# Unravelling the contributions to spin-lattice relaxation in Kramers single-
molecule magnets

Sourav Mondal  Alessandro Lunghi<EMAIL_ADDRESS>School of Physics, AMBER and
CRANN Institute, Trinity College, Dublin 2, Ireland

###### Abstract

The study of how spin interacts with lattice vibrations and relaxes to
equilibrium provides unique insights on its chemical environment and the
relation between electronic structure and molecular composition. Despite its
importance for several disciplines, ranging from magnetic resonance to quantum
technologies, a convincing interpretation of spin dynamics in crystals of
magnetic molecules is still lacking due to the challenging experimental
determination of the correct spin relaxation mechanism. We apply ab initio
spin dynamics to a series of twelve coordination complexes of Co2+ and Dy3+
ions selected among $\sim$240 compounds that largely cover the literature on
single-molecule magnets and well represent different regimes of spin
relaxation. Simulations reveal that the Orbach spin relaxation rate of known
compounds mostly depends on the ions’ zero-field splitting and little on the
details of molecular vibrations. Raman relaxation is instead found to be also
significantly affected by the features of low-energy phonons. These results
provide a complete understanding of the factors limiting spin lifetime in
single-molecule magnets and revisit years of experimental investigations by
making it possible to transparently distinguish Orbach and Raman relaxation
mechanisms.

## Introduction

Coordination compounds of first-row transition metals and lanthanide ions
offer a vast playground for the exploration of electronic and magnetic
properties for applications ranging from catalysisgupta2008catalytic  and
sensorscable2011luminescent  to luminescencebuenzli2015design ;
wegeberg2021luminescent . In particular, molecules showing magnetic properties
due to the presence of unpaired $d$/$f$ electrons are under an intense
scrutiny for applications in the areas of information
storagesessoli1993magnetic , spintronicsrocha2006spin  and quantum
scienceatzori2019second ; gaita2019molecular ; wasielewski2020exploiting .
However, the delivery of molecule-based technologies strongly relies on the
possibility to overcome their short spin lifetime. Similarly to hard
ferromagnetscoey2002permanent , coordination compounds possessing easy-axis
magnetic anisotropy are known to exhibit long spin relaxation time, hence
their name single-molecule magnets (SMMs)coronado2020molecular . Unless
cryogenic temperatures are achieved, the interaction between spin and phonons,
namely the spin-phonon coupling, is the main responsible for magnetic moment
relaxationlunghi2022toward . Early studies sessoli1993magnetic ;
accorsi2006tuning  have shown that spin relaxation time, $\tau$, of SMMs
follows an Arrhenius-like law

$\tau=\tau_{0}e^{U_{eff}/k_{B}T}\>,$ (1)

where for a given temperature, $T$, the pre-exponential factor $\tau_{0}$ set
the relaxation time-scale and the $U_{eff}$ represents an effective magnetic
moment reversal barrier due to the presence of magnetic anisotropy. $U_{eff}$
is intimately connected to the electronic structure of magnetic ions and it
coincides with the energy of the electronic excited state promoting relaxation
through the absorption and emission of a series of phononslunghi2022toward ,
i.e. the Orbach relaxation mechanism. Increasing $U_{eff}$ has been the main
strategy to improve $\tau$ and many efforts have been devoted to engineering
coordination compounds with large zero-field splittingsaccorsi2006tuning ;
rinehart2011exploiting . The most successful strategy employs the use of Co2+
or Dy3+ ions with a strong and axial crystal field as building blocks for
SMMszabala2021single . Record values of 450bunting2018linear  and
1541guo2018magnetic  have been reached for single-ion complexes of Co2+ and
Dy3+, respectively.

Now that the limit in crystal field axiality has been virtually
reachedguo2018magnetic , new strategies towards high-temperature SMMs are
required. Coupling multiple ions stands as one possible route and important
milestones in this direction have been achievedrinehart2011strong ;
gould2022ultrahard ; magott2022intermetallic . Another strategy instead
requires to look at the entire spin-phonon relaxation process to determine
other physical quantities that influence it. Here we pursue the latter
approach.

Figure 1: Experimental correlations and molecular structures. The top left and
top right panels report the experimental $\tau_{0}$ vs $U_{eff}$ for a set of
$\sim$240 single-ion Co2+ and Dy3+ complexes, respectively, individuated by
scraping the literature on single-molecule magnets or from the SIMDAVIS
databaseduan2021data . Black squares are used to identify the twelve molecules
selected for the study. The middle panel reports the molecular geometries of
selected six Co2+ complexes 1-6. The bottom panel reports the molecular
geometries of selected six Dy3+ complexes 7-12. Color codes for atoms: Dy in
purple, Co in indigo, N in light blue, O in red, B in gray, Br in dark green,
C in dark brown, S in yellow, P in gold, Cl in black, H in pale pink.

A striking example of how our limited knowledge of spin-phonon relaxation has
impacted the field of SMMs comes from the visualization of the correlation
between $\tau_{0}$ and $U_{eff}$. The top panel of Fig. 1 reports these two
quantities for $\sim$240 single-ion Co2+ and Dy3+ SMMs that largely cover the
relevant literature until early 2019 and highlights how $\tau_{0}$ spans eight
orders of magnitude and strongly correlates with $U_{eff}$, undercutting the
large values achieved for the latter. This is a result of the fact that
despite the many efforts to control $U_{eff}$, no clear insight on how to
chemically control the pre-exponential factor $\tau_{0}$ is yet available and
no attempts in optimizing it has ever been made. Moreover, as values of
$U_{eff}$ above 30 K had been reported, strong deviations from Eq. 1 have been
observed and attributed to Raman relaxationharman2010slow ; zadrozny2011slow ,
i.e. a process involving spin transitions due to the simultaneous absorption
and emission of two phonons. Only recently theoretical models have been able
to shine some light on the nature of this mechanism, showing that Raman
mechanisms can also present an Arrhenius like behaviour under certain
conditionslunghi2020multiple ; gu2020origins ; briganti2021complete . This
situation poses an important challenge. Although experimental and
computational strategies have been designed to disentangle Orbach and Raman
relaxationpedersen2015design ; rechkemmer2016four , discerning the two
mechanisms is far from trivial and misinterpretation of the fitted parameters
in Eq. 1 has been suggestedpedersen2015design ; novikov2015trigonal ;
rechkemmer2015comprehensive ; rechkemmer2016four ; lunghi2017role ;
gu2020origins . Understanding the nature of these contributions to spin
relaxation and their dependency from chemical structure is a fundamental step
toward controlling spin-phonon coupling and deliver improved SMMs.

Here we aim at providing a deeper understanding of the contributions to spin-
phonon relaxation in single-molecule magnets with the goal of removing
ambiguities in the interpretation of experiments and conclusively establish
what determines the rate of relaxation time. To achieve this we exploit our
recently developed ab initio spin dynamics approach, where the time evolution
of the molecular magnetic moment under the influence of phonons is predicted
from first principles and without the need of any information from the
experiments except for the crystal structurelunghi2017intra ;
lunghi2019phonons ; lunghi2020multiple ; briganti2021complete ;
lunghi2022toward . It has been demonstrated that this method can
quantitatively describe both one- and two-phonon processes, responsible for
Orbach and Raman relaxation mechanisms, across the entire relevant temperature
rangelunghi2022toward . Here we apply this strategy to a total of twelve Co2+
and Dy3+ SMMs. These compounds are selected from all the $\sim$ 240 single-ion
SMMs identified in literatureduan2021data  to sample different regimes of
$\tau_{0}$ vs $U_{eff}$ and provide an unprecedented benchmark for both
simulations and experiments.

We show that the correlation between $\tau_{0}$ and $U_{eff}$ arising from
literature is only virtual and due to years of misinterpretation of Raman
relaxation mechanism as Orbach. Thanks to the access to all the details of
spin relaxation, we demonstrate that the variance in the Orbach relaxation
time among different SMMs is largely determined by the static crystal field
splitting, while Raman relaxation time is also dependent on the details of
molecular vibrations and spin-phonon coupling, providing a revised road map
for the design of improved SMMs.

## Computational Methods

Molecules selection. The SIMDAVIS database has been used to gather a set of
183 Dy3+ single-ion SMMs and their respective relaxation dataduan2021data . A
manual search for the corresponding Co2+ single-ion SMMs published until early
2019 has instead been carried out and resulted in 56 compounds. Table S1
reports all the references, $U_{eff}$ and $\tau_{0}$ for the Cobalt SMMs.
Although our analysis does not account for the literature in its entirety, the
selected compounds represent the entire range of relaxation regimes. Twelve
molecules were selected from this data set following these criteria as closely
as possible: i) reported relaxation data span the entire range of $\tau_{0}$
vs $U_{eff}$, ii) molecules are chemically and structurally diverse. The six
Co2+ and six Dy3+ molecules chosen for this study are
[Co(C3S5)2](Ph4P)2fataftah2014mononuclear  (1),
[Co(SPh)4](Ph4P)zadrozny2011slow  (2), $\beta$-Copavlov2016polymorphism  (3),
$\alpha$-Copavlov2016polymorphism  (4), [Co(PPh3)2Br2]SMM5  (5), [CoL2][
(HNEt3)2]SMM6 , where H2L= 1,2-bis(methane-sulfonamido)benzene (6),
[L2Dy(H2O)5][I]${}_{3}\cdot$L${}_{2}\cdot$H2OSMM7  (7) where L= tBuPO(NHiPr)2,
[Dy(bbpen)Br]SMM8  (8) where H2bbpen=
N,N’-bis(2-hydroxybenzyl)-N,N’-bis(2-methylpyridyl)ethylenediamine),
[Dy(bbpen)Cl]SMM8  (9), Dy[NHPhiPr2]3(THF)2SMM10  (10),
Dy(Bc${}^{\text{Me}}$)3SMM11  (11) where [Bc${}^{\text{Me}}$]− =
dihydrobis(methylimidazolyl)borate, and
Dy[Cp${}_{2}^{\text{ttt}}$][B(C6F5)4]SMM12  (12) where (Cp${}^{\text{ttt}}$ =
C5H5Bu3-1,2,4). The chemical structure of 1-12 is reported in the bottom panel
of Fig. 1. Compounds 1-12 all respect the criteria i)-ii) except for the pairs
3-4 and 8-9, which have instead been chosen to challenge ab initio spin
dynamics over minimal structural variations.

Electronic structure simulations. Cell and geometry optimization and
simulations of $\Gamma$-point phonons have been performed with periodic
density functional theory (pDFT) using the software CP2K.kuhne2020cp2k  Cell
optimization was performed employing a very tight force convergence criteria
of 10−7 a.u. and SCF convergence criteria of 10-10 a.u. for the energy. A
plane wave cutoff of 1000 Ry, DZVP-MOLOPT Gaussian basis sets, and Goedecker-
Tetter Hutter pseudopotentialsgoedecker1996separable  were employed for all
atoms. The Perdew-Burke-Ernzerhof (PBE) functional and DFT-D3 dispersion
corrections were used.perdew1996generalized ; grimme2010consistent .

ORCAneese2020orca  had been used to compute the magnetic properties. Magnetic
properties for Dy3+ ions were computed from the CASSCF calculations employing
active space of seven 4$f$ orbitals with nine electrons (9,7) and by using all
the solutions with multiplicity six, 224 solutions with multiplicity four, and
490 solutions with multiplicity two. Similarly, for Co2+ ions, magnetic
properties were computed from the CASSCF calculations employing active space
of five 3$d$ orbitals with seven electrons (7,5) and by using all the
solutions with multiplicity four, and 40 solutions with multiplicity two. The
RIJCOSX approximation for coulomb integral and the integration grid of GridX6
were used for both the ions. The basis sets DKH-def2-QZVPP for Co atoms, DKH-
def2-SVP for H and SARC2-DKH-QZVP for Dy atoms were used. DKH-def2-TZVPP basis
set has been used for the rest of the atoms present in the systems.

Spin-phonon coupling and relaxation simulations. First order spin-phonon
coupling coefficients $(\partial\hat{H}_{s}/\partial Q_{\alpha})$ are computed
as

$\left(\frac{\partial\hat{H}_{s}}{\partial
Q_{{\alpha}}}\right)=\sum_{i}^{3N}\sqrt{\frac{\hbar}{2\omega_{\alpha}m_{i}}}L_{\alpha
i}\left(\frac{\partial\hat{H}_{s}}{\partial X_{i}}\right)\>.$ (2)

where $Q_{{\alpha}}$ is the displacement vector associated with the
$\alpha$-phonon and $N$ is the number of atoms in the unit cell. $L_{\alpha
i}$ and $\omega_{\alpha}$ are the Hessian matrix eigenvectors and the phonons
angular frequency. Only $\Gamma$-point phonons are used. The first order
derivatives of the spin Hamiltonian with respect to the Cartesian degree of
freedom $X_{i}$, $(\partial\hat{H}_{s}/\partial X_{i})$, is computed by
numerical differentiationlunghi2017intra . Each molecular degree of freedom is
sampled eight times between $\pm$ 0.08 Å. Spin-phonon coupling coefficients
are used to calculate the spin-phonon relaxation time on the basis of Redfield
equations.lunghi2019phonons ; lunghi2020multiple ; lunghi2022toward  First-
and second-order time-dependent perturbation theory have been used to simulate
both one- and two-phonon processes. The software MolForge is used for these
simulations and it is freely available at
github.com/LunghiGroup/MolForgelunghi2022toward . As discussed elsewhere, the
simulation of Kramers systems in zero external field requires the use of the
non-diagonal secular approximation, where population and coherence terms of
the density matrix are not independent from one another. This is achieved by
simulating the dynamics of the entire density matrix for one-phonon
processeslunghi2019phonons ; lunghi2022toward . An equation that accounts for
the dynamics of the entire density matrix under the effect of two-phonon
processes resulting from fourth-order time dependent perturbation theory is
not yet available. However, it is possible to remove the coupling between
population and coherence terms by orienting the molecular easy axis along the
quantization $z$-axis and by applying a small magnetic field to break Kramers
degeneracylunghi2022toward . Here we employ the latter strategy to simulate
Raman relaxation.

## Results

###  Ab Initio Spin Dynamics

The bottom panel of Fig. 1 shows the molecular structure of the six Co2+ and
six Dy3+ molecules chosen for this study. The corresponding values of $\tau$
and $U_{eff}$ are highlighted in the top panel of Fig. 1, showing the large
span of values of these quantities. Compounds 6 and 12 are among the SMMs with
the highest values of $U_{eff}$, while 1 and 10 are reported to have a very
large $\tau_{0}$ and small $U_{eff}$. These compounds also show a varied
coordination chemistry. For instance, 1, 2, 5, and 6 are possess a
tetrahedrally coordinated Co2+ ion, while 3 and 4 show octahedral
coordination. Similarly, molecules 10 and 11 among the Dy3+ complexes show
coordination number of six and molecule 7 shows the coordination number of
seven. On the other hand, compounds 8-9, and 12 show the coordination number
of three and bis-$\eta^{5}$, respectively. The chosen compounds also show a
varied set of ligands and charge states. The Co2+ complexes present a metal
ion coordinated by N or S donor atoms in most cases, except in 5 where Co is
bonded by Br and P donor atoms. Dy3+ also present a varied coordination,
including Oxygen, Nitrogen, halide ions and metal-organic bonds. In most
cases, the molecular unit is charged, except for 5 which is neutral. For
instance, 1, 2, 6 are di-negative anions and 3 and 4 are mono-positive
cations. On the contrary, most of the Dy complexes are neutral, except 7 and
12, which are both cations.

With exception of 6 and 12 that were studied in a previous work
lunghi2020multiple ; lunghi2022toward , all the unit cells of these compounds
were optimized with pDFT, as described in the Computational Methods section.
Electronic structure simulations at the level of CASSCF are then carried out
on all isolated molecular structures with the coordinates fixed to the pDFT
optimized value. The magnetic properties of all the compounds are then
computed by mapping electronic structure results onto effective Hamiltonians.
The effective spin Hamiltonian

$\hat{H}_{S}=D\hat{S}^{2}_{z}+E\left(\hat{S}^{2}_{x}-\hat{S}^{2}_{y}\right)\>,$
(3)

is used to describe the ground state of all Co2+ compounds, while an effective
Crystal Field Hamiltonian is used for Dy3+ compounds

$\hat{H}_{CF}=\sum_{l=2,4,6}\sum_{m=-l}^{l}B_{m}^{l}\hat{O}_{m}^{l}\>,$ (4)

where the operators $\hat{O}_{m}^{l}$ are tesseral function of the total
angular momentum operators, $\vec{\mathbf{J}}$. We will refer to any of the
two operators with $\hat{H}_{0}$ in the following. All the studied Co2+
compounds are found to exhibit a $S=3/2$ ground state with uni-axial
anisotropy ($D<0$), while the Dy3+ compounds show a $M_{j}=\pm 15/2$ ground-
state Kramers doublet (KD) well separated in energy from the excited KDs, see
Table 1 and Table S4 in ESI.

Table 1:  Simulated energy data for the Kramer’s Doublets. $\Delta_{01}$ represents the difference of energy between the first excited state and the ground state Kramers doublets, and $\Delta_{07}$ represents the difference of energy between the last and the ground state Kramers doublets in Dy3+ compounds. System |  $\Delta_{01}$ (cm-1) |  $\Delta_{07}$ (cm-1)
---|---|---
1 | 273.51 | -
2 | 93.44 | -
3 | 199.56 | -
4 | 224.78 | -
5 | 20.06 | -
6 | 198.08 | -
7 | 293.84 | 706.85
8 | 394.41 | 864.21
9 | 370.51 | 810.70
10 | 240.94 | 968.38
11 | 10.55 | 520.95
12 | 451.00 | 1473.01
Figure 2: Spin-phonon relaxation times. Simulated values of $\tau$ are
reported with continuous lines, while experimental values are reported with
solid square symbols. Color code for left panel: 1 (violet), 2 (green), 3
(blue), 4 (turquoise), 5 (red), and 6 (black). Color code for right panel: 7
(violet), 8 (green), 9 (blue), 10 (turquoise), 11 (red), and 12 (black).

Once the eigenstates, $|a\rangle$, and eigenvalues, $E_{a}$, of these
operators have been obtained, spin dynamics can be simulated by computing the
transition rate among different spin states, $W_{ab}$. Spin relaxation in
molecular Kramers systems with large magnetic anisotropy takes contributions
from one- and two-phonon processes. Considering one-phonon processes, the
transition rate, $\hat{W}^{1-ph}_{ba}$, among spin states reads

$\hat{W}^{1-ph}_{ba}=\frac{2\pi}{\hbar^{2}}\sum_{\alpha}|\langle
b|\left(\frac{\partial\hat{H}_{0}}{\partial
Q_{\alpha}}\right)|a\rangle|^{2}G^{1-ph}(\omega_{ba},\omega_{\alpha})\>,$ (5)

where $\hbar\omega_{ba}=E_{b}-E_{a}$ and the term
$\left(\partial\hat{H}_{0}/\partial Q_{\alpha}\right)$ provides the intensity
of the coupling between spin and the $\alpha$-phonon $Q_{\alpha}$. The
function $G^{1-ph}$ reads

$G^{1-Ph}(\omega,\omega_{\alpha})=\delta(\omega-\omega_{\alpha})\bar{n}_{\alpha}+\delta(\omega+\omega_{\alpha})(\bar{n}_{\alpha}+1)\>,$
(6)

where $\bar{n}_{\alpha}=(e^{\hbar\omega_{\alpha}/k_{B}T}-1)^{-1}$ is the
Bose−Einstein distribution accounting for the phonons’ thermal population,
$k_{B}$ is the Boltzmann constant, and the Dirac delta functions enforce
energy conservation during the absorption and emission of phonon by the spin
system, respectively. Eq. 5 accounts for the Orbach relaxation mechanism,
where a series of phonon absorption processes leads the spin from the fully
polarized state $M_{s}=S$ to an excited state with an intermediate value of
$M_{s}$ before the spin can emit phonons back to $M_{s}=-S$, and similarly for
states characterized by the total angular momentum, $J$.

Two-phonon processes provide an alternative pathway of relaxation to
equilibrium, namely the Raman mechanism. We model two-phonon spin-phonon
transitions, $W^{2-ph}_{ba}$, as

$\hat{W}^{2-ph}_{ba}=\frac{2\pi}{\hbar^{2}}\sum_{\alpha\beta}\left|T^{\alpha\beta,+}_{ab}+T^{\beta\alpha,-}_{ab}\right|^{2}G^{2-ph}(\omega_{ba},\omega_{\alpha},\omega_{\beta})\>,$
(7)

where the terms

$T^{\alpha\beta,\pm}_{ab}=\sum_{c}\frac{\langle
a|(\partial\hat{H}_{s}/\partial Q_{{\alpha}})|c\rangle\langle
c|(\partial\hat{H}_{s}/\partial
Q_{{\beta}})|b\rangle}{E_{c}-E_{b}\pm\hbar\omega_{\beta}}$ (8)

involve the contribution of all the spin states $|c\rangle$ at the same time,
often referred to as a virtual state. The function $G^{2-ph}$ fulfills a
similar role as $G^{1-ph}$ for one-phonon processes, and includes
contributions from the Bose-Einstein distribution and imposes energy
conservation. $G^{2-ph}$ accounts for all two-phonon processes, i.e.
absorption of two phonons, emission of two-phonons or absorption of one phonon
and emission of a second one. The latter process is the one that determines
Raman relaxation rate, and in this case $G^{2-ph}$ reads

$G^{2-ph}(\omega,\omega_{\alpha},\omega_{\beta})=\delta(\omega-\omega_{\alpha}+\omega_{\beta})\hat{n}_{\alpha}(\hat{n}_{\beta}+1).$
(9)

All the parameters appearing in Eqs. 5 and 7 are computed from first
principles (see computational methods section). In a nutshel, lattice harmonic
frequencies, $\omega_{\alpha}/2\pi$, and normal modes, $Q_{\alpha}$, are
computed by finite differentiation after geometry optimization with pDFT. All
the parameters appearing in Eqs. 3 and 4, i.e. $D$, $E$, and $B^{l}_{m}$, are
numerically differentiated with respect to the atomic displacements defined by
$Q_{\alpha}$ to obtain the spin-phonon coupling coefficients
$\left(\partial\hat{H}_{s}/\partial Q_{\alpha}\right)$. Once all the matrix
elements $W^{n-ph}_{ba}$ have been computed, $\tau^{-1}$ can be predicted by
simply diagonalizing $W^{n-ph}_{ba}$ and taking the smallest non-zero
eigenvalue. The study of $W^{1-ph}$ provides the Orbach contribution to the
relaxation rate, $\tau^{-1}_{Orbach}$, while $W^{2-ph}$ provides the Raman
contribution, $\tau^{-1}_{Raman}$. The total relaxation time is thus computed
as $\tau^{-1}=\tau^{-1}_{Orbach}+\tau^{-1}_{Raman}$.

Fig. 2 reports the experimental values and the simulations results for $\tau$
as a function of temperature for both Co2+ and Dy3+ complexes. Overall
simulations reproduce experimental results very well and they prove capable of
reproducing trends in relaxation rate for different molecules without any
input from experiments nor adjustable parameters in their equations. The best
results are obtained for 3, 6, 7, 8, 9, and 12 where the deviation between
experiments and simulations is vanishingly small. In particular, the
comparison between 8 and 9 is illustrative of the power of ab initio
simulations, which are shown to be able to predict differences in relaxation
times coming from substituting a Cl- with a Br- as ligands in the first
coordination sphere. Interestingly, in the case of Co2+ compounds with large
$U_{eff}$, such as for 6, 4, and 3, we predict values of $\tau$ that fall
within the same order of magnitude. Although simulations are not able to
perfectly distinguish different Co2+ molecules to this degree of accuracy, the
relaxation time is in good agreement with experimental observations. It is
important to note that the contribution to relaxation coming from dipolar-
mediated cross-relaxation, not included in simulations, is an important factor
that potentially contributes to the residual deviations between simulations
and experiments. Experimental results for diluted compounds or in external
field are often not available (see Table S6) and for instance, the effect of
dipolar relaxation is particular visible in 2, where experimental relaxation
times start flattening out at low temperature. The importance of accounting
for dipolar cross-relaxation in the comparison between relaxation data and
simulations has been recognized for $S=1/2$ systemslunghi2022toward  and
supports the hypothesis that the residual errors for Co2+ SMMs is due to this
effect. This argument is also in agreement with the higher accuracy obtained
in high-$U_{eff}$ Dy SMMs, which are naturally screened from dipolar
relaxation. The largest deviations are observed for 5 and 11. These compounds
have the smallest zero-field splittings and we attribute these somewhat larger
errors to the absence of acoustic and border-zone phonons in our simulations.
The latter are not accounted for by simulating the sole unit-cell phonons and
their absence mostly affects the low-energy vibrational density of states in
resonance with the spin transitions of 5 and 11. Finally, we note that the
simulation of relaxation times in Dy SMMs is particularly sensitive to the
accuracy of the crystal field parameters appearing in $\hat{H}_{CF}$ and thus
to the quality of the modellization of the magnetic ion’s coordination sphere.
Geometries optimized with pDFT and the use of the first coordination sphere
for the simulation of $\hat{H}_{CF}$ were found to be accurate in all cases
except for 7, where the Dy-H2O distances are not well reproduced by DFT and
the inclusion of the second coordination sphere is necessary. Accurate results
were obtained by simulating $\hat{H}_{CF}$ for a model including first and
second coordination spheres with experimental X-ray distances (see Fig. S6).

### Analysis of Spin Relaxation

Now that we have validated our simulations against experimental results, we
are in the position to exploit the full power of ab initio spin dynamics to
disentangle all the contributions to relaxation time. Fig. 3 shows the
decomposition of $\tau$ in terms of Orbach and Raman contributions for
compound 9, as an illustrative example. The results for all the twelve
molecules are reported in Fig S1. In agreement with previous
simulationslunghi2020multiple ; briganti2021complete ; lunghi2022toward ,
Orbach relaxation is found to dominate at high temperature, while Raman
mechanism only becomes relevant at low temperature, where phonons in resonance
with high-energy spin transitions become too unpopulated. Most importantly,
simulations reveal that experimental results for all the molecules were
obtained in a regime strongly influenced by Raman relaxation. The only
exceptions are represented by 8 and 9, where both magnetization decay
experiments and AC magnetometry were employed to sample both the particularly
long Orbach relaxation times and the Raman relaxation ones. Surprisingly, even
in the case of 11, Raman relaxation is found to be the dominant relaxation
mechanism despite the first excited KD is low in energy and amenable to
promoting Orbach relaxation at low temperature. 5 is the only system where the
Orbach relaxation mechanism is found to be dominating in the experimentally
accessible temperature range. In most cases, experimental relaxation times
measured with AC magnetometry falls at the transition stage between Orbach and
Raman-lead regimes, calling for a reinterpretation of the values of the
extracted $U_{eff}$ and $\tau_{0}$ and a new strategies for determining them
in experiments. Ab initio simulations offer such an opportunity.

Figure 3:  Orbach and Raman contributions to relaxation time. Comparison of
experiment (Green square) with the simulated Orbach (blue circle) and Raman
(red triangle up) relaxation for 9. Blue solid line represents the fitting of
Orbach simulation data with the equation:
$\tau_{Orbach}=\tau_{0}\mathrm{exp}(U_{eff}/k_{B}T)$. Similarly, solid red
line represents the fitting of Raman simulation data with the equation:
$\tau_{Raman}=\tau^{\prime}_{0}\mathrm{exp}(W_{eff}/k_{B}T)$. Figure 4:
Interpretation of spin relaxation with Arrhenius laws. The figure reports
$W_{eff}$ as a function of $\tau^{{}^{\prime}}_{0}$ and $U_{eff}$ as a
function of $\tau_{0}$ for the complexes 1–6 (top panel), and for the
complexes 7–12 (bottom panel). Green and red triangles represent the simulated
values for Orbach and Raman values, respectively. The selected compounds 1–12
are labelled with the corresponding index number.

The effective reversal barrier and the pre-exponential factor, $U_{eff}$ and
and $\tau_{0}$, have been extracted from the fitting of the simulated Orbach
data with the Arrhenius expression of Eq. 1, as depicted in Fig. 3. The
results obtained for Raman relaxation follows a more complex mathematical law.
Recent literaturelunghi2020multiple ; gu2020origins ; briganti2021complete
has shown that Raman relaxation is supposed to follow the temperature law

$\tau^{-1}_{Raman}=\sum_{i}(\tau^{\prime}_{0,i})^{-1}~{}\frac{e^{W_{eff,i}/k_{B}T}}{\left(e^{W_{eff,i}/k_{B}T}-1\right)^{2}}\>,$
(10)

where $W_{eff,i}$ corresponds to the energy of the $i$-th pair of degenerate
phonons absorbed and emitted. If only a single pair of phonons contributes to
relaxation, the Raman relaxation time also exhibits an Arrhenius behaviour at
low temperature ($W_{eff,i}\gg k_{B}T$),

$\tau_{Raman}=\tau^{\prime}_{0}e^{W_{eff}/k_{B}T}\>.$ (11)

This behaviour is observed in simulations to a good degree, and we therefore
attempt to fit the values of $\tau^{\prime}_{0}$ and $W_{eff}$ from low-$T$
simulated data, as depicted in Fig. 3.

Fig. 4 reports the values of $\tau_{0}$, $\tau^{\prime}_{0}$, $U_{eff}$ and
$W_{eff}$, for all twelve compounds. The $U_{eff}$ for the Co2+ systems (1–6)
match the energy of the excited KD with $M_{s}=\pm 1/2$. This is in agreement
with the fact that the latter is the only available excited KD able to mediate
Orbach relaxation. The same is not true for Dy3+ complexes, where more than
one excited KD is available. Orbach relaxation for the molecules 7, 8, 9, and
11 is found to be mediated by transitions between ground state ($M_{j}=\pm
15/2$) and the second excited KD, while for 10 and 12, Orbach transitions
occur through the third and fifth excited KD, respectively. Importantly, the
values of $\tau_{0}$ are found to lie in the range $10^{-8}$ \- $10^{-12}$ s,
a much smaller time window with respect to what extracted from literature (see
Fig. 1). Raman relaxation provides an opposite picture. $W_{eff}$ is found to
span a quite narrow range of small values in the order of tens of cm-1. On the
other hand, the values of $\tau^{\prime}_{0}$ span a range of $\sim$5-6 and
$\sim$10 orders of magnitude for Co2+ and Dy3+, respectively. The comparison
between Figs. 1 and 4 makes it clear that the experimental values of $U_{eff}$
and $\tau_{0}$ extracted from literature do not describe the sole Orbach
relaxation mechanism but are deeply affected by Raman relaxation. This is
particularly important for molecules with a reported small value of $U_{eff}$.

The values of $U_{eff}$ compare nicely with the energy of the excited KDs and
present no mystery, but more insights on the other quantities are necessary in
order to understand what regulates them. Let us begin from $\tau_{0}$.
According to Eq. 5, $\tau^{-1}_{0}$ receives contributions from three factors:
i) the density of phonons in resonance with the relevant spin transition, ii)
their coupling with the magnetic moment, as measured by the derivatives of the
effective Hamiltonian coefficients (see Eqs. 3 and 4), and iii) the nature of
the static effective Hamiltonian used to compute the matrix elements of the
spin-phonon coupling operator. Contributions i) and ii) can be easily assessed
by computing the spin-phonon coupling densitylunghi2017intra , i.e. the
average coupling of magnetic moment to phonons with a certain energy
$\hbar\omega_{\alpha}$. This quantity is defined in ESI and reported in Fig.
S4-S5 for all the compounds. Despite the presence of different features among
the twelve compounds, the average values of spin-phonon coupling intensity are
not dramatically different. This suggest that contribution iii) is in facts
largely responsible for the variance of $\tau_{0}$.

Figure 5: Correlation between phonon energies and Raman Arrhenius activation
energy. $W_{eff}$ is plotted as a function of frequency of the first optical
mode at the $\Gamma$-point for complexes 1–6 (red squares) and 7–12 (black
squares). The correlation coefficient for the two quantities is 0.91 and -0.14
for Co2+ and Dy3+, respectively.  Figure 6:  Orbach and Raman relaxation of
compound 12 with artificial phonons and spin-phonon coupling coefficients. The
black continuous line represents the original $\tau$ vs $1/T$ for 12. Symbols
correspond to the simulated values of $\tau$ obtained using the static
effective Hamiltonian of 12 with phonons and spin-phonon coupling of other
molecules: 1 (violet), 2 (green), 3 (blue), 4 (turquoise), 5 (red), and 6
(black). Color code for right panel: 7 (violet), 8 (green), 9 (blue), 10
(turquoise), 11 (red), and 12 (black).

Let us now turn to the analysis of Raman relaxation rates. According to recent
literature, the value of $W_{eff}$ should coincide with the lowest-energy
phonons significantly coupled to the magnetic momentlunghi2020multiple ;
gu2020origins ; briganti2021complete . In our approximation this generally
corresponds to some of the first available optical phonons. Fig. 5 shows the
correlation between the first mode at the $\Gamma$-point and the fitted values
of $W_{eff}$. In the case of Co2+ a good correspondence between the two
quantities is found, validating previous results and further suggesting the
importance of low-energy optical vibrations. Surprisingly, the same degree of
correlation is not observed for Dy3+ compounds, where $W_{eff}$ is found to
span slightly larger values of energy that the first optical mode one.
Moreover, the values extracted for $W_{eff}$ do not clearly corresponds to
peaks in the spin-phonon coupling density. For Dy3+ compounds, several excited
KDs are at play and it is hard to find a simple rationale to this behaviour.
We advance the hypothesis that the presence of very high-energy KDs involved
in the virtual state of Dy3+ compounds promotes the effect of phonons with
higher energy than the first ones available at the $\Gamma$-point. It is still
important to remark that the values of $W_{eff}$ are still in the order of
tens of cm-1, a value commensurate with low-energy optical vibrations and much
lower than the energy of excited KDs. Further analysis shows that none of the
phonons with energy higher than $\sim$100 - 150 cm-1 contribute to spin
relaxation (see Figs. S7) This is in agreement with previous observations that
the main factor in the determination of most important phonons is the Bose-
Einstein populationlunghi2020multiple ; briganti2021complete . The latter
decreases exponentially as the energy of vibrations increases, thus leaving
the lowest-energy available modes to drive spin relaxation.

From a qualitative point of view, Raman relaxation rate depends on similar
quantities to the Orbach one, and $\tau^{\prime}_{0}$ is influenced by both
the spin-phonon coupling density and the nature of static effective
Hamiltonian. As discussed for Orbach relaxation, the differences in the former
quantity across the twelve molecules are not dramatic and cannot account for
the large span of values predicted for $\tau^{\prime}_{0}$. The variance in
values of $\tau^{\prime}_{0}$ must therefore come i) the nature of the
eigenstates of the static effective Hamiltonian used to compute the matrix
elements of the spin-phonon coupling operator, and ii) the energy of the
excited KDs appearing at the denominator of Eq. 7. In both cases, these
quantities are intimately linked to the nature of the static crystal field and
the intensity of the zero-field splitting.

According to this analysis, the static effective Hamiltonian stands out as the
most important contribution to both $\tau_{0}$ and $\tau^{\prime}_{0}$. In
order to provide a conclusive proof of this claim we perform a simulation of
$\tau$ for Dy3+ compounds where the static effective Crystal field of 12 is
used together with the phonons and spin-phonon coupling of the other
compounds. Results for Orbach and Raman relaxation in this artificial
conditions are reported in the top and bottom panels of Fig. 6, respectively.
Orbach rates are found to all fall within one order of magnitude, with the
genuine value of 12 as the slowest one. In the case of Raman relaxation a
similar behaviour is observed, except for a larger variance of relaxation
times, which now span up to two orders of magnitude. The latter value should
be compared with a total variation of $\sim$6 orders of magnitude in normal
conditions (see for instance the values of $\tau$ at 20 K in the right panel
of Fig. 2). We attribute this larger sensitivity to the details of spin-phonon
coupling and the vibrational density of state in Raman relaxation to the fact
that these quantities appear to a power law of four instead of two as in
Orbach relaxation. Performing the same analysis for the Co2+ SMMs we find the
same qualitative behaviour but a larger span of relaxation times, reaching
almost two orders of magnitude for Orbach rates and four orders or magnitude
for Raman relaxation (see Fig. S3). This analysis conclusively demonstrates
that the static effective Hamiltonian is the most important contribution to
$\tau$ across all relaxation regimes. However, the details of spin-phonon
coupling add on top of that to fine-tune the value of $\tau$ and in the case
of Raman relaxation significantly contribute to determining the magnetic
moment lifetime.

## Discussion and Conclusions

Molecular magnetic anisotropy has been identified as a key ingredient for slow
spin relaxation since the very first observation of magnetic hysteresis in a
mixed-valence Mn12 cluster, now 30 years agosessoli1993magnetic . Decades of
success stories of molecular magnetism have marked the synthesis of compounds
with previously unimaginably large zero-field splitting values, reaching
record values of above 2000 Kzabala2021single ; guo2018magnetic . These
results effectively translated into the possibility of stabilizing the
molecular magnetic moment over times scales of hundreds of seconds at
temperatures as high as 80 Kguo2018magnetic . Despite the past success, the
field of single-molecule magnets now find itself at a critical stage. It has
been argued that the strategies that have led to large $U_{eff}$ in single-ion
complexes cannot realistically lead to much further improvement and that new
approaches to increasing $\tau$ must be foundreta2021ab ; briganti2021complete
. We argue that further progress in the design of molecular compounds with
long spin lifetime is to be found in a better understanding of the entire
process of spin relaxation. In this contribution we have highlighted how the
unique focus on $U_{eff}$ has led to overlooking many important aspects of
spin relaxation. The analysis of the pre-exponential factor $\tau_{0}$ is a
striking example of such a situation, where years of information available in
literature have remained unexplored.

In order to move out from this impasse, new tools are needed. In this work we
built on our recent contributions and have further shown how ab initio spin
dynamics simulations provide an effective way to obtain unprecedented
quantitative details on the spin relaxation process of single-molecules
magnets. Here we have fully disentangled the various contributions to both
Orbach and Raman relaxation posing an end to years of debate on the subject.
We have demonstrated that differently from the predictions of the canonical
theory of spin-phonon relaxationgatteschi2006molecular  and from what arise
from literature, there is no simple correlation between $U_{eff}$ and
$\tau_{0}$. Moreover, simulations made it possible to individuate realistic
ranges for both these quantities and those relative to Raman relaxation, when
interpreted as an Arrhenius process. We anticipate that this information will
play a fundamental role in the interpretation of future experiments and will
provide a guide to the assignment of Orbach and Raman relaxation mechanisms.

The unprecedented effort of studying twelve crystals of SMMs made it possible
to take a first glimpse to the correlation between chemical structure and spin
relaxation. Despite the large differences between the selected molecular
compounds, we demonstrated that Orbach spin relaxation rate is mostly
determined by the zero-field splitting. Now that spin relaxation can be
correctly interpreted, it becomes clear that the natural variations of
$\tau_{0}$ are too small to overcome the effect of $U_{eff}$. The little
dependence of the Orbach rate over the details of spin-phonon coupling
suggests that little improvement can be achieved by serendipitously tuning
vibrational modes. As highlighted in Fig. S2, the vibrational density of
states below $\sim$ 1500-1700 cm-1 is densely populated and only a very
delicate tailoring can bring phonons and spin transitions completely out of
resonanceullah2019silico . Similarly, the static zero-field splitting is found
to also strongly affect Raman relaxation rate. However, in this case the
features of spin-phonon coupling and low-energy vibrations significantly
contribute to modulating relaxation time and may offer a way forward to
further improvement. These findings thus shift the attention to Raman
relaxation as the most feasible improvement that can be obtained in single-ion
SMMs, and point to the necessity of better understanding how the low-energy
vibrational structure of coordination compounds can be chemically engineered.
We envision that a tighter synergy between experimental techniques and
simulations hold the key to significant advances in this direction. For
instance, inelastic neutron scatteringgarlatti2020unveiling  and terahertz
spectroscopyatzori2017spin ; albino2019first ; albino2021temperature ;
de2021exploring ; ma2022ligand  have already been used to provide unique
insights on the low energy part of the vibrational spectrum. Similarly, far-
infrared magneto spectroscopy can be used to provide insights on the spin-
phonon coupling of vibrations in close resonance to the spin transitions
responsible for Orbach relaxationrechkemmer2016four ; moseley2018spin ;
blockmon2021spectroscopic ; kragskow2022analysis . Once these techniques are
combined with ab initio simulations, a clear picture of how vibrations couple
with spin is possible.

Last but not least, we would like to stress out that the present findings were
made possible by studying a large number of complexes on the same footing. We
envision that overcoming the traditional approach of studying a single
molecule at the time or homologous small series of compounds can lead to a
much better understanding of structure-properties relations across the
chemical space. Moreover, we have here shown that accurate predictions down to
one order of magnitude of $\tau$ are now possible for a large breadth of
chemical compositions and zero-field splittings. This level of accuracy is
already enough for enabling a blind exploration of the chemical space in
search of new compounds. We envision that the further combination of ab initio
spin dynamics with machine learningjanet2017predicting ; lunghi2020limit ;
lunghi2020multiple ; lunghi2020surfing ; nguyen2022predicting  and high-
throughput strategiesnandy2021computational ; duan2021data  may lead to a
significant reduction in its computational cost, potentially leading to a
paradigm shift in the way we design molecular compounds.

In conclusion, we have provided a full ab initio description of spin
relaxation in twelve single-molecule magnets based on Co2+ and Dy3+ ions that
represent the extent of the chemical space explored so far in this field. Our
simulations made it possible to resolve a conflicting interpretation of
experimental results in terms of Orbach and Raman mechanism and to rationalize
all the main contributions to relaxation. We found that zero-field splitting
is the main figure of merit determining spin relaxation, but the efficiency of
Raman relaxation is also significantly influenced by the details of the low-
energy vibrational spectrum. We anticipate that these results will
significantly inform future synthetic strategies.

Acknowledgements
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. [948493]). Computational resources were provided by the
Trinity College Research IT and the Irish Centre for High-End Computing
(ICHEC). We acknowledge Matteo Briganti for the useful discussions.

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# First design of a superconducting qubit for the QUB-IT experiment

Danilo Labranca  Hervè Atsè Corti  Leonardo Banchi  Alessandro Cidronali
Simone Felicetti  Claudio Gatti  Andrea Giachero  Angelo Nucciotti  Department
of Physics, University of Milano-Bicocca - Milano, Italy  INFN, Sezione di
Milano-Bicocca - Milano, Italy  Department of Information Engineering,
University of Pisa - Pisa, Italy  INFN, Sezione di Firenze - Sesto Fiorentino
(FI), Italy  Department of Physics and Astronomy, University of Florence -
Sesto Fiorentino (FI), Italy  Department of Information Engeneering,
University of Florence - Firenze, Italy  Institute for Complex Systems,
National Research Council (ISC-CNR) - Rome, Italy  Laboratori Nazionali di
Frascati dell’INFN - Frascati (RM), Italy

###### Abstract

Quantum sensing is a rapidly growing field of research which is already
improving sensitivity in fundamental physics experiments. The ability to
control quantum devices to measure physical quantities received a major boost
from superconducting qubits and the improved capacity in engineering and
fabricating this type of devices. The goal of the QUB-IT project is to realize
an itinerant single-photon counter exploiting Quantum Non Demolition (QND)
measurements and entangled qubits, in order to surpass current devices in
terms of efficiency and low dark-count rates. Such a detector has direct
applications in Axion dark-matter experiments (such as QUAX[1]), which require
the photon to travel along a transmission line before being measured. In this
contribution we present the design and simulation of the first superconducting
device consisting of a transmon qubit coupled to a resonator using the Qiskit-
Metal software developed by IBM. Exploiting the Energy Participation Ratio
(EPR) simulation we were able to extract the circuit Hamiltonian parameters,
such as resonant frequencies, anharmonicity and qubit-resonator couplings.

††journal: Nuclear Instruments and Methods in Physics Research A.

##  1 Design

The design and simulation phase is fundamental in order to address the best
circuit parameters for the superconducting qubit device before moving to
manufacturing stage. The aim of our first design is to build a superconducting
qubit that can be controlled and read through a resonator coupled to a
feedline. For this chip we considered a substrate consisting of
$0.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of silicon oxide on top of
$600\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of silicon.
The design of our circuit was developed exploiting Qiskit-Metal [2] toolkit
and it is shown in Fig. 1. The circuit consists of an Xmon-type qubit
capacitively coupled to a quarter wave readout resonator. Each electrode of
the Xmon cross is $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide and
$360\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long. The gap between the cross
and the ground plane is $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide. The
readout resonator is $5.76\text{\,}\mathrm{mm}$ long, with
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ trace width and
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap, matching a characteristic
impedance $Z_{0}=$50\text{\,}\Omega$$. The coupling element is a
$130\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long claw, with
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap to the ground plane. The
resonator is capacitively coupled to the feedline by a
$400\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long coupling leg. The lumped
element circuit equivalent to our design is shown in Fig. 2.

(a)

(b)

Figure 1: Design rendered in Qiskit-Metal GUI. (a) Zoom on the Xmon qubit:
this type of qubit consists of a Josephson Junction shunted by a cross-shaped
capacitance. (b) Xmon qubit capacitively coupled to a quarter wave resonator
for readout. The readout resonator is also capacitively coupled to the
feedline.  Figure 2: Lumped element circuit equivalent of our design. The
resonator is represented as an LC circuit ($L_{r}$, $C_{r}$), while the qubit
is a Josephson Junction shunted by a capacitance ($C_{s}$). $C_{g}$ is the
qubit-resonator coupling capacitance and $C_{k}$ is the resonator-feedline
coupling capacitance.

##  2 Simulations

The simulations are performed exploiting Ansys Q3D and Ansys HFSS111ANSYS
Electronics Desktop 2021 R2. The former is used to extract the capacitance
values for circuit elements. The simulated capacitances are
$C_{s}=$98.19\text{\,}\mathrm{fF}$$, for the qubit shunt capacitance,
$C_{g}=$4.40\text{\,}\mathrm{fF}$$ and
$C_{k}=$8.62\text{\,}\mathrm{f}\mathrm{F}$$ for the qubit-resonator and
resonator-feedline coupling capacitances respectively. The lumped element
equivalent inductance and capacitance for a quarter wave resonator can be
calculated as:

$\displaystyle\centering C_{r}\@add@centering$
$\displaystyle=\frac{\pi}{4\omega_{r}Z_{0}}\approx$499\text{\,}\mathrm{fF}$$
(1) $\displaystyle L_{r}$
$\displaystyle=\frac{1}{C_{r}\omega_{r}^{2}}\approx$2.03\text{\,}\mathrm{nH}$$
(2)

from [3].
The chosen value for the Josephson junction inductance is
$L_{j}=$11\text{\,}\mathrm{nH}$$, matching a critical current of
$I_{c}=$29.92\text{\,}\mathrm{nA}$$. With these circuit parameters we expect a
Josephson energy $E_{j}=$14.86\text{\,}\mathrm{GHz}$$ and a capacitve energy
$E_{c}=$188.80\text{\,}\mathrm{MHz}$$, resulting in a ratio
$E_{j}/E_{c}\approx 79$ in the transmon regime. The qubit and readout
resonator resonant frequencies are
$\omega_{01}/2\pi=$4.55\text{\,}\mathrm{GHz}$$ and
$\omega_{r}/2\pi=$5.01\text{\,}\mathrm{GHz}$$ respectively, leading to a
detuning of $\Delta_{0}/2\pi\approx$457\text{\,}\mathrm{MHz}$$. The qubit-
resonator coupling strength can be calculated as ([4]):

$g_{n,n+1}=\sqrt{n+1}\frac{2\beta
eV_{rms}}{\hbar}\left(\frac{E_{j}}{32E_{c}}\right)^{\frac{1}{4}}$ (3)

with $\beta=C_{g}/(C_{g}+C_{s})$, $V_{rms}=\sqrt{\hbar\omega_{r}/2C_{r}}$ and
$e$ elementary charge, leading to $g_{01}/2\pi=$47.38\text{\,}\mathrm{MHz}$$.
We can then calculate the total dispersive shift as ([4]):

$\chi=\chi_{01}-\frac{\chi_{12}}{2}$ (4)

where $\chi_{ij}=g_{ij}^{2}/(\omega_{ij}-\omega_{r})$. The total dispersive
shift value is $\chi/2\pi=$-1.44\text{\,}\mathrm{MHz}$$, being larger than the
resonator resonance width $\kappa/2\pi=$1.12\text{\,}\mathrm{MHz}$$ allows the
qubit state readout. Fig. 3 shows the expected transmission on the feedline as
a function of frequency for the qubit in the ground or excited state.
An estimate of the qubit relaxation time can be calculated as follows ([5]):

$T_{1}=\frac{\Delta_{0}^{2}}{g_{01}^{2}}\frac{Q}{\omega_{r}}\approx$13\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$ (5)

where $Q\approx 4432$ (calculated as in [6]) is the resonator quality factor.
Ansys HFSS is used to perform the eigenmode simulation in order to compute the
resonant frequencies of the circuit and exploit the Energy Participation Ratio
(EPR) simulation [7, 8]. From the EPR analysis we can extract the qubit and
resonator frequencies, anharmonicities and the total dispersive shift. The
simulated qubit and resonator frequencies are
$\omega_{01}^{EPR}/2\pi=$4.43\text{\,}\mathrm{GHz}$$ and
$\omega_{r}^{EPR}/2\pi=$5.17\text{\,}\mathrm{GHz}$$ respectively. These values
match the expected frequencies within $2.6\%$ and $3.2\%$ respectively. The
qubit anharmonicity extracted from the EPR simulation is
$\alpha^{EPR}=$-193.43\text{\,}\mathrm{MHz}$$ and matches the expected $E_{c}$
within $2.5\%$. The simulated value for the total dispersive shift is
$\chi^{EPR}/2\pi=$-1.37\text{\,}\mathrm{MHz}$$, within $4.9\%$ the expected
value.

Figure 3: Feedline transmission as a function of frequency when the qubit is
in the ground (blue) or excited state (red).

##  3 Conclusions

We were able to simulate one possible design for the first chip fabrication,
comparing the results with the expected circuit features. The simulations
agree with the calculated values within a few percents. It should be noted
that longer and more accurate simulations could further reduce these gaps.
Before moving to the manufacturing stage, a similar design with inductive
resonator-feedline coupling will be evaluated, as well as different qubit and
Josephson junction parameters. Higher values for the anharmonicity and longer
qubit relaxation time will be pursued by accurately tuning couplings and
frequencies. Further parameter tuning is ongoing to optimize the qubit
readout. A simulation-data comparison will be performed as soon as the first
chip is produced in order to validate the design procedure.

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