Seebeck Effect in Nanomagnets

TL;DR

This paper develops a theoretical framework for the Seebeck effect in nanomagnets, highlighting the influence of spin accumulation and transverse temperature gradients on thermopower at nanoscale dimensions.

Contribution

It introduces a comprehensive theory accounting for spin accumulation and transverse effects in nanomagnets, supported by ab initio calculations on magnetic alloys.

Findings

Spin accumulation significantly impacts thermopower in nanomagnets.

Transverse temperature gradients modify the Seebeck effect.

Ab initio results confirm theoretical predictions.

Abstract

We present a theory of the Seebeck effect in nanomagnets with dimensions smaller than the spin diffusion length, showing that the spin accumulation generated by a temperature gradient strongly affects the thermopower. We also identify a correction arising from the transverse temperature gradient induced by the anomalous Ettingshausen effect and an induced spin-heat accumulation gradient. The relevance of these effects for nanoscale magnets is illustrated by ab initio calculations on dilute magnetic alloys.

Tables

Table 1. Table 1: Computed spin-resolved and charge thermopowers as defined in the text for magnetic Cu 0.99 (Mn 0.5 Ir 0.5 ) 0.01 and Cu 0.99 Mn 0.01 as well as non-magnetic Cu 0.99 Ir 0.01 dilute alloys. The conventional spin Seebeck coefficient is shown for comparison. All quantities are calculated at 300 K in units of μ 𝜇 \mu V/K.

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
$S_{xx}$	-6.14	-4.26	-7.09
${\Sigma }_{xx}$	0.85	-2.69	-7.09
${\Sigma }_{xx}^s$	-7.72	-4.33	0.00

Equations

$$(\nabla \mu /e{)}_{{}_{j=0}}=S\nabla T.$$

$$S=\frac{{\sigma }^{+}{S}^{+}+{\sigma }^{-}{S}^{-}}{{\sigma }^{+}+{\sigma }^{-}},$$

$${\mathbf{j}}^{\pm }$$

$${\mathbf{q}}^{\pm }$$

$$\left(\begin{array}{c}\mathbf{j}\\ {\mathbf{j}}^s\\ \mathbf{q}\\ {\mathbf{q}}^s\end{array}\right)=\left(\begin{array}{cccc}\hat{\sigma }& {\hat{\sigma }}^s& \hat{\sigma }\hat{S}T& \hat{\sigma }{\hat{S}}^{s}T\\ {\hat{\sigma }}^s& \hat{\sigma }& \hat{\sigma }{\hat{S}}^{s}T& \hat{\sigma }\hat{S}T\\ \hat{\sigma }\hat{S}T& \hat{\sigma }{\hat{S}}^{s}T& \hat{\kappa }T& {\hat{\kappa }}^{s}T\\ \hat{\sigma }{\hat{S}}^{s}T& \hat{\sigma }\hat{S}T& {\hat{\kappa }}^{s}T& \hat{\kappa }T\end{array}\right)\left(\begin{array}{c}\nabla \mu /e\\ \nabla {\mu }^s/2e\\ -\nabla T/T\\ -\nabla T^s/2T\end{array}\right)$$

$$\nabla \mu =\frac{1}{2}(\nabla {\mu }^{+}+\nabla {\mu }^{-}),\nabla T=\frac{1}{2}(\nabla T^{+}+\nabla T^{-})$$

$$\nabla {\mu }^s=\frac{1}{2}(\nabla {\mu }^{+}-\nabla {\mu }^{-}),\nabla T^s=\frac{1}{2}(\nabla T^{+}-\nabla T^{-}).$$

$$\hat{S}={\hat{\sigma }}^{-1}({\hat{\sigma }}^{+}{\hat{S}}^{+}+{\hat{\sigma }}^{-}{\hat{S}}^{-})$$

$${\hat{S}}^s={\hat{\sigma }}^{-1}({\hat{\sigma }}^{+}{\hat{S}}^{+}-{\hat{\sigma }}^{-}{\hat{S}}^{-})$$

$$0=\hat{\sigma }(\nabla \mu /e)+{\hat{\sigma }}^s(\nabla {\mu }^s/2e)-\hat{\sigma }\hat{S}\nabla T,$$

$$0={\hat{\sigma }}^s(\nabla \mu /e)+\hat{\sigma }(\nabla {\mu }^s/2e)-\hat{\sigma }{\hat{S}}^s\nabla T,$$

$$\mathbf{q}=T\hat{\sigma }[\hat{S}(\nabla \mu /e)+{\hat{S}}^s(\nabla {\mu }^s/2e)]-\hat{\kappa }\nabla T.$$

$${\frac{\nabla \mu }{e}|}_{{}_{j=0}}=\hat{\Sigma }\nabla T.$$

$$\hat{\Sigma }={\left(\hat{\sigma }-{\hat{\sigma }}^s{\hat{\sigma }}^{-1}{\hat{\sigma }}^s\right)}^{-1}\left(\hat{\sigma }\hat{S}-{\hat{\sigma }}^s{\hat{S}}^s\right).$$

$$\hat{\Sigma }=({\hat{S}}^{+}+{\hat{S}}^{-})/2.$$

$${\frac{\nabla {\mu }^s}{2e}|}_{{}_{j=0}}={\hat{\Sigma }}^s\nabla T,$$

$${\hat{\Sigma }}^s={\left(\hat{\sigma }-{\hat{\sigma }}^s{\hat{\sigma }}^{-1}{\hat{\sigma }}^s\right)}^{-1}(\hat{\sigma }{\hat{S}}^s-{\hat{\sigma }}^s\hat{S}),$$

$${\hat{\Sigma }}^s=({\hat{S}}^{+}-{\hat{S}}^{-})/2.$$

$$({\nabla }_x\mu /e{)}_{{}_{j=0}}={\Sigma }_{xx}{\nabla }_{x}T.$$

$$\hat{\sigma}^{(s)}=\left(\begin{array}[]{ccc}\sigma_{xx}^{(s)}&-\sigma_{yx}^{(s)}&0\\ \sigma_{yx}^{(s)}&\sigma_{xx}^{(s)}&0\\ 0&0&\sigma_{zz}^{(s)}\end{array}\right)\ ,$$

$$\mathbf{q}=\left[T\hat{\sigma }(\hat{S}\hat{\Sigma }+{\hat{S}}^s{\hat{\Sigma }}^s)-\hat{\kappa }\right]\nabla T,$$

$${\nabla }_y{T}_{\mathrm{ind}}=-\frac{A_{yx}}{A_{yy}}{\nabla }_x{T}_{\mathrm{ext}},$$

$$\hat{A}=T\hat{\sigma }(\hat{S}\hat{\Sigma }+{\hat{S}}^s{\hat{\Sigma }}^s)-\hat{\kappa }.$$

$$({\nabla }_x\mu /e{)}_{{}_{j=0}}=\left[{\Sigma }_{xx}-{\Sigma }_{xy}\frac{A_{yx}}{A_{yy}}\right]{\nabla }_x{T}_{\mathrm{ext}}.$$

$$(\nabla \mu /e{)}_{{}_{j=0}}=\hat{\Sigma }\nabla T+{\hat{\Sigma }}^s\nabla T^s/2,$$

$$({\nabla \mu }^s/2e{)}_{{}_{j=0}}={\hat{\Sigma }}^s\nabla T+\hat{\Sigma }\nabla T^s/2.$$

$$\mathbf{q}=\hat{A}\nabla T+\hat{B}\nabla T^s/2\text{and}{\mathbf{q}}^s=\hat{B}\nabla T+\hat{A}\nabla T^s/2,$$

$$\hat{B}=T\hat{\sigma }(\hat{S}{\hat{\Sigma }}^s+{\hat{S}}^s\hat{\Sigma })-{\hat{\kappa }}^s,$$

$$({\nabla }_x\mu /e{)}_{{}_{j=0}}$$

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I Introduction

Spin caloritronics Bauer2010 ; Bauer2012 ; Boona2014 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 Bauer2012 . 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 Slachter2010 and the reciprocal Peltier Flipse2012 ; Goennenwein2012 effect. Another group of phenomena is caused by SOC and belongs to relativistic spin caloritronics Bauer2012 including the anomalous Miyasato2007 and spin Cheng2008 ; Liu2010 ; Ma2010 ; Tauber2012 ; Bose2019 Nernst effects.

The Seebeck effect Seebeck1822 or thermopower stands for the generation of an electromotive force or gradient of the electrochemical potential $\mu$ by temperature gradients $\nabla T$. The Seebeck coefficient $S$ parameterized the proportionality when the charge current $j$ vanishes:

[TABLE]

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

[TABLE]

where $\sigma^{\pm}$ and $S^{\pm}$ 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 Comment_1 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 Eq. (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 Watzmann_2016 by their temperature dependence.

II Theory

In the two-current model of spin transport in a single-domain magnet Mott36 ; Fert68 ; Son87 , extended to include heat transport, the charge ($\bm{j}$) and heat ($\bm{q}$) current densities read

[TABLE]

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” Eqs. (3) and (4) can be rewritten as

[TABLE]

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

[TABLE]

and the gradients of the spin $\mu^{s}=\mu^{+}-\mu^{-}$ Son87 ; Johnson87 ; Johnson88 ; Valet93 and spin-heat $T^{s}=T^{+}-T^{-}$ accumulations Hatami2007 ; Heikkila2010_PRB ; Heikkila2010_SSC ; Bauer2012 ; Dejene2013 ; Marun2014 ; Wong2015

[TABLE]

Finally, the tensors

[TABLE]

and

[TABLE]

in Eq. (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 Eq. (2).

III Results

In the following we apply Eq. (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 Hatami2010 . We focus first on longitudinal transport and disregard $\bm{\nabla}T^{s}$. 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:

[TABLE]

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

[TABLE]

From Eqs. (10) and (11), we find

[TABLE]

When the spin accumulation in Eq. (10) vanishes we recover $\hat{\Sigma}\rightarrow\hat{S}$. Equation (14) involves only directly measurable material parameters Comment_2 , but the physics is clearer in the compact expression

[TABLE]

The spin polarization of the Seebeck coefficient

[TABLE]

reads

[TABLE]

or

[TABLE]

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 Miyasato2007 and planar Pu2006 Nernst effects. The diagonal and off-diagonal elements of $\hat{\Sigma}^{s}$ describe the spin-dependent Seebeck effect Slachter2010 ; Bauer2012 , as well as (also in non-magnetic systems) the spin and planar-spin Nernst effects Cheng2008 ; Liu2010 ; Ma2010 ; Tauber2012 ; Bose2019 , respectively. We do not address here anomalous and Hall transport in the purely charge and heat sectors of Eq. (5).

III.1 Longitudinal spin accumulation

A temperature gradient in $x$ direction $\bm{\nabla}T\parallel\bm{e}_{x}$ induces the voltage in the same direction:

[TABLE]

In order to assess the importance of the difference between Eqs. (14) and (15) and the conventional thermopower Eq. (2) we carried out first-principles transport calculations for the ternary alloys Cu1-v(Mn1-wIrw)v, where $w\in[0,1]$ and the total impurity concentration is fixed to $v=1$ at.% Tauber2013 . We calculate the transport properties from the solutions of the linearized Boltzmann equation with collision terms calculated for isolated impurities Mertig99 ; Gradhand10 . We disregard spin-flip scattering Gradhand10 , 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 Gradhand09 . Figure 2 summarizes the calculated room-temperature (charge) thermopower Eqs. (8) or (14) and (15) and their spin-resolved counterparts, Eqs. (17) and (18). Table 1 contains additional information for the binary alloys Cu(Mn) and Cu(Ir) with $w=0$ or $w=1$ in Fig. 2, respectively. Here we implicitly assume an applied magnetic field that orders all localized moments.

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 Eq. (8). Even though a spin-accumulation gradient suppresses the Seebeck effect, an opposite sign of $S_{xx}^{+}$ and $S_{xx}^{-}$ can enhance $\Sigma_{xx}^{s}$ beyond the microscopic as well as macroscopic thermopower. Indeed, Hu et al. Hu2014 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.

III.2 Hall transport

In the presence of spin-orbit interactions the applied temperature gradient $\bm{\nabla}T_{\mathrm{ext}}$ induces anomalous Hall currents. When the electron-phonon coupling is weak, the spin-orbit interaction can, for example, induces transverse temperature gradients. In a cubic magnet the charge and spin conductivity tensors are antisymmetric. With magnetization and spin quantization axis along $z$:

[TABLE]

and analogous expressions hold for $\hat{S}$ and $\hat{S}^{s}$. A charge current in the $x$ direction generates a transverse heat current that heats and cools opposite edges, respectively. A transverse temperature gradient $\bm{\nabla}T_{\mathrm{ind}}\parallel\bm{e}_{y}$ is signature of this anomalous Ettingshausen effect Hu2013 gradient. From Eqs. (12), (13), and (16)

[TABLE]

where $\bm{\nabla}T=\bm{e}_{x}\nabla_{x}T_{\mathrm{ext}}+\bm{e}_{y}\nabla_{y}T_{\mathrm{ind}}$. Assuming weak electron-phonon scattering, the heat cannot escape the electron systems and $q_{y}=0$. Equation (21) then leads to

[TABLE]

where $A_{yx}$ and $A_{yy}$ are components of the tensor

[TABLE]

Consequently, Eq. (13) leads to a correction to the thermopower

[TABLE]

However, this effect should be small Gradhand10_2 ; Lowitzer2011 ; Gradhand2011 ; Gradhand2012 for all but the heaviest elements but may become observable when $\Sigma_{xx}$ vanishes, which according to Fig. 2 should occur at around $w=0.125$.

III.3 Spin temperature gradient

At low temperatures, the spin temperature gradient $\bm{\nabla}T^{s}$ may persist over length scales smaller but of the same order as the spin accumulation Dejene2013 . From Eqs. (3), (15), and (18) it follows

[TABLE]

[TABLE]

Starting with Eq. (5) and employing Eqs. (25) and (26) for the heat and spin-heat current densities we obtain

[TABLE]

where

[TABLE]

and $\hat{A}$ is defined by Eq. (23). With $\bm{\nabla}T^{s}=\bm{e}_{y}\nabla_{y}T_{\mathrm{in}}^{s}$ and $\bm{\nabla}T=\bm{e}_{x}\nabla_{x}T_{\mathrm{ex}}+\bm{e}_{y}\nabla_{y}T_{\mathrm{in}}$ we find

[TABLE]

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

III.4 Spin diffusion length and mean free path

Our first-principles calculation are carried out for bulk dilute alloys based on Cu and in the single site approximation of spin-conserving impurity scattering. The Hall effects are therefore purely extrinsic. This is an approximation that holds on length scales smaller than various spin diffusion lengths $l_{\mathrm{sf}}$. On the other hand, the Boltzmann equation approach is valid when the sample is larger than the elastic scattering mean free path $l$, so our results should be directly applicable for sample lengths $L$ that fulfill $l$ $<L\leq l_{\mathrm{sf}}.$ According to Refs. Gradhand10_2, and Gradhand2012, , for the ternary alloy Cu(Mn0.5Ir0.5) with impurity concentration of $1$ at.% the present results hold on length scales $26$ nm $<L\leq 60$ nm and $100$ nm $<L\leq 400$ nm for Cu(Mn). On the other hand, for nonmagnetic Cu(Ir) the applicability is limited to a smaller interval $10$ nm $<L\leq 16$ nm. We believe that while the results outside these strict limits may not be quantitatively reliable, they still give useful insights into trends.

IV Summary and outlook

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(Mn1-wIrw) with $w<0.5$ the spin diffusion length is $l_{\mathrm{sf}}>60$ 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(Mn1-wIrw) 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 $\Sigma_{xx}$ vanishes, for example for Cu(Mn1-wIrw) alloys at $w\approx 0.125$.

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 $l_{\mathrm{sf}}.$ It would still be incomplete, since the relaxation of heat to the lattice by electron-phonon interactions and spin-heat by electron-electron scattering Heikkila2010_PRB ; Heikkila2010_SSC are not included. We therefore propose to proceed pragmatically: The regime $l<l_{\mathrm{sf}}<L$ 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.

V Acknowledgements

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). M.G. 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.