Second harmonic generation in Dirac/Weyl semimetals with broken tilt inversion symmetry

Yang Gao; Yang Gao; Bin Ge; Bin Ge

doi:10.1364/OE.414524

1. Introduction

Nonlinear optics is a subject associated with the various nonlinear phenomena of media in strong light fields and its application. Nonlinear optics response of materials with linear energy dispersion has seen a renewed impetus. For a two-dimensional materials system, there has been a surge of interest in nonlinear optical responses of graphene [1–12]. Due to the spatial inversion symmetry of the monolayer graphene, the strong third-order nonlinear processes are allowed in the electric dipole approximation [1,3,7,10]. Furthermore, the second-order nonlinear response in graphene can be observed beyond the electric dipole moment approximation [4,5,9,11]. For the few-layer graphene, the electric dipole allows second-harmonic generation due to the inversion symmetry broken [13,14]. For a three-dimensional materials system, the Weyl semimetals can be realized from a Dirac system by breaking either time-reversal symmetry or inversion symmetry, which provides an intrinsic source for nonlinear technology.

In recent years, people have paid much attention to the research of nonlinear response described by topological quantities involving the Berry connection/curvature of the Dirac/Weyl semimetals system in theory [15–24]. Many of them have also been verified by experiments [25–27]. For the second-order nonlinear optical response in Weyl semimetals, a nonvanishing photocurrent requires broken spatial inversion symmetry in the absence of external magnetic fields [15,16,19–21,24]. However, after considering the external magnetic field, the second harmonic generation is linear with the magnetic field can be induced in Weyl semimetals with broken time-reversal symmetry. The chiral magnetic effect and the optical activity have contributed to the second nonlinear optical response [16–18]. The second-order nonlinear Hall effect was calculated within the Boltzmann approach at a low-frequency electric field and was predicted [28]. Moreover, the zero and double frequency component of the Hall conductivity proportional to the so-called Berry dipole was obtained in the two/three-dimensional Dirac/Weyl semimetals model. The strong nonlinear Hall effect can be measured in anisotropic or tilted Weyl semimetals with broken spatial inversion symmetry [29–37]. Based on the previous studies, a nonvanishing second-order nonlinear optical response in Weyl semimetals requires broken spatial inversion symmetry in the absence of external magnetic fields. Theoretically, the third-order optical and the nonlinear plasmonic responses of three-dimensional massless Dirac/Weyl Fermions have been studied in recently, and very strong optical nonlinearities which are similar to graphene have been reported [38–40]. In addition, the terahertz high harmonic generation in the Dirac Semimetal Cd$_3$As$_2$ have been studied by both experimental and theoretical methods, thus open a new pathway to develop a novel frequency convertor in terahertz frequency based on Dirac semimetals [41–43]. The linear energy-momentum dispersion near the Dirac/Weyl points is crucial for nonlinear optical effects based on the above works.

In this work, we theoretically study the second harmonic generation(SHG) in tilted Dirac/Weyl semimetals with broken tilt inversion symmetry in the absence of an external magnetic field. The SHG process will not relate to the topological effect of Weyl semimetals. Our studies focus on the effects of tilt of Dirac/Weyl nodes, paying attention to the Dirac/Weyl semimetals with broken tilt inversion symmetry. The employed method is based on the quantum mechanical density matrix formalism, as described in Refs. [9] and [10]. We find two contributions to the conductivity in the low-frequency region, one coming from the intraband transitions and describing by Drude-like effects, and the other from the interband-intraband transitions due to the linear energy dispersion of Weyl semimetals near the Dirac/Weyl points. While for a high-frequency incident field, the appearance of prominent resonant peaks in the nonlinear conductance originates from the two-photon absorption process. The most interesting aspect of the second harmonic conductivity is that it does not depend on the chirality of the Weyl node, which is different from the second-order nonlinear Hall effect in Weyl semimetals [37]. It is found that Weyl semimetals have a very high nonlinear susceptibility, and an optimal tilt of the Weyl node for the maximum nonlinear current has been predicted.

2. Linear response of Weyl semimetals

We consider a 3D Weyl semimetal with a tilt in the z-direction within a continuum model Hamiltonian, which takes the form [44–46]

(1)$$H_0(\boldsymbol{k}) =\chi\hbar v_F\boldsymbol{k}\cdot\boldsymbol{\sigma}+ \hbar v_Ft_{\chi,z}k_z\sigma_0 ,$$

where $v_F$ is the Fermi velocity, $\chi =\pm$ defines the chirality of the valley, $\boldsymbol{k}$ is the momentum, $\sigma _0$ is a $2\times 2$ identity matrix and $\boldsymbol {\sigma }$ are the Pauli matrices. We use $t_{\chi ,z}$ ($t_z\in [0,1)$) to describe the tilt of Weyl cones. The case with parameter $t_{+,z}=-t_{-,z}=t_z$ respects tilt inversion symmetry and the case with parameter $t_{+,z}=t_{-,z}=t_z$ breaks tilt inversion symmetry [46]. The energy bands of the model are $\varepsilon _{\boldsymbol {k}s}=\hbar v_F\left (t_{\chi ,z}k_z+s k\right )$, and $s=\pm 1$ for conduction and valence bands, respectively. The corresponding eigenstates are given by

(2)$$\Psi_{\boldsymbol{k},s}(\boldsymbol{r})=\langle\boldsymbol{r}|\boldsymbol{k},s\rangle =\frac{e^{i\boldsymbol{k}\cdot\boldsymbol{r}}}{\sqrt{2V}}\left( \begin{array}{cc} \sqrt{1+s\cos \theta_{\boldsymbol{k}}} \\ s\sqrt{1-s\cos \theta_{\boldsymbol{k}}} e^{i\phi_{\boldsymbol{k}}}\end{array} \right),$$

where $\cos \theta _{\boldsymbol {k}}=\frac {d_z(\boldsymbol {k})}{d(\boldsymbol {k})}$ and $\tan \phi _{\boldsymbol {k}}=\frac {d_y(\boldsymbol {k})}{d_x(\boldsymbol {k})}$, and $V$ is the quantization volume.

In the presence of external ac fields, the interaction Hamiltonian $H^{\textrm {opt}}$ of electron-photon system in the velocity gauge can be obtained by the replacement $\hat {\boldsymbol {p}} \Rightarrow \hat {\boldsymbol {p}}+\frac {e}{c}\boldsymbol {A}$. Here $\boldsymbol {A}$ is the vector potential and the electric field $\boldsymbol {E}=-\frac {1}{c}\frac {\partial \boldsymbol {A}}{\partial t}$. With keeping the linear terms in $\boldsymbol {A}$, the Hamiltonian (1) which becomes

(3)$$H=H_0+H^{\textrm{opt}}=H_0+\frac{e}{c}\hat{\boldsymbol{v}}\cdot\boldsymbol{A} .$$

The length gauge has also been used [9–11]. In this paper, we will consider the vector potentials corresponding to transverse electromagnetic fields with frequency $\omega$, wave vector $\boldsymbol {q}$ and polarization vector $\hat {\boldsymbol {e}}$, i.e.,

(4)$$\boldsymbol{A}=\hat{\boldsymbol{e}}A(\omega)e^{{-}i\omega t+i\boldsymbol{q}\cdot \boldsymbol{r}}+c.c.$$

The current operator is given by $\hat {\boldsymbol {j}}=e\hat {\boldsymbol {v}}$, where $e=-|e|$, and $\hat {\boldsymbol {v}}=i\hbar ^{-1}[H,\hat {\boldsymbol {r}}]$ is the velocity operator. The Fourier component of the current is thus [9,47]

(5)$$\boldsymbol{j}(\omega)=\sum_{mn} \boldsymbol{j}_{mn}^{(\boldsymbol{q})}\rho_{nm}^{(1)}(\omega),$$

where $\boldsymbol {j}_{mn}^{(\boldsymbol {q})}=\langle m|e^{-i\boldsymbol {q}\cdot \boldsymbol {r}}\hat {\boldsymbol {j}}|n\rangle$. The density matrix $\rho _{nm}$ can be calculated from a collisionless quantum kinetic equation

(6)$$\dot{\rho}_{nm}={-}\frac{i}{\hbar}(\varepsilon_{\boldsymbol{k}_n}-\varepsilon_{\boldsymbol{k}_m})\rho_{nm}-\frac{i}{\hbar}[H^{\textrm{opt}}(\omega),\rho]_{nm}.$$

We seek a perturbative solution of the form $\rho _{nm}=\rho _{nm}^{(0)}+\rho _{nm}^{(1)}+\rho _{nm}^{(2)}+\cdots$, where $\rho _{nm}^{(1)}\propto E$ and $\rho _{nm}^{(2)}\propto E^2$. Solving the kinetic Eq. (6) in the first order we get

(7)$${\rho}_{nm}^{(1)}(\omega)=\frac{H^{\textrm{opt}}_{nm}(\omega)[\rho^{(0)}_{m m}-\rho^{(0)}_{n n}]}{\hbar\omega-\varepsilon_{nm}},$$

where $\varepsilon _{nm}=\varepsilon _{\boldsymbol {k}_n}-\varepsilon _{\boldsymbol {k}_m}$. To the leading order of the wave vector, the corresponding current is

(8)$$\boldsymbol{j}(\omega) =\frac{e^2}{c}\sum_{mn}\frac{\hat{\boldsymbol{v}}_{mn}(\hat{\boldsymbol{v}}_{nm}\cdot \boldsymbol{A}(\omega) )}{\hbar\omega-\varepsilon_{nm}} \left[\rho_{mm}^{(0)}-\rho_{nn}^{(0)}\right] .$$

The 0-th order density matrix is $\rho _{nn}^{(0)}=f_n$, where $f_n$ is the Fermi distribution $f(\varepsilon _{\boldsymbol {k}n})=1/[e^{(\varepsilon _{\boldsymbol {k}n}-\mu )/k_BT}+1]$, $k_B$ the Boltzmann constant, $T$ the temperature, and $\mu$ the chemical potential. We have to evaluate the optical conductivity separately the intraband contribution $m=n$ and interband contribution $m\neq n$, one can easily obtain the Drude and interband frequency dependent complex conductivity are [9]

(9)$$\sigma_{\alpha\beta}^{\textrm{intra}}(\omega) =\frac{ie^2}{\hbar^2\omega}\sum_n \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_\alpha} \left(-\frac{\partial f_n}{\partial \varepsilon_{\boldsymbol{k}_n}}\right) \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_\beta},$$

and

(10)$$\sigma_{\alpha\beta}^{\textrm{inter}}(\omega) =i e^2\hbar\sum_{mn}\frac{f_n-f_m}{\varepsilon_{\boldsymbol{k}_m}-\varepsilon_{\boldsymbol{k}_n}} \frac{\langle m|\hat{\boldsymbol{v}}_\alpha|n\rangle\langle n|\hat{\boldsymbol{v}}_\beta|m\rangle } {\hbar\omega-(\varepsilon_{\boldsymbol{k}_n}-\varepsilon_{\boldsymbol{k}_m})},$$

here $\alpha$, $\beta$ denote Cartesian coordinate components.

By substituting $\omega \rightarrow \omega +i\gamma$ due to the electronic relaxation processes, Fig. 1 shows the conductivity $\sigma (\omega )$ for small tilt parameter $t_z$ is in agreement with Ref. [48]. These results are evaluated from the analytic formula given in Appendix A. The intraband contribution [see Fig. 1(a)] has a Drude form, and the interband contribution [see Fig. 1(b)] has a modified steplike (logarithmic) singularity at $\Omega =2$ in its real (imaginary) part [48]. In the numerical calculation, the parameters are taken as the tilt $t_{+,z}=0.2$, the relaxation constant $\Gamma =\hbar \gamma /\mu =0.005$, the cutoff $\Lambda /\mu =10$ and the temperature $T=0$.

Fig. 1. The real and imaginary parts of the first-order conductivity $\sigma (\omega )$ for a Weyl point of chirality $\chi =+$ as a function of the frequency $\Omega =\hbar \omega /\mu$ at the tilt parameter $t_{+,z}=0.2$, the relaxation constant $\Gamma =\hbar \gamma /\mu =0.005$, the cutoff $\Lambda /\mu =10$ and the temperature $T=0$.

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3. Second order response of Weyl semimetals

In this section, we calculate the second harmonic current of the Weyl semimetals system. In the presence of photon perturbation fields, with the use of quantum kinetic Eq. (6), it is straightforward to apply second-order perturbation theory, the general solution of the second-order density matrix is

(11)$${\rho}_{nm}^{(2)}(2\omega) =\frac{1}{2\hbar\omega-\varepsilon_{nm}} \sum_l H^{\textrm{opt}}_{nl}(\omega)H^{\textrm{opt}}_{lm}(\omega) \left[\frac{\rho^{(0)}_{m m}-\rho^{(0)}_{ll}}{\hbar\omega-\varepsilon_{lm}} -\frac{\rho^{(0)}_{ll}-\rho^{(0)}_{n n}}{\hbar\omega-\varepsilon_{nl}} \right].$$

Also from Eq. (5), the current at the double frequency can be calculated as

(12)$$\boldsymbol{j}(2\omega)=\sum_{mn} \boldsymbol{j}_{mn}^{(2\boldsymbol{q})}\rho_{nm}^{(2)}(2\omega),$$

where $\boldsymbol {j}_{mn}^{(2\boldsymbol {q})}=\langle m|e^{-2i\boldsymbol {q}\cdot \boldsymbol {r}} \hat {\boldsymbol {j}}|n\rangle$. With in the dipole approximation, we rewrite the second harmonic current into the final expression as

(13)$$\boldsymbol{j}(2\omega) =\frac{e^3}{c^2} \sum_{mnl}\frac{ \hat{\boldsymbol{v}}_{mn}( \hat{\boldsymbol{v}}_{nl}\cdot \boldsymbol{A}(\omega))( \hat{\boldsymbol{v}}_{lm}\cdot \boldsymbol{A}(\omega))}{2\hbar\omega-\varepsilon_{nm}} \left[\frac{\rho^{(0)}_{m m}-\rho^{(0)}_{ll}}{\hbar\omega-\varepsilon_{lm}} -\frac{\rho^{(0)}_{ll}-\rho^{(0)}_{n n}}{\hbar\omega-\varepsilon_{nl}} \right].$$

the factor $( \hat {\boldsymbol {v}}_{nl}\cdot \boldsymbol {A}(\omega ))( \hat {\boldsymbol {v}}_{lm}\cdot \boldsymbol {A}(\omega ))$ is the sum of interband and intraband contribution. In order to avoid possible confusion, we also evaluate the conductivity separately the intraband contribution $m=n=l$ and all mixed interband-intraband contributions: $m=n=-l$, $l=m=-n$ and $l=n=-m$. With this procedures all the second order conductivity tensor can be obtained:

(14)$$\sigma^{(\textrm{I})}_{\alpha\beta\gamma}(2\omega) =\frac{e^3}{2\hbar^3\omega^2}\sum_{n} \frac{\partial f_n}{\partial \varepsilon_{\boldsymbol{k}_n}} \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_\gamma} \frac{\partial^2\varepsilon_{\boldsymbol{k}_n}}{\partial k_\alpha \partial k_\beta},$$

(15)$$\sigma^{(\textrm{II})}_{\alpha\beta\gamma}(2\omega) =\frac{e^3}{\omega^2}\sum_{mn}\frac{(\hat{v}_{\alpha})_{nn} (\hat{v}_{\beta})_{nm}(\hat{v}_{\gamma})_{mn}} {\left[\hbar\omega-\varepsilon_{mn}\right]\left[\hbar\omega+\varepsilon_{mn}\right]} \left(f_n-f_m\right),$$

(16)$$\sigma^{(\textrm{III})}_{\alpha\beta\gamma}(2\omega) =\frac{e^3}{\hbar\omega^2}\sum_{mn}\frac{(\hat{v}_{\alpha})_{nm}(\hat{v}_{\beta})_{mn}}{2\hbar\omega-\varepsilon_{mn}} \frac{\partial (f_m-f_n)}{\partial k_\gamma},$$

(17)$$\sigma^{(\textrm{IV})}_{\alpha\beta\gamma}(2\omega) =\frac{e^3}{\omega^2}\sum_{mn}\frac{(\hat{v}_{\alpha})_{nm}(\hat{v}_{\beta})_{mn}} {[2\hbar\omega-\varepsilon_{mn}][\hbar\omega-\varepsilon_{mn}]} [(\hat{v}_{\gamma})_{mm}-(\hat{v}_{\gamma})_{nn}]\left(f_m-f_n\right).$$

We notice that in Eq. (1), the time-reversal symmetry is broken by the tilt. For $t_z\neq 0$ we find that the $\alpha \alpha z$, $\alpha z\alpha$ and $z\alpha \alpha$ $(\alpha =x,y,z)$ components of the nonlinear conductivity tensor are nonzero and all other components equal to zero.

As an example of the second-order response of Weyl semimetals, now we can calculate the nonlinear conductivity tensor $\sigma ^{(\textrm {I})}_{zzz}(2\omega )$, which is controlled by the integral of the type:

(18)$$\sigma^{(\textrm{I})}_{zzz}(2\omega) ={-}\frac{e^3 v_F^2}{2\hbar\omega^2}\int\frac{d^3k}{(2\pi)^3}\left(t_z+\frac{k_z}{k}\right)\frac{k_x^2+k_y^2}{k^3}\delta(\varepsilon_{\boldsymbol{k}n}-\mu).$$

In this work, we consider the n-doped Weyl semimetals with a positive chemical potential $\mu$. As we can see the nonlinear conductivity tensor $\sigma ^{(\textrm {I})}_{zzz}(2\omega )$ is even with respect to $k_x$ and $k_y$ but odd with $k_z$. From the above results, it is important to notice that the $\sigma ^{(\textrm {I})}_{zzz}(2\omega )$ becomes exactly zero when $t_z=0$. The presence of the tilt makes the Weyl semimetals with ellipsoidal Fermi surface, thus the tilt $t_z$ is performs a very important function to get a finite value upon integration. This is, the tilt $t_z$ is crucial in controlling the SHG process. The conductivity tensor $\sigma _{zzz}(2\omega )$ of Eqs. (14)–(17) are worked out in detail in Appendix B. By a straightforward calculation, it is easy to see that $\sigma _{zzz}(2\omega )$ is an odd function of $t_z$, and we can check that the second harmonic conductivity do not depend on the chirality $\chi$. The final results can be obtained by summing over a pair of Weyl nodes with opposite chirality. For the case with tilt inversion symmetry $t_{+,z}=-t_{-,z}=t_z$, $\sigma _{zzz}^{\chi =+}(2\omega )+\sigma _{zzz}^{\chi =-}(2\omega )=0$. And for the case with broken tilt inversion symmetry $t_{+,z}=t_{-,z}=t_z$, $\sigma _{zzz}^{\chi =+}(2\omega )+\sigma _{zzz}^{\chi =-}(2\omega )=2\sigma _{zzz}(2\omega )$.

In fact, the lifetime for electronic transitions in Weyl semimetals is finite. By substituting $\omega \rightarrow \omega +i\gamma$, Fig. 2 shows the nonzero component of the nonlinear conductivity tensor $\sigma _{zzz}(2\omega )$ for the SHG process as a function of the energy of the photons $\hbar \omega$ of the incident light. In the low-frequency region ($\Omega \ll 1$), the main contribution to nonlinear conductivity is classical, and the behavior of the nonlinear conductivity tensor $\sigma ^{(\textrm {I})}_{zzz}$ of the Weyl semimetals in the Drude-like model is given in Fig. 2(a). The term $\sigma ^{(\textrm {I})}_{zzz}$ is purely classical, which containing the product of two intraband contributions. Different from the linear response regime[Fig. 1(a)], the real parts of $\sigma ^{(\textrm {I})}_{zzz}$ exhibition positive and negative amplitudes because of the complex denominator function $(2\omega +i\gamma )(\omega +i\gamma )$ contribution to the SHG processes. The term $\sigma ^{(\textrm {II})}_{zzz}$ is purely quantum, which containing the product of two interband contributions. The weak resonance is seen at the frequencies of the single-photon ($\Omega \approx 2$) interband transitions corresponding to nonresonant excitation[Fig. 2(b)]. We notice that the resonance located at ${2}/(1+t_z)<\Omega <{2}/(1-t_z)$ due to the tilt of the Weyl cone, and the absolute value of single-photon interband transitions is smaller than the classical term. Furthermore, $\sigma ^{(\textrm {II})}_{zzz}$ diverge as $1/\omega ^2$ in the low-frequency region. $\sigma ^{(\textrm {III})}_{zzz}$ and $\sigma ^{(\textrm {IV})}_{zzz}$ containing the product of one intraband and one interband contributions. Firstly, let us discuss the case of the two-photon absorption properties of Weyl semimetals. The nonlinear conductivity $\sigma ^{(\textrm {III})}_{zzz}$ has a sharp resonance at ${1}/(1+t_z)<\Omega <{1}/(1-t_z)$ corresponding to the two-photon absorption edge and diverges at low-frequency region[Fig. 2(c)]. Secondly, the contribution to $\sigma ^{(\textrm {IV})}_{zzz}$ containing summands with the single and two-photon interband transitions[Eq. (17)], and the resonance peaks are located at ${1}/(1+t_z)<\Omega <{1}/(1-t_z)$ and ${2}/(1+t_z)<\Omega <{2}/(1-t_z)$ corresponding to nonresonant excitation [Fig. 2(d)], respectively. Moreover, at low frequencies, the interband-intraband contributions $\sigma ^{(\textrm {II})}_{zzz}$, $\sigma ^{(\textrm {III})}_{zzz}$ and $\sigma ^{(\textrm {IV})}_{zzz}$ diverge as $1/\omega ^2$[see Eq. (15)–(17)]. For simplicity, we consider the zero temperature situation. With increasing the temperature, the resonant peaks in the nonlinear conductivity decrease and broaden. This behavior is reminiscent of usual Fermi liquids, where the thermal fluctuations enhance the scattering of electrons. Furthermore, at high temperatures, not all electron states of the low-lying subbands are fully occupied due to the thermal excitation. Thus their combined effects result in the decrease of the conductivity with temperature [49,50].

Fig. 2. The nonlinear conductivity tensor $\sigma _{zzz}(2\omega )$ for the process of SHG as a function of the incident photon energy at $\Gamma =0.05$. The other parameters are the same as those of Fig. 1.

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The total second-order conductivity $\sigma ^{(2)}_{zzz}=\sigma ^{(\textrm {I})}_{zzz}+\sigma ^{(\textrm {II})}_{zzz}+\sigma ^{(\textrm {III})}_{zzz}+\sigma ^{(\textrm {IV})}_{zzz}$. The frequency dependence of $\sigma ^{(2)}_{zzz}$ for the different $t_z$ are shown in Fig. 3(a) and (b). As discussed in the above, due to the linear energy dispersion of Weyl semimetals near the Weyl points, we obtain two contributions to the conductivity $\sigma ^{(2)}_{zzz}$ in the low-frequency region. One coming from the intraband transitions described by the Drude-like effects[see Eq. (14)], and the other coming from the interband-intraband transitions[see the prefactor ($1/\omega ^2$) in Eq. (15)–(17)]. By increasing the tilt of Weyl cones, the absolute value of conductivity $|\sigma ^{(2)}_{zzz}|$ increased firstly, then decreased[Fig. 3(a)]. In the high-frequency region($\Omega \approx 1,2$), the mixed interband-intraband transitions have a strong resonance at $\hbar \omega =\mu$ and a weaker singularity at $\hbar \omega =2\mu$ which corresponding to the two-photons and one-photon processes match the chemical potential gap $2\mu$. The absolute values of the three resonances to the second-order conductivity decreases rapidly with increasing the incident photon energy. Moreover, the interband transitions, which correspond to the one-photon process($\hbar \omega =2\mu$), do not significantly contribute to the total second harmonic conductivity. Since $|\sigma ^{(2)}_{zzz}|$ firstly increases and then decreases with the increasing of the tilt of Weyl cones at high-frequency region($\Omega \approx 1$)[Fig. 3(b)], there is an optimal tilt for the maximum nonlinear conductivity[Fig. 3(c)]. The frequency dependence of $\sigma ^{(2)}_{zzz}$ for the different $\Gamma$ are shown in Fig. 3(d). With decreasing the relaxation constant, the resonant peaks in the nonlinear conductivity increase. This is a universal phenomenon of the actual relaxation dynamics of electrons [49,50].

Fig. 3. The nonlinear conductivity tensor $|\sigma _{zzz}(2\omega )|$ for the process of SHG at (a) low and (b) high frequencies for different tilt $t_z$. (c) $|\sigma _{zzz}(2\omega )|$ as a function of $t_z$ for different $\Omega$ at $\Gamma =0.05$. (d) $|\sigma _{zzz}(2\omega )|$ as a function of $\Omega$ for different $\Gamma$ at $t_z=0.1$. The other parameters are the same as those of Fig. 1.

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Figure 4 shows the SHG optical conductivity as a function of the energy of the incident light. The parameters are taken as $\mu =50$meV, the tilt $t_z$=0.2, the cutoff $\Lambda =1$eV, and a reasonable value for the dephasing rate $\Gamma \simeq 1$meV. Indeed, in the case of Dirac/Weyl semimetals in air, the second-order current in the material is of order $j_{2\omega }^z=\sigma ^{(2)}_{zzz}E_{\omega ,z}^2$. Compared with the experimental results in literatures, the external electric field $E\simeq 2.5\times 10^4$V/m, a laser beam is focused to a spot size of 10-$\mu$m diameter in a crystal. We obtain $j_{2\omega }^z\simeq 5.7\times 10^{-2}$A at $t_z=0.2$ at low frequencies $(\hbar \omega \simeq \Gamma \ll \mu )$ with two-band transitions in our method. And $j_{2\omega }^z\simeq 1.43\times 10^{-4}$A near the high-frequency region $(\hbar \omega \simeq \mu )$. The nonlinear optical polarizability $\chi _{zzz}$ for the process of SHG are given by

(19)$$\chi_{zzz}(2\omega)=\frac{\sigma_{zzz}(2\omega)}{2i\omega\epsilon_0},$$

where $\epsilon _0$ is vacuum permittivity. Our calculated $\chi _{zzz}$ is $5.43\times 10^3$pm/V at high-frequency region. The SHG process in Dirac/Weyl semimetals with broken tilt inversion symmetry originating from the optical interband transitions of the tilted energy band is different from the case considered in Ref. [25], in which the optical properties arise from the crystals (TaAs) without inversion symmetry. The interaction Hamiltonian in the length gauge has been used in their works [25]. However, our solutions are in the same order of magnitude with the experimental results. In the low-frequency region, we find $\chi _{zzz}\simeq 2.2\times 10^7$pm/V. The nonlinear optical polarizability of a Dirac/Weyl semimetal in the low-frequency region is thus about 10000 times larger than it in the high-frequency region. This result is surprisingly high, perhaps the highest among known three-dimensional materials. The Dirac/Weyl semimetals can be used as the source for nonlinear technology.

Fig. 4. The nonlinear conductivity tensor $\sigma _{zzz}(2\omega )$ for the process of SHG as a function of the incident photon energy. The other parameters are taken as $\mu =50$meV, $t_z$=0.2, $\Gamma =1$meV, and $\Lambda =1$eV [42].

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4. Conclusions

In conclusion, we have investigated the second-order nonlinear response of Dirac/Weyl semimetals with broken tilt inversion symmetry in the absence of an external magnetic field under the electric dipole approximation by using the full quantum-mechanical theory. Based on the analytic formula for the second-order nonlinear conductivity, we find two contributions to the conductivity in the low-frequency region, one coming from the intraband transitions and the other from the interband-intraband transitions due to the linear energy dispersion of Weyl semimetals near the Dirac/Weyl points. In the high-frequency region, the appearance of prominent resonant peaks in the nonlinear conductance originates from the two-photon absorption process. The nonlinear conductivity for each node is independent of the chirality.

Appendix

A. Calculation of the first-order conductivity

In this work, we focus on the n-doped Weyl semimetals with a positive chemical potential $\mu$.

Contribution of intraband transitions

In this case, the matrix elements $\hat {\boldsymbol {v}}_{nn}$ of the velocity operator reduce to

(20)$$(\hat{v}_{x})_{nn}=s_nv_F\sin \theta \cos \phi, (\hat{v}_{y})_{nn}=s_nv_F\sin \theta \sin \phi, (\hat{v}_{z})_{nn}=v_F(t_z+s_n\cos \theta).$$

Therefore we obtain

(21)$$\begin{aligned}&\sigma_{xx}^{\textrm{intra}}(\omega)=\sigma_{yy}^{\textrm{intra}}(\omega)\\ =&\frac{ie^2}{\hbar^2\omega}\sum_n \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_x} \left(-\frac{\partial f_n}{\partial \varepsilon_{\boldsymbol{k}_n}}\right) \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_x},\\ =&\frac{ie^2}{\hbar^2\omega(2\pi)^3}\int \frac{k^2 \sin\theta dk d\theta d\phi}{\hbar v_F\left(1+t_z\cos\theta\right)} (\hbar v_F \sin\theta\cos\phi)^2 \delta\left(k-k_F\right)\\ =&\frac{i e^2\mu^2}{8\pi^2\hbar^3 v_F\omega}\left[-\frac{2}{(t_z^2-1)t_z^2}+\frac{1}{t_z^3}\ln\frac{1-t_z}{1+t_z} \right], \end{aligned}$$

where $k_F=\frac {\mu }{\hbar v_F\left (1+t_z\cos \theta \right )}$. Similarly

(22)$$\sigma_{zz}^{\textrm{intra}}(\omega) =\frac{i e^2\mu^2}{4\pi^2\hbar^3 v_F\omega}\left[-\frac{2}{t_z^2}+\ln\frac{1+t_z}{1-t_z} \right].$$

When the tilt parameter $t_z\rightarrow 0$, all diagonal components have the same form:

(23)$$\sigma_{xx}^{\textrm{intra}}(\omega)=\sigma_{yy}^{\textrm{intra}}(\omega)=\sigma_{zz}^{\textrm{intra}}(\omega) =\frac{ie^2\mu^2}{6\pi^2\hbar^3v_F\omega}.$$

Contribution of interband transitions

In this case, the matrix elements $\hat {\boldsymbol {v}}_{nm}$ of the velocity operator reduce to

(24)$$\begin{aligned}&(\hat{v}_{x})_{nm}=v_F(is_m\sin \phi- \cos\theta\cos \phi), (\hat{v}_{y})_{nm}=v_F({-}is_m\cos \phi-\cos \theta\sin \phi),\\ &(\hat{v}_{z})_{nm}=v_F|\sin \theta|. \end{aligned}$$

When $m\neq n$, we obtain

(25)$$\begin{aligned}&\sigma_{xx}^{\textrm{inter}}(\omega)=\sigma_{yy}^{\textrm{inter}}(\omega)\\ =&i e^2\hbar\sum_{mn}\frac{f_n-f_m}{\varepsilon_{\boldsymbol{k}_m}-\varepsilon_{\boldsymbol{k}_n}} \frac{\langle m|\hat{\boldsymbol{v}}_\alpha|n\rangle\langle n|\hat{\boldsymbol{v}}_\beta|m\rangle } {\hbar\omega-(\varepsilon_{\boldsymbol{k}_n}-\varepsilon_{\boldsymbol{k}_m})}\\ =&\frac{i e^2 v_F}{(2\pi)^3}\int \frac{k^2\sin\theta dk d\theta d\phi }{\left(1+t_z\cos\theta\right)} \left[ \frac{\cos^2 \theta \cos^2 \phi +\sin^2 \phi} {\hbar\omega+2\hbar v_F k} +\frac{\cos^2 \theta \cos^2 \phi +\sin^2 \phi}{\hbar\omega-2\hbar v_F k}\right]\\ &\times\frac{1}{2\hbar v_F k}\Theta\left(k-k_F\right)\\ =&\frac{i e^2}{16\pi^2\hbar^2 v_F}\int_{{-}1}^1 dx\frac{x^2 +1}{1+t_zx} \int_{k_F}^\infty dk \left[ \frac{\hbar v_Fk} {\hbar\omega+2\hbar v_F k} +\frac{\hbar v_Fk}{\hbar\omega-2\hbar v_F k}\right]. \end{aligned}$$

Similarly

(26)$$\sigma_{zz}^{\textrm{inter}}(\omega) =\frac{i e^2}{8\pi^2\hbar^2 v_F}\int_{{-}1}^1 dx\frac{1-x^2}{1+t_zx} \int_{k_F}^\infty dk \left[ \frac{\hbar v_Fk} {\hbar\omega+2\hbar v_F k} +\frac{\hbar v_Fk}{\hbar\omega-2\hbar v_F k}\right].$$

When the tilt parameter $t_z\rightarrow 0$, all diagonal components have the same form:

(27)$$\sigma_{xx}^{\textrm{inter}}(\omega)=\sigma_{yy}^{\textrm{inter}}(\omega)=\sigma_{zz}^{\textrm{inter}}(\omega) ={-}\frac{i e^2}{24\pi^2\hbar^2 v_F}\hbar\omega\ln\frac{4\Lambda^2-(\hbar\omega)^2}{4\mu^2-(\hbar\omega)^2}.$$

where $\Lambda$ is the cutoff.

B. Calculation of the second-order conductivity

In this section, we calculate the different contributions to the second-order conductivity defined by Eqs. (14)–(17).

(28)$$\begin{aligned}\sigma^{(\textrm{I})}_{zzz}(2\omega) &=\frac{e^3}{2\hbar^3\omega^2}\sum_{n} \frac{\partial f_n}{\partial \varepsilon_{\boldsymbol{k}_n}} \frac{\partial\varepsilon_{\boldsymbol{k}_n}}{\partial k_z} \frac{\partial^2\varepsilon_{\boldsymbol{k}_n}}{\partial k_z \partial k_z}\\ &={-}\frac{e^3}{16\pi^3\hbar^3\omega^2}\int \frac{d^3k}{\hbar v_F\left(1+t_z\cos\theta\right)}\delta\left(k-k_F\right) \hbar v_F\left(t_z+\frac{k_z}{k}\right) \hbar v_F\frac{k_x^2+k_y^2}{k^3}\\ &=\frac{ e^3\mu}{8\pi^2\hbar^3\omega^2}\left[\frac{4t_z^2-6}{t_z^3}-\frac{3(t_z^2-1)}{t_z^4}\ln\frac{1+t_z}{1-t_z} \right]. \end{aligned}$$

And

(29)$$\begin{aligned}\sigma^{(\textrm{II})}_{zzz}(2\omega) &=\frac{e^3}{\omega^2}\sum_{mn}\frac{(\hat{v}_{z})_{nn} (\hat{v}_{z})_{nm}(\hat{v}_{z})_{mn}} {\left[\hbar\omega-\varepsilon_{mn}\right]\left[\hbar\omega+\varepsilon_{mn}\right]} \left(f_n-f_m\right)\\ &=\frac{e^3}{4\pi^3\omega^2}\int \frac{ k^2\sin\theta dk d\theta d\phi}{\hbar v_F\left(1+t_z\cos\theta\right)} \frac{v_F^3\cos \theta\sin^2 \theta }{\left[\hbar\omega+2\hbar v_Fk\right]\left[\hbar\omega-2\hbar v_Fk\right]} \Theta\left(k-k_F\right)\\ &=\frac{e^3}{2\pi^2\hbar^3\omega^2}\int_{{-}1}^1 dx \frac{x( 1- x^2)}{ \left(1+t_zx\right)} \int_{k_F}^\infty dk \frac{(\hbar v_Fk)^2} {\left(\hbar\omega+2\hbar v_Fk\right)\left(\hbar\omega-2\hbar v_Fk\right)}. \end{aligned}$$

Similarly, we can obtain

(30)$$\sigma^{(\textrm{III})}_{zzz}(2\omega) =\frac{e^3\mu^2}{8\pi^2\hbar^3\omega^2}\int_{{-}1}^1 dx \frac{(x^2-1)(t_z+x)}{\left(1+t_zx\right)^3} \left[\frac{ 1}{\hbar\omega+\hbar v_Fk_F} -\frac{1}{\hbar\omega-\hbar v_Fk_F}\right],$$

And

(31)$$\begin{aligned}\sigma^{(\textrm{IV})}_{zzz}(2\omega) =&\frac{e^3}{2\pi^2\hbar^3\omega^2}\int_{{-}1}^1 dx \frac{(x^2-1)x}{\left(1+t_zx\right)} \int_{k_F}^\infty dk\left[\frac{(\hbar v_F k)^2} {[2\hbar\omega-2\hbar v_Fk][\hbar\omega-2\hbar v_Fk]}\right.\\ &\left.+\frac{(\hbar v_F k)^2} {[2\hbar\omega+2\hbar v_Fk][\hbar\omega+2\hbar v_Fk]}\right]. \end{aligned}$$

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Second harmonic generation in Dirac/Weyl semimetals with broken tilt inversion symmetry

Abstract

1. Introduction

2. Linear response of Weyl semimetals

3. Second order response of Weyl semimetals

4. Conclusions

Appendix

A. Calculation of the first-order conductivity

Contribution of intraband transitions

Contribution of interband transitions

B. Calculation of the second-order conductivity

Disclosures

References

Cited By

Figures (4)

Equations (31)

Optics Express