Current-induced second harmonic generation in inversion-symmetric Dirac and Weyl semimetals
Abstract
Second harmonic generation (SHG) is a fundamental nonlinear optical phenomenon widely used both for experimental probes of materials and for application to optical devices. Even-order nonlinear optical responses including SHG generally require breaking of inversion symmetry, and thus have been utilized to study noncentrosymmetric materials. Here, we study theoretically the SHG in inversion-symmetric Dirac and Weyl semimetals under a DC current which breaks the inversion symmetry by creating a nonequilibrium steady state. Based on analytic and numerical calculations, we find that Dirac and Weyl semimetals exhibit strong SHG upon application of finite current. Our experimental estimation for a Dirac semimetal CdAs and a magnetic Weyl semimetal CoSnS suggests that the induced susceptibility for practical applied current densities can reach with mid-IR or far-IR light. This value is 10-10 times larger than those of typical nonlinear optical materials. We also discuss experimental approaches to observe the current-induced SHG and comment on current-induced SHG in other topological semimetals in connection with recent experiments.
Introduction.— Intense light incident on materials induces various nonlinear optical responses (NLORs) reflecting the details of material properties Bloembergen (1996); Boyd (2008). The study of NLORs remains an important topics in condensed matter studies since NLORs not only give a rich information of symmetry information about materials but also yield useful optical devices. In recent years, a close relationship between the NLORs and the notion of band geometry has been revealed Moore and Orenstein (2010); Deyo et al. (2009); Sodemann and Fu (2015); Sipe and Shkrebtii (2000); Young and Rappe (2012); Morimoto and Nagaosa (2016). In particular, three-dimensional (3D) topological materials can support novel NLORs Hosur (2011); Chan et al. (2017); Wu et al. (2017); de Juan et al. (2017); Golub and Ivchenko (2018); Patankar et al. (2018). Among these, inversion-symmetry-broken topological semimetals (SMs) are attracting keen attention as recent optical measurements of TaAs, which is an inversion-symmetry-broken Weyl semimetal (WSM), reported strong second harmonic generation (SHG) with signal 100 times larger than a typical value in GaAs Wu et al. (2017); Patankar et al. (2018), and other strong nonlinear optical properties as well Osterhoudt et al. (2019); Sirica et al. (2019). From the theoretical side, various interesting nonlinear optical phenomena have been proposed Hosur (2011); Chan et al. (2017); de Juan et al. (2017), including a quantization of the circular photogalvanic effect that originates from the topological properties of WSMs de Juan et al. (2017); Flicker et al. (2018).
On the other hand, there are various topological SMs preserving inversion symmetry which are also intensively studied. One example is topological Dirac semimetals (DSMs), such as CdAs Rosenberg and Harman (1959); Wang et al. (2013); Crassee et al. (2018) or NaBi Wang et al. (2012), where the Dirac point is protected by crystalline symmetry. The other example is inversion-symmetric magnetic WSMs, such as CoSnS Liu et al. (2019) or MnSn Kuroda et al. (2017), where the time-reversal symmetry is broken instead of the inversion symmetry. In these materials, odd-order NLORs are only allowed, where the dominant effect is the third-order NLOR. However, once the inversion symmetry is broken by applying a suitable perturbation, inversion-symmetric materials can also exhibit even order NLORs.
Motivated by this idea and by general interest in the creation of nonequilibrium states with new or amplified responses, we investigate the creation of second-order NLORs in inversion-symmetric Dirac/Weyl SMs. For the inversion-symmetry breaking perturbation, we consider DC electric field which makes the electron distribution asymmetric in momentum space and induces finite current, resulting in broken inversion symmetry. In this study, we focus on SHG, which is a phenomenon that injected light with frequency is converted into light with doubled frequency as schematically shown in Fig. 1. SHG from current-driven materials has been called current-induced SHG (CISHG) and studied theoretically Khurgin (1995); Wu et al. (2012); Cheng et al. (2014) and experimentally in several materials, such as Si Aktsipetrov et al. (2009), GaAs Ruzicka et al. (2012), graphene Bykov et al. (2012); An et al. (2013) and superconducting NbN Nakamura et al. (2020). This device geometry is similar to what is used to measure photoconductivity in inversion-symmetric insulators, where an applied DC electric field leads to a current pulse under illumination. In metals, this DC field will already produce some current, so the most visible optical consequence of the field-induced symmetry breaking is now CISHG.
Yet, CISHG in topological materials, especially Dirac/Weyl SMs, has not been explored so far. Here, we study CISHG in Dirac/Weyl SMs by taking two complementary approaches. One is analytic calculation with ideal Weyl (Dirac) Hamiltonians and the other is numerical, based on tight-binding models. The results of both approaches are consistent and show that inversion-symmetric Dirac/Weyl SMs support a divergently large CISHG when the Fermi level is located near the Dirac/Weyl points. Based on our results, we estimate the order of the nonlinear susceptibility , characterizing the strength of the CISHG. Considering the realistic parameters corresponding to the materials, a Dirac SM, CdAs, and a Weyl SM, CoSnS, and find that it can reach for practical applied current densities. These values are 10-10 times larger than those of typical nonlinear materials Wu et al. (2017); Bergfeld and Daum (2003); Haislmaier et al. (2013). We also address the experimental methods to observe the CISHG and the possibility of CISHG in other topological SMs.
Methods.— SHG is characterized by the response tensor defined via where [] is the Fourier component of the time-dependent current [electric field ] proportional to []. The indices run over and the sum over repeated indices is implied throughout this paper. From the standard time-dependent perturbation theory Moss et al. (1990); Ghahramani et al. (1991); Yang et al. , we have the following expression for the SHG response tensor
(1) |
where
(2) | ||||
(3) | ||||
(4) | ||||
(5) |
with , , , , , and where the integration is performed over the entire Brillouin zone. Here, represents the -th band of the Hamiltonian (the implicit sum over repeated indices is also taken for the band indices and ) and is a distribution function of electrons. In equilibrium which is the Fermi distribution function with inverse temperature . The subscript 2p (1p) in Eq. (1) denotes the contribution of two-photon (one-photon) resonance.
To calculate the response tensor of CISHG, we need the distribution function of a nonequilibrium steady state (NESS) carrying finite current. To obtain it, we use the Boltzmann equation with the relaxation time approximation under a static electric field , which is where denotes the relaxation time Yu et al. (2014); Sodemann and Fu (2015); Morimoto et al. (2016); Yasuda et al. (2016). Solving this equation recursively, we obtain the distribution function for NESS as
(6) |
with . We use in Eq.(2)-(5) to calculate the current-induced SHG. The example of under the electric field in -direction is shown in the right panel of Fig. 2 (b) and the equilibrium distribution is also shown in the left panel of Fig. 2 (b) for reference. From these figures, we can see the distribution function is deformed and asymmetric in the -direction under the electric field.
Analytic results with Weyl Hamiltonian. — To study the CISHG in inversion-symmetric Dirac/Weyl SMs, we take two complementary approaches. One approach is based on a simple Weyl Hamiltonian
(7) |
where represents Pauli matrices ( identity matrix), , ( and correspond the hopping amplitude and the lattice constant respectively), , and . The Hamiltonian represents a single Weyl () or anti-Weyl () node located at [The band structure is shown in Fig. (2) (a)]. This Hamiltonian is very simple, but the low-energy physics of Dirac/Weyl SMs are well-described by this Hamiltonian Armitage et al. (2018). WSMs have pairs of Weyl and anti-Weyl nodes in the band structure and they locate at different points. On the other hand, DSMs support Weyl and anti-Weyl nodes at the same point, which is called a Dirac node. In the following, we start from calculation of the SHG of a single (anti-)Weyl node and then sum up contributions from all the Weyl nodes FN2 . For simplicity, we assume that the electric fields are applied in the -direction, i.e. .
By a straightforward calculation with the Weyl Hamiltonian (7) shown in Supplemental Material, we can evaluate Eq. (1) analytically at zero temperature. Considering the symmetry, the independent non-zero components are only -, - and -components Bloembergen (1996); Boyd (2008); FNB . The -component of the response tensor from a single Weyl node is given as
(8) |
with
(9) | |||
(10) |
The other independent components are given as and with . The differences between the components are only numerical factors and their qualitative behaviors are same. Thus, we focus on the -component below.
Using these results, we can obtain the CISHG response tensor for Dirac/Weyl SMs. Since the above results do not depend on the position and the chirality of the Weyl nodes, we can calculate the response tensor of Dirac/Weyl SMs simply by multiplying the number of Weyl nodes considering the degeneracy. Therefore, assuming that Weyl (Dirac) SMs have two Weyl (Dirac) nodes FN3 , the response tensors for Weyl and Dirac SMs are and respectively. The real and imaginary part of are shown in Figs. 2 (c) and (d). From these figures and Eq. (8)-(10), we can see that the SHG spectra have a large peak around (two-photon resonance) and a small peak around (one-photon resonance). The height (weight) of the two peaks is proportional to for , leading to diverging enhancement. We note that the power of is different from that in the graphene case discussed in the previous study Cheng et al. (2014) because of the different dimensionality FN1 . Our results suggest that Dirac/Weyl SMs where the Fermi level is near the Dirac/Weyl points can show strong SHG FNA .
Numerical results with tight-binding models.— Let us move on to the numerical calculation with a tight-binding Hamiltonian describing DSMs. This approach is closer to real materials than the previous approach because we consider multiple (in this model, four) bands and take the nonlinearity and periodicity of the band structure into account. We use the following tight-binding model
(11) |
with
(12) | ||||
(13) | ||||
(14) | ||||
(15) |
introduced in Ref. Yang and Nagaosa (2014). As shown in the topological phase diagram [Fig. 3 (c)], this model hosts several topological phases. In particular, the topological DSM phase is realized in a wide range of parameters. In this phase, the energy dispersion has a pair of Dirac points located on -axis as shown in Fig. 3 (a). These Dirac cones are protected by the rotational symmetry and topologically robust, which is also the case in the typical topological DSM material, CdAs Yang and Nagaosa (2014).
Using this model, we calculate the SHG response tensor under the -directed electric field at finite temperature FN4 . By numerical calculations, we obtain the SHG spectra shown in Figs. 3 (d) and (e). First, we can find strong peaks around and they show a divergent behavior as . These signatures are consistent with our analytic results shown in Figs. (2) (c) and (d) FN9 . These findings strongly suggest that Dirac/Weyl SMs generally support large CISHG. The other feature in the spectra is the appearance of a large peak at when -. This behavior reflects the van Hove singularity at . Indeed, the joint density of states (JDOS) FN8 shows a singularity at as shown in Fig. 3 (b) FN7 .
In addition to Dirac SMs, we also carried out a tight-binding calculation for Weyl SMs (See Supplemental Material). We study a two-band tight-binding model describing Weyl SMs and obtain qualitatively similar results to those of Dirac SMs. Therefore, strong CISHG in Weyl SMs is also supported by both analytic and numerical calculations.
Discussion.— Our calculation suggests that Dirac and Weyl SMs show very strong CISHG. To connect these results with experiments, we estimate the strength of CISHG. First of all, we need to specify an experimental setup to give an estimate because the achievable electric field depends on the type of experiment. We propose two kinds of experimental setup shown in Fig. 4. One is a standard SHG measurement under a DC bias voltage. The other is a THz pump SHG measurement, where the pump frequency is low enough to be seen as a static field. The former approach is static and thus should be easier than the other one, which is time-resolved. On the other hand, the latter approach is advantageous for applying a strong electric field because very strong THz fields such as 1-80 MV/cm has been achieved Fülöp et al. (2020).
Next, we estimate the strength of electric fields inside the material. For the DC bias case, the experimental control parameter is current density rather than field strength. Following Ohm’s law, the internal electric field is given as , where and are the current density and the conductivity, respectively. For the THz pump case, we have to take into account the mismatch of impedance. The internal electric field is represented as with the external pump field and the refractive index . In the THz regime, the refractive index is given as where is the vacuum permittivity and is the pump field frequency. To be specific, we consider two materials: a Dirac semimetal, CdAs and a Weyl semimetal, CoSnS. Using the low temperature conductivity of these materials FN5 and assuming , and THz as typical values, we obtain for the DC bias case and for the THz pump case in CdAs (CoSnS).
To estimate the strength of CISHG, we evaluate the nonlinear susceptibility . The susceptibility takes the largest value when the probe frequency is resonant to the Fermi energy, i.e. , and we consider this resonant case below. The Fermi energy of CdAs (CoSnS) is typically 100 (50) meV Liu et al. (2014); Crassee et al. (2018); Liu et al. (2019) and thus the frequency of probe light is 24 (12) THz, which corresponds to the wavelength 12.5 (25) m in the mid-IR (far-IR) regime. Using the formula (8) with the parameters for CdAs (CoSnS) FN6 , we obtain () pmV in the DC bias case and () pmV in the THz pump case. Compared to the typical value of the susceptibility of SHG, such as pmV of TaAs (the fundamental wavelength 800 nm) Wu et al. (2017), pmV of GaAs ( 810 nm) Bergfeld and Daum (2003), 15-19 pmV of BiFeO ( 1.55 m) Haislmaier et al. (2013), the values evaluated above are very large and suggest that CdAs and CoSnS are promising candidates showing very strong CISHG. For the DC bias case, the response is relatively small because the internal electric fields are small, but CISHG of CdAs can be comparable to SHG of TaAs due to its longer relaxation time. Remarkably, the responses of both CdAs and CoSnS in the THz pump case is 10 times larger than that of TaAs, which has the largest , and 10 times larger than that of BiFeO.
Next, we mention other frequency regimes. In the THz regime, the response is expected to be much larger than the mid-IR regime since the CISHG becomes divergently large with . The analytic result [Eq. (8)] indicates that the resonant response at - meV (i.e. 0.24-2.4 THz) is roughly 10-100 times larger than that at meV. This enhancement is expected to be realized by changing the doping level. For example, the sample of CdAs with (i.e. at the Dirac point) has been fabricated as used in Ref. Liu et al. (2014). In the higher frequency regime, such as near-IR or visible regime, the effect of the Dirac cones becomes smaller, while van Hove singularity points due to merging of the Dirac cones give rise to a large CISHG response. This contribution can be comparable to the contribution of Dirac points as shown in Figs. 3 (d) and (e).
We comment on the possibility of the CISHG in other topological SMs. Since our analytic results are only based on the simple Weyl point Hamiltonian without any assumption about symmetry, the similar CISHG can occur even in inversion-symmetry-broken Weyl SMs, such as TaAs. Such materials are expected to show a large CISHG in addition to the original SHG, and these two contributions are separable via changing the applied electric field. Very recent experiments Sirica et al. (2020) suggest that indeed the CISHG component is detectable in TaAs using an optically pumped current, which is found to change the symmetry of SHG in the plane perpendicular to that material’s polar axis; our model predicts that the signal induced in CdAs should be much stronger because its relaxation time is at least an order of magnitude longer. Furthermore, our tight-binding calculation suggests that the van Hove singularities can be an origin of a large CISHG while they are not divergent like the CISHG from Weyl nodes. Thus, since topological nodal SMs have van Hove singularities protected by its topology, they are also expected to be candidate materials showing strong CISHG.
In this paper, we have shown that Dirac/Weyl SMs with inversion symmetry show very strong CISHG, and inversion-breaking topological SMs may also be expected to show strong CISHG on top of the zero-current ordinary SHG. These results suggest that topological SMs have value as a nonlinear optical material whose SHG intensity is controllable from zero to very large value by electric current. Moreover, the SHG is also controlled by changing the direction of the current. This high degree of control can provide a new route to realize switchable nonlinear optical devices.
Acknowledgements.
We thank Daniel E. Parker for valuable discussions. This work is supported by the Quantum Materials Program (JWO, JEM) and a Simons Investigatorship (JEM). KT thanks JSPS for support from Overseas Research Fellowship. TM was supported by JST PRESTO (JPMJPR19L9) and JST CREST (JPMJCR19T3).References
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Supplemental Material: “Current-induced second harmonic generation in inversion-symmetric Dirac and Weyl semimetals”
S1. Derivation of the response tensor from Weyl Hamiltonian
In this section, we calculate the SHG response tensor under finite current in -direction using the Weyl Hamiltonian where and its eigenvectors and satisfy and with . We denote the response tensor from a single Weyl point as where the indices and run over . This consists of four parts , and as shown in Eqs. (1)-(5) in the main text. Since for the Weyl Hamiltonian, and are zero. Taking a sum for the band indices and assuming , we obtain the simpler form of and . For , they are written as
(S1) | ||||
(S2) |
For , they are given as
(S3) | ||||
(S4) |
For the definitions of , , , , and , see the main text. By a straightforward calculation, we can check that the final results do not depend on the chirality and thus we assume below.
In the following, we evaluate the quantities given by Eqs. (S1) and (S2). First, due to the symmetry, it turns out that non-zero independent components of the tensor are only -, - and -components. To calculate them, we use
The distribution function under the electric field in -direction is given as and we truncate the distribution function truncated up to the second order of . Using this form of the distribution function, the terms of the 0-th and the 2-nd order of vanishes due to the symmetry and only the 1-st order term remains. Therefore, the distribution functions in Eqs. (S1) and (S2) are replaced as . The derivative is calculated as where
where and the temperature is zero. To clarify the calculation process, we consider the most simple one, the -component, and show the calculation explicitly. Using the above results, is written down as
(S5) |
Here, we assume that is sufficiently small that the resonant factor behaves like and taking the factor into the integrand in the last line of Eq.(S5). To clarify the physical dimension, we didimentionalize several quantities in Eq. (S5). For this purpose, we factorize as where and are constants with dimension of energy and length respectively. In lattice models, and correspond to the hopping amplitude and the lattice constant. Using these quantities, we obtain
(S6) |
with and . Then, the problem is reduced to evaluate the integral , which is defined as
with . To evaluate it, we change the variables as , , and . Then, we can perform the integration as
(S7) |
Applying Eq. (S7) to Eq. (S6), we obtain
(S8) |
For the one-photon contribution , it turns out that because the second term of Eq. (S2) is equal to . Thus, we obtain
(S9) |
This simplification only occurs for the -component. For the - and -components, we have to calculate the one-photon contribution itself respectively. Then, we finally obtain , the sum of the one-photon contribution [Eq. (S8)] and the two-photon contribution [Eq. (S9)], given as Eq. (8) in the main text.
The other components, the - and -components, are also calculated in the same manner. For the Weyl Hamiltonian, the -component and the -component are the sum of the two-photon () and one-photon () components and each component is given as
(S10) | ||||