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# Revealing fingerprints of valence excitons in x-ray absorption spectra with
the Bethe-Salpeter equation

Nasrin Farahani1 ID Daria Popova-Gorelova1,2 ID<EMAIL_ADDRESS>1I. Institute for Theoretical Physics and Centre for Free-Electron Laser
Science, Universität Hamburg, Notkestr. 9, 22607 Hamburg, Germany
2The Hamburg Centre for Ultrafast Imaging (CUI),Luruper Chaussee 149, 22607
Hamburg, Germany

###### Abstract

The Bethe-Salpeter equation (BSE) is a powerful theoretical approach that is
capable to accurately treat electron-hole interactions in materials in an
excited state. We developed an ab initio framework based on the BSE to
describe a pump-probe experiment, in which an x-ray pulse probes solid-state
valence excitons by means of x-ray absorption spectroscopy. Our theoretical
framework is of relevance for an accurate modeling of pump-probe experiments
of photo-excited materials that utilize novel capabilities offered by x-ray
science.

Excitons, bound electron-hole pairs, can be created in a material due to
absorption of light and are responsible for energy-conversion processes.
Excitonic effects can play an essential role for electronic excitations in
solids and influence their dynamics. Despite the crucial role of excitons for
functionality of optoelectronic devices, their properties are not completely
understood. X-ray absorption spectroscopy (XAS) is an element-specific
technique, in which an x-ray pulse resonantly excites an electron from a core
orbital into unoccupied states of a material or a molecule [1, 2, 3]. Its
ability to deliver information about electronic structure of matter made it a
powerful method for studying materials in an excited state [4, 5, 6, 7].
Especially, it has been applied to investigate dynamics of excitons in
functional materials [8, 9, 10, 11, 12, 13]. Accurate interpretation of
features in XAS spectra and mapping them to properties of a photo-excited
material relies on an accurate ab initio modelling of corresponding pump- and
probe-induced processes. Several theoretical approaches were recently applied
to describe XAS of excitons in materials using model Hamiltonians [14, 15] or
using Time-Dependent Density Functional Theory [9, 12] as well as approaches
applicable to molecular systems are being developed [16, 17, 18, 19]. In this
work, we present an ab initio framework based on the Bethe-Salpeter equation
(BSE) [20, 21, 22] to describe an experiment in which a pump pulse creates a
valence exciton in a material and a probe x-ray pulse probes the exciton by
inducing a transition of a core electron into unoccupied states below Fermi
level. Our framework allows for advanced modelling of XAS spectra of photo-
excited materials.

The Bethe-Salpeter equation (BSE) is a state-of-the-art ab initio many-body
approach that can accurately treat excited-states in matter including
excitonic effects. On the one hand, it has been proven to be a powerful tool
for calculations of optical excitations in solids and for the description of
valence excitons. On the other hand, this method substantially improves
calculations of XAS in solid-state materials since excitonic effects can also
play an important role for core-excited states. It is common to use the BSE
for calculations of either optical absorption spectra, when an initial state
is a ground state and final states are optically-excited states [23, 24, 25,
26], or x-ray absorption spectra, when an initial state is a ground state and
final states are core-excited states [27, 28, 29]. In a recent study, the BSE
was employed to describe an optical-pump–optical-probe experiment that
involves valence-excited states [30]. In our study, we employ the BSE to
describe initial states that are optically-excited states and to describe
final states that are core-excited states.

We illustrate our results with calculations for 4H-SiC that is known to have a
strong excitonic effect [31, 32, 33]. One advantage of this material is that
its optical absorption spectra have been extensively investigated [34, 35, 36,
37, 38, 39, 40, 41, 42, 43] and its XAS has been measured [44, 45], which
allows us to verify the accuracy of the BSE calculations. The other advantage
is that it has two types of atoms in the unit cell and we can compare results
for Si and C K-edges.

We derive XAS of a photo-excited material within the second-order time-
dependent perturbation theory and second quantization formalism [46] in
Supplementary material (SM) [47]. We show that the absorption cross section in
the pump-probe experiment is proportional to

$\displaystyle\sigma\propto\sum_{\lambda_{F}}\left|\sum_{\lambda_{I}}\frac{\int
d^{3}r_{1}\langle\Psi_{\lambda_{F}}|\hat{\psi}^{\dagger}(\mathbf{r}_{1})e^{i\mathbf{k}_{x}\cdot\mathbf{r}_{1}}(\bm{\epsilon}_{x}\cdot\hat{\mathbf{p}})\hat{\psi}(\mathbf{r}_{1})|\Psi_{\lambda_{I}}\rangle}{\Gamma_{F}-i(E_{\lambda_{I}}-E_{\lambda_{F}}+\omega_{x})}\frac{\int
d^{3}r_{2}\langle\Psi_{\lambda_{I}}|\hat{\psi}^{\dagger}(\mathbf{r}_{2})(\bm{\epsilon}_{o}\cdot\hat{\mathbf{p}})\hat{\psi}(\mathbf{r}_{2})|\Psi_{0}\rangle}{\Gamma_{I}-i(E_{0}-E_{\lambda_{I}}+\omega_{o})}\right|^{2},$
(1)

where this and all the following expressions are in atomic units.
$|\Psi_{0}\rangle$ is the ground state a many-electron system with the energy
$E_{0}$, $|\Psi_{\lambda_{I}}\rangle$ is its optically-excited state with
eigenenergy $E_{\lambda_{I}}$, $|\Psi_{\lambda_{F}}\rangle$ is its core-
excited state with the eigenenergy $E_{\lambda_{F}}$ after absorption of an
x-ray photon. $\Gamma_{I}$ and $\Gamma_{F}$ are life-time broadenings of the
states $\lambda_{I}$ and $\lambda_{F}$. $\mathbf{k}_{x}$ is the wave vector of
the x-ray pulse. $\omega_{x}$, $\bm{\epsilon}_{x}$, and $\omega_{o}$,
$\bm{\epsilon}_{o}$ are the photon energies and polarizations of an x-ray
probe pulse and an optical pump pulse, correspondingly. $c$ is the speed of
light, $\hat{\mathbf{p}}$ is the momentum operator and $\psi^{\dagger}$,
$\psi$ are field operators [48]. Within the BSE formalism, a many-body wave
function are eigenfunctions of a two-particle excitonic Hamiltonian that are
represented as [49, 50, 51]

$\displaystyle|\Psi_{\lambda}\rangle=\sum_{e,h,{\mathbf{k}}}A^{\lambda}_{eh{\mathbf{k}}}|e_{\mathbf{k}}h_{\mathbf{k}}\rangle,$
(2)

where $|e_{\mathbf{k}}h_{\mathbf{k}}\rangle$ is a basis state, where an
electron occupies a one-particle state $|\phi_{e{\mathbf{k}}}\rangle$ and a
hole occupies a one-particle state $|\phi_{h{\mathbf{k}}}\rangle$. We can also
represent this state as
$|e_{\mathbf{k}}h_{\mathbf{k}}\rangle=\hat{c}_{e{\mathbf{k}}}^{\dagger}\hat{c}_{h{\mathbf{k}}}|\Psi_{0}\rangle$,
where $\hat{c}_{e{\mathbf{k}}}^{\dagger}$ ($\hat{c}_{h{\mathbf{k}}}$) creates
(annihilates) an electron from the one-particle state
$|\phi_{e{\mathbf{k}}}\rangle$ ($|\phi_{h{\mathbf{k}}}\rangle$) of the many-
body ground-state wave function $|\Psi_{0}\rangle$. If many-body wave
functions can be represented as a product state of the Fock space and one-
particle wave functions form an orthonormal basis set, then the field
operators can be represented as
$\hat{\psi}(\mathbf{r})=\sum_{\mu,{\mathbf{k}}}\hat{c}_{\mu{\mathbf{k}}}\phi_{\mu{\mathbf{k}}}(\mathbf{r})$
and
$\hat{\psi}^{\dagger}(\mathbf{r})=\sum_{\mu,{\mathbf{k}}}\hat{c}^{\dagger}_{\mu{\mathbf{k}}}\phi^{\dagger}_{\mu{\mathbf{k}}}(\mathbf{r})$
[52]. Within a standard application of the BSE for solid-state materials,
$\phi_{\mu{\mathbf{k}}}(\mathbf{r})$ are Kohn-Sham orbitals, where $\mu$ is a
band and a ${\mathbf{k}}$-point index. We now assume that the wave functions
of both the initial state $\lambda_{I}$ and the final state $\lambda_{F}$ that
enter Eq. (1) are the solutions of the BSE equation and derive the following
expression for the absorption cross section in SM [47]

$\displaystyle\sigma=\frac{1}{c}\sum_{\lambda_{F}}\left|\sum_{\lambda_{I}}w_{\lambda_{I}}\frac{\sum_{e,v,\mu,{\mathbf{k}}}A^{*\lambda_{F}}_{e\mu{\mathbf{k}}}A^{\lambda_{I}}_{ev{\mathbf{k}}}\int
d^{3}r\phi_{v{\mathbf{k}}}^{\dagger}(\mathbf{r})e^{i\mathbf{k}_{x}\cdot\mathbf{r}}(\bm{\epsilon}_{x}\cdot\hat{\mathbf{p}})\phi_{\mu{\mathbf{k}}}(\mathbf{r})}{\Gamma_{F}-i(E_{\lambda_{I}}-E_{\lambda_{F}}+\omega_{x})}\right|^{2}.$
(3)

Here, $\mu$ are core states and $v$ are valence states, $w_{\lambda_{I}}$ is a
“weight” associated with a valence-excited state $\lambda_{I}$

$\displaystyle w_{\lambda_{I}}$ $\displaystyle=\sqrt{\frac{2\pi
I_{o}}{c\omega_{o}^{2}}}\frac{\sum_{e,h,{\mathbf{k}}}A^{*\lambda_{I}}_{eh{\mathbf{k}}}\int
d^{3}r\phi_{e{\mathbf{k}}}^{\dagger}(\mathbf{r})(\bm{\epsilon}_{o}\cdot\hat{\mathbf{p}})\phi_{h{\mathbf{k}}}(\mathbf{r})}{\Gamma_{I}-i(E_{0}-E_{\lambda_{I}}+\omega_{o})},$
(4)

where $I_{o}$ is the intensity of the pump pulse. The expression in Eq. (3)
can be intuitively understood as the cross section resulting due to x-ray-
induced transitions from states $\lambda_{I}$ to states $\lambda_{F}$ weighted
by a probability amplitude of the optical excitation into the state
$\lambda_{I}$.

Our assumption that the final state can be described by the BSE wave function
in Eq. (2) implies that there is one exciton in the final state. This applies
to experiments, in which an x-ray-pulse energy is below the x-ray absorption
threshold (XAT) and an electron from a core shell fills the hole in valence
bands as shown in Fig. 1. Such experiments carry information about holes of a
photo-excited material. Since the hole is coupled to the electron, the XAS
spectra are also affected by the presence of an electron in the conduction
states. This effect is covered by our formalism.

We perform the BSE+G0W0 calculations for 4H-SiC implemented in exciting
software package [28, 29] to calculate both valence-excited states [53, 54,
55, 56, 57] and core-excited states [58, 59, 60, 61]. exciting is an
electronic-structure density-functional-theory (DFT) package that employs all-
electron full-potential linearized augmented plane-wave method combined with
local orbital approach (LAPW+lo) [62]. This method allows for the treatment of
core orbitals in solid-state materials [63]. A core state
$|\phi_{\mu{\mathbf{k}}}\rangle$ has a ${\mathbf{k}}$-point index, since core
states are represented as periodic Bloch functions within the LAPW+lo method.
We use the generalized gradient approximation (GGA)[64] in the implementation
of Perdew, Burke, and Ernzerhof [65] for exchange-correlation functional
calculation in the DFT calculation. G0W0 approximation was applied on top of
the DFT calculation to correct quasi-particle energies that enter the BSE
equation [66, 67]. We performed calculations on a $4\times 4\times 4$
$\mathbf{k}$-point grid, the number of unoccupied bands considered in G0W0
calculations were 300. Valence-excited and core-excited BSE calculations were
performed using 18 unoccupied bands, with Lorentzian broadening widths of 0.48
eV and 0.32 eV, respectively. With the two BSE calculations, we obtain the
necessary components for the evaluation of XAS spectra with Eq. (3): the
coefficients $A^{\lambda_{F}}_{e\mu{\mathbf{k}}}$ and
$A^{\lambda_{I}}_{ev{\mathbf{k}}}$, the energies $E_{\lambda_{F}}$ and
$E_{\lambda_{I}}$, and the momentum matrix elements between the Kohn-Sham
orbitals.

We first analyze the optical absorption spectrum of 4H-SiC calculated with the
BSE. The absorption cross section averaged over polarizations is proportional
to the imaginary part of the dielectric function $\varepsilon$ that is shown
in Fig. 2(a). It agrees with the experimental data reported in Refs. [42, 43],
(see SM [47] for the data comparison). We also show
$\operatorname{Im}(\varepsilon)$ calculated with the DFT for comparison in
Fig. 2(a).

Figure 1: Scheme of the considered pump-probe experiment. First, a pump pulse
creates a ”valence exciton”, where a red circle bounded to a yellow circle
represents the created exciton. Second, an x-ray probe pulse creates a ”core
exciton” by inducing a transition of an electron from a core-shell to a
vacancy in a valence band.

The significant difference between the DFT and the BSE+G0W0 optical spectra in
the energy region 4–6 eV indicate that the excitonic effects are particularly
strong for excited states with these energies. For this reason, we further
consider the photon energy of a pump pulse to be $4.6$ eV.

The properties of a pump pulse determine the shape of created excitons. In
Figs. 2(b)-(f), we analyze how the structure of excitons depends on the pump-
pulse polarization $\bm{\epsilon}_{o}$. Figs. 2(b) - (d) show
$|w_{\lambda_{I}}|^{2}$ for different valence-excited states $\lambda_{I}$ as
a function of their energy $E_{\lambda_{I}}$. $|w_{\lambda_{I}}|^{2}$ is
proportional to the probability of a transition to a $\lambda_{I}$ state.
Figs. 2(e) - (f) show the averaged distribution of the hole density of created
excitons $\rho_{h}$. Details of the calculation of $\rho_{h}$ and the crystal
structure of 4H-SiC are provided in SM [47].

Comparing Figs. 2(b) and (c), we see that a pump pulse with polarization along
the $\mathbf{a}$ and a pump pulse with a polarization along the $\mathbf{b}$
lattice vectors excite same excitonic states $\lambda_{I}$ with the same
probability. We also obtain that the corresponding hole distributions shown in
Fig. 2(e) are identical for these pump-pulse polarizations. They are
predominantly of a Carbon $p$-type character aligned along the both
$\mathbf{a}$ and $\mathbf{b}$ directions slightly bent towards the
$\mathbf{c}$ vector. The excitons’ hole distributions for $\bm{\epsilon}_{o}$
parallel to the $\mathbf{c}$ lattice vector are predominantly of a Carbon
$p$-type character pointing along the $\mathbf{c}$ direction.

Figure 2: (a) Imaginary part of the dielectric function of 4H-SiC. Spectrum
calculated by means of BSE+G0W0 is shown with a solid line. Spectrum
calculated by means of DFT is shown in the gray filled curve. The dotted line
illustrates the photon energy of the pump pulse. (b)-(d) Weights
$w_{\lambda_{I}}$ of valence-excited states $\lambda_{I}$ as a function of
their energy $E_{\lambda_{I}}$ for the pump-pulse photon energy
$\omega_{o}=4.6$ eV and polarization of a pump pulse along (b) the
$\mathbf{a}$ lattice vector; (c) the $\mathbf{b}$ lattice vector and (d) the
$\mathbf{c}$ lattice vector. (e)-(f) Averaged distribution of the excitons’
holes in real space for a pump pulse polarized (e) perpendicular to the
$\mathbf{c}$ lattice vector and (f) along $\mathbf{c}$ lattice vector. We use
the VESTA package [68] for visualization.

We show the results of our calculations of Carbon K-edge and Si K-edge x-ray
spectra of an optically-excited 4H-SiC probed by an x-ray pulse with an energy
below XAT in Fig. 3. We also show the XAS from 4H-SiC in the ground state that
is in a good agreement with experimental data [44, 45, 47]. We find that the
presence of an exciton leads to an appearance of an additional peak in the
spectra below the XAT. Such an additional peak in the XAS signal after a
photo-excitation has been observed in experimental studies [8, 9, 10, 12]. It
appears due to the creation of a coupled electron-hole pair. The hole of the
pair provides an additional channel for an electron transition from a core to
a state near the Fermi energy. The position of the peak at an energy lower
than XAT can be intuitively understood in an one-particle picture. In such a
picture, the energy of the peak would be given by the energy difference
between a core orbital and a valence band, which is lower than the energy of
transitions to conduction bands involved in a ground-state XAS (see Fig. 1).

Figure 3: Carbon and Silicon K-edges XAS spectra. Red and blue peaks show XAS
spectra of optically-excited 4H-SiC at Carbon and Silicon Kedges,
correspondingly, for different polarizations of a pump and a probe pulse.
Black curves shaded in rectangles show XAS spectra of a ground-state 4H-SiC
depending on the probe-pulse polarization. (a)–(i) Carbon K-edge XAS spectra
for (a) $\bm{\epsilon}_{o}\parallel\mathbf{a}$,
$\bm{\epsilon}_{x}\parallel\mathbf{a}$; (b)
$\bm{\epsilon}_{o}\parallel\mathbf{a}$,
$\bm{\epsilon}_{x}\parallel\mathbf{b}$; (c)
$\bm{\epsilon}_{o}\parallel\mathbf{a}$,
$\bm{\epsilon}_{x}\parallel\mathbf{c}$; (d)
$\bm{\epsilon}_{o}\parallel\mathbf{b}$,
$\bm{\epsilon}_{x}\parallel\mathbf{a}$; (e)
$\bm{\epsilon}_{o}\parallel\mathbf{b}$,
$\bm{\epsilon}_{x}\parallel\mathbf{b}$; (f)
$\bm{\epsilon}_{o}\parallel\mathbf{b}$;
$\bm{\epsilon}_{x}\parallel\mathbf{c}$; (g)
$\bm{\epsilon}_{o}\parallel\mathbf{c}$;
$\bm{\epsilon}_{x}\parallel\mathbf{a}$; (h)
$\bm{\epsilon}_{o}\parallel\mathbf{c}$,
$\bm{\epsilon}_{x}\parallel\mathbf{b}$; (i)
$\bm{\epsilon}_{o}\parallel\mathbf{c}$,
$\bm{\epsilon}_{x}\parallel\mathbf{c}$. (j)–(l) Silicon K-edge XAS spectra for
the polarization of pump and probe being parallel to (j)
$\bm{\epsilon}_{o}\parallel\mathbf{a}$;
$\bm{\epsilon}_{x}\parallel\mathbf{a}$; (k)
$\bm{\epsilon}_{o}\parallel\mathbf{c}$,
$\bm{\epsilon}_{x}\parallel\mathbf{b}$; (l)
$\bm{\epsilon}_{o}\parallel\mathbf{c}$,
$\bm{\epsilon}_{x}\parallel\mathbf{c}$. The relative intensity of the
additional peak and the ground-state XAS is arbitrary as it depends on the
intensity of a pump pulse.

If the energy of an x-ray pulse is above the XAT, then a final state in an
optical-pump–x-ray-probe experiment is a doubly-excited excitonic state with
one core-excited and one valence-excited exciton. Such transitions are beyond
our formalism and that is why we do not observe any changes in the XAS spectra
above the XAT. In an experiment, it would also be challenging to resolve the
change of the XAS spectra above the threshold due to a strong background from
a ground-state XAS signal. In our study, we focus on the signal below the XAT,
which does not suffer from a background.

XAS spectra of 4H-SiC excited by a pump pulse with polarizations
$\bm{\epsilon}_{o}\parallel\mathbf{a}$ are shown in Figs. 3(a)-(c). K-edge XAS
spectra are sensitive to a $p$-type contribution to a wave function of
unoccupied states, since they are governed by transitions from an $1s$
orbital. In agreement with our observation that the hole is of a predominantly
$p$-type character pointing symmetrically in the $\mathbf{a}$ and $\mathbf{b}$
directions, we observe that intensive and identical peaks appear for the probe
pulses polarized along the $\mathbf{a}$ and along $\mathbf{b}$ directions. A
small peak appears in the case of $\bm{\epsilon}_{x}\parallel\mathbf{c}$ as
well, which is because the hole has also a non-zero projection along the
$\mathbf{c}$ direction (see also the two-dimensional (2D) cuts of the density
in the SM [47]). The intensity of the peaks for a pump pulse polarized along
the $\mathbf{b}$ direction in Fig. 3(d)-(f) are the same as that in Fig.
3(a)-(c) in agreement with the fact that the averaged hole distributions are
identical for the pump pulses polarized along $\mathbf{a}$ and along
$\mathbf{b}$.

XAS signals for 4H-SiC excited by an the optical pulse polarized along the
$\mathbf{c}$ direction are shown in Fig. 3(g)-(i). The strongest XAS signal in
this case is for the probe pulse polarized along the $\mathbf{c}$ direction.
This indicates that the hole distribution is predominantly aligned along the
$\mathbf{c}$ vector, which agrees with our observation in Fig. 2(f). The hole
distribution also has a small contribution of a $p$-type character centred on
Si atoms, which is visible on 2D density cuts in SM [47]. And this is why we
observe the additional peak in the calculated Si K-edge XAS spectra in Fig.
3(j)-(l) as well, but find that Si K-edge XAS spectra are overall much less
intensive than the Carbon K-edge XAS spectra. These results demonstrate that
there is a direct correspondence between the shape of the excitons’ hole
distribution and the dependence of the intensity of the corresponding feature
in the XAS spectra on polarization of the x-ray probe pulse.

In conclusion, we developed an ab initio framework to describe optical-
pump–x-ray-probe experiments in solid-state materials that treats excitonic
effects at the BSE level, the state-of-the-art many-body approach. This way,
it is capable to accurately describe modern experiments, where excited-state
dynamics in functional materials or complexes are probed by means of x-ray
pulses [69, 13]. Using the example of 4H-SiC, we demonstrated that a shape of
an exciton can strongly depend on the excitation properties, which can
eventually affect exciton transport. XAS is a powerful tool to reveal details
about the shape of a positive charge of a photo-excited material with an
element- and orbital-specificity.

## Acknowledgement

We gratefully acknowledge the financial support of the Volkswagen Foundation.
D P.-G. acknowledges support from the Cluster of Excellence ’CUI: Advanced
Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 -
project ID 390715994.

## References

  * De Groot and Kotani [2008] F. De Groot and A. Kotani,  _Core level spectroscopy of solids_ (CRC press, 2008).
  * Yano and Yachandra [2009] J. Yano and V. K. Yachandra, X-ray absorption spectroscopy, Photosynthesis research 102, 241 (2009).
  * Baumgarten [2012] L. Baumgarten, X-ray absorption spectroscopy, Scattering Methods for Condensed Matter Research: Towards Novel Applications at Future Sources, Jülich (Germany) 33, F4.1 (2012).
  * Ramasesha  _et al._ [2016] K. Ramasesha, S. R. Leone, and D. M. Neumark, Real-time probing of electron dynamics using attosecond time-resolved spectroscopy, Annual Review of Physical Chemistry 67, 41 (2016).
  * Santomauro  _et al._ [2017] F. G. Santomauro, J. Grilj, L. Mewes, G. Nedelcu, S. Yakunin, T. Rossi, G. Capano, A. Al Haddad, J. Budarz, D. Kinschel, D. S. Ferreira, G. Rossi, M. Gutierrez Tovar, D. Grolimund, M. Samson, Valerieand Nachtegaal, G. Smolentsev, M. Kovalenko V, and M. Chergui, Localized holes and delocalized electrons in photoexcited inorganic perovskites: Watching each atomic actor by picosecond x-ray absorption spectroscopy, Structural Dynamics 4 (2017).
  * Fracchia  _et al._ [2018] M. Fracchia, P. Ghigna, A. Vertova, S. Rondinini, and A. Minguzzi, Time-resolved x-ray absorption spectroscopy in (photo) electrochemistry, Surfaces 1, 138 (2018).
  * Huang  _et al._ [2021] N. Huang, H. Deng, B. Liu, D. Wang, and Z. Zhao, Features and futures of x-ray free-electron lasers, The Innovation 2, 100097 (2021).
  * Garratt  _et al._ [2021] D. Garratt, L. Misiekis, D. Wood, E. Witting-Larsen, M. Matthews, O. Alexander, P. Ye, S. Jarosch, A. Bakulin, T. Penfold, and J. Marangos, Ultrafast exciton dynamics in poly(3-hexylthiophene) probed with time resolved x-ray absorption spectroscopy at the carbon k-edge,  _2021 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference_ , 2021 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference , jsiii_2_4 (2021).
  * Garratt  _et al._ [2022] D. Garratt, L. Misiekis, D. Wood, E. W. Larsen, M. Matthews, O. Alexander, P. Ye, S. Jarosch, C. Ferchaud, C. Strüber, A. S. Johnson, A. A. Bakulin, and J. P. Penfold, T. J. Marangos, Direct observation of ultrafast exciton localization in an organic semiconductor with soft x-ray transient absorption spectroscopy, Nature Communications 13, 3414 (2022).
  * Hillyard  _et al._ [2009] P. Hillyard, S. Kuchibhatla, T. Glover, M. Hertlein, N. Huse, P. Nachimuthu, L. Saraf, S. Thevuthasan, and K. Gaffney, Atomic resolution mapping of the excited-state electronic structure of cu2o with time-resolved x-ray absorption spectroscopy, Physical Review B 80, 125210 (2009).
  * Palmieri  _et al._ [2020] T. Palmieri, E. Baldini, A. Steinhoff, A. Akrap, M. Kollár, E. Horváth, L. Forró, F. Jahnke, and M. Chergui, Mahan excitons in room-temperature methylammonium lead bromide perovskites, Nature communications 11, 850 (2020).
  * Zürch  _et al._ [2017] M. Zürch, H.-T. Chang, L. J. Borja, P. M. Kraus, S. K. Cushing, A. Gandman, C. J. Kaplan, M. H. Oh, J. S. Prell, D. Prendergast, C. D. Pemmaraju, D. M. Neumark, and S. R. Leone, Direct and simultaneous observation of ultrafast electron and hole dynamics in germanium, Nature Communications 8, 15734 (2017).
  * Garratt  _et al._ [2024] D. Garratt, M. Matthews, and J. Marangos, Toward ultrafast soft x-ray spectroscopy of organic photovoltaic devices, Structural Dynamics 11, 010901 (2024).
  * Malakhov  _et al._ [2023] M. Malakhov, G. Cistaro, F. Mart\́text{in}n, and A. Picón, Exciton migration in two-dimensional materials, arXiv e-prints , arXiv:2309.01190 (2023).
  * Hansen  _et al._ [2023] T. Hansen, T. Bezriadina, and D. Popova-Gorelova, Theoretical description of attosecond x-ray absorption spectroscopy of frenkel exciton dynamics, Molecules 28, 4502 (2023).
  * Dutoi  _et al._ [2013] A. D. Dutoi, K. Gokhberg, and L. S. Cederbaum, Time-resolved pump-probe spectroscopy to follow valence electronic motion in molecules: Theory, Phys. Rev. A 88, 013419 (2013).
  * Skeidsvoll  _et al._ [2020] A. S. Skeidsvoll, A. Balbi, and H. Koch, Time-dependent coupled-cluster theory for ultrafast transient-absorption spectroscopy, Phys. Rev. A 102, 023115 (2020).
  * Khalili  _et al._ [2020] K. Khalili, L. Inhester, C. Arnold, A. S. Gertsen, J. W. Andreasen, and R. Santra, Simulation of time-resolved x-ray absorption spectroscopy of ultrafast dynamics in particle-hole-excited 4‐(2-thienyl)-2,1,3-benzothiadiazole, Structural Dynamics 7, 044101 (2020).
  * Rott  _et al._ [2021] F. Rott, M. Reduzzi, T. Schnappinger, Y. Kobayashi, K. F. Chang, H. Timmers, D. M. Neumark, R. d. Vivie-Riedle, and S. R. Leone, Ultrafast strong-field dissociation of vinyl bromide: An attosecond transient absorption spectroscopy and non-adiabatic molecular dynamics study, Structural Dynamics 8, 034104 (2021).
  * Salpeter and Bethe [1951] E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Physical Review 84, 1232 (1951).
  * Strinati [1988] G. Strinati, Application of the green’s functions method to the study of the optical properties of semiconductors, La Rivista del Nuovo Cimento 11, 1 (1988).
  * Onida  _et al._ [2002] G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body green’s-function approaches, Reviews of modern physics 74, 601 (2002).
  * Hirose  _et al._ [2015] D. Hirose, Y. Noguchi, and O. Sugino, All-electron gw+ bethe-salpeter calculations on small molecules, Physical Review B 91, 205111 (2015).
  * Gui  _et al._ [2018] X. Gui, C. Holzer, and W. Klopper, Accuracy assessment of gw starting points for calculating molecular excitation energies using the bethe–salpeter formalism, Journal of Chemical Theory and Computation 14, 2127 (2018).
  * Bruneval  _et al._ [2015] F. Bruneval, S. M. Hamed, and J. B. Neaton, A systematic benchmark of the ab initio bethe-salpeter equation approach for low-lying optical excitations of small organic molecules, The Journal of Chemical Physics 142 (2015).
  * Jacquemin  _et al._ [2017] D. Jacquemin, I. Duchemin, and X. Blase, Is the bethe-salpeter formalism accurate for excitation energies? comparisons with td-dft, caspt2, and eom-ccsd, The Journal of Physical Chemistry Letters 8, 1524 (2017).
  * Vorwerk  _et al._ [2020] C. Vorwerk, F. Sottile, and C. Draxl, Excitation pathways in resonant inelastic x-ray scattering of solids, Physical Review Research 2, 042003 (2020).
  * Gulans  _et al._ [2014] A. Gulans, S. Kontur, C. Meisenbichler, D. Nabok, P. Pavone, S. Rigamonti, S. Sagmeister, U. Werner, and C. Draxl, Exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory, Journal of Physics: Condensed Matter 26, 363202 (2014).
  * Draxl and Cocchi [2020] C. Draxl and C. Cocchi, exciting core-level spectroscopy, International Tables for Crystallography 1 (2020).
  * Sangalli  _et al._ [2023] D. Sangalli, M. D’Alessandro, and C. Attaccalite, Exciton-exciton transitions involving strongly bound excitons: An ab initio approach, Physical Review B 107, 205203 (2023).
  * Bockstedte  _et al._ [2010] M. Bockstedte, A. Marini, O. Pankratov, and A. Rubio, Many-body effects in the excitation spectrum of a defect in sic, Physical review letters 105, 026401 (2010).
  * Klahold  _et al._ [2018] W. M. Klahold, W. J. Choyke, and R. P. Devaty, High resolution optical spectroscopy of free exciton and electronic band structure near the fundamental gap in 4h sic, in  _Materials Science Forum_, Vol. 924 (Trans Tech Publ, 2018) pp. 239–244.
  * Klahold  _et al._ [2020] W. Klahold, W. Choyke, and R. Devaty, Band structure properties, phonons, and exciton fine structure in 4 h-sic measured by wavelength-modulated absorption and low-temperature photoluminescence, Physical Review B 102, 205203 (2020).
  * Zhang and Kioupakis [2023] X. Zhang and E. Kioupakis, Phonon-assisted optical absorption of sic polytypes from first principles, Physical Review B 107, 115207 (2023).
  * Demmouche and Coutinho [2018] K. Demmouche and J. Coutinho, Electronic exchange-correlation, many-body effect issues on first-principles calculations of bulk sic polytypes, International Journal of Modern Physics B 32, 1850328 (2018).
  * Peng-Shou  _et al._ [2004] X. Peng-Shou, X. Chang-Kun, P. Hai-Bin, and X. Fa-Qiang, Theoretical study on the band structure and optical properties of 4h-sic, Chinese Physics 13, 2126 (2004).
  * de Oliveira  _et al._ [2003] A. de Oliveira, J. Freitas Jr, W. Moore, A. Silva, I. Pepe, J. Almeida, J. Osório-Guillén, R. Ahuja, C. Persson, K. Järrendahl, O. Lindquist, N. Edwards, and Q. Wahab, Spectroscopy studies of 4h-sic, Materials Research 6, 43 (2003).
  * Adolph  _et al._ [1997] B. Adolph, K. Tenelsen, V. Gavrilenko, and F. Bechstedt, Optical and loss spectra of sic polytypes from ab initio calculations, Physical Review B 55, 1422 (1997).
  * Ahuja  _et al._ [2002] R. Ahuja, A. Ferreira da Silva, C. Persson, J. Osorio-Guillen, I. Pepe, K. Järrendahl, O. Lindquist, N. Edwards, Q. Wahab, and B. Johansson, Optical properties of 4h–sic, Journal of applied physics 91, 2099 (2002).
  * Ching  _et al._ [2006] W. Ching, Y.-N. Xu, P. Rulis, and L. Ouyang, The electronic structure and spectroscopic properties of 3c, 2h, 4h, 6h, 15r and 21r polymorphs of sic, Materials Science and Engineering: A 422, 147 (2006).
  * Gao [2015] S.-P. Gao, Band gaps and dielectric functions of cubic and hexagonal diamond polytypes calculated by many-body perturbation theory, physica status solidi (b) 252, 235 (2015).
  * Lindquist  _et al._ [2001] O. Lindquist, K. Järrendahl, S. Peters, J. Zettler, C. Cobet, N. Esser, D. Aspnes, A. Henry, and N. Edwards, Ordinary and extraordinary dielectric functions of 4h–and 6h–sic from 3.5 to 9.0 ev, Applied Physics Letters 78, 2715 (2001).
  * Harris [1995] G. L. Harris,  _Properties of silicon carbide_, EMIS datareviews series (INSPEC, Institution of Electrical Engineers, 1995).
  * Tallarida  _et al._ [2006] M. Tallarida, D. Schmeißer, F. Zheng, and F. Himpsel, X-ray absorption and photoemission spectroscopy of 3c-and 4h-sic, Surface science 600, 3879 (2006).
  * Lüning  _et al._ [1999] J. Lüning, S. Eisebitt, J.-E. Rubensson, C. Ellmers, and W. Eberhardt, Electronic structure of silicon carbide polytypes studied by soft x-ray spectroscopy, Physical Review B 59, 10573 (1999).
  * Santra [2008] R. Santra, Concepts in x-ray physics, Journal of Physics B: Atomic, Molecular and Optical Physics 42, 023001 (2008).
  * [47] See Supplemental Material at [URL will be inserted by publisher] for details of derivation of presented equations, definition of an average hole distribution of an exciton and its two-dimensional-plane cuts, and theoretical data that can be directly compared to experimental data showing optical and x-ray absorption spectra .
  * Fetter and Walecka [2012] A. L. Fetter and J. D. Walecka,  _Quantum theory of many-particle systems_ (Courier Corporation, 2012).
  * Quintela  _et al._ [2022] M. Quintela, J. Henriques, and N. Peres, Theoretical methods for excitonic physics in two-dimensional materials, physica status solidi (b) 259 (2022).
  * Karni  _et al._ [2021] O. Karni, E. Barré, V. Pareek, J. D. Georgaras, M. K. Man, C. Sahoo, D. R. Bacon, X. Zhu, H. B. Ribeiro, A. L. O’Beirne, J. Hu, A. Al-Mahboob, M. M. M. Abdelrasoul, N. S. Chan, A. Karmakar, A. J. Winchester, B. Kim, K. Watanabe, T. Taniguchi, K. Barmak, J. Madéo, F. H. d. Jornada, T. F. Heinz, and K. M. Dani, Moiré-localized interlayer exciton wavefunctions captured by imaging its electron and hole constituents, arXiv preprint arXiv:2108.01933 (2021).
  * Man  _et al._ [2021] M. K. Man, J. Madéo, C. Sahoo, K. Xie, M. Campbell, V. Pareek, A. Karmakar, E. L. Wong, A. Al-Mahboob, N. S. Chan, D. R. Bacon, X. Zhu, M. M. M. Abdelrasoul, X. Li, T. F. Heinz, F. H. da Jornada, T. C. Cao, and K. M. Dani, Experimental measurement of the intrinsic excitonic wave function, Science advances 7, eabg0192 (2021).
  * Cistaro  _et al._ [2022] G. Cistaro, M. Malakhov, J. J. Esteve-Paredes, A. J. Ur\́text{in}a-Álvarez, R. E. Silva, F. Martin, J. J. Palacios, and A. Picon, Theoretical approach for electron dynamics and ultrafast spectroscopy (edus), Journal of Chemical Theory and Computation 19, 333 (2022).
  * Sagmeister and Ambrosch-Draxl [2009] S. Sagmeister and C. Ambrosch-Draxl, Time-dependent density functional theory versus bethe–salpeter equation: an all-electron study, Physical Chemistry Chemical Physics 11, 4451 (2009).
  * Puschnig and Ambrosch-Draxl [2002] P. Puschnig and C. Ambrosch-Draxl, Optical absorption spectra of semiconductors and insulators including electron-hole correlations: An ab initio study within the lapw method, Physical Review B 66, 165105 (2002).
  * Leng  _et al._ [2016] X. Leng, F. Jin, M. Wei, and Y. Ma, Gw method and bethe–salpeter equation for calculating electronic excitations, Wiley Interdisciplinary Reviews: Computational Molecular Science 6, 532 (2016).
  * Vorwerk  _et al._ [2019] C. Vorwerk, B. Aurich, C. Cocchi, and C. Draxl, Bethe-salpeter equation for absorption and scattering spectroscopy: implementation in the exciting code, Electronic Structure 1, 037001 (2019).
  * Henneke  _et al._ [2020] F. Henneke, L. Lin, C. Vorwerk, C. Draxl, R. Klein, and C. Yang, Fast optical absorption spectra calculations for periodic solid state systems, Communications in Applied Mathematics and Computational Science 15, 89 (2020).
  * Vorwerk  _et al._ [2022] C. Vorwerk, F. Sottile, and C. Draxl, All-electron many-body approach to resonant inelastic x-ray scattering, Physical Chemistry Chemical Physics 24, 17439 (2022).
  * Urquiza  _et al._ [2023] M. L. Urquiza, M. Gatti, and F. Sottile, Pseudopotential bethe-salpeter calculations for shallow-core x-ray absorption near-edge structures: Excitonic effects in $\alpha$\- al 2 o 3, Physical Review B 107, 205148 (2023).
  * Olovsson  _et al._ [2009] W. Olovsson, I. Tanaka, T. Mizoguchi, P. Puschnig, and C. Ambrosch-Draxl, All-electron bethe-salpeter calculations for shallow-core x-ray absorption near-edge structures, Physical Review B 79, 041102 (2009).
  * de Groot  _et al._ [2021] F. M. de Groot, H. Elnaggar, F. Frati, R.-p. Wang, M. U. Delgado-Jaime, M. van Veenendaal, J. Fernandez-Rodriguez, M. W. Haverkort, R. J. Green, G. van der Laan, Y. Kvashnin, A. Hariki, H. Ikeno, H. Ramanantoanina, C. Daul, B. Delley, M. Odelius, M. Lundberg, O. Kuhn, S. I. Bokarev, E. Shirley, J. Vinson, K. Gilmore, M. Stener, G. Fronzoni, P. Decleva, P. Kruger, M. Retegan, Y. Joly, C. Vorwerk, C. Draxl, J. Rehr, and A. Tanaka, 2p x-ray absorption spectroscopy of 3d transition metal systems, Journal of Electron Spectroscopy and Related Phenomena 249, 147061 (2021).
  * Kutepov [2021] A. L. Kutepov, Atomic forces from dirac–kohn–sham equations: implementation in flexible (apw+ lo/lapw)+ lo basis set, Journal of Physics: Condensed Matter 33, 235503 (2021).
  * Unzog  _et al._ [2022] M. Unzog, A. Tal, and G. Kresse, X-ray absorption using the projector augmented-wave method and the bethe-salpeter equation, Physical Review B 106, 155133 (2022).
  * Stampfl  _et al._ [2001] C. Stampfl, W. Mannstadt, R. Asahi, and A. J. Freeman, Electronic structure and physical properties of early transition metal mononitrides: Density-functional theory lda, gga, and screened-exchange lda flapw calculations, Physical Review B 63, 155106 (2001).
  * Yuan  _et al._ [2018] J.-H. Yuan, Q. Chen, L. R. Fonseca, M. Xu, K.-H. Xue, and X.-S. Miao, Gga-1/2 self-energy correction for accurate band structure calculations: the case of resistive switching oxides, Journal of Physics Communications 2, 105005 (2018).
  * Nabok  _et al._ [2016] D. Nabok, A. Gulans, and C. Draxl, Accurate all-electron g 0 w 0 quasiparticle energies employing the full-potential augmented plane-wave method, Physical Review B 94, 035118 (2016).
  * Salas-Illanes  _et al._ [2022] N. Salas-Illanes, D. Nabok, and C. Draxl, Electronic structure of representative band-gap materials by all-electron quasiparticle self-consistent g w calculations, Physical Review B 106, 045143 (2022).
  * Momma and Izumi [2008] K. Momma and F. Izumi, Vesta: a three-dimensional visualization system for electronic and structural analysis, Journal of Applied crystallography 41, 653 (2008).
  * Wernet [2019] P. Wernet, Chemical interactions and dynamics with femtosecond x-ray spectroscopy and the role of x-ray free-electron lasers, Philosophical Transactions of the Royal Society A 377, 20170464 (2019).


# Thermoelectric performance of spin Seebeck effect in Fe3O4/Pt-based thin
film heterostructures

R. Ramos<EMAIL_ADDRESS>WPI Advanced Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  Spin Quantum
Rectification Project, ERATO, Japan Science and Technology Agency, Sendai
980-8577, Japan  A. Anadón  Instituto de Nanociencia de Aragón, Universidad de
Zaragoza, E-50018 Zaragoza, Spain  Departamento de Física de la Materia
Condensada, Universidad de Zaragoza, E-50009 Zaragoza, Spain  I. Lucas
Instituto de Nanociencia de Aragón, Universidad de Zaragoza, E-50018 Zaragoza,
Spain  Departamento de Física de la Materia Condensada, Universidad de
Zaragoza, E-50009 Zaragoza, Spain  K. Uchida  Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  PRESTO, Japan Science and
Technology Agency, Saitama 332-0012, Japan  P. A. Algarabel  Departamento de
Física de la Materia Condensada, Universidad de Zaragoza, E-50009 Zaragoza,
Spain  Instituto de Ciencia de Materiales de Aragón, Universidad de Zaragoza
and Consejo Superior de Investigaciones Científicas, 50009 Zaragoza, Spain  L.
Morellón  Instituto de Nanociencia de Aragón, Universidad de Zaragoza, E-50018
Zaragoza, Spain  Departamento de Física de la Materia Condensada, Universidad
de Zaragoza, E-50009 Zaragoza, Spain  M. H. Aguirre  Instituto de Nanociencia
de Aragón, Universidad de Zaragoza, E-50018 Zaragoza, Spain  Departamento de
Física de la Materia Condensada, Universidad de Zaragoza, E-50009 Zaragoza,
Spain  Laboratorio de Microscopías Avanzadas, Universidad de Zaragoza, E-50018
Zaragoza, Spain  E. Saitoh  WPI Advanced Institute for Materials Research,
Tohoku University, Sendai 980-8577, Japan  Spin Quantum Rectification Project,
ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan  Institute
for Materials Research, Tohoku University, Sendai 980-8577, Japan  Advanced
Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan  M.
R. Ibarra  Instituto de Nanociencia de Aragón, Universidad de Zaragoza,
E-50018 Zaragoza, Spain  Departamento de Física de la Materia Condensada,
Universidad de Zaragoza, E-50009 Zaragoza, Spain  Laboratorio de Microscopías
Avanzadas, Universidad de Zaragoza, E-50018 Zaragoza, Spain

###### Abstract

We report a systematic study on the thermoelectric performance of spin Seebeck
devices based on Fe3O4/Pt junction systems. We explore two types of device
geometries: a spin Hall thermopile and spin Seebeck multilayer structures. The
spin Hall thermopile increases the sensitivity of the spin Seebeck effect,
while the increase in the sample internal resistance has a detrimental effect
on the output power. We found that the spin Seebeck multilayers can overcome
this limitation since the multilayers exhibit the enhancement of the
thermoelectric voltage and the reduction of the internal resistance
simultaneously, therefore resulting in significant power enhancement. This
result demonstrates that the multilayer structures are useful for improving
the thermoelectric performance of the spin Seebeck effect.

Thermoelectrics is one of the promising energy harvesting technologies for
waste heat management and recovery, having potential applications in
thermoelectric generation or temperature sensing. Thermoelectric generation
usually relies on the Seebeck effect, a phenomenon that refers to the
generation of an electromotive force parallel to the direction of an applied
temperature gradient in electrically conducting materials [see Fig. 1(a)].
In spintronics,Wolf et al. (2001); Žutić et al. (2004) a different
thermoelectric principle was recently discovered. The principle is based on a
spin counterpart of the Seebeck effect, in analogy named the spin Seebeck
effect (SSE).Uchida et al. (2008) The SSE refers to the generation of spin
currents Maekawa et al. (2013) in a ferromagnetic material (F) upon
application of a temperature gradient. The spin current induced by the SSE is
injected into a non-magnetic metal (N) in contact with F and converted into an
electromotive force by means of the inverse spin Hall effect (ISHE) in
suitable N materialsSaitoh et al. (2006); Valenzuela and Tinkham (2006);
Kimura et al. (2007); Costache et al. (2006) [Fig. 1(b)]. The SSE has been
observed in a range of ferromagnetic materials: metals,Uchida et al. (2008)
semiconductors Jaworski et al. (2010, 2011) and insulators.Uchida et al.
(2010a); Meier et al. (2013); Niizeki et al. (2015) Thus, the SSE has been
experimentally established as a general transport phenomenon in ferromagnets.
Since the discovery of the SSE, the study of the interaction among spin,
charge, and heat currents has been strongly invigorated, which is the main
focus of spin caloritronics.Bauer et al. (2012); Boona et al. (2014) This has
resulted in the discovery of various thermospin effects, such as the spin
Peltier effect Flipse et al. (2014) and the spin dependent versions of the
ordinary Seebeck Slachter et al. (2010) and Peltier Flipse et al. (2012)
effects among others.Le Breton et al. (2011); Yu et al. (2010); An et al.
(2013)
Theoretically, the SSE Xiao et al. (2010); Adachi et al. (2011, 2013) is
understood as an effect resulting from the thermal non-equilibrium between
magnons in F and conduction electrons in N, generating a spin current at the
F/N interface proportional to the effective temperature difference between
magnons in F and electrons in N. Theories from other view points have also
been developed,Hoffman et al. (2013); Zhang and Zhang (2012); Rezende et al.
(2014) where roles of magnon spin currents inside the ferromagnet has been
highlighted.Zhang and Zhang (2012); Rezende et al. (2014) This is supported by
several recent experiments: the dependence of SSE voltage with the thickness
of F,Kehlberger et al. (2015) the enhancement of the SSE voltage in magnetic
multilayers,Ramos et al. (2015) and the suppression of the SSE at high
magnetic fields.Kikkawa et al. (2015); Jin et al. (2015)
The SSE has potential advantages over conventional thermoelectric technologies
for thermal energy harvesting. The experimental geometry, having paths for
heat and electric currents independent and perpendicular to each other, allows
to have two different materials that can be optimised independently and it is
also advantageous for the implementation of thin film devices over large
surfaces, for example, simply by thin film coating.Kirihara et al. (2012)
Moreover, the observation of the SSE in magnetic insulators implies the
potential to generate pure spin currents with lesser dissipation losses due to
mobile charge carriers and further expanding the range of possible materials
where spin mediated thermoelectric conversion can be studied. However, a main
disadvantage of the SSE is the low magnitude of the thermoelectric output.
Strategies to overcome this limitation are currently being explored. One
possibility is to increase the spin current detection efficiency by taking
advantage of the spin Hall angle characteristics of different N materials,Liu
et al. (2012); Pai et al. (2012); Niimi et al. (2012); Laczkowski et al.
(2014) as it has been shown in the spin Hall thermopiles.Uchida et al. (2012)
A different approach can be directed towards increasing the thermal spin
current generation, as recently shown in the SSE of magnetic multilayers.Ramos
et al. (2015)
In this paper, we have investigated the thermoelectric performance of two
types of SSE devices: spin Hall thermopiles Uchida et al. (2012) and magnetic
multilayers formed by repeated growth of $n$ number of [F/N] bilayers,
hereafter referred to as [F/N]n. We have performed room temperature
measurements of the thermoelectric voltage and power output characteristics
for both the structures. The materials chosen to form the basic F/N bilayer
structure are magnetite, Fe3O4, as the F layer and platinum, Pt, as the N
layer. Magnetite is a ferrimagnetic oxide, commonly employed in
spintronicsWolf et al. (2001); Žutić et al. (2004) due to its half-metallic
character and high Curie temperature.Dedkov et al. (2002); Walz (2002) These
properties have inspired several studies of its magnetotransport properties.Li
et al. (1998); Ramos et al. (2008, 2014); Wu et al. (2010, 2012) Platinum is a
material commonly used for spin current detection, due to its relatively large
spin Hall angle.Hoffmann (2013); Sinova et al. (2015)
The Fe3O4 films were grown on magnesium oxide, MgO(001), substrates by pulsed
laser deposition using a KrF excimer laser with 248 nm wavelength, 10 Hz
repetition rate, and 3 $\times$ 109 W/cm2 irradiance in an ultrahigh-vacuum
chamber (UHV). The Pt films were deposited in the same UHV chamber by DC
magnetron sputtering. Further details on the growth can be found in Ref. 47.
The film thickness was measured by x-ray reflectivity and its structural
quality was confirmed by x-ray diffraction and transmission electron
microscopy (TEM). The film cross sections were prepared by Focused Ion Beam
and measured by high angle annular dark field (HAADF) scanning transmission
electron microscopy (STEM). The measurements were carried out in a probe
aberration corrected FEI Titan 60-300 operated at 300 kV. The fabrication of
the spin Hall thermopile structure was carried out using optical lithography,
following a two-step process consisting of Ar milling of the Fe3O4/Pt bilayer
and fabrication of Al interconnection wires by metal evaporation and lift-off.
The SSE experiments were performed using the longitudinal configuration.Uchida
et al. (2010b) The sample dimensions are $L_{x}$ = 2 mm, $L_{y}$ = 7 mm and
$L_{z}$ = 0.5 mm, except for the spin Hall thermopile structure in which the
in plane sample geometry is defined by the lithography pattern. Figure 2(a)
shows a schematic illustration of the longitudinal SSE measurement geometry.
The sample is placed between two AlN plates with high thermal conductivity,
where the top plate has a resistive heater and the lower plate works as a heat
sink. By passing an electric current to the heater, a temperature gradient
($\nabla T$) is applied in the $z$ direction. This generates a temperature
difference ($\Delta T$) between the bottom and top of the sample, which are
stabilized at temperatures $T$ and $T+\Delta T$, respectively. All the
measurements were carried out at $T$ = 300 K. The temperature difference is
monitored by two thermocouples connected differentially. The SSE voltage in
the Pt film is measured along the $y$ direction with a nanovoltmeter, while a
magnetic field is applied in the $x$ direction.
The measurement mechanism is as follows. In the longitudinal configuration, a
spin current is injected into the Pt layer with the spatial direction
$\mathbf{J}_{\text{S}}$ perpendicular to the Fe3O4/Pt interface (parallel to
$\nabla T$) and the spin-polarization vector $\bm{\sigma}$ (parallel to the
magnetization of Fe3O4). Then, an electric field ($\mathbf{E}_{\text{ISHE}}$)
is generated in Pt due to the ISHE, which can be expressed as:

$\mathbf{{E}}_{\text{ISHE}}=\theta_{\text{SH}}\rho(\mathbf{J}_{\text{S}}\times\bm{\sigma}),$
(1)

where $\theta_{\text{SH}}$ and $\rho$ denote the spin Hall angle and electric
resistivity of Pt, respectively. By measuring the voltage $V$ between the ends
of the Pt film, we can electrically detect the SSE by means of the ISHE as
$V_{\text{ISHE}}=E_{\text{ISHE}}L_{y}$. Although the longitudinal
configuration is widely used to measure the SSE in ferrimagnetic insulators,
in samples with electrically conductive F layers there can be an additional
contribution of the anomalous Nernst effect (ANE) in itinerant
ferromagnets.Ramos et al. (2014) Since Fe3O4 is electrically conductive at
room temperature, we need to evaluate the contribution from the ANE of the
Fe3O4 film to the measured SSE voltage in the Fe3O4/Pt system. We have
previously shown that by considering simple equivalent circuit model,Ramos et
al. (2013) the following expression can be obtained:
$E_{y}=\frac{r}{1+r}E_{\text{ANE}}$, which gives the electric field in the
Fe3O4/Pt bilayer as a function of the ANE-induced electric field
($E_{\text{ANE}}$) and the shunting parameter
$r=\frac{\rho_{\text{Pt}}}{\rho_{\text{Fe${}_{3}$O${}_{4}$}}}\frac{t_{\text{Fe${}_{3}$O${}_{4}$}}}{t_{\text{Pt}}}$,
with $\rho_{\text{Pt}}$ ($\rho_{\text{Fe${}_{3}$O${}_{4}$}}$) and
$t_{\text{Pt}}$ ($t_{\text{Fe${}_{3}$O${}_{4}$}}$) being the resistivity and
thickness of the Pt (Fe3O4) layer, respectively. In the case of the Fe3O4/Pt
system, the ANE of the Fe3O4 film is strongly suppressed due to the fact that
the resistivity of Fe3O4 is two orders of magnitude greater than that of Pt.
For instance, in the Fe3O4/Pt bilayer with $t_{\text{Fe${}_{3}$O${}_{4}$}}$ =
50 nm and $t_{\text{Pt}}$ = 5 nm, used for the spin Hall thermopile, we
obtained that the ANE signal in the Fe3O4/Pt bilayer is reduced to 10 % of the
ANE in a Fe3O4 single film. This gives a contribution from the ANE of the
Fe3O4 film to the SSE measured in the Fe3O4/Pt bilayer system, which is of
only about 2 % of the measured voltage. Therefore, the SSE dominates the
thermoelectric voltage in our Fe3O4/Pt systems.
First, we show the SSE in the spin Hall thermopile structure in comparison to
that in a single Fe3O4/Pt bilayer. In analogy to thermopile structures
commonly used in conventional thermoelectric modules, where the output voltage
is increased by connecting a number of thermocouple junctions of materials
with dissimilar Seebeck coefficients electrically in series and thermally in
parallel,Snyder and Toberer (2008) the spin Hall thermopiles improve the
electrical detection of the SSE by enhancing the ISHE voltage. This is
achieved by using two materials with different spin Hall angles. These are
placed alternatively on top of the ferromagnet forming an array and connected
in series forming a zig-zag structure [see Fig. 2(b)].
In this study, we fabricated the spin Hall thermopile comprising a set of 6
parallel Fe3O4/Pt wires. The wires were formed by patterning the Fe3O4/Pt
bilayer film. This is due to the fact that the ANE suppression depends on the
relative resistance values between the Fe3O4 and Pt layers, therefore we
cannot deposit an array of Pt wires on a plain Fe3O4 film, as the Pt wires
will become comparatively more resistive and the ANE contribution, with
respect to the Fe3O4/Pt bilayer, will change. The Fe3O4/Pt wires are then
electrically connected in series forming a zig-zag structure using a metal
with negligible spin Hall angle Hoffmann (2013) and therefore no ISHE
contribution (130 nm of Al in our case). In this spin Hall thermopile, the SSE
signal is generated by the Fe3O4/Pt wires and the Al wires are only used for
electrical connection. The width of the Fe3O4/Pt and Al wires are 200 $\mu$m
and 100 $\mu$m, respectively and their length is 5 mm except for the rightmost
and leftmost Fe3O4/Pt wires, which have a length of 6 mm including electrode
pads [see Fig. 2(b) for an optical image of the thermopile].
Figure 2(c) shows the comparative results for the field dependence of the
transversal voltage $V$ for the single Fe3O4/Pt bilayer sample and the
Fe3O4/Pt spin Hall thermopile at $\Delta T$ = 1 K. We clearly observe a strong
enhancement of the SSE voltage in the spin Hall thermopile, as expected from
the series connection of the Fe3O4/Pt elements, effectively increasing the
length of the sample. Figure 2(d) shows $V$ at $H$ = 7 kOe as a function of
$\Delta T$, thus confirming the linear dependence between $V$ and $\Delta T$.
From the slope of these curves we can extract the magnitude of the SSE
thermopower of the single bilayer sample and spin Hall thermopile, obtaining
the values of 10 $\mu$V/K and 50 $\mu$V/K, respectively. The SSE voltage per
one Fe3O4/Pt wire in the spin Hall thermopile is estimated to be 8.2 $\mu$V/K
and a value of the electric field per unit temperature difference ($V$/$\Delta
TL$) of 1.6 mV/Km is obtained. This value is consistent with the electric
field ($V$/$\Delta TL_{y}$) measured in the Fe3O4/Pt bilayer, therefore,
showing that the increase in voltage is proportional to the increase in the
effective length of the sample. The thermoelectric voltage of the spin Hall
thermopile can be further enhanced by increasing the integration density of
wires.Uchida et al. (2012)
The electric power output characteristics of the SSE in the Fe3O4/Pt single
bilayer and spin Hall thermopile were also investigated. This is performed by
attaching a variable load resistance ($R_{\text{L}}$) to the samples and
measuring the voltage drop across $R_{\text{L}}$, as schematically shown in
the inset of Fig. 3. Then, we estimate the output power as
$V^{2}_{\text{L}}$/$R_{\text{L}}$. Figure 3 shows the comparative results
obtained for the single bilayer and spin Hall thermopile as a function of
$R_{\text{L}}$; the output power is maximized when $R_{\text{L}}$ equals the
internal resistance of the sample, as expected from impedance-matching
criterion for maximum power transfer and also in agreement with previous
reports.Uchida et al. (2014) We can see that, despite the increased voltage
response in the spin Hall thermopile structure, there is no advantage in terms
of extractable electrical power output from the device when compared to the
response of the single Fe3O4/Pt bilayer. This is a consequence of the fact
that the voltage enhancement is due to the increase in effective length of the
sample, which is associated to an increase in the internal resistance.Uchida
et al. (2014)
In the case of ANE-based thermopile structures Sakuraba et al. (2013) it was
proposed that, if the ANE voltage is independent of sample thickness, the
performance can be improved by reducing the internal resistance of the
devices, by just increasing the ferromagnet thickness. However, this approach
cannot be used in the case of spin Hall thermopiles, since increasing the
thickness of the N layers will result in a reduction of the ISHE voltage
Kikkawa et al. (2013). Therefore, in order to increase the electric output
power of the SSE, we must follow a different approach. To realize the
enhancement of the output power of the SSE, we have investigated the
thermoelectric performance of the SSE multilayers ([F/N]n) formed by repeated
growth of $n$ number of F/N bilayers [Fig. 4(a)]. This type of structure has
recently been reported to present an enhancement of the transversal
thermoelectric voltage in various magnetic multilayer systems based on
oxide/metal,Ramos et al. (2015) all-metallic,Lee et al. (2015); Uchida et al.
(2015) and all-oxide Shiomi et al. (2015) junctions. The advantage of the
multilayer structure is that along with the voltage enhancement, the internal
resistance of the structure decreases with increasing the layer number, due to
parallel connection between the layers. Therefore, a significant enhancement
of the extractable electric power is expected.
Here, we show the measured results of voltage and output power characteristics
of the SSE in [Fe3O4(23)/Pt(7)]n multilayer structures, with $n$ = 1, 6 and
12. The structural quality of the multilayers was confirmed by TEM. A STEM-
HAADF image of the [Fe3O4/Pt]12 multilayer shows well defined interfaces [see
Fig. 4(b)]. As shown in Fig. 4(c), the voltage in the multilayer structure
with $n$ = 6 and 12 presents an enhancement of about one order of magnitude in
comparison with that in the single bilayer with $n$ = 1. The observed
enhancement of the SSE voltage in the Fe3O4/Pt multilayer systems can be
explained as a result of an increase in the thermally generated spin current
flowing across the multilayers.Ramos et al. (2015) This enhancement can be
measured due to the presence of Pt interlayers that detect the spin current
enhancement within the multilayer. Importantly, since the resistance of the
multilayer decreases with increasing the number of the Fe3O4/Pt layers [see
inset of Fig 4(c)], it is expected that the power as well as voltage is
enhanced. The power output characteristics of the multilayer samples are shown
in Fig. 4(d). We found that the power largely increases in the multilayer
systems, reaching more than two orders of magnitude enhancement compared to
the single bilayer case: the power is estimated to be 0.01, 1 and 2 pW/K2 for
$n$ = 1, 6 and 12, respectively.
In summary, we have performed thermoelectric voltage and power measurements of
the spin Hall thermopile and SSE multilayer structures based on Fe3O4/Pt
junction systems. Our results show that the spin Hall thermopile can enhance
the voltage, but cannot improve the power output due to the increase of the
internal resistance of the devices. However, the voltage increase is
beneficial to increase the sensitivity of the SSE and might have possible
sensor applications. In contrast, the multilayer SSE devices significantly
enhance both thermoelectric voltage and power compared to the conventional
bilayer systems, leading to an efficiency improvement in SSE-based
thermoelectric generation. Although the obtained power is still small, we
believe that the SSE in multilayers can be further enhanced by careful choice
of materials and device geometry. Since the observed SSE enhancement in
multilayers is expected to depend on spin transport properties across the
multilayers, using materials with longer spin transport characteristic lengths
might lead to further enhancement in the SSE.
The authors thank S. Maekawa, H. Adachi, T. Kikkawa, J. Shiomi, T. Oyake, A.
Kirihara, and M. Ishida for valuable discussions. The microscopy works were
conducted in the Laboratorio de Microscopías Avanzadas at Instituto de
Nanociencia de Aragón, Universidad de Zaragoza. This work was supported by the
Spanish Ministry of Science (through projects PRI-PIBJP-2011-0794,
MAT2011-27553-C02, including FEDER funding), the Aragón Regional Government
(project E26), Thermo-spintronic Marie-Curie CIG (Grant Agreement No. 304043),
JST-PRESTO “Phase Interfaces for Highly Efficient Energy Utilization” from
JST, Japan, Grant-in-Aid for Scientific Research on Innovative Areas “Nano
Spin Conversion Science” (26103005), Grant-in-Aid for Challenging Exploratory
Research (26600067), Grant-in-Aid for Scientific Research (A) (15H02012) from
MEXT, Japan, NEC Corporation, and The Noguchi Institute.

## References

  * Wolf et al. (2001) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001).
  * Žutić et al. (2004) I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
  * Uchida et al. (2008) K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008).
  * Maekawa et al. (2013) S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E. Saitoh, J. Phys. Soc. Jpn. 82, 102002 (2013).
  * Saitoh et al. (2006) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006).
  * Valenzuela and Tinkham (2006) S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).
  * Kimura et al. (2007) T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007).
  * Costache et al. (2006) M. V. Costache, M. Sladkov, S. M. Watts, C. H. van Der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006).
  * Jaworski et al. (2010) C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat Mater 9, 898 (2010).
  * Jaworski et al. (2011) C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, R. C. Myers, and J. P. Heremans, Phys. Rev. Lett. 106, 186601 (2011).
  * Uchida et al. (2010a) K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, et al., Nat. Mater. 9, 894 (2010a).
  * Meier et al. (2013) D. Meier, T. Kuschel, L. Shen, A. Gupta, T. Kikkawa, K. Uchida, E. Saitoh, J.-M. Schmalhorst, and G. Reiss, Physical Review B 87, 054421 (2013).
  * Niizeki et al. (2015) T. Niizeki, T. Kikkawa, K. Uchida, M. Oka, K. Z. Suzuki, H. Yanagihara, E. Kita, and E. Saitoh, AIP Advances 5, 053603 (2015).
  * Bauer et al. (2012) G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012).
  * Boona et al. (2014) S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014).
  * Flipse et al. (2014) J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. V. Wees, Phys. Rev. Lett. 113, 027601 (2014).
  * Slachter et al. (2010) A. Slachter, F. L. Bakker, J.-P. Adam, and B. J. van Wees, Nat. Phys. 6, 879 (2010).
  * Flipse et al. (2012) J. Flipse, F. L. Bakker, a. Slachter, F. K. Dejene, and B. J. van Wees, Nat. Nanotechnol. 7, 166 (2012).
  * Le Breton et al. (2011) J.-C. Le Breton, S. Sharma, H. Saito, S. Yuasa, and R. Jansen, Nature 475, 82 (2011).
  * Yu et al. (2010) H. Yu, S. Granville, D. P. Yu, and J.-P. Ansermet, Phys. Rev. Lett. 104, 146601 (2010).
  * An et al. (2013) T. An, V. I. Vasyuchka, K. Uchida, A. V. Chumak, K. Yamaguchi, K. Harii, J. Ohe, M. B. Jungfleisch, Y. Kajiwara, H. Adachi, et al., Nat. Mat. 12, 549 (2013).
  * Xiao et al. (2010) J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010).
  * Adachi et al. (2011) H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011).
  * Adachi et al. (2013) H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013).
  * Hoffman et al. (2013) S. Hoffman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013).
  * Zhang and Zhang (2012) S. S.-L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012).
  * Rezende et al. (2014) S. M. Rezende, R. L. Rodríguez-Suárez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Phys. Rev. B 89, 014416 (2014).
  * Kehlberger et al. (2015) A. Kehlberger, U. Ritzmann, D. Hinzke, E.-J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jungfleisch, et al., Phys. Rev. Lett. 115, 096602 (2015).
  * Ramos et al. (2015) R. Ramos, T. Kikkawa, M. H. Aguirre, I. Lucas, A. Anadón, T. Oyake, K. Uchida, and H. Adachi, Phys. Rev. B 92, 220407(R) (2015).
  * Kikkawa et al. (2015) T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 064413 (2015).
  * Jin et al. (2015) H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015).
  * Kirihara et al. (2012) A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, 11, 686 (2012).
  * Liu et al. (2012) L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012).
  * Pai et al. (2012) C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012).
  * Niimi et al. (2012) Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, and Y. Otani, Phys. Rev. Lett. 109, 156602 (2012).
  * Laczkowski et al. (2014) P. Laczkowski, J.-C. Rojas-Sánchez, W. Savero-Torres, H. Jaffrès, N. Reyren, C. Deranlot, L. Notin, C. Beigné, a. Marty, J.-P. Attané, et al., Appl. Phys. Lett. 104, 142403 (2014).
  * Uchida et al. (2012) K. Uchida, T. Nonaka, T. Yoshino, T. Kikkawa, D. Kikuchi, and E. Saitoh, Appl. Phys. Express 5, 093001 (2012).
  * Dedkov et al. (2002) Y. S. Dedkov, U. Rüdiger, and G. Güntherodt, Phys. Rev. B 65, 64417 (2002).
  * Walz (2002) F. Walz, J. Phys.: Condens. Matter 14, R285 (2002).
  * Li et al. (1998) X. W. Li, a. Gupta, G. Xiao, W. Qian, and V. P. Dravid, Appl. Phys. Lett. 73, 3282 (1998).
  * Ramos et al. (2008) R. Ramos, S. K. Arora, and I. V. Shvets, Phys. Rev. B 78, 214402 (2008).
  * Ramos et al. (2014) R. Ramos, M. H. Aguirre, A. Anadón, J. Blasco, I. Lucas, K. Uchida, P. A. Algarabel, L. Morellón, E. Saitoh, and M. R. Ibarra, Physical Review B 90, 054422 (2014).
  * Wu et al. (2010) H.-C. Wu, M. Abid, B. S. Chun, R. Ramos, O. N. Mryasov, and I. V. Shvets, Nano Lett. 10, 1132 (2010).
  * Wu et al. (2012) H.-C. Wu, R. Ramos, R. G. S. Sofin, Z.-M. Liao, M. Abid, and I. V. Shvets, Appl. Phys. Lett. 101, 052402 (2012).
  * Hoffmann (2013) A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013).
  * Sinova et al. (2015) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).
  * Orna et al. (2010) J. Orna, P. A. Algarabel, L. Morellón, J. A. Pardo, J. M. de Teresa, R. López Antón, F. Bartolomé, L. M. García, J. Bartolomé, J. C. Cezar, et al., Phys. Rev. B 81, 144420 (2010).
  * Uchida et al. (2010b) K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010b).
  * Ramos et al. (2013) R. Ramos, T. Kikkawa, K. Uchida, H. Adachi, I. Lucas, M. H. Aguirre, P. Algarabel, L. Morellón, S. Maekawa, E. Saitoh, et al., Appl. Phys. Lett. 102, 072413 (2013).
  * Snyder and Toberer (2008) G. J. Snyder and E. S. Toberer, Nat. Mat. 7, 105 (2008).
  * Uchida et al. (2014) K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, and E. Saitoh, J. Phys.: Condens. Matter 26, 343202 (2014).
  * Sakuraba et al. (2013) Y. Sakuraba, K. Hasegawa, M. Mizuguchi, T. Kubota, S. Mizukami, T. Miyazaki, and K. Takanashi, Appl. Phys. Express 6, 033003 (2013).
  * Kikkawa et al. (2013) T. Kikkawa, K. Uchida, S. Daimon, Y. Shiomi, H. Adachi, Z. Qiu, D. Hou, X.-F. Jin, S. Maekawa, and E. Saitoh, Phys. Rev. B 88, 214403 (2013).
  * Lee et al. (2015) K.-D. Lee, D.-J. Kim, H. Yeon Lee, S.-H. Kim, J.-H. Lee, K.-M. Lee, J.-R. Jeong, K.-S. Lee, H.-S. Song, J.-W. Sohn, et al., Sci. Rep. 5, 10249 (2015).
  * Uchida et al. (2015) K. Uchida, T. Kikkawa, T. Seki, T. Oyake, J. Shiomi, K. Takanashi, and E. Saitoh, Phys. Rev. B 92, 094414 (2015).
  * Shiomi et al. (2015) Y. Shiomi, Y. Handa, T. Kikkawa, and E. Saitoh, Appl. Phys. Lett. 106, 232403 (2015).

Figure 1: (Color online) Schematic illustrations of (a) the conventional
Seebeck effect and (b) the spin Seebeck effect (SSE) with the induced inverse
spin Hall effect (ISHE) in a non-magnetic metal attached to a ferromagnetic or
ferrimagnetic material. $\mathbf{E}_{\text{S}}$ and $\mathbf{E}_{\text{ISHE}}$
denote the electric fields generated by the Seebeck effect and ISHE,
respectively. $\mathbf{H}$, $\mathbf{M}$, $\nabla T$, and
$\mathbf{J}_{\text{S}}$ denote the magnetic field, magnetization, temperature
gradient, and spatial direction of the spin current, respectively.

Figure 2: (Color online) Schematic illustrations of the longitudinal SSE
measurement configurations (a) in a ferromagnetic insulator(F)/non-magnetic
metal(N) bilayer system and (b) in a spin Hall thermopile (an optical image of
the thermopile sample is also shown). (c) Magnetic field $H$ dependence of the
SSE voltage $V$ response at the temperature difference of $\Delta T$ = 1 K in
the simple bilayer and spin Hall thermopile samples comprising a
Fe3O4(50)/Pt(5) (thickness in nm) junction. (d) $\Delta T$ dependence of $V$
at $H$ = 7 kOe.

Figure 3: (Color online) Power output ($V^{2}_{\text{L}}$/$R_{\text{L}}$) as a
function of the load resistance $R_{\text{L}}$ obtained for the single bilayer
and spin Hall thermopile based on the Fe3O4(50)/Pt(5) junction. $V_{\text{L}}$
denotes the maximum output power is obtained in the impedance matching
condition (see empty arrows). The inset shows the schematic illustration of
the measurement configuration to determine the electrical power output
characteristics of the SSE.

Figure 4: (Color online) (a) A schematic illustration of the [F/N]n
multilayers, $n$ denotes the number of Fe3O4/Pt bilayers. (b) A scanning
transmission electron microscopy image of the cross section of a
[Fe3O4(23)/Pt(7)]12 sample. (c) $H$ dependence of $V$/$\Delta T$ in the
[Fe3O4(23)/Pt(7)]n multilayers for $n$ = 1, 6 and 12. The inset to (c) shows
the resistance of the multilayers as a function of $n$. (d)
$V^{2}_{\text{L}}$/$R_{\text{L}}$ in the [Fe3O4(23)/Pt(7)]n multilayers as a
function of $R_{\text{L}}$. Arrows represent the impedance matching condition.