# Angular momentum transfer via relativistic spin-lattice coupling from first
principles.

Sergiy Mankovsky  Department of Chemistry/Phys. Chemistry, LMU Munich,
Butenandtstrasse 11, D-81377 Munich, Germany  Svitlana Polesya  Department of
Chemistry/Phys. Chemistry, LMU Munich, Butenandtstrasse 11, D-81377 Munich,
Germany  Hannah Lange  Department of Chemistry/Phys. Chemistry, LMU Munich,
Butenandtstrasse 11, D-81377 Munich, Germany  Markus Weißenhofer  Department
of Physics, University of Konstanz, DE-78457 Konstanz, Germany  Ulrich Nowak
Department of Physics, University of Konstanz, DE-78457 Konstanz, Germany
Hubert Ebert  Department of Chemistry/Phys. Chemistry, LMU Munich,
Butenandtstrasse 11, D-81377 Munich, Germany

###### Abstract

The transfer and control of angular momentum is a key aspect for spintronic
applications. Only recently, it was shown that it is possible to transfer
angular momentum from the spin system to the lattice on ultrashort time
scales. In an attempt to contribute to the understanding of angular momentum
transfer between spin and lattice degrees of freedom we present a scheme to
calculate fully-relativistic spin-lattice coupling parameters from first-
principles. By treating changes in the spin configuration and atomic positions
at the same level, closed expressions for the atomic spin-lattice coupling
parameters can be derived in a coherent manner up to any order. Analyzing the
properties of these parameters, in particular their dependence on spin-orbit
coupling, we find that even in bcc Fe the leading term for the angular
momentum exchange between the spin system and the lattice is a Dzyaloshiskii-
Moriya-type interaction, which is due to the symmetry breaking distortion of
the lattice.

###### pacs:

71.15.-m,71.55.Ak, 75.30.Ds

††preprint: APS/123-QED

Spintronics is an emerging field aiming for the development of future
nanoelectronic devices. A key aspect is the transport and control of angular
momentum Hirohata  _et al._ (2020). While the focus has long been on spin-
polarized electrons to carry the angular momentum, newer lines of research
include the magnonic spin as angular momentum carrier, opening perspectives
for insulator spintronics. Understanding the flow of angular momentum is also
vital for the progress of ultrafast magnetization switching. In certain
ferrimagnets, a single laser pulse can switch the magnetization orientation on
a sub-picosecond time scale Radu  _et al._ (2011), due to the exchange of spin
angular momentum between the two magnetic sublattices Mentink  _et al._
(2012); Wienholdt  _et al._ (2013). However, recent work on ultrafast
demagnetization in ferromagnets has demonstrated that angular momentum can
also be transferred from the spin system to the lattice on similar time scales
Tauchert  _et al._ (2022). In the lattice, the spin angular momentum is
absorbed in terms of phonons carrying the angular momentum till — on larger
times scales — the macroscopic Einstein-deHaas effect sets in Dornes  _et al._
(2019). These findings add another piece to the mysteries of spintronics and
ultrafast phenomena, namely the understanding of the microscopic mechanisms
that transfer angular momentum between the spin system and the lattice.

The calculation of spin-lattice coupling terms, including the exchange of
angular momentum between spins and lattice degrees of freedom, is only at its
beginning Garanin and Chudnovsky (2015); Mentink  _et al._ (2019); Streib  _et
al._ (2019); Rückriegel  _et al._ (2020). The development of new tools for the
quantitative calculation of spin lattice dynamics — so-called molecular-spin
dynamics simulations Ma  _et al._ (2008); Perera  _et al._ (2016); Aßmann and
Nowak (2019); Hellsvik  _et al._ (2019); Strungaru  _et al._ (2021) — is
delayed by the fact that a systematic derivation of proper spin-lattice
parameters is still missing. For a pure spin model the calculation of exchange
coupling parameters $J_{ij}$ of the Heisenberg Hamiltonian on a first-
principles level by means of the so-called Lichtenstein formula Liechtenstein
_et al._ (1987) is now a well established approach to supply the necessary
input for Monte Carlo Polesya  _et al._ (2010) as well as spin-dynamics
simulations Mryasov (2005); Skubic  _et al._ (2008). Corresponding extensions
of this computational scheme are now available to account for the full
tensorial form of the interaction parameters Udvardi  _et al._ (2003); Ebert
and Mankovsky (2009) as well as their extension to a multi-site formulation
Mankovsky  _et al._ (2020).

Including the lattice degrees of freedom is much more challenging. A practical
scheme to calculate microscopic spin-lattice coupling (SLC) parameters
quantitatively and on the basis of electronic structure calculations has been
suggested so far only by Hellsvik et al. Hellsvik  _et al._ (2019) by applying
the Lichtenstein formula as well as its relativistic generalization Sadhukhan
_et al._ (2022) for a system with one atom moved gradually from its
equilibrium position.

In this letter we present and exploit an improved, fully-relativistic scheme
that treats changes to the spin configuration and atomic positions on the same
level. This allows to derive closed expressions for the atomic SLC parameters
in a coherent way up to any order. First numerical results are presented for
the three-site terms of bcc Fe. Surprisingly, even in a bcc crystal the
leading term for the exchange of spin angular momentum with the lattice is a
Dzyaloshiskii-Moriya-type interaction emerging due to the symmetry breaking
distortion of the lattice.

To describe the coupling of spin and spatial degrees of freedom we adopt an
atomistic approach and start with the expansion of a phenomenological spin-
lattice Hamiltonian

$\displaystyle{\cal H}_{sl}$ $\displaystyle=$
$\displaystyle-\sum_{i,j,\alpha,\beta}J_{ij}^{\alpha\beta}e_{i}^{\alpha}e_{j}^{\beta}-\sum_{i,j,\alpha,\beta}\sum_{k,\mu}J_{ij,k}^{\alpha\beta,\mu}e_{i}^{\alpha}e_{j}^{\beta}u^{\mu}_{k}$
(1)
$\displaystyle\qquad\qquad\quad-\sum_{i,j}\sum_{k,l}J_{ij,kl}^{\alpha\beta,\mu\nu}e_{i}^{\alpha}e_{j}^{\beta}u^{\mu}_{k}u^{\nu}_{l}\;,$

that can be seen as a lattice extension of a Heisenberg spin Hamiltonian.
Accordingly, the spin and lattice degrees of freedom are represented by the
orientation vectors $\hat{e}_{i}$ of the magnetic moments $\vec{m}_{i}$ and
displacement vectors $\vec{u}_{i}$ for each atomic site $i$. In Eq. (1) we
omit pure lattice terms that involve the force constants Hellsvik  _et al._
(2019) as we focus here on the magnetic part of the Hamiltonian. The pure spin
part (term 1) has been restricted to its two-site contributions with the
corresponding coupling parameters denoted two-site SSC below. An extension of
this term to higher order can be done in a straight forward manner Mankovsky
_et al._ (2020). Also, the spin-lattice coupling has been restricted to three
and four-site terms (terms 2 and 3). As relativistic effects are taken into
account, the exchange interactions are described in tensorial form,
$J_{ij}^{\alpha\beta}$, instead of the simpler isotropic form $J_{ij}$ of the
standard Heisenberg spin Hamiltonian. The Hamiltonian in Eq. (1) is similar in
form to the one discussed by Hellsvik et al. Hellsvik  _et al._ (2019),
potentially providing a suitable basis for advanced molecular-spin dynamics
simulations.

In previous works expressions for the exchange coupling parameters $J_{ij}$
Liechtenstein  _et al._ (1987) or $J_{ij}^{\alpha\beta}$ Udvardi  _et al._
(2003); Ebert and Mankovsky (2009), respectively, have been derived by mapping
the free energy landscape ${\cal F}(\\{\hat{e}_{i}\\})$ obtained from first-
principles electronic structure calculations on the Heisenberg spin
Hamiltonian. Here, we follow the same strategy by mapping the free energy
landscape ${\cal F}(\\{\hat{e}_{i}\\},\\{\vec{u}_{i}\\})$ accounting for its
dependence on the spin configuration $\\{\hat{e}_{i}\\}$ as well as atomic
displacements $\\{\vec{u}_{i}\\}$ on the same footing. Making use of the
magnetic force theorem the change in free energy $\Delta{\cal F}$ induced by
changes of the spin configuration $\\{\hat{e}_{i}\\}$ with respect to a
suitable reference system and simultaneous finite atomic displacements
$\\{\vec{u}_{i}\\}$ can be written in terms of corresponding changes to the
single-particle energies,

$\displaystyle\Delta{\cal F}=\int^{E_{F}}\\!\\!\\!dE(E-E_{F})\Delta
n(E)=-\\!\\!\int^{E_{F}}\\!\\!\\!dE\Delta N(E)\,,\;$ (2)

where $E_{F}$ is the Fermi energy and $\Delta n(E)$ and $\Delta N(E)$ are
corresponding changes to the density of states (DOS) $n(E)$ and integrated
density of states (NOS) $N(E)$, respectively.

$\Delta N(E)$ can be evaluated efficiently Liechtenstein  _et al._ (1987);
Udvardi  _et al._ (2003); Ebert and Mankovsky (2009) via the so-called Lloyd
formula when the underlying electronic structure is described by means of the
multiple scattering or Korringa-Kohn-Rostoker (KKR) formalism Ebert  _et al._
(2011). Adopting this approach we find

$\displaystyle\Delta{\cal F}$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\mbox{Im\,
Tr\,}\int^{E_{F}}dE\,\left(\mbox{ln}\,\underline{\underline{\tau}}(E)-\mbox{ln}\,\underline{\underline{\tau}}^{0}(E)\right)\;,$
(3)

with the so-called scattering path operator
$\underline{\underline{\tau}}^{(0)}(E)$, where the double underlinement
indicates matrices with respect to site and spin-angular momentum indices
Ebert  _et al._ (2011). Within the KKR formalism these super matrices,
characterizing the reference $\underline{\underline{\tau}}^{(0)}(E)$ and
perturbed $\underline{\underline{\tau}}(E)$ systems, respectively, are given
by

$\displaystyle\underline{\underline{\tau}}^{(0)}(E)$ $\displaystyle=$
$\displaystyle\Big{[}\underline{\underline{m}}^{(0)}(E)-\underline{\underline{G}}(E)\Big{]}^{-1}\;,$
(4)

with $\underline{\underline{G}}(E)$ the structure Green function and
$\underline{\underline{m}}^{(0)}(E)=[\underline{\underline{t}}^{(0)}(E)]^{-1}$
the inverse of the corresponding site-diagonal scattering matrix that carries
all site-specific information depending on $\\{\hat{e}_{i}\\}$ and
$\\{\vec{u}_{i}\\}$ Ebert  _et al._ (2011).

Considering a ferromagnetic reference state ($\hat{e}_{i}=\hat{z}$) with all
atoms in their equilibrium positions ($\vec{u}_{i}=0$) the perturbed state is
characterized by finite spin tiltings $\delta\hat{e}_{i}$ and finite atomic
displacements of the atoms $\vec{u}_{i}$ for the sites $i$. Writing for site
$i$ the resulting changes in the inverse $t$-matrix as
$\Delta^{s}_{\mu}\underline{m}_{i}=\underline{m}_{i}(\delta\hat{e}_{i}^{\mu})-\underline{m}^{0}_{i}$
and
$\Delta_{\nu}^{u}\underline{m}_{i}=\underline{m}_{i}({u}_{i}^{\nu})-\underline{m}^{0}_{i}$
allows to replace the integrand in Eq. (3) by

$\displaystyle\mbox{ln}\,\underline{\underline{\tau}}-\mbox{ln}\,\underline{\underline{\tau}}^{0}$
$\displaystyle=$
$\displaystyle-\ln\Big{(}1+\underline{\underline{\tau}}\,[\Delta^{s}_{\mu}\underline{\underline{m}}_{i}+\Delta_{\nu}^{u}\underline{\underline{m}}_{j}+...]\Big{)}\;,$
(5)

where all site-dependent changes in the spin configuration $\\{\hat{e}_{i}\\}$
and atomic positions $\\{\vec{u}_{i}\\}$ are accounted for in a one-to-one
manner by the various terms on the right hand side. This implies in particular
that the matrices $\Delta^{s(u)}_{\mu(\nu)}\underline{\underline{m}}_{i}$ in
Eq. (5) are site-diagonal and have non-zero blocks only for site $i$. Due to
the use of the magnetic force theorem these blocks may be written in terms of
the spin tiltings $\delta\hat{e}_{i}^{\mu}$ and atomic displacements of the
atoms ${u}_{i}^{\nu}$ together with the corresponding auxiliary matrices
$\underline{T}^{\mu}_{i}$ and ${\cal U}_{i}^{\nu}$, respectively, as

$\displaystyle\Delta^{s}_{\mu}{\underline{m}}_{i}$ $\displaystyle=$
$\displaystyle\delta\hat{e}^{\mu}_{i}\,\underline{T}^{\mu}_{i}\,,$ (6)
$\displaystyle\Delta^{u}_{\nu}\underline{m}_{i}$ $\displaystyle=$
$\displaystyle u^{\nu}_{i}\underline{\cal U}_{i}^{\nu}\,.$ (7)

Inserting these expressions into Eq. (5) and the result in turn into Eq. (3)
allows us, in a straight forward way, to calculate the parameters of the spin-
lattice Hamiltonian as the derivatives of the free energy with respect to
tilting angles and displacements. With this we derived a new scheme to obtain
systematically SLC terms up to any order.

In the following we will restrict ourselves to the third-order SLC parameters,
which are linear with respect to the displacements (for more details including
the fourth-order SLC parameters see the Supplemental Materials (SM) Sup ). One
can write the three-site expression as:

$\displaystyle{\cal J}^{\alpha\beta,\mu}_{ij,k}$ $\displaystyle=$
$\displaystyle\frac{\partial^{3}{\cal F}}{\partial e^{\alpha}_{i}\,\partial
e^{\beta}_{j}\,\partial u^{\mu}_{k}}=\frac{1}{2\pi}\mbox{Im\,
Tr\,}\int^{E_{F}}dE\,\,$ (8) $\displaystyle\times$
$\displaystyle\Big{[}\underline{T}^{\alpha}_{i}\,\underline{\tau}_{ij}\underline{T}^{\beta}_{j}\,\underline{\tau}_{jk}\underline{\cal
U}^{\mu}_{k}\,\underline{\tau}_{ki}+\underline{T}^{\alpha}_{i}\,\underline{\tau}_{ik}\underline{\cal
U}^{\mu}_{k}\,\underline{\tau}_{kj}\underline{T}^{\beta}_{j}\,\underline{\tau}_{ji}\Big{]}\;\;$

We will call all these terms three-site SLC terms in the following, even if
site indices are identical. The prefactor $1/2$ occurs to avoid double
counting of the identical terms upon summations in Eq. (1) over indices $i$
and $j$.

Below, we present our first results for the SLC parameters for bcc Fe based on
a ferromagnetic reference system with its magnetization $\vec{M}$ in
$z$-direction. Furthermore, to check the validity of our new approach for the
calculation of ${\cal J}^{\alpha\beta,\mu}_{ij,k}$ we performed also
conventional super-cell calculations for the two-site SSC parameters
$J^{\alpha\beta}_{ij}(u^{\mu}_{k})$ as a function of displacement
$u^{\mu}_{k}$ of atom $k$. These calculations have been done for a $2\times
2\times 2$ super cell implying a periodic displacement $u^{\mu}_{k}$. For the
comparison, we multiply our SLC parameters ${\cal J}^{\alpha\beta,\mu}_{ij,k}$
with $u^{\mu}_{k}$ and compare with the SSC parameters
$J^{\alpha\beta}_{ij}(u^{\mu}_{k})$ for varying $u^{\mu}_{k}$, see Fig. 1 and
for a further comparison our SM Sup . If not otherwise noted, we restrict
ourselves to displacments of atom $k$ along the $x$-axis,
$\vec{u}_{k}||\hat{x}$.

Since we are especially interested in the exchange of angular momentum between
the spin and lattice degrees of freedom we focus on the SOC-driven elements of
the three-site SLC tensor ${\cal J}^{\alpha\beta,\mu}_{ij,k}$, which give rise
to magneto-crystalline anisotropies (MCA) and Dzyaloshinskii-Moriya
interactions (DMI). The interatomic three-site SLC parameters represent a
’non-local’ contribution to the MCA Udvardi  _et al._ (2003) as well as DMI
induced by a displacement of atom $k$. Fig. 1 shows, as an example, the
nearest-neighbor SLC products ${\cal J}^{xy,x}_{ij,i}\cdot u^{x}_{i}$ compared
with the two-site SSC interaction parameters for four groups of atoms as
sketched in the inset of Fig. 1. Obviously, one finds good agreement between
the different approaches for small amplitudes of the displacement $u^{x}_{i}$.
Here one should stress that the present approach gives direct access to the
SLC parameters of any order with respect to the atomic displacement or spin
tilting in an extremely efficient way.

Figure 1:  SLC parameters versus distortion for bcc Fe: Comparison of the
products ${\cal J}^{xy,x}_{ij,i}\cdot u^{x}_{i}$ (solid lines) with the
corresponding SLC terms $\Delta
J^{xy}_{ij}(u^{x}_{i})={J}^{xy}_{ij}(u^{x}_{i})-{J}^{xy}_{ij}(0)$ calculated
for a distorted system with the super cell technique (dotted lines) for one
atom $i$ displaced by $u^{x}_{i}$ along the $x$-axis. The inset shows the
labeling of the nearest neighbor atoms used in the figure.

The anti-symmetric part of the off-diagonal SLC parameters, ${\cal J}^{{\rm
off-a},\mu}_{ij,k}=\frac{1}{2}({\cal J}^{xy,\mu}_{ij,k}-{\cal
J}^{yx,\mu}_{ij,k})$, can be interpreted Udvardi  _et al._ (2003) as the DMI,
$D^{z}_{ij}$, induced by the symmetry-breaking displacement of atom $k$ and
one can define a Dzyaloshinskii-Moriya-like spin lattice coupling (DSLC),
${\cal D}^{z,\mu}_{ij,k}={\cal J}^{{\rm off-a},\mu}_{ij,k}$. Note that the
conventional DMI vanishes for the non-distorted bcc Fe lattice due to
inversion symmetry. Furthermore, for symmetry reasons, all anti-symmetric off-
diagonal elements of the three-site SLC are equal to zero in the case of a
displacement of atom $k$, positioned at the same distance from atoms $i$ and
$j$, along $\hat{z}$, implying ${\cal D}^{z,z}_{ij,k}=0$. This does not apply
for the other components, ${\cal D}^{x,z}_{ij,k}$ and ${\cal D}^{y,z}_{ij,k}$.

A further analysis of our SLC parameters in this context is shown in Fig. 2,
again for $i=k$ which implies that the displacement along $x$ direction is
applied to one of the interacting atoms. Results for $k\neq j(i)$ are shown in
the SM Sup . Different components of the SCL parameters are plotted as a
function of the distance $r_{ij}$. The absolute values of the DSLC parameters
$|\vec{\cal D}|^{\mu=x}_{ij,k}$ show a rather slow decay with the distance
$r_{ij}$. These parameters can take different values for the same distance,
which is a result of the symmetry imposed vanishing of certain components of
the DMI-like SLC for some $\vec{r}_{ij}$ directions, which depend in turn also
on the direction of the displacement $\vec{u}$. The isotropic SLC parameters
${\cal J}^{\mathrm{iso},\mu=x}_{ij,j}$, which have only a weak dependence on
the SOC, are about one order of magnitude larger than the DSLC. All other SOC-
driven parameters shown in Fig. 2, characterizing the displacement-induced
contributions to MCA, are much smaller than the DSLC.

Figure 2:  Magnitude of site-off-diagonal and site-diagonal SLC parameters:
DMI $|\vec{\cal D}^{x}_{ij,j}|$ and isotropic SLC ${\cal
J}^{\mathrm{iso},x}_{ij,j}$ (top), anti-symmetric diagonal components ${\cal
J}^{\mathrm{dia-a},x}_{ij,j}$ and ${\cal J}^{\mathrm{dia-a},x}_{ii,k}$
(middle), and symmetric off-diagonal components ${\cal
J}^{\mathrm{off-s},x}_{ij,j}$ and ${\cal J}^{\mathrm{off-s},x}_{ii,k}$
(bottom).

Calculating the ${\cal D}^{\nu,\mu}_{ij,k}$ parameters for ordered FePt in the
CuAu structure, with Fe atoms on sites $i$ and $j$, we demonstrate in the SM
Sup  a key role of the SOC on the atoms mediating these interactions. It
follows that the DSLC parameters ${\cal D}^{\nu,\mu}_{ij,k}$ (both, for $k=j$
and $k\neq j$) exhibit only weak dependence on SOC strength of Fe sites, while
they decrease by about an order of magnitude when the SOC on Pt sites goes to
zero, indicating the key role of the atoms mediating the coupling between the
atoms $i$ and $j$ for the DSLC magnitude. A more detailed analysis of the
properties of three-site SLC parameters is given in the SM Sup . The role of
the dipolar contribution to the SLC is not discussed here as it is expected to
play a minor role for bcc Fe.

a)  b)  c)

Figure 3:  a) Longitudinal (top) and perpendicular (bottom) forces (with
respect to $\vec{r}_{ij}$) emerging from the isotropic, DMI-type and MCA-type
contributions to the SLC parameters. The central spin at $\vec{r}=(0,0,0)$ is
tilted in $y$-direction while all the others point in $z$-direction
(transparent arrows in b) and c)). b) Directions of the forces with color
coding indicating their perpendicular components with red being high and grey
being low. c) Directions of the perpendicular forces with the same color
coding as in b).

In an attempt to estimate the forces that act on the atoms for a non-collinear
spin configuration we assume naively that these are given via the gradient of
our spin-lattice Hamiltonian, Eq. (1), with respect to the displacements,
$\vec{F}=-\frac{\partial{\cal H}_{\rm{sl}}}{\partial\vec{u}_{k}}$, see also
Skubic  _et al._ (2008). We note, however, that this spin-lattice Hamiltonian
does not conserve the total (spin plus lattice) angular momentum which
questions this approach. However, within this approach we are able to
calculate the forces acting on neighboring atoms of a central spin which is
tilted in $y$-direction while all the other spins are aligned in
$x$-direction. Because of this non-collinear spin configuration forces act on
the neighboring atoms which are shown in Fig. 3. These forces are decomposed
into their longitudinal and perpendicular parts (with respect to the distance
vector $\vec{r}_{ij}$) and whether they come from isotropic, DMI-type or MCA-
type contributions to the SLC parameters. The main contribution to the
longitudinal forces, which do not transfer angular momentum, comes once again
from the isotropic Heisenberg interaction. However, the largest perpendicular
— and with that angular momentum transferring — component is once again due to
the DMI-type interaction. Plugging in numbers for the mass of an Fe atom we
note that the latter is sufficient to accelerate a free Fe atom within 100 fs
to a velocitiy of about 1 m/s.

The role of the DMI-like SLC parameters for the spin-lattice angular momentum
transfer can also be discussed in terms of magnon-phonon interactions
Rückriegel  _et al._ (2014); Garanin and Chudnovsky (2015); Streib  _et al._
(2019); Rückriegel  _et al._ (2020). With this in mind, we represent the
second term of the Hamiltonian in Eq. (1), which is linear with respect to
displacement $\vec{u}_{k}$, in terms of isotropic and anisotropic interactions
focusing on the DSLC contribution

$\displaystyle{\cal H}_{\rm{me-DMI}}$ $\displaystyle=$
$\displaystyle\frac{1}{S^{2}}\sum_{i,j}\sum_{k,\mu}\vec{\cal
D}_{ij,k}^{\mu}\cdot(\hat{s}_{i}\times\hat{s}_{j})u^{\mu}_{k}\,.$ (9)

${\cal H}_{\rm{me-DMI}}$ is now given in a form using spin operators
$\hat{s}^{\alpha}_{i}$ instead of spin orientation vectors
$\hat{e}^{\alpha}_{i}$, where $S$ is the maximal value for spin moment per
atom. For the sake of simplicity, we consider again a ferromagnetic system
with one atom per unit cell as, e.g., bcc Fe, and with the magnetization $M$
along $z$ direction. To describe the magnon-phonon interactions, it is
customary to introduce spin raising and lowering operators
$\hat{s}^{\pm}_{i}=\hat{s}^{x}_{i}\pm i\hat{s}^{y}_{i}$. Using the Holstein-
Primakoff transformation Holstein and Primakoff (1940); Sup , local spin
fluctuations can be represented in terms of canonical boson operators,
$\hat{b}^{\dagger}_{i}$ and $\hat{b}_{i}$. Performing Fourier transformations
for the spin operators and the atomic displacements the Hamiltonian ${\cal
H}_{\rm{me-DMI}}$ can be reduced to the momentum representation form Sup

$\displaystyle{\cal H}_{\rm{me-DMI}}$ $\displaystyle=$
$\displaystyle\frac{2i}{\sqrt{2S}}\sum_{\vec{q},\mu}\bigg{[}{\cal
D}_{\vec{q}}^{-,\mu}\hat{b}_{\vec{q}}-{\cal
D}_{-\vec{q}}^{+,\mu}\hat{b}^{\dagger}_{-\vec{q}}\bigg{]}{u}^{\mu}_{\vec{q}}\;$
(10)
$\displaystyle-\;\frac{2i}{\sqrt{N}}\frac{1}{S}\sum_{\vec{k},\vec{k}^{\prime},\mu}{\cal
D}_{\vec{k},\vec{k}^{\prime}}^{z,\mu}\,\hat{b}^{\dagger}_{\vec{k}}\hat{b}_{\vec{k}^{\prime}}\,{u}^{\mu}_{(\vec{k}^{\prime}-\vec{k})}\,,$

where
${u}^{\mu}_{\vec{q}}=\sum_{\lambda}\epsilon^{\mu}_{\lambda,\vec{q}}X_{\lambda,\vec{q}}$
is the eigenvector corresponding to the phonon mode $(\lambda,\vec{q})$ with
the polarization $\epsilon^{\mu}_{\lambda,\vec{q}}$. In the second
quantization form for the phonon modes $X_{\lambda,\vec{q}}$ can be given in
terms of the phonon creation and annihilation operators, respectively Böttger
(1983); Sup . Following the conclusions in Ref. Rückriegel  _et al._ (2020),
the first term in Eq. (10), which is determined by the DSLC components ${\cal
D}_{ij,k}^{x,\mu}$ and ${\cal D}_{ij,k}^{y,\mu}$, describes the magnon-phonon
scattering that allows for an exchange of angular momentum. On the other hand,
the DSLC component ${\cal D}_{ij,k}^{z,\mu}$ (with $z$ the magnetization
direction) contributes to the magnon-number conserving scattering
characterized by the energy transfer only (see Ref. Rückriegel  _et al._ ,
2020).

The upper part of Fig. 4 shows the Fourier transforms ${\cal
D}_{\vec{q}}^{x,\mu}$ of the SLC parameters for bcc Fe, given by the
expression:

$\displaystyle{\cal D}_{\vec{q}}^{\nu,\mu}$ $\displaystyle=$
$\displaystyle\sum_{j,k}{\cal
D}_{ij,k}^{\nu,\mu}e^{i\vec{q}\cdot(\vec{R}_{j}-\vec{R}_{i})}e^{-i\vec{q}\cdot(\vec{R}_{k}-\vec{R}_{i})}\;,$
(11)

which are plotted for $\vec{q}$ along the high-symmetry lines in the Brillouin
zone (see also SM Sup ). One finds a strong dependence on the direction and
absolute value of wave vector $\vec{q}$. Moreover, this dependence is
different for different components of the atomic displacements, which reflects
in turn the difference of the magnon-phonon interactions for the transverse
and longitudinal phonon modes.

Figure 4:  Imaginary part $\mbox{Im}\,{\cal D}_{\vec{q}}^{x,\mu}$ (top) and
$\mathrm{Im}(J_{\vec{q}}^{xy,\mu}+J_{\vec{q}}^{yx,\mu})/2$ (bottom) of the SLC
parameters for bcc Fe, with $\mu=x,y,z$, plotted for $\vec{q}$ along high-
symmetry lines of the Brillouin zone. The real part is zero in all cases.

In addition, the lower part of Fig. 4 shows the Fourier transforms of the
cite-diagonal SLC parameters
$\mathrm{Im}(J_{\vec{q}}^{xy,\mu}+J_{\vec{q}}^{yx,\mu})/2$, with $\mu=x,y,z$,
which can be seen as a local contribution to the magnetic anisotropy induced
by atomic displacements. Comparing the results confirm the dominating role of
the DSLC for the magnon-phonon hybridization.

The contribution of the first term in the DSLC Hamiltonian ${\cal H}_{\rm{me-
DMI}}$ in Eq. (10) to the phonon angular momentum dynamics can also be
characterized by the corresponding torque on the lattice produced by the spin
system Rückriegel  _et al._ (2020). In particular, the torque on the phonon
spin entering the equation of motion for the phonon angular momentum
$\vec{\cal L}_{ph}=\sum_{k}\vec{u}_{k}\times\vec{\pi}_{k}$ is given by the
expression Rückriegel  _et al._ (2020)

$\displaystyle\vec{\cal T}_{\rm{me}}$ $\displaystyle=$
$\displaystyle-\sum_{k}\vec{u}_{k}\times\frac{\partial{\cal
H}_{\rm{me}}}{\partial\vec{u}_{k}}$ (12)

with the $\vec{u}_{i}$ and $\vec{\pi}_{i}$ the displacement and linear
momentum operators, respectively.

Thus, the one magnon - one phonon contribution to the torque, $\vec{\cal
T}_{\rm{me}}$, originating from the first term of the DSLC Hamiltonian ${\cal
H}_{\rm{me-DMI}}$ in Eq. (10), is given by the expression (see SM Sup )

$\displaystyle{\cal T}^{\gamma}_{\rm{me-DMI}}$ $\displaystyle=$
$\displaystyle\frac{2i}{\sqrt{2S}}\sum_{\vec{q},\lambda}\bigg{[}{\Gamma}_{\lambda,\vec{q}}^{-,\gamma}\hat{b}_{\vec{q}}-{\Gamma}_{\lambda,-\vec{q}}^{+,\gamma}\hat{b}^{\dagger}_{-\vec{q}}\bigg{]}X_{\lambda,\vec{q}}\,,$
(13)

Here, the interaction vertices ${\Gamma}_{\lambda,\vec{q}}^{\pm,\gamma}$ are
defined as

$\displaystyle{\Gamma}_{\lambda,\vec{q}}^{\pm,\gamma}$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha\beta\gamma}\bigg{[}\epsilon^{\alpha}_{\lambda,\vec{q}}{\cal
D}_{\vec{q}}^{\pm,\beta}-\epsilon^{\beta}_{\lambda,\vec{q}}{\cal
D}_{\vec{q}}^{\pm,\alpha}\bigg{]}\,.$ (14)

with $\epsilon_{\alpha\beta\gamma}$ the Levi-Civita symbol. As one can see
they are fully determined by the DSLC parameters discussed above.

In summary, we present a scheme to calculate microscopic and relativistic
spin-lattice coupling parameters from first-principles. The perturbation due
to a lattice distortion is treated on the same footing as the distortion due
to spin tilting, giving access to SLC parameters up to any order for these
perturbations. In linear order of the displacements our parameters can be
compared with spin-spin coupling parameters obtained via super-cell
calculations with an applied distortion Hellsvik  _et al._ (2019) and good
agreement is found. Analyzing the properties of these parameters, in
particular their dependence on spin-orbit coupling, we find that even in bcc
Fe the leading term which is responsibla for the exchange of angular momentum
between the spin system and the lattice is a Dzyaloshiskii-Moriya-type
interaction, which emerges due to the symmetry breaking distortion of the
lattice. Our findings, hence, stress the importance of relativistic effects
for the transfer of angular momentum from magnonic excitations to circularily
polarized phonons Tauchert  _et al._ (2022); Garanin and Chudnovsky (2015);
Mentink  _et al._ (2019); Streib  _et al._ (2019); Rückriegel  _et al._ (2020)

## Acknowledgements

The work in Konstanz was supported by the DFG via SFB 1432 and Project No. NO
290/5-1.

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[1,2,3,4,5]A. Giachero

1]National Institute of Standards and Technology, Boulder, 80305, Colorado,
USA

2]Department of Physics, University of Colorado, Boulder, 80309, Colorado, USA

3]Department of Physics, University of Milano Bicocca, Milan, I-20126, Italy

4]INFN - Milano Bicocca, Milan, I-20126, Italy

5]Bicocca Quantum Technologies (BiQuTe) Centre, Milan, I-20126, Italy

# Kinetic inductance traveling wave amplifier designs for practical microwave
readout applications

<EMAIL_ADDRESS>M. Visser  J. Wheeler  L. Howe  J. Gao  J.
Austermann  J. Hubmayr  A. Nucciotti  J. Ullom  [  [  [  [  [

###### Abstract

A Kinetic Inductance Traveling Wave amplifier (KIT) utilizes the nonlinear
kinetic inductance of superconducting films, particularly Niobium Titanium
Nitride (NbTiN), for parametric amplification. These amplifiers achieve
remarkable performance in terms of gain, bandwidth, compression power, and
frequently approach the quantum limit for noise. However, most KIT
demonstrations have been isolated from practical device readout systems. Using
a KIT as the first amplifier in the readout chain of an unoptimized microwave
SQUID multiplexer coupled to a transition-edge sensor microcalorimeter we see
an initial improvement in the flux noise [1]. One challenge in KIT integration
is the considerable microwave pump power required to drive the non-linearity.
To address this, we have initiated efforts to reduce the pump power by using
thinner NbTiN films and an inverted microstrip transmission line design. In
this article, we present the new transmission line design, fabrication
procedure, and initial device characterization – including gain and added
noise. These devices exhibit over 10 dB of gain with a 3 dB bandwidth of
approximately 5.5–7.25 GHz, a maximum practical gain of 12 dB and typical gain
ripple under 4 dB peak-to-peak. We observe an appreciable impedance mismatch
in the NbTiN transmission line, which is likely the source of the majority of
the gain ripple. Finally we perform an initial noise characterization and
demonstrate system-added noise of three quanta or less over nearly the entire
3 dB bandwidth.

###### keywords:

Quantum noise, parametric amplifier, traveling wave, microwave kinetic
inductance detector, detector array read out, qubit read out

##  1 Introduction

Fundamental physics experiments operating at microwave frequencies require
ultra-sensitive readout schemes that greatly benefit from a amplification
chain with the lowest possible noise, high gain, and large bandwidth. These
requirements are essential for the readout of large arrays of detectors
without compromising the information conveyed by the signal. Although
semiconductor amplifiers, such as high-electron-mobility transistor (HEMT)
amplifiers, provide large gain and bandwidths [2], their noise is often $\sim
10–40$ times the Standard Quantum Limit (SQL) [3] at microwave frequencies.
Josephson parametric amplifiers [4, 5] (JPAs) offer noise performance at or
below the SQL. However, their bandwidth is limited to a few hundred megahertz
[6] – significantly limiting the number of devices they can practically read
out.

An emerging technology for achieving these requirements is that of the
broadband TWPA, such as the Josephson TWPA (JTWPA) [7] or the kinetic
inductance TWPA (KITWPA) [8], or KIT. These devices reach noise levels near
the SQL with high gain over a larger bandwidth (a few GHz). A TWPA amplifier
generally consists of a long transmission line designed to exploit a specific
nonlinearity in a superconducting circuit. A large pump tone modulates these
nonlinear elements, coupling the pump ($f_{p}$) to a signal ($f_{s}$) and
idler ($f_{i}$) tone via frequency conversion. In the Four-Wave-Mixing (4WM)
case $2f_{p}=f_{s}+f_{i}$, while for three-wave-mixing (3WM)
$f_{p}=f_{s}+f_{i}$, i.e. abundant pump photons are exchanged for signal (and
idler) photons, resulting exponential signal gain as photons interact with the
modulated nonlinearity. In the case of JTWPA, the nonlinearity used is the
Josephson inductance[9], while a KIT uses the superconducting nonlinear
kinetic inductance [10]. Compared with JTWPAs, KITs are simple to fabricate
and require only few lithography and etching steps, without overlapping
structures. Additionally, KIT amplifiers based on Niobium Titanium Nitride
(NbTiN) thin films provide a high dynamic range (-$60$ dBm), gain (20 dB), and
operate near the SQL [11]. KITs have been successfully used to read out
superconducting qubits [12], Microwave Kinetic Inductance Detectors (MKIDs)
[13], Transition Edge Sensor (TESs) [1] and also showed impressive performance
operating at 4 K [14].

However, contemporary KITs require a strong pump, typically around -30 dBm at
the KIT input [11]. Isolating this power from the device under test is
challenging, potentially resulting in excess noise and a degradation of
readout channel performance. These devices are based on 20-30 nm NbTiN films
with a kinetic inductance, $L_{k}$, in the range of 7-10 pH/sq [11, 15].
Reduction of the pump power may be realized by using a thinner superconducting
film with a higher kinetic inductance [16]. In this study, we present
preliminary results on KITs designed with a target $L_{k}$ of 35 pH/sq adapted
from the previous coplanar waveguide geometry [11] to an inverted microstrip
[15].

Figure 1:  Cross-sectional side (a) and top side (b) views of a stub-loaded
inverted micro-strip line. Relative dimensions are not to scale.

##  2 Transmission line design

The kinetic inductance of a superconducting transmission line under dc current
bias $I_{dc}$ is $L_{k}(I)=L_{d}\left(1+{I^{2}}/{I_{*}^{2}}\right)$, with
$L_{d}=L_{0}\left(1+I_{dc}^{2}/{I_{*}^{2}}\right)$ [11]. Here $I_{*}$ is an
intrinsic material parameter that controls the scale of the kinetic inductance
nonlinearity [17]. $I$ ($I_{dc}$) is the rf (dc) current, $L_{0}$ is the NbTiN
kinetic inductance at zero dc current, $L_{d}$ is the line inductance under
nonzero dc bias and zero rf current ($I=0$).
$\varepsilon=2I_{dc}/(I_{*}^{2}+I_{dc}^{2})$ describes the 3WM process, while
and $\xi=1/(I_{*}^{2}+I_{dc}^{2})$ describes 4WM.

The scaling current $I_{*}$ is directly proportional to the cross-sectional
area of the film ($A=t\cdot w$, see Fig. 1), thus decreasing $t$ reduces
$I_{*}$. Moreover, reducing the film thickness $t$ results in an increase in
$L_{0}$, so the required pump power (dc bias) to achieve a given $L_{k}$
decreases quadratically (linearly) with $t$. Thicknesses of 5 and 10 nm yield
kinetic inductance values of approximately 100 pH/sq and 30 pH/sq,
respectively – along with scaling currents of about 0.6 mA and 3 mA,
respectively[16]. These values should be compared with the previous KIT
device: $L_{k}=10\,$pH/sq and $I_{*}=7$ mA [11, 14].

Using an Inverted Microstrip (IMS) geometry, we recently implemented a $t=10$
nm thick NbTiN film to realize $L_{k}$ of around 35 pH/sq. The $\alpha$-Si
thickness is $d=100$ nm. The sky (ground) plane is made of a thick Nb layer
($t=100$ nm, $L_{k}$ negligible) deposited on top of the $\alpha$-Si. The
microstrip center line and finger widths are $w=1\,\mu$m while the spacing
between adjacent fingers is $s=1\,\mu$m, resulting in a elementary cell length
of $w+s=2\,\mu$m (Fig. 1). The line impedance is determined by the inductance
(capacitance) per unit length $\mathcal{L}$ ($\mathcal{C}$) via
$Z_{0}=\sqrt{\mathcal{L}/\mathcal{C}}$. With these dimensions, electromagnetic
simulations provided a finger length of $18\,\mu$m ($6.5\,\mu$m) for matching
$Z_{0}=50\,\mathrm{\Omega}$ ($Z_{0}=80\,\mathrm{\Omega}$). The full KIT line
is made from a string of 1200 super-cells, each composed of 30 unloaded-cells
and 6 loaded-cells. This gives a stop-band around 10.5–11.00 GHz and, since
the gain profile is centered near $f_{p}/2$ and the pump is applied just above
the stop-band, this centers the gain near 6 GHz. The total transmission line
length is 8.64 cm.

Beyond shortening the total line length (and thus overall device size) via
increasing the kinetic inductance and, as consequence, the inductance per unit
length and the stub-to-ground capacitance per unit length, the IMS approach
also benefits from significantly higher fabrication yield relative to previous
devices implementing a Coplanar Waveguide (CPW) transmission line. Elimination
of the potential for a center-ground short (lithographic) failure common in
the CPW geometry resulted in a 100% device yield with the new IMS devices. On
the other hand, the presence of a non-vacuum dielectric could potentially
increase loss and noise, including those sourced from two-level system (TLS)
interactions.

##  3 Device Fabrication

The devices are fabricated on a 76.2 mm, high-resistivity, intrinsically-
doped, float-zone Si wafer. Immediately before loading into the vacuum chamber
the native oxide is stripped using HF [18]. As for previous designs, the NbTiN
is reactively co-sputtered from Ti and Nb targets in an Ar:N2 atmosphere at
500∘C with the rates tuned to maximize the superconducting transition
temperature. After the NbTiN growth, the substrate is cooled to room
temperature without breaking vacuum and a 50 nm layer of Al is deposited. The
wafer is then patterned using an i-line stepper. The Al is etched away using
Transene A etchant, which does not attack the NbTiN beneath, only leaving Al
in the bondpad regions. The wafer is then re-patterned and the NbTiN is etched
using a CHF3 based plasma in an ICP-RIE to define the center strip of the
inverted microstrip. This etch does not quickly trench into the Si which could
complicate later step coverage, and leaves a surface compatible with high $Q$
microwave devices [19]. An insulating layer of $\alpha$-Si is deposited at
room temperature using ICP-PECVD, and then the ground plane of 100 nm of Nb is
sputtered. The ground plane near the bondpads and transition from microstrip
to CPW is patterned and etched using a CF4-based RIE, which is selective
versus the underlying $\alpha$-Si. Finally, the vias in the $\alpha$-Si are
patterned and etched in an SF6 RIE with the etch stopping on the Al covering
the bond-pads.

##  4 Characterization Measurements

On each wafer, we produce both KIT amplifier dies and diagnostic chips. The
diagnostic chips are composed of lumped element resonator arrays to directly
measure the NbTiN kinetic inductance and $\alpha$-Si permittivity. For the
$L_{k}$ measurement we use an interdigitated capacitor and NbTiN straight
segment to form the resonator, while for the permittivity measurement we use a
parallel plate capacitor. The KIT dc bias is supplied to the devices using two
bias-tees, while the pump signal is attenuated by 10 dB at 4 K and then
delivered to the KIT package with a directional coupler. We measure a KIT
critical current of 0.38 mA and a scaling current of 2.1 mA, which are
compatible with values obtained in an earlier study with the same material
[16]. These values are lower compared to those measured for the $t=20$ nm
NbTiN CPW version [11] ($I_{c}=2.4$ mA and $I_{*}=7$ mA) as expected.

Gain profile measurements are performed by determining the ratio between the
forward transmission $S_{21}$ with fixed dc bias and the pump on and off. In
total, eight devices were tested and all exhibited gain ranging between 10–20
dB (Fig. 2) and centered near 6 GHz. During tune-up of the gain profile
$I_{dc}$ ranged from 0.12 to 0.24 mA, and the pump frequency is typically
$f_{p}=12.6$ GHz. Higher gain is possible with increased dc bias and pump
power at the expense of a larger ripple.

Figure 2:  (left) Gain measured for two different configurations: 20 dB gain
has been obtained with $I_{dc}=0.24\,$mA, $f_{p}=12.662$ GHz and KIT-input
pump power around -31 dBm, while 12 dB gain has been obtained with
$I_{dc}=0.13\,$mA, $f_{p}=12.666$ GHz and -35 dBm pump power. Note the 20 dB
gain settings result in significant levels of intermodulation products and
higher order parametric processes, making operation in this regime less
stable. Time-domain reflectometer measurement for eight amplifiers. The
characteristic impedance is approximately $50\,\mathrm{\Omega}$ for all parts
related to the characterization system, dropping down to $35\,\mathrm{\Omega}$
when the signal travels through the amplifiers. The impedance mismatch arises
from a significantly higher $\varepsilon_{r}$ than was measured in an earlier
process run and has been corrected for the next device revision.

Typical pump powers are comparable to the previous 20 nm NbTiN CPW version
(-30 dBm on-chip pump power) [11]. However, the current devices have a much
thinner NbTiN film so the required pump power is expected to be lower than the
CPW devices and currently under investigation. We also discover an impedance
mismatch of the microstrip line using a Time-domain Reflectometry (TDR)
measurement which yields a characteristic impedance of $Z_{0}\sim
35\,\mathrm{\Omega}$ (Fig. 2, right). With the diagnostic chips we measured
the NbTiN film kinetic inductance to be $L_{k}=30$ pH/sq, and the $\alpha$-Si
permittivity to be $\varepsilon=9.6$. The latter is substantially discrepant
with the expected value from past measurements. Simulations performed with the
measured $L_{k}$ and $\varepsilon$, yielded a characteristic impedance around
$Z_{0}=39\,\mathrm{\Omega}$.

Figure 3:  A simplified dilution refrigerator schematic (a) used for the noise
characterization. The KIT operates with gain $G_{K}$ and added noise $N_{K}$,
and its output is amplified at 4.45 K by a commercial HEMT amplifier with gain
$G_{K}\sim 38$ dB and added noise $N_{H}=3$–10 quanta (compatible with [11])
and again at room temperature with $G_{W}\sim 26$ dB. We may neglect $N_{W}$
as the effective signal temperature at this stage is $\mathcal{O}(10^{7})$ K
and the amplifier’s noise figure is $\lesssim 6$. Use of a 20 dB isolator at
0.05 K, between the KIT output and HEMT input, allows us to neglect the HEMT’s
noise as an additional term in the calculation of the noise power at the KIT
input for each switch position. (b) Estimated system-added noise (left
vertical axis) and KIT gain (right vertical axis) as a function of the
frequency. The darker yellow line shows the smoothed gain profile.

To characterize the system-added noise with the KIT as the first-stage
amplifier we use a modified $y$-factor (“hot/cold”) method using three coaxial
lines with attenuators at three different temperature stages {20, 20, 30} dB
at {4.45, 0.95, 0.05} K. These three noise temperatures are connected to the
KIT via a cryogenic micro-electromechanical system (MEMS) switch. A simplified
schematic circuit for the noise characterization is shown in Fig. 3(a). Each
noise input line is terminated at room temperature so consideration of the
entire chain moving from 293 K to the KIT input is necessary to obtain the
correct effective temperatures. After a full loss characterization of each
component (all at base temperatures) we obtain mean input temperatures of
{3.41, 0.55, 0.16} K, or {11.0, 1.8, 0.7} quanta across the KIT bandwidth.

To estimate the KIT-on system-added noise we measure the system output power
level using a Spectrum Analyzer (SA) configured in zero-span mode with 1 MHz
resolution and video bandwidths. We set the SA center frequency, gather ten
traces, and actuate the MEMS switch between each of its three positions –
after which the center frequency is stepped and the procedure is repeated. At
each SA frequency we perform a linear fit to the three-point curve
corresponding to the system output noise level at each switch position. We
then extract an estimate for the system-added noise, $N_{\Sigma}$ (units of
quanta), via

$N_{out}^{s}=G_{c}(N_{in}^{s}+N_{in}^{i}+N_{\Sigma}).$ (1)

$G_{c}$ is the total chain gain and $N_{in}^{s,i}$ is the input noise quanta
at the signal or idler frequencies. We use the high-gain approximation of
$G_{K}\approx G_{K}-1$ to simplify our analysis [11] and, leave $G_{c}$ as a
free parameter in the fit. In this technique we neglect the HEMT-added noise,
$N_{H}$ (measured concurrently to be $N_{H}=3$–10 quanta across the KIT
bandwidth). Doing so effectively attributes the HEMT contribution to the total
system-added noise as belonging to the KIT-added noise. In the case where
$N_{H}$, when referred to the KIT input, is much less than the SQL (in our
setup with 20 dB isolation and a KIT gain of 13 dB the HEMT noise at the KIT
input is far below the SQL) this measurement yields an upper bound for the
estimated system-added noise with the KIT as the first-stage amplifier. Fig
3(b) shows the estimated system-added noise and demonstrates a 2.5 GHz 3 dB
bandwidth (5.6–7.1 GHz) over which $N_{\Sigma}\leq 3$ quanta (860 mK at 6
GHz). As the experimental setup is quite similar to that of [11], and we
measure a comparable HEMT noise, these results indicate the new inverted
microstrip KIT design operates with a similar proximity to the SQL as previous
CPW devices.

##  5 Conclusion and Future Plans

We designed and produced a stub-loaded inverted microstrip Kinetic Inductance
Traveling Wave Parametric Amplifier, utilizing a superconducting material with
significantly higher kinetic inductance than prior devices. The increased
kinetic inductance should lead to a reduction in the required pump power and
dc bias for amplifier operation and will be carefully characterized in next-
generation devices. Although the device’s characteristic impedance is not
well-matched to 50 $\Omega$, we achieve gain over 10 dB and a system-added
noise around 3 quanta (860 mK) at 6 GHz. These promising results prompted the
development of a new design optimized for a matched characteristic impedance,
which has already been fabricated and is currently being characterized.
Integration of the second generation KITs with rf-SQUID multiplexer readout
devices used for TES detector readout, and assessment of the readout system
performance is currently underway. Future steps towards improving these
devices’ practical use in detector readout chains involve implementation of
on-chip bias-tees and directional couplers/diplexers. Such on-chip
superconducting bias circuits minimize loss and reduce the total amplifier
footprint; making them much easier to use in already complex detector systems.
These compact, low-noise amplifiers, could be advantageous also for MKIDs,
particularly in single-photon counting applications [20], and for MMCs [21],
which can be read out using rf-SQUID microwave multiplexing similar to those
currently is use for TES readout.

Acknowledgments This work is supported by the National Aeronautics and Space
Administration (NASA) under Grant No. NNH18ZDA001N-APRA, the Department of
Energy (DOE) Accelerator and Detector Research Program under Grant No.
89243020SSC000058, and DARTWARS, a project funded by the European Union’s
H2020-MSCA under Grant No. 101027746. The work is also supported by the
Italian National Quantum Science and Technology Institute through the PNRR MUR
Project under Grant PE0000023-NQSTI.

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