Effect of magnetic anisotropy on spin-dependent thermoelectric effects in nanoscopic systems

Conventional and spin-related thermoelectric effects in electronic transport through a nanoscopic system exhibiting magnetic anisotropy, with both uniaxial and transverse components, are studied theoretically in the linear-response regime. In particular, a magnetic tunnel junction with a large-spin impurity, either a magnetic atom or a magnetic molecule, embedded in the barrier is considered as an example. Owing to magnetic interaction with the impurity, conduction electrons traversing the junction can scatter on the impurity, which effectively can lead to angular momentum and energy exchange between the electrons and the impurity. As we show, such processes have a profound effect on the thermoelectric response of the system. Specifically, we present a detailed analysis of charge, spin, and thermal conductance, together with the Seebeck and spin Seebeck coefficients (thermopowers). Since the scattering mechanism also involves processes when electrons are inelastically scattered back to the same electrode, one can expect the flow of spin and energy also in the absence of charge transport through the junction. This, in turn, results in a finite spin thermopower, and the magnetic anisotropy plays a key role for this effect to occur.

Figure 1

Influence of magnetic anisotropy on the spectrum of an exemplary large-spin ($S=5/2$) impurity. (a) For an isotropic spin, all $2S+1$ spin states are energetically degenerate. A conduction electron can flip its spin orientation when scattered on the impurity, which results in transfer of the quantum of angular momentum $\hslash$ between the electron and impurity. The impurity undergoes then a transition between two neighboring spin states, $|\Delta S_z|=1$. These transitions are schematically marked by the double-sided gray arrows. However, spin-orbit interaction in the presence of low symmetry of the impurity usually leads to a spin anisotropy (b)–(e). In the simplest case, the magnetic anisotropy can be uniaxial (b), (c), which can be further distinguished into (b) magnetic anisotropy of the “easy-axis” type, the spin prefers orientation along a specific axis (easy axis, indicated by a vertical dashed bold line) without favoring any of the two orientations, which leads to an energy barrier for spin reversal; and (c) magnetic anisotropy of the “easy-plane” type, the spin prefers orientation close to a specific plane (or in the plane for an integer $S$), often referred to as the easy plane (indicated by a color plane perpendicular to the easy axis $z$). The uniaxial anisotropy is very often also accompanied by a transverse component (d), (e), which introduces a mixing of the spin states. Due to such a mixing, depicted by thin color lines connecting states in (d) and (e), spin-flip scattering of conduction electrons can in principle transfer the impurity between any two time-reversed (TR) spin states. As an illustration, possible transitions from one specific state are marked in (d) and (e) on the right side, with the states in the left column being time reversed with respect to the states in the right column.

Figure 2

Systematic classification of the thermoelectric effects considered in this work. Generally, in a biased system (with electric, spin, and thermal bias) a flow of charge, spin, and heat occurs, and the relation between a specific bias and currents is determined by the relevant conductances (indicated by thin lines). These currents are not completely independent. For instance, under isothermal conditions, $\delta T=0$, a charge (spin) current is accompanied by a heat flow [a phenomenon known as the Peltier and spin Peltier effects (dashed arrows)], and characterized by the Peltier and spin Peltier coefficients $\Pi$ and ${\Pi }_{\text{S}}$, respectively. On the other hand, in the absence of a charge (spin) flow $I_{\text{C}}=0$ ($I_{\text{S}}=0$), a thermal bias can lead to charge (spin) accumulation, resulting in an electric and spin bias, respectively. This phenomenon is referred to as the Seebeck and spin Seebeck effect (solid arrows), and is quantified by the thermopower $\mathcal{S}$ (or Seebeck coefficient) and spin thermopower ${\mathcal{S}}_{\text{S}}$ (or spin Seebeck coefficient). We note that all the transport coefficients presented in the above figure are experimentally measurable. More detailed discussion can be found in Refs. [2, 31].

Figure 3

Schematic illustration of the model system under consideration: a magnetic tunnel junction with a large-spin impurity embedded in the tunnel barrier. Transport of electrons in such a junction can be driven by an external electric bias $\delta V$, spin bias $\delta V_{\text{S}}$, and the difference $\delta T$ between electrodes' temperatures $T_q[q=L\left(\text{eft}\right),R\left(\text{ight}\right)]$. For conceptual simplicity, we assume here $\delta V=\delta V_{\text{S}}=0$. The thermal bias $\delta T$ is included via the Fermi-Dirac distribution functions $f^q\left(\omega \right)$, resulting in a different smearing of densities of occupied states around the equilibrium electrochemical potential ${\mu }_0$ in each electrode. Apart from direct tunneling between the electrodes [an example for spin-up electrons indicated by (i)], electrons can be also scattered by the impurity spin. As a result of such a process, an electron can either tunnel to an opposite electrode (ii) or return to the same electrode (iii), and its spin can be conserved or flipped [as shown for instance in (ii) and (iii)].

Figure 4

(a) Elements of the generalized conductance matrix $\mathit{G}$ of a bare tunnel junction (i.e., in the absence of spin impurity ${\alpha }_{\text{ex}}=0$) in the parallel (left) and antiparallel (right) magnetic configurations, presented as a function of the position of the bottom edge of the conduction band in the right electrode $E_R$ for a constant value of $E_L=-1$ eV, and for $P_L=P_R\equiv P=0.5$. The conductances are expressed in the units as indicated. The corresponding magnetoconductance is shown in (b). Although the conductances are plotted for $T=1$ K, they remain constant within the temperature range of interest, that is, up to 100 K.

Figure 5

Thermal conductance (a), thermopower $\mathcal{S}$ (c), (d), and spin thermopower ${\mathcal{S}}_{\text{S}}$ (e), (f) calculated as a functions of temperature $T$ for indicated values of $E_R$. Note that these values of $E_R$ correspond to the points indicated in Fig. 4. Whereas the heat conductance is presented for the parallel magnetic configuration, the thermopower and spin thermopower are shown for both parallel and antiparallel configurations. The corresponding heat magnetoconductance HMC is presented in (b). The other parameters as in Fig. 4.

Figure 6

The effect of a spin impurity with $S=2$ on the electrical conductance of a magnetic tunnel junction. Left panel [(a)–(c)]: Conductances $G$ and $G^{\text{m}}$ in the parallel (P) magnetic configuration, together with the corresponding magnetoconductances MC and ${\mathrm{MC}}^{\text{m}}$, shown for $T=0.1$ K as a function of the parameter ${\alpha }_{\text{ex}}$ describing the electron tunneling with scattering on the impurity. Thin lines serve only as a guide for eyes. Three distinctive cases regarding the uniaxial magnetic anisotropy constant $D$ (with $E=0$) are considered: (a) an isotropic impurity ($D=0$), and an anisotropic impurity with the spin anisotropy of (b) the easy-axis type ($D>0$), as well as (c) the easy-plane type ($D<0$) with $\left|D\right|=100\mu \mathrm{eV}$. Graphical depiction of the impurity energy spectrum for each of these three cases is shown below. Right panel [(d)–(k)]: Dependence of the conductances in the parallel magnetic configuration, scaled to the corresponding values at $T=0.1$ K, and of the corresponding magnetoconductances on temperature for selected values of ${\alpha }_{\text{ex}}$ for $D>0$ (d)–(g) and $D<0$ (h)–(k). Vertical dashed lines represent the temperature equivalent to the zero-field splitting: (d)–(g) $\text{ZFS}=3D\approx 3.5$ K and (h)–(k) $\text{ZFS}=\left|D\right|\approx 1.16$ K. We note that for $D=0$ the relevant conductances are constant within the temperature range under consideration. Other parameters: ${\alpha }_{\text{d}}=1$ and $P_L=P_R\equiv P=0.5$.

Figure 7

Decomposition of the diagonal $G$ (a), (b) and $G_{\text{S}}$ (c), (d), and nondiagonal $G^{\text{m}}$ (e), (f) elements of the generalized conductance matrix $\mathit{G}$ into spin-conserving (long-dashed lines marked as ‘sc’) and spin-flip [short-dashed and dashed-dotted lines marked as ‘sf(1)’ and ‘sf(2)’] contributions shown as a function of temperature $T$ for ${\alpha }_{\text{ex}}=0.5$ and parallel magnetic configuration. Bold solid lines stand for the sum of all the contributions. Both the easy-axis ($D>0$, left panel) and easy-plane ($D<0$, right panel) types of the uniaxial magnetic anisotropy are analyzed. Furthermore, horizontal thin lines representing the spin-isotropic case ($D=0$) have been added for comparison. Note the break in the conductance scales, which has been introduced to highlight the qualitative changes of the contributions under consideration. Vertical lines, like in the right panel of Fig. 6, illustrate the relevant zero-field splittings. For detailed definitions of the contributions, see Eqs. (45, 46, 47, 48, 49). All other parameters as in Fig. 6.

Figure 8

Thermal conductance $\kappa$ (a)–(c), thermopower $\mathcal{S}$ (d)–(f), and spin thermopower ${\mathcal{S}}_{\text{S}}$ (g)–(l) plotted as functions of temperature $T$ for several values of ${\alpha }_{\text{ex}}$. Similarly as in the left panel of Fig. 6, different columns correspond to (from left) $D=0, D>0$, and $D<0$. Unlike $\kappa$ and $\mathcal{S}$, which are monotonic functions of $T$ in both magnetic configurations, and thus only the parallel one is shown, ${\mathcal{S}}_{\text{S}}$ is a monotonic function of $T$ only in the antiparallel configuration (j)–(l), whereas in the parallel one (g)–(i) it is nonmonotonic and it can even change its sign with increasing $T$. Vertical lines, like in the right panel of Fig. 6, illustrate the relevant zero-field splittings. All parameters as in Fig. 6.

Figure 9

The influence of transverse magnetic anisotropy on the transport and thermoelectric properties of a magnetic tunnel junction with a spin impurity. Both integer ($S=2$, left side) and half-integer ($S=5/2$, right side) impurity spin number are considered, and the uniaxial anisotropy of the easy-axis type is assumed ($D=100\mu \mathrm{eV}$). Top panel [(a), (b)]: Graphical illustration of the idea of how the transverse anisotropy term in the impurity Hamiltonian (10) mixes different states $|S_z\rangle$ of the $z$ component of the impurity spin. Bottom panel [(c)–(n)]: Elements of the generalized conductance matrix $G$ [(c)–(f)] and $G^{\text{m}}$ [(g)–(j)], shown as a function of temperature $T$ for the parallel and antiparallel magnetic configurations. The solid lines illustrate the situation of $E=0$. (k)–(n) Temperature dependence of the spin thermopower ${\mathcal{S}}_{\text{S}}$ for the values of $E/D$ as indicated in (c). Note that both parallel and antiparallel magnetic configurations are considered. Other parameters: ${\alpha }_{\text{d}}=1, {\alpha }_{\text{ex}}=0.5$, and $P_L=P_R\equiv P=0.5$.

Figure 10

Electrical conductances $G$ (a)–(d) and $G^{\text{m}}$ (e)–(h) plotted vs temperature $T$ for several values of the spin-polarization parameter $P_R$ of the right electrode and $P_L=0.5$. Similarly as in Fig. 9, two representative values of the impurity spin number are considered, and the conductances are shown for the parallel and antiparallel magnetic configurations. Apart from $E/D=0.3$, all other parameters are assumed the same as in Fig. 9.

Figure 11

Analogous to Fig. 10, except that now the heat conductance $\kappa$ (a) and the corresponding heat magnetoconductance HMC (b), together with the thermopower $\mathcal{S}$ (c), (d) and spin thermopower ${\mathcal{S}}_{\text{S}}$ (e), (f) are shown. Whereas $\kappa$ is plotted only for the parallel magnetic configuration, $\mathcal{S}$ and ${\mathcal{S}}_{\text{S}}$ are presented for both parallel and antiparallel configurations. Bottom panel: In order to trace the sign change of the spin thermopower in (e) from the negative one for small $P_R$ (see bold solid line for the limit of $P_R=0$) to the positive one for large $P_R$ (see dashed-double-dotted line for the limit of $P_R=1$), in the range $P_R=〈0.4,0.6〉$ with the interval $\Delta P_R=0.01$ we plot a series of curves (thin lines). The value of $P_R$ for each of these curves is color coded, with a black line representing $P_R=0.5$.

Figure 12

The effect of switching off the two-electrode tunneling processes on thermoelectric characteristics of a model system with an impurity of spin $S=2$ (with $D=100\mu \mathrm{eV}$ and $E=0$). Top panel [(a)–(f)]: The left column presents the temperature dependence of $G$ (a), $G^{\text{m}}$ (b), and spin $G_{\text{S}}$ (c) conductances in the parallel magnetic configuration for several values of the parameter $\eta$ (see the main text). In the right column, heat conductance $\kappa$ (d) in the parallel magnetic configuration and spin thermopower ${\mathcal{S}}_{\text{S}}$ (e), (f) in both the configurations are plotted. The bottom panel shows the spin magnetoconductance MC (g) and heat magnetoconductance HMC (h). Other parameters: ${\alpha }_{\text{d}}={\alpha }_{\text{ex}}=1$ and $P_L=P_R=0.5$.

Figure 13

Decomposition of spin $G_{\text{S}}$ (a) and heat $\kappa$ (b) conductances into different contributions [see Eqs. (48) and (50), respectively], shown as functions of temperature $T$ for $\eta =0$, with all other parameters as in Fig. 12. In (c) an example mechanism based on single-electrode electron tunneling processes that leads to transport of spin and energy through the junction is depicted. Recall that the left electrode has higher temperature than the right one $T_L>T_R$. The rightmost panel represents energy spectra of the spin impurity with the occupation probabilities for all spin states schematically marked with the dots.

Figure 14

Analysis of the influence of the uniaxial magnetic anisotropy (quantified by $D$) on the spin-dependent thermoelectric characteristics of the system in the absence of the charge transfer between electrodes. Two different cases of the impurity spin number $S$ are presented: for $S$ being integer (a)–(d) and for $S$ being half-integer (e)–(h). Spin $G_{\text{S}}$ (a), (e) and heat $\kappa$ (b), (f) conductances in the parallel magnetic configuration, as well as spin thermopower ${\mathcal{S}}_{\text{S}}$ both in the parallel (c), (g) and antiparallel (d), (h) configurations are plotted as a function of temperature $T$. Note that the solid line represents here the case of a spin-isotropic impurity ($D=0$), whereas the dashed (dashed-dotted) lines refer to a spin-anisotropic impurity with the uniaxial anisotropy of the easy-axis (easy-plane) type. Moreover, in the bottom panel ${\left({\mathcal{S}}_{\text{S}}\right)}_{\text{P}}={\left({\mathcal{S}}_{\text{S}}\right)}_{\text{AP}}=0$ for $D=0$. Remaining parameters: $\eta =0, {\alpha }_{\text{ex}}=1, E=0$, and $P_L=P_R=0.5$.

Figure 15

Analogous to Fig. 14, but now the effect of transverse magnetic anisotropy $E$ is investigated for two different types of the uniaxial magnetic anisotropy $D>0$ and $D<0$. Except $\left|D\right|=100\mu \mathrm{eV}$, all other parameters as in Fig. 14.

Figure 16

Temperature evolution of the spin conductance $G_{\text{S}}$ (a), heat conductance $\kappa$ (b), and spin thermopower ${\mathcal{S}}_{\text{S}}$ (c), (d) shown in the case of the junction including the spin impurity with $S=2$ ($D=100\mu \mathrm{eV}$ and $E/D=0.3$) for selected values of the spin-polarization parameter of the right electrode $P_R$ ($P_L=0.5$). All the quantities in question are presented for the parallel magnetic configuration, and ${\mathcal{S}}_{\text{S}}$ additionally also in the antiparallel one. Bottom panel [(e)–(h)]: Analogous to the top panel, except that now the quantities under discussion are plotted as functions of the spin polarizations of both electrodes $P_L$ and $P_R$, for $T=2$ K. Dashed line in (h) separates visually the region of negative ${\mathcal{S}}_{\text{S}}$ (above the line) from the region of positive ${\mathcal{S}}_{\text{S}}$ (below the line). Other parameters: $\eta =0$ and ${\alpha }_{\text{ex}}=1$.