# Theory of a Single Magnetic Impurity on a Thin Metal Film
in Proximity to a Superconductor

Jon Ortuzar  CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain  Jose
Ignacio Pascual  CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain
Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain  F. Sebastian
Bergeret  Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU,
E-20018 Donostia-San Sebastián, Spain  Donostia International Physics Center
(DIPC), 20018 Donostia-San Sebastian, Spain  Miguel A. Cazalilla  Donostia
International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain
Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain

(August 28, 2024)

###### Abstract

We argue that the formation of Yu-Shiba-Rusinov excitations in proximitized
thin films is largely mediated by a type of Andreev-bound state named after de
Gennes and Saint-James. This is shown by studying an experimentally motivated
model and computing the overlap of the wave functions of these two subgap
states. We find the overlap stays close to unity even as the system moves away
from weak coupling across the parity-changing quantum phase transition. Based
on this observation, we introduce a single-site model of the bound state
coupled to a quantum spin. The adequacy of this description is assessed by
reintroducing the coupling to the continuum as a weak perturbation and
studying its scaling flow using Anderson’s poor man’s scaling.

##  I Introduction

The presence of impurities on superconductors results in subgap bound states
known as Yu-Shiba-Rusinov (YSR) states Yu (1965); Shiba (1968); Rusinov (1969)
(see e.g. Ref. Balatsky  _et al._ , 2006 for a review). These excitations can
be probed using scanning tunneling spectroscopy (STS) and appear as narrow
resonances in tunneling spectra Ji  _et al._ (2008); Heinrich  _et al._
(2018). YSR states were originally discovered as solutions to the scattering
problem of a magnetic impurity in bulk superconductors, by treating the
magnetic exchange with the impurity as a classical Zeeman field that couples
to the local spin density of quasi-particles Yu (1965); Shiba (1968); Rusinov
(1969); Balatsky  _et al._ (2006). However, this approximation does not take
into account the quantum nature of the impurity spin, which can give rise to
many-particle effects such as the Kondo effect Kondo (1964) and it is
determinant when e.g. describing the spin carried by the YSR excitations
Skurativska  _et al._ (2023).

A fully quantum-mechanical treatment of this problem aimed at providing a
comprehensive description of experiments Rubio-Verdú  _et al._ (2021), often
requires the use of sophisticated but numerically costly methods such as the
numerical renormalization group (NRG) Satori  _et al._ (1992); žitko  _et al._
(2011) or continuous time MonteCarlo Odobesko  _et al._ (2020).

Figure 1: Panel (a) shows a sketch of the studied system: a magnetic impurity
interacting via exchange with a proximitized thin metal film. Panels (b) and
(c) are the local density of states (LDoS) in the bulk of the superconductor
and in the thin proximitized metallic film. Panel (d) is a convoluted Heinrich
_et al._ (2018) spectroscopic measurement using STM of a thin (about 4 mono-
layers) proximitized Au film on V($100$)

In recent years, single-site models Affleck  _et al._ (2000); Vecino  _et al._
(2003); von Oppen and Franke (2021) have emerged as a computationally
affordable approach to treat some of the quantum many-particle aspects of the
YSR problem von Oppen and Franke (2021). These models have already been
successfully used to explain some spectral features observed in recent
experiments Trivini  _et al._ (2023); Liebhaber  _et al._ (2022); Machida  _et
al._ (2022). Moreover, it has been also applied to explain the complex many-
body physics of a magnetic molecule on a clean gold film proximitized by a
superconducting substrateTrivini  _et al._ (2023).

Proximitized systems have been studied mainly in the diffusive limit using the
Usadel formalism Usadel (1970). This approach predicts the decay of the
proximity effect as well as spectroscopic features such as the closing of the
gap and the formation of a minigap Zhou  _et al._ (1998); le Sueur  _et al._
(2008). Experiments with diffussive systems Gupta  _et al._ (2004); Wei  _et
al._ (2019); Serrier-Garcia  _et al._ (2013); le Sueur  _et al._ (2008) have
clearly confirmed those predictions. However, the systems studied in this
context are mesoscopic in size and the experimental probes that have been
employed cannot resolve the behavior of a single magnetic impurity.

On the other hand, thanks to currently available growth techniques, it is
possible to grow clean metallic overlayers with thicknesses of few atomic
layers on top of superconductors Trivini  _et al._ (2023); Vaxevani  _et al._
(2022); Schneider  _et al._ (2022); Liu  _et al._ (2022). These novel hybrid
systems open the door to otherwise impossible on-surface synthesis, and may
allow one day the study of self-organized spin chains Zhao  _et al._ (2022);
Mishra  _et al._ (2021); Hieulle  _et al._ (2021) as well as other, more
complex, molecular structures Mishra  _et al._ (2020) on superconductors. Such
systems are clearly not in the diffusive limit and have to be described within
the ballistic limit. In this case, subgap bound states appear in the normal
region and extend into the superconductor over distances of the order of the
coherence length. The existece of such states has been known for some time,
since the work of de Gennes and Saint James de Gennes and Saint-James (1963);
Kulik (1969).

The aim of this work is to extend the application of the single-site model von
Oppen and Franke (2021) to describe the complexity of a single magnetic
impurities interacting with a thin metallic film in proximity to a
superconductor. This is a problem of much interest to a number of recent
experiments Trivini  _et al._ (2023); Cortés-del Río  _et al._ (2021); Island
_et al._ (2017). Below, we first study the system treating the magnetic
impurity as a classical spin in the ballistic limit where there is a single de
Gennes-Saint James (dGSJ) bound state in the gap. We find that a large overlap
exists between the wavefunctions of the dGSJ and YSR states. Motivated by this
result, we propose that the single-site model is a relevant simplified model
for complex system consisting of the magnetic impurity on the proximitized
thin film. The model can be solved exactly and also provides a computationally
cheap way to treat the many-particle effects associated with the quantum spin
of the impurity. The adequacy of the single-site model for the system of
interest here is assessed by means of a ”poor man’s” scaling analysis. To this
end, we introduce a Hamiltonian consisting of a single-site model perturbed by
an impurity-mediated coupling to the continuum of other excitations. Under
certain conditions we find that, as the high-energy continuum states are
integrated out, the impurity remains most strongly coupled to the single site
describing the dGSJ state.

The structure of the article is the following: In Sec. II we describe the
system and the approximations used. Section III is divided in two subsections
with the first focusing on the YSR states resulting from the interaction of
the magnetic impurity with the proximitized film. This study is undertaken
assuming the spin of the magnetic impurity can be treated classically. In the
second subsection, we describe the calculation of the wave function overlap
between the YSR and dGSJ states. In Sec. IV, we introduce the single-site
model for the magnetic impurity on a proximitized film. Finally, in section V,
we argue that the single-site model provides an accurate description of this
system using Poor man’s scalingAnderson (1970). The most technical details of
the calculations have been relegated to the Appendices.

##  II  System and Model

Fig. 1 shows a schematic picture of the system studied in this work, which is
motivated by experiments reported in Ref. Trivini  _et al._ , 2023 and Refs.
Vaxevani  _et al._ , 2022; Schneider  _et al._ , 2022; Liu  _et al._ , 2022.
The system consists of a magnetic impurity on top of a thin normal metal film
(N) in proximity to a superconductor (S). The superconductor occupies the
half-space $x>0$, while the N film corresponds to $-a<x<0$. The system is
translationally invariant in the $(y,z)$-plane, so it is convenient to
describe the electron wave function as $\psi(x,\mathbf{k_{\parallel}})$, where
$\mathbf{k_{\parallel}}$ is the component of the momentum vector parallel to
the S/N interface at $x=0$. We assume a perfect S/N interface with no Fermi
surface mismatch or potential barrier, such that the N region acts as a cavity
for electrons with energy $E<\Delta$: they undergo Andreev retro-reflections
at the S/N interface and normal specular reflections at the interface with
vacuum. According to the Bohr-Sommerfeld quantization rule, the phase
accumulated along a closed classical trajectory must be a multiple of $2\pi$.
In the N/S system under consideration, a closed trajectory consists of two
Andreev retro-reflections at the S/N interface and two normal reflections at
$x=-a$. Thus,

$\frac{2aE}{\hbar
v_{F}\cos\varphi}-\cos^{-1}\left(\frac{E}{\Delta}\right)=n\pi$ (1)

where $\cos^{-1}(E/\Delta)$ is the phase shift associated to each AR, and
$\cos\varphi={k_{\parallel}/k_{F}}$. Eq. (1), determines the subgap bound
states, also known as De Gennes-Saint James (dGSJ) states de Gennes and Saint-
James (1963). It is valid for clean N-layers with a mean free path larger than
the thickness $a$, and it describes a continuum of subgap states de Gennes and
Saint-James (1963); DE GENNES (1964).

For the STM experiments of interest to us here, assuming specular
tunnelingMcMillan (1968), the decay of the wavefunction of these excitations
in vacuum is determined by the metal work-function. This energy scale is of
the order of one electron-volt and therefore much larger than the
superconducting gap. Therefore, in vacuum, the tail of the dGSJ wave function
is essentially indistinguishable from that of an electron at the Fermi level
in the normal state, and excitations with finite $\bm{k}_{\parallel}$
penetrate less into the vacuum. As a result, when probed with a STM in the
tunneling regime, excitations with large $|\bm{k}_{\parallel}|$ are filtered
out Kieselmann (1987a); Arnold (1978); McMillan (1968) and dGSJ states are
observed as narrow subgap resonances made of dGSJ quasi-particles with
$\bm{k}_{\parallel}\approx\bm{0}$ Trivini  _et al._ (2023). Moreover, a small
amount of disorder will randomize trajectories with $\cos\varphi<a/l$, where
$l$ is the mean free path, suppressing the coherence of such trajectories. A
magnetic impurity on top of the proximitized film has compact and anisotropic
orbitals that typically couple to several scattering channels from the
substrate. However, since the dGSJ quasi-particles with
$\bm{k}_{\parallel}\simeq\bm{0}$ penetrate farther into the vacuum, they are
also expected to contribute substantially to the most strongly coupled
scattering channel. Thus, one can effectively approximate the tunneling
problem using a one-dimensional model which neglects the motion parallel to
the surface:

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{J}\;,$ (2)

where

$\begin{split}\mathcal{H}_{0}=&\sum_{\sigma}\int_{-a}^{\infty}dx\,\psi^{\dagger}_{\sigma}(x)\left[-\frac{\hbar^{2}}{2m^{*}}\partial^{2}_{x}-E_{F}\right]\psi_{\sigma}(x^{\prime})\\\
&+\int_{0}^{\infty}dx\,\Delta\>\psi^{\dagger}_{\uparrow}(x)\psi_{\downarrow}(x)+\mathrm{h.c.}\;,\end{split}$
(3)

and

$\mathcal{H}_{J}=\sum_{\sigma\sigma^{\prime}}J\psi^{\dagger}_{0\sigma}\bm{S}\cdot\bm{s}_{\sigma\sigma^{\prime}}\psi_{0\sigma^{\prime}}\;.$
(4)

Here, $\psi_{\sigma}(x)$ ($\psi^{{\dagger}}_{\sigma}(x)$) represents the
annihilation (creation) operator for an electron with spin
$\sigma=\uparrow,\downarrow$ in the metal-superconductor substrate. $H_{0}$
describes a proximitized thin film of thickness $a>0$. The first term contains
the kinetic energy and chemical potential $E_{F}$, and the second term is the
s-wave pairing potential. The pairing potential is not self-consistently
calculated. Corrections due to self-consistency result in a spatially non-
uniform pairing potential $\Delta(x)$, but they have only a small effect on
the spectral properties of the dGSJ states Arnold (1978); Kieselmann (1987b).
The magnetic exchange with the impurity is described by $H_{J}$, with $\bm{s}$
denoting the electron-spin Pauli matrices and $\bm{S}$ denoting the impurity
spin operator. The operators $\psi_{0\sigma}$ ($\psi^{{\dagger}}_{0\sigma}$)
annihilate (create) electrons at the position of the impurity. For the one-
dimensional model introduced above, $\psi_{0\sigma}=\psi_{\sigma}(x=-a)$. In
the following section, we analyze this model using the approach of Yu, Shiba,
and Rusinov (YSR) Yu (1965); Shiba (1968); Rusinov (1969), where the impurity
spin $\bm{S}$ is treated as a classical vector.

##  III YSR in Proximitized Thin Films

In the previous section, we have derived the equation that determines the
spectrum of subgap states (cf Eq. 1) using the Bohr-Sommerfeld semiclassical
approximation. As explained above, we will focus on the one-dimensional case,
which corresponds to $\cos\varphi=1$ in Eq. (1). To deal the coupling to the
magnetic impurity, we solve the model described by Eqs. (2)-(4). To this end,
we use Green’s functions (GFs) and follow the approach outlined in Ref.
Arnold, 1978. The technical details of the calculation are described in
Appendix A. From the knowledge of the retarded GFs, $G(\omega+i\eta,x,x)$, the
local density of states (LDoS) $\rho(\omega,x)$ of the system is obtained by
using $\rho(\omega,x)=-\tfrac{1}{\pi}\imaginary\>G(\omega+i\eta,x,x)$.

Fig. 2 (a) shows the LDoS on the surface as a function of film thickness. As
we increase the thickness, new dGSJ states enter the gap. The GF also has
poles with a finite imaginary part outside the superconducting gap that
correspond to states in the continuum (i.e. above the superconductor gap), and
give rise to McMillan-Rowell-Tomasch oscillations Rowell and McMillan (1966);
Tomasch (1966). From here on, we focus our discussion on thin films with a
single subgap bound state. In the following subsection, we tackle the coupling
to the magnetic impurity.

###  III.1 YSR states

The GF for the S/N system provides the starting point for calculating the
spectral properties of the YSR excitations. The properties of the latter can
be obtained by solving the following integral equation:

$\begin{split}G_{YSR}(x,x^{\prime})=&G(x,x^{\prime})+G(x,-a)V\times\\\
&(1-VG(-a,-a))^{-1}G(-a,x^{\prime})\;.\end{split}$ (5)

Here, $G(x,x^{\prime})$ is the GF obtained in the previous subsection. The
scattering potential for a spin-$S$ impurity in the Nambu notation is
$V=JS\sigma_{z}\tau_{0}$, assuming that the impurity (classical) spin points
along the $z$-axis.

Fig 2 (b) shows the evolution of the YSR state as a function of the exchange
coupling $\alpha=\nu_{0}\pi JS$, with $\nu_{0}$ being the normal metal DoS
defined so that the quantum phase transition (QPT), where the energy of the
YSR state crosses the center of the gap, happens for $\alpha=1$Yu (1965);
Shiba (1968); Rusinov (1969). Note that the exchange coupling splits dGSJ
state into two states (spin up and down), one of which shifts to higher energy
while the other shifts to lower energy, see Fig. 2(c). As $J$ increases beyond
a certain value, the higher energy state disappears into the continuum. From
this point on, the energy of the remaining subgap state behaves similarly to a
YSR in a bulk superconductor Yu (1965); Shiba (1968); Rusinov (1969). For
thicker films, with more than one dGSJ state, the behavior is similar: each
bound state splits in two, shifting in opposite directions depending on their
spin projection, with more excited states eventually merging in the continuum
and disappearing.

The transmutation of the dGSJ into the YSR state can be regarded as a
consequence of a spectral reorganization taking place around $|\omega|=\Delta$
caused by AR (see Fig. 1c). Using an analogy to semiconductor physics, YSR
states appear in a superconductor because the coherence ”peak” behavior
$\sim(\omega^{2}-\Delta^{2})^{-1/2}$ (cf. Fig.1 b) resembles a van Hove
singularity at the bottom (top) of the conduction (valence) band of a one-
dimensional insulator. Bound states appear due to the infinitesimal attraction
provided by the magnetic impurity Dirac-delta potential. However, in a
proximitized film, AR reorganizes the spectral weight by removing the van
Hove-like singularity while shifting most of its spectral weight to the dGSJ
state (cf. Fig.1 c). Together with the localization of the dGSJ states at the
surface, this enables the transmutation of one of the dGSJ states per spin
into a YSR. Thus, a large overlap of the wavefunctions of YSR and dGSJ states
is expected, as explicitly demonstrated in the following subsection.

Figure 2: (a) Evolution of the dGSJ states as a function of the thickness of
the metallic layer. (b) Evolution of the YSR state for a fixed metallic layer
thickness as a function of the exchange coupling. (c) Zoom-in of the evolution
of the YSR state. (d) Wavefunction of the SJdG and YSR states averaged over
distances $\gg k^{-1}_{F}$.

###  III.2 Overlap between SJdG and YSR States

In this section, we compute the overlap of the YSR and the dGSJ states as a
function of the exchange coupling $J$. This can be achieved by using the GF
obtained from the scattering solution of the problem with and without magnetic
impurity. The square of the overlap is computed from the following integral
involving the residue of the two GFs:

$\displaystyle|\Theta|^{2}$ $\displaystyle=\int
dx\left[u_{dGSJ}(x)u^{*}_{YSR}(x)+v_{dGSJ}(x)v^{*}_{YSR}(x)\right]$
$\displaystyle=\int
dxdx^{\prime}\,\mathrm{Tr}\left\\{\Res\>G(x,x^{\prime}))\Res\>G_{YSR}(x^{\prime},x)\right\\}.$
(6)

Here $\Res G_{YRS}$ ($\Res G$) is the residue of the Nambu GF matrix at the
YSR (dGSJ) pole with spin up.

In Fig. 3 (a) and (b) we show the behavior of the overlap $\Theta$ as a
function of exchange coupling $J$ for different values of film thickness
(which determines the dGSJ state energy). To check our results beyond the
leading order in $\Delta/E_{F}$, we also compute the overlap by solving the
Bogoliubov-de Gennes equations for a one-dimensional tight-biding chain
containing up to $2000$ sites. The results are shown in Fig. 3(c) as a
function of $J$ normalized to the critical value $J_{c}$ where the system
undergoes the parity-changing quantum phase transition Yu (1965); Shiba
(1968); Rusinov (1969); Balatsky  _et al._ (2006).

Figure 3: Overlap between the YSR wavefunction and the SJdG wavefunction.
Panels (a) and (b) show the calculation done from the continuous model for
different SJdG bound state energies and values of $E_{F}/\Delta$. Panel (c)
shows the same calculation done with a tight-binding model.

The overlap between the YSR and dGSJ states decreases as the exchange coupling
increases, but it remains close to unity even across the quantum phase
transition. It is worth noting that the energy of the YSR excitation shifts
away from that of the dGSJ state as the exchange coupling is increased. The
significant overlap between the two states suggests that the YSR state
primarily descends from the dGSJ state, with a minor contribution from the
continuum states of the proximitized film. Therefore, in a first
approximation, the coupling with the magnetic impurity can be described by
replacing the proximitized film with a single level representing the dGSJ
state.

##  IV Single-site Model

Motivated by the results of the previous section, we introduce a simplified
model that replaces the entire proximitized film with a single site
representing the dGSJ state. As we show below, this model is useful for
analyzing the coupling between the dGSJ state and a quantum spin. The
Hamiltonian of the single site is given by:

$H_{0}=\sum_{\sigma}E_{s}\left(\gamma^{{\dagger}}_{\sigma}\gamma_{\sigma}-\frac{1}{2}\right),$
(7)

where $\gamma_{\sigma}$ ($\gamma^{{\dagger}}_{\sigma}$) are the annihilation
(creation) operators for a dGSJ quasi-particle with spin
$\sigma=\uparrow,\downarrow$, and $E_{s}$ is the eigenvalue of the BdG
Hamiltonian (in the absence of magnetic impurity). As explained in Appendix B,
this Hamiltonian can be recast in terms of electron operators
$d_{\sigma},d^{{\dagger}}_{\sigma}$ as follows:

$H_{0}=U\sum_{\sigma}n_{\sigma}+\left[\Delta_{s}d_{\downarrow}d_{\uparrow}+\mathrm{h.c.}\right],$
(8)

where $n_{\sigma}=d^{{\dagger}}_{\sigma}d_{\sigma}$; $U$ and $\Delta_{s}$ are
effective scattering and pairing potentials, respectively. In terms of $U$ and
$\Delta_{s}$, $E_{s}=\sqrt{U^{2}+\Delta^{2}_{s}}$. Without loss of generality,
below we discuss the particle-hole symmetric case where $U=0$ and therefore
$E_{s}=\Delta_{s}$

Next, we introduce the coupling to the impurity. To make contact with the
classical description employed in the previous section, we first discuss the
Ising limit of the exchange coupling, i.e.

$H^{\mathrm{Ising}}_{J}=J^{\parallel}_{dd}S^{z}\left(n_{\uparrow}-n_{\downarrow}\right),$
(9)

where $J^{\parallel}_{dd}>0$ is the exchange coupling with the dGSJ quasi-
particle. This model reproduces the most salient features of the YSR states
described above. To begin with, note that, besides the fermion parity
$P=\prod_{\sigma}(-1)^{n_{\sigma}}=\pm 1$, the impurity spin operator $S^{z}$
is also conserved in this limit, i.e.
$\left[S^{z},H_{0}+H^{\mathrm{Ising}}_{J}\right]=0$. Thus, the ground state is
doubly degenerate corresponding to the two possible orientations of the
classical vector $\bm{S}=\pm S\bm{\hat{z}}$: For $J_{dd}<J_{c}=2\Delta_{s}$
the ground state is one of the two following states
$\\{|BCS\rangle\otimes|\pm\tfrac{1}{2}\rangle\\}$ with $P=+1$ and
$\gamma_{\sigma}|BCS\rangle=0$. For $J_{dd}>J_{c}$, the ground state is one in
$\\{|\uparrow\rangle\otimes|-\tfrac{1}{2}\rangle,|\downarrow\rangle\otimes|+\tfrac{1}{2}\rangle\\}$
with $P=-1$ and $|\sigma\rangle=\gamma^{{\dagger}}_{\sigma}|BCS\rangle$. The
YSR excitation is a transition between these two ground states of opposite
parity with excitation energy von Oppen and Franke (2021); Trivini  _et al._
(2023) $|\Delta_{s}-J_{dd}/2|$. In addition, the odd parity sector of the
Hilbert space also contains the following two states:
$\\{|\uparrow\rangle\otimes|+\frac{1}{2}\rangle,|\downarrow\rangle\otimes|-\frac{1}{2}\rangle\\}$
with excitation energy equal to $\Delta_{s}+J_{dd}/2$. For small $J_{dd}$, a
transition from the ground state with $P=+1$ to these states corresponds to
the second subgap peak in the LDoS of the classical approach that shifts up in
energy with increasing exchange and eventually disappears into the continuum,
see Fig 2 (c).

Next, we generalize Eq. (9) by adding the spin-flip term, which allows the
impurity spin to fluctuate:

$H^{d}_{J}=J^{\parallel}_{dd}S^{z}\left(n_{\uparrow}-n_{\downarrow}\right)+J^{\perp}_{dd}(S^{+}d^{{\dagger}}_{\downarrow}d_{\uparrow}+\mathrm{h.c.})$
(10)

As argued in Refs. von Oppen and Franke, 2021; Vecino  _et al._ , 2003, the
single-site model provides an economical and fully quantum-mechanical
description of YSR spectra in superconductors which compares well with the
results obtained using sophisticated but computationally expensive methods
like the Numerical Renormalization Group (NRG) žitko  _et al._ (2011). The
accuracy of this description in the present system will be addressed in the
following section.

The spin-flip term, $\propto J^{\perp}_{dd}>0$, has important consequences for
the spectrum of the model. In the weak coupling limit, i.e. for
$J^{\parallel}_{dd}+2J^{\perp}_{dd}<2\Delta_{s}$, (assuming an unbiased
preparation of the system) the ground state is described by the following
density matrix:

$\rho_{GS}=\frac{1}{2}\left[|BCS,+\tfrac{1}{2}\rangle\langle\langle+\tfrac{1}{2},BCS|+|BCS,-\tfrac{1}{2}\rangle\langle-\tfrac{1}{2},BCS|\right],$
(11)

On the other hand, in the strong coupling limit where
$J^{\parallel}_{dd}+2J^{\perp}_{dd}>2\Delta_{s}$, the ground state is a
singlet:

$|GS\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle\otimes|+\tfrac{1}{2}\rangle+|\downarrow\rangle\otimes|-\tfrac{1}{2}\rangle\right).$
(12)

that is, a pure state resulting from the quantum superposition of the two
ground states of the Ising limit of the model. In weak and strong-coupling
regimes, unlike the conventional classical approach of YSR Yu (1965); Shiba
(1968); Rusinov (1969), the quantum model predicts that YSR excitations carry
no spin polarization Skurativska  _et al._ (2023).

Finally, since in the original model (cf. Eq. 4) the energy of the YSR does
not grow without bound as the exchange with the magnetic impurity $J$ becomes
arbitrarily large, the couplings $J^{\perp}_{dd},J^{\parallel}_{dd}$ cannot be
much larger than $\Delta_{s}$ in the single-site model. Note that, for large
$J^{\perp}_{dd},J^{\parallel}_{dd}$ the energy of the YSR grows like
$\max\\{J^{\perp}_{dd},J^{\parallel}_{dd}\\}$. Thus, for the energy of the YSR
to remain within the gap, the exchange couplings of the single-site model must
saturate to an upper bound so that
$\max\\{J^{\perp}_{dd},J^{\parallel}_{dd}\\}\lesssim\Delta_{s}$. Therefore,
they must be regarded as renormalized exchange interactions, which are also
the result of the spectral reorganization and localization of excitations with
energy $\sim\Delta_{s}$ caused by Andreev reflection at the S/N interface.

##  V Scaling Approach

In order to investigate the accuracy of the single-site model, we reintroduce
the coupling to the continuum of excitations as a perturbation. Whether this
perturbation changes the low-energy spectrum substantially or not can be
assessed using the poor man’s scaling method Anderson (1970), as we describe
in the following.

In the single-site model, the effective exchange coupling of the impurity and
dGSJ quasi-particle is $J_{dd}=J^{\perp}_{dd}=J^{\parallel}_{dd}$, where, for
the sake of simplicity, we assume an isotropic coupling. Our conclusions also
apply to the anisotropic case with small modifications. Through the exchange
interaction with the magnetic impurity, the dGSJ quasi-particles can also
couple to the continuum of excitations of the proximitized film. Let us
introduce the following modified exchange coupling which, besides the coupling
to the dGSJ, describes an impurity-mediated coupling of the dGSJ-site to the
continuum, and will be treated below as a perturbation:

$\begin{split}H^{dc}_{J}&=\sum_{\sigma\sigma^{\prime}}\left(J_{dd}\>d^{{\dagger}}_{\sigma}\bm{s}_{\sigma\sigma^{\prime}}d_{\sigma}+J_{\Phi\Phi}\>\Phi^{{\dagger}}_{0\sigma}\bm{s}_{\sigma\sigma^{\prime}}\Phi_{0\sigma^{\prime}}\right)\cdot\bm{S}\\\
&\qquad+J_{d\Phi}\sum_{\sigma\sigma^{\prime}}\left(d^{{\dagger}}_{\sigma}\bm{s}_{\sigma\sigma^{\prime}}\Phi_{0\sigma^{\prime}}+\Phi^{{\dagger}}_{0\sigma}\bm{s}_{\sigma\sigma^{\prime}}d_{\sigma^{\prime}}\right)\cdot\bm{S}.\end{split}$
(13)

The operators $\Phi_{0\sigma},\Phi^{{\dagger}}_{0\sigma}$ are the annihilation
and creation operators for electrons in the continuum at the position of
magnetic impurity. Phenomenologically, we have assumed different couplings for
the various processes involving the scattering of the dGSJ and the continuum
excitations by the impurity. These couplings can be calculated from first
principles. However, they depend on microscopic details of the matrix elements
of the impurity orbitals and the continuum of both subgap and outer-gap
excitations which are difficult to model. For this reason, we treat their bare
values as free parameters in the analysis below.

We carry out the poor man’s scaling analysis Anderson (1970) of the model (13)
by integrating out the high energy degrees of freedom from the continuum with
energies of the order of the bandwidth $D\sim E_{F}$. Since these band-edge
modes exhibit vanishing superconducting correlations because their energies
are well above the gap, the calculations do not differ much from those of the
standard Kondo scaling of a magnetic impurity Anderson (1970). Some details
are provided in Appendix C. In what follows, we focus on the discussion of the
solutions to the scaling equations, which read

$\displaystyle\frac{dg_{\Phi\Phi}}{d\ell}$ $\displaystyle=g^{2}_{\Phi\Phi},$
(14) $\displaystyle\frac{dg_{d\Phi}}{d\ell}$
$\displaystyle=g_{d\Phi}g_{\Phi\Phi},$ (15)
$\displaystyle\frac{dg_{dd}}{d\ell}$ $\displaystyle=g^{2}_{d\Phi}.$ (16)

Here $g_{dd}=2\nu_{0}J_{dd}$, $g_{d\Phi}=2\nu_{0}J_{d\Phi}$, and
$g_{\Phi\Phi}=2\nu_{0}J_{\Phi\Phi}$ are dimensionless couplings, $\nu_{0}\sim
1/D$ being the mean density of continuum states. The scaling variable $\ell$
is defined such that the bandwidth is reduced according to
$D(\ell)=De^{-\ell}\to 0$ as $\ell\to+\infty$, where $D\sim E_{F}$.

As the bandwidth of the system is reduced, the above scaling equations imply
that the renormalization of $g_{dd}$ and $g_{d\Phi}$ is driven by the growth
of $g_{\Phi\Phi}$. Indeed, Eq. (14) for $g_{\Phi\Phi}$ is mathematically
identical to the scaling equation for the exchange coupling of a magnetic
impurity in a normal metal (Kondo scaling). It can be readily solved by the
ansatz $g_{\Phi\Phi}(\ell)=(\ell^{*}-\ell)^{-1}$, where
$\ell^{*}=1/g_{\Phi\Phi}(0)$. Like the ordinary Kondo scaling, $\ell^{*}$
corresponds to the logarithmic scale where $g_{\Phi\Phi}(\ell)$ diverges and
the perturbative renormalization breaks down. This happens when the bandwidth
becomes of the order of a “Kondo temperature”, $T^{\Phi}_{K}$, i.e. for
$\ell^{*}=\log(D/T^{\Phi}_{K})$. Hence, $g_{\Phi\Phi}(\ell^{*})\sim 1$ leads
to $T^{\Phi}_{K}=De^{1/(2\nu_{0}J_{\Phi\Phi})}$. Note that
$T^{\Phi}_{K}\gg\Delta_{s}$ would imply that the continuum states at energies
much higher than the superconducting gap are strongly coupled to the magnetic
impurity. In this situation, the single-site description as introduced above
breaks down. In the classical approach, such a strong coupling to the
continuum should result in substantial suppression of the overlap between the
YSR and dSGJ states.

Indeed, the wavefunction overlap $\Theta$ (cf. Fig. 3) can be used to obtain a
rough estimate the ratios of the bare couplings $g_{d\Phi}(0)/g_{dd}(0)$, and
$g_{\Phi\Phi}(0)/g_{dd}(0)$. To this end, we first notice that $g_{dd}\sim
J_{dd}$, $g_{d\Phi}\sim J_{d\Phi}$, and $g_{\Phi\Phi}(0)\sim J_{\Phi\Phi}$
contain matrix elements with zero, one, and two powers of the continuum
orbitals, respectively (recall that the exchange couplings are second order in
the matrix element describing the tunneling between the impurity magnetic
orbital and the metallic host states). Let $\gamma=1-|\Theta|$ measure the
degree of admixture of the YSR state with the continuum; $\gamma$ will be
enhanced by quantum fluctuations relative to the estimates provided by the
classical approach (cf. Sec. III). Nonetheless, we expect $\gamma$ to remain
much smaller than one. Thus, $g_{dd}(0)\sim\gamma^{0}$, $g_{d\Phi}\sim\gamma$
and $g_{\Phi\Phi}\sim\gamma^{2}$, to leading order in $\gamma$. Furthermore,
$g_{dd}(0)=2\nu_{0}J_{dd}\sim\Delta_{s}/D\sim\Delta/D\ll 1$ according to the
discussion at the end of the previous section.

Next, we proceed to obtain solutions to the scaling equations using the above
estimates for the initial conditions of the flow. Concerning the solutions of
(15) and (16), we notice that (15) is solved by the ansatz
$g_{d\Phi}(\ell)=r_{d\Phi}/(\ell^{*}-\ell)$ with
$r_{d\Phi}=g_{d\Phi}(0)/g_{\Phi\Phi}(0)$. Introducing this result into Eq.
(14) and integrating, we obtain the following renormalized coupling between
the impurity and the dSGJ:

$g_{dd}(\ell)=g_{dd}(0)+\frac{g^{2}_{d\Phi}(0)}{g_{\Phi\Phi}(0)}\frac{(\ell/\ell^{*})}{1-(\ell/\ell^{*})}$
(17)

Using
$g^{2}_{d\Phi}(0)/g_{\Phi\Phi}(0)=\gamma^{2}g^{2}_{dd}(0)/[\gamma^{2}g_{dd}(0)]\simeq\gamma^{0}g_{dd}(0)$,
the above expression simplifies to:

$g_{dd}(\ell)\simeq\frac{g_{dd}(0)}{1-(\ell/\ell^{*})}.$ (18)

which needs to be compared with the behavior of the renormalized coupling to
the continuum after setting $g_{\Phi\Phi}(0)\simeq\gamma^{2}g_{dd}(0)$:

$g_{\Phi\Phi}(\ell)\simeq\frac{\gamma^{2}g_{dd}(0)}{1-(\ell/\ell^{*})}.$ (19)

Note that both couplings diverge at $\ell^{*}=\log(D/T^{\Phi}_{K})$ with
$T^{\Phi}_{K}\simeq De^{-1/2(\nu_{0}\gamma^{2}J_{dd})}\ll\Delta$ if $\gamma\ll
1$, which is consistent with what was discussed above. For instance, if we
choose $\gamma\approx 0.2$ (corresponding to $\Theta\approx 0.8$), then

$\frac{g_{\Phi\Phi}(\ell)}{g_{dd}(\ell)}\simeq\gamma^{2}\ll 1.$ (20)

Thus, as the continuum states are integrated out, the impurity remains most
strongly coupled to the single site describing the dGSJ quasi-particle and
therefore the single-site model remains an accurate description of the
magnetic impurity on the proximitized thin film.

Let us close this section by pointing out some potential problems with the
scaling analysis described above. First of all, like the original poor man’s
scaling Anderson (1970), the equations are obtained perturbatively. Therefore,
the solutions to the scaling equations are valid provided the couplings remain
small compared to unity. This is not a problem under the above assumptions
because the scale where the couplings diverge $\ell^{*}$ is much smaller than
the superconductor gap and the scaling must be stopped at the scale of
$\Delta$. As we get closer to the gap scale, the superconducting correlations
cannot be neglected, and taking them into account will modify the flows of the
renormalized couplings. Nevertheless, we should interpret the above analysis
as providing information on the tendency of the high-energy continuum states
to couple to the impurity in the presence of the coupling to the dGSJ state.
In order to follow the renormalization of the coupling to the continuum from
high to low energies, it would be desirable to carry out calculations using
the NRG and starting from a more microscopic description of the system, e.g.
using model parameters obtained from first principle calculations. Such
calculation should provide a more quantitative assessment of the accuracy the
single-site model introduced in this work for proximitized films.

##  VI Conclusions

We have studied the YSR excitations in a thin metal film proximitized by a
superconductor. This has been carried out by introducing a one-dimensional
model of the metal film/superconductor substrate. We have discussed the
spectrum of this model, which consists of subgap bound states known as de
Gennes-Saint James (dGSJ) states. We have shown that Andreeev-reflection at
the metal/superconductor interface leads to a substantial spectral
reorganization around and below the gap energy. Next, the spectrum of the
system when a magnetic impurity is deposited on the metal film has been also
described. Treating the impurity spin as a classical vector, we have found
there is substantial overlap of the wavefunctions of the Yu-Shiba-Rusinov
(YSR) and the dGSJ states. Motivated by these results, a single-site model has
been introduced. This model replaces the complexity of the proximitized film
with a single-site that represents the dGSJ quasi-particle excitation and is
coupled to the impurity with an effecive change coupling. The single-site
model is exactly solvable and allows us to go beyond the classical description
of the impurity by treating its spin quantum mechanically. Finally, we have
addressed the accuracy of the single-site model by phenomenologically re-
introducing the coupling to the continuum of excitations of the proximitized
film as a perturbation and using the poor man’s scaling method: Under
conditions suggested by the findings of the classical approach, we have shown
that the exchange coupling with the site that describes the dGSJ quasi-
particle excitation remains the dominant coupling under scaling. Thus, the
continuum of excitations of the proximitized film can be neglected in a first
approximation, and the YSR states can be regarded as resulting from the
exchange interaction of the magnetic (quantum) impurity with the dGSJ quasi-
particles.

The approach used here can be generalized to treat impurities with higher spin
and account for single-ion as well as magnetic exchange anisotropies. Our
results provide theoretical support for the model used to analyze the STS
spectra reported in Ref. Trivini  _et al._ , 2023. In addition, since the
single-site model introduced here is computationally cheaper than more
sophisticated numerical methods like the numerical renormalization group (NRG)
žitko  _et al._ (2011) or continuous-time Montecarlo Odobesko  _et al._
(2020), it can be used to model more complex systems such as chains or other
nanostructures of magnetic impurities on proximitized films, which would be
otherwise rather intractable by those methods. For this reason, we also
believe it is worth revisiting the system studied here using much more
sophisticated numerical tools, in order to quantitatively assess the
limitations of the single-site model as introduced in this work.

###### Acknowledgements.

We acknowledge financial support from Grants No. PID2019-107338RB-C61, No.
CEX2020-001038-M, No. PID2020-112811GB-I00, and PID2020-114252GB-I00, funded
by MCIN/AEI/ 10.13039/501100011033, from the Diputación Foral de Guipuzcoa,
the ELKARTEK project BRTA QUANTUM (no. KK-2022/00041), and from the European
Union (EU) through the Horizon 2020 FET-Open projects SPRING (No. 863098) and
SUPERTED (No. 800923), and the European Regional Development Fund (ERDF).
M.A.C. has been supported by Ikerbasque, Basque Foundation for Science, and
MCIN Grant No. PID2020-120614GB-I00 (ENACT). F.S.B. thanks Prof. Björn
Trauzettel for his hospitality at Würzburg University, and the A. v. Humboldt
Foundation for financial support. J.O. acknowledges the scholarship
PRE_2022_2_00950 from the Basque Government. The authors also thank Katerina
Vaxevani and Stefano Trivini for their help measuring the experimental
spectrum on Fig 1 (d) and several discussions.

##  Appendix A Two-layer Green’s Functions

As discussed in Ref. Feuchtwang, 1974, the GFs of a composite system (3) can
be obtained from the GFs of the constituent subsystems. We denote the GFs of
each subsystem as $g_{i}(x,x^{\prime})$, with $i=N,S$. Next, we impose the
following boundary conditions:

$\displaystyle\dfrac{dg_{i}(x,x^{\prime})}{dx}\Big{|}_{x=0}$
$\displaystyle=0,\qquad\dfrac{dg_{i}(x,x^{\prime})}{dx^{\prime}}\Big{|}_{x^{\prime}=0}=0,$
(21) $\displaystyle\dfrac{dg_{N}(x,x^{\prime})}{dx}\Big{|}_{x=-a}$
$\displaystyle=0,\qquad\dfrac{dg_{N}(x,x^{\prime})}{dx^{\prime}}\Big{|}_{x^{\prime}=-a}=0,$
(22) $\displaystyle g_{S}(x\to+\infty,x^{\prime})$ $\displaystyle=0,\qquad
g_{S}(x,x^{\prime}\to+\infty)=0,$ (23) $\displaystyle\lim_{\delta\to
0^{+}}\,\tau_{3}\dfrac{dg_{i}(x,x^{\prime})}{dx}\Big{|}^{x=x^{\prime}+\delta}_{x=x^{\prime}-\delta}$
$\displaystyle=\dfrac{2m}{\hbar^{2}},\qquad\lim_{\delta\to
0^{+}}\,\tau_{3}\dfrac{dg_{i}(x,x^{\prime})}{dx^{\prime}}\Big{|}^{x^{\prime}=x+\delta}_{x^{\prime}=x-\delta}=\dfrac{2m}{\hbar^{2}}.$
(24)

We assume a zero derivative at the vacuum interface, except for the semi-
infinite superconductor at $x,x^{\prime}\rightarrow\infty$, for which the GFs
are assumed to vanish.

Using the above boundary conditions, and assuming continuity of the full GF
and its derivative at the N/S interface, we obtain the following relations:

$G(x,x^{\prime})=\left\\{\begin{aligned}
&g_{S}(x,x^{\prime})\theta(x^{\prime})\mp
g_{S}(x,0)[g_{S}(0,0)+g_{N}(0,0)]^{-1}g_{S,N}(0,x^{\prime})\text{ , for
}x>0,x^{\prime}\gtrless 0\\\ &g_{N}(x,x^{\prime})\theta(-x^{\prime})\mp
g_{N}(x,0)[g_{S}(0,0)+g_{N}(0,0)]^{-1}g_{S,N}(0,x^{\prime})\text{ , for
}x<0,x\gtrless 0\end{aligned}\right.\;.$ (25)

The GFs for the isolated system are easy to calculate from Bogoliubov-de
Gennes equation McMillan (1968). After that, one can get the dressed GF from
(25), see [Arnold]. The GF shows a pole with the following energy
distribution:

$\dfrac{\omega}{\sqrt{\Delta^{2}-\omega^{2}}}\tan\left(\frac{2ma}{\hbar^{2}k_{F}}\omega\right)=1$
(26)

Note that by expanding the tangent around zero to the first order we arrive at
the same solution as obtained from a semi-classical argument Eq. (1).

##  Appendix B Single site Hamiltonian for a proximitized superconductor

By solving the Bogoliubov-de Gennes equations for the proximitized thin film,
the electron field operator at the position of the magnetic
impurity,$\phi_{0\sigma}$, can be written as follows:

$\psi_{0\sigma}=u_{0}\gamma_{\sigma}+\sigma
v^{*}_{0}\gamma^{\dagger}_{-\sigma}+\Phi_{o\sigma}.$ (27)

We start by changing the basis on the unperturbed Hamiltonian (3), where first
two terms described the dGSJ quasi-particle and $\Phi_{0\sigma}$ describes the
modes in the continuum. We now introduce a rotation for the operators creating
the discrete state:

$\begin{pmatrix}d_{\sigma}\\\
d^{\dagger}_{-\sigma}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\gamma_{0\sigma}\\\
\gamma^{\dagger}_{0-\sigma}\end{pmatrix}\;.$ (28)

This rotation leads to (8), where $U=E_{s}\cos 2\theta$ and
$\Delta_{s}=E_{s}\sin 2\theta$. Furthermore, requiring that

$\psi_{\uparrow}=\sqrt{Z}d_{\uparrow}++\Phi_{0\uparrow}=u_{0}\gamma_{\sigma}+\sigma
v^{*}_{0}\gamma^{\dagger}_{-\sigma}+\Phi_{0\uparrow}\;.$ (29)

Hence, $\tan\theta=-v_{0}/u_{0}$ and $Z=u_{0}^{2}+v_{0}^{2}$, where
$U=E_{s}(u_{0}^{2}-v_{0}^{2})$ and $\Delta_{s}=2E_{s}u_{0}v_{0}$.

##  Appendix C Calculation of the scaling equations

In order to perturbatively obtain the scaling equations of the model
introduced in Sec. V, we consider an expansion of the partition function of
the system, i.e.

$Z(D)=Z_{0}(D)\big{\langle}\mathcal{T}\exp\left[-\int^{\beta}_{0}H^{dc}_{J}(\tau)\right]\big{\rangle}_{0},$
(30)

in powers of the couplings $J_{dd},J_{d\Phi}$ and $J_{\Phi\Phi}$. In the above
expression for $Z(D)$ $Z_{0}(D)=\mathrm{Tr}\>e^{\beta H_{0}}$ is the partition
function of the system without magnetic impurity at inverse absolute
temperature $\beta=(k_{B}T)^{-1}$; $\langle\ldots\rangle_{0}$ is the
expectation value over the non-interacting grand canonical ensemble defined by
$H_{0}$. The operator $H_{J}(\tau)=e^{H_{0}\tau}H_{J}e^{-H_{0}\tau}$, where
$H_{J}$ is given in Eq. (13), describes the magnetic exchange with the
impurity in the interaction representation and $\mathcal{T}$ is the imaginary
time-ordering symbol. We have also introduced a parameter,
$D\leavevmode\nobreak\ E_{F}$, which is the bandwidth of the composite thin
film and superconductor system.

Following Anderson Anderson (1970), we shall use perturbation theory to obtain
a map onto a system with smaller bandwidth $D^{\prime}=D-\delta D<D$.
Associated with the bandwidths $D$ and $D^{\prime}$, there are also the
following characteristic (imaginary) time scale (in units where $\hbar=1$)
$\tau_{c}=D^{-1}$ and $\tau^{\prime}_{c}=(D^{\prime})^{-1}>\tau_{c}$. The
lowest order terms of the perturbation series for the system with bandwidth
$D$ read:

$Z(D)=Z_{0}(D)\left\\{1-\int d\tau\,\langle
H^{dc}_{J}(\tau)\rangle_{0}+\frac{1}{2!}\int\limits_{|\tau-\tau^{\prime}|>\tau_{c}=D^{-1}}d\tau
d\tau^{\prime}\,\langle
T\left[H^{dc}_{J}(\tau)H^{dc}_{J}(\tau^{\prime})\right]\rangle_{0}+\cdots\right\\},$
(31)

where we have made explicit the constraints on $\tau$ imposed by the finite
bandwidth of the continuum of states described by $\Phi_{\sigma}$ and
$\Phi^{{\dagger}}_{\sigma}$.

Next, let us integrate out the high energy degrees of freedom contained
$\Phi_{0\sigma}$ and $\Phi^{{\dagger}}_{0\sigma}$ (recall that
$d_{\sigma},d^{{\dagger}}_{\sigma}$ describe a low-energy subgap state and it
cannot be integrated out). Such degrees of freedom involve excitations with
energies $\sim D$ above the ground state and therefore determine the short
imaginary time behavior of the Green’s functions for $\Phi_{\sigma}$. Note
that, since at excitation energies $\sim D$ Bogoliubov quasi-particles either
behave as electrons or holes (in other words, either $u\to 0$ or $v\to 0$),
the anomalous GFs involving the operator $\Phi_{0}$, ie.
$\langle\mathcal{T}\left[\Phi_{0\uparrow}(\tau)\Phi_{0\downarrow}(\tau^{\prime})\right]\rangle_{0}$,
etc, vanish for $|\tau-\tau^{\prime}|\simeq\tau^{-1}_{c}$. Thus, in the above
perturbation series, for $|\tau-\tau^{\prime}|\sim\tau_{c}$, we need to
consider only normal correlations, which take the familiar Fermi liquid form:

$\langle\mathcal{T}\left[\Phi_{0\sigma}(\tau)\Phi^{{\dagger}}_{0\sigma^{\prime}}(\tau^{\prime})\right]\rangle_{0}\simeq\frac{\nu_{0}\delta_{\sigma\sigma^{\prime}}}{(\tau-\tau^{\prime})}$
(32)

for $|\tau^{\prime}-\tau|\simeq\tau^{-1}_{c}$, where $\nu_{0}$ is the (mean)
density of states of the normal state. Thus, the first non-constant
contribution to the scaling of the couplings stems from the second order term.
We first split the integrals over $\tau,\tau^{\prime}$ according to:

$\int\limits_{|\tau-\tau^{\prime}|>\tau_{c}=D^{-1}}d\tau
d\tau^{\prime}\,\ldots=\int\limits_{|\tau-\tau^{\prime}|>\tau^{\prime}_{c}=(D^{\prime})^{-1}}d\tau
d\tau^{\prime}\,\ldots+\int\limits_{\tau^{\prime}_{c}=(D^{\prime})^{-1}>|\tau-\tau^{\prime}|>\tau_{c}=D^{-1}}d\tau
d\tau^{\prime}\,\ldots$ (33)

and consider the terms in the second term for which
$\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}$. Expanding the second order
term in powers, corrections to the couplings contained in the first order term
are generated at $O(J^{2}_{d\Phi})$, $O(J_{d\Phi}J_{\Phi\Phi})$ and
$O(J^{2}_{\Phi\Phi})$. We explicity evaluate below the $O(J^{2}_{d\Phi})$
term. The calculations for the remaining terms are similar and not reproduced
here. Einstein’s convention of repeated index summation is used throughout:

$\displaystyle O(J^{2}_{d\Phi})$
$\displaystyle=\frac{J^{2}_{d\Phi}(D)}{2!}\int\limits_{\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}}d\tau
d\tau^{\prime}\,\left\\{\langle\mathcal{T}\left[S^{a}(\tau)S^{b}(\tau^{\prime})\right]\rangle_{0}\left(s^{a}_{\sigma\sigma^{\prime}}s^{b}_{\lambda\lambda^{\prime}}\right)\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)\Phi_{0\sigma^{\prime}}(\tau)\Phi^{{\dagger}}_{0\lambda}(\tau^{\prime})d_{\lambda^{\prime}}(\tau^{\prime})\right]\rangle_{0}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\langle\mathcal{T}\left[S^{a}(\tau)S^{b}(\tau^{\prime})\right]\rangle_{0}\left(s^{a}_{\sigma\sigma^{\prime}}s^{b}_{\lambda\lambda^{\prime}}\right)\langle\mathcal{T}\left[\Phi^{{\dagger}}_{0\sigma}(\tau)d_{\sigma^{\prime}}(\tau)d^{{\dagger}}_{\lambda}(\tau^{\prime})\Phi_{0\lambda^{\prime}}(\tau^{\prime})\right]\rangle_{0}\right\\}$
(34)
$\displaystyle=\frac{J^{2}_{d\Phi}(D)\nu_{0}}{2!}\int\limits_{\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}}d\tau
d\tau^{\prime}\,\left\\{\langle\mathcal{T}\left[S^{a}(\tau)S^{b}(\tau^{\prime})\right]\rangle_{0}\frac{\left(s^{a}_{\sigma\lambda}s^{b}_{\lambda\lambda^{\prime}}\right)}{(\tau-\tau^{\prime})}\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)d_{\lambda^{\prime}}(\tau^{\prime})\right]\rangle_{0}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\langle\mathcal{T}\left[S^{a}(\tau)S^{b}(\tau^{\prime})\right]\rangle_{0}\frac{\left(s^{b}_{\lambda\sigma}s^{a}_{\sigma\sigma^{\prime}}\right)}{(\tau-\tau^{\prime})}\langle\mathcal{T}\left[d_{\sigma^{\prime}}(\tau)d^{{\dagger}}_{\lambda}(\tau^{\prime})\right]\rangle_{0}\right\\}$
(35)
$\displaystyle=-\frac{J^{2}_{d\Phi}(D)\nu_{0}}{4}\int\limits_{\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}}d\tau
d\tau^{\prime}\,\frac{i\epsilon^{abc}\left[s^{a},s^{b}\right]_{\sigma\sigma^{\prime}}}{|\tau-\tau^{\prime}|}\langle
T\left[S^{c}(\tau)\right]\rangle_{0}\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)d_{\sigma^{\prime}}(\tau^{\prime})\right]\rangle_{0}$
(36)
$\displaystyle=-\frac{J^{2}_{d\Phi}(D)\nu_{0}}{2}\int\limits_{\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}}d\tau
d\tau^{\prime}\,\frac{\left(\epsilon^{abc}\epsilon^{abf}s^{f}_{\sigma\sigma^{\prime}}\right)}{|\tau-\tau^{\prime}|}\langle
T\left[S^{c}(\tau)\right]\rangle_{0}\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)d_{\sigma^{\prime}}(\tau^{\prime})\right]\rangle_{0}$
(37)
$\displaystyle=-\frac{J^{2}_{d\Phi}(D)\nu_{0}}{2}\int\limits_{\tau^{\prime}_{c}>|\tau-\tau^{\prime}|>\tau_{c}}d\tau
d\tau^{\prime}\,\frac{1}{|\tau-\tau^{\prime}|}\,\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)S^{c}(\tau)s^{c}_{\sigma\sigma^{\prime}}d_{\sigma^{\prime}}(\tau^{\prime})\right]\rangle_{0}$
(38)

In the above derivation we have used the following results:
$\epsilon^{abc}\epsilon^{abd}=2\delta_{cd}$ and

$\displaystyle\mathcal{T}\left[S^{a}(\tau)S^{b}(\tau^{\prime})\right]$
$\displaystyle=\theta(\tau-\tau^{\prime})S^{a}S^{b}+\theta(\tau^{\prime}-\tau)S^{b}S^{a}$
(39)
$\displaystyle=\frac{1}{2}(S^{a}S^{b}-S^{b}S^{a})\left[\theta(\tau-\tau^{\prime})-\theta(\tau^{\prime}-\tau)\right]+\frac{1}{2}(S^{a}S^{b}+S^{b}S^{a})$
(40)
$\displaystyle=\frac{i}{2}\epsilon^{abc}S^{c}\mathrm{sgn}(\tau-\tau^{\prime})+\left\\{S^{a},S^{b}\right\\}$
(41)

because $S^{a}(\tau)=e^{H_{0}\tau}S^{a}e^{-H_{0}\tau}=S^{a}$. As noted above,
the operators describing the dGSJ quasi-particle have time dynamics varying on
the scale of $\Delta^{-1}\ll\tau^{\prime}_{c}$, which is very slow compared to
the fast degrees of freedom being integrated out from $\Phi_{0c}$ and
$\Phi^{{\dagger}}_{c}$. Introducing $\tau_{-}=\tau-\tau^{\prime}$ and
$\tau_{+}=(\tau+\tau^{\prime})/2$. Thus, the term proportional to
$\\{S^{a},S^{b}\\}$ drops because it is multiplied by $\tau_{-}^{-1}$ rather
than $|\tau_{-}|^{-1}$ and the integral over $\tau_{-}$ of former vanishes to
leading order. Thus, to leading order in $\tau_{-}$, we are left with

$\displaystyle O(J^{2}_{d\Phi})$ $\displaystyle=-J^{2}_{d\Phi}(D)\nu_{0}\int
d\tau_{+}\,\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau_{+})\bm{S}(\tau_{+})\cdot\bm{s}_{\sigma\sigma^{\prime}}(\tau_{+})d_{\sigma}(\tau_{+})\right]\rangle_{0}\,\int\limits_{\tau^{\prime}_{c}>|\tau_{-}|>\tau_{c}}\frac{d\tau_{-}}{|\tau_{-}|}$
(42) $\displaystyle=-2\nu_{0}\frac{\delta D}{D}J^{2}_{d\Phi}(D)\int
d\tau\,\langle\mathcal{T}\left[d^{{\dagger}}_{\sigma}(\tau)\bm{S}(\tau)\cdot\bm{s}_{\sigma\sigma^{\prime}}(\tau)d_{\sigma}(\tau)\right]\rangle_{0}.$
(43)

In the last expression, we have evaluated the integral over $\tau_{-}$ using

$\int\limits_{\tau^{\prime}_{c}>|\tau_{-}|>\tau_{c}}\frac{d\tau_{-}}{|\tau_{-}|}=2\log\left(\frac{\tau^{\prime}_{c}}{\tau_{c}}\right)=2\log\left(\frac{D}{D^{\prime}}\right)=-2\log\left(\frac{D-\delta
D}{D}\right)\simeq\frac{2\delta D}{D},$ (44)

and replaced $\tau_{+}\to\tau$. Notice that the resulting expression in Eq.
(43) takes the same form as the contribution $\propto J_{dd}$ in the first
order term of (31). This leads to the following recursion relation:

$\displaystyle J_{dd}(D-\delta D)$
$\displaystyle=J_{dd}(D)+2\nu_{0}J^{2}_{d\Phi}(D)\frac{\delta D}{D}$ (45)

Assuming the couplings are continuous functions of the cut-off $D$, the
recursion relation becomes a differential equation:

$D\frac{dJ_{dd}(D)}{dD}=-2\nu_{0}J^{2}_{d\Phi},$ (47)

which implies that $J_{cc}$ increases with decreasing bandwdith $D$.

Similarly, we can tackle the terms at $O(J_{d\Phi}J_{\Phi\Phi})$ and
$O(J_{\Phi\Phi})$ (note the latter one is the only one present in the standard
poor man’s scaling treatment of the Kondo model). From those terms, the
following differential equations are obtained:

$\displaystyle D\frac{dJ_{d\Phi}(D)}{dD}$
$\displaystyle=-2\nu_{0}J_{d\Phi}J_{\Phi\Phi},$ (48) $\displaystyle
D\frac{dJ_{dd}(D)}{dD}$ $\displaystyle=-2\nu_{0}J^{2}_{\Phi\Phi}.$ (49)

It is convenient to introduce a new scaling variable defined by the
differential equation:

$\frac{dD}{D}=d\ell\Rightarrow D(\ell)=D_{0}e^{-\ell}.$ (50)

Thus, as $\ell\to+\infty$ $D(\ell)\to 0$. Furthermore, if we define the
dimensionless couplings $g_{dd}=2\nu_{0}J_{dd}$,
$g_{d\Phi}=2\nu_{0}J_{d\Phi}$, and $g_{\Phi\Phi}=2\nu_{0}J_{\Phi\Phi}$, we
finally arrive at the scaling equations (14) to (16) discussed in Sec. V.

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# Lightweight Low-Noise Linear Isolator Integrating Phase-Engineered Temporal
Loops

###### Abstract

The quest for efficient and versatile microwave and optical isolators has
recently spawned several novel space-time-modulated isolator structures.
However, such space-time isolators suffer from a large profile and complex
architecture caused by progressive nonreciprocal space-time coupling
properties. To overcome these limitations, we propose a nonmagnetic phase-
engineered temporal-loop-based isolator featuring large isolation levels, weak
undesired time harmonics, and a low profile. The proposed isolator is composed
of two temporal loops that provide desired constructive and destructive
interferences of different time harmonics. Furthermore, these two loops are
designed in a way to assure that the circulation and reflection of different
time harmonics strengthen a low insertion loss unidirectional signal
transmission. An experimental demonstration of the proposed time-modulated
isolator at microwave frequencies is provided, featuring strong unidirectional
wave transmission through the isolator with more than 27 dB contrast between
the forward and backward waves across a fractional bandwidth of $14.3\%$. The
proposed isolator outperforms the nonlinear-based and transistor-based
isolators by featuring a highly linear response with OP${}_{\text{1dB}}$ of
higher than $31$ dBm, high power rating of more than $47$ dBm, and a low noise
figure of $3.4$ dB.

###### keywords:

Isolator, Time modulation, Electromagnetics, Nonreciprocity,
Telecommunications

Sajjad Taravati George V. Eleftheriades

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering,
University of Toronto, Toronto, Ontario M5S 3G4, Canada
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Isolators are components that block electromagnetic wave signals in one
direction but allow them to pass in the opposite direction. Such a critical
task is required in numerous applications, including protecting microwave,
millimeter wave, terahertz and optical sources from back reflections
disturbing the source operation, or mitigating multi-path interference in
communication systems. Additionally, isolators generally improve the
designability of the overall system as they eliminate spurious interferences,
interactions between different components and undesired signal routing.
Isolators can be realized by the simultaneous breaking of time-reversal
symmetry and spatial inversion [1, 2, 3]. Conventional techniques and
structures for the realization of isolators include magnetically biased two-
dimensional electron gas systems [4], gyroelectric waveguides [5], transistor-
loaded transmission lines [6, 7, 8, 9, 10, 11, 12], magnetic ferrites [1, 13,
14, 15, 16, 17, 18, 19], nonlinearity [20], and space-time-modulation [21, 22,
23, 24]. Although these approaches have their own unique advantages and
applications, they suffer from distinct limitations and disadvantages that
restrict their applications. For instance, magnetically biased isolators
require bulky magnets and are incompatible with integrated circuit technology
[25]. An alternative approach is transistor-based isolators [8, 10, 12] which
eliminate the bulky magnet and are compatible with integrated-circuit
technology but suffer from limited power handling and noise performance, and
are of limited availability at high frequencies. Nonlinear isolators may be a
good choice for some applications but they only provide isolation to high
power signals, while they pass low-level signals quasi-reciprocally [26].

Time modulation has recently attracted a surge of interest thanks to its
distinctive capabilities in non-magnetic nonreciprocal signal transmission and
frequency generation [27, 28, 3, 29, 30, 31, 32, 33]. As a result, diversified
applications of time modulation techniques have recently been discovered,
including signal amplification [34, 35, 36], unidirectional beam splitting
[37], pure frequency conversion [38], multi-functional and nonreciprocal
metasurfaces [39, 40, 41, 42, 43], circulators [28, 44], nonreciprocal filters
[45], and multi-functional antennas [46, 47, 48]. The linear non-magnetic
nonreciprocity undergoing time modulation has been shown to be a prominent
alternative to conventional cumbersome ferrite-based, transistor-loaded and
nonlinear nonreciprocal devices [49, 50, 51, 24, 52, 53, 54, 46]. Such dynamic
meta-structures may be designed to introduce linear, highly efficient and
reconfigurable isolators.

The advent and development of space-time-modulated isolators over the past
decade has opened up an intriguing research domain for the realization of
versatile and efficient electromagnetic isolation at different frequencies. As
a result of such discoveries, several unique space-time-modulated isolators
have been proposed ranging from acoustics [54, 36] and microwaves [51, 23, 24]
to optics and photonics [21]. However, the isolation provided by the space-
time modulation requires an optically long structure so that an efficient
progressive unidirectional coupling between the fundamental space-time
harmonic and higher-order space-time harmonics occurs [21, 23, 24]. This in
turn leads to a complex and cumbersome architecture, which results in a
complex and costly fabrication. From the practical point of view, a space-
time-modulated isolator should be implemented using a set of subwavelength-
spaced varactors emulating a homogeneous space-time medium, where the length
of the medium is in the order of the temporal modulation frequency so that a
desired unidirectional coherency between the incident wave signal and the
modulation signal occurs. As a consequence, a space-time-modulated isolator
may require more than forty varactors [51, 47, 23, 24], which leads to long
and cumbersome devices.

Here, we propose a lightweight, low-profile, and non-magnetic isolator formed
by two time-modulated loops, each of which comprising phase-shifted temporal
transmission lines. The lightweight and low-profile properties originate from
the fact that it requires only four varactors. This is in contrast to space-
time-modulated isolators where a progressive unidirectional coherence is
required; Here the isolation is achieved by proper constructive and
destructive interferences at the two ports of the isolator. We show that by
proper design of the band structure of these phase-shifted temporal
transmission lines, a proper signal isolation between the forward and backward
waves can be achieved. The frequency and phase transition in a time-varying
transmission line is achieved by varactors excited by a time-varying voltage.
The isolator possesses a lightweight and low-profile structure, and exhibits
more than 32 dB isolation with very weak spurious time harmonics. Such
promising results show that this isolator can be used as a suitable
alternative to conventional bulky magnetic microwave isolators.

##  2 Results and Discussion

Figure 1 depicts the architecture of the time-modulated isolator. This
isolator is formed by two time-varying loops, themselves composed of two
phase-shifted temporal transmission lines. The left temporal loop is composed
of two (upper and lower) temporal transmission lines, where the capacitance of
the transmission lines is modulated in time, incurring a $\phi_{1}$ phase
difference between the upper and lower temporal transmission lines, i.e.,

$C_{1}(t)=C_{\text{av-1}}+\delta_{1}\cos(\Omega t),$ (1a)
$C_{1}^{\prime}(t)=C_{\text{av-1}}+\delta_{1}\cos(\Omega t+\phi_{1}).$ (1b)

Furthermore, the right temporal loop is composed of two (upper and lower)
temporal transmission lines, where the capacitance of the transmission lines
is modulated in time and incurring a $\pi$ phase difference between the upper
and lower temporal transmission lines, i.e.,

$C_{2}(t)=C_{\text{av-2}}+\delta_{2}\cos(\Omega t+\phi_{2}),$ (1c)
$C_{2}^{\prime}(t)=C_{\text{av-2}}+\delta_{2}\cos(\Omega t+\phi_{2}+\pi).$
(1d)

In (1a) to (1d), $C_{\text{av-1}}$ and $C_{\text{av-2}}$ are the average
permitivitties of the first and second loops, respectively, $\delta_{1}$ and
$\delta_{2}$ denote the modulation amplitudes of the first and second loops,
respectively, and $\Omega$ is the temporal modulation frequency. Here, the
right temporal loop assumes a stronger modulation amplitude, that is,
$\delta_{2}>>\delta_{1}$. We will show that by proper design of the phase and
amplitude of the time modulation of the two loops, an appropriate constructive
interference for the forward wave incidence (left to right propagation) and a
proper destructive interference for the backward wave incidence (right to left
propagation) may be achieved through the three (gray-colored in Fig. 1) nodes
of the architecture, i.e., at the input, middle and output of the structure.

To best understand the effect of the phase and amplitude of the time
modulation on the output voltage of the temporal transmission lines, we first
investigate the variation of the voltage through a temporal-varactor-loaded
transmission line. Figure 2 illustrates a transmission line loaded by a phase-
shifted temporal varactor. The time-varying capacitance of the transmission
line is expressed as

$C_{\text{eq}}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi),$ (2)

where $C_{\text{av}}$ is the average capacitance of the time-varying varactor.
The total capacitance of the transmission line reads
$C_{\text{tot}}(t)=C_{0}C_{\text{eq}}(t)$, where $C_{0}$ is the mean
capacitance per unit length of the transmission line. It may be shown that, by
proper engineering of the dispersion of the transmission line, the output
voltage acquires a frequency transition (up/down-conversion) accompanied by a
phase transition. Such frequency and phase transitions are shown in the
dispersion diagram in the right side of Fig. 2, where a frequency up-
conversion (from $\omega_{0}$ to $\omega_{0}+\Omega$) is accompanied by the
phase addition of $\phi$.

We assume the dispersion bands of the structure are engineered so that only
the fundamental and the first higher-order time-harmonics are excited, whereas
all higher-order time-harmonics are suppressed. Hence, the voltage is defined
based on the superposition of the $n=0$ and $n=-1$ space-time harmonics
fields, i.e.,

$V_{\text{S}}(z,t)=a_{0}(z)e^{-i\left(k_{0}z-\omega_{0}t\right)}+a_{1}(z)e^{-i\left(k_{0}z-(\omega_{0}+\Omega)t\right)}.$
(3)

Since the transmission line is modulated only in time, and not in space, we
consider an identical wave number $k_{0}$ for both the fundamental and the
higher-order harmonics. As it is shown in Fig. 2, and as a result of the
temporal sinusoidally periodic capacitance (in Eq. (2)) the dispersion diagram
of the temporal transmission line is periodic with respect to $\omega$ axis
[55], that is $k(\omega_{0}+\Omega)=k(\omega_{0})=k_{0}$. As a result of such
temporal periodicity, electromagnetic transitions are vertical in the
dispersion diagram (Fig. 2). The unknown spatially variant amplitudes
$a_{0}(z)$ and $a_{1}(z)$ are to be found through satisfying both the
Telegrapher’s equations and initial conditions at $z=0$. Considering a
lossless transmission line with time-varying capacitance, the Telegrapher’s
equations read

$\frac{\partial V(z,t)}{\partial z}=-L_{0}\frac{\partial I(z,t)}{\partial t},$
(4a) $\frac{\partial I(z,t)}{\partial
z}=-C_{0}\frac{\partial[C_{\text{eq}}(t)V(z,t)]}{\partial t},$ (4b)

where $L_{0}$ and $C_{0}$ are the inductance and capacitance per unit length
of the unloaded transmission line, respectively. Equations (4a) and (4b) yield

$\displaystyle\frac{\partial^{2}V(z,t)}{\partial
z^{2}}=\dfrac{1}{v_{\text{p}}^{2}}\frac{\partial^{2}[C_{\text{eq}}(t)V(z,t)]}{\partial
t^{2}},$ (5)

where $v_{\text{p}}=1/\sqrt{L_{0}C_{0}}$.

The up-conversion assumes the initial conditions of $a_{0}(0)=V_{0}$ and
$a_{1}(0)=0$, which gives (see Supplementary Information)

$a_{0}(z)=V_{0}\cos\left(\dfrac{\delta k_{1}}{4C_{\text{av}}}z\right),$ (6a)
$a_{1}(z)=V_{0}\dfrac{k_{1}}{k_{0}}e^{+i\phi}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right).$ (6b)

Equations (6a) and (6b) show that the change of the frequency and phase of the
up-converted signal corresponds to the frequency and phase of the temporal
modulation signal, $\Omega$ and $+\phi$. In addition, the maximum amplitude of
the up-converted signal $a_{1}(z)$ occurs at $\delta
k_{1}/C_{\text{av}}=2\pi$. This reveals that one can control the amplitude of
the up-converted signal by changing the modulation amplitude $\delta$.
Following the same procedure, we can determine the input and output voltages
for the down-conversion. The down-conversion assumes the initial conditions of
$a_{0}(0)=0$ and
$a_{1}(0)=V_{1}=\text{Max}[a_{1}(z)]=V_{0}k_{1}/k_{0}exp(+i\phi_{0})$, which
gives

$a_{1}(z)=V_{1}\cos\left(\dfrac{\delta k_{1}}{4C_{\text{av}}}z\right),$ (7a)
and $a_{0}(z)=V_{1}\dfrac{k_{0}}{k_{1}}e^{-i\phi}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right)=V_{0}e^{i(\phi_{0}-\phi)}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right).$ (7b)

Equations (7a) and (7b) show that the change of the frequency and phase of the
down-converted signal correspond to the frequency and phase of the temporal
modulation signal, $\Omega$ and $-\phi$.

Figure 3 shows a schematic representation of the dispersion diagram for
asymmetric frequency-phase transitions in a temporal transmission line. Here,
a frequency up-conversion from $\omega_{0}$ to $\omega_{0}+\Omega$ is
accompanied by a positive additive phase, i.e., $+\phi_{1}$ in the forward
transmission and $+\phi_{2}$ in the backward transmission. However, a
frequency down-conversion from $\omega_{0}+\Omega$ to $\omega_{0}$ is
accompanied by a negative additive phase, i.e., $-\phi_{2}$ in the forward
transmission and $-\phi_{1}$ in the backward transmission.

Figures 4 sketches a general representation of the architecture of the
proposed time-varying isolator. The isolator is composed of two temporal
loops, each of which formed by two sinusoidally time-varying transmission
lines. In order to achieve the desired constructive and destructive
interferences at the two ports of the isolator, for both forward and backward
signal injections, two power splitters/combiners possessing two quarter-
wavelength arms are utilized at the two ends of the isolator. In addition, a
half-wavelength transmission line is placed at the middle of the isolator to
separate the two loops from each other and provide desired constructive and
destructive interferences at different time harmonics. We aim to achieve a
unidirectional signal transmission by proper engineering of the forward
(Figures 4) and backward ( 4) frequency and phase transitions in the proposed
temporal isolator. In the forward signal transmission in Fig. 4, the signal at
$\omega_{0}$ is injected to port 1 of the isolator, passes through the two
loops of the isolator and generates first higher-order time harmonics. Here,
the fundamental time harmonic $\omega_{0}$ at the upper and lower arms of the
right-side loop acquire identical phases, and therefore combine coherently at
port 2 of the isolator. In contrast, the first higher-order time harmonics
acquire different phase shifts at the upper and lower arms of the right-side
loop, i.e., with $\pi$ phase difference, and hence produce a null at port 2 of
the isolator. As a result, due to the constructive interference of the
fundamental time harmonic $\omega_{0}$, and destructive interference of the
first higher-order time harmonics ($\omega_{+1}$ and $\omega_{-1}$) at the
common port of the right-side power combiner, a full transmission of the
fundamental time harmonic occurs while undesired harmonics are highly
suppressed.

For the backward signal injection, shown in Fig. 4, the input signal at
$\omega_{0}$ is injected to port 2 of the isolator (on the right side of the
isolator). The injected signal passes through the right side loop of the
isolator and generates first higher-order time harmonics but with different
phases at the upper and lower arms of the isolator, i.e., with $\pi$ phase
difference for the first higher-order time harmonics ($\omega_{+1}$ and
$\omega_{-1}$), while the fundamental time harmonic $\omega_{0}$ at the upper
and lower arms of the right-side loop acquire identical phases. As a
consequence, a destructive interference and suppression of the fundamental
time harmonic $\omega_{0}$ occurs by the half-wavelength long middle
interconnector. One should note that since the $\omega_{+1}$ and $\omega_{-1}$
time harmonics acquire $\pi$ phase difference at the upper and lower arms of
the right-side loop, the half-wavelength long middle interconnector provides a
constructive interference for them. However, these two harmonics
($\omega_{+1}$ and $\omega_{-1}$) will generate a complete null at the port 1
of the isolator (output port for backward transmission). We shall stress that,
as a result of the large asymmetry in the amplitudes of the two loops of the
isolator ($\delta_{1}<<\delta_{2}$), the $\omega_{0}$ harmonic that is
suppressed by the half-wavelength long middle interconnector in the backward
transmission, cannot be regenerated by the left-side loop which possesses a
small modulation amplitude. Thus, none of the time harmonics can pass through
the isolator in the backward direction.

Such a contrast between the forward and backward transmissions is based on the
asymmetry of the structure, where the two temporal loops possess different
modulation amplitudes. Here, the modulation amplitude of the right loop is
much larger than the left loop. Hence, the right loop produces much stronger
time harmonics and the left loop can be used for fine tuning of the isolator
parameters, e.g. the amplitude of the transmitted signals. The half-wavelength
long middle interconnector and the quarter-wavelength long arms of the two
power splitters are essential parts of this architecture, providing the
required destructive and constructive interferences at the fundamental and
first higher and lower order time harmonics.

Figure 5 shows the implementation of the time-modulated isolator by four
phase-shifted time-varying varactors. Here, four varactors are used for the
creation of phase-shifted time modulation. Two power splitters feed the four
temporal transmission lines, and eight fixed capacitances are considered for
decoupling of the DC bias and low frequency modulation signal from the
incident and transmitted high frequency incident microwave signal.

Figures 6 and 6 show photos of the top and bottom layers of the fabricated
lightweight low-profile magnetless time-modulated isolator. This isolator is
designed at the center frequency of 1.48 GHz with the modulation frequency of
$\Omega=2\pi\times 103.8$ MHz and modulation power of 20 dBm. Port 1 and Port
2 are the main ports of the isolator for incidence of the microwave signal,
while two other ports, i.e., Mod. 1 input and Mod. 2 input, are considered for
the modulation signal injection with different phase shifts.

We next demonstrate the experimental results for the scattering parameters of
the isolator for different bias voltages when the modulation is off
($\Omega=0$). Figure 7 shows the measurement setup for the input matching of
the two main ports of the isolator using an E8361C Agilent vector network
analyzer. Figure 7 plots the input matching of two ports of the isolator for
equal dc voltages for all varactors ($|S_{11}|=|S_{22}|$). We next measure the
input matching of the two ports of the isolator where the loops are supplied
by unequal dc voltages. Figure 7 plots the input matching where the left loop
is fed by the dc voltage of $V_{\text{L}}=1V$ and the right loop is fed by the
dc voltage of $V_{\text{R}}=7V$. It may be seen from this figure that a
supplied dc voltage of 7V for the right loop has opened up a desired input
matching ($|S_{11}|<-13|$ dB) around 1.5 GHz, whereas a much lower supplied dc
voltage of 1V for the left loop has not opened up a desired input matching
($|S_{11}|>-2$ dB). This is due to the fact that the capacitance of the
varactors in each loop vary with the supplied voltage and introduces different
impedances, i.e., the higher the dc voltage, the lower the capacitance of the
varactors. Figure 7 plots the input matching for another set of unequal dc
voltages for the right and left loops, i.e., $V_{\text{L}}=2V$ and
$V_{\text{R}}=9V$. Figures 7\- 7 show that the operation frequency and input
matching of the two ports of the isolator can be controlled individually by
the supplied dc voltage of the varactors that determines the average
capacitance of the varactors. To achieve a desired isolation at a given
frequency, we shall ensure that both ports of the isolator are matched through
the dc bias voltage of varactors.

Figure 8 shows a photo of the experimental set-up for the measurement of the
unidirectional signal transmission through the temporal isolator, where
$\Omega=2\pi\times 103.8$ MHz. Figure 8 plots the experimental results for
forward wave incidence and transmission, while Fig. 8 plots the experimental
results for backward wave incidence and transmission. It may be seen from the
experimental results that the contrast between the forward and backward
transmitted waves is more than 27 dB, whereas the undesired time harmonics are
highly suppressed.

Table 1 summarizes the performance of the fabricated prototype and compares
its performance with those of recently proposed isolators. In this table, NF
stands for noise figure, IL stands for insertion loss, RL stands for return
loss, Isol. denotes the isolation level, PH stands for power handling, and FBW
is an acronym for the fractional bandwidth which is equal to the frequency
bandwidth over the center frequency. Additionally, OP${}_{\text{1dB}}$ is the
output 1 dB compression point which shows the prominent linearity of the
isolator. Table 1 lists isolators realized through different technologies,
i.e., space-time (ST) modulation, magnetic-ferrite-loaded waveguides,
nonlinearity, and transistor-loaded transmission lines. The proposed isolator
can be realized in a much more compact fashion, and may be integrated into
chip technology at high frequencies thanks to the availability of variable
capacitors at such frequencies [56]. In addition, in contrast to transistor-
based isolators [6, 51] where nonlinear and low power handling (PH)
transistors are placed in series with the incident signal, our isolator is
endowed by high power handling as the varactors are placed in parallel to the
incident signal. In our experiment, we applied up to +33 dBm input signal, and
the power handling is expected to exceed $47$ dBm.

Additionally, the proposed time-modulated isolator exhibits a much lower noise
figure of 3.4 dB in comparison to two transistor-based isolators, 13 dB NF of
Ref. [6] and 6.7 dB NF of Ref. [51]. It can be argued that one of the main
reasons that transistor-based circulators never became main stream is their
poor noise performance. However, varactor diodes introduce much lower noise to
the system [57]. There exist several potential sources of noise, that is,
thermal noise of the series resistance of a varactor, shot noise of the
forward current, shot noise of the reverse current, ionisation noise and
reverse-breakdown noise [58]. Nevertheless, it has been proved, theoretically
and experimentally, that varactor-based parametric amplifiers and circuits
introduce much lower noise in comparison to other techniques [59, 57, 60, 61].

By leveraging the linear response of the time modulation, the proposed
temporal isolator exhibits an outstanding linear response, where the
OP${}_{\text{1dB}}$ is more than 31 dBm. Figure 9 plots the simulation (using
Keysight’s Advanced Design System commercial software) and experimental
results for the output power $P_{\text{O}}$ versus input power $P_{\text{I}}$
which shows the output signal follows the input signal linearly.

##  3 Conclusions

In this study, we have successfully demonstrated a phase-engineered temporal-
loop-based isolator. This isolator is featuring large isolation levels, weak
undesired time harmonics and a low profile. The experimental demonstration of
the proposed time-modulated isolator is carried out at microwave frequencies.
The experimental results show strong unidirectional wave transmission through
the isolator with more than 27 dB contrast between the forward and backward
waves across a fractional bandwidth of $14.3\%$. Such an isolator outperforms
the nonlinear-based and transistor-based isolators by featuring a highly
linear response with OP${}_{\text{1dB}}$ of higher than $31$ dBm, high power
rating of more than $47$ dBm, and a low noise figure of $3.4$ dB.

##  4 Experimental Section

This isolator is designed at the center frequency of 1.48 GHz with the
modulation frequency of $\Omega=2\pi\times 103.8$ MHz and modulation power of
20 dBm. The isolator is implemented on a RT6010 substrate with permittivity
$\epsilon_{\text{r}}=10.2$, thickness $h=50$ mil and $\tan\delta=0.0023$. The
two 50 Ohm coplanar waveguide (CPW) transmission lines at the back of the
structure possess a width of W=45 mils and a gap of G=20 mils. We have used
four BB837 varactors manufactured by Infineon Technologies for realizing the
phase-shifted time modulation. We utilized two ZFBT-6GQ+ minicircuits bias-
tees.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

This work is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC).

Conflict of Interest

The authors declare no conflict of interests.

Author contributions

S.T. carried out the analytical modeling, numerical simulations, sample
fabrication, and measurements. G.V.E. planned, coordinated, and supervised the
work. All authors discussed the theoretical and experimental aspects and
interpreted the results. All authors contributed to the preparation and
writing of the manuscript. Correspondence and requests for materials should be
addressed to Sajjad Taravati (email: sajjad.taravati@utoronto.ca).

Data Availability

The data that supports the findings of this study are available in the
supplementary material of this article.

Figure 1: Schematic representation of the magnetless isolator formed by two
time-modulated loops. Figure 2: Forward and backward frequency-phase
transitions in a temporal transmission line. Figure 3: Schematic
representation of forward and backward frequency-phase transitions in two
cascaded temporal transmission line sections.

Figure 4: Engineered frequency and phase transitions in the proposed temporal
isolator leading to unidirectional signal transmission. (a) Forward signal
injection, where the injected signal passes through the isolator. (b) Backward
signal injection, where the injected signal cannot pass through the isolator.
Figure 5: Implementation of the time-modulated isolator by four phase-shifted
time-varying varactors.

Figure 6: Photos of the fabricated lightweight low-profile magnetless time-
modulated isolator. (a) Top layer. (b) Bottom layer.

Figure 7: Experimental results for the input matching of the isolator for
different bias voltages when the modulation is off ($\Omega=0$). (a)
Measurement setup. (b) Equal dc voltages on all varactors. (c) Unequal dc
voltage for the left and right varactors, i.e., $V_{\text{L}}=1V$ and
$V_{\text{R}}=7V$. (d) Unequal dc voltage for the left and right varactors,
i.e., $V_{\text{L}}=2V$ and $V_{\text{R}}=9V$. The dots on top of the (b)-(d)
plots highlight the variation of the operation frequency by changing the dc
bias voltage of the varactor-loaded loops.

Figure 8: Experimental demonstration. (a) A photo of the measurement setup. (b) Forward transmission. (c) Backward transmission. Figure 9: Simulation and experimental results for linearity of the proposed temporal isolator. Table 1: Summary of the efficiency of the time-modulated isolator and comparison with previously reported isolators. Isolator |  Technology |  Size |  NF (dB) |  FBW ($\%$) |  Isol. (dB) |  IL (dB) |  RL (dB) |  OP${}_{\text{1dB}}$ (dBm) |  Properties
---|---|---|---|---|---|---|---|---|---
This work |  Time modulation |  Small, $<<\lambda_{\text{c}}^{2}$ ($53\times 55~{}\text{mm}^{2}$ at 1.48 GHz) |  3.4 |  14.2 |  $>27$ |  $<3.4$ |  $>16.3$ |  $>31$ |  lightweight, low-profile, low noise, linear, high PH RF bias
Ref. [23] |  Space-time modulation |  Moderate, $\approx\lambda_{\text{c}}^{2}/2$ ($117\times 56~{}\text{mm}^{2}$ at 1.5 GHz) |  4.8 |  6 |  20 |  $<2$ |  $>30$ |  $>21$ |  linear, Large profile, RF bias
Ref. [24] |  Space-time modulation (self-biased) |  Moderate, $\approx\lambda_{\text{c}}^{2}$ ($184\times 62~{}\text{mm}^{2}$ at 2 GHz) |  4.6 |  7.4 |  25.4 |  $<2$ |  $>25$ |  $>22$ |  Moder. FBW, No RF bias, linear
Ref. [14] |  Ferrite-loaded waveguide |  Moderate, $\approx\lambda_{\text{c}}^{2}/2$ ($30\times 6~{}\text{mm}^{2}$ at 15.4 GHz) |  N/A |  19 |  22 |  4.6 |  N/A |  N/A |  Cumbersome, magnetic bias
Ref. [62] |  Ferrite-loaded waveguide |  Large, $>>\lambda_{\text{c}}^{2}$ ($83\times 33~{}\text{mm}^{2}$ at 33.25 GHz) |  N/A |  40 |  25 |  1.5 |  15 |  33 |  Heavy, cumbersome
Ref. [63] |  Ferrite-loaded waveguide |  Large, $>>\lambda_{\text{c}}^{2}$ ($81\times 33~{}\text{mm}^{2}$ at 33.25 GHz) |  N/A |  40 |  27 |  1.5 |  8 |  34 |  Heavy and cumbersome
Ref. [20] |  Nonlinearity |  Moderate, $\approx\lambda_{\text{c}}^{2}$ at 0.75 GHz |  N/A |  5.33 |  25 |  $<12$ |  N/A |  N/A |  Narrow-band
Ref. [6] |  Transistor (200-nm HBT) |  Small, $<<\lambda_{\text{c}}^{2}$ ($0.64\times 0.43~{}\text{mm}^{2}$ at 33.25 GHz) |  13 |  18 |  30 |  2 |  $>10$ |  N/A |  Very low PH, high NF
Ref. [51] |  Transistor (180-nm CMOS) |  Small, $<<\lambda_{\text{c}}^{2}$ ($0.66\times 0.435~{}\text{mm}^{2}$ at 24 GHz) |  6.7 |  1 |  36.8 |  1.8 |  $>15$ |  $>13$ |  Very low PH, high NF, narrow-band

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