[2]Andrei D<EMAIL_ADDRESS>
# Superconductor-insulator transition in capacitively coupled superconducting
nanowires

Alex Latyshev  I.E. Tamm Department of Theoretical Physics, P.N. Lebedev
Physical Institute, 119991 Moscow, Russia  National Research University Higher
School of Economics, 101000 Moscow, Russia  Andrew G. Semenov  Department of
Physics, Moscow Pedagogical State University, 119435 Moscow, Russia  Institute
for Quantum Materials and Technologies, Karlsruhe Institute of Technology
(KIT), 76021 Karlsruhe, Germany

###### Abstract

We investigate superconductor-insulator quantum phase transitions in ultrathin
capacitively coupled superconducting nanowires with proliferating quantum
phase slips. We derive a set of coupled Berezinskii-Kosterlitz-Thouless-like
renormalization group equations demonstrating that interaction between quantum
phase slips in one of the wires gets modified due to the effect of plasma
modes propagating in another wire. As a result, the superconductor-insulator
phase transition in each of the wires is controlled not only by its own
parameters but also by those of the neighboring wire as well as by mutual
capacitance. We argue that superconducting nanowires with properly chosen
parameters may turn insulating once they are brought sufficiently close to
each other.

###### keywords:

quantum phase transitions; RG equations; quantum phase slips

##  1 Introduction

Quantum fluctuations dominate the physics of superconducting nanowires at
sufficiently low temperatures making their behavior markedly different from
that of bulk superconductors [1, 2, 3, 4]. Many interesting properties of such
nanowires are attributed to the effect of quantum phase slips (QPS) which
correspond to fluctuation-induced local temporal suppression of the
superconducting order parameter inside the wire accompanied by the phase
slippage process and quantum fluctuations of the voltage in the form of
pulses. By applying a bias current one breaks the symmetry between positive
and negative voltage pulses and, as a result, a superconducting nanowire
acquires a non-vanishing electric resistance down to lowest temperatures [5,
6]. This effect was directly observed in a number of experiments [7, 8, 9,
10].

Likewise, quantum phase slips in superconducting nanowires yield shot noise of
the voltage [11] which originates from the process of quantum tunneling of
magnetic flux quanta across the wire. One can also proceed beyond the voltage-
voltage correlator and evaluate all cumulants of the voltage operator, thus
deriving full counting statistics of quantum phase slips [12]. This theory
enables one to obtain a complete description of superconducting fluctuations
in such nanowires. Interesting QPS-related effects also occur in
superconducting nanorings which can be employed, e.g., for possible
realization of superconducting qubits [13]. Such effects were investigated
theoretically [14] and observed experimentally [15, 16].

Each quantum phase slip generates sound-like plasma modes [17] which propagate
along the wire and interact with other quantum phase slips. The exchange of
such Mooij-Schön plasmons produces logarithmic in space-time interaction
between different QPS which magnitude is controlled by the wire diameter
(cross section) [5]. For sufficiently thick wires this interaction is strong
and quantum phase slips are bound in close pairs. Accordingly, the (linear)
resistance of such wires tends to zero at $T\to 0$, thus demonstrating a
superconducting-like behavior in this limit. On the other hand, inter-QPS
interaction in ultrathin wires is weak, quantum phase slips are unbound and
the superconducting phase fluctuates strongly along the wire. In this case the
wire looses long scale superconducting properties, its total resistance
remains non-zero and even tends to increase with decreasing temperature thus
indicating an insulating behavior at $T\to 0$. At zero temperature the
transition between these two types of behavior comes as a quantum phase
transition (QPT) driven by the wire diameter [5]. Below we will also refer to
this QPT as superconductor-insulator transition (SIT).

In this work we are going to show that this SIT can be substantially modified
in a system of capacitively coupled superconducting nanowires even without any
direct electric contact between them. In our previous work [18] we already
elucidated some non-local QPS-related effects in such nanowires which yield
non-equilibrium voltage fluctuations in the system which exhibit a non-trivial
dependence on frequency and bias current. Here we will demonstrate that
quantum fluctuations in one of the two wires effectively ”add up” to those of
another one, thereby shifting QPT in each of the wires in a way to increase
the parameter range for the insulating phase. Qualitatively the same effect is
expected to occur in a single superconducting nanowire that has the form of a
meander frequently used in experiments.

##  2 The model

We first consider the system of two long parallel to each other
superconducting nanowires, as it is schematically shown in Figure 1a.

Figure 1: The systems under consideration: a) Two capacitively coupled
superconducting nanowires and b) Superconducting nanowire in the form of a
meander.

The wires are described by geometric capacitances $C_{1}$ and $C_{2}$ (per
unit wire length) and kinetic inductances $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ (times length) effectively representing the two transmission
lines. Capacitive coupling between these two nanowires is accounted for by the
mutual capacitance $C_{m}$. The corresponding contribution to the system
Hamiltonian that keeps track of both electric and magnetic energies in these
coupled transmission lines reads

$\hat{H}_{TL}=\frac{1}{2}\sum_{i,j=1,2}\int
dx(\mathcal{L}^{-1}_{ij}\hat{\Phi}_{i}(x)\hat{\Phi}_{j}(x)+(1/\Phi^{2}_{0})C^{-1}_{ij}(\nabla\hat{\chi}_{i}(x)\nabla\hat{\chi}_{j}(x)),$
(1)

where $x$ is the coordinate along the wires, $\mathcal{L}_{ij}$ and $C_{ij}$
denote the matrix elements of the inductance and capacitance matrices

$\check{\mathcal{L}}=\left[\begin{array}[]{crl}\mathcal{L}_{1}&0\\\
0&\mathcal{L}_{2}\\\
\end{array}\right],\quad\check{C}=\left[\begin{array}[]{crl}C_{1}&C_{m}\\\
C_{m}&C_{2}\\\ \end{array}\right]$ (2)

and $\Phi_{0}=\pi/e$ is the superconducting flux quantum. Note that for the
sake of simplicity here and below we set Planck constant $\hbar$, speed of
light $c$ and Boltzmann constant $k_{B}$ equal to unity.

The Hamiltonian (1) is expressed in terms of the dual operators
$\hat{\chi}(x)$ and $\hat{\Phi}(x)$ [14] which obey the canonical commutation
relation

$[\hat{\Phi}(x),\hat{\chi}(x^{\prime})]=-i\Phi_{0}\delta(x-x^{\prime})$ (3)

and are related to the charge density and the local phase operators,
respectively $\hat{Q}(x)$ and $\hat{\varphi}(x)$, by means of the following
equations

$\hat{Q}(x)=\frac{1}{\Phi_{0}}\nabla\hat{\chi}(x),\;\;\;\;\;\;\hat{\varphi}=2e\int^{x}_{0}dx^{\prime}\hat{\Phi}(x^{\prime}).$
(4)

Physically, $\hat{\Phi}_{i}(x)$ represents the magnetic flux operator, while
the operator $\hat{\chi}_{i}(x)$ is proportional to that for the total charge
$\hat{q}_{i}(x)$ that has passed through the point $x$ of the $i$-th wire up
to the some time moment $t$, i.e.
$\hat{q}_{i}(x)=-\hat{\chi}_{i}(x)/\Phi_{0}$.

Provided the wires are thick enough the low energy Hamiltonian in Eq. (1) is
sufficient. However, for thinner wires one should also account for the effect
of quantum phase slips. The corresponding contribution to the total
Hamiltonian for our system can be expressed in the form [14]

$\hat{H}_{QPS}=-\sum_{j=1,2}\gamma_{j}\int dx\cos(\hat{\chi}_{j}(x)),$ (5)

where

$\gamma_{j}\sim(g_{j\xi}\Delta/\xi)\exp(-ag_{j\xi}),\quad j=1,2$ (6)

denote the QPS amplitudes per unit wire length [6], $g_{j\xi}=R_{q}/R_{j\xi}$
is dimensionless conductance of the $j$-th wire segment of length equal to the
superconducting coherence length $\xi$ (here and below $R_{q}=2\pi/e^{2}\simeq
25.8$ K$\Omega$ is the quantum resistance unit and $R_{j\xi}$ is the normal
state resistance of the corresponding wire segment), $\Delta$ is the
superconducting order parameter and $a\sim 1$ is a numerical prefactor. We
also note that the Hamiltonian (5) describes tunneling of the magnetic flux
quantum $\Phi_{0}$ across the wire and can be viewed as a linear combination
of creation ($e^{i\hat{\chi_{i}}}$) and annihilation ($e^{-i\hat{\chi_{i}}}$)
operators for the flux quantum $\Phi_{0}$ .

It is obvious from Eq. (4) that QPS events cause redistribution of charges
inside the wire and generate pairs of voltage pulses moving simultaneously in
the opposite directions (cf., Figure 1a)

$\hat{V}_{i}(t)=1/\Phi_{0}\sum_{j=1,2}C^{-1}_{ij}(\nabla\hat{\chi}_{j}(x_{1},t)-\nabla\hat{\chi}_{j}(x_{2},t)).$
(7)

Clearly, in the presence of capacitive coupling quantum phase slips in one of
the wires also generate voltage pulses in another one.

To summarize the above considerations, the total Hamiltonian for our system is
defined as a sum of the two terms in Eqs. (1) and (5),

$\hat{H}=\hat{H}_{TL}+\hat{H}_{QPS},$ (8)

representing an effective sine-Gordon model that will be treated below.

##  3 Quantum phase transitions: renormalization group analysis

In order to quantitatively describe QPT in coupled superconducting wires we
will employ the renormalization group (RG) analysis. This approach is well
developed and was successfully applied to a variety of problems in condensed
matter theory, such as, e.g., the problem of weak Coulomb blockade in tunnel
[19, 20, 21, 22] and non-tunnel [23, 24, 25] barriers between normal metals or
that of a dissipative phase transition in resistively shunted Josephson
junctions [19, 26, 27, 28]. In the case of superconducting nanowires QPT was
described [5] with the aid of RG equations equivalent to those initially
developed for two-dimensional superconducting films [29] which exhibit
classical Berezinskii-Kosterlitz-Thouless (BKT) phase transition driven by
temperature. In contrast, quantum SIT in quasi-one dimensional superconducting
wires [5] with geometric capacitance $C$ and kinetic inductance
${\mathcal{L}}$ is controlled by the parameter [5]

$\lambda=\frac{R_{q}}{8}\sqrt{\frac{C}{\mathcal{L}}}$ (9)

proportional to the square root of the wire cross section $s$.

It follows immediately from the analysis of Ref. [5] that provided the two
superconducting wires depicted in Figure 1a are decoupled from each other,
i.e. for $C_{m}\to 0$, one should expect two independent QPT to occur in these
two wires respectively at $\lambda_{1}=2$ and at $\lambda_{2}=2$ where,
according to Eq. (9), we define
$\lambda_{1,2}=(R_{q}/8)\sqrt{C_{1,2}/{\mathcal{L}}_{1,2}}$. The task at hand
is to investigate the effect of capacitive coupling between the wires on these
two QPT.

For this purpose let us express the grand partition function of our system
${\mathcal{Z}}={\rm Tr}\exp(-\hat{H}/T)$ in terms of the path integral

${\mathcal{Z}}=\int D\chi_{1}\int D\chi_{2}\exp(-S[\chi_{1},\chi_{2}]),$ (10)

where

$S=\frac{1}{2\Phi^{2}_{0}}\sum_{i,j=1,2}\int
dxd\tau\Big{(}\xi\Delta\mathcal{L}_{ij}\partial_{\tau}\chi_{i}\partial_{\tau}\chi_{j}+\frac{1}{\xi\Delta}C^{-1}_{ij}\partial_{x}\chi_{i}\partial_{x}\chi_{j}\Big{)}-\sum_{i=1,2}y_{i}\int
dxd\tau\cos\chi_{i}$ (11)

is the effective action corresponding to the Hamiltonian (8) and

$y_{i}=\gamma_{i}\xi/\Delta\sim g_{j\xi}\exp(-ag_{j\xi})\ll 1$ (12)

denote effective fugacity for the gas of quantum phase slips in the $i$-th
wire. Note that, having in mind that the QPS core size in $x$\- and
$\tau$-directions is respectively $x_{0}\sim\xi$ and
$\tau_{0}\sim\Delta^{-1}$, in Eq. (11) for the sake of convenience we rescaled
the spatial coordinate in units of $x_{0}$, i.e. $x\to x\xi$ and the time
coordinate in units of $\tau_{0}$, i.e. $\tau\to\tau/\Delta$.

In the spirit of Wilson’s RG approach we routinely divide the $\chi$-variables
into fast and slow components $\chi_{i}=\chi_{i}^{f}+\chi_{i}^{s}$, where

$\displaystyle\chi^{f}_{i}(x,\tau)=\int_{\Lambda<\omega^{2}+q^{2}<\Lambda+\delta\Lambda}\frac{d\omega
dq}{2\pi}\chi_{\omega,q}e^{i\omega\tau+iqx},\;\;\;\chi^{s}_{i}(x,\tau)=\int_{\omega^{2}+q^{2}<\Lambda}\frac{d\omega
dq}{2\pi}\chi_{\omega,q}e^{i\omega\tau+iqx}.$ (13)

Setting $\delta\Lambda/\Lambda\ll 1$, expanding in the fast field components
$\chi_{i}^{f}$ and integrating them out we proceed perturbatively in $y_{1,2}$
and observe that in order to account for the leading order corrections it is
necessary to evaluate the matrix Green function at coincident points which
reads

$\check{G}^{f}(0,0)=\Phi^{2}_{0}\int\frac{d\omega
dq}{(2\pi)^{2}}\Big{(}\xi\Delta\check{\mathcal{L}}\omega^{2}+\frac{1}{\xi\Delta}\check{C}^{-1}q^{2}\Big{)}^{-1}=2(\delta\Lambda/\Lambda)\check{\lambda},$
(14)

where $\check{\lambda}=(R_{q}/8)\check{\mathcal{V}}\check{C}$ and
$\check{\mathcal{V}}=(\check{C}\check{\mathcal{L}})^{-1/2}$ is the velocity
matrix for plasmon modes propagating along the wires. The matrix
$\check{\lambda}$ has the form

$\check{\lambda}=\frac{1}{\sqrt{\frac{1}{v^{2}_{1}}+\frac{1}{v^{2}_{2}}+\frac{2\sqrt{1-\frac{C_{m}^{2}}{C_{1}C_{2}}}}{v_{1}v_{2}}}}\left[\begin{array}[]{crl}\lambda_{1}\left(\frac{1}{v_{1}}+\frac{\sqrt{1-\frac{C_{m}^{2}}{C_{1}C_{2}}}}{v_{2}}\right)&R_{q}C_{m}/8\\\
R_{q}C_{m}/8&\lambda_{2}\left(\frac{1}{v_{2}}+\frac{\sqrt{1-\frac{C_{m}^{2}}{C_{1}C_{2}}}}{v_{1}}\right)\\\
\end{array}\right],$ (15)

where $v_{i}=1/\sqrt{C_{i}\mathcal{L}_{i}}$ is the velocity of the Mooij-Schön
modes in the $i$-th wire in the absence of capacitive coupling between the
wires, i.e. for $C_{m}\to 0$.

Following the standard procedure [29] and proceeding to bigger and bigger
scales $\Lambda$, we eventually arrive at the following RG equations for the
QPS fugacities $y_{1}$ and $y_{2}$:

$\frac{dy_{i}}{d\log\Lambda}=(2-\lambda_{ii})y_{i},\quad i=1,2,$ (16)

where $\lambda_{11}$ and $\lambda_{22}$ are diagonal elements of the matrix
$\check{\lambda}$ (15). Note that here we restrict our RG analysis to the
lowest order in $y_{1,2}$ which is sufficient for our purposes. As long as one
keeps only the linear in $y_{1,2}$ terms in the RG equations all other
parameters of our problem, e.g., $\lambda_{ii}$, remain unrenormalized.

As it can be observed from Eqs. (16), our system exhibits two BKT-like QPT at
$\lambda_{11}=2$ and $\lambda_{22}=2$. In the limit $C_{m}\to 0$ the wires are
independent from each other, $\lambda_{11(22)}\to\lambda_{1(2)}$ and these QPT
obviously reduce to that predicted in Ref. [5]. However, for non-zero
capacitive coupling between the wires the two QPT occur at the values of
$\lambda_{1,2}$ exceeding 2. For the first wire the corresponding phase
transition point is fixed by the condition

$\lambda_{1}=2\frac{\sqrt{1+\frac{v^{2}_{1}}{v^{2}_{2}}+2\frac{v_{1}}{v_{2}}\sqrt{1-\frac{C_{m}^{2}}{C_{1}C_{2}}}}}{1+\frac{v_{1}}{v_{2}}\sqrt{1-\frac{C_{m}^{2}}{C_{1}C_{2}}}}.$
(17)

The same condition for the second wire is obtained from Eq. (17) by
interchanging the indices $1\leftrightarrow 2$.

The above results allow to conclude that in the presence of capacitive
coupling SIT in both wires occurs at larger values of $\lambda_{1,2}$ than in
the absence of such coupling. In other words, quantum fluctuations in one of
these wires effectively decrease superconducting properties of the other one.

It follows from Eq. (17) that the magnitude of such mutual influence depends
on the ratio of the plasmon velocities in the two wires $v_{1}/v_{2}$ and on
the strength of the capacitive coupling controlled by $C_{m}$. Provided the
wire cross sections $s_{1}$ and $s_{2}$ differ strongly the plasmon velocities
$v_{i}\propto\sqrt{s_{i}}$ also differ considerably. Assume, for instance,
that the first wire is much thinner than the second one. In this limit we have
$v_{1}\ll v_{2}$ and, hence, the QPT condition (17) in the first wire remains
almost unaffected for any capacitive coupling strength. If, on the contrary,
the first wire is much thicker than the second one, then one has $v_{1}\gg
v_{2}$ and the condition (17) reduces to $\lambda_{1}\simeq
2/\sqrt{1-C_{m}^{2}/(C_{1}C_{2})}$ demonstrating that the critical value
$\lambda_{1}$ can exceed 2 considerably for sufficiently large values $C_{m}$.

Figure 2: a) Critical surfaces corresponding to SIT at $\lambda_{11}=2$ and
$\lambda_{22}$=2. b) Phase diagram for two capacitively coupled
superconducting nanowires with $\lambda_{1}=2.01$ and $\lambda_{2}=2.03$. Both
curves $\lambda_{11}(C_{m})$ and $\lambda_{22}(C_{m})$ decrease and cross the
critical line $\lambda_{c}=2$ with increasing mutual capacitance $C_{m}$.

It is obvious that the strength of capacitive coupling depends on the distance
between the wires. At large distances this coupling is negligible $C_{m}\to
0$, whereas as the wires get closer to each other the value $C_{m}$ increases
and, hence, their mutual influence increases as well. Let us choose the wire
parameters in such a way that for $C_{m}=0$ both these wires remain in the
superconducting phase being relatively close to SIT. In this case the
parameters $\lambda_{1}$ and $\lambda_{2}$ should be just slightly bigger than
2. Moving the wires closer to each other we ”turn on” capacitive coupling
between them, thus, decreasing both values $\lambda_{1}$ and $\lambda_{2}$
below 2. As a result, two superconducting wires become insulating as soon as
they are brought sufficiently close to each other. This remarkable physical
phenomenon is illustrated by the phase diagram in Figure 2b.

In order to complete this part of our analysis we point out that transport
properties can be investigated in exactly the same manner as it was done,
e.g., in Ref. [5] in the case of a single nanowire. Generalization of the
technique [5] to the case of two capacitively coupled superconducting
nanowires is straightforward. For a linear resistance of the $i$-th wire
$R_{i}(T)$ and for $\lambda_{ii}>2$ (or for any $\lambda_{ii}$ at sufficiently
high temperatures) we obtain

$R_{i}(T)\propto\gamma^{2}_{i}T^{2\lambda_{ii}-3},\quad i=1,2.$ (18)

##  4 Extension to other geometries

The effects discussed here can be observed in a variety of structures
involving superconducting nanowires. For instance, superconducting nanowires
in the form of a meander (see Figure 1b) are frequently employed in
experiments, see, e.g., Ref. [30]. In this case different segments of the wire
are parallel to each other being close enough to develop electromagnetic
coupling. Having in mind the above analysis one expects that the wire of such
a geometry would be ”less superconducting” than the same wire that has the
form of a straight line.

For an illustration, let us mimic the behavior of the wire depicted in Figure
1b by considering three identical parallel to each other capacitively coupled
superconducting nanowires. For simplicity we will assume the nearest neighbor
interaction, i.e. the second (central) nanowire is coupled to both the first
and the third nanowires via the mutual capacitance $C_{m}$ whereas the latter
two are decoupled from each other. We again assume that the wires are thin
enough and quantum phase slips may proliferate in each of these wires.

Quantum properties of this system are described by the same effective action
(11) where the inductance and capacitance matrices now take the form

$\displaystyle\check{\mathcal{L}}=\left[\begin{array}[]{crl}\mathcal{L}&0&0\\\
0&\mathcal{L}&0\\\ 0&0&\mathcal{L}\\\
\end{array}\right],\quad\check{C}=\left[\begin{array}[]{crl}C&C_{m}&0\\\
C_{m}&C&C_{m}\\\ 0&C_{m}&C\\\ \end{array}\right],$ (25)

and the summation runs over the indices $i,j=1,2,3$. Proceeding along the same
lines as in the previous section we again arrive at Eq. (14), where the
diagonal elements of the matrix $\check{\lambda}$ now read

$\displaystyle\lambda_{22}=\frac{\lambda}{2}\left(\sqrt{1-\sqrt{2}\frac{C_{m}}{C}}+\sqrt{1+\sqrt{2}\frac{C_{m}}{C}}\right),$
(26)
$\displaystyle\lambda_{11}=\lambda_{33}=\frac{\lambda}{2}\left(1+\frac{1}{2}\left(\sqrt{1-\sqrt{2}\frac{C_{m}}{C}}+\sqrt{1+\sqrt{2}\frac{C_{m}}{C}}\right)\right)$
(27)

and the QPS interaction parameter $\lambda$ is defined in Eq. (9). We again
arrive at the RG equations of the form (16) (now with $i=1,2,3$). Being
combined with Eqs. (26), (27) these RG equations demonstrate that in the
presence of capacitive coupling SET occur at $\lambda_{ii}=2$ implying
$\lambda>2$ for each of the three wires. This observation is fully consistent
with our previous results derived for two coupled nanowires.

Furthermore, the RG equation (16) with $i=2$ combined with Eq. (26) also
describes the effect of interacting quantum phase slips and QPT in the wire
having the form of a meander (Figure 1b). In this case, within the
approximation of the nearest neighbor capacitive interaction between the wire
segments QPT occurs at

$\lambda=\frac{4}{\sqrt{1-\sqrt{2}\frac{C_{m}}{C}}+\sqrt{1+\sqrt{2}\frac{C_{m}}{C}}},$
(28)

i.e. the critical value of the parameter $\lambda$ exceeds 2 as soon as the
mutual capacitance $C_{m}$ differs from zero. As it is clear from Eqs. (26),
(27), the approximation of the nearest neighbor interaction appears to be well
justified in the limit $C_{m}\ll C$. For stronger interactions with $C_{m}\sim
C$ this approximation most likely becomes insufficient for a quantitative
analysis. However, on a qualitative level our key observations should hold
also in this case: A nanowire in the form of a straight line with $\lambda$
slightly exceeding the critical value 2 should demonstrate superconducting-
like behavior with $R(T)\propto T^{2\lambda-3}$ [5] whereas the wire with
exactly the same parameters may turn insulating provided it has the form of a
meander with capacitive coupling between its segments.

##  5 Results and Discussion

We have analyzed the effect of quantum fluctuations in capacitively coupled
superconducting nanowires. We have demonstrated that plasma modes propagating
in one such nanowire play the role of an effective quantum environment for
another one modifying the logarithmic interaction between quantum phase slips
in this wire. As a result, the superconductor-insulator quantum phase
transition gets shifted in a way to increase the parameter range for the
insulating phase. Hence, superconducting nanowires may turn insulating
provided they are brought close enough to each other. It would be interesting
to observe this effect in forthcoming experiments with superconducting
nanowires.

We acknowledge partial support by RFBR Grant No. 18-02-00586. AGS acknowledges
support by the Russian Science Foundation (project No. 19-72-10101).

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# Magnons and fundamental magnetic interactions in a ferromagnetic monolayer:
The case of Ni monolayer

Khalil Zakeri<EMAIL_ADDRESS>Heisenberg Spin-dynamics Group,
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany  Albrecht von Faber  Heisenberg Spin-
dynamics Group, Physikalisches Institut, Karlsruhe Institute of Technology,
Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany  Arthur Ernst  Institute for
Theoretical Physics, Johannes Kepler University, Altenberger Strasse 69,
A-4040 Linz, Austria  Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2,
D-06120 Halle, Germany

The experimental investigations of the magnetic interactions in an atomically
thin magnetic layer are essential to understand the physics of low-dimensional
magnets. The full spectrum of collective magnetic excitations (magnons) would
provide an access to these fundamental interactions on the atomic scale. Here
in order to be able to excite the magnons by means of spin-polarized electrons
we couple a Ni monolayer to one and two atomic layers of Co and probe the full
experimental magnon dispersion relation up to the Brillouin zone boundary.
Comparing to the results of ab initio calculations we quantify the complex
pattern of the magnetic exchange interaction in the Ni monolayer. We show that
although the magnons in this system are rather stiff, the Heisenberg exchange
coupling between the Ni spins is weak. We unravel the origin of the observed
large magnon stiffness constant being a consequence of the small spin density
of the Ni atoms.
The collective excitations of magnetic solids can be described by their
representative quasi-particles, called magnons. Of particular interest are the
high wavevector magnons, since they are governed by the magnetic exchange
interaction between neighboring spins [1]. This interaction, also known as
Heisenberg exchnage interaction (HEI), is indispensable for understanding many
phenomena in magnetism [2, 3, 4]. The Heisenberg spin Hamiltonian describing
this fundamental interaction is usually given by
$\mathcal{H_{\mathrm{HEI}}}=-\sum_{i\neq
j}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$, where the exchange parameter
$J_{ij}$, describes the interaction between atomic spins $\mathbf{S}_{i}$ and
$\mathbf{S}_{j}$ sitting on sites $i$ and $j$. The pattern of HEI in low-
dimensional itinerant magnets can be very complex [5, 6, 7, 8, 9, 6].
Fortunately, probing the full magnon dispersion relation provides a direct and
unambiguous way of resolving the complex pattern of HEI [9, 10, 11].

Several experimental techniques based on the inelastic scattering of neutrons,
photons and electrons have been successfully implemented to probe magnons in
bulk and ultrathin films of Fe and Co down to the monolayer and even
submonolayer regime. However, so far it has been challenging to probe the full
magnon dispersion relation in Ni, in particular a Ni monolayer. This may have
several reasons. (i) The inelastic scattering cross-section of neutrons as
bulk probes by magnons scales with the magnetic form factor which, in turn, is
directly related to the spin density of the unit cell. Owing to its rather
small magnetic moment and its itinerant magnetism character, Ni has not been
in the favor of the neutron scattering experiments [12]. The magnon spectrum
of bulk Ni has only been measured over a very small range of momentum [13, 14,
15]. (ii) Likewise, resonant inelastic x-ray scattering experiments have been
limited by effects associated with the x-ray fluorescence [16]. In the most
recent experiments only a small fraction of the Brillouin zone (BZ) could be
covered [17]. (iii) Theoretical calculations of the inelastic electron
scattering cross-section have revealed that due to the small exchange
splitting and the presence of Stoner excitations at low energies it is
practically not feasible to measure the magnon dispersion relation in Ni thin
films and at Ni surfaces when using electrons as probing particles [18]. All
these together have made the full magnon dispersion relation in Ni
inaccessible. Given the fact that the neutron and x-ray scattering techniques
do not have the monolayer sensitivity the experimental magnon spectrum as well
as the magnetic exchange parameters in a Ni monolayer have remained hitherto
fully unknown. Meanwhile, however, the theory of magnetic excitations has been
well advanced [2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In order to
verify the validity of the theoretical approaches, the authors had to compare
their results to the only one available set of the experimental data, which
covers a very small fraction of the Brillouin zone [13, 14, 15].

The magnons and the fundamental magnetic interactions in the Ni monolayer are
also of great interest in the context of unconventional topological
superconductivity in the Bi/Ni bilayer, where the interfacial Ni magnons are
proposed to be responsible for the superconducting pairing mechanism being of
$d_{xy}\pm id_{x^{2}+y^{2}}$ character [29].

Besides their importance for the fundamental understanding of the physics of
low-dimensional solids, magnons are also of great interest to the field of
magnonics, where the idea is to use the magnons as information carriers [30,
31, 32, 33].

In this Letter by performing spin-polarized high resolution electron energy-
loss spectroscopy (SPHREELS) experiments on specifically designing multilayer
structures we probe, for the first time, the full dispersion relation of the
magnon mode, which is partially localized in the Ni monolayer. We show that
the Ni magnons are rather stiff and show a notable dispersion relation up to
the surface BZ boundary. This is in sharp contrast to the results of the Fe
monolayer on the same substrate or on W(110) and Pd(001), where the magnons
are found to be rather soft. Comparing the results to those of first-
principles calculations we shall comment on the observed large magnon
stiffness constant. Moreover, we will provide the complex pattern of HEI in
the Ni monolayer.

We have shown earlier that in multilayer structures the spatial localization
of the magnons depends on the pattern of $J_{ij}$ [9, 34]. Under some
circumstances one may selectively excite magnons which are localized either
predominantly at the surface or interface, or almost equally at both.
Moreover, previous investigations have revealed that the excitation cross-
section of magnons with electrons is the highest when the sample surface is
composed of Co atoms [35, 36, 37, 38, 39, 40, 34]. Hence, in order to
substantially enhance the magnon excitation cross-section we cover the Ni
monolayer with one and two atomic layers of Co and designed the following
multilayers: Co/Ni/Ir(001) and 2Co/Ni/Ir(001). A similar idea has recently
been used to probe the standing spin waves with a parallel momentum
$q_{\parallel}<0.3$ Å-1 in Co/Ni multiyers composed of several atomic layers
[41, 42].

Figure 1: (a) Typical SPHREELS spectra recorded on the Co/Ni/Ir(001) structure
at $|q_{\parallel}|=0.7$ Å-1. The spectra were recorded at the incident energy
of $E_{i}=10$ eV and at room temperature. The red and blue spectra, denoted by
$I_{\downarrow}$ and $I_{\uparrow}$, were recorded with the spin polarization
vector of the incident electron beam being parallel and antiparallel to the
magnetization $\mathbf{M}$, respectively. The difference spectrum
$I_{\downarrow}-I_{\uparrow}$ is shown by the green color. The scattering
geometry is schematically illustrated in the inset. The energy and wavevector
of the incident (scattered) beam are shown by $E_{i}$ and $\mathbf{k_{i}}$
($E_{f}$ and $\mathbf{k_{f}}$), respectively. (b) A series of difference
spectra recorded for different magnon wavevectors $|q_{\parallel}|$ ranging
from 0.25 to 0.90 Å-1. The excitation energy (peak position) is marked by the
black circles.

All the sample preparation and magnon spectroscopy experiments were performed
under ultrahigh vacuum conditions. We first examine the Co/Ni/Ir(001)
epitaxial system. The surface of Ir(001) was cleaned using the standard
cleaning procedure, described in details in Refs. [43, 44, 34]. The procedure
leads to a well-ordered (1$\times$5) reconstructed surface. The Ni and Co
monolayers were epitaxially grown by molecular beam epitaxy at room
temperature. Figure 1(a) shows typical SPHREELS spectra recorded on the
Co/Ni/Ir(001) structure at a wavevector of $|q_{\parallel}|=|\Delta
k_{\parallel}|=|k_{i\parallel}-k_{f\parallel}|=0.7$ Å-1, where
$k_{i\parallel}$ and $k_{f\parallel}$ denote the parallel momentum of the
incoming and outgoing beam, respectively [see the inset of Fig. 1(a)]. The
spectra were recorded for the two possible spin orientations of the incoming
electron beam, i.e., parallel ($I_{\downarrow}$, red color) and antiparallel
($I_{\uparrow}$, blue) to the sample magnetization $\mathbf{M}$. Due to the
conservation of the total angular momentum during the scattering event, the
magnons are only excited by incidence of minority electrons (electrons with
their spin parallel to $\mathbf{M}$). Hence, the difference spectrum
$I_{\downarrow}-I_{\uparrow}$ includes all the information regarding the
magnons excitation energy (or frequency) [45, 46] and lifetime [47, 48, 49].
In this experiment $\mathbf{M}$ was parallel to the
$[\overline{1}10]$-direction and the magnon wavevector $q_{\parallel}$ was
along the $[110]$-direction. This corresponds to the $\bar{\Gamma}$–$\bar{\rm
X}$ of the surface BZ. The magnon dispersion relation was probed along this
symmetry direction. Different magnon wavevectors were achieved by changing the
scattering angles $\theta_{i}$ and $\theta_{f}$. A series of difference
spectra recorded for different values of $|q_{\parallel}|$ ranging from 0.25
to 0.90 Å-1 is shown in Fig. 1(b). The results clearly indicate that the
measured magnon mode exhibits a rather stiff dispersion, similar to the
standing spin waves in thick ferromagnetic layers [50].

Figure 2: The magnon dispersion relation of (a) Co/Ni/Ir(001) and (b)
2Co/Ni/Ir(001). The calculated magnon dispersion relation is shown for two
cases: without (light-blue) and with (dark-blue) considering the
reconstruction of the Ir surface. The lower insets show a schematic
representation of the structures. The upper inset in (a) shows the surface BZ.

The resulting magnon dispersion relation is summarized in Fig. 2(a),
representing the the acoustic magnon mode of the system. In order to
experimentally verify the partial localization of this mode in the Ni
monolayer and to ensure that the dispersion relation is governed by the
exchange parameters in both Ni and Co layer, we added another Co monolayer on
top of the structure and probed, once more, the magnon dispersion relation.
The results of the 2Co/Ni/Ir(001) multilayer are shown in Fig. 2(b). The very
similar dispersion relation for the two systems is an indication that the
probed magnon mode is partially localized in the Ni monolayer. However,
comparing the results to those of Fe monolayer on W(110) [51, 7, 52, 53] and
Pd(001) [54, 55], or those of buried Fe monolayer on the same substrate [56,
34], one realizes that in the present case the magnon dispersion relation is
very stiff. This is surprising, since it has been found theoretically that
both HEI and the Curie-temperature of Ni are much smaller than those of Fe and
Co [2, 3, 19, 27].

In order to quantify the strength of HEI and to unravel the origin of the
probed magnon mode we resort to the first-principles calculations of the
magnetic exchange parameters in both Co/Ni/Ir(001) and 2Co/Ni/Ir(001) systems.
Our first-principles calculations are based on density functional theory and
the fully relativistic Korringa-Kohn-Rostoker electronic structure method. In
this approach the values of $J_{ij}$ are computed within the framework of the
magnetic force theorem [57]. The experimental lattice parameters were used as
the input of the first-principles calculations. The calculations provide most
of the magnetic parameters of the systems including the matrix of $J_{ij}$.
The magnon dispersion relation was then computed based on the $J_{ij}$-matrix.
The results of calculations for the two systems are summarized in Fig. 2. The
calculated magnon dispersion relation agrees well with the experimental
results. Moreover, similar to the experiment the calculations indicate that
the acoustic magnon mode of the two systems is very similar. To unravel the
origin of different magnon modes we also calculated the magnon Bloch spectral
function (BSF) and the magnon density of states (DOS). When these quantities
are projected onto different layers they would provide an access to the
spatial localization of different magnon modes. In Fig. 3 we provide the
magnon BSF and DOS for the Co/Ni/Ir(001) structure (the results of
2Co/Ni/I(001) are presented in Supplemental Figure S1 [58]). The projected BSF
of the two magnon modes, shown in Fig. 3, indicates that the spectral weight
of the acoustic magnon mode near the high symmetry $\bar{\rm X}$ and $\bar{\rm
M}$ points as well as along the $\bar{\rm X}$–$\bar{\rm M}$ path is the
highest when magnon BSF is projected onto the Ni monolayer. This means that in
this part of BZ this mode is almost entirely localized in the Ni layer.
Likewise, the partial magnon DOS of the Ni monolayer [Fig.3(b)] is maximum in
this range of energy. While above a magnon energy of about 250 meV both the
magnon BSF and DOS are zero, the low energy part of the magnon spectrum shows
a finite spectral weight in the Ni monolayer. The projected magnon BSF and DOS
onto the Co layer indicate that except for the high-symmetry $\bar{\rm X}$ and
$\bar{\rm M}$ points and along the $\bar{\rm X}$–$\bar{\rm M}$ direction, the
magnons have a considerable spectral weight in the Co layer. The main
conclusion of the results presented in Fig. 3 is that the experimentally
probed magnon mode describes magnons, which have a finite spectral weight in
both layers. The larger spectral weight of this mode in the Co layer makes it
easily accessible to the electrons as probing particles. Hence, this mode is
very efficiently excited in this system. A similar conclusion can be drawn for
the 2Co/Ni/Ir(001) system [58].

Figure 3: The magnon Bloch spectral function (BSF) and the magnon density of states (DOS) for the Co/Ni/Ir(001) system, projected onto the Ni monolayer [(a) and (b)] and onto the Co layer [(c) and (d)]. For the sake of clarity, the data in (b) are multiplied by a factor of 2. Table 1: The magnetic exchange parameters (in meV) and the magnetic moments (in $\mu_{B}$) as calculated by our first-principles calculations. Co1 and Co2 refer to the first and second Co atomic layer on top of the Ni monolayer. For the sake of simplicity only the HEI in the Ni monolayer up to the third nearest neighbors (NNN) are presented. Multilayer system | $J^{NiNi}_{N\parallel}$ | $J^{NiCo_{1}}_{N\perp}$ | $J^{NiNi}_{NN\parallel}$ | $J^{NiNi}_{NNN\parallel}$ | $J^{NiCo_{1}}_{NN\perp}$ | $J^{NiCo_{1}}_{NNN\perp}$ | $J^{NiCo_{2}}_{N\perp}$ | $J^{NiCo_{2}}_{NN\perp}$ | $J^{NiCo_{2}}_{NNN\perp}$ | $\mu_{Ni}$ | $\mu_{Co_{1}}$ | $\mu_{Co_{2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Co/Ni/Ir(001)–1$\times$1 | 1.24 | 6.48 | 0.29 | -0.047 | 0.2 | -0.22 | – | – | – | 0.52 | 1.98 | –
Co/Ni/Ir(001)–1$\times$5 | 1.03 | 5.71 | 0.164 | 0.002 | 0.24 | -0.25 | – | – | – | 0.55 | 1.95 | –
2Co/Ni/Ir(001)–1$\times$1 | 2.75 | 6.28 | -0.001 | 0.275 | 0.6 | -0.21 | -1.13 | 0.11 | -1.06 | 0.57 | 1.84 | 1.87
2Co/Ni/Ir(001)–1$\times$5 | 1.84 | 4.93 | 0.008 | 0.178 | 0.55 | -0.21 | -0.69 | 0.24 | -0.79 | 0.56 | 1.81 | 1.88

Our first-principles calculations indicate that the nearest neighbor
intralayer and interlayer interaction with (without) considering the surface
reconstruction are only $J_{N\parallel}=1.03$ (1.24) meV and $J_{N\perp}=5.71$
(6.48) meV, respectively. Likewise, the interaction between spins located at
larger distances are also rather weak. The second nearest neighbor intralayer
and interlayer interaction with (without) considering the surface
reconstruction are only $J_{NN\parallel}=0.164$ (0.29) meV and
$J_{NN\perp}=0.24$ (0.2) meV, respectively. The values of $J_{ij}$ and the
magnetic moments are provided in Tab. 1. The fact that the HEI in Ni is weak
is in agreement with the calculations for bulk fcc Ni [19, 21, 22, 23, 25, 26,
27]. Note that due to the lack of the full magnon spectrum, the exchnage
parameters in Ni are not known experimentally. The main conclusion of the
results presented in Tab. 1 is that the exchange parameters in the Ni
monolayer are rather small. Moreover, the pattern of the exchange interaction
is complex and includes both ferromagnetic (positive) and antiferromagnetic
(negative) exchange constants. Owing to the weak HEI in the Ni monolayer one
would expect a larger magnon BSF and DOS of the acoustic magnon mode in this
layer. However, it is important to consider that the magnetic moment of Ni is
by a factor of about 3.5 smaller than that of the Co. This means that the
dynamic component of the magnetization (which somewhat scales with the static
magnetic moment) is smaller in the Ni layer. Hence, one observes a larger
magnon BSF and DOS when they are projected onto the Co layer. Note that the
magnon BSF and DOS exhibit finite values in the Ni monolayer. The large magnon
BSF and DOS in the Co layer helps to easily excite this magnon mode.

Near the $\bar{\Gamma}$–point the magnon dispersion relation may be
approximated by $\mathcal{E}(q_{\parallel})\simeq Dq_{\parallel}^{2}$, where
$D$ denotes the magnon exchange stiffness constant. Fitting the data for
$q_{\parallel}<0.4$ Å-1, we find a value of $356\pm 5$ meVÅ2, in reasonable
agreement with the experimental [13, 15, 14] and theoretical [19, 21, 22, 23,
25, 26, 27] values of the bulk Ni. Due to the weak HEI in the Ni monolayer,
this is somewhat surprising. In the following we shed light on the origin of
the stiff magnon mode in the Ni monolayer.

Within the adiabatic formalism (linear spin-wave theory) the magnon dispersion
relation is given by $\mathcal{E}(q)=\frac{2g\mu_{B}}{\mu_{i}}\Sigma_{j\neq
0}J_{0,j}\left[1-\exp i(\mathbf{q}\cdot\mathbf{R}_{0j})\right]$, where $g=2$
is the $g$-factor, $\mu_{B}$ is the Bohr magneton, $\mu_{i}$ is the magnetic
moment of the origin site, $\mathbf{R}_{0j}$ represents the displacement
vector of site $j$ with respect to the origin and $\mathbf{q}$ denotes the
three dimensional vector of the magnon momentum. A careful investigation of
the electronic DOS reveals that the Ni atoms possess a low spin density. A low
spin density leads to both a weak exchange interaction as well as a small
magnetic moment [0.55 (0.52) $\mu_{B}$ for the reconstructed (unreconstructed)
Ir(001)]. The fact that the magnetic moment of Ni is much smaller than that of
the Co has experimentally been verified by probing both bulk samples [59] as
well Co/Ni multilayers [60, 61, 62]. Since the exchnage stiffens scales
inversely with $\mu_{i}$, a weak HEI in Ni can lead to a large magnon exchange
stiffness, comparable to that of the Co and Fe. Hence, assuming the same
exchange constants for Ni and Co is not valid, even though for the small
magnon momentum the dispersion relation of the two systems might be very
similar [41, 42].

Recently, the emergence of topological superconductivity in the Bi/Ni bilayer
is attributed to the magnetic excitations at the interface and their coupling
to the surface state electrons [29]. We anticipate that our quantitative
results on the values of HEI in the Ni monolayer would contribute to a
quantitative understanding of superconducting mechanism in this system.
Moreover, they would provide guidelines for tuning superconductivity in
similar bilayer structures in which the magnons play a decisive role in the
pairing mechanism [63, 64].

In conclusion, we prepared atomically architectured multilayers composed of Ni
and Co epitaxial monolayers on Ir(001). Owing to the large magnetic moment of
the surface Co monolayer and the fact that the excitation cross-section at Co
surfaces is high, the magnons in such designed multilayers can be very
efficiently excited by spin-polarized electron scattering experiments, e.g.,
SPHREELS. The acoustic magnon mode of these systems was found to be localized
in both the Co as well as in the Ni monolayers. The full experimental acoustic
magnon dispersion relation, probed up to the surface BZ, enabled us to
quantitatively resolve the complex pattern of HEI in the Ni monolayer. We
observed a rather weak exchange coupling within the Ni monolayer, even though
the magnon exchange stiffness is rather large (much larger than that of an Fe
monolayer on various substrates). The large magnon stiffness constant is a
consequence of the low spin density of Ni atoms, as confirmed by our first-
principles calculations. In addition to the fact that our results resolve the
long-standing question regarding the quantitative values of HEI in the Ni
monolayer, they are also of the interest to the field of topological
superconductivity in magnetic/topological materials heterostructures.

## Acknowledgments

Financial support by the Deutsche Forschungsgemeinschaft (DFG) through the DFG
Grants No. ZA 902/7-1 and No. ZA 902/8-1 is acknowledged. Kh.Z. thanks the
Physikalisches Institut for hosting the group and providing the necessary
infrastructure.

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