Seebeck effect in nanomagnets

Spin caloritronics [1–3] addresses the coupling between the spin and heat transport in small structures and devices. The effects addressed so far can be categorized into several groups [2]. The first group covers phenomena whose origin is not connected to spin–orbit coupling (SOC). Nonrelativistic spin caloritronics in magnetic conductors addresses thermoelectric effects in which motion of electrons in a thermal gradient drives spin transport, such as the spin-dependent Seebeck [4] and the reciprocal Peltier [5, 6] effect. Another group of phenomena is caused by SOC and belongs to relativistic spin caloritronics [2] including the anomalous [7] and spin [8–12] Nernst effects.

The Seebeck effect [13] or thermopower stands for the generation of an electromotive force or gradient of the electrochemical potential μ by temperature gradients ∇T. The Seebeck coefficient S parameterizes the proportionality when the charge current j vanishes:

$$\begin{equation}{(\mathbf{\nabla }\mu /e)}_{j=0}=S\mathbf{\nabla }T.\end{equation} \tag{ 1 }$$

In the two-current model for spin-polarized systems, the thermopower of a magnetic metal reads

$$\begin{equation}S=\frac{{\sigma }^{+}{S}^{+}+{\sigma }^{-}{S}^{-}}{{\sigma }^{+}+{\sigma }^{-}},\end{equation} \tag{ 2 }$$

where σ^± and S^± are the spin-resolved longitudinal conductivities and Seebeck coefficients, respectively.

Here, we study the Seebeck effect in nanoscale magnets on scales equal or less than their spin diffusion length [14] as in figure 1. Thermal baths on both sides of the sample drive a heat current in the x direction. Since no charge current flows, a thermovoltage builds up at the sample edges that can be observed non-invasively by tunnel junctions or scanning probes. Note that metallic contacts can detect the thermovoltage at zero-current bias conditions, but this requires additional modelling of the interfaces. We show in the following that in the presence of a thermally generated spin accumulation the thermopower differs from equation (2). We then focus on dilute ternary alloys of a Cu host with magnetic Mn and nonmagnetic Ir impurities. By varying the alloy concentrations we may tune to the unpolarized case S⁺ = S⁻, as well as to spin-dependent S⁺ and S⁻ parameters with equal or opposite signs. The single-electron thermoelectric effects considered here can be distinguished from collective magnon drag effects [15] by their temperature dependence.

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In the two-current model of spin transport in a single-domain magnet [16–18], extended to include heat transport, the charge ( j ) and heat ( q ) current densities read

$$\begin{equation}{\boldsymbol{j}}^{\pm }={\hat{\sigma }}^{\pm }(\mathbf{\nabla }{\mu }^{\pm }/e)-{\hat{\sigma }}^{\pm }{\hat{S}}^{\pm }\mathbf{\nabla }{T}^{\pm },\end{equation} \tag{ 3 }$$

$$\begin{equation}{\boldsymbol{q}}^{\pm }={\hat{\sigma }}^{\pm }{\hat{S}}^{\pm }T(\mathbf{\nabla }{\mu }^{\pm }/e)-{\hat{\kappa }}^{\pm }\mathbf{\nabla }{T}^{\pm },\end{equation} \tag{ 4 }$$

where ${\hat{\sigma }}^{\pm }$, ${\hat{S}}^{\pm }$, and ${\hat{\kappa }}^{\pm }$ are the spin-resolved electric conductivity, Seebeck coefficient, and heat conductivity, respectively. All transport coefficients are tensors that reflect crystalline symmetry and SOC. The 'four-current model' equations (3) and (4) can be rewritten as

$$\begin{equation}\left(\begin{matrix}\hfill \boldsymbol{j}\hfill \\ \hfill {\boldsymbol{j}}^{\text{s}}\hfill \\ \hfill \boldsymbol{q}\hfill \\ \hfill {\boldsymbol{q}}^{\text{s}}\hfill \end{matrix}\right)=\left(\begin{matrix}\hfill \hat{\sigma }\hfill & \hfill {\hat{\sigma }}^{\text{s}}\hfill & \hfill \hat{\sigma }\hat{S}T\hfill & \hfill \hat{\sigma }{\hat{S}}^{\text{s}}T\hfill \\ \hfill {\hat{\sigma }}^{\text{s}}\hfill & \hfill \hat{\sigma }\hfill & \hfill \hat{\sigma }{\hat{S}}^{\text{s}}T\hfill & \hfill \hat{\sigma }\hat{S}T\hfill \\ \hfill \hat{\sigma }\hat{S}T\hfill & \hfill \hat{\sigma }{\hat{S}}^{\text{s}}T\hfill & \hfill \hat{\kappa }T\hfill & \hfill {\hat{\kappa }}^{\text{s}}T\hfill \\ \hfill \hat{\sigma }{\hat{S}}^{\text{s}}T\hfill & \hfill \hat{\sigma }\hat{S}T\hfill & \hfill {\hat{\kappa }}^{\text{s}}T\hfill & \hfill \hat{\kappa }T\hfill \end{matrix}\right)\left(\begin{matrix}\hfill \mathbf{\nabla }\mu /e\hfill \\ \hfill \mathbf{\nabla }{\mu }^{\text{s}}/2e\hfill \\ \hfill -\mathbf{\nabla }T/T\hfill \\ \hfill -\mathbf{\nabla }{T}^{\text{s}}/2T\hfill \end{matrix}\right)\end{equation} \tag{ 5 }$$

in terms of the charge j = j ⁺ + j ⁻, spin j ^s = j ⁺ − j ⁻, heat q = q ⁺ + q ⁻, and spin-heat q ^s = q ⁺ − q ⁻ current densities. Here, we introduced the conductivity tensors for charge $\hat{\sigma }={\hat{\sigma }}^{+}+{\hat{\sigma }}^{-}$, spin ${\hat{\sigma }}^{\text{s}}={\hat{\sigma }}^{+}-{\hat{\sigma }}^{-}$, heat $\hat{\kappa }={\hat{\kappa }}^{+}+{\hat{\kappa }}^{-}$, and spin heat ${\hat{\kappa }}^{\text{s}}={\hat{\kappa }}^{+}-{\hat{\kappa }}^{-}$. The driving forces are

$$\begin{equation}\mathbf{\nabla }\mu =\frac{1}{2}(\mathbf{\nabla }{\mu }^{+}+\mathbf{\nabla }{\mu }^{-}),\quad \mathbf{\nabla }T=\frac{1}{2}(\mathbf{\nabla }{T}^{+}+\mathbf{\nabla }{T}^{-})\end{equation} \tag{ 6 }$$

and the gradients of the spin μ^s = μ⁺ − μ⁻ [18–21] and spin-heat T^s = T⁺ − T⁻ accumulations [2, 22–27]

$$\begin{equation}{\mu }^{\text{s}}=\frac{1}{2}(\mathbf{\nabla }{\mu }^{+}-\mathbf{\nabla }{\mu }^{-}),\qquad \mathbf{\nabla }{T}^{\text{s}}=\frac{1}{2}(\mathbf{\nabla }{T}^{+}-\mathbf{\nabla }{T}^{-}).\end{equation} \tag{ 7 }$$

Finally, the tensors

$$\begin{equation}\hat{S}={\hat{\sigma }}^{-1}({\hat{\sigma }}^{+}{\hat{S}}^{+}+{\hat{\sigma }}^{-}{\hat{S}}^{-})\end{equation} \tag{ 8 }$$

and

$$\begin{equation}{\hat{S}}^{\text{s}}={\hat{\sigma }}^{-1}({\hat{\sigma }}^{+}{\hat{S}}^{+}-{\hat{\sigma }}^{-}{\hat{S}}^{-})\end{equation} \tag{ 9 }$$

in equation (5) describe the charge and spin-dependent Seebeck coefficients, respectively. In cubic systems the diagonal component S_ii , where i is the Cartesian component of the applied temperature gradient, reduces to the scalar thermopower equation (2).

In the following we apply equation (5) to the Seebeck effect in nanoscale magnets assuming their size to be smaller than the spin diffusion length. In this case the spin-flip scattering may be disregarded [28]. We focus first on longitudinal transport and disregard ∇ T^s. However, we also discuss transverse (Hall) effects as well as the spin temperature gradient below. We adopt open-circuit conditions for charge and spin transport under a temperature gradient. Charge currents and, since we disregard spin-relaxation, spin currents vanish everywhere in the sample:

$$\begin{equation}0=\hat{\sigma }(\mathbf{\nabla }\mu /e)+{\hat{\sigma }}^{\text{s}}(\mathbf{\nabla }{\mu }^{\text{s}}/2e)-\hat{\sigma }\hat{S}\mathbf{\nabla }T,\end{equation} \tag{ 10 }$$

$$\begin{equation}0={\hat{\sigma }}^{\text{s}}(\mathbf{\nabla }\mu /e)+\hat{\sigma }(\mathbf{\nabla }{\mu }^{\text{s}}/2e)-\hat{\sigma }{\hat{S}}^{\text{s}}\mathbf{\nabla }T,\end{equation} \tag{ 11 }$$

$$\begin{equation}\boldsymbol{q}=T\hat{\sigma }[\hat{S}(\mathbf{\nabla }\mu /e)+{\hat{S}}^{\text{s}}(\mathbf{\nabla }{\mu }^{\text{s}}/2e)]-\hat{\kappa }\mathbf{\nabla }T.\end{equation} \tag{ 12 }$$

The thermopower now differs from the conventional expression given by equation (2). Let us introduce the tensor $\hat{{\Sigma}}$ as

$$\begin{equation}{\left.\frac{\mathbf{\nabla }\mu }{e}\right\vert }_{j=0}=\hat{{\Sigma}}\mathbf{\nabla }T.\end{equation} \tag{ 13 }$$

From equations (10) and (11), we find

$$\begin{equation}\hat{{\Sigma}}={\left(\hat{\sigma }-{\hat{\sigma }}^{\text{s}}{\hat{\sigma }}^{-1}{\hat{\sigma }}^{\text{s}}\right)}^{-1}\left(\hat{\sigma }\hat{S}-{\hat{\sigma }}^{\text{s}}{\hat{S}}^{\text{s}}\right).\end{equation} \tag{ 14 }$$

When the spin accumulation in equation (10) vanishes we recover $\hat{{\Sigma}}\to \hat{S}$. Equation (14) involves only directly measurable material parameters [29], but the physics is clearer in the compact expression

$$\begin{equation}\hat{{\Sigma}}=({\hat{S}}^{+}+{\hat{S}}^{-})/2.\end{equation} \tag{ 15 }$$

The spin polarization of the Seebeck coefficient

$$\begin{equation}{\left.\frac{\mathbf{\nabla }{\mu }^{\text{s}}}{2e}\right\vert }_{j=0}={\hat{{\Sigma}}}^{\text{s}}\mathbf{\nabla }T,\end{equation} \tag{ 16 }$$

reads

$$\begin{equation}{\hat{{\Sigma}}}^{\text{s}}={\left(\hat{\sigma }-{\hat{\sigma }}^{\text{s}}{\hat{\sigma }}^{-1}{\hat{\sigma }}^{\text{s}}\right)}^{-1}(\hat{\sigma }{\hat{S}}^{\text{s}}-{\hat{\sigma }}^{\text{s}}\hat{S}),\end{equation} \tag{ 17 }$$

or

$$\begin{equation}{\hat{{\Sigma}}}^{\text{s}}=({\hat{S}}^{+}-{\hat{S}}^{-})/2.\end{equation} \tag{ 18 }$$

The diagonal elements of $\hat{{\Sigma}}$ govern the thermovoltage in the direction of the temperature gradient. The off-diagonal elements of $\hat{{\Sigma}}$ represent transverse thermoelectric phenomena such as the anomalous [7] and planar [30] Nernst effects. The diagonal and off-diagonal elements of ${\hat{{\Sigma}}}^{\text{s}}$ describe the spin-dependent Seebeck effect [2, 4], as well as (also in non-magnetic systems) the spin and planar-spin Nernst effects [8–12], respectively. We do not address here anomalous and Hall transport in the purely charge and heat sectors of equation (5).

3.1. Longitudinal spin accumulation

A temperature gradient in x direction ∇ T ∥ e _x induces the voltage in the same direction:

$$\begin{equation}{({\mathbf{\nabla }}_{x}\mu /e)}_{j=0}={{\Sigma}}_{xx}{\mathbf{\nabla }}_{x}T.\end{equation} \tag{ 19 }$$

In order to assess the importance of the difference between equations (14) and (15) and the conventional thermopower equation (2) we carried out first-principles transport calculations for the ternary alloys Cu_1−v (Mn_1−w Ir_w ) _v , where w ∈ [0, 1] and the total impurity concentration is fixed to v = 1 at.% [31]. We have chosen this system as a generic example where the Cu host ensures a reasonably large spin diffusion length and the two impurities allow us to scale easily between the different limits of strong spin-dependent scattering induced by magnetism for Cu(Mn) and SOC for Cu(Ir). The derived expressions equally apply to more conventional ferromagnets as long the dimensions of the sample is comparable or smaller than the spin diffusion length. We calculate the transport properties from the solutions of the linearized Boltzmann equation with collision terms calculated for isolated impurities [32, 33]. We disregard spin-flip scattering [33], which limits the size of the systems for which our results hold (see below). We calculate the electronic structure of the Cu host by the relativistic Korringa–Kohn–Rostoker method [34]. Figure 2 summarizes the calculated room-temperature (charge) thermopower equation (8) or (14) and (15) and their spin-resolved counterparts, equations (17) and (18). Table 1 contains additional information for the binary alloys Cu(Mn) and Cu(Ir) with w = 0 or w = 1 in figure 2, respectively. Here we implicitly assume an applied magnetic field that orders all localized moments.

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Table 1. Computed spin-resolved and charge thermopowers as defined in the text for magnetic Cu_0.99Mn_0.01 and Cu_0.99(Mn_0.5Ir_0.5)_0.01 as well as non-magnetic Cu_0.99Ir_0.01 dilute alloys. The conventional spin Seebeck coefficient is shown for comparison. All quantities are calculated at 300 K in units of μV K⁻¹.

System	Cu0.99Mn0.01	Cu0.99(Mn0.5Ir0.5)0.01	Cu0.99Ir0.01
${S}_{xx}^{+}$	−6.87	−7.01	−7.09
${S}_{xx}^{-}$	8.57	1.64	−7.09
Sxx	−6.14	−4.26	−7.09
Σxx	0.85	−2.69	−7.09
${{\Sigma}}_{xx}^{\text{s}}$	−7.72	−4.33	0.00

We observe large differences (even sign changes) between ${S}_{xx}^{+}$ and ${S}_{xx}^{-}$ that causes significant differences between ${{\Sigma}}_{xx}=({S}_{xx}^{+}+{S}_{xx}^{-})/2$ and the macroscopic S_xx . The complicated behavior of the latter is caused by the weighting of S⁺ and S⁻ by the corresponding conductivities, see equation (8). Even though a spin-accumulation gradient suppresses the Seebeck effect, an opposite sign of ${S}_{xx}^{+}$ and ${S}_{xx}^{-}$ can enhance ${{\Sigma}}_{xx}^{\text{s}}$ beyond the microscopic as well as macroscopic thermopower. Indeed, Hu et al [35] observed a spin-dependent Seebeck effect that is larger than the charge Seebeck effect in CoFeAl. Our calculations illustrate that the spin-dependent Seebeck effect can be engineered and maximized by doping a host material with impurities.

3.2. Hall transport

In the presence of spin–orbit interactions the applied temperature gradient ∇ T_ext induces anomalous Hall currents. When the electron–phonon coupling is weak, the spin–orbit interaction can, for example, induce transverse temperature gradients. In a cubic magnet the charge and spin conductivity tensors are antisymmetric. With magnetization and spin quantization axis along z:

$$\begin{equation}{\hat{\sigma }}^{(\text{s})}=\left(\begin{matrix}\hfill {\sigma }_{xx}^{(\text{s})}\hfill & \hfill -{\sigma }_{yx}^{(\text{s})}\hfill & \hfill 0\hfill \\ \hfill {\sigma }_{yx}^{(\text{s})}\hfill & \hfill {\sigma }_{xx}^{(\text{s})}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\sigma }_{zz}^{(\text{s})}\hfill \end{matrix}\right),\end{equation} \tag{ 20 }$$

and analogous expressions hold for $\hat{S}$ and ${\hat{S}}^{\text{s}}$. A charge current in the x direction generates a transverse heat current that heats and cools opposite edges, respectively. A transverse temperature gradient ∇ T_ind ∥ e _y is signature of this anomalous Ettingshausen effect [36] gradient. From equations (12), (13), and (16)

$$\begin{equation}\boldsymbol{q}=\left[T\hat{\sigma }(\hat{S}\hat{{\Sigma}}+{\hat{S}}^{\text{s}}{\hat{{\Sigma}}}^{\text{s}})-\hat{\kappa }\right]\mathbf{\nabla }T,\end{equation} \tag{ 21 }$$

where ∇ T = e _x ∇_x T_ext + e _y ∇_y T_ind. Assuming weak electron-phonon scattering, the heat cannot escape the electron systems and q_y = 0. equation (21) then leads to

$$\begin{equation}{\mathbf{\nabla }}_{y}{T}_{\text{ind}}=-\frac{{A}_{yx}}{{A}_{yy}}{\mathbf{\nabla }}_{x}{T}_{\text{ext}},\end{equation} \tag{ 22 }$$

where A_yx and A_yy are components of the tensor

$$\begin{equation}\hat{A}=T\hat{\sigma }(\hat{S}\hat{{\Sigma}}+{\hat{S}}^{\text{s}}{\hat{{\Sigma}}}^{\text{s}})-\hat{\kappa }.\end{equation} \tag{ 23 }$$

Consequently, equation (13) leads to a correction to the thermopower

$$\begin{equation}{({\mathbf{\nabla }}_{x}\mu /e)}_{j=0}=\left[{{\Sigma}}_{xx}-{{\Sigma}}_{xy}\frac{{A}_{yx}}{{A}_{yy}}\right]{\mathbf{\nabla }}_{x}{T}_{\text{ext}}.\end{equation} \tag{ 24 }$$

However, this effect should be small [37–40] for all but the heaviest elements but may become observable when Σ_xx vanishes, which according to figure 2 should occur at around w = 0.125.

3.3. Spin temperature gradient

At low temperatures, the spin temperature gradient ∇ T^s may persist over length scales smaller but of the same order as the spin accumulation [25]. From equations (3), (15), and (18) it follows

$$\begin{equation}{(\mathbf{\nabla }\mu /e)}_{j=0}=\hat{{\Sigma}}\mathbf{\nabla }T+{\hat{{\Sigma}}}^{\text{s}}\mathbf{\nabla }{T}^{\text{s}}/2,\end{equation} \tag{ 25 }$$

$$\begin{equation}{({\mathbf{\nabla }\mu }^{\text{s}}/2e)}_{j=0}={\hat{{\Sigma}}}^{\text{s}}\mathbf{\nabla }T+\hat{{\Sigma}}\mathbf{\nabla }{T}^{\text{s}}/2.\end{equation} \tag{ 26 }$$

Starting with equation (5) and employing equations (25) and (26) for the heat and spin-heat current densities we obtain

$$\begin{equation}\boldsymbol{q}=\hat{A}\mathbf{\nabla }T+\hat{B}\mathbf{\nabla }{T}^{\text{s}}/2\quad \text{and}\quad {\boldsymbol{q}}^{\text{s}}=\hat{B}\mathbf{\nabla }T+\hat{A}\mathbf{\nabla }{T}^{\text{s}}/2\enspace ,\end{equation} \tag{ 27 }$$

where

$$\begin{equation}\hat{B}=T\hat{\sigma }(\hat{S}{\hat{{\Sigma}}}^{\text{s}}+{\hat{S}}^{\text{s}}\hat{{\Sigma}})-{\hat{\kappa }}^{\text{s}},\end{equation} \tag{ 28 }$$

and $\hat{A}$ is defined by equation (23). With $\mathbf{\nabla }{T}^{\text{s}}={\boldsymbol{e}}_{y}{\mathbf{\nabla }}_{y}{T}_{\text{in}}^{\text{s}}$ and ∇ T = e _x ∇_x T_ex + e _y ∇_y T_in we find

$$\begin{align}\hfill {({\mathbf{\nabla }}_{x}\mu /e)}_{j=0}& ={{\Sigma}}_{xx}{\mathbf{\nabla }}_{x}{T}_{\text{ex}}+{{\Sigma}}_{xy}{\mathbf{\nabla }}_{y}{T}_{\text{in}}+{{\Sigma}}_{xy}^{\text{s}}{\mathbf{\nabla }}_{y}{T}_{\mathrm{i}\mathrm{n}}^{\text{s}}/2\hfill \\ \hfill & =\left[{{\Sigma}}_{xx}-{{\Sigma}}_{xy}\frac{{A}_{yy}{A}_{yx}-{B}_{yy}{B}_{yx}}{{A}_{yy}{A}_{yy}-{B}_{yy}{B}_{yy}}\right.\hfill \\ \hfill & \left.\quad -{{\Sigma}}_{xy}^{\text{s}}\frac{{A}_{yy}{B}_{yx}-{B}_{yy}{A}_{yx}}{{A}_{yy}{A}_{yy}-{B}_{yy}{B}_{yy}}\right]{\mathbf{\nabla }}_{x}{T}_{\text{ex}}\hfill \end{align} \tag{ 29 }$$

assuming again q_y = 0 and ${q}_{y}^{\text{s}}=0$. Similar to equation (24), the Hall corrections in equation (29) should be significant only when Σ_xx vanishes for w = 0.125. However, experimentally it might be difficult to separate the thermopowers equations (29) and (24).

3.4. Spin diffusion length and mean free path

In summary, we derived expressions for the thermopower valid for ordered magnetic alloys for sample sizes that do not exceed the spin diffusion lengths (that have to be calculated separately). We focus on dilute alloys of Cu with Mn and Ir impurities. For 1% ternary alloys Cu(Mn_1−w Ir_w ) with w < 0.5 the spin diffusion length is l_sf > 60 nm. In this regime the spin and charge accumulations induced by an applied temperature gradient strongly affect each other. By ab initio calculations of the transport properties of Cu(Mn_1−w Ir_w ) alloys, we predict thermopowers that drastically differ from the bulk value even changing sign. Relativistic Hall effects generate spin accumulations normal to the applied temperature gradient that become significant when the longitudinal thermopower Σ_xx vanishes, for example for Cu(Mn_1−w Ir_w ) alloys at w ≈ 0.125.

After having established the principle existence of the various corrections to the conventional transport description it would be natural to move forward to describe extended thin films. A first-principles version of the Boltzmann equation including all electronic spin non-conserving scatterings in extended films is possible, but very expensive for large l_sf. It would still be incomplete, since the relaxation of heat to the lattice by electron-phonon interactions and spin-heat by electron–electron scattering [23, 24] are not included. We therefore propose to proceed pragmatically: The regime l < l_sf < L is accessible to spin-heat diffusion equations that can be parameterized by first-principles material-dependent parameters as presented here and relaxation lengths that may be determined otherwise, such as by fitting to experimental results.

This work was partially supported by the Deutsche Forschungsgemeinschaft via SFB 762 and the priority program SPP 1538 as well as JSPS Grants-in-Aid for Scientific Research (KAKENHI Grant No. 19H00645). MG acknowledges financial support from the Leverhulme Trust via an Early Career Research Fellowship (ECF-2013-538) and a visiting professorship at the Centre for Dynamics and Topology of the Johannes-Gutenberg-University Mainz.

The data that support the findings of this study are available upon reasonable request from the authors.