# Thermoelectric signatures of order-parameter symmetries in iron-based
superconducting tunnel junctions

Claudio Guarcello  0000-0002-3683-2509 Corresponding author:
<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello”, Università di
Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy  INFN,
Sezione di Napoli Gruppo Collegato di Salerno, Complesso Universitario di
Monte S. Angelo, I-80126 Napoli, Italy  Alessandro Braggio
0000-0003-2119-1160<EMAIL_ADDRESS>NEST, Istituto Nanoscienze-
CNR and Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy
Francesco Giazotto  0000-0002-1571-137X<EMAIL_ADDRESS>NEST,
Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro
12, I-56127 Pisa, Italy  Roberta Citro  0000-0002-3896-4759<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, Via Giovanni
Paolo II, 132, I-84084 Fisciano (SA), Italy  INFN, Sezione di Napoli Gruppo
Collegato di Salerno, Complesso Universitario di Monte S. Angelo, I-80126
Napoli, Italy  CNR-SPIN c/o Universitá degli Studi di Salerno, I-84084
Fisciano (Sa), Italy

###### Abstract

Thermoelectrical properties are frequently used to characterize the materials
and endow the free energy from wasted heat for useful purposes. Here, we show
that linear thermoelectric effects in tunnel junctions with Fe-based
superconductors, not only address the dominance between particle and hole
states, but even provide information about the superconducting order parameter
symmetry. In particular, we observe that nodal order parameters present a
maximal thermoelectric effect at lower temperatures than for nodeless cases.
Furthermore, we show also that superconducting tunnel junctions between Fe-
based and BCS superconductors could provide a thermoelectric efficiency ZT
exceeding 6 with a linear Seebeck coefficient around $S\approx
800\;\mu\text{V/K}$ at a few Kelvin. These results pave the way to novel
thermoelectric machines based on multi-band superconductors.

_Introduction_ $-$ Physical systems based on hybrid superconducting junctions
have demonstrated a great potential for energy management issues [1, 2, 3, 4].
Recently, they also attracted interest for their unexpectedly good
thermoelectric (TE) performance [5, 6, 7, 8, 9, 10, 11, 12], finding also a
role in different quantum technology applications [13, 14, 15]. In a two-
terminal system, a necessary condition for thermoelectricity in the linear
regime, i.e., for a small voltage $\delta V$ and a small temperature bias
$\delta T$, is breaking of the particle-hole (PH) symmetry. A sizable linear
TE effect, much larger than that commonly found in metallic structures, has
been recently reported in SC-ferromagnet tunnel junctions (TJs) [16, 5, 6],
due to the spin-dependent effective breaking of PH symmetry.

Here, we show a robust linear TE effect in hybrid superconducting junctions
with a multiband SC, namely, an Fe-based SC (FeSC). Our results, beyond
proving a linear TE effect with unconventional SCs, establish TE phenomena as
a potential probe of the superconducting order parameter symmetries. In fact,
one of the central problems for unconventional SCs is the nature of the
pairing mechanism, which is tightly connected to the order parameter
symmetries [17, 18, 19, 20]. Pairing symmetry is known to be $d$-wave in
cuprates, but is still unresolved for the FeSCs. They are unique among
unconventional SCs, since different ordering phenomena are present in a multi-
orbital scenario [21, 22, 23]. It was first theoretically predicted that FeAs-
based high-temperature SCs have a sign change of the order parameter on the
Fermi surface [21]. Experimental evidence for FeSC seems to be favourable to
$s_{\pm}$ pairing, implying that the electron- and hole-like bands both
develop an $s$-wave superconducting state with opposite sign in order
parameter [21, 24]. Point contact Andreev reflection spectroscopy was also
been applied to FeSCs to probe the order parameter symmetry [25, 26]. However,
the results of these studies are not completely conclusive due to the
complexity of the Andreev reflection spectra of a normal metal/multiband-SC
interface.

Figure 1: (a) Cartoon of the thermally-biased TJ formed by an FeSC and a
normal or a $s$-wave superconducting electrode. (b) Electron-like ($+$, blue
curve) and hole-like ($-$, red curve) band contributions to the DoS,
$\mathcal{N}_{\rm{FeSC}}^{\pm}\left(\varepsilon\right)$, see Eq. (S3). The
hopping parameters, in units of $\left|t_{1}\right|$, are
$(t_{1},t_{2},t_{3},t_{4})=(-1,1.3,-0.85,-0.85)$. The chemical potential is
taken at half filling $\mu=1.54$, and the gap size is assumed to be
$\Delta_{0}^{\text{FeSC}}=0.1$. The inset shows a magnifications of the total
DoS, $\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$, in the energy range
$\left|\varepsilon\right|\leq 2\Delta_{0}^{\small\text{FeSC}}$.

Notably, bulk thermoelectrical measurements of FeSC have been reported for the
normal state [27], although the analysis is quite intricate due to the
competition of different mechanisms, such as phonon- [28] and magnon-drag
phenomena [29], in a multiband setting. The linear TE properties of a TJ
between a FeSC and a normal metal allow identifying the strong intrinsic PH
asymmetry in the FeSC density of states (DoS) in a quite direct way, _also
for_ the superconducting phase. We will also potentially discriminate between
different order parameter symmetries. Furthermore, if the normal-metal is
replaced by a Bardeen–Cooper–Schrieffer (BCS) SC, we observe also astounding
TE figures of merit, which may be relevant for energy harvesting applications
and quantum technologies [14, 30, 31, 15].

_Thermoelectrical transport_ $-$ In order to address the physics of the FeSC,
we focus on the linear TE properties of a TJ between FeSC and a normal metal
(or a SC), see Fig. 1(a). The linear-response coefficients of charge ($I$) and
heat ($\dot{Q}$) currents can be expressed in terms of the Onsager matrix [9]

$\begin{pmatrix}I\\\ \dot{Q}\end{pmatrix}=\begin{pmatrix}\sigma&\alpha\\\
\alpha&\kappa T\\\ \end{pmatrix}\begin{pmatrix}\delta V\\\ \delta
T/T\end{pmatrix},$ (1)

where we assumed time-reversal symmetry satisfied and a small voltage
(temperature) bias $\delta V$ ($\delta T$). Here, $\alpha$ is the TE
coefficient, while $\sigma$ and $\kappa$ are the electric and thermal
conductances, respectively. At the lowest order of tunneling, it is easy to
express those linear coefficients as

$\begin{pmatrix}\sigma\\\ \alpha\\\
\kappa\end{pmatrix}=\frac{G_{T}}{e}\\!\bigintss_{-\infty}^{\infty}\begin{pmatrix}e\\\
\varepsilon\\\
\varepsilon^{2}\big{/}eT\end{pmatrix}\frac{\mathcal{N}_{L}\left(\varepsilon\right)\mathcal{N}_{R}\left(\varepsilon\right)d\varepsilon}{4k_{\text{B}}T\cosh^{2}\\!\left(\varepsilon/2k_{\text{B}}T\right)}$
(2)

in terms of the lead DoS, $\mathcal{N}_{j}\left(\epsilon\right)$ with $j=L,R$,
where $G_{T}$ is the normal-state conductance of the junction, $-e$ is the
electron charge, and $k_{\text{B}}$ is the Boltzmann constant. In our
analysis, we take into account two different cases, i.e., a junction formed
between a FeSC tunnel coupled with a normal lead, i.e., with
$\mathcal{N}_{R}(\epsilon)=1$ being the energy-independent normalized DoS in
Eq. (2), or alternatively with an $s$-wave SC, with
$\mathcal{N}_{R}\left(\epsilon\right)=\left|\text{Re}\left[\frac{\epsilon+i\Gamma}{\sqrt{(\epsilon+i\Gamma)^{2}-\Delta^{2}\left(T\right)}}\right]\right|$,
where $\Gamma=\gamma\Delta_{0}$ is the phenomenological Dynes parameter [33],
$\Delta_{0}=1.764k_{\text{B}}T_{c}$, and $T_{c}$ is the critical temperature
of the BCS SC. In this case, it is also implicitly assumed that the Josephson
coupling [34, 35] between the two superconducting leads is strongly suppressed
[36]. However, let us point out that most of the results shown in the
following, particularly those in which we will look at the effect of the order
parameter symmetry on the TE response, are obtained by considering an FeSC-I-N
junction, where the Josephson effect is not even present.

To calculate the DoS of FeSC we rely on the two-orbital, four-band tight-
binding approach by Raghu _et. al_ [1], which is the minimal model [38]. The
diagonalization of this tight-binding Hamiltonian model leads to eigenvalues
that can be written in a compact form as

$\varepsilon_{\mathbf{k}\pm}^{d}=\sqrt{\left(\varepsilon_{\mathbf{k}\pm}^{0}\right)^{2}+\left|\vec{d}_{\mathbf{k},g}\right|^{2}}.$
(3)

Here
$\varepsilon_{\mathbf{k}\pm}^{0}=\xi_{\mathbf{k}+}-\mu\pm\sqrt{\xi^{2}_{\mathbf{k}xy}+\xi^{2}_{\mathbf{k}-}}$,
$\mu$ is the chemical potential, and
$\vec{d}_{\mathbf{k},g}=\big{(}0,0,s_{x^{2}y^{2}}\big{)}$, where
$s_{x^{2}y^{2}}\equiv s_{\pm}=\Delta_{0}^{\text{FeSC}}\cos k_{x}\cos k_{y}$,
$\Delta_{0}^{\text{FeSC}}$ is the gap size, and
$\left|\vec{d}_{\mathbf{k}}\right|^{2}$ represents the effective amplitude of
the pairing interactions [2]. For the explicit expressions of
$\xi_{\mathbf{k}\pm}$ and $\xi_{\mathbf{k}xy}$ we refer the reader to Ref
[40]. We adopted the $s_{\pm}$-wave state, which is the mostly accepted FeSC
pairing state [41, 42]. However, the multiband character of FeSCs offers also
chances for more exotic pairing states [2, 42], and thus we will also discuss
TE properties with other order parameter symmetries in the following.
Hereafter, we took the interorbital hopping parameter $\left|t_{1}\right|$ as
a standard unit of energy, and temperature will be measured in units of
$\left|t_{1}\right|/k_{B}$ [43].

Figure 2: FeSC-I-N junction: (a) and (b) Seebeck coefficient,
$S(T,\Delta_{0}^{\text{FeSC}})$, and figure of merit,
ZT$(T,\Delta_{0}^{\text{FeSC}})$ at $\mu=1.54$. (c) and (d) $S(T,\mu)$ and
ZT$(T,\mu)$, at $\Delta_{0}^{\text{FeSC}}=0.1$. The legends in panels (c) and
(d) refer also to panels (a) and (b), respectively, and the red dashed line
marks the condition $(\Delta_{0}^{\text{FeSC}},\mu)=(0.1,1.54)$. FeSC-I-S
junction: (e) and (f) $S(T,\Delta_{0})$ and ZT$(T,\Delta_{0})$, at $\mu=1.54$,
$\Delta_{0}^{\text{FeSC}}=0.1$, and $\gamma=10^{-4}$. The dashed lines in
panels (c) and (f) mark the values $\Delta^{th}_{0}=1.764T$. A cartoon in the
top-right corner of each panel helps to recognise the type of junction
considered at a glance.

Finally, the FeSC total DoS turns out to be the sum of an electron-like [$+$,
blue curve in Fig. 1(b)] and a hole-like [$-$, red curve in Fig. 1(b)] band
contribution [44],
$\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)=\mathcal{N}_{\rm{FeSC}}^{+}\left(\varepsilon\right)+\mathcal{N}_{\rm{FeSC}}^{-}\left(\varepsilon\right)$,
where

$\mathcal{N}_{\rm{FeSC}}^{\pm}\left(\varepsilon\right)\\!=\\!\sum_{\mathbf{k}}\frac{\varepsilon+\varepsilon_{\mathbf{k}\pm}^{0}}{2\varepsilon_{\mathbf{k}\pm}^{d}}\left\\{\delta\left[\varepsilon_{\mathbf{k}\pm}^{d}-\varepsilon\right]-\delta\left[\varepsilon_{\mathbf{k}\pm}^{d}+\varepsilon\right]\right\\}\\!.\\!\\!\\!$
(4)

Note also that the superconducting instability opens the gap symmetrically
around the chemical potential, as illustrated in the inset of Fig. 1(b) (see
also Ref [40]).

_Figures of merit_ $-$ In order to quantify the TE performance, it is usual to
consider the Seebeck coefficient $S=-\alpha/(\sigma T)$ and thermodynamic
efficiency with the dimensionless figure of merit $\text{ZT}=S^{2}\sigma
T/\left[\kappa-\alpha^{2}/(\sigma T)\right]$ [45]. A large value of ZT means
better thermodynamic efficiency and if it tends to infinity, the efficiency of
the device tends to the Carnot limit. In the following, we show how $S$ and ZT
of an FeSC-insulator-normal metal (FeSC-I-N) junction depend on the
temperature, considering possible different values of
$\Delta_{0}^{\text{FeSC}}$ [46] and changing the doping level, $\mu$ [47].

Figures 2(a) and (b) collect the $S(T,\Delta_{0}^{\text{FeSC}})$ and
ZT$(T,\Delta_{0}^{\text{FeSC}})$ maps at the half-filling condition
$\mu=1.54$. For a given $\Delta_{0}^{\text{FeSC}}$, both the Seebeck
coefficient and the TE efficiency behave non-monotonically, with a clear
maximum that shifts towards gradually higher temperatures as
$\Delta_{0}^{\text{FeSC}}$ increases. Indeed, $\Delta_{0}^{\text{FeSC}}$ is
the energy scale that mainly influences the optimal temperature that maximizes
the TE effect. Furthermore, by increasing $\Delta_{0}^{\text{FeSC}}$ the
subgap states reduce, correspondingly requiring higher energies to achieve the
same TE effect. The use of FeSCs is a further advantage over conventional SC
configurations, since it allows operation at higher temperatures that provide
a larger Seebeck coefficient.

Figures 2(c) and (d) show what happens when changing the doping level, keeping
fixed $\Delta_{0}^{\text{FeSC}}=0.1$ [48]. The FeSC DoS dependence on $\mu$ is
illustrated in Ref [40]. The $S(T,\mu)$ map still reveals single-peaked
profiles, but modifying the doping we observe the inversion of the Seebeck
coefficient sign around $\mu\sim 0.75$, below (above) which $S>0$ ($S<0$) for
the hole (electron) DoS contribution dominates. We note that the point at
which the Seebeck coefficient changes sign differs from the half-filling
condition. The reason of that is the lack of symmetry between particle-like
and hole-like bands for an FeSC in the energy window determined by the working
temperatures.

In the cases discussed so far, we achieve Seebeck coefficients up to
$\left|S\right|\sim 150~{}\mu\text{V/K}$ reaching also TE efficiencies of
$\text{ZT}\sim 0.5$. In the case of an undoped FeSC, $\mu=1.54$, with
$\Delta_{0}^{\text{FeSC}}=0.1$, i.e., the red dashed lines in Fig. 2(a-d), the
highest TE efficiency is reached at $T\simeq 1.6\times 10^{-3}$, which may be
a temperature around $2.8\;\text{K}$.

We remark that the maximum Seebeck coefficient we obtain is several orders of
magnitude larger than that usually found in metallic structures at the same
temperatures. However, the thermoelectricity of the FeSC-I-N junction
outperforms magnetic TJs [49] and is quite well comparable with hybrid SCs-
ferromagnets TJs [50, 51] and quantum-dot setup [52].
It is noteworthy to show that, if we replace the normal metal with a BCS SC
with a gap $\Delta_{0}$, the linear thermoelectricity can be further enhanced.
This effect can be ascribed to the additional contribution of the conventional
DoS peaks intertwined with the multi-band character of the FeSC. Thus, in Fig.
2(e) and (f) we present the $S(T,\Delta_{0})$ and ZT$(T,\Delta_{0})$ maps of a
FeSC-insulator-SC (FeSC-I-S) TJ. In this case, we assume a specific FeSC with
a given gap $\Delta_{0}^{\text{FeSC}}=0.1$ and we explore the TE response at
different values of the BCS superconducting gap $\Delta_{0}$. A region of the
$(T,\Delta_{0})$ parameter space emerges in which both Seebeck coefficient and
TE efficiency increase significantly, even reaching the values
$\left|S\right|\sim 870~{}\mu\text{V/K}$ and ZT$\sim 6.5$ at
$(T,\Delta_{0})_{max}\simeq(0.63,5.3)\times 10^{-3}$. In natural units, these
quantities correspond to $T\simeq 1.8\;\text{K}$ for a BCS SC with
$T_{c}\simeq 5.2\;\text{K}$ [53].

Figure 3: FeSC-I-N junction: (a) DoS,
$\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$, (b) Seebeck coefficient,
$S(T)$, and (c) figure of merit, ZT$(T)$ by changing the symmetry of the order
parameter, at $\mu=1.54$ and $\Delta_{0}^{\text{FeSC}}=0.1$. The inset in (a)
illustrates the low-energy behaviour of the DoSs. The legend in panel (b)
refers to all panels.

_Order parameter symmetry detection_ $-$ We show here that TE figures of merit
are also a powerful tool for addressing order-parameter symmetry (OPS). We
compare the temperature dependence of $S$ and ZT of an FeSC-I-N TJ, with
$\mu=1.54$ and $\Delta_{0}^{\text{FeSC}}=0.1$, taking into account different
OPSs: we cover the three possible $s$-wave symmetries, i.e., the constant-gap
case $s_{0}$, $s_{x^{2}y^{2}}=\Delta_{0}^{\text{FeSC}}\cos k_{x}\cos k_{y}$,
and $s_{x^{2}+y^{2}}=\Delta_{0}^{\text{FeSC}}(\cos k_{x}+\cos k_{y})/2$, and
the two $d$-wave symmetries, i.e., $d_{xy}=\Delta_{0}^{\text{FeSC}}\sin
k_{x}\sin k_{y}$ and $d_{x^{2}-y^{2}}=\Delta_{0}^{\text{FeSC}}(\cos k_{x}-\cos
k_{y})/2$ [54]. Since it will be useful later on, we recall that the
$d_{x^{2}-y^{2}}$, $d_{xy}$, and $s_{x^{2}+y^{2}}$ OPSs are nodal, while the
others are nodeless [2, 3].

Figure 3 demonstrates that the TE figures of merit can provide valuable clues
for determining the OPS of the system. Starting from the top of this figure,
panel (a) takes a closer look at the different DoSs in play, with the inset
serving to highlight the low-energy region. Figures 3(b)-(c) illustrate the
Seebeck coefficient $S(T)$ and the TE efficiency ZT$(T)$, both showing, in a
semilog scale, bell-shaped, single-peaked profiles for each symmetry
considered. It immediately stands out that the “position” of these peaks
depends strongly on the OPS: indeed, for the $s_{0}$ and $s_{x^{2}y^{2}}$
cases, both $S$ and ZT are peaked roughly around $T^{\rm peak}\sim 10^{-3}$,
while the other symmetries give $S$ and ZT peaks centred on temperatures an
order of magnitude lower. To give realistic numbers (here the subscript
distinguishes symmetries), the $S$ peak for nodeless OPS is located at
$T_{x^{2}y^{2}}^{\rm peak}\simeq 3.3\;\text{K}$(blue), $T_{0}^{\rm peak}\simeq
1.4\;\text{K}$(violet), whereas for nodal cases one finds $T_{xy}^{\rm
peak}\simeq 0.13\;\text{K}$(yellow), $T_{x^{2}+y^{2}}^{\rm peak}\simeq
0.12\;\text{K}$(green), and $T_{x^{2}-y^{2}}^{\rm peak}\simeq
0.10\;\text{K}$(red). To grasp this result, we recall that the energy window
relevant for calculating the TE coefficients scales with temperature, i.e.,
$\left|\varepsilon\right|\sim T$ [40]. For instance, the energies considered
in the inset of Fig. 3(a) are essentially those where one should focus if
$T\sim 10^{-4}$. Here, it is evident that only the $d_{xy}$, $s_{x^{2}+y^{2}}$
and $d_{x^{2}-y^{2}}$ DoSs (i.e., those showing ZTs peaked at these
temperatures) are clearly non-zero. We see that nodeless FeSC pairings present
maximal thermoelectricity (absolute value of the Seebeck coefficient) at
relatively high temperatures. Instead, for the nodal cases, due to the
presence of low energy states in the gap, the maximal thermoelectricity is
observed at much lower temperature regimes. This result is quite robust
against variations of the gap amplitude, hopping parameter, and chemical
potential values [40]. Furthermore, we also see that the first two DoSs, i.e.,
the yellow and green curves, are unbalanced toward the hole side
($\varepsilon<0$), unlike the third, i.e., the red curve, which appears to be
slightly unbalanced toward particle side ($\varepsilon>0$). This is directly
reflected in the sign of $S$, which immediately tells us the PH asymmetry of
the FeSC DoS and, therefore, $S>0$ ($S<0$) in the former (latter) case. This
is clearly confirmed also by looking the PH asymmetry of the DoSs on the
larger energy scale considered in Fig. 3(a). It is evident, for instance, that
the $s_{0}$ and $s_{x^{2}y^{2}}$ OPSs, i.e., the violet and blue curves,
respectively, are unbalanced to the right, i.e., the particle contribution
dominates, thereby making $S$ negative.

We emphasize that, in principle, the measurement of the PH asymmetry could be
addressed by directly measuring the tunneling differential conductance.
However, a systematic experimental asymmetry in the bias polarization of the
junction cannot be easily excluded loosing sensitivity for small PH asymmetry.
The linear thermoelectricity much safely returns this information in an
independent way. Yet, we stress that the thermoelectrical signatures discussed
in this Letter, being associated with the quasiparticle tunneling in the
junction, are not affected by any phonon- or magnon-drag effects, which
instead usually influence the bulk TE properties in the normal phase [27].

_Conclusions_ $-$ To summarize, we have demonstrated that an FeSC TJ can show
sizable TE efficiency, and that both the TE figure of merit and the Seebeck
coefficient are found to be non-monotonic, single-peaked functions of
temperature. Moreover, they can provide details on the underlying symmetry of
the order parameter addressing the PH asymmetry of the DoS. In particular, we
demonstrated in FeSC-I-N junction that the position of both the ZT and $S$
peaks allows us to clearly distinguish nodal from nodeless symmetries.
Furthermore, the sign of $S$ provides further information on the asymmetry of
PH, distinguishing cases where the TE efficiency is not discriminating, such
as for the two $d$-waves symmetries. Our results also establish the relevance
of multiband SCs for a novel generation of TE devices.

As a closing remark, we observe that the proposed approach may be used for
studying other quantum materials [42]. Multi-orbital pairing approaches have
been widely used also to shed light in other multiband SCs, such as ruthenates
and nickelates, and insights about Hund-metal. Therefore, the TE-based
investigation of tunneling junctions presented in this Letter complements the
actual experimental techniques, finding a fertile ground for the study of
novel quantum materials.

###### Acknowledgements.

F.G. acknowledges PNRR MUR project PE0000023-NQSTI for partial financial
support. F.G and A.B. acknowledge the EU’s Horizon 2020 research and
innovation program under Grant Agreement No. 964398 (SUPERGATE) and No.
101057977 (SPECTRUM) for partial financial support. A.B. acknowledges the
Royal Society through the International Exchanges between the UK and Italy
(Grants No. IEC R2 212041).

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  * Note [8] The Raghu’s approach entails two pairing gaps, one for each orbital, $\Delta_{1,2}$, that satisfy the condition $\Delta_{1}(k_{x},k_{y})=\Delta_{2}(k_{y},k_{x})$ for all the pairing symmetries described above, except for $d_{x^{2}-y^{2}}$ giving $\Delta_{1}(k_{x},k_{y})=-\Delta_{2}(k_{x},k_{y})$ [2, 3]. In the latter case, the eigenvalues and the DoS are not reduced to the simple expressions given in Eqs. (S1)$-$(S3), but we use the general expression of the spectral function given in Ref. [2] with the eigenvalues presented in Ref [3].
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Supplemental Material: Thermoelectric signatures of order-parameter symmetries
in iron-based superconducting tunnel junctions

##  S-.1 Model for an FeSC

Figure S1: DoSs, $\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$, at
different values of $\mu\in[0-2.5]$, $\Delta_{0}^{\text{FeSC}}=0.1$, and
$(t_{1},t_{2},t_{3},t_{4})=(-1,1.3,-0.85-0.85)$. The insets sketch a zoom of
both $\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$ and
$\mathcal{N}^{\pm}_{\rm{FeSC}}\left(\varepsilon\right)$, see the solid, dashed
($+$), and dot-dashed lines ($-$), respectively, in the energy range
$\left|\varepsilon\right|\lesssim 2\Delta_{0}^{\small\text{FeSC}}$. which is
highlighted by a gray shaded area in the main plot.

We use the two-orbital, four-band tight-binding model developed by Raghu _et.
al_ [1] (see also Refs. [2, 3] for more details). The diagonalization of this
tight-binding Hamiltonian model leads to eigenvalues that can be written in a
compact form as

$\varepsilon_{\mathbf{k}\pm}^{d}=\sqrt{\left(\varepsilon_{\mathbf{k}\pm}^{0}\right)^{2}+\left|\vec{d}_{\mathbf{k},g}\right|^{2}}\qquad\qquad\text{where}\qquad\qquad\varepsilon_{\mathbf{k}\pm}^{0}=\xi_{\mathbf{k}+}-\mu\pm\sqrt{\xi^{2}_{\mathbf{k}xy}+\xi^{2}_{\mathbf{k}-}}.$
(S1)

Here $\mu$ is the chemical potential and
$\vec{d}_{\mathbf{k},g}=\big{(}0,0,s_{x^{2}y^{2}}\big{)}$, where
$s_{x^{2}y^{2}}=\Delta_{0}^{\text{FeSC}}\cos k_{x}\cos k_{y}$,
$\Delta_{0}^{\text{FeSC}}$ is the gap size. In Eq. (S1), we take into account
the kinetic energy terms $\xi_{\mathbf{k}\alpha\beta}$ of a particle with
momentum $\mathbf{k}$ changing the orbital from $\beta$ to $\alpha$, given by

$\displaystyle\xi_{\mathbf{k}\pm}$ $\displaystyle\\!=\\!$
$\displaystyle(\xi_{\mathbf{k}xx}\pm\xi_{\mathbf{k}yy})/2$ (S2)
$\displaystyle\xi_{\mathbf{k}xx}$ $\displaystyle\\!=\\!$
$\displaystyle-2t_{1}\cos(k_{x})\\!-\\!2t_{2}\cos(k_{y})\\!-\\!4t_{3}\cos(k_{x})\cos(k_{y}),$
$\displaystyle\xi_{\mathbf{k}yy}$ $\displaystyle\\!=\\!$
$\displaystyle-2t_{2}\cos(k_{x})\\!-\\!2t_{1}\cos(k_{y})\\!-\\!4t_{3}\cos(k_{x})\cos(k_{y}),$
$\displaystyle\xi_{\mathbf{k}xy}$ $\displaystyle\\!=\\!$
$\displaystyle-4t_{4}\sin(k_{x})\sin(k_{y}).$

The coefficients $t_{1}$ and $t_{2}$ are the intraorbital nearest-neighbor
hopping amplitudes, while $t_{3}$ and $t_{4}$ are the intraorbital and
interorbital next-nearest-neighbor hopping amplitudes, respectively [4].
Different sets of hopping parameters correspond to different Fe-based
superconductors (FeSC) (as in the manuscript, energies and temperatures are in
units of $\left|t_{1}\right|$ and $\left|t_{1}\right|/k_{B}$, respectively),
i.e., $(t_{1},t_{2},t_{3},t_{4})=(-1,1.3,-0.85-0.85)$ for iron pnictdes [1],
or $(t_{1},t_{2},t_{3},t_{4})=(-1,1.5,-1.2-0.95)$ for iron selenides [5, 6].
In this article, we focus primarily on iron pnictdes, although, as we show in
the following, the main claims about the capability of thermoelectric (TE)
measurements to resolve the pairing symmetry remain valid even using the
hopping parameter values for selenides (even if for this kind of FeSC more
sophisticated modelling has proven to be more effective [7]).

##  S-.2 Density of states of a FeSC

For the sake of convenience, we rewrite the FeSC density of states (DoS)
presented in the main text as

$\mathcal{N}_{\rm{FeSC}}^{\pm}\left(\varepsilon\right)=\frac{\eta}{\pi^{3}}\iint_{0}^{\pi}dk_{x}dk_{y}\frac{\varepsilon+\varepsilon_{\mathbf{k}\pm}^{0}}{2\varepsilon_{\mathbf{k}\pm}^{d}}\left[\frac{1}{\left(\varepsilon-\varepsilon_{\mathbf{k}\pm}^{d}\right)^{\\!2}+\eta^{2}}-\frac{1}{\left(\varepsilon+\varepsilon_{\mathbf{k}\pm}^{d}\right)^{\\!2}+\eta^{2}}\right],$
(S3)

assuming a Lorentzian energy broadening like i.e.,
$\delta\left(\varepsilon-\varepsilon_{\mathbf{k}\pm}^{d}\right)=\eta\Big{/}\left\\{\pi\left[\left(\varepsilon-\varepsilon_{\mathbf{k}\pm}^{d}\right)^{2}+\eta^{2}\right]\right\\}$
[2] (we set $\eta=0.01$ in line with Ref. [8]).

Figure S2: (a) $N^{-}(\mu)-N^{+}(\mu)$, where
$N^{\pm}(\mu)=\pm\int_{0}^{\pm\infty}\mathcal{N}_{\rm{FeSC}}\left(\varepsilon,\mu\right)d\varepsilon$,
as a function of $\mu$. (b) Normalized weight functions,
$W_{x}\left(\varepsilon,T\right)$, in Eqs. (S4)-(S5) as a function of
$\varepsilon/T$.

The behavior of the DoS, $\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$, at
different values of $\mu\in[0-2.5]$ and $\Delta_{0}^{\text{FeSC}}=0.1$ is
shown in Fig. S1. We clearly observe that by changing the doping level the DoS
shifts, but leaving the gap opening around $\varepsilon=0$. Moreover, as the
doping increases, the DoSs gradually change from fully gap to gapless
behaviour, as expected according to Ref. [2] (see insets in Fig. S1 where a
zoom is reported); in fact, for a large enough doping one finds a fully gapped
hole-like Fermi surface, giving a gap in the DoS at low energies, and a
partially gapped electron-like Fermi surface, giving a DOS growing
quasilinearly around $\varepsilon=0$. In order to illustrate this behavior, we
include in the insets of Fig. S1 the DoSs
$\mathcal{N}_{\rm{FeSC}}\left(\varepsilon\right)$ (solid line),
$\mathcal{N}^{+}_{\rm{FeSC}}\left(\varepsilon\right)$ (dashed line), and
$\mathcal{N}^{+}_{\rm{FeSC}}\left(\varepsilon\right)$ (dot-dashed line) within
the energy range $\left|\varepsilon\right|\lesssim
2\Delta_{0}^{\small\text{FeSC}}$. It is evident that for $\mu=0,$ and $0.5$,
see Fig. S1(a-b), the electron-like contribution does not enter in play at low
energies, and in fact the hole DoS contribution dominates (as discussed in the
main text) the TE response of the system. For $\mu=1$, see Fig. S1(c), the
electronic component starts to play a role and for $\mu=1.5$, see Fig. S1(d),
both the electrons and holes contributions effectively partake in the overall
DoS and appear clearly gapped. The last two panels, for $\mu=2$ and $2.5$,
demonstrate that the quasi-linear behaviour at low energies of the DoS is
entirely ascribable to the electronic component, the hole being still gapped.
The sign of the TE coefficient roughly reconcile with this behaviour since for
hole-dominated case it is typically positive where for electron-dominated is
more negative.
It is also interesting to look at the behaviour of the quantity
$N^{\pm}(\mu)=\pm\int_{0}^{\pm\infty}\mathcal{N}_{\rm{FeSC}}\left(\varepsilon,\mu\right)d\varepsilon$;
in particular, in Fig. S2(a) we focus on the difference
$N^{-}(\mu)-N^{+}(\mu)$ as a function of $\mu$. We recall that the undoped
compound, having the half-filled, two electrons per site configuration,
requires $\mu=1.54$, namely, the value largely used in the manuscript, at
which $N^{-}-N^{+}=0$. For $\mu$ lower (higher) than the half-filling value,
the positive (negative) energy portion of DoS predominates, being
$N^{-}-N^{+}<0$ ($N^{-}-N^{+}>0$). A change of slope is evident for the
threshold value $\mu^{th}\sim 1.15$; this is due to the fact that, increasing
the doping level, the electron-like band starts to be strongly occupied (e.g.,
see right panels of Fig. S1). This mechanism explains why for
$\mu\sim\mu^{th}$ there is a strong change in the "movement" of the TE peak
with respect to the doping illustrated in Figs. 2(c) and (d) of the main text.
In particular, we see that for $\mu\gtrsim\mu^{th}$ ($\mu\lesssim\mu^{th}$)
the TE peak shifts towards lower (higher) temperatures increasing the electron
doping. At lowest doping the thermoelectricity changes sign since it is hole
dominated and electronic-like band is almost completely depleted.

##  S-.3 TE coefficients of a tunneling barrier

Figure S3: FeSC-I-N junction: (a), (b), and (c)
$\alpha(T,\Delta_{0}^{\text{FeSC}})$, $\sigma(T,\Delta_{0}^{\text{FeSC}})$,
and $\kappa(T,\Delta_{0}^{\text{FeSC}})$ at $\mu=1.54$; (d), (e), and (f)
$\alpha(T,\mu)$, $\sigma(T,\mu)$, and $\kappa(T,\mu)$ at
$\Delta_{0}^{\text{FeSC}}=0.1$. FeSC-I-S junction: (g), (h), and (i)
$\alpha(T,\Delta_{0})$, $\sigma(T,\Delta_{0})$, and $\kappa(T,\Delta_{0})$ at
$\mu=1.54$ and $\Delta_{0}^{\text{FeSC}}=0.1$. A cartoon in the bottom-left
corner of each panel helps to recognise the type of junction considered at a
glance.

It is useful to express the TE coefficient, $\alpha$, and the electric and the
thermal conductances, $\sigma$ and $\kappa$,

$\begin{pmatrix}\sigma\\\ \alpha\\\
\kappa\end{pmatrix}=\frac{G_{T}}{e}\\!\bigintss_{-\infty}^{\infty}\begin{pmatrix}e\\\
\varepsilon\\\
\varepsilon^{2}\big{/}eT\end{pmatrix}\frac{\mathcal{N}_{L}\left(\varepsilon\right)\mathcal{N}_{R}\left(\varepsilon\right)d\varepsilon}{4k_{\text{B}}T\cosh^{2}\\!\left(\varepsilon/2k_{\text{B}}T\right)},$
(S4)

in such a way as to group all terms excluding DoSs into "weight functions",
that is

$x=\frac{G_{T}}{e}\int_{-\infty}^{\infty}W_{x}(\varepsilon,T)\mathcal{N}_{L}(\varepsilon)\mathcal{N}_{R}(\varepsilon)d\varepsilon\qquad\text{with}\qquad
x=(\sigma,\alpha,\kappa).$ (S5)

The behavior of the weight functions, $W_{x}\left(\varepsilon,T\right)$ with
$x=(\sigma,\alpha,\kappa)$, normalised to their maximum value, is shown in
Fig. S2(b). We immediately note that all three functions quickly go to zero
for $\left|\varepsilon\right|\gtrsim 10\;T$, thus limiting the temperature
range relevant for the calculation of linear TE coefficients. We also observe
that $\sigma$ and $\kappa$ are even functions of energy, unlike $\alpha$,
which is odd; from this characteristic derives the dependence of the sign of
the Seebeck coefficient, $S=-\alpha/(\sigma T)$, on the predominant
particle/hole contribution.

In Fig. S3 we show how TE coefficient, $\alpha$, and the electric and the
thermal conductances, $\sigma$ and $\kappa$, vary in the case discussed in
Fig. 2 of the main text (note, in Fig. S3 we are showing dimensionless
normalised quantities). In particular: panels (a), (b), and (c) show
$\alpha(T,\Delta_{0}^{\text{FeSC}})$, $\sigma(T,\Delta_{0}^{\text{FeSC}})$,
and $\kappa(T,\Delta_{0}^{\text{FeSC}})$ at $\mu=1.54$, while panels (d), (e),
and (f) show $\alpha(T,\mu)$, $\sigma(T,\mu)$, and $\kappa(T,\mu)$ at
$\Delta_{0}^{\text{FeSC}}=0.1$, in the case of a FeSC-I-N junction. Instead,
panels (g), (h), and (i) $\alpha(T,\Delta_{0})$, $\sigma(T,\Delta_{0})$, and
$\kappa(T,\Delta_{0})$ at $\mu=1.54$ and $\Delta_{0}^{\text{FeSC}}=0.1$, in
the case of a FeSC-I-S junction. This figure allows us to definitively state
that, taken individually, these quantities are unable to provide the
information that instead emerges clearly when they are recombined to form the
Seebeck coefficient, $S$, and figure of merit, ZT, as shown in Fig. 2 of the
article. Finally, we observe also that the black region in Fig. S3(d)
indicates negative $\alpha$ values, giving $S>0$.

##  S-.4 TE and other FeSC order parameter symmetries

Figure S4: TE efficiency in the case of $d_{xy}=\Delta_{0}^{\text{FeSC}}\sin
k_{x}\sin k_{y}$ pairing symmetry. FeSC-I-N junction: (a) and (b) Seebeck
coefficient, $S(T,\Delta_{0}^{\text{FeSC}})$, and figure of merit,
ZT$(T,\Delta_{0}^{\text{FeSC}})$ at $\mu=1.54$. (c) and (d) $S(T,\mu)$ and
ZT$(T,\mu)$, at $\Delta_{0}^{\text{FeSC}}=0.1$. The legends in panels (c) and
(d) refer also to panels (a) and (b), respectively, and the red dashed line
marks the $(\Delta_{0}^{\text{FeSC}},\mu)=(0.1,1.54)$ case. FeSC-I-S junction:
(e) and (f) $S(T,\Delta_{0})$ and ZT$(T,\Delta_{0})$, at $\mu=1.54$,
$\Delta_{0}^{\text{FeSC}}=0.1$, and $\gamma=10^{-4}$. The dashed lines in
panels (c) and (f) marks the values $\Delta^{th}_{0}=1.764T$. A cartoon in the
top-right corner of each panel helps to recognise the type of junction
considered at a glance.

For the sake of completeness, we also show how the Seebeck coefficient and TE
efficiency behave considering a $d_{xy}$ pairing symmetry. As can be seen from
Fig. S4, the main characteristics discussed in the manuscript for
$s_{\pm}$-symmetry are also evident here, with a clear shift towards lower
temperatures, in line with what was discussed in Fig. 3 in the main text for a
nodal case. Another quite evident difference is that the sign of $S$ is
reversed from what is shown in Fig. 2 of the main article, see Figs.
S4(a)-(c)-(e): this indicates that where hole contributions for
$s_{\pm}$-symmetry is predominant, particle contributions now prevail (and
vice versa). In the case of an FeSC-I-N junction, we also note that the
threshold doping value around which the sign of $S$ changes is somewhat higher
than in the case shown in the article (the switch occurs roughly at $\mu\sim
1.25$), see Fig. S4(b). Finally, we observe that the maximum values obtained
for $S$ and ZT in this case are practically the same as those obtained in the
case of $s_{\pm}$ pairing symmetry. However, considering an FeSC-I-S junction,
the region of the parameter space where these maxima are reached is different,
being the ZT peak in Fig. S4(f) located at
$(T,\Delta_{0})_{max}\simeq(0.23,2.1)\times 10^{-4}$; in other words, in this
case the maximum TE efficiency would be attainable for a superconducting
tunnel junction at $T\simeq 40\;\text{mK}$ formed between a FeSC and a SC with
$T_{c}\simeq 210\;\text{mK}$.

Finally, Fig. S5 show that the TE of a tunnel junction resolves better between
nodal and nodeless cases increasing the gap value, see panels (a) and (b) for
$\Delta_{0}^{\text{FeSC}}=0.05$ and $0.2$, respectively. Furthermore, we
tested the results also with the hopping parameters of FeSC selenides, see
Fig. S5(c) for $\Delta_{0}^{\text{FeSC}}=0.1$ and $\mu=1.54$.

Figure S5: FeSC-I-N junction: Temperature dependence of Seebeck coefficient,
$S(T)$, (upper panels) and figure of merit, ZT$(T)$, (lower panels) by
changing the symmetry of the order parameter, at a fixed $\mu=1.54$ and
setting (a) $\Delta_{0}^{\text{FeSC}}=0.05$ and (b)
$\Delta_{0}^{\text{FeSC}}=0.2$, while in (c) we set
$\Delta_{0}^{\text{FeSC}}=0.1$ in the specific case of
$(t_{1},t_{2},t_{3},t_{4})=(-1,1.5,-1.2-0.95)$ for iron selenides [6]. The
legend in panel (a) refers to all panels.

Figure S6: FeSC-I-N junction: Temperature dependence of Seebeck coefficient,
$S(T)$, (upper panels) and figure of merit, ZT$(T)$, (lower panels) by
changing the symmetry of the order parameter, at a fixed
$\Delta_{0}^{\text{FeSC}}=0.1$ and setting (a) $\mu=1.45$ and (b) $\mu=1.54$,
while in (c) we set $\mu=1.65$. The legend in panel (a) refers to all panels.

##  S-.5 Robustness to chemical potential variations

The level of doping usually considered in Fe-based superconductors, like
FeSeTe, doesn’t change dramatically the position of the chemical potential and
we expect that the impact of $\mu$ on TE performance is not strong at changing
the doping level by a few percent. We have verified that for a level doping of
$3\%$ compared to the half-filling the chemical potential is roughly shifted
by $1.45$ to $1.65$ and this does not affect the results. In any case, just to
get an idea of how Fig. 3 of the main text changes in the case of a doped
system, we show below the curves of $S(T)$ and ZT$(T)$ at different order
parameter symmetries, for $\mu=\\{1.45,1.54,1.65\\}$. It is evident that the
result is qualitatively robust to possible doping, although quantitatively
there may be some differences.

##  S-.6 Power factor

A complete TE characterization of the device requires also looking at the
power that the system produces as an energy harvester. This can be quantified
analyzing the electrical conductance $\sigma$, or more precisely the power
factor $\text{PF}=\sigma\,S^{2}$, where $S$ is the Seebeck coefficient [9].
Indeed, while large values of ZT imply a high efficiency, this often happens
at the cost of low power, or alternatively if PF is very low, it may be
experimentally difficult to measure $S$, because the voltage bias produced by
the temperature gradient may drastically drop when a small load is applied to
the system to measure such a voltage. However, defining the quantities of
interest for our work, we focused on the intensive TE figures of merit: in
fact, both the Seebeck coefficient and the ZT do not depend on the conductance
of the tunnel barrier, $G_{T}$. The case of the power factor, which depends on
$G_{T}$ being directly proportional to $\sigma$, is different.

In Fig. S7 we report the comparison between $\left|S\right|(T)$ and
$\text{PF}(T)$, the latter normalised to the tunnel conductance of the
junction. In particular, we focus on the cases of interest for our work, i.e.,
an FeSC-I-N junction, see panel (a), and an FeSC-I-S junction, see panel (b),
in the case of an undoped FeSC, $\mu=1.54$, with
$\Delta_{0}^{\text{FeSC}}=0.1$, i.e., the red dashed lines in Fig. 2(a-d) of
the main text, while the value of $\Delta_{0}$ in (b) is chosen in such a way
as to give the maximum of $\left|S\right|$. We immediately notice that
$\text{PF}(T)$, despite the quantitative differences, is qualitatively the
same in the two cases, and that the power factor could actually be quite low
at the maximum Seebeck coefficient. However, for our purpose the crucial point
is if the effect is still measurable. For an FeSC-I-N junction, i.e., in the
cases described in Fig. 3 of the main text by comparing the various
symmetries, it is evident that at the temperature where $\left|S\right|$
peaks, the power factor is $\text{PF}\sim
5\;\text{pW}\,\text{K}^{-2}\,\Omega$. The case of the FeSC-I-S junction is
different: in fact, the peak of $\left|S\right|$ occurs at lower temperatures,
where the power factor is quite low ($\text{PF}\sim
0.2\;\text{pW}\,\text{K}^{-2}\,\Omega$). In the latter case, a compromise
between a not-too-low working temperature and a non-zero power factor may be
necessary to measure ZT. We will return to the calculation of PF also in the
next section.

Figure S7: Power factor (normalised to the tunnel conductance $G_{T}$ of the
junction), PF, and Seebeck coefficient, $|S(T)|$ , as a function of
temperature for an FeSC-I-N (a) and an FeSC-I-S junction (b). The values of
other parameters are $\mu=1.54$ and $\Delta_{0}^{\text{FeSC}}=0.1$, while for
(b) the value of $\Delta_{0}$ is chosen in such a way as to give the maximum
of $S$. A cartoon in the top-right corner of each panel helps to recognize the
type of junction considered at a glance.

##  S-.7 Relation between the superconducting gaps at the optimal TE
efficiency

Figure S8: FeSC-I-S junction: (a) ZT$(T,\Delta_{0})$, at $\mu=1.54$ and
$\Delta_{0}^{\text{FeSC}}=0.1$. The black dashed line mark the values
$\Delta^{th}_{0}=1.764T$, while the red dot-dashed line is used to find the
optimal TE point at different $\Delta_{0}^{\text{FeSC}}$ values, still within
the green shaded area. A cartoon in the top-left corner helps to recognize the
type of junction considered at a glance. (b-e) Temperature dependence of both
the power factor, PF, (normalised to the tunnel conductance $G_{T}$ of the
junction) and the ZT at the $\Delta_{0}$ values along the red dot-dashed line
in (a), $\mu=1.54$, and $\Delta_{0}^{\text{FeSC}}=\\{0.05,0.1,0.2,0.3\\}$.
Figure S9: FeSC-I-S junction: (a) $\Delta_{0}$ at the ZT maximum,
$\Delta_{0}^{\text{ZT}_{\text{max}}}$, and (b) PF (normalized to $G_{T}$) at
the ZT maximum, $\text{PF}^{\text{ZT}_{\text{max}}}$, as a function of
$\Delta_{0}^{\text{FeSC}}$. The red dashed curve in (a) represents the
function
$\Delta_{0}^{\text{ZT}_{\text{max}}}=k(\Delta_{0}^{\text{FeSC}})^{2}$, with
$k\sim 1/2$, that appears to fit our results very well.

One could in principle expect some correlation between the size of the gaps
and the position of the peaks of the TE figures of merit, although the
relation between these quantities is hard to guess given the nonlinear nature
of the expressions involved. However, we realised that the relation connecting
the superconducting gaps at the optimal TE point is not simply linear, but is
instead a “functional”; in particular, we found, at the maximum ZT, a
quadratic dependence of $\Delta_{0}$ on $\Delta_{0}^{\text{FeSC}}$. The
procedure that we follow to achieve this result is briefly outlined below.

First, we started looking at Fig. 2(f) of the main text, which is reported
even in the panel (a) of Fig. S8, that is obtained for
$\Delta_{0}^{\text{FeSC}}=01$. We argued that at a different
$\Delta_{0}^{\text{FeSC}}$ the “island” of high ZT values would still emerge,
but in another position within the highlighted green shaded area of the
$(T,\Delta_{0})$-parameter space, which is bounded by the black dotted line.
The latter is a threshold for TE effects, i.e., it is the temperature above
which the gap of the conventional superconductor is fully closed; thus, it
does not dependent of the Fe-SC parameters. It is therefore reasonable to
expect that the region of the $(T,\Delta_{0})$-parameter space in which to
expect the maximum TE effects is just “before” this threshold. We looked for
the optimal TE conditions by “staying” along the red dot-dashed line, which is
just parallel to the black dashed line and passing through the ZT maximum. In
this way, for each $\Delta_{0}^{\text{FeSC}}$ it is possible to find the
“position” of the ZT peak without scanning the whole
$(T,\Delta_{0})$-parameter space, thus avoiding a highly time-consuming
numerical task. Panels (b-e) of Fig. S8 are obtained in this way, setting four
distinct values of $\Delta_{0}^{\text{FeSC}}=\\{0.05,0.1,0.2,0.3\\}$; in each
panel, we present both ZT$(T)$ (blue curve) and $\text{PF}(T)$ (red curve).
All these plots show a ZT$(T)$ peaked profile, reaching values $\gtrsim 6$. At
a given $\Delta_{0}^{\text{FeSC}}$, it is therefore easy to extract the
position of the ZT peak (and therefore the
$\Delta_{0}^{\text{ZT}_{\text{max}}}$ value) that gives the maximum TE
efficiency. All that remains is to collect the pairs
$(\Delta_{0}^{\text{FeSC}},\Delta_{0}^{\text{ZT}_{\text{max}}})$ at different
$\Delta_{0}^{\text{FeSC}}$ values.

In Fig. S9 we report in (a) the $\Delta_{0}$ at the ZT maximum,
$\Delta_{0}^{\text{ZT}_{\text{max}}}$, and in (b) the PF at the ZT maximum,
$\text{PF}^{\text{ZT}_{\text{max}}}$, (normalized to $G_{T}$) at different
values of $\Delta_{0}^{\text{FeSC}}$. The red dashed curve in (a) describes
the function
$\Delta_{0}^{\text{ZT}_{\text{max}}}=k(\Delta_{0}^{\text{FeSC}})^{2}$, with
$k\sim 1/2$, that appears to fit our results very well. In (b), it is evident
that $\text{PF}^{\text{ZT}_{\text{max}}}$ grows linearly with
$\Delta_{0}^{\text{FeSC}}$.

In conclusion, we derived a quadratic dependence of the superconducting gap
that gives high TE performance on the Fe-based superconducting gap. The reason
for this dependence is not immediately apparent, and the question could be
investigated further in more detail at a later stage in a following work.

## References

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# Implementation of synthetic fast-ion loss detector and imaging heavy ion
beam probe diagnostics in the 3D hybrid kinetic-MHD code MEGA

P. Oyola<EMAIL_ADDRESS>Department of Atomic, Molecular and Nuclear Physics,
Universidad de Sevilla, Sevilla, 41012, Spain  J. Gonzalez-Martin  Centro
Nacional de Aceleradores (CNA) CSIC, 41092 Seville, Spain  Department of
Mechanical Engineering and Manufacturing, Universidad de Sevilla, Sevilla,
41092, Spain  M. Garcia-Munoz  Department of Atomic, Molecular and Nuclear
Physics, Universidad de Sevilla, Sevilla, 41012, Spain  Centro Nacional de
Aceleradores (CNA) CSIC, 41092 Seville, Spain  J. Galdon-Quiroga  Max Planck
Institute for Plasma Physics, 85748 Garching, Germany  G. Birkenmeier  Max
Planck Institute for Plasma Physics, 85748 Garching, Germany  Physics
Department E28, Technical Univerty Munich, 85748 Garching, Germany  E. Viezzer
Department of Atomic, Molecular and Nuclear Physics, Universidad de Sevilla,
Sevilla, 41012, Spain  Centro Nacional de Aceleradores (CNA) CSIC, 41092
Seville, Spain  J. Dominguez-Palacios  Department of Atomic, Molecular and
Nuclear Physics, Universidad de Sevilla, Sevilla, 41012, Spain  Centro
Nacional de Aceleradores (CNA) CSIC, 41092 Seville, Spain  J. Rueda-Rueda
Department of Atomic, Molecular and Nuclear Physics, Universidad de Sevilla,
Sevilla, 41012, Spain  J. F. Rivero-Rodriguez  Centro Nacional de Aceleradores
(CNA) CSIC, 41092 Seville, Spain  Department of Mechanical Engineering and
Manufacturing, Universidad de Sevilla, Sevilla, 41092, Spain  Y. Todo
National Institute for Fusion Science, 509-5292 Toki, Japan  ASDEX Upgrade
team  See author list of H. Meyer et al. Nucl. Fusion59, 112014 (2019)

###### Abstract

A synthetic Fast-Ion Loss Detector (FILD) and an imaging Heavy Ion Beam Probe
(i-HIBP) have been implemented in the 3D hybrid kinetic-magnetohydrodynamic
code MEGA. First synthetic measurements from these two diagnostics have been
obtained for neutral beam injection (NBI) driven Alfvén Eigenmode (AE)
simulated with MEGA. The synthetic fast-ion losses show a strong correlation
with the AE amplitude. This correlation is observed in the phase-space,
represented in coordinates $(P_{\phi},E)$, being toroidal canonical momentum
and energy, respectively. Fast-ion losses and the energy exchange diagrams of
the confined population are connected with lines of constant $E^{\prime}$, a
linear combination of $E$ and $P_{\phi}$. First i-HIBP synthetic signals also
have been computed for the simulated AE, showing displacements in the
strikeline of the order of $\sim$1\text{\,}\mathrm{m}\mathrm{m}$$, above the
expected resolution in the i-HIBP scintillator of
$\sim$100\text{\,}\mu\mathrm{m}$$.

††preprint: AIP

##  I Introduction

In magnetically confined fusion plasmas, Alfvén Eigenmodes (AEs) can be
excited by various fast-ion sources, such as neutral beam injection (NBI) and
fusion-born alpha particles. In turn, AEs can enhance the fast-ion transport
and can lead to fast-ion losses towards the first wall in fusion devices.
These uncontrolled fast-ion losses against the wall can result in hazardous
heat loads in future reactors, like ITER[1, 2].

Previous work has been carried out to characterize AEs experimentally[3, 4,
5], focused on the identification of the fast-ion transport induced by these
modes and the poloidal structures of the modes. In the ASDEX Upgrade (AUG)
tokamak, the poloidal array of fast-ion loss detectors[6, 7, 8] (FILD) have
detected that the fast-ion losses are correlated with the AEs amplitude and
frequency, demonstrating the AE and fast-ion interaction. Accurate
characterization of these losses and the mode structures plays a key role in
understanding the fast-ion confinement[9, 10, 11, 12].

In this work, these experiments are modelled with the 3D non-linear hybrid
kinetic-MHD code MEGA[13]. Two synthetic diagnostics have been developed in
MEGA, providing further insight into the Alfvénic activity, by studying two
fundamental key points: the fast-ion loss and the radial structures.

These synthetic diagnostics are based on two of the diagnostic systems in AUG:
the poloidal array of Fast-Ion Loss Detector (FILD) and the imaging Heavy-Ion
Beam Probe[14, 15, 16] (i-HIBP). For the first, a realistic 3D wall for the
AUG tokamak has been implemented in MEGA showing a correlation between the
fast-ion losses towards the wall and the AE activity in the simulations. For
the latter, a predictive study of the synthetic signal demonstrates the
capability of i-HIBP for measuring an AE located between midradius and the
edge of the plasma.

This paper is organized as follows. The model implemented in the code MEGA is
briefly described in Sec. II where the implementation of the realistic 3D wall
is described. The implementation of the synthetic diagnostic for i-HIBP is
described in Sec. III. Sec. IV is devoted to the analysis of MEGA simulations
showing the fast-ion losses and the synthetic signal for the i-HIBP
diagnostic. A summary is given in Sec. V.

##  II MEGA and the realistic 3D wall

MEGA is a numerical code that computes the self-consistent evolution of a bulk
plasma and the fast-ion population in realistic 3D configurations using
cylindrical coordinates. In this code, the bulk plasma is modelled using the
complete non-linear single-fluid resistive-MHD equations[13, 17]. Coupling
between bulk plasma and fast-ion population is done via the current density in
the momentum balance equation.

The set of MHD equations is spatially discretized using $4^{th}$ order finite
differences on a cylindrical grid, covering the full tokamak geometry. The
fast-ion distribution is sampled by markers covering the 5D reduced phase-
space $(\bm{X},p_{\parallel},\mu)$, $\bm{X}$ being the guiding-center
position; $p_{\parallel}$, parallel momentum; and $\mu$ the magnetic dipole
moment. These markers are evolved using the gyro-kinetic equations with Finite
Larmor Radius (FLR) corrections. The fast-ion distribution is evolved using
particle-in-cell with the $\delta f$ method[18].

The cylindrical grid resolution has been chosen to be
$(N_{R},N_{\phi},N_{z})=(128,64,256)$, allowing the evaluation up to the $n=5$
toroidal mode number. Higher toroidal mode numbers are filtered out to reduce
numerical noise. The time evolution is obtained using an explicit $4^{th}$
order Runge-Kutta scheme.

Figure 1: Density, temperature and toroidal rotation used as starting point
for MEGA simulations, taken from the AUG tokamak discharge $\\#34570$
($t=$3.53\text{\,}\mathrm{s}$$). The reconstructed $q$-profile is shown in
dashed lines.

Experimental profiles and magnetic reconstructions from AUG pulse $\\#34570$
are used as inputs for the MEGA simulations. In Fig. 1 the initial profiles
are presented, corresponding to a discharge at the AUG tokamak. The initial
fast-ion distribution is given by a modelled off-axis NBI slowing-down
distribution using a Gaussian term[8] for the spatial dependence. The initial
distribution function used in this work is:

$F_{\text{phase-space}}\propto
e^{-\frac{\left(\rho-\rho_{0}\right)^{2}}{2\left(\Delta\rho\right)^{2}}}\frac{1}{v^{3}+v^{3}_{\text{crit}}}\text{erfc}\left(\frac{v-v_{\text{birth}}}{\Delta
v}\right)\times\\\
e^{-\frac{\left(\Lambda-\Lambda_{0}\right)^{2}}{2\left(\Delta\Lambda_{0}\right)^{2}}}$
(1)

being $\rho$ the normalized poloidal magnetic flux. Spatial parameters have
been fixed to $\rho_{0}=0.4$, $\Delta\rho=0.15$ in this work.
$v_{\text{birth}}$ is the birth velocity, has been set to
$93\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, and $\Delta v=0.05\cdot
v_{\text{birth}}$. For the pitch-angle, a Gaussian dependence in
$\Lambda\equiv 1-\lambda^{2}=\frac{\mu B}{E}$ is introduced. The pitch-angle
parameters has been set to $\Lambda_{0}=0.55$ and $\Delta\Lambda_{0}=0.20$ in
this work. Finally, $v_{\text{crit}}$ is the critical velocity[19].

The parameter regulating the fast-ion density,
$\beta_{\text{fi}}=\frac{p_{\text{fi}}}{B_{\text{axis}}^{2}/2\mu_{0}}$, is set
to $\beta_{\text{fi}}=0.67\%$, being $p_{\text{fi}}$ the fast-ion pressure.
This value corresponds to the NBI6 for the same discharge. Note that this work
does not intend to reproduce the Toroidal Alfvén Eigenmodes (TAEs) observed in
the experiment, but to provide a suitable simulation case to test the newly
implemented synthetic diagnostics. A detailed comparison to the experiment is
ongoing[20].

Figure 2: Map of the wall (here represented for $\phi=0^{o}$) used in MEGA to
obtain the fast-ion losses. Green and red regions indicate inside/outside
simulation domain for the FI markers. The first wall is used as limit where
fast-ion markers are captured. Solid lines represent flux surfaces. In blue,
the separatrix is indicated.

A 3D wall has been implemented in MEGA, allowing to stop the evolution of
fast-ion markers when they reach the first wall. The mapping of the wall to
the cylindrical grid (where the MHD equations are solved) allows for a fast
implementation of the 3D wall, without a significant impact on the simulation
efficiency (less than $1\%$). In Fig. 2, a poloidal cut of the mapped wall in
MEGA is shown. The regions marked in yellow (some of them lie behind the 2D
wall model represented by the thicker black lines) determine where the fast-
ion evolution will be stopped and considered as fast-ion losses (FIL).

##  III Synthetic i-HIBP diagnostic

The i-HIBP diagnostic injects a heavy-neutral primary beam
(${}^{133}\text{Cs}$ or ${}^{85,87}\text{Rb}$) into the plasma that ionizes
due to multiple processes. These ionized particles, forming the secondary
beam, start a gyromotion until reaching a scintillator plate. The signal on
the scintillator translates into a two-fold information: the intensity of the
strike line provides the plasma density ($n_{e}$); and the strike line
position and shape provides information on the magnetic and electric fields,
$\bm{B}$ and $\bm{E}$, respectively[16].

The synthetic diagnostic of i-HIBP, the new i-HIBPsim code, is based on
kinetic simulations for the two main species, the heavy-neutrals (primary
beam) and heavy-ions (secondary beam). Markers are launched at the injection
port and tracked into the plasma using a Boris leap-frog scheme[21]. In this
work, an infinitely small beam is used, i.e., the width of the beam and
divergences are set to zero. The secondary beam birth distribution is obtained
by using a beam attenuation model:

$\dot{W}_{j}=-\sum_{k\in\text{reactions}}W_{j}n_{k}\langle\sigma
v\rangle_{k},$ (2)

where the sum is over all possible reactions that attenuate the beam; $n_{k}$
is the secondary reactant density; and $\langle\sigma v\rangle_{k}$ is the
reaction rate of the $k^{\text{th}}$ reaction. Only two reactions are
considered to generate the attenuation of the primary: the electron-impact
ionization[22] and the charge-exchange reactions with main-ions[23, 24, 25].
Single-ionization step is implemented in the simulation code as the
recombination via charge-exchange (i.e., $Cs^{+}+D^{0}\rightarrow
Cs^{0}+D^{+}$) is expected to be much smaller, since the neutral density in
AUG is of the order[26] of $n_{0}\sim 10^{16}\ \text{m}^{-3}$. Impurities
induced ionizaation reaction rates, as extrapolated from lithium in [27], are
negligible compared to the main ion charge-exchange and electron-impact
ionization rates. This, combined with the typical impurity concentration
($\sim 1\%$ after the boronization[28]), makes this interaction negligible,
compared collisions with electrons and main ions.

The secondary beam will travel following gyroorbits until hitting the
scintillator, determined via a ray-triangle algorithm[29]. The beam-
attenuation equation is also used to determine the secondary beam flux into
the scintillator. A single-step ionization is used for the secondary beam
considering only the electron-impact ionization[30, 31]. Charge-exchange
recombination for the secondary beam is not taken into account, as for the
primary beam.

Markers evolve in a fully 3D input electromagnetic field, allowing for a
direct connection with MEGA. The electromagnetic perturbations computed by
MEGA for a certain plasma phenomena, can be used to feed the synthetic
diagnostic and obtain the predicted signal.

In previous experiments in fusion devices, like TJ-II, a similar diagnostic,
the HIBP, has been used to detect and characterize the poloidal mode numbers
and structure of AE[32]. The scintillator-based i-HIBP will provide the high
spatial resolution measurements, as shown in Fig. 5 (up to
$100\text{\,}\mu\mathrm{m}$ in the scintillator).

##  IV Simulation results

The off-axis fast-ion distribution produces in the plasma a TAE located at the
midradius ($\rho_{\text{pol}}=0.70$). This example case, using the realistic
3D wall, already suggests an important fast-ion loss mechanism, via the wave-
particle resonances. The location of the AE, as shown in Fig. 3(b), is close
enough to the edge for the heavy-ions in i-HIBP to feel the perturbations. The
green lines represent trajectories for ${}^{85}\mathrm{Rb}$ under the
perturbed magnetic field, proving the possible range of detection.

Figure 3: (a) In solid lines, evolution of the energy associated to the
toroidal $\left|n\right|$ mode numbers. In dashed lines, the FIL flux
associated to the region in velocity-space $\mu\in\left(3.0,5.0\right)\cdot
10^{-15}\rm{J/T}$. (b) Poloidal representation of the perturbation at
$t=$0.07\text{\,}\mathrm{m}\mathrm{s}$$. Flux surfaces corresponding to
$\rho_{\text{pol}}=0.70,1.00$ as well as the 2D wall structures, have been
represented for visual guidance. In green, ${}^{85}\mathrm{Rb}$ orbits
obtained with i-HIBPsim and discussed in Section IV.2.

The implementation of an off-axis initial fast-ion spatial distribution leads
to Alfvénic activity closer to the plasma edge. In Fig. 3(a), the toroidal
mode energy in the bulk plasma is shown on a logarithmic scale. The
$\left|n\right|=4$ mode shows the largest growth and dominates the plasma
phenomena. In dashed, the fast-ion flux for given
$\mu\in\left(3.0,5.0\right)\cdot 10^{-15}\ \text{J/T}$ interval has been
represented. Note that to avoid the inclusion of unrealistic fast-ion prompt
losses in the simulation the quantity $\left|\delta w_{j}\right|N_{j}$ has
been presented instead, where $\left|\delta w_{j}\right|$ is the differential
weight evolution with respect to the equilibrium, and $N_{j}$ is the number of
particles represented initially by the marker. The markers strongly
interacting with the mode have a higher $\left|\delta w_{j}\right|$, hence
allowing us to focus on fast-ion loss induced by the mode.

In Fig. 3(b) the poloidal structure of the mode at
$t=$0.07\text{\,}\mathrm{m}\mathrm{s}$$ is shown. The mode is located around
the surface $\rho_{\text{pol}}=0.7$, superimposed for visual guidance. A
Fourier transform of its time evolution shows that the frequency of the mode
is $f=$102\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

###  IV.1 Synthetic fast-ion losses

In the AUG tokamak, strong TAE-coherent fast-ion losses have been detected
with the fast-ion loss detector poloidal array[11]. The induced losses can be
explained by the magnetic perturbation producing an open trajectory, without a
net energy exchange; or via a power exchange with the mode resulting in an
orbit kick away from the confined region.

Fast-ion losses in MEGA simulations have been obtained using a self-consistent
approach, capturing the fast-ion markers during the simulations. To identify
whether these losses are produced by a significant interaction with the AE,
the phase-space, represented in variables $\left(P_{\phi},E,\mu\right)$, being
toroidal canonical momentum, energy and magnetic moment, respectively. In Fig.
4 (a-b) the phase-space for $\mu\in\left(3.0,5.0\right)\cdot 10^{-15}\rm{J/T}$
is shown. For the confined population (a) the instantaneous energy exchange is
presented, indicating the region where the mode is interacting the strongest
with the fast-ions. For the FIL (b) the impinging flux onto the 3D wall is
presented.

Figure 4: (a-b) Slice of the velocity space for the fast-ion, corresponding to
$\mu\in\left(3.0,5.0\right)\cdot 10^{-15}\rm{J/T}$. (a) Instantaneous power
exchange of the confined fast-ion population at
$t=$0.059\text{\,}\mathrm{m}\mathrm{s}$$, superimposed with the lines of
constant $E^{\prime}$. Dashed blue lines represents the divertor surface
projected on the velocity space. (b) Fast-ion loss flux (number of fast-ion
hitting the wall) for times $t>0.060\rm{ms}$. (c) Toroidal plane of the fast-
ion losses $t>0.060\rm{ms}$, where important structures have been represented.
Here $\theta=0$ represents the midplane and $\theta=-\pi$, the lower divertor.

In the presence of a wave with a given low and constant frequency,
$\omega_{n}$, and toroidal mode number $n$, the conserved quantity[4] is
$E^{\prime}=E-\frac{\omega_{n}}{n}P_{\phi}$. Contour lines with this quantity
have been superimposed in both Fig. 4(a,b). Fast-ions that drive the AE, drift
away and hit the wall following constant $E^{\prime}$ lines. As a visual
guidance, the divertor region is represented in both figures as cyan dashed
line, which is the region with the largest heat loads. In Fig. 4(c), the FIL
flux is represented in the angular plane only below the midplane ($\theta=0$),
since it is the region receiving most of the FIL flux.

A simulation with only the toroidal mode numbers $\left|n\right|=0,4$, i.e.,
filtering the rest of the Fourier components, shows that both the confined
population energy exchange is still present in the same location. The FIL in
these simulations do not differ significantly from the multi-n simulation. We
can conclude that the fast-ion transport and losses are dominated by their
interaction with the toroidal mode number $n=4$.

This analysis suggests that the fast-ion losses are predominately expulsed
from the plasma due to the energy exchange with the AE during the linear
growth phase. This would play a central role in the interpretation of the FILD
experiments in the AUG tokamak, since it connects the fast-ion loss
measurements to an energy exchange with the mode.

###  IV.2 Synthetic i-HIBP signal

The simulated AE is used here to construct the i-HIBP synthetic signal. The
radial structures, as shown in Fig. 3(b) are located in an off-axis location,
$\rho_{\text{pol}}\approx 0.55-0.75$. The total mode amplitude is of the order
of $\delta B_{r}/B_{\text{axis}}\sim 10^{-3}$, typical from NBI-driven AE
experiments in the AUG tokamak, allowing us to obtain realistic estimates of
the i-HIBP signal.

Figure 5: Strikeline representation for the equilibrium case for the
${}^{87}\mathrm{Rb}$ case. In white dots, the origin points of the secondary
has been displayed. The convention for strike line perturbations is defined in
the figure: perpendicular to the strike line (mostly in the X-direction) is
positive to the right.

The signal on the scintillator has been obtained by tracking both the primary
and secondary beam using the i-HIBPsim synthetic diagnostic, described in
Section III. The long gyro-radius (of the order of
$20\text{\,}\mathrm{c}\mathrm{m}$ in the current setup, with
$E=$70\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$$ for the
${}^{85,87}\mathrm{Rb}$, $E=$50\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$$ for
the ${}^{133}\mathrm{Cs}$; and $B=$2.5\text{\,}\mathrm{T}$$) of the heavy-ion
takes them further within the plasma where electromagnetic perturbations
modify their orbits before reaching the scintillator. In Fig 5, the strikeline
on the scintillator for the ${}^{87}\mathrm{Rb}$ is shown for the baseline
scenario, i.e., without perturbations. For visual guidance, points with the
birth $\rho_{\text{pol}}$ location are superimposed, and the sign convention
for the line perturbation is presented.

Simulations with and without the AE perturbations have been carried out in
order to assess the changes in the strike line on the scintillator. The
perturbation case has been analyzed with and without the generated AE electric
field ($\delta
E_{r}\sim$6\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{m}$$), in order to
isolate the impact of the electric field on the strike pattern. Two key
parameters are studied to determine the impact on the strike line: the
perpendicular displacement of the strikeline (perpendicular to the case
without perturbation), $\delta_{\bot}$ in Fig. 6(a); the modification of the
intensity in the strike line due to the density and temperature perturbations,
in Fig. 6(b). Both species, ${}^{85,87}\mathrm{Rb}$ and ${}^{133}\mathrm{Cs}$,
available as separate sources in the AUG i-HIBP diagnostic, have been used in
the analysis.

Figure 6: Impact on the strike line of the simulated TAE. (a) Perpendicular
displacement (with respect to the one defined in Fig. 5) of the strikeline due
to magnetic perturbation alone (solid lines) and including both electric and
magnetic perturbations (dashed lines). (b) Relative deviation in the intensity
seen in the strikeline with respect to the equilibrium. Dashed and solid lines
are superimposed in this case. For ${}^{133}\mathrm{Cs}$ case, relative
variation reaches up to $\sim 75\%$.

The relevant comparison in the strikeline is the case without electric field
(solid lines) and with the electric field (dashed lines). For both species,
the displacement caused only by the magnetic field perturbation lies in the
range of $\sim$40\text{\,}\mu\mathrm{m}$$, while the electrostatic potential
induces a deviation up to $\sim$2\text{\,}\mathrm{m}\mathrm{m}$$, above the
expected optical resolution ($\sim$100\text{\,}\mu\mathrm{m}$$) on the
scintillator. For the intensity pattern variation, Fig. 6(b), the deviation
due to the electric perturbation is negligible. The relative pattern variation
is an order of magnitude different between $\mathrm{Cs}$ and $\mathrm{Rb}$
beams, being $\approx 75\%$ and $\approx 15\%$ respectively. This systematic
deviation in intensity points to the possibility of measuring $\delta n_{e}$
due to Alfvénic modes in i-HIBP.

##  V Conclusions

In this work, the synthetic diagnostics for fast-ion losses and the i-HIBP
diagnostic have been developed and applied to a MEGA simulation. The synthetic
fast-ion loss diagnostic shows already promising results, connecting the fast-
ion losses to the interaction with Alfvénic phenomena, through $E^{\prime}$
lines. This methodology can now be extended to the rigorous study of plasma
instabilities and understanding the fast-ion losses associated to them.

The preliminary study of the synthetic i-HIBP signal for the simulated AE
shows that the radial structures may be resolved by the i-HIBP diagnostic. The
impact on the strikeline due to the electric perturbation induced by the AEs
will be measurable with the i-HIBP scintillator
($\approx$2\text{\,}\mathrm{m}\mathrm{m}$$).

## Acknowledgements and data availability

This work received funding from the European Starting Grant (ERC) from project
3D-FIREFLUC and from the Spanish Ministry of Science under Grant No.
FPU19/02267. This work has been carried out within the framework of the
EUROfusion Consortium and has received funding from the Euratom research and
training programme 2014-2018 and 2019-2020 under grant agreement No 633053.
The views and opinions expressed herein do not necessarily reflect those of
the European Commission

## Data availability statement

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

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