Phenomenological analysis of transverse thermoelectric generation and cooling performance in magnetic/thermoelectric hybrid systems

By solving Eq. (1) with the aforementioned boundary conditions, we obtain the transverse thermopower for the STTG system as

where $S_{\text{TE(M)}}$ is the Seebeck coefficient of the thermoelectric (magnetic) material and $r\equiv L_{\text{TE}}^{y}L_{\text{TE}}^{z}/(L_{\text{M}}^{y}L_{\text{M}}^{z})$ is the size ratio between the thermoelectric and magnetic materials; see Methods in Ref. Zou2020, for further details of the derivation. Equation (5) shows that the transverse thermopower for the STTG system can be enhanced owing to the superposition of the ANE contribution in the magnetic layer (first term) and the Seebeck-driven AHE: the STTG contribution (second term). Importantly, the second term can be designed by the combination of the thermoelectric and magnetic layers as well as their dimensions, i.e., $r$. With increasing $r$, the second term becomes effective and approaches $\tan\theta_{\text{H}}(S_{\text{M}}-S_{\text{TE}})$, where $\theta_{\text{H}}\equiv\tan^{-1}(\rho_{\text{AHE}}/\rho_{\text{M}})$ is the anomalous Hall angle of the magnetic layer. Therefore, a large second term needs large values of $\theta_{\text{H}}$, $S_{\text{TE}}$, and $r$ because $S_{\text{TE}}\gg S_{\text{M}}$ for typical materials.

We demonstrate the behavior of $S_{\text{tot}}^{y}$ using the parameters of typical thermoelectric materials (n-type Si(1), n-type Si(2), p-type Si, and $\text{Bi}_{2}\text{Te}_{3}$) and magnetic materials (Ni, $\text{Co}_{2}\text{MnGa}$, $\text{L}1_{0}$-ordered FePt, NdFeB, and $\text{SmCo}_{5}$) shown in Table 1, where n-type Si(1) and (2) have different transport properties and hereafter "type" is omitted. $\text{Co}_{2}\text{MnGa}$ is a Heusler ferromagnet showing large AHE and ANE.Sakai2018NatPhysCo2MnGa ; Reichlova2018APLCo2MnGafilm ; SakurabaCMG , $\text{L}1_{0}$-ordered FePt is often used in spintronics because it has a strong perpendicular magnetic anisotropy in a thin film form. Seki2008NmatFePt1 ; Seki2011APLFePt2 NdFeB and $\text{SmCo}_{5}$ are rare-earth permanent magnets in practical use, which are known to exhibit substantially large AHE. Miura2019APLSmCo5 In the previous work by Zhou et al., Ni, $\text{Co}_{2}\text{MnGa}$, and FePt were used for the experimental demonstration of STTG. Zou2020 Figure 2(a) [2(b)] shows $S_{\text{tot}}^{y}$ as a function of $r$ for the combination of the thermoelectric materials and Ni ($\text{Co}_{2}\text{MnGa}$). In the case of Ni with negative $\tan\theta_{\text{H}}$, the second term of Eq. (5) is negative (positive) for n(p)-Si because of negative (positive) $S_{\text{TE}}$ and can be larger than $S_{\text{ANE}}$ owing to large $S_{\text{TE}}$ when $r$ is large. We also find that the $r$ value at which the Seebeck-driven contribution is apparent with respect to $S_{\text{ANE}}$ for n-Si(1) and p-Si is larger than that for n-Si(2) and $\text{Bi}_{2}\text{Te}_{3}$ due to small $\rho_{\text{TE}}$ of n-Si(2) and $\text{Bi}_{2}\text{Te}_{3}$. As shown in Fig. 2(b), the second term of Eq. (5) for $\text{Co}_{2}\text{MnGa}$ with positive $\tan\theta_{\text{H}}$ has the opposite sign to that for Ni for the same thermoelectric material. Importantly, when the magnetic material with large $\tan\theta_{\text{H}}$ is used, we can obtain the transverse thermopower of $>100\ \mu\text{V }\text{K}^{-1}$, which is more than an order of magnitude larger than $S_{\text{ANE}}$ of $\text{Co}_{2}\text{MnGa}$. The STTG contribution can further be increased in proportion to $\tan\theta_{\text{H}}$, as exemplified in the inset to Fig. 2(b). In Figs. 2(c)-2(e), we show the behavior of $S_{\text{tot}}^{y}$ as a function of $r$ for $\text{L}1_{0}$-ordered FePt [Fig. 2(c)], NdFeB [Fig. 2(d)], and $\text{SmCo}_{5}$ [Fig. 2(e)], which enable STTG in the absence of external magnetic fields owing to their large coercive force and remanent magnetization.Miura2019APLSmCo5 ; Zou2020 We find that NdFeB with relatively large $\tan\theta_{\text{H}}$ exhibits a larger STTG contribution than FePt and $\text{SmCo}_{5}$, although $|S_{\text{ANE}}|$ of NdFeB is smaller than that of FePt and $\text{SmCo}_{5}$.Miura2019APLSmCo5 These demonstrations show that the transverse thermopower induced by STTG can be more than an order of magnitude larger than that induced by ANE by optimizing the combination of the thermoelectric and magnetic materials as well as their dimensions.

Here, we show that the insulator layer in Fig. 1 is important to obtain large STTG. For the system in which thermoelectric and magnetic materials are directly connected in the $z$ direction, we can derive the transverse thermopower by solving Eq. (1) with the boundary condition based on the Maxwell’s equations: $E_{\text{TE}}^{x(y)}=E_{\text{M}}^{x(y)}$. When the closed circuit is formed in the $x$ direction, the equivalent circuit in the $y$ direction gives $V_{\text{M}}^{y}=(I_{\text{M}}^{y}+I_{\text{TE}}^{y})R_{\text{load}}$. The transverse thermopower in this case is calculated as

where we assume $L_{\text{TE}}^{x(y)}=L_{\text{M}}^{x(y)}$ and the linear temperature gradient along the $x$ direction for simplicity. Here, $\rho_{\text{eff}}^{y}$ is the effective transverse resistivity of the STTG system Zou2020 :

Since the shunting factor $1/[1+r(\rho_{\text{eff}}^{y}/\rho_{\text{TE}})]\leq 1$, we obtain smaller transverse thermopower when the thermoelectric and magnetic materials are directly connected: $S_{\text{shunt}}^{y}\leq S_{\text{tot}}^{y}$. In the following, therefore, we focus only on the case shown in Fig. 1.

We next discuss the power factor (PF) for the STTG system. To derive PF,Zou2020 we first calculate the maximum output power generated by the transverse thermoelectric voltage, $P_{\text{out}}\equiv(V_{\text{M}}^{y})^{2}/R_{\text{load}}$, with respect to the load resistance. We calculate the maximum power $(P_{\text{out}})_{\text{max}}$ as follows:

where we used $\Delta T\simeq-L^{x}\nabla_{x}T$ with $\nabla_{x}T\simeq\nabla_{x}T_{\text{TE}}\simeq\nabla_{x}T_{\text{M}}$. We then normalize $(P_{\text{out}})_{\text{max}}$ by the temperature gradient and the volume of the magnetic material and obtain PF for the STTG system as

This expression becomes equivalent to the power factor for ANE, ${S_{\text{ANE}}}^{2}/\rho_{\text{M}}$, by taking the limit $r\to 0$. Since the second term of Eq. (7) is usually small, the parameter dependence of PF is determined mainly by $S_{\text{tot}}^{y}$. Although PF in Eq. (9) can be one or two orders of magnitude larger than that for ANE with appropriate choice and design of materials in a similar manner to $S_{\text{tot}}^{y}$, the difference between the volume of the magnetic material and the total volume of the STTG system should be taken into account to discuss the thermoelectric performance and efficiency of STTG.

Now, we are in a position to discuss the figure of merit for the transverse thermoelectric conversion in the STTG system. We maximize the efficiency $\eta\equiv P_{\text{out}}/(Q_{\text{TE}}^{x}|_{T=T_{\text{h}}}+Q_{\text{M}}^{x}|_{T=T_{\text{h}}})$ with respect to the load resistance $R_{\text{load}}$, where $Q_{\text{TE(M)}}^{x}|_{T=T_{\text{h}}}$ is the heat current from the hot reservoir in the thermoelectric (magnetic) material in the $x$ direction. We then obtain the maximum efficiency as

where $\eta_{\text{c}}=1-T_{\text{c}}/T_{\text{h}}$ is the Carnot efficiency and $\overline{T}=(T_{\text{h}}+T_{\text{c}})/2$ is the average temperature of the hot and cold reservoirs. Here

is the isothermal figure of merit for the STTG system, where

is the effective thermal conductivity of the STTG system with $\kappa_{\text{TE(M)}}$ being the thermal conductivity of the thermoelectric (magnetic) material. Here, $Z_{\text{tot}}\overline{T}\leq 1$, which is a characteristic of the isothermal figure of merit for the transverse thermoelectric conversion.Harman1962JAPTheoryNernst In the same manner as $S_{\text{tot}}^{y}$ and PF, $Z_{\text{tot}}\overline{T}$ reduces to the isothermal figure of merit for ANE Harman1962JAPTheoryNernst when $r\to 0$.

Although $Z_{\text{tot}}\overline{T}$ is a complicated function of parameters of thermoelectric and magnetic materials, $Z_{\text{tot}}\overline{T}$ can be enhanced for a combination of a magnetic material with large $\tan\theta_{\text{H}}$ and a thermoelectric material with large figure of merit for the Seebeck effect, $Z_{\text{TE}}\overline{T}={S_{\text{TE}}}^{2}\overline{T}/(\rho_{\text{TE}}\kappa_{\text{TE}})$, as discussed in Ref. Zou2020, . This can be intuitively understood with naive approximations as follows. The second term of $\rho_{\text{eff}}^{y}$ [Eq. (7)] and the third term of $\kappa_{\text{eff}}^{x}$ [Eq. (12)] are negligibly small for usual materials. In addition, we assume $\rho_{\text{M}}\kappa_{\text{M}}\ll\rho_{\text{TE}}\kappa_{\text{TE}}$ and consider only the STTG contribution in $S_{\text{tot}}^{y}$. With these approximations, we find that the figure of merit for STTG takes its maximum of $Z_{\text{tot}}\overline{T}=Z_{\text{TE}}\overline{T}(\tan\theta_{\text{H}})^{2}/4$ at $r=\rho_{\text{TE}}/\rho_{\text{M}}$. This approximation means that a maximum value of the figure of merit for STTG is mainly determined by $\tan\theta_{\text{H}}$ and $Z_{\text{TE}}\overline{T}$.

The behaviors of $Z_{\text{tot}}\overline{T}$ are exemplified in Fig. 3 (note that the aforementioned approximations are not applied here). Figure 3(a) shows $Z_{\text{tot}}\overline{T}$ for $\text{Co}_{2}\text{MnGa}$ at $\overline{T}=300\text{ K}$ as a function of $r$ and $S_{\text{TE}}$. Although $S_{\text{TE}}$ is treated as a parameter, ${S_{\text{TE}}}^{2}/\rho_{\text{TE}}$ and $\kappa_{\text{TE}}$ are fixed at the values for $\text{Bi}_{2}\text{Te}_{3}$, i.e., $Z_{\text{TE}}\overline{T}=\text{const.}$, where the black curve in Fig. 3(a) shows the case for the $\text{Co}_{2}\text{MnGa}$/$\text{Bi}_{2}\text{Te}_{3}$ junction with the parameters shown in Table 1. By choosing optimal $r$, $Z_{\text{tot}}\overline{T}$ can be larger than the figure of merit for ANE in $\text{Co}_{2}\text{MnGa}$ both for positive and negative $S_{\text{TE}}$ values. The difference in the maximum $Z_{\text{tot}}\overline{T}$ between positive and negative $S_{\text{TE}}$ values is attributed to the $S_{\text{ANE}}$ offset in Eq. (5); the situation depends on the sign and magnitude of $S_{\text{ANE}}$. As shown in Fig. 3(b), when $\tan\theta_{\text{H}}=0.5$ is assumed, the maximum value of $Z_{\text{tot}}\overline{T}$ is dramatically improved in comparison with the figure of merit for ANE in $\text{Co}_{2}\text{MnGa}$. Here, due to the dominant contribution of STTG, the significant improvement of $Z_{\text{tot}}\overline{T}$ appears both for positive and negative $S_{\text{TE}}$ values. This demonstration shows that not only $S_{\text{tot}}^{y}$ and PF but also $Z_{\text{tot}}\overline{T}$ can be enhanced by AHE driven by the Seebeck effect.

Finally, we mention that the STTG system also works as a transverse thermoelectric temperature modulator by replacing the load resistance with an external battery in Fig. 1. The external battery induces a charge current in the $y$ direction, and the charge current is bent in the $x$ direction by AHE. When the charge current in the $x$ direction flows in the closed circuit comprising the thermoelectric and magnetic materials, heat is generated or absorbed at the junctions by the Peltier effect. This is the reciprocal process of STTG. The resultant temperature gradient under the adiabatic condition in the $x$ direction is calculated as follows. Since we use the same boundary conditions for $T_{\text{TE(M)}}$, the temperature gradients in the thermoelectric and magnetic materials are the same as Eqs. (3) and (4), respectively. By solving Eq. (1) with the boundary conditions $I_{\text{TE}}^{x}+I_{\text{M}}^{x}=0$, $V_{\text{TE}}^{x}=V_{\text{M}}^{x}$, we obtain $I_{\text{M}}^{x}$ and $I_{\text{M}}^{y}$ as follows:

where $R_{\text{TE(M)}}^{x}=\rho_{\text{TE(M)}}L_{\text{TE(M)}}^{x}/(L_{\text{TE(M)}}^{y}L_{\text{TE(M)}}^{z})$ is the resistance of the thermoelectric (magnetic) material in the $x$ direction, $R_{\text{AHE}}=\rho_{\text{AHE}}L_{\text{M}}^{y}/(L_{\text{M}}^{z}L_{\text{M}}^{x})$ is the anomalous Hall resistance, and $R_{\text{eff}}^{y}=\rho_{\text{eff}}^{y}L_{\text{M}}^{y}/(L_{\text{M}}^{z}L_{\text{M}}^{x})$ is the effective resistance in the magnetic material in the $y$ direction. Using Eq. (13) and $Q_{\text{TE}}^{x}+Q_{\text{M}}^{x}=0$, we obtain the resultant temperature gradient under the adiabatic condition in the $x$ direction as follows:

where we assumed $Z_{\text{tot}}T\ll 1$ and the linear temperature gradient in the $x$ direction for simplicity. Here, $\Pi_{\text{tot}}^{y}$ is the transverse charge-to-heat conversion coefficient for the STTG system defined as

where $q_{\text{TE(M)}}^{x}$ is the heat current density in the thermoelectric (magnetic) material in the $x$ direction. We note that, in the definition of $\Pi_{\text{tot}}^{y}$, the aforementioned adiabatic condition in the $x$ direction is not assumed. The coefficient satisfies the reciprocal relation $\Pi_{\text{tot}}^{y}=TS_{\text{tot}}^{y}$ (see the right longitudinal axis in Fig. 2). By taking the limit $r\to 0$, $\Pi_{\text{tot}}^{y}$ reduces to the anomalous Ettingshausen coefficient, $\Pi_{\text{AEE}}=TS_{\text{ANE}}$, and the temperature gradient in Eq. (15) reduces to that for the anomalous Ettingshausen effect.Seki2018JphysDFePtfilm ; Seki2018APL

We define the coefficient of performance (COP) for the STTG system as

where $V_{\text{ext}}=-V_{\text{M}}^{y}$ is the voltage of the external battery and $K_{\text{eff}}^{x}=\kappa_{\text{eff}}^{x}L_{\text{M}}^{y}L_{\text{M}}^{z}/L_{\text{M}}^{x}$ is the effective thermal conductance of the STTG device. Here, we used Eqs. (2)-(4) and (13) for the numerator, and Eq. (14) for the denominator. To cool the STTG system, the heat current should be absorbed from the cold reservoir, $Q_{\text{TE}}^{x}|_{T=T_{\text{c}}}+Q_{\text{M}}^{x}|_{T=T_{\text{c}}}\leq 0$, and this gives the following condition for the temperature difference:

We show the upper bound of $\Delta T/\overline{T}$ in Fig. 4(a); this is the achievable $\Delta T/\overline{T}$ for a given $Z_{\text{tot}}\overline{T}$. Figure 4(b) shows the $|I_{\text{M}}^{y}|$ dependence of COP and the cooling power density defined as $-(Q_{\text{TE}}^{x}|_{T=T_{\text{c}}}+Q_{\text{M}}^{x}|_{T=T_{\text{c}}})/L_{\text{M}}^{y}L_{\text{M}}^{z}$. The behaviors are similar to those for the conventional Peltier and Ettingshausen effects.Mobarak2021PRAppl The maximum COP with respect to $I_{\text{M}}^{y}$ is calculated to be

where $\text{COP}_{\text{limit}}=T_{\text{c}}/\Delta T$ is the achievable limit of COP. Equation (19) reduces to the maximum COP for the anomalous Ettingshausen effectHarman1962JAPCOP when $r\to 0$. $\text{COP}_{\text{max}}$ for the STTG system can be larger than that for the anomalous Ettingshausen effect alone; its behavior is similar to that of $Z_{\text{tot}}\overline{T}$ [compare Fig. 4(c) with 3(a)].