# Superconductor-ferromagnet hybrids for non-reciprocal electronics and
detectors

Zhuoran Geng1, Alberto Hijano2,3, Stefan Ilić2, Maxim Ilyn2, Ilari J.
Maasilta1, Alessandro Monfardini4, Maria Spies5, Elia Strambini5, Pauli
Virtanen1, Martino Calvo4, Carmen González-Orellána2, Ari P. Helenius1, Sara
Khorshidian5, Clodoaldo I. L. de Araujo5,6, Florence Levy-Bertrand4, Celia
Rogero2,7, F. Giazotto5, F. Sebastián Bergeret2,7, and Tero T. Heikkilä1 1
Department of Physics and Nanoscience Center, University of Jyväskylä, P.O.
Box 35 (YFL), FI-40014 University of Jyväskylä, Finland  2 Centro de Física de
Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU, E-20018 Donostia-San
Sebastián, Spain  3 Department of Condensed Matter Physics, University of the
Basque Country UPV/EHU, 48080 Bilbao, Spain  4 Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  5 NEST, Istituto
Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy  6
Laboratory of Spintronics and Nanomagnetism (LabSpiN), Departamento de Física,
Universidade Federal de Viçosa, Viçosa, 36570-900, Minas Gerais, Brazil  7
Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian,
Spain<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We review the use of hybrid thin films composed of superconductors and
ferromagnets for creating non-reciprocal electronic components and self-biased
detectors of electromagnetic radiation. We begin by introducing the theory
behind these effects, as well as discussing various potential materials that
can be used in the fabrication of these components. We then proceed with a
detailed discussion on the fabrication and characterization of Al/EuS/Cu and
EuS/Al/Co-based detectors, along with their noise analysis. Additionally, we
suggest some approaches for multiplexing such self-biased detectors.

###### Contents

  1. 1 Introduction
  2. 2 Fundamentals
    1. 2.1 Non-reciprocal transport in hybrid superconductor/ferromagnet structures
      1. 2.1.1 S/FI layer structures
      2. 2.1.2 Interface transport model
      3. 2.1.3 Material specific implementation
    2. 2.2 Materials and device preparation
      1. 2.2.1 Device Fabrication
      2. 2.2.2 EuS properties
    3. 2.3 Tunnel characteristics
      1. 2.3.1 Characterization and optimization of the exchange splitting
      2. 2.3.2 Spin selective tunnel junctions
      3. 2.3.3 Thermoelectricity in spin selective tunnel junctions
  3. 3 Applications
    1. 3.1 X-ray detection
      1. 3.1.1 Linear theory
      2. 3.1.2 Numerical results
      3. 3.1.3 X-ray detector design
    2. 3.2 THz detection
    3. 3.3 Multiplexing read-out of self-powered detectors
      1. 3.3.1 Principle and design
      2. 3.3.2 Planar superinductances
      3. 3.3.3 3-D lumped-element capacitance
  4. 4 Discussion and outlook

##  1 Introduction

Conventional superconducting electronics [1] rely on a combination of
supercurrent and quasiparticle current transport across superconducting wires
and different types of weak links. These combinations enable various
functionalities, such as magnetometry [2], current or voltage amplifiers [3],
voltage standards [4], and detectors based on resistance [5] or inductance [6]
that depend on the nonequilibrium state of the system. Compared to their
semiconductor counterparts, superconducting electronics lack a basic element:
non-reciprocal devices such as diodes or thermoelectric elements. The absence
of non-reciprocity can be attributed to the intrinsic electron-hole symmetry
of the superconducting state. However, this symmetry can be broken using
combinations of magnetic and superconducting elements [7, 8], which allow, in
principle, the achievement of strong non-reciprocity or thermoelectric figure
of merit. These phenomena can be employed to create superconducting spintronic
tunnel diodes [9], building blocks for superconducting logic and cryogenic
memory, or novel types of detectors, such as the superconductor–ferromagnet
thermoelectric detector (SFTED)[10], with applications in astrophysics for the
detection of the cosmic microwave background [11], and terahertz-radiation
sensing used, for example, in security imaging [12]. Remarkably, in the SFTED
the absorbed radiation directly generates the desired measurement signal,
without the need for a separate bias current or voltage.

This review is divided into two parts. The first part focuses on the
fundamentals of the building blocks for the SFTED, namely superconductor-
ferromagnetic-insulator (S/FI) bilayers. We begin with a brief account of the
underlying theory, focusing on aspects of the physics that are important for
functionalities. The full theory for the underlying transport phenomena has
been summarized elsewhere [8, 13]. We then discuss the main material
combinations used so far, along with their basic properties, and the
characterization of the superconducting state in the presence of the magnetic
proximity effect, as well as the basic non-reciprocal current-voltage
characteristics and thermoelectric signals. The second part focuses on the
realization of radiation sensors operating in two different ranges of detected
electromagnetic signals, and their readout. We provide an outlook and present
some open challenges in the last section.

##  2 Fundamentals

In this first part of the article, we concentrate on the fundamental aspects
related to the fabrication of the SFTED. It comprises three sections that
present the theory underlying thermoelectricity and non-reciprocal transport
in superconductor-ferromagnetic insulator systems, the fabrication process of
these building blocks, and their spectral characterization.

###  2.1 Non-reciprocal transport in hybrid superconductor/ferromagnet
structures

The aim of this work is to identify and realize two types of functionalities
in an electronic device at low temperatures. The first of them is non-
reciprocity. Non-reciprocal electronic transport in solid-state devices
implies an asymmetry in the current-voltage characteristic: $I(V)\neq-I(-V)$.
Non-reciprocity is associated with inversion symmetry breaking, meaning that
non-reciprocal elements usually consist of hybrid structures involving
different materials.

A typical example is provided by p-n diodes, which consist of p-type and
n-type semiconducting layers in contact with each other. In addition to the
broken inversion symmetry, the non-reciprocal electron currents require the
breaking of the electron-hole symmetry resulting from the n- and p-doping of
the two semiconducting layers. The working principle of a p-n diode can be
understood from the sketch in Fig. 1(a), which shows a positively biased p-n
junction. In the p-n junction, free electrons from the n-region diffuse into
the p-region and vice versa for the holes. The regions near the interface lose
their charge neutrality, with the negatively charged acceptor dopant atoms
remaining in the p-region and the positively charged donor dopants remaining
in the n-region, forming the depletion layer. This charged region establishes
a built-in potential that counteracts quasiparticle diffusion, leading to zero
net current at equilibrium. A forward bias allows the majority carriers of
each region to cross the depletion layer and be injected into the nearby
region, where they recombine with opposite charge quasiparticles.

For several electronics-based quantum technologies, the interesting operating
regime lies at sub-Kelvin temperatures. This poses certain challenges to
traditional semiconductor technology, and substantial effort has been
dedicated to scaling this technology down to the lowest temperatures [14, 15,
16]. Due to these challenges, research into new alternatives for non-
reciprocal electronic transport at low temperatures has intensified.

Figure 1:  Density of states (solid line) and charge carrier density (shaded
region) in thermal equilibrium for a (a) p-n junction and (b) spin-split S/N
tunnel junction. The dashed lines in panel (b) represent the density of states
of the filtered states. Here $\mu_{p}$ and $\mu_{n}$ are the quasi-chemical
potentials of the p- and n-regions, and $\mathrm{N_{A}}$ and $\mathrm{N_{D}}$
are the holes and electrons supplied by the acceptor and donor impurities,
respectively.

An alternative realization of non-reciprocity to traditional semiconducting
devices can be achieved by combining two types of correlated interacting
electronic phases: superconductivity and magnetism. This review concentrates
on hybrid heterostructures formed using these elements. We illustrate the
working principle of a superconducting spintronic tunnel diode in Fig. 1(b).
There, the spin-split superconducting state along with spin-polarized
tunneling conspire to provide unidirectional current. The quasiparticles pass
through the insulating barrier due to quantum tunneling. In the absence of
spin splitting, at $T=0$, there is no tunneling current until the applied bias
is greater than the superconducting gap $e|V|>\Delta$, since the chemical
potential difference must provide enough energy to create an excitation in the
superconductor. If particle-hole symmetry is broken due to spin-splitting and
spin filtering, for instance, the bias required for a finite current in the
forward bias configuration $eV>\Delta-h$ is smaller than in the backward bias
$eV<-\Delta-h$ ($h$ is the exchange field), so that the charge transport is
non-reciprocal.

The second functionality we discuss is thermoelectricity, where linear
electric excitation leads to heat currents, or temperature differences can be
associated with charge currents. In other words, the linear response charge
and heat currents $I,\dot{Q}$ can be expressed in terms of bias voltage $V$
and temperature difference $\Delta T$ as

$\begin{pmatrix}I\\\ \dot{Q}\end{pmatrix}=\begin{pmatrix}G&\alpha\\\
\alpha^{\prime}&G_{\rm th}T\end{pmatrix}\begin{pmatrix}V\\\ \Delta
T/T\end{pmatrix},$ (1)

where $G$ and $G_{\rm th}$ are charge and heat conductances, respectively, and
$\alpha$ is the thermoelectric coefficient. By Onsager symmetry,
$\alpha^{\prime}$ is obtained from $\alpha$ through the time-reversal
transformation. In general, $\alpha$ characterizes the degree of electron-hole
symmetry breaking in the spectrum. When $\alpha^{\prime}=\alpha$ (appropriate
for this paper), the thermoelectric effects are characterized by the
dimensionless figure of merit $ZT=\alpha^{2}/(G_{\rm th}GT-\alpha^{2})$. In
ordinary metals with a large Fermi energy ($E_{F}$) it is of the order of
$o(k_{B}T/E_{F})$, and therefore very small at sub-Kelvin temperatures.
Semiconductor materials exhibit the most significant thermoelectric effects at
around room temperature [17]. However, due to their 10-100 meV scale energy
gap, semiconductors tend to freeze out at sub-Kelvin temperatures, and the
thermoelectric efficiency is lost. Similar to the diode functionality, sub-
Kelvin thermoelectricity can be realized through the combination of
superconductors and ferromagnets, as detailed in the following sections.

Figure 2: Schematic idea for biasless detection based on thermoelectricity.
Absorbed light with power $P_{\gamma}$ heats up one part of the
heterostructure, creating a temperature difference $\Delta T$, which results
in a thermoelectric current $I=\alpha\Delta T/T\propto P_{\gamma}$.

Both non-reciprocity and thermoelectricity can be utilized in detectors. The
majority of existing superconducting detector types, such as transition edge
sensor (TES) [18, 5] and kinetic inductance detector (KID) [19] are based on
probing how the response of the material changes upon irradiation. This
requires the use of additional probing lines, which can also introduce added
noise. Moreover, such detectors tend to be rather non-linear and therefore
they can saturate relatively quickly. Conversely, using a diode or a
thermoelectric element as a sensor enables converting the detected radiation
directly into a current or voltage. Hence, such detectors are self-biased by
the radiation (as in Fig. 2).

In the following, we describe how the spin-dependent scattering at an
interface between a superconductor (S) and a ferromagnetic insulator (FI)
leads to two key phenomena underlying both the non-reciprocity and
thermoelectric transport in such systems: spin splitting of the density of
states and spin-polarized transport.

####  2.1.1 S/FI layer structures

The magnetic proximity effect concept in hybrid systems, comprising
ferromagnetic insulators and superconductors, was introduced early on by de
Gennes [20] within the realm of superconductivity theory. The first
experimental observations of spin splitting associated with the interfacial
exchange interaction in superconductors were, however, obtained much later in
Refs. [21, 22] using EuO/Al systems, followed by observations in EuS/Al
systems, also at zero external field [23]. Subsequently, other material
combinations have been explored, and the theoretical understanding has been
refined.

Figure 3:  Spin-dependent scattering and tunnelling through a ferromagnetic
insulator (FI) sandwiched between two conductors (L and R). In a simplified
band model, [24] the electron bands of the FI exhibit a gap $E_{g}$, and are
spin split by an intrinsic exchange field $h_{\mathrm{F}I}$. Electron
scattering from $L$ to $R$ at Fermi level $E_{F}$ is spin-dependent: the
reflection coefficients $r$ acquire a spin-dependent phase shift $\Theta$, and
the transmission is polarized to favor one spin direction. Top right: The
magnetic proximity effect can also be interpreted through interaction with
local moments at the FI interface. The physics is greatly influence by the
interface quality.

Electron transport at ferromagnetic insulator interfaces is explained by a
scattering model as introduced in Ref. [24] and further discussed in [25, 26].
Figure 3 illustrates this with a ferromagnetic insulator layer between two
conductors. At the FI interface, an incident electron with spin
$\sigma=\uparrow,\downarrow$ can reflect back, exhibiting a spin-sensitive
amplitude given by $r_{\uparrow/\downarrow}=r_{\uparrow/\downarrow}^{0}e^{\pm
i\Theta/2}$. In a thicker FI layer, while the reflection probabilities for
both spins are nearly one, the amplitudes differ by a scattering phase shift
$e^{i\Theta}$, where $\Theta$ is called the spin-mixing angle. If conductor L
is a clean thin film of thickness $d$, electron eigenstates in it are given by
a quantization condition $e^{2ik_{z}(\epsilon_{\uparrow/\downarrow})d\pm
i\Theta/2+i\phi_{0}}=1$, where $\phi_{0}$ is a spin-independent phase shift.
Consequently, $\Theta$ induces a spin splitting

$\displaystyle\bar{h}$
$\displaystyle=\frac{\epsilon_{\uparrow}-\epsilon_{\downarrow}}{2}\approx\frac{\Theta}{4d}\left|\frac{dk_{z}}{d\epsilon}\right|^{-1}=\frac{v_{F}\cos(\alpha)\Theta}{4d}\,,$
(2)

where $k_{z}=k_{F}\cos\alpha=\sqrt{2m^{*}\epsilon-k_{\parallel}^{2}}$ is the
incident momentum of the electron hitting the FI interface at angle $\alpha$,
$|\Theta|\ll 1$, and $m^{*}$ is the effective mass. The splitting depends
inversely on the thickness $d${, as predicted by de Gennes [20]. A scattering
calculation in such a band model, [24] assuming the FI has a band gap $E_{g}$
and an internal exchange field $h_{\mathrm{FI}}$, gives the spin-mixing angle
$\Theta\approx
2[\sqrt{E_{F}/E_{g}}+\sqrt{E_{g}/E_{F}}]^{-1}(h_{\mathrm{FI}}/E_{g})\cos\alpha=\Theta_{0}\cos\alpha$
for $E_{g}\gtrsim{}E_{F}$, which is proportional to $h_{\rm FI}$. On the other
hand, the transmission amplitudes
$t_{\uparrow/\downarrow}\propto{}\exp\bigl{[}-d_{\mathrm{FI}}\sqrt{2m^{*}(E_{g}\pm{}h_{\mathrm{FI}})+k_{\parallel}^{2}}\bigr{]}$
are spin-dependent and the spin-filtering polarization
$\mathcal{P}=(|t_{\uparrow}|^{2}-|t_{\downarrow}|^{2})/(|t_{\uparrow}|^{2}+|t_{\downarrow}|^{2})$
increases by increasing the FI thickness $d_{\mathrm{FI}}$, which is also
observed [27, 28, 29, 30]. Qualitatively, this simplified model predicts then
both the presence of the magnetic proximity effect and the spin-polarized
tunneling, even though its quantitative accuracy is limited [24, 31].

The effect of FI interfaces on superconductors can also be described by
modeling the exchange interaction between the conduction electrons and the
localized magnetic moments $\bm{S}_{\bm{r}}$ at the FI interface, [32, 26]

$\displaystyle H_{\rm ex}=-J_{\rm
ex}\sum_{\bm{r}\alpha\beta}\Psi^{\dagger}_{\alpha}(\bm{r})(\bm{S}_{\bm{r}}\cdot\bm{\sigma})_{\alpha\beta}\Psi_{\beta}(\bm{r})\,,$
(3)

where $J_{\rm ex}$ is an effective parameter describing the interaction
strength. Semiclassical theory suggests the generation of a localized
interfacial Zeeman field for conduction electrons, represented as $h_{\rm
int}(z)=h_{\rm ex}^{\prime}\delta(z)$, complemented by spin-flip scattering.
[26] Such $\delta$-function model can match experiments reasonably well [33].
Here $h_{\mathrm{ex}}^{\prime}=-J_{ex}n_{m}\langle{S}\rangle/\hbar$, with
$n_{m}$ the 2D surface density of magnetic moments and $\langle{S}\rangle$
their average spin. [34] In certain limits, this model can be mapped to the
scattering model, as the scattering matrices can be incorporated to boundary
conditions of semiclassical transport equations [25, 24, 35, 36, 37]. These
models allow for a convenient description of FI/superconductor(S) hybrid
systems, also in cases where the superconductors are not ideal, e.g.,
containing potential or magnetic impurities.

One way to probe the magnetic interface properties experimentally is via spin
transport measurements [38, 39]. In the scattering model, these properties are
often characterized by the complex-valued spin-mixing conductance
$G_{\uparrow\downarrow}$. The spin-dependent current incoming to the interface
is
$I_{\mathrm{in}}^{\sigma\sigma^{\prime}}\propto(a_{\mathrm{in}}^{\sigma^{\prime}})^{*}a_{\mathrm{in}}^{\sigma}$,
where $a_{\mathrm{in}}$ are the incoming spin-dependent scattering amplitudes.
The difference compared to the reflected current is [36, 38]

$\displaystyle
I_{\mathrm{in}}^{\sigma\sigma^{\prime}}-I_{\mathrm{refl}}^{\sigma\sigma^{\prime}}=[1-(r_{\sigma})^{*}r_{\sigma^{\prime}}]I_{\mathrm{in}}^{\sigma\sigma^{\prime}}=g_{\sigma\sigma^{\prime}}I_{\mathrm{in}}^{\sigma\sigma^{\prime}}\,.$
(4)

The spin-mixing conductance
$G_{\uparrow\downarrow}=\frac{e^{2}}{2\pi\hbar}\sum_{nn^{\prime}}[\delta_{nn^{\prime}}-(r^{nn^{\prime}}_{\uparrow})^{*}r^{nn^{\prime}}_{\downarrow}]$
is defined by summing $g_{\uparrow\downarrow}$ over all scattering channels
$n$, and describes the absorbed spin current transverse to the magnetization
of the FI. In the simple interface scattering model, and assuming complete
reflection $|r_{\sigma}|=1$, we have $g_{\sigma\sigma}=0$ and
$g_{\uparrow\downarrow}=g^{r}_{\uparrow\downarrow}+ig^{i}_{\uparrow\downarrow}$,
$g^{r}_{\uparrow\downarrow}=1-\cos\Theta$,
$g^{i}_{\uparrow\downarrow}=\sin\Theta$. For $|\Theta|\ll 1$,
$g_{\uparrow\downarrow}\approx{}i\Theta$ is imaginary, and the spin-mixing
conductance per square area is found to be [35, 37, 34]

$\displaystyle\mathop{\mathrm{Im}}G_{\uparrow\downarrow}/\mathcal{A}\approx\frac{e^{2}}{\hbar}N_{F}h_{\rm
ex}^{\prime}\approx\frac{e^{2}}{\hbar}\frac{k_{F}^{2}}{16\pi^{2}}\Theta_{0}\,,$
(5)

where $N_{F}$ is the Fermi level density of states per spin. That the
imaginary part of $G_{\uparrow\downarrow}$ is associated with the interfacial
exchange holds also more generally [35, 34]. Physically, the absorbed spin
current produces spin torque acting on the FI magnetization, leading to
phenomena observable for example in ferromagnetic resonance experiments; see
Ref. [38] for a review. Furthermore, in conjunction with the spin-Hall effect,
it provides an observable contribution to the magnetoresistance.[40, 41, 39]

The magnetic proximity effect in S/FI bilayers can also be observed [21, 31,
29, 33] by probing the density of states (DOS) $N_{\sigma}(E)$ in S, as the
effective exchange field $\bar{h}$ from (2) splits the BCS peaks in the DOS.
In ideal superconductors, $N_{\sigma}=N_{\mathrm{BCS},\sigma}$, the splitting
is

$\displaystyle
N_{\mathrm{BCS},\sigma=\uparrow/\downarrow}(E)=\mathop{\mathrm{Re}}\frac{-i(E\pm\bar{h}+i0^{+})}{\sqrt{\Delta^{2}-(E\pm\bar{h}+i0^{+})^{2}}}\,.$
(6)

Although the splitting $\bar{h}$ in the ballistic model depends on the
momentum direction, it is expected to be averaged by disorder of the bulk and
the interface, and by Fermi liquid interactions [24, 31, 26, 42]. For thin
films, in this case one expects [26]

$\displaystyle\bar{h}\approx\frac{h_{\rm ex}^{\prime}}{d}$ (7)

which scales similarly with film thickness $d$ as in the ballistic limit.
Experimental evidence is compatible with momentum independent $\bar{h}$ [31].

Experimentally, $N_{\sigma}$ can be probed via spin-polarized tunneling [29,
30]. Tunneling current through a spin-polarized contact to a spin-split
superconductor is

$\displaystyle I(V)$
$\displaystyle=\sum_{\sigma=\uparrow,\downarrow}\frac{G_{\sigma}}{e}\int_{-\infty}^{\infty}dE\,N_{\sigma}(E)[f_{S}(E)-f_{N}(E+eV)]\,,$
(8)

where $f_{N,S}$ are Fermi distribution functions on the superconductor/normal
sides of the tunnel junction, and $G_{\sigma}$ are the spin-dependent
tunneling conductances. In S/FI/N junctions, where the ferromagnetic insulator
also serves as the tunnel barrier, the spin-dependent conductances
$G_{\sigma}\propto{}|t_{\sigma}|^{2}$ inherit spin-dependence from the
tunneling amplitudes $t_{\uparrow/\downarrow}$. In F/I/S/FI structures, where
the tunnel electrode itself is ferromagnetic, the spin-dependence in
$G_{\sigma}\propto{}N_{F,\sigma}$ is influenced by its spin-dependent density
of states. However, interface properties generally play a significant role as
well [43, 30].

Usually, experimental $I-V$ characteristics do not match Eq. (8) with the
ideal BCS density of states of Eq. (6). These discrepancies can be addressed
by considering the non-idealities of the superconductor that alter
$N_{\sigma}$, as discussed below. Other contributing factors may include the
structure of the tunnel junction, such as pinholes and other irregularities,
as well as features of the experimental setup like voltage fluctuations [44].
To account for these features phenomenologically, one can introduce the Dynes
parameter $\Gamma>0$ and replace $E$ with $E+i\Gamma$ in Eq. (6). We return to
these points in Sec. 2.3.

The phase diagram of thin spin-split superconducting films has been
extensively studied, theoretically [45, 46, 47, 26] and experimentally [48,
49, 26]. A main feature at low temperatures is the presence of a first-order
transition between superconducting and normal states, which in the ideal case
occurs at the Chandrasekhar–Clogston value of the exchange field
$h_{c}=\Delta/\sqrt{2}$. This is expected to be affected by factors such as
Fermi liquid interactions, and short spin-flip $\tau_{sf}$ or spin-orbit
$\tau_{so}$ scattering times in the superconductor which generally reduce the
effect of the exchange field [47, 50, 13] and also change the tunneling DOS.

Figure 4:  (a) Critical exchange field for different thicknesses $d$ of the
superconductor and (b) critical thickness for different interfacial exchange
fields $h_{\rm ex}^{\prime}$. The dashed lines represent the temperatures at
which the transition is of first order. The inset in panel (a) shows effective
exchange field $\bar{h}$ (full line) and the thin film approximation (7)
(dashed line) as a function of the thickness of the S layer for $h_{\rm
ex}^{\prime}=0.5\Delta_{0}\xi_{S}$.

The thickness dependence of the average exchange field (2) and (7) implies
there is a critical thickness $d_{c}$ of the S film, defined by
$\bar{h}(d_{c})=h_{c}$, below which the film is not superconducting.
Prediction from a semiclassical model [51] is shown in Fig. 4(a) for the
critical interfacial exchange field, and in Fig. 4(b) for the critical
thickness. Inset of Fig. 4(a) shows $\bar{h}(d)$, which is well approximated
by Eq. (7) for thicknesses $d$ smaller than the coherence length $\xi_{S}$.
For $d\gtrsim 3\xi_{S}$, it becomes weak enough allowing superconductivity at
zero temperature for any value of the interfacial field.

The magnetic properties of the FI also have important consequences for the
physics. Generally, the form of the magnetic hysteresis is determined by them,
and magnetic details of the interface can matter for the temperature
dependence. Additionally, for bilayers with lateral dimensions surpassing the
domain sizes in the FI, spatial variations in the exchange field introduce
complexities that must be considered[52, 26, 53, 54, 33, 51, 55]. Moreover,
the magnetic fringe fields of the FI lead to orbital depairing in the
superconductor, generally suppressing superconductivity and rounding sharp
superconducting features in its density of states.

There are numerous additional factors that influence the experimental
outcomes. The physics is very sensitive to the interface quality: on the one
hand, conduction electrons that scatter back before interacting with localized
FI magnetic moments, do not acquire magnetic phase shifts or spin splitting.
On the other hand, e.g. subgap bound states at the interface are predicted
also to enhance the spin mixing. [56, 57] Hence, the behavior of the magnetic
proximity effect is sensitive to the materials and growth conditions
influencing the interface physics. We return to these points in Sec. 2.1.3 and
Sec. 2.2.

####  2.1.2 Interface transport model

The interplay of spin-splitting and spin-filtering in FI/S structures is
predicted to lead to electron-hole symmetry breaking [7, 58, 8, 13]. Namely,
in Eq. (8), due to the combination of spin-dependence of the conductance
($G_{\uparrow}\neq{}G_{\downarrow}$), and the spin-dependent asymmetry of the
density of states ($N_{\sigma}(-E)=N_{-\sigma}(E)\neq{}N_{\sigma}(E)$), it is
possible to have $I(-V)\neq{}-I(V)$. These features enable both thermoelectric
effects and non-reciprocal charge transport.

Figure 5:  (a) $I(V)$ curve of a N/FI/S junction for different values of the
polarization $\mathcal{P}$. (b) Rectification coefficient $R=I_{\rm
sym}/I_{\rm asym}$. Parameters used in the plots: $h=0.3\Delta$,
$T_{N}=T_{S}=0.1\Delta$, $\Gamma=10^{-3}\Delta$.

##### Thermoelectricity.

For tunnel junctions, main features of spin-split superconductor
thermoelectric effects are expected to follow Eq. (8). Experiments [59, 60]
indeed match its predictions well. For small temperature differences between
the normal and superconducting sides of the tunnel junction, $\Delta
T=T_{S}-T_{N}\ll T$, low voltages $eV<\Delta-h$, and low temperatures
$k_{B}T\ll\Delta,h$, Eq. (8) reduces to [7, 61]

$I(V,\Delta T)=-\alpha\frac{\Delta T}{T}+I_{S}\left(e^{eV/k_{B}T}-1\right)\\\
+I_{S}(\mathcal{P}-1)\left(\cosh\frac{eV}{k_{B}T}-1\right).$ (9)

The first term describes the thermoelectric effect. Here,
$\mathcal{P}=(G_{\uparrow}-G_{\downarrow})/(G_{\uparrow}+G_{\downarrow})$ is
the spin polarization of the tunnel junction, and $\alpha$ is the
thermoelectric coefficient, which in the ideal case (6) has the value
$\alpha\approx\mathcal{P}(G_{T}/e)\sqrt{2\pi\tilde{\Delta}}e^{-\tilde{\Delta}}[\Delta\sinh(\tilde{h})-h\cosh{\tilde{h}}]$,
[7] where $\tilde{\Delta}=\Delta/k_{B}T$ and $\tilde{h}=h/k_{B}T$. Moreover,
$I_{S}=G_{T}/eK_{1}(\tilde{\Delta})e^{\tilde{h}}$, where $K_{1}$ is a modified
Bessel function.

Thermoelectric applications generally require, in addition to the description
of the electrical circuit where the thermoelectric element is embedded, also
modeling the induced temperature difference $\Delta T$. Generally, it depends
on the various possible thermal conductance channels. Tunneling of electrons
is one of these and can also be described in the above approach, [8, 13] but
it is often not the most significant contribution. In particular, the
electron-phonon coupling often dominates in a wide temperature range, see
Refs. [62, 13] for a review. Generally, all heat conductances are
exponentially suppressed in the superconducting state, due to the presence of
the superconducting gap. This is why, in the presence of radiation with
frequency larger than the gap, superconductors tend to heat up more than
normal metals. A combined electrothermal theory for S/FI junction detector
applications is discussed in more detail in Sec. 3.1.

##### Rectification.

The second and the third term in Eq. (9) describe the non-reciprocal charge
transport in the form of a non-ideal Shockley diode equation. Namely, the
second term is the ideal Shockley contribution, while the third term describes
deviations from ideal behavior, and vanishes at perfect polarization
$\mathcal{P}=1$. The I-V characteristic is shown in Fig. 5(a). The
rectification can be characterized by decomposing the current into
antisymmetric $I_{\rm asym}=\frac{1}{2}|I(V)-I(-V)|$ and symmetric $I_{\rm
sym}=\frac{1}{2}|I(V)+I(-V)|$ parts in voltage, and defining the rectification
coefficient

$\mathcal{R}=\frac{I_{\rm sym}}{I_{\rm asym}}.$ (10)

The maximum of $\mathcal{R}(V)$ is determined by the spin-polarization of the
junction, as shown in Fig. 5(b), which provides a way to estimate the
polarization parameter from the experimental $I(V)$ curves.

In the presence of an AC voltage $V_{ac}$, N/FI/S junctions produce a
rectified DC current, which at high frequencies and low temperatures
($\hbar\omega\gg\Delta,h,eV_{ac}$ and $k_{B}T\ll\Delta-h$) has the form [61]

$I_{rc}=\frac{\mathcal{P}G_{T}}{2e}\frac{e^{2}V_{ac}^{2}}{\hbar^{2}\omega^{2}}h.$
(11)

Note that the sign of the rectified current is determined by the relative sign
of $h$ and $\mathcal{P}$, namely,
$\text{sign}(I_{rc})=\text{sign}(h\mathcal{P})$. Similarly, for the current
produced by the thermoelectric effect, $I_{th}=-\alpha\frac{\Delta T}{T}$, we
have $\text{sign}(I_{th})=-\text{sign}(\Delta T\alpha)=-\text{sign}(\Delta
T\mathcal{P}h)$. Therefore, $I_{th}$ and $I_{rc}$, have the same sign if
$\Delta T<0$, meaning that the normal side of the junction is at a higher
temperature. On the other hand, if the superconducting side of the junction is
at a higher temperature ($\Delta T>0$), the two currents have opposite signs,
so the rectification and thermoelectric effects compete. Therefore, to produce
the maximal signal using FI/S junctions in e.g. radiation detectors, special
care must be taken to optimize the ratio of rectification and thermoelectric
effects, which depends on material-specific parameters and junction geometry.

The results above assume ideal I-V characteristics. As noted in Sec. 2.1.1,
several sources of nonideality can be phenomenologically described by
introducing the Dynes parameter $\Gamma$ in the superconducting density of
states. It enables subgap conduction channels, which can provide significant
contributions at very low temperatures when
$\frac{\Gamma}{\Delta}\gtrsim\frac{\Delta}{k_{B}T}e^{-\Delta/k_{B}T}$. In
particular, the correction to the tunneling current due to small $\Gamma$ at
$k_{B}T,h,eV\ll\Delta$ is [61]

$\delta I=G_{T}\frac{\Gamma
V}{\Delta}\bigg{(}1+\frac{3}{2}\frac{eVh}{\Delta^{2}}\bigg{)},$ (12)

yielding the rectification coefficient of $R\approx
3\mathcal{P}eVh/(2\Delta^{2})$. Similarly, nonzero $\Gamma$ also affects the
thermoelectric response. At low temperatures $k_{B}T\ll\Delta-h$ and for small
$\Gamma$ the thermoelectric coefficient obtains an extra contribution

$\delta\alpha=\frac{G_{T}h\pi^{2}k_{B}^{2}T^{2}\Delta^{2}\Gamma}{e(\Delta^{2}-h^{2})^{5/2}}.$
(13)

On the other hand, the thermoelectric voltage (or Seebeck coefficient),
represented as $-\alpha k_{B}\Delta T/(eG)$, generally decreases due to $G$
increasing more than $\alpha$ [7]. Besides the Dynes parameter, somewhat
similar effects on the thermoelectric coefficients are provided by spin-flip
scattering as described in [63].

##### Detector yield.

The efficiency of conversion of absorbed radiation can be characterized by the
quantum yield, i.e., the number $n_{\rm th}(\omega)$ of electrons transported
across the junction per absorbed photon at frequency $\omega$. For steady-
state thermoelectric conversion, the estimation can be made using Eq. (9) and
a heat balance model

$a_{S/N}P_{\gamma}=G_{\rm th}^{S/N}(T_{S}-T_{\rm bath})\pm G_{\rm
th}(T_{S}-T_{N}).$ (14)

This describes fractions $a_{S}$ and $a_{N}$ of the absorbed power
$P_{\gamma}=\hbar\omega/\delta t$ distributed on the S and N sides in time
$\delta t$, and relaxing via junction and parasitic thermal conductances
$G_{\rm th}$ and $G_{\rm th}^{S/N}$. Now, assume that $k_{B}T\ll\Delta,h$ so
that $G_{\rm th}^{S}\ll{}G_{\rm th}\ll{}G_{\rm th}^{N}$, [62, 13] and that
$\hbar\omega\gtrsim 2\Delta$ so that absorption $a_{S}$ is non-negligible.
Using the low-temperature values of the junction transport coefficients,

$n_{\rm th}=I_{\rm th}\delta t/e\approx
a_{S}\mathcal{P}\frac{\hbar\omega}{\Delta-h}\mathop{\mathrm{sgn}}(h),$ (15)

for $\Gamma\lesssim{}e^{-\Delta/k_{B}T}\Delta^{2}/h$. In the opposite limit of
large $\Gamma$, $n_{\rm th}\approx 3a_{S}\mathcal{P}\hbar\omega
h/(\Delta^{2}-h^{2})$ which is smaller by only a factor of $h/\Delta$. Hence,
absorption of a single photon fairly generally transports $n_{\rm th}>1$
electrons across the junction.

A similar estimate can be made for the rectification effect, assuming that
fraction $a>0$ of the incoming power $P_{\gamma}$ generates an AC voltage
across the N/FI/S junction, $aP_{\gamma}=V_{ac}^{2}/(2\tilde{R})$. Here
$\tilde{R}=R/(1+RG_{T})$ is the parallel resistance of the junction and a
radiation absorber of resistance $R$. Using Eq. (11),

$n_{\rm rc}=I_{\rm rc}\delta
t/e=a\mathcal{P}\frac{h}{\hbar\omega}G_{T}\tilde{R}.$ (16)

Here $n_{rc}<1$ at $\hbar\omega\gtrsim\Delta$, indicating that the
thermoelectric effect can dominate AC rectification, and lead to better device
performance.

####  2.1.3 Material specific implementation

The results above highlight that for practical purposes it is desirable to
find suitable superconductor/ferromagnet combinations. Thermoelectricity and
non-reciprocal effects are maximized by having, on one hand, a good spin-
filter efficiency and, on the other hand, a sizable spin-splitting of the
superconductor. Both of these effects depend crucially on interfacial
parameters. As discussed in the next section, very good experimental control
over spin split superconductivity has been achieved in EuS/Al systems [29, 64,
31, 65, 33, 66, 9]. Crucial for the splitting is the size of the imaginary
part of the mixing conductance, as explained in section 2.1.1. This value can
be notably large at EuS/metal interfaces, as demonstrated in spin Hall
magnetoresistance experiments on EuS/Pt systems [67], which found $h_{\rm
ex}^{\prime}\approx{}0.8\,\mathrm{meV\,nm}$, as extracted from measured
$\mathop{\mathrm{Im}}G_{\uparrow\downarrow}$ using Eq. (5). This value
corresponds to the superconducting gap $\Delta_{\mathrm{Al}}$ of aluminum with
a film thickness of $d\approx 4\,\mathrm{nm}$. In EuS/Al the induced effective
exchange field can range from values much smaller than $\Delta_{\mathrm{Al}}$,
up to such large values that superconductivity is fully suppressed, as
observed in [68].

Recent experiments also suggest a finite induced exchange field in YIG/NbN
[69] and YIG/Nb [70, 71], even though the spin-mixing conductance of YIG/metal
interface is dominated by the real part and not the imaginary part, in
contrast to EuS based structures [67].

With respect to spin-selective transport, Eu-chalcogenides serve as highly
efficient spin-filtering barriers. Efficiencies larger than 95% have been
achieved in EuSe barriers, whereas 80% spin-filter efficiency has been
achieved in EuS barriers [72]. However, a fundamental challenge arises when
using these magnetic barriers in several of the devices discussed herein:
achieving high spin filter efficiency necessitates thick barriers, which
subsequently exhibit substantial resistances. ”One way to circumvent this
issue is to utilize a ferromagnet as the normal metal electrode in lieu of a
magnetic barrier. [60, 9]. Indeed the thermoelectric effect has been observed
in EuS/Al/Co [9], EuS/Al/Fe [60], and in Al/Fe tunnel junctions [59]. In the
latter, an external magnetic field was applied to provide the spin-splitting
in the superconductor. A spin filtering efficiency of 75% was reported in
NbN/GdN/NbN Josephson junctions [73], and a sizable interfacial exchange
coupling was seen in Nb/GdN samples [74].

Table 1:  Summary of reported values for spin filtering efficiency (barrier
polarization, $\mathcal{P}$) and exchange splittings ($\bar{h}$) across
different experimental studies. Color coding indicates  ferromagnets and
spin-split superconductors.

  * Material combination | Barrier polarization $\mathcal{P}$ | Exchange splitting $\bar{h}$ @ applied field $B$ | Reference
---|---|---|---
EuO/Al/AlO3/Al | no spin-filter barrier | 1 T @ 0.1 T …1.73 T @ 0.4 T | [21]
Au/EuS/Al | 0.8 | 1.6 T @ 0 T | [23]
Al/EuS/Al | 0.6-0.85 | 1.9-2.6 T @ 0 T | [64]
Ag/EuSe/Al |  $>$ 0.97 | 0 T @ 0 T | [75]
EuSe/Al/AlO3/Ag | no spin-filter barrier | 4 T @ 0.6 T | [75]
NbN/GdN/NbN | 0.75 | $-$ | [73]
NbN/GdN/TiN | 0.97 | 1.4 T @ 0 T | [76]
EuS/Al/Fe | 0.15-0.20 | 1.5 T @ 0.4 T | [60]
Cu/EuS/Al | 0.48 | 1.83 T @ 0.1 T | [9]
EuS/Al/AlOx/Co | 0.3 | 1.67 T @ 0 T | [9]
EuS/Al/AlOx/Al | no spin-filter barrier | 2.52 T @ 0 T | [51]

Table 1 summarizes various experimental studies on FI/S structures, providing
values for the spin filtering efficiency and the spin-splitting field.

###  2.2 Materials and device preparation

For cryogenic thermoelectric detectors and non-reciprocal electronics that
utilize spin-polarized tunneling junctions with exchange-split superconducting
electrodes, there are application-dependent considerations that for example
require maximizing the amplitude of the thermoelectric effect, which depends
on the splitting of DoS, or setting limits on the tunnel junction resistance
to achieve rapid readout. This implies having good control on the properties
of the devices constituent material as well as the growth parameters of the
heterostructures. In this section, we focus on the elements concerning the
interface between a superconducting thin film and a magnetic insulator or
semiconductor that creates a spin-split density of states. Application-
dependent issues are discussed in the corresponding sections. In the early
works of Meservey and Tedrow, many different material combinations were
tested, including chalcogenides of rare-earth elements as well as very thin
metallic layers interfaced with Al and V layers (for a review see [29]). The
effect of the exchange splitting is usually explored via tunneling
spectroscopy. Spin splittings corresponding to effective magnetic fields of
the order of several Tesla have been achieved, in particular using
ferromagnetic insulators. The splitting decreases with the increase in
thickness of the superconducting film in close accordance with the predictions
of the de Gennes’ theory [20]. In the case of metallic magnetic layers
adjacent to superconducting films the situation is more subtle since Cooper
pairs can leak into the magnetic region leading, in general, to a suppression
of superconductivity. For this reason, in order to obtain a good splitting, it
is more convenient to use magnetic insulators. Many different ferromagnetic
insulators (FI), including transition metal oxides or rare earth oxides, such
as gadolinium gallium garnet (Gd3Ga5O12) and yttrium iron garnet (Y3Fe5O12)
were tested. However, EuS was the first FI where spin filtering was determined
[23], which has implications in magnetic tunnel junctions and spin valves
[66], as well as in superconducting electronics, and radiation sensors [64,
60, 77, 23, 78, 51].

To obtain reliable and reproducible amplitudes of the exchange splitting, it
has been demonstrated that the quality of the interfaces is crucial [64].
Introducing a thin insulating layer between the magnetic and superconducting
films results in a complete quenching of the exchange interaction [64].
Moreover, in many examples where lithography-based nano-fabrication was used
to create devices with exchange split DOS, especially based on the combination
of Al and EuS [49, 79], the exposure of EuS to atmosphere before the
deposition of the superconductor prevents maintaining the exchange splitting
without external magnetic field even after the initial saturation of the
magnetization below the Curie temperature of the EuS. These findings suggest
that the EuS was partially oxidized during the lithographic nanofabrication,
which causes formation on the interface of the paramagnetic Eu-based compound
with zero magnetic remanence. In contrast, in several references of devices
prepared by Moodera and coworkers, a zero-field exchange split DoS was
demonstrated in the absence of any external magnetic field [80, 33]. The
explanation lies in the method of device fabrication, which is in-situ shadow
mask evaporation. In the early works of Meservey and Tedrow, the crossed-wire
tunneling junctions used in tunneling spectroscopy experiments were created by
subsequent deposition of the FI and S materials using different shadow masks
in a high vacuum environment (1x10-8 mbar) and preventing the exposition of
the layers to the air [21, 64]. More recently, it has been shown that the EuS
sublimation temperature also plays a significant role, with a trend toward Eu-
rich compounds at higher sublimation temperatures [81]. This modification to
the chemical composition is mirrored in the magnetic properties, with an
increase in magnetic moment and Curie temperature. Therefore, the need for
precise control of the composition and cleanliness of the interface between
the superconductor and the magnetic layer places constraints on the
fabrication techniques for tunneling junctions.

To illustrate the growth of the required heterostructures for fabricating
cryogenic thermoelectric detectors or non-reciprocal electronics, in the
subsequent paragraph, we detail the shadow mask fabrication procedure of the
EuS/Al-base devices described in Ref. [51], which was similar to the one
summarized long time ago in the review by Meservey and Tedrow[29]. To
complement it, we discuss those aspects relevant to understand and control the
device performance. These example potentially serve as a foundational elements
for a SFTED.

Figure 6: Schematic representation of the different steps involved in the
hybrid heterostructure growth.

####  2.2.1 Device Fabrication

As we mentioned before, in Ref. [51] the procedure to fabricate X/AlOx/Al/EuS
tunnel junctions was essentially shadow mask fabrication. Here, X denotes a
metallic layer, either superconducting or magnetic. The sequence is
schematically represented in Fig. 6, and it includes: i) the deposition of the
magnetic EuS on the insulating substrate via electron-beam (e-beam)
sublimation from stoichiometric powder; ii) the growing of the superconducting
Al electrode through the shadow mask; iii) the formation of the tunneling
barrier via the oxidation in the oxygen plasma; iv) the deposition of the
counter electrodes X through another shadow mask; and v) finally deposition of
the capping layer to protect the devices from the atmosphere. All the process
is done without breaking the vacuum at any time, and therefore preventing the
undesirable oxidation or contamination of the layers and interfaces.

The substrates for fabricating cryogenic thermoelectric detectors or non-
reciprocal electronics must be good insulators, such as high-quality polished
silica or SiN. Thin film of EuS layer is deposited as a continuous film by
employing a Knudsen cell (Tungsten crucible) filled with commercially
available stoichiometric EuS powder, as Smoes and coworkers demonstrated in
1977 [82]. During the sublimation process, the crucible temperature can be
estimated via the power vs temperature calibration curve and the growth rate
is calibrated using a quartz microbalance, or monitored via a flux monitor.
This is relevant for sample preparation because EuS powder decomposes during
evaporation and the ratio Eu:S in the gas phase varies with increasing
temperature, which affects its magnetic behaviour [82]. Superconducting and
magnetic metallic wires, Al and Co in example of Fig. 6 must be deposited over
the EuS layer in the same vacuum chamber, i.e. without breaking the vacuum, in
order to avoid the oxidation of the EuS surface. There are several methods to
deposit aluminum layers in vacuum environments, being the thermal evaporation
used in Ref. [51] one of the most convenient, because of its reproducibility,
its cleanness and its precise control of film thickness and composition.
Although for the SFTED, the second metallic electrode needs to be a
ferromagnet, as for example Co, for the characterization of the spin-splitting
induced by the EuS in the Al wire, it is convenient to first grow an
Al/AlOx/Al/EuS structure [23, 64, 33, 51].

Regarding oxide barrier fabrication, the conventional method to grow the AlOx
barrier is thermal oxidation, where aluminum is directly exposed to air (or
pure molecular oxygen), resulting in many instances in amorphous layers that
lead to inhomogeneous interfaces [83]. Moreover this method is self-limited in
the barrier thickness, being only possible to grow barriers of around 2 nm
[84]. Among alternative methods to produce tunnel barriers, the most popular
is the oxygen glow discharge plasma oxidation that employs atomic oxygen [85]
instead of molecular one to produce much thicker barriers in shorter time.
However, this is an aggressive technique because the oxygen atoms that arrive
to the sample have high kinetic energy and can create defects in the AlOx
layer. This drawbacks can be overcome by Inductively Coupled Plasma sources,
which produce barriers with very low amount of pinholes and high
reproducibility [86, 87].

Typical tunneling spectroscopy measurements of the Al/AlOx/Al/EuS junctions
shown in Fig. 7 revealed the spin-split DoS of the bottom Al electrode even in
the absence of an external magnetic field. The characteristic splitting of the
differential conductance ($dI/dV$) curves becomes more pronounced when the
magnetization of the sample is saturated, and remains almost intact when the
field is reduced back to zero. Work [81] discussed the possibility of a
systematic control of the exchange coupling strength via tuning of the
parameters of sublimation of EuS. The control of the effective exchange field
is essential to optimize the SFTED.

####  2.2.2 EuS properties

Early works aimed at the growth of thin films of EuS unveiled a strong
variation in their chemical composition associated with different deposition
conditions [88, 89]. Since the evaporation of sulfur, which is required for
Molecular Beam Epitaxy (MBE), is technically challenging, the preferred
technique for growing EuS is sublimation from stoichiometric powder. Detailed
study of the thermodynamics of this process showed that solid EuS decomposes
due to the heating and the proportion of Eu to S in the gas phase changes
slowly with increasing temperature from nearly 1:1 ratio at 1400 K to the Eu-
reach mixture at temperatures above 2000 K [82]. Usually, the temperature of
sublimation is kept high to obtain higher growth rate. Excess of Eu in the
deposited films is compensated by means of heating of the substrate. For
instance, it has been demonstrated that increasing the temperature to 630 K
results in near-stoichiometric films, while cooling the substrate promotes the
growth of Eu-rich films [88, 89, 90]. In the pioneering works of Moodera’s
group on the fabrication of the exchange-coupled Al/EuS bilayers, EuS (and Al)
are usually deposited on the substrates cooled down with liquid nitrogen
because it results in higher magnetic remanence (more rectangular hysteresis
loops), which is an advantage for the applications that require persistence of
the exchange-coupling between the EuS and Al in zero magnetic field. A recent
work, Ref. [81], has shown that deposition of the EuS on the substrates kept
at room temperature or cooled with liquid nitrogen has to be accompanied by
the control of the sublimation temperature in order to limit the exchange
coupling strength in the Al/EuS - based devices. It was found that the
interaction gets stronger with the increase in Eu concentration. It quickly
surpasses the critical value, quenching the superconductivity of the thin Al
films, once the Eu/S ratio becomes higher than unity [81]. Control of the
power of the e-beam evaporators allows for precise manipulation of the EuS
sublimation temperature, which opens up a way to the tuning of the exchange
coupling. For instance, in the series of samples listed in Table 2, effective
exchange field variations of up to a factor of three are obtained by tuning
the layer thicknesses and/or the growth temperatures.

The capability to control the exchange coupling strength in the Al/EuS
bilayers enables the validation of the theoretical description of the
proximity effect for hybrid S/FI systems presented in Section
2.1.1.Furthermore, by tuning the exchange coupling strength to be near the
critical value, it becomes possible to investigate the magnetization reversal
process in EuS films across a temperature range below 1.2 $K$. This range is
typically inaccessible for the majority of magnetic measurement systems [51].
Figure 7 (c) presents typical results for tunneling spectroscopy of the
Al/AlOx/Al/EuS tunnel junction. In this spectrum, the characteristic splitting
induced by the exchange interaction between the superconductor and the
magnetic insulator is observable within a narrow range of external magnetic
fields. This behaviour was interpreted in Ref. [51] as decreasing of the
average effective field in the Al/EuS bilayer close to the coercive field and
quenching of the superconductivity in the Al wire interfaced with EuS, once
the applied field saturates the magnetization of the magnetic layer. Thorough
measurements of the transport properties of the Al wire coupled to EuS showed
that magnetization reversal causes step-wise changes of the resistance. This
indicates local quenching of the superconductivity. Phase diagram built for
the range of temperatures from 30 mK to 1.2 K and within the $\lvert
B\rvert\lesssim$ 50 mT allowed for tracking a variation of the coercive field
with temperature. This unveiled a number of critical temperatures that we
attributed to the presence of small superparamagnetic particles of EuS [51].
These findings were correlated with results of magnetic characterization by
means of XAS/XMCD spectroscopy, magnetometry and low-temperature ferromagnetic
resonance (FMR) [91, 81]. For instance, it was observed that sulfur-rich and
near stoichiometric polycristalline EuS films with moderate exchange coupling
strengths show very broad and shallow peaks of FMR, meanwhile the Eu-rich
films possess sharper resonance but their coupling to thin Al films causes
quenching of the superconductivity. Besides the preparation of polycrystalline
films, growth of textured and single crystal epitaxial EuS layers was reported
[89, 92, 93, 94, 95, 96]. The temperature of the substrate and the density of
dislocations were crucial to controlling the stoichiometry of EuS [89, 92].
Small lattice mismatch and suitable range of the substrate thermal stability
made possible the growth of very high-quality epitaxial EuS/InAs(001) films
[93]. These layers were employed to rigorously study their magnetic and
electronic properties and to grow the fully epitaxial Al/EuS/InAs(001)
heterostructures [94]. Together with results obtained in the GdN/NbN system
[97], these works make up a solid base for understanding the exchange coupling
phenomena in the epitaxial S/FI layers.

An alternative approach to the fabrication of heterostructures, which may
overcome the challenges associated with traditional growth methods, involves
utilizing two-dimensional (2D) materials. Van der Waals stacking of magnetic
insulator and superconductor layers, as reported in several recent studies,
has produced high-quality samples exhibiting a diverse array of intriguing
properties. This showcases the significant potential of this technique [98,
99]. Inherently, 2D materials exhibit a sharp interface between adjacent
layers in their stacks. This permits the engineering of heterostructure
interfaces with a level of precision only achievable with the most advanced
Molecular Beam Epitaxy systems [100]. Due to minimal chemical intermixing
between layers, it is possible to combine distinct slabs of various 2D
materials and even single atomic layers in the case of graphene or hexagonal
boron nitride. This precision opens up a way to study the phenomena typically
unobserved in bulkier heterostructures. For example, a rich assortment of
strongly correlated electronic states is evident in twisted bilayer graphene
[101]. On the other hand, sharp well-defined interfaces between dissimilar 2D
materials enable engineering of interface-dependent effects, like exchange
splitting of the electronic density of states in the superconductors [99]. For
instance, a heterostructure that integrates an insulating layer of the
ferromagnetic CrBr3 and superconducting NbSe2 showcases these capabilities
[98]. While some 2D materials exhibit strong reactivity and chemical
instability, a possibility of obtaining single layers via exfoliation is
widely accepted as a feature of the 2D materials, which facilitates building
of the heterostructures by means of stacking of the layers via mechanical
micromanipulations. Nevertheless, exfoliation produces flakes of variable
thickness that requires evaluation and selection procedure incompatible with
automatic fabrication processes. Therefore, technologies for growing
macroscopic (wafer-size) single-layer films and the protocols of transferring
of these films have being developed for a number of most relevant 2D materials
[100, 102]. Such advancements pave the way for implementing 2D materials in
the applications and creates incentives for further research activity in the
field. However, several challenges still need to be addressed to fully
leverage the benefits of this new group of materials. In terms of materials
science, significant work is required to broaden the selection of materials
available as large, high-quality wafer-size single layers. While progress is
being made in developing growth processes for semiconducting and
superconducting materials, exfoliation remains the most common method for
obtaining single-layer magnetic 2D material[100, 102]. An alternative to
wafer-scale production of 2D materials could involve the development of
moderate-temperature growth techniques for 2D layers, which could be
compatible with standard lithography-based nanofabrication [103]. On the front
of device physics and technology, significant efforts will be required to
adapt fabrication processes to the new materials and address common challenges
associated with implementing electronic devices, such as establishing
electrical contacts to the 2D layers [104, 105]. Specific challenges related
to realizing superconducting electronic devices with non-reciprocal transport
properties include determining the spin-filtering efficiency and the strength
of spin splitting due to exchange coupling in heterostructures of 2D
superconductors and 2D magnetic insulators. This topic requires further
experimental investigation.

###  2.3 Tunnel characteristics

In this section, we summarize the low-temperature characterization methods,
which provide information on spin-splitting and polarization of the tunneling
transport and how to improve it. We also discuss experiments on the
thermoelectric and rectification effects in different junctions based on S/FI
elements.

####  2.3.1 Characterization and optimization of the exchange splitting

Table 2: Overview of the investigated samples with varying layer thicknesses
and growth temperatures.

Samples | Structure |  $\Delta_{1}$ ($\mu$eV) |  $\Delta_{2}$ ($\mu$eV) |  $\bar{h}$ ($\mu$eV) | $\bar{h}/\Delta_{2}$
---|---|---|---|---|---
S1 | Al (10nm)/ AlOx/ Al (12nm)/ EuS (19nm)/ silica | 254 | 131 | 68 | 0.5
S2 | Al (10nm)/ AlOx/ Al (10nm)/ EuS (11.5nm)/ silica |  |  | 118 |
S3 | Al (12nm)/ AlOx/ Al (12nm)/ EuS (16.5nm)/ silica | 200 | 275 | 150 | 0.57
S4 | Al (12nm)/ AlOx/ Al (6nm)/ EuS (10nm)/ silicaa |  |  | $\bar{h}>\bar{h}_{cr}$ |

aEuS and Al grown at cold $T$ (150 K)

The properties of the ferromagnetic superconductor generated with a FI/S
bilayer are heavily influenced by factors such as operating temperatures and
external magnetic fields as well as material characteristics including the
quality of the interface between the two layers, the thicknesses of layers and
the material selection for the FI and S layer [51]. Different materials
combinations are discussed in Sec. 2.1.3. In the most typical case of using Al
as the superconductor, the critical temperature is relatively low ($T_{C}\sim
1-2$ K, depending on the film thickness), but also the spin-orbit interactions
(SOI) are weak due to the low atomic mass of Al. In the presence of disorder,
SOI leads to spin relaxation [8] and therefore the spin-splitting essential
for non-reciprocal electronic properties of the tunnel junction is clearest in
metals with weak SOI. By contrast, the disadvantage of operating below 1 K is
not significant for detection applications that typically need to operate at
ultra-low temperatures to minimize thermal noise.

The FI that pairs optimally with Al is EuS due to a minimal lattice mismatch
making ideal Al/EuS interfaces so that a strong ferromagnetic exchange
interactions can be induced in the Al. Such interactions generate an effective
Zeeman splitting ($\bar{h}$) in the DoS comparable to the Al superconducting
gap ($\Delta_{0}\simeq 200$ $\mu$eV), i.e., equivalent to the Zeeman splitting
induced by an external magnetic field of a few Tesla. For the fabrication of
optimal FI/S bilayers two experimental parameters are crucial: the quality of
the FI/S interface to induce strong exchange interactions in the S film and
the thickness of the S that facilitates such interactions through the film
thickness $d$. It has been demonstrated [51] that for $d\geq 3\xi_{S}$ the
induced $\bar{h}$ is almost negligible, determining the upper limit to the S
thickness. On the other hand, to maximize $\bar{h}$ the Al thicknesses ($d$)
can be lowered [31] down to a critical thickness ($d_{c}$) below which the
ferromagnetic correlations increase up to the Chandrasekhar-Clogston limit and
kill superconductivity as described in Sec. 2.1.1. Using Eq. (7), in thin
layers such a lower limit can be estimated from the Zeeman splitting of a
sample as:

$d_{c}=\frac{\sqrt{2}d\bar{h}(d)}{\Delta_{0}}\simeq\frac{\sqrt{2}h_{\rm
ex}^{\prime}}{\Delta_{0}}.$ (17)

It is worth noting that from this equation, $d_{c}$ depends only on the
intrinsic parameters $h_{\rm ex}^{\prime}$ and $\Delta_{0}$. As described in
Sec. 2.1.3, for an Al/EuS bilayer with $\Delta_{\mathrm{Al}}\simeq 200$
$\mu$eV and $h_{\rm ex}^{\prime}\simeq 0.8$ meV nm it is possible to estimate
$d_{c}\simeq 4$ nm. This limit has been numerically evaluated and demonstrated
by comparing the tunneling spectroscopy of samples made with Al/EuS bilayers
of different thicknesses [51].

Besides the direct magnetic proximity effect, external magnetic fields can
also be exploited to enhance $\bar{h}$. Interestingly, a large increase is
typically observed even at small fields for which the simple additional Zeeman
term would be negligible. This suggests a superparamagnetic behavior in EuS
since the spins at the Al interface surface likely have a weak coupling with
the bulk EuS. [33]. This effect has already been observed in early experiments
[60, 33, 66] and in some samples can be strong enough to induce a S to N phase
transition.

Figure 7:  Tunneling spectroscopy of sample S3-e. (a) First magnetization of
the EuS. Before the application of any magnetic field ($B=0$, black arrow), a
clear $\bar{h}$ is observed (broad peak around $V=\pm 0.4mV$) in the tunneling
conductance. It is a signature of the magnetization of the individual EuS
grains not being aligned with respect to one another. Sweeping from 0 to 30 mT
said broad peak quickly splits into two (the magnetization directions of the
individual EuS grains are being aligned) and then disappears above around 16
mT (dashed line) where the superconductivity of one Al layer is quenched (the
bottom Al layer which is in contact with the EuS). Measurement done at $T=30$
mK. (b) Evolution of the peaks in (a) taken at $B=0$ (black), $10$ mT (green)
and $20$ mT (blue), indicated in (a) by arrows. (c) Magnetic field sweep of
the tunneling spectroscopy at $T=30$ mK. After the first polarization of the
EuS layer, a spin-splitting in the tunneling spectroscopy is observed between
around +8 mT and -12 mT. The split peaks move slightly towards each other
throughout the sweep. Here, the surface spins of the EuS are dependent on and
therewith behaving like the EuS grains which are responsible for the
narrowing. Beyond mentioned applied magnetic field strengths, the spin-
splitting disappears. Now, the $\bar{h}$ is again so large that it quenches
the superconducting state of the bottom Al layer. The superconductivity of the
top Al layer is preserved, as can be concluded from the remaining peaks and
the gap. Throughout the sweep of the $B$ field, the peaks shift almost
linearly in $B$ as represented by the fit lines: (solid line)
$eV_{1}=\Delta_{1}+\Delta_{2}+\bar{h}=580\mu eV+35\mu_{B}B$ and (dashed line)
$eV_{2}=\Delta_{1}+\Delta_{2}-\bar{h}=290\mu eV-35\mu_{B}B$. From the
difference between the two curves, it is possible to estimate
$\bar{h}(B)=145\mu eV+35\mu_{B}B$ and to extract $\Delta_{1}\simeq 240\mu$eV.
From the tunneling spectroscopy at 15 mT it is possible to evaluate
$\Delta_{2}\simeq 200\mu$eV (d) Tunneling spectroscopy at no applied magnetic
field: before the first magnetization (black) and after magnetization (blue).
Data extracted from (a) and (c) respectively, as indicated by the arrows. (e)
Magnetic field sweep of the tunneling spectroscopy at $T=1$ K. At a higher
temperature, the size of the superconducting gaps ($\Delta_{1,2}$) decreases
and the spin-split peaks are only visible when the external magnetic field is
opposed to the polarization of the EuS layer. The surface spins have enough
energy at this temperature to be independent of the EuS grains, explaining why
the spin-splitting increases with increasing applied magnetic field. A third,
so called matching peak located between the superconducting gap and the spin-
split peaks is likewise observed. With increasing negative $B$ field the peaks
shift almost linearly in $B$ as represented by the fit lines: (brown solid
line) $eV_{1}=\Delta_{1}+\Delta_{2}+\bar{h}=340\mu eV-172\mu_{B}B$; (blue
dashed line) $eV_{2}=\Delta_{1}+\Delta_{2}-\bar{h}=250\mu eV+172\mu_{B}B$;
(green dash-dotted line) $eV_{3}=\Delta_{1}-\Delta_{2}+\bar{h}=140\mu
eV-172\mu_{B}B$. With a simple linear combinations of the three fit functions
it is possible to estimate $\bar{h}(B)=45\mu eV-172\mu_{B}B$,
$\Delta_{1}\simeq 195\mu$eV and $\Delta_{2}\simeq 100\mu$eV with the
assumption of a negligible dependence of $\Delta_{1,2}$ on $B$. (f) Tunneling
spectroscopy at $B=-9.5$ mT, taken from panel (e). The spin-split peaks (brown
and blue) and the matching peak (green) are visible (indicated by a circle of
the respective color as corresponding to the fit lines in (e)).

Characterization of the magnetic proximity effect was also necessary in the
SuperTED project and was quantified from the tunneling spectroscopy of
Al/AlOx/Al/EuS samples of different thicknesses. [51] Table 2 summarizes
selected measured samples and the extracted values for the order parameter of
the Al tunnel electrode ($\Delta_{1}$) and of the Al layer adjacent to the EuS
($\Delta_{2}$), as well as the induced exchange splitting $\bar{h}$, and the
ratio $\bar{h}/\Delta_{2}$. In Fig. 7 we show an example of tunneling
spectroscopy of an Al/EuS bilayer in which the Al thickness was optimized to
set $\bar{h}$ just below the maximal Chandrasekhar-Clogston limit. It
corresponds to sample S3-e in Table 2 and it is partially reported in ref.
[51]. The EuS/Al bilayer is in tunnel contact with an Al probe that allows us
to quantify $\bar{h}$ from the amplitude of the splitting observed in the
energy spectrum of the tunneling conductance. Notably, after the cooldown of
the device and before the magnetization of the EuS layer a small splitting was
already visible in the tunneling spectroscopy as shown in the black line of
Fig. 7b. This indicates the presence of magnetic domains with sizes larger
than the superconducting coherence length $\xi_{0}$ in the EuS film as
described in previous works [33]. Domain sizes much below $\xi_{0}$ would lead
to a vanishing average spin splitting. Upon the application of an external
magnetic field ($B$) the magnetic domains start to align and the observed
splitting increases as shown in Fig. 7a up to an in-plane field of 16 mT.
Above this field the secondary peak at the higher energy suddenly disappears
as a consequence of the S to N transition induced in the Al/EuS bilayer by the
increased $\bar{h}(B)$.

The superparamagnetic component of the EuS is further confirmed by measuring
the tunneling spectroscopy at lower magnetic fields as shown in Fig. 7c. The
magnetic field response is hysteretic: By tracing $B$ back a N to S transition
can be observed at $B<9$ mT with the re-appearance of the secondary peak. The
splitting then decreases by lowering $B$ down to -12 mT. Below -12 mT the EuS
layer reverses its polarization and $\bar{h}$ increases again above the
Chandrasekhar-Clogston limit with the resulting quenching of
superconductivity. By reversing the sweep this picture reverts as expected for
a ferromagnetic system (data not shown). Interestingly, by following the
position of the two peaks in B we notice an almost linear decreasing of the
splitting (as underlined by the black fit lines in Fig. 7c) consistent with a
linear field dependence of $\bar{h}=\bar{h}(0)+35\mu_{B}B$. $\bar{h}(0)\simeq
145$ $\mu$eV represents the intrinsic exchange splitting induced by the
ferromagnetism of the EuS while the linear part in $B$ is the
superparamagnetic component ($35\gg 2$) and corresponds to the action of the
external magnetic field on the polarization of the surface spins interacting
with the Al layer and weakly coupled to the bulk EuS. In addition, the action
of $B$ ($<9$ mT) applied anti-parallel with respect to the EuS polarization
results in a decreasing of $\bar{h}(B)$.

The tunneling spectroscopy of the junction at higher temperatures reveals an
even richer and non-trivial dependence of the bilayer superconductivity on the
external magnetic field. This dependence is especially relevant for extending
the temperature range of the detectors. In particular, as shown in Fig. 7d,
the tunneling spectroscopy measured at 1 K is characterized by a normal state
for the EuS/Al bilayer for most of the magnetic fields explored including at
zero magnetic field, while only when the polarization of $B$ is anti-parallel
with respect to the polarization of the EuS layer ($-2$ mT $>B>-10$ mT), the
bilayer is in the superconducting state. This behavior can be explained by an
increase of $d_{C}(T,\bar{h})$ overcoming the physical thickness of the device
at higher temperature as estimated in Fig. 4(b) and the monotonic relation
with $\bar{h}$. In contrast to the low-temperature measurement, now the
exchange splitting is enhanced by the external magnetic field
$\bar{h}(B)=\bar{h}(0)-172\mu_{B}B$ (notice the opposite slope in $B$) with a
stronger superparamagnetism (larger slope) and a weaker intrinsic effect
$\bar{h}(0)\simeq 45$ $\mu$eV. This high-temperature scenario is likely
related with a ferromagnetic to paramagnetic transition of the EuS surface
spin pattern expected to have a transition around $\simeq 900$ mK [33], much
below the transition temperature of the bulk EuS ($\simeq 17$ K). As a result
of this transition, the surface spin alignment, determining the spin splitting
in the Al, is only weakly coupled to the magnetization of the bulk EuS.
Notably, at high temperature, the tunneling spectroscopy is enriched by a
third peak representing the matching peak appearing at
$eV\simeq\Delta_{1}-\Delta_{2}+\bar{h}$. The almost linear evolution of the
three peaks in $B$ is reported in Fig. 7d and underlined by three fit lines.

Finally, similarly to the low temperature spectroscopy, by repeating the
measurement in the opposite field direction the features are reversed with
respect to the magnetic field (data not shown) confirming the magnetic
hysteresis of the device and the ferromagnetism of the EuS layer.

Figure 8: Differential conductance as a function of in-plane external magnetic
field for sample S1-b. Subgap states and delta vary with $B$. The inset shows
the field dependence of the zero bias conductance $G_{0}(B)$ normalized
respect to the normal state conductance $G_{N}$.

Within the same batch of samples some devices (S1) showed a weaker $\bar{h}$
so that superconductivity could survive at larger in-plane magnetic fields. In
Fig. 8 we show the evolution of the tunneling spectroscopy at large magnetic
fields, up to 400 mT. Also for these devices a clear superparamagnetic effect
is visible in the enhancement of the Zeeman splitting at large magnetic
fields. Moreover, increasing the magnetic field also lowers the amplitude of
the peaks and increases the zero bias conductivity, a phenomenon that can be
described with an enhancement of spin-scattering and/or inelastic scattering
events typically described by a generic Dynes parameter $\Gamma$ [106, 59].
The evolution of the zero-bias conductance ($G_{0}$) compared to the normal
state conductance ($G_{N}$) is shown in the inset of Fig. 8. Although the
precise origin of the magnetic field dependence is not clear from this data, a
possible explanation is the orbital depairing from the combination of the
external field and the stray fields of the EuS.

Figure 9: (a) and (b) Differential conductance measured for typical spin
selective tunnel junction made of EuS/Al/AlOx/Co at different bias voltages
$V$ and magnetic fields. The field was swept from -140 to +140 mT. (b) Traces
selected from (a) showing: a persistency of the spin polarization and
filtering also in the absence of magnetic field ($B=0$ blue dots); the maximal
asymmetry at 28 mT (red dots); no asymmetry and spin filtering at the coercive
field of EuS ($B=6$ mT green dots); and an almost flat tunneling spectroscopy
corresponding to the quenching of superconductivity at large magnetic fields
($B=140$ mT green dots). (c) Rectification coefficient $R=I_{\rm sym}/I_{\rm
asym}$ extracted from the same IV characteristics used in panel (b). Notably,
the $R(V)$ curves provide an alternative method to quantify the asymmetry of
the junction and the degree of spin polarization as described in Sec. 2.1.2.
(d) Field dependence of the Dynes parameter $\Gamma=G_{0}/G_{N}$ extracted
from the tunneling spectroscopy of three similar samples of EuS/Al/AlOx/Co.
Notably, the increase of $\Gamma$ at $B\simeq 6$ mT corresponds to the
switching field of the EuS and Co layers, while at large fields ($|B|>100$mT)
$\Gamma$ tends to unity as a consequence of the quenching of the
superconducting layer.  Figure 10:  (a) Differential conductance of an
Al/AlOx/Fe tunnel junction as a function of voltage bias $V$ for different
magnetic fields $B$. (b) False-color scanning electron microscopy image of a
region of the Al/AlOx/Fe sample. The image displays the measurement
configuration for thermoelectric measurements. (c) Thermoelectric current
$I_{th}$ as a function of magnetic field $B$ measured at a base temperature
$T_{0}$ = 100 mK, together with theoretical fitting. (d) Differential
conductance of the tunnel junction EuS/Al/AlOx/Fe as a function of the applied
bias $V$ for various applied fields. (e) Thermoelectric current $I_{th}$
measured for the sample in (d) as a function of the applied magnetic field
$\mu_{0}H$ at different base temperatures $T_{0}$. Data adapted from [59, 60].

####  2.3.2 Spin selective tunnel junctions

As shown in Sec. 2.1, to obtain non-reciprocal transport from a FI/S based
tunnel junction a spin selective probe is required. There are two primary
methods to achieve this: (i) by using a FI as a spin selective tunnel barrier
in S/FI/N [9, 23, 64, 75, 107] or S/FI/S [64, 107, 108, 77, 109] junctions
using a ferromagnetic probe in FI/S/I/F [9, 110] and S/FI/F [111]
configurations. For both configurations, clear non-reciprocal transport has
been observed with an $I-V$ characteristic of the junction similar to an ideal
semiconducting pn diode for small voltage bias ($V<\Delta_{0}$), with
corrections dependent on the non-ideal spin polarization ($1-\mathcal{P}$) as
described in Eq. (9). The non-reciprocity is also visible in the asymmetry of
the tunneling spectroscopy as depicted in Fig. 9a and b showing the
differential conductance measured at different magnetic fields and bias
voltages for an EuS/Al/AlOx/Co tunnel junction.

From the amplitude of the asymmetric peaks it is possible to extract the spin
polarization of the junction $\mathcal{P}\simeq 0.2$ at 28 mT and
$\mathcal{P}\simeq 0.15$ at 0 mT, then showing a permanence of spin-
polarization even at zero field. Unlike tunnel barriers that employ non-
ferromagnetic probes, the zero bias conductivity ($G_{0}$) is not much smaller
than the normal-state conductance $G_{N}$, an effect similarly observed in
prior experiments [112] and similar devices. Such subgap conductance depends
also on the applied external magnetic field as shown in Fig. 9d, showing
$\Gamma(B)$ evaluated from the ratio between the normal-state and zero-bias
conductances for three similar devices. Notably, for all the devices
$\Gamma(B)$ shows a similar trend with a minimum at low magnetic fields
parallel to the polarization of the two ferromagnetic layers while a local
maximum appears at the coercive field up to the global maximum of unity above
100 mT and corresponding to the quenching of superconductivity.

The origin of this subgap conductivity is still under debate. For example,
stray fields from the Co layer as well as pin-holes in the AlOx barrier can
impact the superconductivity of the Al layer. Independent of the microscopic
origin of the subgap conductance, the net effect on the device performance is
a reduction of non-reciprocal charge transport that, to the first order, is
linear in $\Gamma$ as shown in Eq. 12 while it is less detrimental (by a
factor $hT^{2}/\Delta^{3}$) for thermoelectricity [7] as shown in Eq. 13.

####  2.3.3 Thermoelectricity in spin selective tunnel junctions

Thermoelectricity in spin selective tunnel junctions has been observed, for
the first time, in Al/AlOx/Fe junctions [59]. Here the Zeeman splitting of the
Al layer was induced by an external in-plane magnetic field as shown in the
tunneling spectroscopy measurement of Fig. 10a. The Fe contact was heated by a
Joule current and thermoelectric current generated from such a thermal
gradient was measured with a third Cu contact using a circuit scheme shown in
Fig. 10b. With this configuration a thermocurrent up to 100 pA was observed at
$B=1$ Tesla, 100 mK bath temperature and $\simeq$100 mK thermal gradient (data
reported in Fig. 10c extracted form Ref. [59]). Notably, a sizable
thermoelectric signal was visible only above 0.5 T, a field at which the
tunneling spectroscopy shows an asymmetry, confirming the intimate relation
between thermoelectricity and non-reciprocity. From the measurement a Seebeck
coefficient up to $\simeq 100$ $\mu V/K$ was estimated. The measured effects
were in a very good agreement with the tunneling theory of [7] discussed in
Sec. 2.1.2. One year later, the same group performed a similar experiment on a
EuS/Al/AlOx/Fe junction [60]. Thanks to the presence of the additional EuS
layer a larger thermometric signal (up to $0.7$ nA) was visible already at
small magnetic fields due to the strong exchange interaction induced by the FI
layer as shown in Fig. 10e. However, thermoelectricity was not visible at zero
magnetic field consistently with the weak ferromagnetic response of the EuS as
observed also from the tunneling spectroscopy in Fig. 10d. This was likely due
to the poor interface between EuS and Al resulting from the nanostructuring
procedure that requires exposing EuS to air before the growth of the Al layer.

After adding an antenna to the setup of Fig. 10, the system could in principle
be used as a thermoelectric detector. There the effect of transverse
rectification in detection is likely negligible due to the fact that the
lateral size of the junction is much smaller than the characteristic length
estimated in [9]. However, the operation of the detector is likely hampered by
the necessity of applying a relatively large magnetic field to reveal the
magnetic proximity effect.

##  3 Applications

In this second part, we explore the applications of S/FI structures in
radiation detection. In the first section, we describe how an SFTED can be
used as a microcalorimeter for X-ray detection. The second section is devoted
to THz detection, whereas in the third section, we discuss possible
multiplexing read-out.

###  3.1 X-ray detection

A microcalorimeter is a microscale thermal detector that thermalizes the
energy of an absorbed particle, creating a temperature deviation which is then
measured by a sensor. Modern microcalorimeters should possess high energy
resolution, broad bandwidth, minimal dissipation, straightforward fabrication,
and the capability for operation in a sensor-array with readout-multiplexing.
The superconductor-ferromagnet heterojunction is an attractive sensing device
for cryogenic microcalorimeter applications[113]. The pronounced
thermoelectric effect[7, 59] in this device converts the absorbed energy into
an electrical signal without requiring bias power, fundamentally reducing the
heat dissipation and wiring complexity of the detector. Theoretical studies
indicate that a superconductor-ferromagnet thermoelectric detector (SFTED) has
the potential to be a swift cryogenic microcalorimeter. Its energy resolution
could rival state-of-the-art ultrasensitive detectors, such as the transition-
edge sensor (TES)[18, 114], kinetic inductance detector (KID)[19], and the
superconducting tunnel junction (STJ)[115].

This section reviews the linear microcalorimeter theory of the SFTED as
presented in Refs. [10, 113, 116]. We also address the time-domain analytical
models and the pulse excitation solutions for the SFTED[117]. Furthermore,
design considerations and optimizations of the detector are discussed with
numerical examples using practical parameters.

Figure 11: Schematic of a (a) superconductor-ferromagnetic thermoelectric
detector (SFTED). (b) and (c) are the thermal model and the electrical circuit
of the SFTED.

####  3.1.1 Linear theory

A simplified schematic of the SFTED is presented in Fig. 11. Consider an X-ray
photon with energy $E_{pt}$ impacting a superconducting absorber. The thin
superconducting electrode of the junction is assumed to be thermalized with
the bulk absorber, thus sharing the same electron temperature $T_{\rm abs}$,
which jumps above the phonon bath temperature $T_{b}$ after the absorption of
an X-ray photon, and then decays exponentially due to the heat flow into the
bath. Conversely, the normal electrode of the junction remains at the bath
temperature $T_{b}$.

A finite temperature difference, $\Delta T=T_{\rm abs}-T_{b}$, induces a
thermoelectric voltage $V_{J}$ across the junction, simultaneously creating a
heat current $\dot{Q}_{J}(\Delta T,V_{J})$ [see panel (b)] and a
thermoelectric charge current $I_{J}(\Delta T,V_{J})$ [see panel (c)],
transporting energy and charge between the hot and the cold electrodes, until
the detector is restored back to its quiescent state $\Delta T=0$.

The detector’s thermal and electrical balance govern $\dot{Q}_{J}(\Delta
T,V_{J})$ and $I_{J}(\Delta T,V_{J})$, and within the small-signal regime
$\Delta T<T_{b}$, the balance can be expressed in a linearized form using the
state variables $\Delta T$ and $V_{J}$ [10, 116] as:

$\displaystyle C_{\rm abs}\frac{d\Delta T}{dt}$
$\displaystyle=\dot{Q}_{in}-G_{th}^{\rm tot}\Delta T+\alpha V_{J}$ (18)
$\displaystyle I_{J}$ $\displaystyle=\alpha\frac{\Delta
T}{T_{b}}-\frac{1}{R_{J}}V_{J},$

where $C_{\rm abs}$ is the heat capacity of the absorber111Due to the large
volume difference between the absorber and the thin superconducting electrode
($<20$ nm) in practical devices, we assume that the absorber heat capacity
dominates and omit the contribution of the electrode., $R_{J}\equiv\partial
V_{J}/\partial I_{J}$ and $\alpha\equiv T_{b}\partial I_{J}/\partial\Delta T$
are the dynamic resistance and thermoelectric coefficient [7] of the junction
at $V_{J}=0$ and $\Delta T=0$, respectively. In Fig. 11(b)’s thermal model,
$G_{th}^{\rm tot}=G_{th}+G_{ep}$ represents the total thermal conductance
between the sensing elements — the absorber and superconducting electrode —
and the bath, where $G_{th}\equiv\partial\dot{Q}_{J}/\partial\Delta T$ and
$G_{ep}\equiv\partial\dot{Q}_{ep}/\partial\Delta T$ are the thermal
conductances through the tunnel barrier of the junction and through the
electron-phonon coupling in the superconductors, respectively.

In the electrical model from Fig. 11(c), the thermoelectric current $I_{J}$ at
frequency $\omega$ is given by $I_{J}(\omega)=[i\omega C+1/(i\omega L)]V_{J}$,
with $L$ and $C$ as the inductance and capacitance of the lumped circuit. As a
result, Eqs. (18) read as $\dot{Q}_{in}(\omega)=Y_{th}(\omega)\Delta T-\alpha
V_{J}$ and $\alpha\Delta T/T_{b}=Y_{el}(\omega)V_{J}$ in frequency domain,
where $Y_{th}=i\omega C_{\rm abs}+G_{th}^{\rm tot}$ and
$Y_{el}=1/R_{J}+i\omega C+1/(i\omega L)$ are the total thermal and electrical
admittances [10].

Thus, the voltage and current responsivity of the detector, considering
capacitive and inductive loads, can be defined as:

$\lambda_{V}\equiv\frac{V_{J}}{\dot{Q}_{in}}=\frac{\alpha}{Y_{th}Y_{el}T_{b}-\alpha^{2}},\quad\lambda_{I}\equiv\frac{I_{L}}{\dot{Q}_{in}}=\frac{\lambda_{V}}{i\omega
L},$ (19)

where $I_{L}$ is the current in the inductor.

The noise equivalent power (NEP) is a key metric to evaluate the performance
of a detector, particularly important for bolometric applications. It is
defined as the input radiation power in 1 Hz bandwidth required by the
detector to generate a signal equal to its noise. For a SFTED, the total
$\mathrm{NEP}_{\rm
tot}^{2}=\mathrm{NEP}_{amp}^{2}+\mathrm{NEP}_{\textrm{TED}}^{2}$, where
$\mathrm{NEP}_{amp}$ is induced by the noise from the amplification chain that
is used to readout the detector signal, and $\mathrm{NEP}_{\textrm{TED}}$
arises from the junction noise combined with the thermodynamic fluctuation
noise (TFN) [10]:

$\mathrm{NEP}^{2}_{\textrm{TED}}=\frac{4k_{B}T_{b}^{2}G_{th}^{\rm
tot}}{\mathrm{ZT}}\big{[}1+(1+\mathrm{ZT})\tau_{th}^{2}\omega^{2}\big{]},$
(20)

where $\tau_{th}=C_{\rm abs}/G_{th}^{\rm tot}$ is the thermal time constant
and ZT is the thermoelectric figure of merit reading as
$\mathrm{ZT}(\omega)=\alpha\lambda_{V}(\omega)$. Here it is worth noting that
a ZT value exceeding one enhances the NEP of SFTED over a standard bolometer
at low frequency, and this comes about because of the direct negative
electrothermal effect on the noise [116].

Furthermore, it has been shown [10] that the zero-frequency
NEP${}_{\textrm{TED}}$ can be optimized with respect to $R_{T}$ yielding a
condition $G_{th}/G_{ep}=\sqrt{1+zT_{i}}$ where
$zT_{i}=\alpha/(G_{th}T_{b}/R_{J}-\alpha^{2})$ is the intrinsic figure of
merit of the SFTED. As a result, the optimal detector intrinsic zero-frequency
noise equivalent power reads as:

$\mathrm{NEP}^{2}_{\textrm{TED,opt}}=\frac{4k_{B}T_{b}^{2}G_{ep}}{\mathrm{zT_{i}}}\big{(}1+\sqrt{1+\mathrm{zT_{i}}}\big{)}^{2},$
(21)

which is an important result particularly in guiding the design of SFTED for
the bolometric applications.

For calorimetric applications, the performance of the detector is widely
benchmarked by the energy resolution $\Delta E$. In the small-signal limit it
is [5]

$\Delta
E=\bigg{(}\int_{0}^{\infty}d\omega\frac{2}{\pi\mathrm{NEP}^{2}}\bigg{)}^{-1/2},$
(22)

obtained by applying the optimal filter and assuming an infinite bandwidth
amplifier. For SFTED, the intrinsic energy resolution $\Delta
E_{\mathrm{TED}}$ (omitting the amplifier noise) can be simplified to [113]:

$(\Delta E_{\mathrm{TED}})^{2}=4k_{B}T_{b}^{2}C_{\rm
abs}\frac{\sqrt{1+\mathrm{ZT}}}{\mathrm{ZT}}.$ (23)

As one can see, the intrinsic energy resolution of the SFTED can go below the
level of a TFN-limited feedback-less calorimeter ($\Delta
E_{\mathrm{TFN}}^{2}=4k_{B}T_{b}^{2}C_{\rm abs}$) when $\mathrm{ZT}\geq 1.6$.

Furthermore, [117] shows how Eqs. (18) can be solved analytically in time-
domain, providing a direct description of the potentially complex behavior of
the signal pulse of SFTED and a guide for designing and optimizing the
detector and readout components. For the sake of simplicity, here we only
revisit a case that the detector signal is readout from the inductive load,
i.e., by a cryogenic current amplifier based on Superconducting QUantum
Interference Device (SQUID), and the capacitance in Fig. 11(c) is omitted
($C\rightarrow 0$).

Because of the strong similarity between the thermoelectric effect and the
electrothermal feedback of a transition edge sensor (TES) [5], Eqs. (18) can
be rearranged to a first-order differential matrix format [117] analogous to
those of the TES [5], reading as

$\frac{d}{dt}\begin{pmatrix}I_{L}\\\ \Delta
T\end{pmatrix}=\begin{pmatrix}-\tau_{el}^{-1}&\frac{\mathcal{L}_{I}G_{th}^{\rm
tot}}{\alpha L}\\\ -\frac{\alpha R_{J}}{C_{\rm
abs}}&-\tau_{I}^{-1}\end{pmatrix}\begin{pmatrix}I_{L}\\\ \Delta
T\end{pmatrix},$ (24)

where we have defined $\mathcal{L}_{I}=\alpha^{2}R_{J}/(G_{th}^{\rm
tot}T_{b})$ as an analog of the constant current-bias low-frequency loop gain
of the TES, $\tau_{I}=\tau_{th}/(1-\mathcal{L}_{I})$ as the constant current
thermal time constant, and $\tau_{el}=L/R_{J}$ as the electrical time
constant. It is worth mentioning that, despite the similarity, the main
differences between Eqs. (24) and those of TES [5] are in the definition of
$\mathcal{L}_{I}$, and that the thermoelectric $\alpha$ appears in a dual role
both within $\mathcal{L}_{I}$ and as the analog of the DC current of the TES.
In particular, the thermoelectric factor $\alpha$ has a unit of current,
making $\mathcal{L}_{I}$ correctly dimensionless. One should also note that it
is totally different from the dimensionless logarithmic temperature
sensitivity of the resistance for the TES, which is also typically denoted by
$\alpha$.

Equations (24) have analytical solutions [117], which yield two time
constants, denoted as
$\tau_{\pm}^{-1}=(\tau_{el}^{-1}+\tau_{I}^{-1}\pm\sqrt{\Delta})/2$, where
$\Delta=(\tau_{el}^{-1}-\tau_{I}^{-1})^{2}-4\mathcal{L}_{I}(\tau_{el}\tau_{th})^{-1}$.
These time constants signify the rise ($\tau_{+}$) and decay ($\tau_{-}$) time
of the SFTED current pulse in response to a delta-impulse absorption event.
The general stability condition of a microcalorimeter detector requires that
the signal pulse decays back to the quiescent state either with oscillation
(underdampled case $\Delta<0$) or without (critically damped $\Delta=0$ or
overdampled $\Delta>0$), requiring the real part of $\tau_{\pm}$ to be
positive. An unbiased SFTED calorimeter is always stable because the loop gain
satisfies $\mathcal{L}_{I}<1$ [10, 117]. However, an oscillatory signal
extends the recovery time and significantly slows down the detector, therefore
a more restrictive exponential pulse decay condition $\Delta\geq 0$ is desired
for the SFTED.

####  3.1.2 Numerical results

Figure 12: Color scaled detector energy resolution $\Delta E$ as a function of
bath temperature $T_{b}$ and tunnel resistance $R_{T}$ (a) without the read-
out noise and (b) with a current read-out that has current noise
$\sqrt{S_{I}}=60$ fA/$\sqrt{Hz}$. The black dotted lines denote four thermal
time constant contour lines. The total noise equivalent power NEPtot at 10 kHz
of the detector read out by (c) voltage pre-amplifier with voltage noise
spectral density $\sqrt{S_{V}}=0.3$ nV/$\sqrt{Hz}$ and (d) current pre-
amplifier with current noise $\sqrt{S_{I}}=60$ fA/$\sqrt{Hz}$. The white
dashed lines denote where the detectors intrinsic NEPTED equals the NEPamp
contributed from the respective read-out, i.e.
$\mathrm{NEP}_{\mathrm{TED}}=\sqrt{S_{V}}/|\lambda_{V}|$ is satisfied in panel
(c) and $\mathrm{NEP}_{\mathrm{TED}}=\sqrt{S_{I}}/|\lambda_{I}|$ is satisfied
in panel (d).

It is a non-trivial task to achieve the best performance of a detector. This
is also the case when designing a SFTED. In particular, the non-ideality of
the fabricated junction and the noise of the readout circuit and the amplifier
will degrade the performance of a SFTED compared to the optimal operation
regime. These topics have been analyzed and discussed in previous studies [10,
113, 116, 117]. Here we revisit the key conclusions with numerical examples
that incorporate practical detector parameters. To achieve this, we employed
numerical computations based on Eq.(22) to determine energy resolution, as
demonstrated in Fig.12 and Fig.13. Furthermore, we examine the transient
pulses by numerically solving the eigenfunction Eq.(24), as illustrated in
Fig.14.

The most important parameters for designing a SFTED include: the tunneling
resistance $R_{T}$, the exchange field $h$, the polarization $P$ and the
broadening parameter $\Gamma$ of the tunnel junction, the volume $V_{\rm abs}$
and heat capacitance $C_{\rm abs}$ of the absorber, the noise power spectra
density (PSD) of the read-out amplifier, and the temperature $T_{b}$ of the
bath.

Among these parameters, the tunneling resistance $R_{T}$ and the bath
temperature $T_{b}$ are of paramount importance to the optimization of the
detector, as they are technically relatively easy to adjust. $T_{b}$ can be
controlled by the cryogenic apparatus on-the-fly, whereas $R_{T}$ can be tuned
in detector fabrication by modifying the area of the junction and/or the
oxidation time of the tunnel barrier [87]. In Fig. 12, the theoretical energy
resolution and the thermal time constant of a SFTED are demonstrated as
functions of $R_{T}$ and $T_{b}$, when a set of realistic parameters,
$h=0.3\Delta_{0}$, $P=0.5$, $\Gamma=10^{-3}\Delta_{0}$, $C=10$ nF, $L=2.6$
$\mu$H and $V_{\rm abs}=10^{4}$ $\mu$m3 are selected to represent a real
device.

The intrinsic energy resolution of the SFTED $\Delta E_{\textrm{TED}}$ is
presented in panel (a) together with four contour dotted-lines denoting
different thermal time constants as functions of $R_{T}$ and $T_{b}$. Optimal
SFTED performance is achieved around a temperature of $0.2$ K with a tunneling
resistance under $100$ $\Omega$. Increasing or decreasing the temperature
leads to a reduced figure of merit $ZT$ (see Fig. 2 panel (b) of Ref. [116]),
whereas larger $R_{T}$ degrades $\Delta E$ by limiting the bandwidth of the
detector.

The electrical signal from a cryogenic microcalorimeter is typically too small
to be directly read out by room temperature electronics. Consequently, a low-
temperature pre-amplification of either the voltage or the current signal is
required. Typically, voltage read-out works better for high-impedance sensors,
whereas low-impedance sensors prefer current read-out. For a SFTED, it is not
immediately obvious which type of read-out works better as $R_{T}$ can be
adjusted via fabrication, and the total energy resolution $\Delta E$ depends
on the noise of the amplifier.

The total noise equivalent power NEPtot of the detector at 10 kHz as functions
of $T_{b}$ and $R_{T}$ are plotted in panel (c) when using a state-of-the-art
cryoHEMT [118] as voltage read-out ($\sqrt{S_{V}}=0.3$ nV/$\sqrt{{\rm Hz}}$,
assuming $L\rightarrow\infty$) and in panel (d) for using a transformer
coupled SQUID [119, 120] as current read-out ($\sqrt{S_{I}}=60$ fA/$\sqrt{\rm
Hz}$, assuming $C\rightarrow\infty$). In both panels, the white dashed lines
signify the condition $\textrm{NEP}_{\textrm{TED}}=\textrm{NEP}_{\rm amp}$
such that the amplifier noise dominates the NEPtot on the right (left) side of
the dashed line when using a voltage (current) read-out.

When compared to the intrinsic energy resolution presented in panel (a), one
can find that a SFTED with $R_{T}$ less than $100\textrm{ $\Omega$}$ with
$T_{b}$ between $0.2\sim 0.25$ K provides the best performance [$\Delta
E_{\textrm{TED}}<2$ eV with $>10$ kHz bandwidth, see panel (a)] for a voltage
read-out. Although with higher $R_{T}$ the noise is dominated by the
detector’s intrinsic noise and a smaller NEPtot is obtained [see panel (c)],
it leads to an undesired overall performance because of either a degraded
intrinsic energy resolution [see panel (a)] or a large thermal time constant.
As for a current read-out, it can be seen that the read-out noise matches to
that from a SFTED with $R_{T}$ around $10\sim 40\textrm{ $\Omega$}$ at $T_{b}$
between $0.2\sim 0.25$ with a performance similar to the case of the voltage
read-out. As an example, panel (b) shows the total energy resolution $\Delta
E$ of a SFTED using a current read-out with noise of $\sqrt{S_{I}}=60\textrm{
fA/$\sqrt{Hz}$}$. One finds that the energy resolution degrades drastically in
the parameter region with higher $R_{T}$, the effect being stronger for low
$T_{b}$. Clearly, optimizing between $T_{b}$ and $R_{T}$ is critical.

In general, current read-out offers a larger bandwidth, has lower dissipation
[121] and good multiplexing schemes [114]. Hence it is preferable to be used
with SFTED. As a result, to demonstrate the influence of other detector
parameters in the following section, we consider a current read-out and set
the detector to have $R_{T}=40\textrm{ $\Omega$}$ at a temperature of $T=0.23$
K, based on the above discussion.

Figure 13: Total energy resolution $\Delta E$ (including amplifier noise)
plotted as a function of (a) exchange field $h$, (b) polarization $P$, (c)
broadening parameter $\Gamma$ and (d) absorber volume $V_{\rm abs}$. The three
colored lines demonstrate $\Delta E$ with three different current read-out
noise levels, and the black dashed line demonstrates the intrinsic resolution.
Panel (d) is computed with materials parameters for an Sn absorber.

The total energy resolution, including the amplifier noise, is demonstrated as
a function of the exchange field $h$ and the polarization $P$ in Fig. 13
panels (a) and (b), respectively, with different read-out noise levels denoted
by different colors. As discussed in Sec. 2.1.1, $h$ and $P$ characterize the
ferromagnetic-superconductor junction. They are influenced by both the
ferromagnetic insulator and the ferromagnetic electrode materials, as well as
by junction fabrication conditions. Optimal resolution appears to be achieved
when $h=0.4-0.5$ and $P$ is maximized, which aligns with findings from Refs.
[10, 113].

The broadening parameter $\Gamma$ of the tunnel junction, describing the
broadening of the superconducting DOS and the excess sub-gap current, also
influences the energy resolution, as discussed in Ref. [116]. As shown in Fig.
13 (c), a higher $\Gamma$ leads to a degraded energy resolution. To achieve,
for example, $\Delta E<2$ eV with an amplifier with $\sqrt{S_{I}}=100$
fA/$\sqrt{Hz}$, $\Gamma$ needs to be smaller than $10^{-2}\Delta_{0}$.

The absorber of a microcalorimeter absorbs the incident photon, and transfers
and relaxes its energy to the electron and phonon systems within $10^{-9}$ s,
the precise time depending on the absorber material and size [122, 123, 124].
In Fig. 13(d) we demonstrate the dependence of $\Delta E$ on the volume
$V_{\rm abs}$ of a superconducting Sn absorber. This takes advantage of the
weak electron-phonon coupling in a superconductor, reducing heat loss to the
bulk phonons [125]. In contrast, a normal-metal absorber would often
necessitate a more intricate micro-machined membrane or beam support structure
to confine the absorbed energy in the phonon system [114]. Superconducting
absorbers have been used in microcalorimetry applications [126, 127], in
particular together with superconducting tunnel junction sensors [128, 129].
Generally speaking, the volume $V_{\rm abs}$ of the absorber should typically
be determined by the desired X-ray energy range of the application such that
the temperature change of the sensor induced by the absorption would stay in
the linear response range of the detector, i.e., $E_{\textrm{x-ray}}/C_{\rm
abs}<0.1T$ where $E_{\textrm{x-ray}}$ is the energy of the X-ray photon. For
example, a Sn absorber with a volume $\sim 10^{-14}$ m3 has $C_{\rm abs}\sim
10^{14}$ J/K [130], and thus can be used for the detection of an X-ray photon
with keV energy. Moreover, it is also worth mentioning that the SFTED likely
has weaker constraints on $V_{\rm abs}$ than a TES, as the transition to the
non-linear regime is not as abrupt.

Figure 14: (a) Current-temperature pulse cycles under different operational
conditions, denoted by the color of the lines. (b) Signal current pulse under
different operational conditions corresponding to panel (a).

We now illustrate the transient evolution of the detector signal in Fig. 14.
Panel (a) displays the pulse cycles in the signal current-temperature space
($I_{L}$-$\Delta T_{J}$) under four distinct conditions, whereas the
corresponding current pulse signals in time domain are plotted in panel (b).
With these examples, SFTED has a desirable response when $\Delta>0$
(overdamping) and $\tau_{I}>\tau_{el}$, triggering faster pulses (blue
curves). In contrast, two undesired conditions are also demonstrated with the
orange ($\Delta>0$ but $\tau_{I}<\tau_{el}$) and red ($\Delta<0$) curves, both
leading to slow response and recovery due to the reverse self-biasing [117],
which should be avoided. On the other hand, the green curve represents the
condition $\Delta=0$ wherein double roots emerge from Eq. (24). Such a
condition is often considered as an optimized compromise between the energy
resolution and the slew rate requirement (finite detector rise-time) of the
read-out electronics [5].

####  3.1.3 X-ray detector design

In the operation of an X-ray microcalorimeter, detecting a photon typically
involves the detector to first absorb the energy of the incident photon,
converting it to thermal excitations, and then to confine the excitations
temporarily, creating a finite temperature excursion pulse. At the same time,
these excitations need to be transduced to a measurable electrical signal. For
a practical device, these functions are carried out by the absorber, the
thermal isolation and the sensing structures, respectively [114]. Figure 15
shows an example of a real SFTED microcalorimeter, designed for X-ray
detection.

Figure 15: Fake-color optical micrograph of a SFTED calorimeter, in which the
material compositions are illustrated with different colors. A superconductor-
ferromagnet heterojunction area is marked with the white dashed line,
consisting of layers of Si/SiN/EuS/Al/AlOx/Co from the bottom to the top. A Sn
absorber is placed to the side of the junction, contacting directly the Al
lead with AlOx layer removed via chemical etching.

In this device, the sensing structure of the detector is a superconductor-
ferromagnet heterojunction, consisisting of layers of EuS/Al/AlOx/Co (from
bottom to top), on the top of a nitridized Si substrate. Such a tunnel
junction is fabricated by depositing a $200\textrm{ $\mu$m}$-wide
ferromagnetic (Co) electrode on the top of and orthogonal to a $200\textrm{
$\mu$m}$-wide superconducting (Al) electrode with a thin plasma oxidized AlOx
layer in between serving as the insulating tunnel barrier. The junction
parameters are optimized for the detection of X-rays based on the discussion
presented in previous sections.

The absorber structure is a $350$ nm thick superconducting Sn square (with an
area of $100\textrm{ $\mu$m}\times 100\textrm{ $\mu$m}$) providing 90% and 13%
absorption efficiencies for 1 keV and 6 keV photons, respectively [131]. It
contacts directly the superconducting Al electrode (with the insulating
barrier AlOx removed chemically) and is placed laterally next to the junction
area. Such a superconducting absorber configuration has been adopted before by
superconducting tunnel junction (STJ) [128] and TES detectors [132], and has
been extensively studied both numerically and experimentally [129].

Lastly, to confine the quasiparticles in the Sn absorber and Al electrode (the
thermal excitations in an SFTED) and to minimize their excess escape, the Al
electrode is patterned into an island with a lateral size of about the size of
the absorber+junction, separated from the remaining leads, as illustrated in
Fig. 15. In addition, a superconducting Nb lead is used to contact to the Al
island via a set of narrow fingers. Due to the larger superconducting energy
gap of Nb (typically $>1$ meV) contrasting to that of Al (typically $\sim 0.2$
meV), such a lead provides an electrical connection, while effectively
trapping the quasiparticles inside the Al island [129]. On the other hand,
because the temperature difference $\Delta T$ between the superconducting and
ferromagnetic electrodes produces the electrical signal [10] in a SFTED, the
Co electrode is kept with a large area of $200\textrm{ $\mu$m}\times 4\textrm{
mm}$, helping it to thermalize to the bath.

###  3.2 THz detection

The SFTED detector can be applied to measure electromagnetic radiation in a
wide range of the spectrum. Besides X-rays, we consider here the measurement
in the THz range. We define in this context the THz range as the range of
frequencies comprised between 0.1 and 3 THz. In terms of wavelengths this is
equivalent to a range of 0.1 to 3 mm. This region of the electromagnetic
spectrum is particularly interesting for present and future astrophysics
applications. Specifically, this includes observations of galaxy clusters,
primordial galaxies, cold interstellar dust, and star-forming regions in our
galaxy as well as neighboring galaxies [133]. High-precision observations will
in particular be needed in the coming years to be combined and complement with
the exceptional JWST (James Webb Space Telescope) infrared data.

To enhance the sensitivity of the SFTED junctions to incoming THz radiation,
we have investigated several options. Among them, in decreasing order of
fabrication complexity:

  * •

Implementing a higher gap superconducting (e.g. Nb) planar antenna, feeding
the SFTED sensor. From the fabrication point-of-view, this would have required
adding a further metal layer (S2), and realising high-quality metal-on-metal
contacts. On top of that, in order to properly match the typical antenna
impedance with the SFTED junction, a lithography and etching of the SFTED
structure would have been required.

  * •

Patterning the SFTED junction into an impedance-matched absorber. The focusing
of the THz radiation is achieved by adopting a micro-lens on the back of the
silicon substrate. We stress that the dielectric substrate is transparent to
THz radiation, allowing back-illumination. This approach is compatible with
the existing layers, but would have required an additional, delicate
lithography and etching of the SFTED junction structure.

  * •

Based on the previously available shadow masks, post-patterning a non-
optimised absorber in the existing S wire leading to the SFTED junction. This
only requires a lithography on the superconducting (Aluminium) wire, away from
the SFTED junction.

After discussions, we concluded that the last option was the safest for an
initial proof-of-concept. For this purpose, a photosensitive resist, type
S1805, is spun onto the chip and then baked. With UV lithography the pattern
is subsequently realized. For both the development and the etching of the
Aluminium, the developer solution MF319 is employed.

Figure 16 shows a picture of the SFTED junction environment that has been
fabricated and tested, together with a schematic of the optical cryogenics
measurement setup.

Figure 16: THz detector testing setup (top view) and picture of the SFTED
junction. The cryostat optics is described in [134].

The preliminary results of the testing under illumination at 150 GHz and 110
GHz along the InPol (parallel to the S and the absorber wires) and CrossPol
(parallel to the N wire) electric field polarisation directions are shown in
Figs. 17 and 18.

Figure 17: Conductance-voltage characteristics of the SFTED junction shown in
Fig. 16 under different illumination conditions. In particular, varying the
direction of the polarisation and the frequency of the millimetre-wave source.
$H~{}=~{}20$ mT along the direction of the superconductor (S) and $T$ = 120
mK.

In our initial measurement, we traced the SFTED junction’s conductance-voltage
characteristic under various illumination conditions. This measurement has
been taken under an applied magnetic field of 20 mT along the direction of the
superconductor (S). The millimeter-wave radiation distinctly impacts the
curves, especially the residual conductance within the superconducting gap.

Figure 18: Zero bias current flowing into the N electrode as a function of the
150 GHz illumination conditions (polarisation, and ON/OFF). $H$ = 20 mT along
the direction of the superconductor (S) and $T$ = 120 mK.

In a subsequent experiment, we left the detector unbiased and observed the
current flowing into the N electrode. This represents the standard way for a
SuperTED detector. We plot in Fig. 18 the time-traces of the current,
expressed in nA, while switching ON/OFF the millimetre-wave source, and
switching its polarisation between InPol and CrossPol. The ON/OFF steps,
despite the presence of an offset current, represent the first hint of
detection of 150 GHz photons from an un-biased SFTED junction, possibly
related to the thermoelectric effect. Although the signal has a low S/N ratio,
it appears to be roughly 50 % stronger in the InPol (absorber, S) direction
than in the CrossPol (N) direction. This is in nice agreement with the 3D
electromagnetic simulations that we have run on the structure shown in Fig.
16. These simulations predict the expected absorption of the metallic
structure in the range 100-300 GHz.

Based on this first preliminary detection, the SuperTED detector, after
decisive optimisations, might have the potential to compete with the existing
cryogenic sensors operating at THz frequencies, i.e. Kinetic Inductance
Detectors (KID), Transition Edge Sensors (TES), high-impedance bolometers and
others. To our knowledge the preliminary data shown here is the first
demonstration of biasless superconductor-based radiation sensing. However,
further work is still needed for its analysis and optimization.

###  3.3 Multiplexing read-out of self-powered detectors

The thermoelectric detectors TED are expected to generate a current upon
absorbing electromagnetic radiation. In view of addressing hundreds of
detectors at cryogenic temperature, we developed an idea for a possible highly
multiplexed readout. Such a readout would solve both the complexity and the
thermal load created by the hundreds of wires needed to address individual
detectors. The readout is based on planar superconducting resonators biased by
the DC-like current generated by the TEDs.

####  3.3.1 Principle and design

A superconducting resonator can be modelled as an LC-circuit in which part of
the inductance is coming from the kinetic inductance $L_{k}$ of the
superfluid: the entire superfluid condensate opposes changes in the current
direction. The resonator resonates at a frequency $f=1/(2\pi\sqrt{LC})$, where
$C$ is the capacitance and $L=L_{K}+L_{G}$ is the total inductance, i.e., the
sum of the kinetic and geometric inductances. State-of-the-art superconducting
resonators have a quality factor of millions. In thin films at low temperature
the superfluid contribution to the sheet kinetic inductance is [135]:

$\displaystyle L_{K}=\frac{m}{2n_{s}e^{2}d}$ (25)

where $m$ is the effective electron mass, $d$ is the superconductor thickness,
$n_{s}$ is the superfluid density and $e$ is the elementary charge. Equation
(25) highlights a key principle for the readout scheme: a decrease of the
superfluid density $n_{s}$ results in an increase of the kinetic inductance
$L_{K}$, and thus in a change of the resonance frequency. Since the
circulating current in a superconductor decreases the superfluid density
$n_{s}$ [136], the superconducting resonator frequency can be adjusted using a
small bias current. The first demonstration of this concept, for one
resonator, has been achieved by M. R. Vissers and collaborators in 2015 [137].

The current generated by a thermoelectric detector will be sent into a
superconducting resonator. In turn, the resonance frequency will vary with the
current. For a current $I$ small compared to the critical current $I_{*}$ the
relative frequency shift is

$\displaystyle\frac{\delta
f}{f}=-\frac{\alpha}{2}\left(\frac{I}{I_{*}}\right)^{2},$ (26)

where $\alpha$ is the fraction of the kinetic inductance over the total
inductance (kinetic plus geometric) at zero bias current. Detection of a small
current then requires a small critical current $I_{*}$. For this purpose, two
parameters can be optimized: the material dependent critical current density
$J_{*}=I_{*}/S$, and the cross section $S$ of the superconducting wire. Small
critical current density can be achieved especially in disordered
superconductor materials, such as granular aluminium, for which $J_{*}\sim
3\times 10^{10}$ A$\cdot$m-2 [138, 139]. Using electronic e-beam lithography
nanowire width as small as 100 nm can be obtained for superconducting film
thickness of the order of 50 nm. Eventually, we estimate that for
superconducting granular aluminum it is possible to achieve critical current
as low as 150 $\mu$A. For a resonance at 1 GHz such critical current leads to
a frequency shift of 14 kHz for an injected current of 800 pA. In principle,
it is thus possible to detect the 800 pA generated by the thermoelectric
detectors for typical millimeter detection application.

The general problem is to design a resonator that achieves a high quality
factor even when connected to detectors with low impedance (less than one
Ohm). To achieve this, the dc-current lines must act as, or go through, a low-
pass filter to allow the dc-current (DC-kHz) to flow from the detector to the
resonator but prevent the resonating ac-current (GHz range) to leak out of the
resonator. We have explored different options, including planar
superinductances [140], 3-D lumped-element capacitors or inductive-grounded
coplanar waveguide configurations. We describe in the following the first two
options.

Figure 19: Design of the iKID readout scheme with two different materials
design (a) Resonators (in blue) are to connect to the detectors (not shown)
via the dc-current lines (in red). The blue horizontal line is a readout line
to monitor the resonance frequencies. (b) Zoom on a resonator.

####  3.3.2 Planar superinductances

We use two different granular aluminium (grAl) compositions: a low kinetic
inductance grAl for the resonator part (in blue in Fig. 19) and a high kinetic
inductance grAl (in red in Fig. 19) for the two wires connecting the resonator
to the detector (superinductances). Figure 19 shows the design of the readout
scheme for current bias kinetic inductance detectors (iKIDs). We designed
thirteen resonators. For testing purposes, only eight were connected to DC-
current ports. The readout line (blue horizontal line) allows monitoring
simultaneously the resonance frequencies. The resonance frequencies are
adjusted by varying the total length of the resonators. The electromagnetic
simulations foresee resonance frequencies in the 5-7 GHz range and quality
factor in the 3-8$\times 10^{4}$ range.

Figure 20: Picture of the iKID readout scheme prototype with two different
materials. The matrix of thirteen resonators is in the center. Only the
readout line is clearly visible: it is the horizontal line. The bottom PCB-
board is connected to the resonators and to the cables.

An iKID prototype chip was realised and packaged in collaboration with I. Pop
and P. Paluch from Karlsruhe Institute of Technology (KIT), Germany. Figure 20
shows the prototype inserted in a specially designed sample holder with an ad-
hoc PCB-board. The PCB-board (at the bottom) is connected on one side to the
superinductances via micro-bonding and on the other side to input DC cables.
The prototype was cooled down at 100 mK with cables connected to a dc-current
source at room temperature. The measurements demonstrate a sensitivity of the
resonance frequency upon the circulating current but most probably due to
Joule heating instead of kinetic inductance variation as the quality factors
were concomitantly reduced. The Joule heating may be due to the high resistive
contacts between the two different materials.

####  3.3.3 3-D lumped-element capacitance

In this case, both ends of the MUX resonator are grounded (at high frequency)
by dielectric parallel-plate capacitors. The size of each capacitor is of
about $100\times 500$ $\mu m^{2}$, smaller than the resonator, assuming a 30
nm thick alumina as dielectric with $\epsilon\approx 10$. In Fig. 21 we show
the electromagnetic simulation of the model. The resonance exhibits a
promising quality factor of the order of about $4\cdot 10^{3}$, paving the way
to achieving a multiplexing factor, i.e. the number of detectors that can be
read out on a single wire, of hundreds to thousands.

Figure 21: Electromagnetic simulation of a promising configuration of iKID
based on 3-D lumped-element capacitance. (a) 3D-model. The dielectric is a 30
nm thick alumina. The metal, green and pink, is aluminum. (b) Current
distribution at the resonance. (c) Electrical response of the KID. The quality
factor of the resonance is of about $4\cdot 10^{3}$. (d) Electrical response
of the DC-lines. The DC-lines are low pass filters.

In summary, based on the first prototype results, thoughts on the new design
and previous experience with disordered superconductors [141], we conclude
that an iKID should be realised using a single (patterned) layer of granular
Aluminium near the insulating transition. This should be capped with a 30 nm
thick alumina layer and a second superconducting metal layer, e.g. standard
aluminium, to realise the low-pass filter. The quality factor of the
resonating readout circuits will allow achieving a multiplexing factor of
hundreds to thousands.

##  4 Discussion and outlook

In this review, we have presented the fundamentals of a new type of a detector
based on thermoelectric and non-reciprocal effects. These arise from the
interplay of two phenomena: The spin-splitting of the density of states of a
superconductor via the magnetic proximity effect, and spin polarization in
transport through tunnel barriers.

In the first part of the text we focus on studying the hybrid thin film
structure composed of the ferromagnetic insulator EuS adjacent to a
superconducting Al. As discussed in Sec. 2.1-2.3, such EuS/Al-based devices
can be used beyond detectors as non-reciprocal elements for low temperature
electronics, such as diodes[9, 142]. We can also envisage more elaborated
circuits including mixers, reverse current regulators, voltage clamping, and
other passive superconducting electronics[1], with the additional control
provided by magnetic fields.

Optimizing these devices requires increasing spin polarization using new
combinations of materials with higher spin-filter efficiency and reducing
their size. An interesting future direction is to use magnetic two-dimensional
van der Waals materials either as sources of the magnetic proximity effect or
as spin filtering tunneling barriers. Interestingly, the non-reciprocity of
transport in these devices is a new way to characterize the magnetic proximity
effect and spin polarization, an alternative to tunnel spectroscopy. In
particular, when the induced spin splitting field cannot be directly resolved
from the presence of the BCS peaks in tunneling characteristics, its presence
can be determined from the non-reciprocal characteristics of the device.

In this text we have concentrated on the non-reciprocal quasiparticle
tunneling characteristics. Besides that, in the presence of either spin-orbit
coupling or position dependent exchange field, also the supercurrent becomes
non-reciprocal [143]. Such effects form a complementary way of characterizing
the magnetic proximity effect.

In the second part, we concentrate on the properties of the SFTED.
Specifically, in Sec. 3.1, we review the microcalorimetry theory of the SFTED
detectors and determine their noise-equivalent power and energy resolution as
a function of the system parameters. We also discuss in this section possible
designs of the SFTED for the X-ray detection. Section 3.2 briefly discusses
the THz range accessible for SFTED, and present preliminary measurements of a
SFTED junction. Finally, in Sec. 3.3 we discuss possibilities of read-out and
multiplexing.

The detector work also highlights challenges in fabricating the SFTED device.
It would be desirable to have a detector controllable only with a rather small
magnetic field, as the coil used to create it may complicate coupling of the
radiation into the detector. Ideally no field would be required at all and one
would rely on the intrinsic magnetization of the ferromagnetic insulators.
However, for example for EuS the Curie temperature is rather low (about 14-17
K, depending on doping), and it needs to be magnetized after the sample has
been refrigerated to the base temperature. Besides coils, this could perhaps
be done using permanent magnets aligned with the sample.

To minimize the required magnetic field, and thereby to ensure optimal
magnetic proximity effect, we have prepared the EuS/Al systems using in-situ
shadow mass patterning without breaking the vacuum. Such a process results to
relatively large junctions, with lateral dimensions of the order of a few
hundred $\mu$m2. Optimal THz detectors would constitute planar antennas with
small-size absorbers and small-size junctions (much smaller than the
wavelength), and further size reduction is desired. In the X-ray regime the
energy resolution is mostly determined by the relaxation time of the radiation
absorber, which is inevitably significant, so the mere size of the junction is
not a problem. However, in that regime the detection requires fairly
transparent junctions, i.e., with a low normal-state tunneling resistance
$R_{T}$ (see Fig. 12). This requirement is not straightforward to realize in
large junctions without the presence of pinholes and the resulting large
number of subgap states, quantified with the parameter $\Gamma$. Ideal
detector would also require rather low $\Gamma\lesssim 10^{-2}\Delta$ as
indicated in Fig. 13. Moreover, an improved energy resolution for the
detection of keV X-rays could possibly be obtained by suspending the absorber
to prevent the premature escape of athermal and thermal phonons[144, 145].
Such design has been widely adopted for transition edge sensors[114]. On the
other hand, the junctions themselves should not reside in the suspended
region, as maximizing the temperature difference would require thermalizing
the normal part of the junction as well as possible.

We are confident that these technical challenges have a solution. For example,
it may be better to pay the price of a sub-optimal FI/S interface to get a
nanostructured detector. Once such detectors are in place, SFTED-based
detectors can rival existing superconducting detectors such as transition edge
sensors and kinetic inductance detectors. Subsequantly, the next step would be
to construct a large number of detectors and replace the current read-out with
a multiplexing technique, such as the nanowire based multiplexing discussed in
Sec. 3.3. One of the issues to be handled in such an SFTED detector array
would be the dipolar coupling between the magnetizations. However, as the
performance of the detector does not rely on the possibility of (locally)
switching magnetization from one state to another, the whole array could be
controlled with a single magnetic field.

Besides detectors and nonreciprocal electronics, superconductor-ferromagnet
multilayers enable other novel phenomena. The spin-split superconductors can
be used to create absolute spin valves[66], thermal logic circuits[146],
coherent caloritronic systems[147, 148, 149], devices mixing nonequilibrium
spin and equilibrium supercurrent transport [150] and thermal quantum
devices[151, 152, 153]. Moreover, a rich dynamical response of the hybrid
systems can be envisaged: The presence of superconductors influence the
magnetic resonance by mediating magnetic interactions (as a consequence of the
magnetic proximity effect) [20, 154] and via the modified spin battery effect.
Moreover, they may give access to coupling different collective modes, those
inherent to magnetism with those relevant for superconductivity [155].

We wish to finally mention that a very large thermoelectric response was
recently observed in fully superconducting tunnel junctions composed of
different superconductors[156, 157]. These systems, although being perfectly
reciprocal, exhibit a _spontaneous_ breaking of the electron-hole symmetry in
the presence of a large thermal bias[158, 159], thereby providing sizable
thermoelectric figures of merit that could be exploited for the implementation
of novel-concept nanoscale thermoelectric machines[156, 157, 160] and
sensitive passive radiation detectors[161].

## Acknowledgements

This work was mainly funded by EU’s Horizon 2020 Research and Innovation
Program under Grant Agreement No. 800923 (SuperTED). F.S.B and C.R also
acknowledge funding from the Spanish MICINN-AEI (Projects No.
PID2020-114252GB-I00 (SPIRIT) and TED2021-130292B-C42), and the Basque
Government (grant IT-1591-22), M.S. and E.S. acknowledge partial funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska Curie Action IF Grant No. 101022473 (SuperCONtacts). C. I. L.
A. acknowledge partial funding from CNPq, CAPES and Fapemig. F.G. acknowledges
the EU’s Horizon 2020 research and innovation program under Grant Agreement
No. 964398 (SUPERGATE) for partial financial support.

The data of the figures in this work can be obtained at
https://doi.org/10.5281/zenodo.7798143.

## References

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