# Nonequilibrium Seebeck and spin Seebeck effects in nanoscale junctions

Anand Manaparambil<EMAIL_ADDRESS>Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, Uniwersytetu
Poznańskiego 2, 61-614 Poznań, Poland  Ireneusz Weymann  Institute of
Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland

###### Abstract

The spin-resolved thermoelectric transport properties of correlated nanoscale
junctions, consisting of a quantum dot/molecule asymmetrically coupled to
external ferromagnetic contacts, are studied theoretically in the far-from-
equilibrium regime. One of the leads is assumed to be strongly coupled to the
quantum dot resulting in the development of the Kondo effect. The spin-
dependent current flowing through the system, as well as the thermoelectric
properties, are calculated by performing a perturbation expansion with respect
to the weakly coupled electrode, while the Kondo correlations are captured
accurately by using the numerical renormalization group method. In particular,
we determine the differential and nonequilibrium Seebeck effects of the
considered system in different magnetic configurations and uncover the crucial
role of spin-dependent tunneling on the device performance. Moreover, by
allowing for spin accumulation in the leads, which gives rise to finite spin
bias, we shed light on the behavior of the nonequilibrium spin Seebeck effect.

##  I Introduction

Quantum transport through nanoscale systems, such as quantum dots, molecular
junctions and nanowires, has been under tremendous research interest due to
promising applications of such nanostructures in nanoelectronics, spintronics
and spin-caloritronics [1, 2, 3, 4]. Due to the strong electron-electron
interactions and a characteristic discrete density of states, these systems
can exhibit large thermoelectric figure-of-merit and are excellent candidates
for nanoscale heat engines [5, 6, 7, 8, 9]. As far as more fundamental aspects
are concerned, correlated nanoscale systems allow one to explore fascinating
many-body phenomena that are not present in bulk materials. One of such
phenomena is the Kondo effect, which can drastically change the system’s
transport properties at low temperatures by giving rise to a universal
enhancement of the conductance to its maximum [10, 11, 12]. Moreover, in
addition to voltage-biased setups’ investigations, the emergence of Kondo
correlations can be probed in the presence of a temperature gradient, where
thermoelectric transport properties reveal the important physics [5, 6, 7]. In
fact, the thermopower of the quantum dot systems have been shown to contain
the signatures of the Kondo effect. Specifically, the sign changes in the
temperature dependence of the thermopower with the onset of Kondo correlations
have been identified in both the theoretical [12] and experimental [13, 14,
15] studies.

Furthermore, other interesting properties arise when the electrodes are
magnetic, making such nanoscale systems important for spin nanoelectronics
applications [3, 4]. It turns out that ferromagnetism of the leads can compete
with the Kondo correlations giving rise to an interplay between ferromagnet-
induced exchange field and the Kondo behavior [16, 17, 18, 19]. This interplay
has been revealed in theoretical studies on thermoelectric properties of
strongly-correlated molecular and quantum dot systems with ferromagnetic
contacts [20, 21]. From theoretical point of view, accurate description of
low-temperature transport behavior of correlated nanoscale systems with
competing energy scales requires resorting to advanced numerical methods, such
as the numerical renormalization group (NRG) method [22, 23]. Indeed, while
there has been a tremendous progress in complete understanding of transport
properties at equilibrium [24, 25, 26, 27, 28, 29, 30], much less is known in
fully nonequilibrium settings, where standard NRG cannot be applied. The exact
treatment of the nonlinear response regime requires even more sophisticated
numerical techniques [31, 32] and this is why it has been much less explored
[33, 34, 35, 36, 37, 38].

In this work we therefore investigate the nonlinear thermopower of a molecular
magnetic junction and analyze how the spin-resolved transport affects the
nonequilibrium thermoelectric properties of the system. More specifically, we
consider a quantum dot/molecule strongly coupled to one ferromagnetic lead and
weakly coupled to the other nonmagnetic or ferromagnetic lead kept at
different potentials and temperatures, see Fig. 1. We perform a perturbation
expansion in the weak coupling, while the strongly coupled subsystem, where
Kondo correlations may arise, is solved with the aid of the NRG method. This
allows us to extract the signatures of the interplay between the spin-resolved
transport and the Kondo correlations in the Seebeck coefficient in far from
equilibrium settings. Furthermore, we study how different magnetic
configurations of the system affect the differential and nonequilibrium
Seebeck effects. In particular, we show that the Seebeck effect exhibits new
sign changes as a function of the bias voltage which are associated with the
Kondo resonance split by exchange field. These sign changes are found to
extend to the temperature gradients on the order of the Kondo temperature.
Moreover, we also provide a detailed analysis of the nonequilibrium spin
Seebeck coefficient. We believe that our work sheds light on the spin-resolved
nonequilibrium thermopower of correlated nanoscale junctions, in which the
interplay between the Kondo and exchange field correlations is relevant. It
thus provides a better understanding of spin caloritronic nanodevices under
finite temperature and voltage gradients.

The paper is organized as follows: The system Hamiltonian and the theoretical
framework are described in Sec. II. The numerical calculations and the results
are discussed in Sec. III with a short summary and concluding remarks in Sec.
IV.

##  II Theoretical description

###  II.1 Hamiltonian of the system

Figure 1: The schematic of the considered asymmetric tunnel junction with
embedded quantum dot/molecule strongly coupled to a cold ferromagnetic left
lead and weakly coupled to a hot (a) nonmagnetic or (b) ferromagnetic right
lead. The right lead is subject to voltage and temperature gradients, while
the left lead is grounded and kept at zero temperature. The device in (b) can
be in two magnetic configurations: the parallel (P) and antiparallel (AP) one,
as indicated by the arrows.

We consider a nanoscale junction with an embedded quantum dot/molecule, which
is schematically shown in Fig. 1. The quantum dot is assumed to be strongly
coupled to the left ferromagnetic lead and weakly coupled to the right lead,
which can be either nonmagnetic [Fig. 1(a)] or ferromagnetic [Fig. 1(b)]. In
the case of two ferromagnetic electrodes, we will distinguish two magnetic
configurations: the parallel (P) one when the leads magnetic moments point in
the same direction and the antiparallel (AP) one, when the orientation of
magnetic moments is opposite, see Fig. 1(b). It is assumed that there are
finite temperature and voltage gradients applied to the system, with $T_{L}=0$
and $\mu_{L}=0$, whereas $T_{R}=\Delta T$ and $\mu_{R}=-eV$, as shown in Fig.
1, where $T_{\alpha}$ and $\mu_{\alpha}$ are the temperature ($k_{B}\equiv 1$)
and the chemical potential of lead $\alpha$.

With the assumption of weak coupling between the quantum dot and right contact
the system Hamiltonian can be simply written as

$H=H_{L}+H_{R}+H_{T}.$ (1)

$H_{L}$ describes the strongly coupled left subsystem, consisting of the
quantum dot and the left lead, and it is given by

$\displaystyle
H_{L}=\varepsilon_{d}\sum_{\sigma}n_{\sigma}+Un_{\uparrow}n_{\downarrow}+\sum_{k\sigma}\varepsilon_{Lk\sigma}c^{\dagger}_{Lk\sigma}c_{Lk\sigma}$
$\displaystyle+\sum_{k\sigma}t_{Lk\sigma}(d^{\dagger}_{\sigma}c_{Lk\sigma}+c^{\dagger}_{Lk\sigma}d_{\sigma}),$
(2)

where $n_{\sigma}=d^{\dagger}_{\sigma}d_{\sigma}$, with $d^{\dagger}_{\sigma}$
($d_{\sigma}$) being the creation (annihilation) operator on the quantum dot
for an electron of spin $\sigma$, $c_{\alpha k\sigma}$ ($c_{\alpha
k\sigma}^{\dagger}$) annihilates (creates) an electron in the lead $\alpha$
with momentum $k$, spin $\sigma$ and energy $\varepsilon_{\alpha k\sigma}$.
The quantum dot is modeled by a single orbital of energy $\varepsilon_{d}$ and
Coulomb correlations $U$. The hopping matrix elements between the quantum dot
and lead $\alpha$ are denoted by $t_{\alpha k\sigma}$ and give rise to the
level broadening $\Gamma_{\alpha\sigma}=\pi\rho_{\alpha\sigma}|t_{\alpha
k\sigma}|^{2}$, which is assumed to be momentum independent, where
$\rho_{\alpha\sigma}$ is the density of states of lead $\alpha$ for spin
$\sigma$.

The second part of the Hamiltonian describes the right lead and is given by

$H_{R}=\sum_{k\sigma}\varepsilon_{Rk\sigma}c^{\dagger}_{Rk\sigma}c_{Rk\sigma}-e\sum_{k\sigma}\mu_{R\sigma}c^{\dagger}_{Rk\sigma}c_{Rk\sigma},$
(3)

while the last term of $H$ accounts for the hopping between the left and right
subsystems

$H_{T}=\sum_{k\sigma}t_{Rk\sigma}(d^{\dagger}_{\sigma}c_{Rk\sigma}+c^{\dagger}_{Rk\sigma}d_{\sigma}).$
(4)

In the following, we use the lowest-order perturbation theory in $H_{T}$ to
study the spin-dependent electric and thermoelectric properties of the system.

###  II.2 Nonlinear transport coefficients

The electric current flowing through the system in the spin channel $\sigma$
can be expressed as [39, 40]

$\displaystyle I_{\sigma}(V,\Delta T)$ $\displaystyle=$
$\displaystyle-\frac{e\Gamma_{R\sigma}}{\hbar}\int_{-\infty}^{\infty}d\omega\;A_{L\sigma}(\omega)$
(5) $\displaystyle\times[f_{L}(\omega)-f_{R}(\omega-eV)],$

where $f_{\alpha}(\omega)=[1+{\rm exp}(\omega/T_{\alpha})]^{-1}$ is the Fermi-
Dirac distribution function, while $A_{L\sigma}(\omega)$ denotes the spin-
resolved spectral function of the left subsystem. The total current flowing
through the system under potential bias $V$ and temperature gradient $\Delta
T$ is thus $I(V,\Delta T)=\sum_{\sigma}I_{\sigma}(V,\Delta T)$. The spectral
function $A_{L\sigma}(\omega)$ is calculated by means of the NRG method [22,
23, 41], which allows us to include all the correlation effects between the
quantum dot strongly coupled to left contact in a fully nonperturbative
manner. In particular, $A_{L\sigma}(\omega)$ is determined as the imaginary
part of the Fourier transform of the retarded Green’s function of the left
subsystem Hamiltonian $H_{L}$,
${G_{\sigma}(t)=-i\Theta(t)\langle\\{d_{\sigma}(t),d_{\sigma}^{\dagger}(0)\\}\rangle}$.
In NRG calculations, the spectral data is collected in logarithmic bins that
are then broadened to obtain a smooth function.

For the further analysis, it is convenient to express the coupling constants
$\Gamma_{\alpha\sigma}$ by using the spin polarization of the lead $\alpha$,
$p_{\alpha}$, as $\Gamma_{L\sigma}=(1+\sigma p_{L})\Gamma_{L}$ and
$\Gamma_{R\sigma}=(1+\sigma p_{R})\Gamma_{R}$ for the parallel magnetic
configuration, with $\Gamma_{R\sigma}=(1-\sigma p_{R})\Gamma_{R}$ in the case
of the antiparallel configuration of the system. Here,
$\Gamma_{\alpha}=(\Gamma_{\alpha\uparrow}+\Gamma_{\alpha\downarrow})/2$.
Furthermore, in the case when the right lead is nonmagnetic, $p_{R}=0$, while
for both ferromagnetic leads we for simplicity assume $p_{L}=p_{R}\equiv p$.

As far as thermoelectric coefficients are concerned, the differential Seebeck
coefficient can be expressed as [42]

$S_{d}=-\left(\frac{dV}{d\Delta T}\right)_{\\!\\!I}=-\left(\frac{\partial
I}{\partial\Delta T}\right)_{\\!\\!V}\bigg{/}\\!\\!\left(\frac{\partial
I}{\partial\Delta T}\right)_{\\!\\!\Delta T}.$ (6)

Furthermore, the extension of the conventional Seebeck coefficient to the
nonlinear response regime is referred to as the nonequilibrium Seebeck
coefficient $S_{n}$, and it can be defined as [43, 44, 45, 46, 37, 47]

$S_{n}=-\left(\frac{\Delta V}{\Delta T}\right)_{\\!\\!I(V+\Delta V,\Delta
T)=I(V,0)}.$ (7)

The above definitions will be used to describe thermoelectric transport in
different configurations of the system, respectively.

##  III Numerical results and discussion

In this section we present the main numerical results and their discussion. In
our considerations we assume that the left lead is always ferromagnetic, while
the right electrode can be either nonmagnetic or ferromagnetic, cf. Fig. 1.
For the studied setup, the strong coupling to the left contact may give rise
to the Kondo effect [11, 48]. However, it is crucial to realize that the
presence of the spin-dependent hybridization results in a local exchange field
on the quantum dot, which can split the dot orbital level when detuned from
the particle-hole symmetry point, and thus suppress the Kondo resonance. The
magnitude of such exchange field can be estimated from the perturbation
theory, which at zero temperature gives [49],

$\Delta\varepsilon_{\rm
exch}=\frac{2p_{L}\Gamma_{L}}{\pi}\ln\left|\frac{\varepsilon_{d}}{\varepsilon_{d}+U}\right|.$
(8)

The presence of the exchange field and its detrimental effect on the Kondo
phenomenon has been confirmed by various experiments on electronic transport
measurements in quantum dot and molecular systems [17, 18, 50, 51].

We start our considerations with the analysis of electric transport
properties, revealing the effects of the exchange field. Further on, we study
the nonlinear thermoelectric response, first for the case of nonmagnetic right
lead and then for the case of two ferromagnetic leads. In numerical
calculations, we use the following parameters: $U=0.2$, $\Gamma_{L}=0.02$,
$\Gamma_{R}=0.002$, in units of band halfwidth, and $p=0.4$ for the
ferromagnetic leads. For the assumed parameters, the Kondo temperature of the
left subsystem for $\varepsilon_{d}=-U/2$ is equal to [52, 49], $T_{K}\approx
0.035\Gamma_{L}$.

Figure 2: The energy dependence of the spectral functions for the individual
spin channels, (a) $A_{L\uparrow}(\omega)$ and (b) $A_{L\downarrow}(\omega)$
calculated for the strongly coupled left subsystem with orbital energies as
indicated. The zoomed Kondo and split-Kondo peaks are shown in the insets. The
other parameters are: $U=0.2$, $\Gamma_{L}=0.02$, in units of band halfwidth,
and $p=0.4$.

To begin with, it is instructive to analyze the properties of the left
subsystem itself as described by the spectral function. The spectral function
for each individual spin channel is shown in Fig. 2. First of all, one can see
that for $\varepsilon_{d}=-U/2$ there is a pronounced Kondo peak at the Fermi
level for each spin component. However, when detuned from the particle-hole
symmetry, there is a finite exchange-induced splitting, cf. Eq. (8), which
suppresses the Kondo effect when $|\Delta\varepsilon_{\rm exch}|\gtrsim
T_{K}$, with $T_{K}$ denoting the Kondo temperature. Because of that, each
spin component of the spectral function displays a slightly detuned from Fermi
energy side peak, constituting the split Kondo resonance. In addition, the
Hubbard resonances at $\omega\approx\varepsilon_{d}$ and
$\omega\approx\varepsilon_{d}+U$ become affected as well: although their
position is only slightly modified, their magnitude gets strongly spin-
dependent.

Figure 3: The differential conductance for the quantum dot strongly coupled to
ferromagnetic left lead and weakly coupled to (a) nonmagnetic right lead,
ferromagnetic right lead in (b) the parallel magnetic configuration and (c)
the antiparallel magnetic configuration. The parameters are the same as in
Fig. 2 with $\varepsilon_{d}=-U/3$ and different temperature gradients, as
indicated.

The splitting of the Kondo resonance is directly visible in the differential
conductance of the system, which is demonstrated in Fig. 3. This figure
presents the bias voltage dependence of the differential conductance in
different magnetic configurations for various temperature gradients, as
indicated. More specifically, $G$ corresponds to the case when the right lead
is nonmagnetic [cf. Fig. 1(a)], while $G^{P}$ ($G^{AP}$) presents the case of
both ferromagnetic leads in the parallel (antiparallel) alignment [cf. Fig.
1(b)]. When the orbital level is detuned out of the particle-hole symmetry
point, $\varepsilon_{d}=-U/3$, as in the case of Fig. 3, the splitting of the
Kondo peak in the spectral function of the left subsystem becomes revealed in
the differential conductance of the whole system. Let us start with the case
of nonmagnetic right lead, presented in Fig. 3(a). First of all, one can note
a large asymmetry of the differential conductance with respect to the bias
reversal. Moreover, for small temperature gradients, $\Delta T\lesssim T_{K}$,
the split zero-bias anomaly due to the Kondo effect is visible. These features
can be understood by inspecting the behavior of the spectral function around
the Fermi energy, see the insets in Fig. 2. One can note that the split Kondo
peak in $A_{L\uparrow}(\omega<0)$ has smaller weight compared to the split
Kondo peak in $A_{L\downarrow}(\omega>0)$. Because, for low temperature
gradients, for $eV>0$ ($eV<0$) we probe the density of states of the left
subsystem for negative (positive) energies, the above-mentioned asymmetry in
$A_{L\sigma}(\omega)$ gives rise to highly asymmetric behavior of the
differential conductance, see Fig. 3(a), with the peak in the negative voltage
regime more pronounced than the other. Interestingly, when the tunneling to
the right lead becomes spin dependent, in the case of parallel configuration
one observes a rather symmetric behavior of $G^{P}$, with nicely visible split
zero-bias anomaly, see Fig. 3(b). This is due to the fact that the increased
tunneling rate of spin-down electrons due to larger density of states becomes
now reduced since the spin-down electrons are the minority ones in the right
lead. On the other hand, the tunneling of spin-up electrons to the right is
enlarged. As a consequence, the unequal contributions of the currents in each
spin channel become now equalized and the differential conductance in the
parallel configuration exhibits split-Kondo resonance with the side peaks of
comparable height. On the other hand, when the magnetization of the right lead
is flipped, the asymmetric behavior visible in Fig. 3(a) is even further
magnified, see Fig. 3(c). This can be understood by invoking similar arguments
as above, keeping in mind that now the rate of spin-up tunneling to the right
is smaller than that for spin-down electrons. With increase in the temperature
gradient, the Kondo-related behavior gets smeared and finally disappears when
$\Delta T\gtrsim T_{\rm K},|\Delta\varepsilon_{\rm exch}|$.

###  III.1 Effects of exchange field on nonequilibrium thermopower

In this section, we focus on the case where the right lead is nonmagnetic, see
Fig. 1(a). In such a setup it will be possible to observe clear signatures of
ferromagnet-induced exchange field on the thermoelectric properties of the
system subject to temperature and voltage gradients. We first study the case
of the linear response in potential bias with nonlinear temperature gradient
in Sec. III.1.1, while in Sec. III.1.2 the discussion is extended to the case
of nonlinear response regime in both $\Delta T$ and $V$.

####  III.1.1 Zero-bias thermoelectrics with finite temperature gradient

Figure 4: (a) The differential conductance $G$, (b) the differential Seebeck
coefficient $S_{d}$ and (c) the nonequilibrium Seebeck coefficient $S_{n}$ of
the quantum dot strongly coupled to left ferromagnetic lead and weakly
attached to the right nonmagnetic lead plotted as a function of the orbital
energy $\varepsilon_{d}$ and the temperature gradient $\Delta T$. The system
is assumed to be in the linear response regime with respect to the bias
voltage. The other parameters are the same as in Fig. 2.

Figure 4 displays the zero-bias differential conductance $G$, the differential
Seebeck coefficient $S_{\rm d}$ and the nonlinear Seebeck coefficient $S_{\rm
n}$ calculated as a function of orbital level $\varepsilon_{d}$ and finite
temperature gradient $\Delta T$. For low temperature gradients, the
conductance shows considerable increase near three values of
$\varepsilon_{d}$. The peaks for $\varepsilon_{d}\approx 0$ and
$\varepsilon_{d}\approx-U$ correspond to the Hubbard resonances in the
spectral function, whereas the maximum at $\varepsilon_{d}=-U/2$ is due to the
Kondo effect. In fact, in the local moment regime,
$-1\lesssim\varepsilon_{d}/U\lesssim 0$, the Kondo resonance is suppressed by
the exchange field once $|\Delta\varepsilon_{\rm exch}|\gtrsim T_{K}$, i.e.
for values of $\varepsilon_{d}$ away from the particle-hole symmetry point,
cf. Eq. (8). With the increase in the temperature gradient, the Kondo
resonance dies out when $\Delta T>T_{K}$ and the Hubbard peaks get suppressed
when $\Delta T>\Gamma_{L}$, see Fig. 4(a).

In the case of differential and nonlinear Seebeck coefficients presented in
Figs. 4(b) and (c), respectively, we can see an overall antisymmetric behavior
across the particle-hole symmetry point $\varepsilon_{d}=-U/2$. The sign of
the Seebeck coefficient here corresponds to the dominant charge carriers in
transport, holes for $\varepsilon_{d}<-U/2$ and particles for
$\varepsilon_{d}>-U/2$. The differential Seebeck coefficient shows two sign
changes in the local moment regime as a function of the temperature gradient.
The sign change around $\Delta T\approx T_{K}$ originates from the signatures
of the Kondo correlations present in the spectral functions. In the case of
nonlinear Seebeck coefficient, we do not find the corresponding sign changes
because $S_{\rm n}$ can deviate considerably from the linear response Seebeck
coefficient at large $\Delta T$ [29]. Additionally, one can see that both
Seebeck coefficients decay with decreasing $\Delta T$, this behavior can be
described using the Sommerfeld expansion for the linear response Seebeck
coefficient, where

$\left.S(T)\propto\frac{T}{A(\omega=0,T)}\frac{\partial
A}{\partial\omega}\right|_{\omega=0}.$ (9)

We also note that both Seebeck coefficients can possess finite values at even
lower $\Delta T$ inside the local moment regime than outside of it due to the
additional contribution of the Kondo resonance in the spectral function
$A_{L}(\omega)$ at $\omega=0$.

####  III.1.2 The case of nonlinear potential bias and temperature gradients

Figure 5: (a,d) The differential conductance $G$, (b,e) the differential
Seebeck coefficient $S_{d}$ and (c,f) the nonequilibrium Seebeck coefficient
$S_{n}$ as a function of the potential bias $V$ and the temperature gradient
$\Delta T$. The first row corresponds to the particle-hole symmetry point
$\varepsilon_{d}=-U/2$, while the second row shows the case of
$\varepsilon_{d}=-U/3$. The other parameters are the same as in Fig. 4.

Let us now inspect the behavior of the nonequilibrium thermoelectric
coefficients as a function of both potential bias and temperature gradient
shown in Fig. 5, focusing on the $V$ and $\Delta T$ range where Kondo
correlations are important. The first row of the figure corresponds to the
case of particle-hole symmetry, $\varepsilon_{d}=-U/2$, while the second row
presents the results for $\varepsilon_{d}=-U/3$. Consider the first case.
Figure 5(a) depicts the bias and temperature gradient dependence of the
differential conductance $G$. There exist a prominent peak at low $\Delta T$
centered at $V=0$, this is the zero-bias conductance peak characteristic of
the Kondo effect. As the temperature gradient increases, the Kondo peak dies
out and becomes smeared when $\Delta T\gtrsim T_{K}$. The differential and
nonlinear Seebeck coefficients, shown in Figs. 5(b) and (c), exhibit a sign
change with respect to the bias voltage reversal. Moreover, while $S_{d}$
exhibits considerable values around the Kondo peak and becomes suppressed as
$\Delta T$ grows, $S_{n}$ gets enhanced when $\Delta
T\gtrsim(\Gamma_{L}/U)|eV|$.

When the orbital level is detuned out of the particle-hole symmetry point, one
can observe an interesting interplay between the exchange field and Kondo
effect, and its signatures present in the nonlinear thermoelectric
coefficients. First, Fig. 5(d) shows the splitting of the Kondo peak due to
the exchange field present in the strongly correlated subsystem. As observed
in the discussions of Fig. 3(a), the split Kondo peaks are not symmetric, with
the more prominent one in the $eV<0$ regime and both dying off at large
$\Delta T$. Interestingly, the differential and nonlinear Seebeck coefficients
also capture the signatures of the exchange field shown by the split Kondo
peak. In fact, there exist additional sign changes in the nonlinear response
regime with respect to $V$. More specifically, at low $\Delta T$, there is a
sign change at low bias voltages, followed by another one, roughly located
around the split-Kondo peak, see Figs. 5(e) and (f). These sign changes
correspond to the additional energy scale in the system, namely the exchange
field $\Delta\varepsilon_{\rm exch}$. They occur at slightly different
absolute values of $eV$, which is due to the fact that the Kondo resonance in
the local density of states of the left subsystem exhibits an asymmetric
splitting, cf. Fig. 2. With increasing the temperature gradient, we observe
that the right split Kondo peak in the conductance dies out first, accordingly
the regime of positive values of the Seebeck coefficients corresponding to the
right peak disappears around $\Delta T\approx 0.03\,\Gamma_{L}$. Moreover, we
also note that the overall sign change of the thermopower as a function of the
bias voltage is now shifted to negative values of $eV$, as compared to the
case of particle-hole symmetry, see Fig. 5.

Figure 6: (a,b) The differential conductance $G$, (c,d) the differential
Seebeck coefficient $S_{d}$ and (e,f) the nonequilibrium Seebeck coefficient
$S_{n}$ in (first column) the parallel (P) and (second column) antiparallel
(AP) configuration calculated as a function of $\Delta T$ and
$\varepsilon_{d}$ assuming linear response in voltage. The spin polarizations
of both leads are equal to $p=0.4$ and the other parameters are the same as in
Fig. 4.

###  III.2 Effects of different magnetic configurations on nonequilibrium
thermopower

In this section we study the case where the quantum dot is coupled to both
ferromagnetic leads with spin polarization $p=0.4$. The magnetic moments of
the external leads are assumed to be aligned either in parallel or
antiparallel. The focus is on the effects of different magnetic configurations
on nonequilibrium thermoelectric transport properties.

####  III.2.1 The case of zero bias with nonlinear temperature gradient

Figure 7: (a,d) The differential conductance $G$, (b,e) the differential
Seebeck coefficient $S_{d}$ and (c,f) the nonequilibrium Seebeck coefficient
$S_{n}$ as a function of the bias voltage and temperature greadient in the
case of $\varepsilon_{d}=-U/3$. The first (second) row corresponds to the
parallel (antiparallel) magnetic configuration of the system. The other
parameters are the same as in Fig. 6.

The zero-bias thermoelectric properties of the system with two ferromagnetic
leads are shown in Fig. 6. The differential conductance for the parallel
$G^{P}$ and antiparallel $G^{AP}$ configuration of the lead magnetizations is
shown in Figs. 6(a) and (b). The qualitative behavior of both conductances is
similar to the case of nonmagnetic lead on the right, where $G$ shows a region
of high conductance around $\varepsilon_{d}=-U/2$ due to the Kondo effect.
Similarly to the previous case, the exchange field suppresses the linear
response conductance for values of $\varepsilon_{d}$ away from the particle-
symmetry symmetry point. Around $\varepsilon_{d}\approx 0,-U$, there is a rise
in the conductance corresponding to the contribution from the Hubbard peaks.
It is interesting to note that the conductance in the case of parallel
configuration is smaller than that in the antiparallel configuration around
the Kondo resonance, cf. the discussion of Fig. 3, while this situation is
reversed for the resonances at $\varepsilon_{d}\approx 0,-U$.

The Seebeck coefficients $S_{d}^{P}$ and $S_{n}^{P}$ shown in Fig. 6(c) and
(e) for the parallel configuration display very interesting features
corresponding to various energy scales. These coefficients show antisymmetric
behavior across $\varepsilon_{d}=-U/2$ and sign changes as a function of
temperature gradient in the local moment regime
$-1\lesssim\varepsilon_{d}/U\lesssim 0$. Let us first consider the linear
response in $\Delta T$ for $S_{d}^{P}$. In this regime one can relate the
Seebeck coefficient to the conductance through the Mott’s formula. Thus, the
changes of $G^{P}$ as a function of orbital level are reflected in the
corresponding dependence of the thermopower, which shows sign changes as
$\varepsilon_{d}$ is detuned from the particle-hole symmetry point. The first
sign change occurs when detuning is large enough to induce the exchange field
that suppresses the Kondo effect. Further sign change occurs at the onset of
conductance increase (as function of $\varepsilon_{d}$) due to the Hubbard
resonance. This behavior extends to higher $\Delta T$ as long as the thermal
gradient is smaller than the Kondo energy scale (or $\Delta\varepsilon_{\rm
exch}$). Otherwise, another sign change occurs as a function of $\Delta T$,
see Fig. 6(c). Very similar dependence can be observed in Fig. 6(e), which
shows the nonequilibrium Seebeck coefficient $S_{n}^{P}$. The main difference
is present for large $\Delta T$, where $S_{n}^{P}$ takes considerable values
while $S_{d}^{P}$ decreases, as explained earlier.

The situation is completely different in the case of the antiparallel
configuration, where one does not see any additional sign changes, neither in
$S_{d}^{AP}$ nor in $S_{n}^{AP}$, other than the ones present across
$\varepsilon_{d}=-U/2$, see Figs. 6(d) and (f). This can be understood by
realizing that the interplay of exchange field with spin-dependent tunneling
to the right contact hinders the splitting of the Kondo resonance as a
function of the bias voltage. Consequently, one only observes a single
resonance displaced from $V=0$, cf. Fig. 3(c), which results in much more
regular dependence of the differential and nonequilibrium Seebeck
coefficients.

####  III.2.2 The case of nonlinear potential bias and temperature gradient

The nonequilibrium thermoelectric properties of the quantum dot coupled to
both ferromagnetic leads are shown in Fig. 7. The first row corresponds to the
case of parallel configuration of the leads’ magnetizations. The differential
conductance depicted in Fig. 7(a) exhibits the split Kondo anomaly, with side
peaks of similar magnitude located at roughly the same distance from the zero
bias. Both peaks die off with the temperature gradient around $\Delta T\approx
0.05\,\Gamma_{L}$, i.e. when thermal gradient exceeds the Kondo temperature.

At low $\Delta T$ the differential and nonequilibrium Seebeck coefficients
exhibit similar bias voltage dependence to the case presented in Figs. 5(e)
and (f), see Figs. 7(b) and (c). Now, however, the region of negative Seebeck
coefficient is smaller. This can be attributed to the fact that the split
Kondo resonance is more symmetric across the bias reversal in the case of
parallel magnetic configuration, cf. Fig. 3(b). Unlike in the case of
nonmagnetic right lead, the sign changes at finite bias corresponding to the
split Kondo peak persist as long as $\Delta T\lesssim T_{K}$ and disappear
around comparable temperature gradient.

The case of antiparallel magnetic configuration of the system is presented in
the second row of Fig. 7. Consistent with the discussion of Fig. 3(c), the
differential conductance exhibits two conductance peaks but with a large
difference in their magnitudes. The peak in the negative bias regime is far
more pronounced than the miniscule peak one can observe in the positive
regime. Just as in the case of other configurations, the peaks die out with
increasing the temperature gradient but the negative bias peak survives till
larger temperature gradients $\Delta T\approx 0.2\Gamma_{L}$ whereas the
positive bias peak vanishes at temperature gradients as low as $\Delta
T\approx 0.02\Gamma_{L}$.

The Seebeck coefficients $S_{d}^{AP}$ and $S_{n}^{AP}$, shown in Figs. 7(e)
and (f), respectively, demonstrate a similar behavior to the other
configurations only at very low temperature gradients. However, now, instead
of sign changes, one only observes suppression of the Seebeck coefficients at
the corresponding values of the bias voltage associated with the exchange
field. These suppressions extend to temperatures gradients of the order of
$\Delta T\approx 0.03\Gamma_{L}$, see Figs. 7(e) and (f).

Figure 8: The charge Seebeck (first column) and the spin Seebeck (second
column) coefficients under nonlinear temperature gradient $\Delta T$ and
linear response spin bias $V_{s}$ as a function of the orbital level energy
$\varepsilon_{d}$ and $\Delta T$. The first row corresponds to the case of
nonmagnetic right lead, while the second (third) row presents the case of
ferromagnetic right lead in the parallel (antiparallel) magnetic configuration
of the system. The other parameters are the same as in Fig. 6.

###  III.3 Finite spin accumulation and the associated nonequilibrium spin
Seebeck effect

In this section we consider the case when ferromagnetic contacts are
characterized by slow spin relaxation, which can result in a finite spin
accumulation [53, 54]. Such a spin accumulation will induce a spin bias across
the quantum dot. Here, we assume that the spin accumulation and the resulting
spin-dependent chemical potential occurs only in the right lead. Thus, we
define the induced spin bias as, $V_{s}/2=\mu_{R\uparrow}=-\mu_{R\downarrow}$
(keeping $\mu_{L}=0$). The nonequilibrium spin bias across the quantum dot
enables the spin chemical potentials to be tuned separately and thus the
thermal bias induced transport can be different in the separate spin channels.
The system can then exhibit interesting spin caloritronic properties, such as
the spin Seebeck effect in this setup. The spin Seebeck coefficient $S_{s}$
quantifies the magnitude and the direction of the spin current induced in the
presence of a thermal bias [55]. Analogous to the differential Seebeck effect
$S_{d}$, the differential spin Seebeck coefficient $S_{s}$ in the nonlinear
response regime can be defined as

$S_{s}=-\left(\frac{dV_{S}}{d\Delta
T}\right)_{\\!\\!I_{s}}=-\left(\frac{\partial I_{s}}{\partial\Delta
T}\right)_{\\!\\!V_{s}}\bigg{/}\\!\left(\frac{\partial I_{s}}{\partial
V_{s}}\right)_{\\!\\!\Delta T},$ (10)

where $I_{s}=I_{\uparrow}(\mu_{R\uparrow},\Delta
T)-I_{\downarrow}(\mu_{R\downarrow},\Delta T)$ is the net spin current flowing
through the system. This quantity acts as a response over the spin current as
a function of both the spin bias $V_{s}$ and the temperature gradient $\Delta
T$. In addition to the net spin current, there can also exist a charge current
$I=\sum_{\sigma}I_{\sigma}(\mu_{R\sigma},\Delta T)$ flowing across the system
originating solely from the thermal and the spin biases. We define the Seebeck
coefficient that estimates the charge current in the presence of the spin bias
as the charge Seebeck coefficient $S$ [53]. The charge Seebeck coefficient $S$
can thus be defined based on the response of charge current $I$ as

$S=-\left(\frac{dV_{s}}{d\Delta T}\right)_{\\!\\!I}=-\left(\frac{\partial
I}{\partial\Delta T}\right)_{\\!\\!V_{s}}\bigg{/}\\!\left(\frac{\partial
I}{\partial V_{s}}\right)_{\\!\\!\Delta T}.$ (11)

We first discuss the case of linear response in the spin bias $V_{s}$ with
large and finite temperature gradient $\Delta T$, focusing on the differential
spin Seebeck coefficient $S_{s}$ and the charge Seebeck coefficient $S$. It is
pertinent to note that the nonequilibrium equivalent of the spin Seebeck
coefficient $S_{s,n}$ tends to remain undefined in our considerations, since
the magnitude of the spin bias fails to compensate for the thermally induced
spin current in (parts of) the regimes considered. Hence in this paper, we
limit our discussions to the differential spin Seebeck coefficient
$S_{s}\equiv S_{s,d}$ in the case of different configurations. We further
investigate the dependence of $S_{s}$ and $S$ on large and finite spin bias
under applied temperature gradient.

####  III.3.1 The case of zero spin bias with nonlinear temperature gradient

Figure 8 shows the behavior of the charge Seebeck coefficients $S$, $S^{P}$,
$S^{AP}$ and the spin Seebeck coefficients $S_{s}$, $S_{s}^{P}$, $S_{s}^{AP}$
for the case of nonmagnetic right lead, as well as the case of ferromagnetic
lead in the parallel and antiparallel magnetic configurations, respectively.
The first row of Fig. 8 shows the case of right lead with spin polarization
$p=0$, but with finite spin accumulation occurring from the spin-resolved
transport through the quantum dot. Figure 8(a) displays the charge Seebeck $S$
coefficient, which behaves similarly to the differential Seebeck $S_{d}$
presented in Fig. 4 except some points of divergences. At temperature
gradients smaller than $\Gamma_{L}$, there exist two additional sign changes,
both in the local moment regime symmetric across the particle-hole symmetry
point. The points of sign change spread out of the local moment regime for
thermal biases $\Delta T\gtrsim 3\,\Gamma_{L}$. The sign changes of the
Seebeck effect are also accompanied by large divergences in the magnitude of
$S$. The additional sign changes and divergences originate from the behavior
of the denominator in the definition of $S$, cf. Eq. (11). The denominator in
Eq. (11), which can be represented as, $G^{cs}=(\partial I/\partial
V_{s})_{\Delta T}$, is the differential mixed conductance [53] that estimates
the charge current in the presence of a spin bias, which can be either
negative or positive, resulting in its zero crossing points causing the
divergence. From a physical perspective, tuning the temperature gradient in
these specific regimes will result in extraordinary changes in the induced
charge current. Note that the colormaps in Figs. 8(a) and (e) have been
truncated for readability.

The charge Seebeck for the parallel configuration [see Fig. 8(c)] perfectly
recreates the behavior seen in Fig. 6(c). In the case of the parallel
configuration, the relative scaling of the couplings in each spin channels on
the right and left is the same, resulting in a non-negative $G^{cs}$ and,
thus, no divergences. Similarly to Fig. 8(e), the charge Seebeck effect for
the antiparallel configuration, there exist a resemblance to the Seebeck
coefficient discussed in Fig. 6(d), but overlaid by the divergences associated
with $G^{cs}$. In this case, the additional sign changes start from inside the
local moment regime at very low temperature gradients and move out of the
local moment regime monotonously around $\Delta T\approx 10^{-1}\,\Gamma_{L}$.

The differential spin Seebeck coefficient $S_{s}$ shown in panels (b), (d) and
(f) of Fig. 8 for different lead configurations behave antisymmetrically
across the particle-hole symmetry point ($\varepsilon_{d}=-U/2$). There exist
a pronounced spin Seebeck coefficient in the local moment regime for all the
configurations that dies off at $\Delta T\gtrsim 10\,\Gamma_{L}$. Such regions
of considerable spin Seebeck effect have been observed in the linear response
studies of symmetrically coupled quantum dots as a function of the global
temperature $T$ [21, 29]. In addition to the sign change at the particle-hole
symmetry point, at very low $\Delta T$ $S_{s}$ changes sign when moving out of
the local moment regime (i.e., at $\varepsilon_{d}\approx-U,0$). In the case
of the nonmagnetic right lead, the region of sign change outside the local
moment regime extends up to $\Delta T\approx\Gamma_{L}$, whereas for the
antiparallel configuration the sign change extends only up to $\Delta T\approx
0.2\,\Gamma_{L}$. On the other hand, the sign change of the spin Seebeck in
the local moment regime survives at thermal gradients even greater than
$\Delta T\approx 10^{2}\,\Gamma_{L}$ for the parallel configuration.

####  III.3.2 The case of nonlinear spin bias and temperature gradient

Figure 9:  The charge Seebeck (first column) and the spin Seebeck (second
column) coefficients for the orbital level $\varepsilon_{d}=-U/3$ as a
function of the applied spin bias $V_{s}$ and $\Delta T$. The first row
corresponds to the case of nonmagnetic right lead, while the second (third)
row presents the case of ferromagnetic right lead in the parallel
(antiparallel) magnetic configuration of the system. The other parameters are
the same as in Fig. 6.

The dependence of the nonlinear charge Seebeck and the spin Seebeck effect is
shown in Fig. 9 for the case of orbital energy level $\varepsilon_{d}=-U/3$.
The first column in Fig. 9 focuses on the charge Seebeck effect for various
magnetic configurations of the system. For the case of a nonmagnetic right
lead, the charge Seebeck coefficient $S$ changes sign four times as a function
of $V_{s}$ at temperature gradients below $\Delta T\approx 0.5\Gamma_{L}$, see
Fig. 9(a). Two among these sign changes (around $V_{s}\approx 0.001\,U$ and
$V_{s}\approx 0.15\,U$) correspond to the zeros in the mixed conductance
$G^{cs}$, which can be identified from the divergence in $S$ around the sign
changes. The other two sign changes (around $V_{s}\approx-0.05\,U$ and
$V_{s}\approx 0.03\,U$) originate from the zeros of the thermal response
$-({\partial I}/{\partial\Delta T})_{V_{s}}$, i.e. the numerator in the
definition of the charge Seebeck, cf. Eq. (10). As the temperature gradient
increases, the regions of sign change introduced by $G^{c}s$ and the
temperature gradient become larger in the spin bias regime until around
$\Delta T\approx\Gamma_{L}/3$ for the sign change associated with the mixed
conductance and $\Delta T\approx\Gamma_{L}/2$ due to the sign change from the
thermal response. With further increase in the temperature gradient $\Delta T$
the regions of sign change disappear. This happens around $\Delta
T\gtrsim\Gamma_{L}/2$ for the sign change caused by the mixed conductance and
$\Delta T\gtrsim 0.8\Gamma_{L}$ for the sign change due to the thermal
response. The remaining two sign changes at $V_{s}=-U/2$ and $V_{s}=U/2$
correspond to the Hubbard peaks of the quantum dot spectral function. The
region of these sign changes disappears above temperature gradient $\Delta
T\gtrsim 4\,\Gamma_{L}$. At $V_{s}=0$ and very large temperature gradients
(around $\Delta T\gtrsim 10\,\Gamma_{L}$), there exist another sign change
that originates from the zeros of $G^{cs}$. For positive $V_{s}$, this sign
change moves to lower $\Delta T$, while for negative $V_{s}$ this sign change
moves to higher $\Delta T$, see Fig. 9(a).

Figure 9(c) shows the charge Seebeck effect $S^{P}$ corresponding to the
system in parallel configuration of the leads. We observe that there are two
sign changes as a function of the spin bias $V_{s}$. At low temperatures,
$\Delta T\lesssim 0.01\,\Gamma_{L}$, the region of sign change appears between
$V_{s}\approx 0.005\,U$ and $V_{s}\approx U/2$. One can identify that these
sign changes originate solely from the thermal response of the current under
spin bias. With an increase in $\Delta T$, the sign change at $V_{s}\approx
0.005\,U$ crosses over to the negative $V_{s}$ regime and the sign change
around $V\approx U/2$ moves closer to $V_{s}\approx 2\,U/3$, thus increasing
the region of sign change in the spin bias $V_{s}$ regime until around $\Delta
T\approx 0.1\,G_{L}$. On further increase in temperature gradient, the regions
of sign change tend to disappear once $\Delta T\approx 0.2\,G_{L}$. On the
other hand, outside of this regime, the sign of the spin-resolved thermopower
remains positive. We also note that there exist another point of sign change
due to the contribution from the Hubbard peaks in the spectral function.
Unlike in the previous case of $S$, this sign change survives for large
temperature gradients $\Delta T$ and moves closer to $V_{s}=0$ when the
temperature gradient $\Delta T$ is increased $\Delta T\gtrsim\Gamma_{L}$, see
Fig. 9(c).

The charge Seebeck coefficient for the antiparallel configuration $S^{AP}$
does not show any sign change in the local moment regime apart from the
particle-hole symmetry point $\varepsilon_{d}=-U/2$, as seen in Fig. 8(e).
However, as a function of the spin bias $V_{s}$, two points of sign changes
form in the dependence of the charge Seebeck effect $S^{AP}$. One change
occurs in the negative spin bias regime around $V_{s}\approx-0.15\,U$ and the
other one in the positive regime at $V_{s}\approx 0.03\,U$. With increasing
$\Delta T$, these changes move further apart into the negative and positive
spin bias regimes, respectively.

It is important to emphasize that the sign changes observed in the charge
Seebeck coefficient as a function of spin bias $V_{s}$ do not correspond to
the sign changes seen in the Seebeck coefficient as a function of $V$, as
discussed and presented in Fig. 5 and Fig. 7. This is associated with the fact
that the generated current [Eq. (5)] as a function of $V$ scans through each
of the split Kondo resonances shown in Fig. 2 separately, resulting in the
split peaks seen in the differential conductance and the corresponding sign
changes in the Seebeck coefficients. However, as a function of the spin bias
$V_{s}$, the signatures from the split Kondo resonance cannot be identified
directly in the generated current $I$. This is because the spin bias
$\mu_{R\uparrow}-\mu_{R\downarrow}=V_{s}$ scans both split Kondo peaks (see
Fig. 2) simultaneously, and the total current $I$ is rescaled by just relative
couplings of the separate spin channels $\Gamma_{R\sigma}$. Hence, the sign
changes in the charge Seebeck coefficient are solely resulting from the sign
changes in the thermal response and the mixed charge conductance.

The spin Seebeck coefficient in the nonlinear spin bias regime is presented in
the second column of Fig. 9. Panels (b),(d) and (f) show the case of the
nonmagnetic right lead as well as ferromagnetic right lead in the parallel and
antiparallel configuration, respectively. From the discussion of the linear
$V_{s}$ case shown in Fig. 8, we observe that the differential spin Seebeck
coefficient does not change sign inside the local moment regime for all three
configurations. Under finite spin bias $V_{s}$, we can see only one sign
change in the positive spin bias regime around $V\approx U/4$ for all magnetic
configurations. The point of sign change shifts towards the positive regime
with increasing temperature gradient $\Delta T$. The behavior of the spin
Seebeck coefficient is identical for all the configurations apart from slight
differences in the magnitude, meaning that this originates solely from the
properties of the spectral function outside the split Kondo peaks. In the case
of parallel configuration, we observe a small region of additional sign change
around $V_{s}\approx 0.2\,U$ to $V_{s}\approx U/2$. Such a behavior have
already been observed in the nonequilibrium thermopower of similar systems
where it has been attributed to the characteristic behavior of the spectral
function for energies between the Kondo and the Hubbard peak [38].

##  IV Summary

In this paper we have studied the nonequilibrium thermoelectric properties of
the system consisting of a quantum dot/molecule asymmetrically coupled to
external ferromagnetic leads. The strongly coupled ferromagnetic contact
induces an exchange field in the dot that can split and suppress the Kondo
resonance. The emphasis has been put on the signatures of the interplay
between spin-resolved tunneling and strong electron correlations in the
nonequilibrium thermopower of the system. In particular, we have determined
the bias voltage and temperature gradient dependence of the differential and
nonequilibrium Seebeck coefficients. We have observed new signatures in the
Seebeck coefficients corresponding to the Kondo resonance and the regions
where the ferromagnetic contact induced exchange field suppresses the Kondo
effect both in the potential bias and temperature gradient regimes. More
specifically, we have demonstrated that the Seebeck coefficient exhibits new
sign changes as a function of bias voltage, which are associated with the
split Kondo resonance. These sign changes extend to the temperature gradients
on the order of the Kondo temperature. Furthermore, we investigated the
influence of the spin accumulation and the resulting spin bias on the Seebeck
and spin Seebeck coefficients. The nonlinear charge Seebeck coefficient and
the spin Seebeck coefficient showed points of sign changes in the presence of
finite spin and thermal bias, corresponding to the different properties of the
quantum dot spectral function.

###### Acknowledgements.

This work was supported by the Polish National Science Centre from funds
awarded through the decision No. 2017/27/B/ST3/00621. We also acknowledge the
computing time at the Poznań Supercomputing and Networking Center.

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