defined by the electron operator is the sum of three terms Vazifeh and Franz (2013)

 H4b=∑k,jDj(k)c†kΓjck+bjc†kΓbΓjck+b0c†kΓbck. (26)

For a detailed discussion of the phase diagram of this model we refer the reader to Refs. Vazifeh and Franz (2013); Grushin et al. (2015). Here we will highlight the aspects that are relevant to the calculation in the main text.

The first term describes in general a trivial or topological (either weak or strong) topological insulator and respects both time-reversal symmetry () and inversion symmetry (). It is defined through the -matrices and

 Dj(k)=−(tsinkx,tsinky,tsinkz,t∑icoski−M), (27)

in the subspace where the Pauli matrices and act on orbital and spin degrees of freedom respectively and and are identity matrices in the corresponding subspace. The transition between trivial or topological insulator phases is governed by . In the main text we have set , such that at the insulating state corresponds to a strong topological insulator.

The second term defined via the matrix and constant vector breaks and drives de transition to a Weyl semimetal. The absolute value of controls the distance between Weyl nodes. In the main text we have set for Fig. 3 top row and b=(0,0,0) in Fig. 3 bottom row.

The last term is defined via the constant scalar . It breaks and separates the two Weyl nodes in energy. In all of Fig. 3 we set . As for the two-band model the chemical potential is controlled by adding a constant term proportional to and time reversal symmetry can be restored by an appropriate doubling of the model.

Floquet theory derivation for CPGE - In this section, we present an alternative derivation of the stationary photocurrent proportional to the relaxation time by using the Floquet theory. We again consider the two band model for the Weyl fermion defined by , where energy dispersions of the valence and conduction bands are given by and , respectively. In order to study dc current induced by photoexcitation between the two bands, we study the Floquet two band model consisting of the one photon dressed valence band and the bare conduction band, which is given byMorimoto and Nagaosa (2016)

 HF =(Ek,1+ℏω−ieA∗[vxk,12−ivyk,12]ieA[vxk,21+ivyk,21]Ek,2) ≡d0+d⋅σ, (28)

with and the velocity matrix element . These Floquet bands show anticrossing at the optical resonance at which describe steady state under driving. The occupation of the Floquet bands is determined by coupling to a heat bath which we assume to have the Fermi energy between the valence and conduction bands. This enables us to compute steady dc current by using Keldysh Green’s function method. Namely, by using the current operator along the direction in the Floquet formalism,

 ~vz =1ℏ∂HF∂kz≡b0+b⋅σ, (29)

the dc current in the steady state is given byMorimoto and Nagaosa (2016)

 J =e∫dk(2π)3(j1+j2+j3), (30)

with

 j1 =Γ2(−dxby+dybx)d2+Γ24, (31) j2 =(dxbx+dyby)dzd2+Γ24, (32) j3 =(d2z+Γ24)bzd2+Γ24+b0. (33)

The term describes the shift current in the case of linearly polarized light. The term does not lead to the current response proportional to relaxation time; while the factor result in the factor with the resonant wave number , this contribution vanishes after -integration. The term gives the injection current if we consider the term proportional to . In the following, we focus on the term and derive the Berry curvature formula for the injection current. The injection current is obtained by expanding the term up to as

 Jinj =−e3|A|2∫dk(2π)3|vxk,12−ivyk,12|2d2z+Γ24bz. (34)

We note that the term in vanishes after -integral due to the band connectivity. By noticing

 |vxk,12−ivyk,12|2=|vxk,12|2+|vyk,12|2−2Im[vxk,12vyk,21], (35)

we can write

 Jinj =−2πτe3|A|2ℏ∫dk(2π)3[vzk,11−vzk,22]δ(dz) ×[|vxk,12|2+|vyk,12|2−2Im[vxk,12vyk,21]], (36)

where is the relaxation time and . Since the integrand of the first term is odd under the time-reversal symmetry, the first term vanishes after -integration. The second term is described by the Berry curvature by using the identity

 Ωzk=−2Im[vxk,12vyk,21](Ek,1−Ek,2)2. (37)

Thus we obtain the injection current in time reversal-symmetric systems as

 Jinj =−2πe3τ|E|2ℏ∫dk(2π)3[vzk,11−vzk,22]Ωzkδ(Ek,12+ℏω). (38)

Alternatively, since the Berry curvatures for the valence and conduction bands satisfy the relation , this can be rewritten as

 Jinj =−e3τ|E|2h2∫dk[∂(Ek,1−Ek,2)∂kzΩzk]δ(Ek,12+ℏω). (39)

Namely, the nonlinear coefficient (in ) is given by

 βzz =ie32h2∫dk[∂(Ek,1−Ek,2)∂kzΩzk]δ(Ek,12+ℏω). (40)

Thus the sum of the nonlinear conductivities is described by the Berry flux over the surface of the resonance condition in -space as

 Tr[β] =ie32h2∫SdS⋅Ωk, (41)

where denotes the oriented surface element normal to . When the surface surrounds a Weyl point, this leads to quantized injection current as

 Tr[β] =iπe3h2. (42)

Acknowledgements - We acknowledge useful discussions with F. Flicker, B. Fregoso, K. Landsteiner, N. Nagaosa, T. Neupert, J. Orenstein, G. Refael, I. Souza and M. A. H. Vozmediano. This work was supported by the Marie Curie Programme under EC Grant agreements No. 653846 (AGG) and No. 705968 (FJ), by the European Research Council through Grant No. 290846 (FJ), by the Gordon and Betty Moore Foundation’s EPiQS Initiative Theory Center Grant (TM), and NSF DMR-1507141 and a Simons Investigatorship (JEM).

Author contributions - All authors contributed equally to the results presented in this work and the writing of the manuscript.

Data availability - The data that support the findings of this study are available from the corresponding author upon request.

Competing financial interests - The authors declare no competing financial interests.

Correspondence - Correspondence and requests for materials should be addressed to F.J. (email: fernando.dejuan@physics.ox.ac.uk)

## References

• Murakami (2007) Shuichi Murakami, “Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase,” New J. Phys. 9, 356 (2007).
• Wan et al. (2011) Xiangang Wan, Ari M. Turner, Ashvin Vishwanath,  and Sergey Y. Savrasov, “Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates,” Phys. Rev. B 83, 205101 (2011).
• Burkov and Balents (2011) A A Burkov and Leon Balents, “Weyl Semimetal in a Topological Insulator Multilayer,” Phys. Rev. Lett. 107, 127205 (2011).
• Weng et al. (2015) Hongming Weng, Chen Fang, Zhong Fang, B. Andrei Bernevig,  and Xi Dai, “Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides,” Phys. Rev. X 5, 011029 (2015).
• Huang et al. (2015a) Shin-Ming Huang et al., “A weyl fermion semimetal with surface fermi arcs in the transition metal monopnictide taas class,” Nat. Comm. 6, 7373 (2015a).
• Xu et al. (2015a) Su-Yang Xu et al., “Discovery of a weyl fermion semimetal and topological fermi arcs,” Science 349, 613–617 (2015a).
• Huang et al. (2015b) Xiaochun Huang et al., “Observation of the chiral-anomaly-induced negative magnetoresistance in 3d weyl semimetal taas,” Phys. Rev. X 5, 031023 (2015b).
• Lv et al. (2015) B. Q. Lv et al., “Experimental discovery of weyl semimetal taas,” Phys. Rev. X 5, 031013 (2015).
• Yang et al. (2015) LX Yang et al., “Weyl semimetal phase in the non-centrosymmetric compound taas,” Nat. Phys. 11, 728–732 (2015).
• Nair et al. (2008) RR Nair et al., “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
• Fukushima et al. (2008) Kenji Fukushima, Dmitri E. Kharzeev,  and Harmen J. Warringa, “Chiral magnetic effect,” Phys. Rev. D 78, 074033 (2008).
• Aji (2012) Vivek Aji, “Adler-bell-jackiw anomaly in weyl semimetals: Application to pyrochlore iridates,” Phys. Rev. B 85, 241101 (2012).
• Zyuzin et al. (2012) A. A. Zyuzin, Si Wu,  and A. A. Burkov, “Weyl semimetal with broken time reversal and inversion symmetries,” Phys. Rev. B 85, 165110 (2012).
• Grushin (2012) Adolfo G. Grushin, ‘‘Consequences of a condensed matter realization of lorentz-violating qed in weyl semi-metals,” Phys. Rev. D 86, 045001 (2012).
• Goswami and Tewari (2013) Pallab Goswami and Sumanta Tewari, “Axionic field theory of -dimensional weyl semimetals,” Phys. Rev. B 88, 245107 (2013).
• Vazifeh and Franz (2013) M. M. Vazifeh and M. Franz, “Electromagnetic Response of Weyl Semimetals,” Phys. Rev. Lett. 111, 027201 (2013).
• Chen et al. (2013) Y. Chen, Si Wu,  and A. A. Burkov, “Axion response in weyl semimetals,” Phys. Rev. B 88, 125105 (2013).
• Chang and Yang (2015) Ming-Che Chang and Min-Fong Yang, “Chiral magnetic effect in a two-band lattice model of weyl semimetal,” Phys. Rev. B 91, 115203 (2015).
• Goswami et al. (2015) Pallab Goswami, Girish Sharma,  and Sumanta Tewari, “Optical activity as a test for dynamic chiral magnetic effect of weyl semimetals,” Phys. Rev. B 92, 161110 (2015).
• Zhong et al. (2016) Shudan Zhong, Joel E. Moore,  and Ivo Souza, “Gyrotropic magnetic effect and the magnetic moment on the fermi surface,” Phys. Rev. Lett. 116, 077201 (2016).
• Ma and Pesin (2015) Jing Ma and D. A. Pesin, “Chiral magnetic effect and natural optical activity in metals with or without weyl points,” Phys. Rev. B 92, 235205 (2015).
• Son and Spivak (2013) D. T. Son and B. Z. Spivak, “Chiral anomaly and classical negative magnetoresistance of weyl metals,” Phys. Rev. B 88 (2013).
• Parameswaran et al. (2014) S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin,  and A. Vishwanath, “Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals,” Phys. Rev. X 4, 031035 (2014).
• Cortijo (2016) Alberto Cortijo, ‘‘Magnetic-field-induced nonlinear optical responses in inversion symmetric dirac semimetals,” Phys. Rev. B 94, 235123 (2016).
• Xiong et al. (2015) J. Xiong et al., “Evidence for the chiral anomaly in the dirac semimetal na3bi,” Science 350, 413–416 (2015).
• Burkov and Kim (2016) Anton A. Burkov and Yong Baek Kim, “ and chiral anomalies in topological dirac semimetals,” Phys. Rev. Lett. 117, 136602 (2016).
• Yu et al. (2016) Rui Yu, Hongming Weng, Zhong Fang, Hong Ding,  and Xi Dai, “Determining the chirality of weyl fermions from circular dichroism spectra in time-dependent angle-resolved photoemission,” Phys. Rev. B 93, 205133 (2016).
• Kourtis (2016) Stefanos Kourtis, “Bulk spectroscopic measurement of the topological charge of weyl nodes with resonant x rays,” Phys. Rev. B 94, 125132 (2016).
• Olbrich et al. (2009) P. Olbrich et al., “Observation of the orbital circular photogalvanic effect,” Phys. Rev. B 79, 121302 (2009).
• Yuan et al. (2014) Hongtao Yuan et al., “Generation and electric control of spin–valley-coupled circular photogalvanic current in wse2,” Nature Nanotech. 9, 851–857 (2014).
• McIver et al. (2012) JW McIver, D Hsieh, H Steinberg, P Jarillo-Herrero,  and N Gedik, “Control over topological insulator photocurrents with light polarization,” Nature Nanotech. 7, 96–100 (2012).
• Okada et al. (2016) Ken N. Okada et al., “Enhanced photogalvanic current in topological insulators via fermi energy tuning,” Phys. Rev. B 93, 081403 (2016).
• Moore and Orenstein (2010) J. E. Moore and J. Orenstein, “Confinement-induced berry phase and helicity- dependent photocurrents,” Phys. Rev. Lett. 105, 026805 (2010).
• Deyo et al. (2009) E. Deyo, L. E. Golub, E. L. Ivchenko,  and B. Spivak, “Semiclassical theory of the photogalvanic effect in non-centrosymmetric systems,” Preprint at http://arxiv.org/abs/0904.1917  (2009).
• Sodemann and Fu (2015) Inti Sodemann and Liang Fu, “Quantum nonlinear hall effect induced by berry curvature dipole in time-reversal invariant materials,” Phys. Rev. Lett. 115, 216806 (2015).
• Sipe and Shkrebtii (2000) J. E. Sipe and A. I. Shkrebtii, ‘‘Second-order optical response in semiconductors,” Phys. Rev. B 61, 5337–5352 (2000).
• Taguchi et al. (2016) Katsuhisa Taguchi, Tatsushi Imaeda, Masatoshi Sato,  and Yukio Tanaka, “Photovoltaic chiral magnetic effect in weyl semimetals,” Phys. Rev. B 93, 201202 (2016).
• Ishizuka et al. (2016) Hiroaki Ishizuka, Tomoya Hayata, Masahito Ueda,  and Naoto Nagaosa, “Emergent electromagnetic induction and adiabatic charge pumping in noncentrosymmetric weyl semimetals,” Phys. Rev. Lett. 117, 216601 (2016).
• Chan et al. (2017) Ching-Kit Chan, Netanel H. Lindner, Gil Refael,  and Patrick A. Lee, “Photocurrents in weyl semimetals,” Phys. Rev. B 95, 041104 (2017).
• Nielsen and Ninomiya (1981) H.B. Nielsen and M. Ninomiya, ‘‘Absence of neutrinos on a lattice: (i). proof by homotopy theory,” Nuclear Physics B 185, 20 – 40 (1981).
• Belinicher and Sturman (1980) V. I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Usp. Fiz. Nauk 130, 415–458 (1980).
• Huang et al. (2016) Shin-Ming Huang et al., “New type of weyl semimetal with quadratic double weyl fermions,” Proc. Nat. Acad. Sci. 113, 1180 (2016).
• Anderson et al. (2012) Brandon M Anderson, Gediminas Juzeliūnas, Victor M Galitski,  and Ian B Spielman, “Synthetic 3d spin-orbit coupling,” Phys. Rev. Lett. 108, 235301 (2012).
• Behrends et al. (2016) Jan Behrends, Adolfo G. Grushin, Teemu Ojanen,  and Jens H. Bardarson, “Visualizing the chiral anomaly in dirac and weyl semimetals with photoemission spectroscopy,” Phys. Rev. B 93, 075114 (2016).
• Bas et al. (2016) Derek A Bas, Rodrigo A Muniz, Sercan Babakiray, David Lederman, JE Sipe,  and Alan D Bristow, “Photocurrents in bi2se3: bulk versus surface, and injection versus shift currents,” Preprint at http://arxiv.org/abs/1602.01074  (2016).
• Chang et al. (2016) Guoqing Chang et al., “Kramers theorem-enforced weyl fermions: Theory and materials predictions (ag3bo3, tlte2o6 and ag2se related families),” Preprint at http://arxiv.org/abs/1611.07925  (2016).
• Hirayama et al. (2015) Motoaki Hirayama, Ryo Okugawa, Shoji Ishibashi, Shuichi Murakami,  and Takashi Miyake, “Weyl node and spin texture in trigonal tellurium and selenium,” Phys. Rev. Lett. 114, 206401 (2015).
• Ishizaka et al. (2011) K. Ishizaka et al., “Giant rashba-type spin splitting in bulk bitei,” Nat. Mater. 10, 521–526 (2011).
• Volovik and Konyshev (1988) GE Volovik and VA Konyshev, “Properties of superfluid systems with multiple zeros in the spectrum of fermions,” JETP Lett. 47 (1988).
• Fang et al. (2012) Chen Fang, Matthew J. Gilbert, Xi Dai,  and B. Andrei Bernevig, “Multi-weyl topological semimetals stabilized by point group symmetry,” Phys. Rev. Lett. 108, 266802 (2012).
• Bradlyn et al. (2016) Barry Bradlyn et al., “Beyond dirac and weyl fermions: Unconventional quasiparticles in conventional crystals,” Science 353, aaf5037 (2016).
• Carbotte (2016) J. P. Carbotte, “Dirac cone tilt on interband optical background of type-i and type-ii weyl semimetals,” Phys. Rev. B 94, 165111 (2016).
• Xu et al. (2015b) Yong Xu, Fan Zhang,  and Chuanwei Zhang, “Structured weyl points in spin-orbit coupled fermionic superfluids,” Phys. Rev. Lett. 115, 265304 (2015b).
• Soluyanov et al. (2015) Alexey A Soluyanov et al., “Type-ii weyl semimetals,” Nature 527, 495–498 (2015).
• Ashby and Carbotte (2014) Phillip E. C. Ashby and J. P. Carbotte, “Chiral anomaly and optical absorption in weyl semimetals,” Phys. Rev. B 89, 245121 (2014).
• Grushin et al. (2015) Adolfo G. Grushin, Jörn W. F. Venderbos,  and Jens H. Bardarson, “Coexistence of fermi arcs with two-dimensional gapless dirac states,” Phys. Rev. B 91, 121109 (2015).
• Morimoto and Nagaosa (2016) Takahiro Morimoto and Naoto Nagaosa, “Topological nature of nonlinear optical effects in solids,” Science Advances 2, e1501524 (2016).
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