Analysis of model parameter dependences on the second-order nonlinear conductivity in $\mathcal{PT}$-symmetric collinear antiferromagnetic metals with magnetic toroidal moment on zigzag chains

The second-order nonlinear conductivity tensor $\sigma_{\mu\nu\lambda}$ defined as $J_{\mu}=\sigma_{\mu\nu\lambda}E_{\nu}E_{\lambda}$ ($\mu,\nu,\lambda=x,y,z$) is calculated on the basis of the second-order Kubo formula [22]. In the clean limit, the intraband contribution is dominant, which is given by

where $e(>0)$, $\tau$, $\hbar$, and $V$ are the elementary charge, relaxation time, the reduced Planck constant, and the system volume, respectively ¹¹1There is no contribution from the Berry curvature dipole [75] because of the $\mathcal{PT}$ symmetry, while the interband contribution in the $\mathcal{T}$-breaking system [76] is neglected by considering the clean limit.. $f[\varepsilon_{n}({\bm{k}})]$ is the Fermi distribution function for the $n$th-band eigen energy $\varepsilon_{n}({\bm{k}})$. The intraband contribution in Eq. (9) represents the Drude-type one with the dissipation $\tau^{-2}$, whose expression eventually coincides with that obtained by the Boltzmann formalism [51, 52, 22, 53]. Hereafter, we use the scaled $\sigma_{\mu\nu\lambda}$ as $\bar{\sigma}_{\mu\nu\lambda}=\sigma_{\mu\nu\lambda}/(e^{3}\tau^{2}\hbar^{-3})$.

From Eq. (9), one finds the relation $\sigma_{\mu\nu\nu}=\sigma_{\nu\mu\nu}$ by integration by parts. This indicates that the Drude-type nonlinear conductivity $\sigma_{\mu\nu\lambda}$ is the totally symmetric rank-3 tensor with 10 independent components: $\sigma_{xxx}$, $\sigma_{yyy}$, $\sigma_{zzz}$, $\sigma_{xyy}$, $\sigma_{yzz}$, $\sigma_{zxx}$, $\sigma_{xxy}$, $\sigma_{yyz}$, $\sigma_{zzx}$, and $\sigma_{xyz}$. As $\sigma_{\mu\nu\lambda}$ is a third-rank polar time-reversal-odd tensor, i.e., $\sigma_{\mu\nu\lambda}\to-\sigma_{\mu\nu\lambda}$ under $\mathcal{P}$ or $\mathcal{T}$ operation but $\sigma_{\mu\nu\lambda}\to\sigma_{\mu\nu\lambda}$ under $\mathcal{PT}$ operation, it becomes nonzero when both the spatial inversion and time-reversal symmetries are absent. From the multipole viewpoint, above symmetry requirement means that the nonzero tensor components are related to the active odd-parity MT multipoles [54, 55, 56, 42, 57]: three rank-1 MT dipoles $(T_{x},T_{y},T_{z})$ and seven rank-3 MT octupoles $(T_{xyz},T_{x}^{\alpha},T_{y}^{\alpha},T_{z}^{\alpha},T_{x}^{\beta},T_{y}^{\beta},T_{z}^{\beta})$, whose correspondence is given by [39]

where the functional forms of dipoles and octupoles are summarized in Appendix A. The correspondence in Eq. (III.1) is obtained by decomposing $\sigma_{\mu\nu\lambda}$ into the tensor components with the same rotational symmetry to the dipoles and octupoles (See also Appendix A). When the MT dipole and/or MT octupole in Eq. (III.1) are activated in an AFM metal, the corresponding tensor component of $\sigma_{\mu\nu\lambda}$ becomes nonzero. From Eq. (III.1), one finds that MT dipole $T_{\mu}$ is relevant to the longitudinal component $\sigma_{\mu\mu\mu}$ and the transverse components $\sigma_{\mu\nu\nu}$ and $\sigma_{\nu\mu\nu}$ ($\nu\neq\mu$). It means that both nonreciprocal conductivity and nonlinear transverse conductivity are expected to be realized in the presence of the MT dipole, i.e., ferrotoroidal metals [58, 59, 39].

In the present system under the magnetic point-group $m^{\prime}mm$ with the nonzero MT moment $T_{x}^{\rm MF}$, five components $\sigma_{xxx}$, $\sigma_{xyy}$, $\sigma_{yxy}$, $\sigma_{xzz}$, and $\sigma_{zzx}$ can be nonzero, since $T_{x}$, $T_{x}^{\alpha}$, and $T_{x}^{\beta}$ in Eq. (III.1) belong to the totally symmetric irreducible representation [39]. Among them, $\sigma_{xyy}$ and $\sigma_{yxy}$ vanish owing to $k_{y}=0$ in the present two-dimensional system. In addition to the nonzero contribution from the linear conductivity $\sigma_{xx}$, $\sigma_{xxx}$ results in the nonreciprocal current, while $\sigma_{xzz}$ without linear $\sigma_{xz}$ leads to the pure second-order transverse current, respectively.

In the presence of the MT moment $T_{x}^{\rm MF}$, the linear magnetoelectric tensor $\alpha_{\mu\nu}$ in $M_{\mu}=\alpha_{\mu\nu}E_{\nu}$ ($\mu,\nu=x,y,z$) is also finite. We calculate the linear magnetoelectric tensor by the linear response theory as

where $g$ and $\mu_{\rm B}$ are the g factor ($g=2$) and Bohr magneton, respectively. $\sigma_{\mu{\bm{k}}}^{nm}=\braket{n{\bm{k}}}{\sigma_{\mu}}{m{\bm{k}}}$ and $\varv_{\nu{\bm{k}}}^{mn}=\braket{m{\bm{k}}}{\varv_{\nu{\bm{k}}}}{n{\bm{k}}}$ are the matrix elements of spin $\sigma_{\mu}$ and velocity $\varv_{\nu{\bm{k}}}=\partial\mathcal{H}/(\hbar\partial k_{\nu})$ for the eigenstate $\ket{n{\bm{k}}}$. We use the scaled $\bar{\alpha}_{\mu\nu}=\alpha_{\mu\nu}/(e\mu_{\rm B}\hbar)$ in the following discussion.

As $\alpha_{\mu\nu}$ in a $\mathcal{PT}$ symmetric system is relevant to the rank-0–2 odd-parity multipoles: magnetic monopole $M_{0}$, MT dipoles $(T_{x},T_{y},T_{z})$, and magnetic quadrupoles $(M_{u},M_{v},M_{yz},M_{zx},M_{xy})$ (see also Appendix A), the relation is represented as follows [31, 32]:

Since $T_{x}$ and $M_{yz}$ become active for $T_{x}^{\rm AF}\neq 0$ in the present system, $\alpha_{yz}$ and $\alpha_{zy}$ are expected to be nonzero. As $\alpha_{zy}$ is zero due to the two dimensionality, we only consider $\alpha_{yz}$.

For the following discussion, we also present the linear Hall conductivity

We use the scaled value $\bar{\sigma}_{xz}=\sigma_{xz}/(e^{2}\hbar H_{y})$ in the following, where $H_{y}$ is the Zeeman field along the $y$ direction.

We first show the numerical result of the longitudinal nonlinear conductivity $\bar{\sigma}_{xxx}$. Figure 2(a) shows $\bar{\sigma}_{xxx}$ as a function of $T$ for various $\alpha_{1}=0.1$–$0.5$ at $\alpha_{2}=0.1$. The $T$ dependence for different $\alpha_{1}$ is qualitatively similar; $\bar{\sigma}_{xxx}$ is largely enhanced just below $T=T_{{\rm N}}$, and shows maximum with decrease of $T$. While further decreasing $T$, $\bar{\sigma}_{xxx}$ shows the sign change, and then reaches a negative value at the lowest $T$.

The nonzero $\sigma_{xxx}$ is closely related to the formation of the asymmetric band structure under $T_{x}^{\rm MF}\neq 0$, since $\sigma_{xxx}$ has the same symmetry as $T_{x}^{\rm MF}$ [39]. As the asymmetric band modulation is caused by the coupling between $\tilde{T}_{x}^{\rm MF}$ and $\alpha_{1}$, they are indispensable for nonzero $\sigma_{xxx}$. Indeed, $\bar{\sigma}_{xxx}$ vanishes for $\alpha_{1}=0$ or $\tilde{T}_{x}^{\rm MF}=0$. Moreover, $\bar{\sigma}_{xxx}$ is well scaled by $\bar{\sigma}_{xxx}/\alpha_{1}$ at low temperatures $T\lesssim 0.7T_{\rm N}$ for small $\alpha_{1}$. See Sec. V.1 for the essential model parameters in details.

Meanwhile, $\bar{\sigma}_{xxx}$ is not scaled by $\alpha_{1}$ for $0.7\lesssim T/T_{\rm N}\leq 1$ in the region where $\bar{\sigma}_{xxx}$ is drastically enhanced. This is attributed to the rapid increase of $\tilde{T}_{x}^{\rm MF}$ and resultant drastic change of the electronic band structure near the Fermi level. As $\bar{\sigma}_{xxx}$ in Eq. (9) includes the factors $\partial^{2}\varepsilon_{n}({\bm{k}})/\partial k_{x}^{2}$ and $\partial\varepsilon_{n}({\bm{k}})/\partial k_{x}$, the small $X(\bm{k})$ appearing in the denominator of $\partial^{2}\varepsilon_{n}({\bm{k}})/\partial k_{x}^{2}$ and $\partial\varepsilon_{n}({\bm{k}})/\partial k_{x}$ gives a dominant contribution. When considering the small order parameter compared to the ASOI, i.e., $\tilde{T}_{x}^{\rm MF}\lesssim\alpha_{1}$, $X(\bm{k})$ can become small when the Fermi wavenumber $k^{\rm F}_{x}$ satisfies $\tilde{T}_{x}^{\rm MF}\simeq\alpha_{1}\sin k^{\rm F}_{x}$, which results in a large enhancement of $\bar{\sigma}_{xxx}$. Such an enhancement is remarkable when the upper and lower bands are closely located in the paramagnetic state with small $X(\bm{k})$ as shown in Fig. 1(d), which can be realized for small $t_{1}=0.1$ and $\alpha_{2}=0.1$. In short, there are two conditions for large $\bar{\sigma}_{xxx}$: One is the large essential parameters, such as $\alpha_{1}$, $T_{x}^{\rm MF}$, and $J_{\rm AF}$, and the other is to satisfy $\tilde{T}_{x}^{\rm MF}\simeq\alpha_{1}\sin k^{\rm F}_{x}$ in a multi-band system. These conditions can be experimentally controlled by electron/hole doping and temperature.

The sign change of $\bar{\sigma}_{xxx}$ in $T$ dependence is owing to the multiband effect. As shown in Fig. 1(d), the band bottom is shifted in the opposite direction for the upper and lower bands, which means that the opposite sign of the coupling $\alpha_{1}\tilde{T}_{x}^{\rm MF}$ results in the opposite contribution to $\bar{\sigma}_{xxx}$. This is demonstrated by decomposing $\bar{\sigma}_{xxx}$ into the upper- and lower-band contributions, as shown in Fig. 2(b). The results indicate that the dominant contribution of $\bar{\sigma}_{xxx}$ arises from the upper band for $0.9\lesssim T/T_{\rm N}\leq 1$, while that arises from the lower band for $T/T_{\rm N}\lesssim 0.9$. The suppression of the upper-band contribution for low $T$ is because it becomes away from the Fermi level by the development of $T_{x}^{\rm MF}$.

Next, let us discuss the transverse nonlinear conductivity $\bar{\sigma}_{xzz}$. Figure 3 shows the $T$ dependence of $\bar{\sigma}_{xzz}$ for $0.02\leq\alpha_{1},\alpha_{2}\leq 0.1$ with $\alpha_{1}=\alpha_{2}$. The behavior of $\bar{\sigma}_{xzz}$ against $T$ is similar to $\bar{\sigma}_{xxx}$ except for the sign change; $\bar{\sigma}_{xzz}$ becomes nonzero below $T=T_{\rm N}$ and shows the maximum near $T_{\rm N}$. While decreasing $T$, $\bar{\sigma}_{xzz}$ is suppressed and shows an almost constant value.

Similar to $\sigma_{xxx}$, the origin of nonzero $\sigma_{xzz}$ is the asymmetric band modulation under $T_{x}^{\rm MF}\neq 0$ via the effective coupling $\tilde{T}_{x}^{\rm MF}\alpha_{1}$. Besides, we find another contribution from $\alpha_{2}$ for nonzero $\sigma_{xzz}$ in contrast to $\sigma_{xxx}$, where $\bar{\sigma}_{xzz}$ is well scaled by $\alpha_{1}\alpha_{2}^{2}$ as shown in the inset of Fig. 3, as discussed in Sec. V.1. The additional parameter dependence for $\alpha_{2}^{2}$ is owing to an additional symmetry between $k_{z}$ and $k_{z}+\pi$ for $\alpha_{2}=0$, which gives the opposite-sign contribution to $\sigma_{xzz}$ so that totally $\sigma_{xzz}=0$.

We also present another MT-moment-driven phenomena, the magnetoelectric response, and compare its parameter and $T$ dependence to the nonlinear conductivities obtained in the previous section. Figure 4(a) shows the $T$ dependence of $\bar{\alpha}_{yz}$ for $0.02\leq\alpha_{1},\alpha_{2}\leq 0.1$ with $\alpha_{1}=\alpha_{2}$, whose behavior is similar to the transverse nonlinear conductivity $\sigma_{xzz}$ in Fig. 3 except for the sign. $\bar{\alpha}_{yz}$ is nonzero even if $\alpha_{1}=0$ that is different from the nonlinear conductivities, whereas $\alpha_{2}$ and $\tilde{T}_{x}^{\rm MF}$ are essential to obtain the finite $\bar{\alpha}_{yz}$, as detailed in Sec. V.1. As shown in the inset of Fig. 3(a), $\bar{\alpha}_{yz}$ is well scaled as $\bar{\alpha}_{yz}/\alpha_{2}$ for small $\alpha_{2}$.

Moreover, it is noteworthy to comment on the relation between the transverse nonlinear conductivity and a combination of the linear magnetoelectric and Hall coefficients, since the nonlinear transverse transport in the $\mathcal{PT}$-symmetric AFMs can be understood as the Hall transport driven by the induced magnetization through the linear magnetoelectric response at the phenomenological level [14, 21].

We show the $T$ dependence of $\bar{\sigma}_{xz}\bar{\alpha}_{yz}$ in Fig. 4(b) for the same parameters in Fig. 3. The small magnetic field $H_{y}=0.01$ is introduced to mimic the induced magnetization in $\alpha_{yz}$. Compared to the results in Fig. 3 and 4(b), one finds the resemblance between the $T$ dependences of $\bar{\sigma}_{xzz}$ and $\bar{\sigma}_{xz}\bar{\alpha}_{yz}$, both of which are scaled by $\alpha_{1}\alpha_{2}^{2}$. A good qualitative correspondence in these responses indicates that the interpretation of dividing subsequent two linear processes for nonlinear conductivity is reasonable in the present model. The overall quantitative difference $\bar{\sigma}_{xz}\bar{\alpha}_{yz}/\bar{\sigma}_{xzz}\sim 10^{-2}$ may be ascribed to the magnitude of the used internal magnetic field ($H_{y}=0.01$) that should be replaced by the true internal field. However, it is hard to estimate it quantitatively.

We discuss the parameter dependences of the asymmetric band modulation, nonlinear conductivity, and the linear magnetoelectric and Hall coefficients at the level of the microscopic model Hamiltonian. For this purpose, we try to extract the essential parameters for each response from various hoppings, spin-orbit coupling, and internal/external field in the model Hamiltonian based on the method in Refs. 60, 61. In the following, we discuss the important model parameters in each case one by one, and the results are summarized in Table 1. The derivation is shown in Appendix B.

First, the essential parameters for the asymmetric band modulation [60] are given by $\tilde{T}_{x}^{\rm MF}\alpha_{1}$, as shown in Appendix B.1. The result is consistent with the eigenvalues in Eq. (7).

Next, the essential model parameters for $\sigma_{xxx}$ [61] (see also Appendix B.2) are given by

where $F$ and $F^{\prime}$ represent the arbitrary functions. Note that only the even power of $\alpha_{1}$ and $\tilde{T}_{x}^{\rm MF}$ appears in $F$ and $F^{\prime}$. Thus, one finds that the coupling of $\alpha_{1}$ and $\tilde{T}_{x}^{\rm MF}$ is always necessary to induce $\sigma_{xxx}$, which is consistent with the numerical result presented in Sec. IV.1. Moreover, $\sigma_{xxx}$ is closely related to the asymmetric band modulation because both of them are characterized by the same essential model parameters.

Similarly, the essential model parameters of $\sigma_{xzz}$ are given by

where the even power of $\alpha_{1}$, $\alpha_{2}$, and $\tilde{T}_{x}^{\rm MF}$ appears in $F$. Equation (16) shows that the coupling of $\alpha_{1}$ and $\tilde{T}_{x}^{\rm MF}$ is essential to induce $\sigma_{xzz}$ as similar to $\sigma_{xxx}$, which is consistent with the numerical result in Sec. IV.2. Moreover, Eq. (16) indicates that $t_{2}$ and even power of $\alpha_{2}$ are also necessary for $\sigma_{xzz}$ in the present model in Eq. (1).

In a similar way, the essential model parameters to induce $\alpha_{yz}$ and $\sigma_{xz}$ are given by

This indicates that nonzero $\alpha_{yz}\sigma_{xz}$ needs nonzero $\alpha_{1}\alpha_{2}^{2}\tilde{T}_{x}^{\rm MF}$, which shows a good agreement with the condition for $\sigma_{xzz}$. The common essential model-parameter dependence in small parameter region was already confirmed in Secs. IV.2 and IV.3.

It is noteworthy that the above approach to extract the essential model parameters can be straightforwardly applied even when introducing the other model parameters. For example, let us consider the additional interlayer A-B hopping $t_{4}$ in the model Hamiltonian. In this situation, one finds that there is no longer simple correlation between $\sigma_{xzz}$ and $\sigma_{xz}\alpha_{yz}$; the essential model parameters for the former are $\alpha_{1}\tilde{T}_{x}^{\rm MF}$ rather than $\alpha_{1}\alpha_{2}^{2}\tilde{T}_{x}^{\rm MF}$, while those for the latter still remains the same as $\alpha_{1}\alpha_{2}^{2}\tilde{T}_{x}^{\rm MF}$ as discussed in Appendix B. In other words, the factor $\alpha^{2}_{2}t_{2}$ in the square bracket in Eq. (16) is not truly the essential factor. Indeed, the numerical results in the presence of $t_{4}$ give a different temperature dependence from each other, as shown in Appendix C. Thus, the correspondence between $\sigma_{xzz}$ and $\sigma_{xz}\alpha_{yz}$ occurs depending on the hopping in the effective model, which is clarified by performing a procedure in Appendix B.

Finally, we discuss the order estimate of the nonlinear conductivity for $\alpha_{1}=0.5$ and $\alpha_{2}=0.1$ by the ratio $\sigma_{xxx}/(\sigma_{xx})^{2}$ with being independent of the relaxation time in the clean limit. By putting the typical values as $a\sim 0.5$ [nm] and $|t_{2}|=0.2$ eV, we obtain $\sigma_{xxx}/(\sigma_{xx})^{2}\sim 10^{-3}\hbar a^{2}e^{-1}|t_{2}|^{-1}\sim 10^{-18}$ [m³ A^-1] for $T\to 0$ and $10^{-17}$ [m³ A^-1] near $T_{\rm N}$, which is comparable to the value in the 2D nonmagnetic Rashba system under the magnetic field [51]. Further enhancement can be achieved by tuning the model parameters and electron filling.