# Optical and Electronic Properties of Molecular Systems Derived from
Rhodanine

Duvalier Madrid-Úsuga  [<EMAIL_ADDRESS>Carlos A.
Melo-Luna  [  Alberto Insuasty  [  Alejandro Ortiz  [  John H. Reina  [
<EMAIL_ADDRESS>
###### Abstract

Push-Pull functional compounds consisting of dicyanorhodanine derivatives have
attracted a lot of interest because their optical, electronic, and charge
transport properties make them useful as building blocks for organic
photovoltaic implementations. The analysis of the frontier molecular orbitals
shows that the vertical transitions of electronic absorption are characterized
as intramolecular charge transfer; furthermore, we show that the analyzed
compounds exhibit bathochromic displacements when comparing the presence (or
absence) of solvent as an interacting medium. In comparison with materials
defined by their energy of reorganization of electrons (holes) as electron
(hole) transporters, we find a transport hierarchy whereby the molecule
(Z)-2-((1,1-Dicyanomethylene)-5-(4-dimethylamino)benzylidene)-1,3-thiazol-4 is
better at transporting holes than molecule
(Z)-2-((1,1-Dicyanomethylene)-5-(tetrathiafulvalene-2-ylidene)-1,3-thiazol-4.

###### keywords:

American Chemical Society, LaTeX

Universidad del Valle] Centre for Bioinformatics and Photonics—CIBioFi, Calle
13 No. 100-00, Edificio E20, No. 1069, Universidad del Valle, Cali 760032,
Colombia [Universidad del Valle] Department of Physics, Universidad del Valle,
760032 Cali, Colombia Universidad del Valle] Centre for Bioinformatics and
Photonics—CIBioFi, Calle 13 No. 100-00, Edificio E20, No. 1069, Universidad
del Valle, Cali 760032, Colombia [Universidad del Valle] Department of
Physics, Universidad del Valle, 760032 Cali, Colombia Universidad del Norte]
Department of Chemistry and Biology, Universidad del Norte, Km 5 via Puerto
Colombia, 081007 Barranquilla, Colombia Universidad del Valle] Centre for
Bioinformatics and Photonics—CIBioFi, Calle 13 No. 100-00, Edificio E20, No.
1069, Universidad del Valle, Cali 760032, Colombia [Universidad del Valle]
Department of Chemistry, Universidad del Valle, 760032 Cali, Colombia
Universidad del Valle] Centre for Bioinformatics and Photonics—CIBioFi, Calle
13 No. 100-00, Edificio E20, No. 1069, Universidad del Valle, Cali 760032,
Colombia [Universidad del Valle] Department of Physics, Universidad del Valle,
760032 Cali, Colombia IR,NMR,UV

##  1 Introduction

Charge transfer (CT) studies seek to understand the ways in which their
transfer rate of CT depends on the properties of the electron-donor and
electron-acceptor system, solvent, molecular bridge and electronic coupling
between the involved states 1, 2. The different functionality played by these
factors and the way they affect the qualitative and quantitative aspects of
the electron transfer process have been extensively discussed in recent
years3, 4, 5, 6, 7. The need for understanding the processes of electron (ET)
or charge transfer at the molecular level have prompted the study of highly
conjugated molecular systems of the donor-acceptor (D-A) type, given their
unique photo-physical and photo-activated properties 8.

These properties have favoured the application and development of such systems
in areas that comprise non-linear optical materials 9, molecular optical
switches10, and photovoltaic cells11, among others. In the field of
photovoltaics, organic photovoltaic devices with D-$\pi$-A materials have
attracted a lot attention due to their potential in the creation of flexible
and remarkably light solar cells, with low manufacturing cost and high power
conversion efficiency (PCE)12. A key issue to understanding the CT process is
the ability to make quantitative predictions and measurements of the
characteristics associated to the individual molecular systems that allow
useful information for a direct comparison of the electron dynamics inferred
in electron and photochemistry, at the nanometric, molecular and electronic
scales13.

Currently, the increasing availability of kinetic data of CT processes and the
development of computational tools allow the study of different molecular
systems independently of their size, that exhibit better photophysical
properties, and facilitate a direct comparison between theory and
experiment14, 15, 16. These systems consist of covalent bonds of electron-
active chemical species, whether they are electron-donors or electron-
acceptors, which can be either connected directly or through a
$\pi$-conjugated bridge. The derivatives of the Rhodanine implemented in the
synthesis of push-pull systems are an example of these type of systems, and
they have been used as an electron-acceptor fragment in a variety of organic
compounds of interest; for example, in non-linear second order analytical
reactives, and, more recently, as metal-free organic dyes in the manufacturing
of dye-sensitized solar cells (DSSCs)17, 18, 19. For this purpose, the push-
pull molecular system of the donor-rhodanine type is efficiently anchored to
the meso-porous surface of TiO2. The light absorbed by the dye injects
electrons into the conduction band of the TiO2, thus generating an electric
current, while the fundamental state of the dye is regenerated by the
electrolyte20.

The precise prediction of the electron transfer rate in chemical and
biological reactions of this type makes them attractive systems for different
applications in the field of molecular electronics21, 22. In this work, we
present results obtained for new “push-pull” chromophores based on derivatives
of rhodanine, whereby 4-dimethylamine and 2-formyltetratiafulvalene exhibit
the role of electron-donor groups in which the nature of the electron transfer
processes is studied when they are connected to a dicyanorhodanine electron-
acceptor through a small molecular bridge. To determine the most stable
structure, the absorption spectrum and the first electronic state of the
complexes were calculated by means of density functional theory (DFT)
numerical simulation. We study the way the electron transfer in complexes gets
affected by the presence of a solvent that acts as an environment (also in gas
phase), seeking to report on novel quantitative results for such compounds 23,
and explore their potential in the application and design of innovative and
highly efficient donor-acceptor multifunctional devices that exhibit optimal
electronic properties.

##  2 Chromophores and computational details

Here, we consider the molecules
(Z)-2-((1,1-Dicyanomethylene)-5-(4-dimethylamino)benzylidene)-1,3-thiazol-4
(molecule 1), and
(Z)-2-((1,1-Dicyanomethylene)-5-(tetrathiafulvalene-2-ylidene)-1,3-thiazol-4
(molecule 2), as shown in Fig. 1. The geometries obtained for such most stable
conformations (see Supplementary section) were used as input data for the full
optimization of calculations of the ground state by means of the hybrid
functional B3LYP with a base set 6-31G+, using Gaussian 09 24; the
corresponding optimized structures are used for the molecules energy
calculation. The molecules excited states were calculated by means of the
time-dependent density functional theory (TD-DFT), and the results here
reported were carried out with the molecules i) in the gas phase, and ii) by
simulating an environment–methanol as solvent. In order to see the effects
caused by the solvent on the electronic properties of the different compounds,
the working molecules in the solvent were designated as follows: system S1:
Molecule 1 + Methanol and system S2: Molecule 2 + Methanol, while the system
GP1: Molecule 1 in gas phase and the system GP2: Molecule 2 in gas phase.
Additionally, different properties of these molecules, such as higher occupied
molecular orbitals (HOMOs), lower unoccupied molecular orbitals (LUMOs),
energy gap, reorganization energy, Gibbs free energy, and excitation energy
are derived from the computational results. In our theoretical calculations we
take into account the effects due to the solvent, since we aim to make
occurate predictions that compare with the reported experimental spectra. We
consider methanol as solvent with $\epsilon=32.6$, following a Conductor-like
Polarization Continuum Model (C-PCM)25, 26.

Figure 1: Molecular structure of the chromophores under study. Molecule 1 is
the
(Z)-2-((1,1-Dicyanomethylene)-5-(4-dimethylamino)benzylidene)-1,3-thiazol-4,
and molecule 2 is the
(Z)-2-((1,1-Dicyanomethylene)-5-(tetrathiafulvalene-2-ylidene)-1,3-thiazol-4.

##  3 Results and discussion

Electronic Transitions. Computational calculations to optimize the geometry of
molecules 1 and 2 were carried out using DFT-B3LYP/6-31G+; subsequently we
used TD-DFT for determining the transition states that most favoured the
charge transfer and search for a favorable environment in the CT processes.
The systems under study are of the donor-acceptor type in which two different
electron-donor fragments are connected to an electron-acceptor fragment for
generating and effective CT process, which results in a charge separation
state for analyzing the key molecular properties in the calculation of the
charge distributions in these molecules 23.

Figure 2: Spectra for the total density of states (DOS) obtained using
B3LYP/6-31G+ for the systems S1, S2, GP1 and GP2. The blue curve represents
the density of state spectrum, the green lines represent the occupied
molecular orbitals, and the red lines are the virtual molecular orbitals.

D-$\pi$-A molecular systems have two possible mechanisms for charge transfer,
i) super exchange: the charge or electrons transferred do not reside directly
in the molecular bridge and the states occupied by the molecule during this
time are known as virtual excitations; and ii) sequential charge transfer
(hopping): there are real intermediate states that are energetically
accessible, and this (thermally activated) mechanism is generally more
efficient for long-distance electronic transfer processes27.

Here we analyze the frontier molecular orbitals in inder to quantify the
relationship between structural and electronic geometry. For system in solvent
S1, the HOMO is mostly concentrated in the donor (4-dimethylamino group),
while the HOMO-1 and the LUMO are mostly located in the acceptor
(2-(1,1-dicyanomethylene)-1,3-thiazole-4), as seen in Fig. 2. Therefore, the
CT is the charge transfer mixture within the 4-dimethylamino coupled with the
CT from 2-(1,1-dicyanomethylene)-1,3-thiazole-4 to 4-dimethylamino moiety. For
system S2, the HOMO and HOMO-1 are mainly located in the tetrathiafulvalene
moiety, while the LUMO is located in the
2-(1,1-dicyanomethylene)-1,3-thiazole-4 (Fig. 2). Thus, the transitions of
system S2 from the HOMO to the LUMO together with HOMO-1 to LUMO have a more
significant character in the charge transfer with respect to system S1,
reflecting that the tetrathiafulvalene acts as a better electron-donor
fragment than the first one.

When we observe the frontier molecular orbitals of molecules in gas phase, we
see that the behavior described above is maintained. However, their density of
state spectra have very significative changes in the spectral densities
corresponding to each energy level, which indicates a considerable effect due
to the solvent on the electronic and geometric structure of the compounds.

Absorption Spectra. The electronic transition energies and the charge transfer
transitions are calculated using TD-DFT/B3LYP 28, 29. The UV-Vis absorption
spectra for the systems S1, S2, GP1, and GP2, shown in Fig. 3, are compared
with the experimental spectra reported by Insuasty 23 in methanol solution to
validate these computational results. This comparison shows that the spectral
profiles match each other within reasonable accuracy. Here, we observe both
experimental and computational spectra with a Full-Width Half Maximum (FWHM)
of around $100$ nm. Additionally, we observe that the experimental spectrum
shows a maximum absorption wavelength at $439.00$ nm for S1, and the
calculated one is $490.01$ nm. This difference ($\sim 49$ nm) between the
experimental measurements and the calculations takes place by the
overestimation of the energy values in the UV/Vis spectrum, and excitation
energy of the molecular system by the chosen computational method 30, 31.
Moreover, we show the calculated values of $\lambda_{max}$ corresponding to
different solvents. These results confirm the reliability of our method in
agreement with the experimental results, see Table T1 (supplementary
information). We find that system S1 has a strong absorption band at $490.01$
nm, together with other bands of lower energy at $343.37$ nm and $283.00$ nm
corresponding to transitions of the type $\pi-\pi^{*}$, which are associated
to the S2 and S7 states, respectively (see Table 1). The intense high energy
transition at $490.01$ nm is described by the excitation HOMO$\rightarrow$
LUMO (99%), according to the orbital transition diagram (See Supporting
information S1). This high energy transition can be assigned to intramolecular
charge transfer from the 4-dimethylamino electron-donor fragment to the
electron-donor fragment; the low energy transition at $343.37$ nm corresponds
to the transition HOMO$\rightarrow$ LUMO+1 (65%). The electronic transitions
can be seen as a contribution to the intramolecular charge transfer process
from the electron-donor to the electron-acceptor, and like the low energy
transition at $283.00$ nm which is described by HOMO$\rightarrow$ LUMO+3
(75%).

Figure 3: Absorption spectrum of the molecules in solvent (systems S1, S2), and molecules in gas phase (systems GP1 and GP2). Table 1: Wavelengths of the most important simulated transition states $\lambda$, oscillator strengths $f_{os}$, excitation energy $E$, and band gap energy $\Delta E_{H-L}$. System |  States |  $\lambda$ (nm) |  $\Delta E_{H-L}$ (eV) |  $f_{osc}$ |  $E$ (eV) | Composition
---|---|---|---|---|---|---
| $S_{1}$ | 490.01 |  | 0.9727 | 2.530 | HOMO$\rightarrow$LUMO (99%)
S1 | $S_{2}$ | 343.37 | 2.74 | 0.3391 | 3.611 | H-1$\rightarrow$LUMO (%34); HOMO $\rightarrow$ L+1 (65%)
| $S_{7}$ | 283.00 |  | 0.1336 | 4.381 | H-3$\rightarrow$LUMO (16%);HOMO$\rightarrow$L+3 (75%)
| $S_{1}$ | 561.10 |  | 0.1995 | 1.875 | HOMO$\rightarrow$LUMO (99%)
S2 | $S_{4}$ | 388.31 | 2.27 | 0.6559 | 3.193 | H-1$\rightarrow$LUMO (88%); HOMO $\rightarrow$L+1 (10%)
| $S_{6}$ | 325.98 |  | 0.1099 | 3.803 | H-2$\rightarrow$LUMO (92%)
| $S_{10}$ | 294.66 |  | 0.0962 | 4.208 | H-3$\rightarrow$LUMO (80%); H-1 $\rightarrow$ L+1 (13%)
| $S_{1}$ | 427.12 |  | 0.9075 | 2.903 | HOMO$\rightarrow$LUMO (99%)
GP1 | $S_{2}$ | 329.19 | 3.05 | 0.1939 | 3.767 | H-1$\rightarrow$LUMO (70%); HOMO $\rightarrow$ L+1 (29%)
| $S_{7}$ | 281.39 |  | 0.0833 | 4.406 | H-2$\rightarrow$LUMO (76%)
| $S_{1}$ | 671.27 |  | 0.1365 | 1.847 | HOMO$\rightarrow$LUMO (99%)
GP2 | $S_{2}$ | 380.16 | 2.23 | 0.6322 | 3.261 | H-1$\rightarrow$LUMO (98%)
| $S_{3}$ | 297.93 |  | 0.1194 | 4.161 | H-3$\rightarrow$LUMO (79%); H-1 $\rightarrow$ L+1 (11%)

The system S2 shows a band of high energy absorption at $388.31$ nm and bands
of low absorption at $561.10$ nm, $325.98$ nm, and $294.66$ nm. The low energy
transition $S_{1}$ for the system S2 is described by the transition
HOMO$\rightarrow$LUMO (99%), which represents an intramolecular CT from the
2-tetrathiafulvalene electron-donor moiety to the
2-(1,1-dicianometilen)-1,3-tiazol-4 electron-acceptor fragment; for the case
HOMO-1 and LUMO the latter mostly located in the
2-(1,1-dicianometilen)-1,3-tiazol-4 corresponds to the low transition energy
of the state $S_{4}$ described by the transition HOMO-1$\rightarrow$LUMO (88%)
where a CT from electron-donor to electron-acceptor is present. For HOMO-3 and
LUMO both mainly concentrated in 2- (1,1-dicyanomethylene)-1,3-thiazole-4,
associated with the transition HOMO-3$\rightarrow$LUMO (80%), a CT within the
electron-acceptor and not a transfer from the electron-donor to the concise
electron-acceptor takes place.

For the case of the system GP1, comprising the molecule 1 in gas phase, we
observe that it presents a high energy absorption band at $427.12$ nm
corresponding to the transition state of the HOMO$\rightarrow$LUMO ($99$%). In
addition, as in the case of the system S1, it presents two low energy
absorption bands at i) $329.19$ nm, corresponding to the transition state of
the HOMO-1$\rightarrow$LUMO ($70$%) that describes a contribution to the
process of CT in the interior of (2-(1,1-dicyanomethylene)-1,3-thiazol-4),
since the HOMO-1 and LUMO are more concentrated in the acceptor, with a small
CT contribution from the donor to the acceptor (as can be seen in Supporting
information S2); and ii) a low energy band at $281.39$ nm, corresponding to
the transition of the HOMO-2 $\rightarrow$LUMO. By comparing the graphs of
systems S1 and GP1, we obtain that the solvent produces an effective shift or
bathochromic displacement with respect to the wavelength of absorption.

The system GP2 shows a similar behavior to that of system S2. However, a
bathochromic shift is observed in relation to the absorption spectrum when
comparing the systems S2 and GP2. The resulting shift is associated to the
molecule-solvent interaction, since for the system GP2 this presents a high
energy band at $380.16$ nm, and for system S2 in presence of methanol this
occurs at $388.31$ nm. In addition, the system GP2 presents two low energy
bands, one at $671.27$ nm and the other one at $297.93$ nm; they are
associated with the transition states HOMO $\rightarrow$LUMO (99%), and
HOMO-3$\rightarrow$LUMO (79%), respectively. The energy band at $380.16$ nm is
associated to the transition states of HOMO-1 $\rightarrow$LUMO (98%).

The oscillation strength for an electronic transition is proportional to the
transition dipole moment. In general, a large oscillator strength corresponds
to large experimental absorption coefficients or a stronger fluorescence
intensity 32. Table 1 shows that oscillator stregth ($f_{os}$) values that
correspond to the transition states HOMO $\rightarrow$LUMO of systems S1 and
GP1, are similar to each other, as are for systems S2 and GP2. The results
shown in Table 1 reveal that the systems S1 and GP1 correspond to systems that
have a higher absorption capacity when compared to systems S2 and GP2.

Emission Properties. We use TD-DFT, with the hybrid B3LYP and basis set 6-31G+
in order to compute for the structure in an excited state and simulated the
emission spectrum of the systems under study. The maximum emission wavelengths
are shown in Table 2. The transitions $S_{1}\rightarrow S_{0}$ and
$S_{3}\rightarrow S_{0}$ represent fluorescence peaks in the emission
spectrum; in addition, the system S1 has the highest oscillator strength,
which corresponds to a LUMO$\rightarrow$HOMO transition.

Table 2: Emission spectrum results obtained for the systems under study in solvent (S) and gas phase (GP). System | Electronic Transition |  $\lambda_{max}$ (nm) |  $E_{s}$ (eV) |  $f_{os}$ |  Stokes Shift (nm) |  $\tau_{R}$ (ns)
---|---|---|---|---|---|---
S1 | $s_{1}\rightarrow s_{0}$ | 508.90 | 2.436 | 1.0126 | 18.0 | 3.85
S2 | $s_{3}\rightarrow s_{0}$ | 416.78 | 2.975 | 0.5316 | 28.8 | 5.02
GP1 | $s_{1}\rightarrow s_{0}$ | 443.28 | 2.797 | 0.9180 | 16.2 | 3.36
GP2 | $s_{3}\rightarrow s_{0}$ | 415.36 | 2.985 | 0.3792 | 35.2 | 7.29

The results for the excitation energy, oscillator strength and radiative
lifetime are presented in Table 2. We also report the Stokes shift values,
defined as the difference between $\lambda_{max}$ of absorption and
$\lambda_{max}$ of emission spectrum. The Stokes shift gives the energy
difference that exists between the absorption and emission due to the same
levels. This provides information about the probability of radiative and non-
radiative de-excitation between two levels, where the probability of radiative
de-excitation increases with the difference of energy and that of the non-
radiative one decreases. Hence, the first one dominates when the energy levels
are well separated and the second one does it when we have closer levels.
Thus, the radiative lifetime was calculated for the spontaneous emission
spectrum using the Einstein transition probabilities according to the
expression 33, 34:

$\tau_{R}=\frac{c^{3}}{2(E_{fl})^{2}f_{osc}},$ (1)

where $c$ is the speed of light in vacuum, $E_{fl}$ is the fluorescence
excitation energy, and $f_{osc}$ is the oscillator strength.

We conclude, as can be seen from Table 2 that the presence of the solvent
favours the radiative processes for the case of molecule 1, since the Stokes
shift is greater in the presence of methanol ($18.0$ nm) than in the gas phase
($16.2$ nm). However, for the case of molecule 2 this process is favoured in
the gas phase rather than in the presence of the solvent, which is made
evident by the longer radiative lifetime found for the gas phase. It is well
known that short radiative lifetimes lead to a high efficiency of light
emission, while long radiative lifetime facilitates electron and energy
transfer. In our case, the radiative lifetime is shorter for systems with
higher oscillator strength, which leads to an increase in luminescent
efficiency.

The duration of emission ($\tau_{R}$) for the studied molecules have the
following order: $\tau_{R}^{GP2}>\tau_{R}^{S2}>\tau_{R}^{S1}>\tau_{R}^{GP1}$.
This hierarchy indicates that the change of a donor unit strongly decreases
the emission lifetime of the compound in both gas phase and solvent; we find
the highest oscillator strength and the smallest lifetime radiation in the
case of the systems S1 and GP1, which correspond to the molecule 1 under
different environmental conditions. Consequently, molecule 1 represents a good
emission material with high efficiency 35.

The previous results show that molecule 2 exhibit higher energy than molecule
1, in agreement with the absorption and emission spectra values for both in
the gas phase (GP) and the solvent presence (S). Also, we observed that the
radiative times are longer for molecule 2 than for molecule 1. We attribute it
mainly to the fact that in the HOMO-level the molecule 2 shows a charge
distribution in the donor fragment while the molecule-1 shows a probability
distribution in all around the molecular complex, therefore, the charge
transfer process will take more time in the molecule 2 than in 1 as we
calculated. In the molecule 1 case, the electron-donor fragment is the
N,N-dimethylbenzene, which has an aromatic character in its ground state,
i.e., it complies with the Hückel rule of $4N+2\pi$ electrons in the benzene
ring. If the system experiencing photo-excitation, and the phenomenon of
intramolecular charge transfer occurs, the aromaticity is compromised and
competes with the charge separation process acquiring radical-cation
characteristics in such a process. These dynamics increases the radiative
process in the vacuum and within a solvent avoiding the structural variation
counteract this phenomenon. In contrast, the molecule 2, which has the
tetrathiafulvalene unit (TTF) as the electron-donor fragment, this unit is a
non-aromatic fragment of 14 $\pi$ electrons and structural symmetry $C_{2V}$
that deviates from the planarity as shown in Fig. 4 36. In the photoexcitation
process, this donor-fragment is capable of sequentially providing electrons
sequentially until reaching the stable state which implies a dicationic state
in a planar structure with aromatic behavior. This behavior decreases the
radiative processes and improves the interaction with the solvent generating
charge separated species with greater stability.

Figure 4: Molecular structure of the Tetrathiafulvalene (TTF) neutral unit and
the representation as aromatic of the radical cation and dication in the
Hückel sense, showing that while TTF+∙ and TTF2+ have a planar D2h symmetry,
neutral TTF has a boatlike equilibrium structure with C2v symmetry

Charge Transfer Rate. Charge transfer is a crucial process involved in many
physical and biological phenomena such as the photosynthesis 37; this process
can be estimated, in a first approximation, by using the semi-classical theory
of Marcus 38, 39:

$k_{e(h)}=\frac{2\pi}{\hbar}\frac{|V_{e(h)}|^{2}}{\sqrt{4\pi\lambda_{e(h)}k_{B}T}}\exp\left(-\frac{\lambda_{e(h)}}{4k_{B}T}\right),$
(2)

where $V_{e(h)}$ is the electronic coupling between the final and the initial
state for electrons (holes), $\lambda_{e(h)}$ is the reorganization energy for
electron (hole), and $k_{B}$ denotes Boltzmann constant.

For an efficient CT mechanism the reorganization energy of the molecular
system must be small and an electronic coupling between the electron-acceptor
and electron-donor parts is necessary 40. The reorganization energy comprises
two factors, the first one is the internal or intramolecular reorganization
energy ($\lambda_{int}$), and the second one is the external or intermolecular
reorganization energy ($\lambda_{ext}$). The $\lambda_{int}$ accounts for the
structural changes between neutral and ionic states and must be calculated,
while $\lambda_{ext}$ reflects the change in the polarization of the medium
after the CT takes place 41.

The intramolecular reorganization energy $\lambda_{int}$ can be estimated for
the electrons (reorganization energy of electron $\lambda_{e}$) and for the
holes (reorganization energy of holes $\lambda_{h}$), and can be expressed by
the following equation 42, 43, 44:

$\displaystyle\lambda_{e}$
$\displaystyle=\lambda_{1}^{e}+\lambda_{2}^{e}=\left(E_{0}^{-}-E_{-}^{-}\right)+\left(E_{-}^{0}-E_{0}^{0}\right)$
(3) $\displaystyle\lambda_{h}$
$\displaystyle=\lambda_{1}^{h}+\lambda_{2}^{h}=\left(E_{0}^{+}-E_{+}^{+}\right)+\left(E_{+}^{0}-E_{0}^{0}\right),$

where $E_{0}^{+}$($E_{0}^{-}$) is the energy of the cation (anion) calculated
with the optimized structure of the neutral molecule. Similarly,
$E_{+}^{+}$($E_{-}^{-}$) is the energy of cation (anion) calculated with the
optimized cation (anion) structure, $E_{+}^{0}$ ($E_{-}^{0}$) is the energy of
the neutral molecule calculated at the cationic (anionic) state. Finally,
$E_{0}^{0}$ is the energy of the neutral molecule at the ground state.

Table 3: Molecular calculation of the reorganization energy for electrons (holes) $\lambda_{e(h)}$, electronic coupling for the electron (holes) transfer mechanism $V_{e(h)}$, and electron (hole) transfer rate $k_{e(h)}$. System |  $\lambda_{e}$(eV) |  $\lambda_{h}$(eV) |  $V_{e}$(eV) |  $V_{h}$(eV) |  $k_{e}(10^{15}$ s-1) |  $k_{h}(10^{15}$ s-1)
---|---|---|---|---|---|---
S1 | 0.184 | 0.094 | 0.660 | 0.685 | $2.87$ | $5.13$
S2 | 0.340 | 0.219 | 0.700 | 0.620 | $0.53$ | $1.67$
GP1 | 0.271 | 0.151 | 0.655 | 0.520 | $1.01$ | $2.72$
GP2 | 0.374 | 0.221 | 0.720 | 0.630 | $0.37$ | $1.61$

This redistribution of energies for the case of electron reorganization energy
is best observed in Fig. 5. The external reorganization energy $\lambda_{ext}$
explains the nuclear shifts in the surrounding medium and the resulting
electronic effects are much harder to calculate. This is assumed to be, for
many authors, between $0.2$ eV and $0.5$ eV for simple models based on the
dielectric properties of organic matrices 45, 43.

Figure 5: Scheme for the calculation of the reorganization energy for the
electron transfer; $\lambda_{1}$ is the reorganization energy of the neutral
molecule and $\lambda_{2}$ denotes the reorganization energy of the radical
anion.

We calculate the electron reorganization energy using Fig. 5. According to
Marcus model, the rate of electron transfer depends mainly on the energy of
reorganization and the coupling between the donor and the acceptor, in
addition to the general exergonity of the process 42. It is also considered
that for a self-exchange electron transfer process (where interaction with
solvent is not considered) the change in the Gibbs free energy is zero and the
electron transfer rate will only depend intrinsically on the barrier of
activation, marked by the internal and external reorganization energy, and the
electronic coupling parameter $V$. The $\lambda_{int}$ in Eq. (3) for the
self-exchange process has two contributions, arising from the geometric
relaxation along inter-nuclear coordinate upon moving from neutral-state to
the charged-state geometry and vice versa. The electronic coupling $V$,
another key parameter in the CT process, is the geometrically most dependent
element of the kinetic constant because its value depends on the distance
between donor-acceptor and the geometry of the system (orientation of the
orbitals); this parameter is also sensitive to changes in the systems under
study such as solvents, and temperature, among others 46, 47, 48, 49.

Here, we use the generalized Mullinken-Hush method (GMH) 50 to calculate the
electronic couplings, and the operator used in the GMH method is the adiabatic
dipole moment matrix $\mu_{12}$ 50. Under this approach (and in the weak
coupling regime), the electronic coupling for a direct donor-acceptor coupling
is calculated as 51:

$V=\frac{\Delta
E_{12}\mu_{12}}{\sqrt{(\Delta\mu_{1}-\Delta\mu_{2})^{2}+4\mu_{12}^{2}}},$ (4)

where $\Delta E_{12}$ is the orbital energy difference, and $\mu_{12}$ is the
dipole moments difference of the adiabatic states.

The resuls for the reorganization energy, electronic coupling, and the
electron transfer rate are shown in Table 3. As reported in 52, 32, 53, it has
been found that at low values of reorganization energy, the transfer rate is
high. The hole reorganization energy calculated for the systems S2 and GP2 are
smaller than those for the systems S1 and GP1; this implies that the hole
transfer rate is greater in the systems S2 and GP2, and we also note that for
the case of the system S2 compared to the GP2 the hole transport rate is
higher in the presence of methanol than in the case of the gas phase,
confirming this behavior for the case of the system S1 and GP1 where the same
situation is observed, which indicates that methanol does not favour the
transport of holes. Furthermore, the hole reorganization energies
$\lambda_{h}$ for all systems are smaller than that of
N,N’-diphenyl-N,N’-bis(3-methlphenyl)-(1,10-biphenyl)-4,4’-diamine (TPD),
which is a typical hole transport material with $\lambda_{h}=0.290$ eV 54.
This implies that the hole transfer rates of the molecules 1 and 2, in the
condition under study, might be higher than that of TPD. Thus, the molecules 1
and 2 might comprise good hole transport materials from the stand point of the
smaller reorganization energy. On the other hand, we observe that for the case
of the system S2 compared to GP2 the hole transport rate is higher in the
presence of methanol than in the case of the gas phase, confirming this
behavior for the case of the systems S1 and GP1 where the same situation
occurs, which indicates that methanol favours the transport of voids.

The value of $\lambda_{e}$ is smaller for the case of the systems S1 and GP1
than for the systems S2 and GP2: this indicates that the electron transfer
rates for S1 and GP1 will be larger than those due to the systems S2 and GP2
as can be seen in Table 3. In addition, by comparing $\lambda_{e}$ for the
systems S1 and GP1, as well as for S2 and GP2, we see that the following holds
for $\kappa_{e}$: $\kappa_{e}^{S1}>\kappa_{e}^{GP1}$, and
$\kappa_{e}^{S2}>\kappa_{e}^{GP2}$, which indicates that the electron transfer
process is favoured by the presence of solvent in the molecules 1 and 2. In
addition, by comparing the reorganization energies for electron reorganization
and holes, we observe that the values of $\lambda_{h}$ are smaller than those
for $\lambda_{e}$, suggesting that the carrier mobility of the electrons is
larger than that of the holes. Hence, the molecules 1 y 2 can be used as
promising hole transport materials in, e.g., organic light-emitting diodes
from the stand point of the smaller reorganization energy, which can be
corroborated with the $\kappa$ values shown in the Table 3. Finally, given
that $\kappa_{e}$ and $\kappa_{h}$ are greater for the systems S1 and GP1, we
conclude that the molecule 1 is better at transporting charge than the
molecule 2. In addition to the previous analysis, it can be seen that the
molecule 1 is more transport-efficient than the molecule 2.

###  3.1 Conclusions

We have theoretically investigated different optical and electronic properties
for new structures based on 4-dimethylamino and tetrathiafulvalene as
electron-donor groups and dicyanorhodanine as electron-aceptor group, in which
the correlation between structures and electronic dynamics is studied by means
of theoretical chemical calculations. It was observed that the presence of a
solvent with which the molecules 1 and 2 interacts, favours the electronic
transfer process. In addition, the change in the donor group shows that
4-dimethylamino acts as a better electron donor and hole transporter than
tetrathiafulvalene, which can be seen by observing their rates of transfer,
for both electrons and hole. Also, we observe that if compared the transfer
rate of the molecule 1, it is found that this compound is a better promisor
for the transport of holes than electrons ($\kappa_{h}>\kappa_{e}$) in both
cases, in presence of solvent and gas phase. As regards the wavelength of
absorption, this shows a bathochromic effect between the gas phase and the
presence of the solvent. The computational results predict the electronic
properties of the systems S1, S2, GP1 and GP2, and the analysis of the
molecular frontier orbitals shows that the vertical electronic transitions of
absorption of the studied compounds are characterized as intramolecular charge
transfer. In addition, the molecules 1 and 2 can be used as void transport
materials.

The authors acknowledge support by the Colombian Science, Technology and
Innovation Fund-General Royalties System (Fondo CTeI-Sistema General de
Regalías) under contract BPIN 2013000100007.

Contains the optimal geometries, the total energy and the frontier molecular
orbitals involved in the electronic transitions at the TD-DFT B3LYP/6-31G+
levels.

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# Piezoelectric microresonators for sensitive spin detection

Cecile Skoryna Kline  Quantum Engineering Technology Labs and Department of
Electrical and Electronic Engineering, University of Bristol, Woodland Road,
Bristol BS8 1UB, United Kingdom  Jorge Monroy-Ruz  Quantum Engineering
Technology Labs and Department of Electrical and Electronic Engineering,
University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom  Krishna
C. Balram<EMAIL_ADDRESS>Quantum Engineering
Technology Labs and Department of Electrical and Electronic Engineering,
University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom

###### Abstract

Piezoelectric microresonators are indispensable in wireless communications,
and underpin radio frequency filtering in mobile phones. These devices are
usually analyzed in the quasi-(electro)static regime with the magnetic field
effectively ignored. On the other hand, at GHz frequencies and especially in
piezoelectric devices exploiting strong dimensional confinement of acoustic
fields, the surface magnetic fields ($B_{1}$) can be significant. This $B_{1}$
field, which oscillates at GHz frequencies, but is confined to -scale
wavelengths provides a natural route to efficiently interface with nanoscale
spin systems. We show through scaling arguments that $B_{1}{\propto}f^{2}$ for
tightly focused acoustic fields at a given operation frequency $f$. We
demonstrate the existence of these surface magnetic fields in a proof-of-
principle experiment by showing excess power absorption at the focus of a
surface acoustic wave (SAW), when a polished Yttrium-Iron-Garnet (YIG) sphere
is positioned in the evanescent field, and the magnon resonance is tuned
across the SAW transmission. Finally, we outline the prospects for sensitive
spin detection using small mode volume piezoelectric microresonators,
including the feasibility of electrical detection of single spins at cryogenic
temperatures.

††preprint: APS/123-QED

##  I Introduction

Piezoelectric microresonators [1, 2] have revolutionized wireless
communication by enabling small form-factor, high performance radio frequency
(RF) filters that can be compactly packaged into mobile phones. In addition,
these devices have had a broad impact on areas ranging from sensing [3] to
quantum communication [4]. A piezoelectric material enables conversion of RF
electromagnetic fields into acoustic fields, which have wavelengths $\approx$
at GHz frequencies, $10^{5}$ smaller than the cm-scale wavelengths of the RF
fields. This deeply-subwavelength confinement is the key driver for the
majority of applications involving piezoelectric devices.

The constitutive relations for a piezoelectric device [5] relate the stress
($\vec{T}$) induced by an applied electric field ($\vec{E}$). While this is
strictly true for applied DC fields, the equations are extended to RF fields
under the quasi-(electro)static approximation [6], wherein the Poisson
equation for electrostatics is substituted for Maxwell’s equations for the
electromagnetic field, and is solved along with the elastic wave equation to
propagate acoustic fields in piezoelectric devices. The quasistatic
approximation is usually justified because the deeply sub-wavelength
confinement provided by the acoustic field ensures that the far-field
electromagnetic radiation component is minimal. By definition, using this
approximation forces the magnetic field to be strictly zero.

On the other hand, it is known that piezoelectric devices radiate
electromagnetically [7] and that this radiation presents a limit on the
achievable mechanical quality factor ($Q_{m}$) in piezoelectric resonators
[8]. While the far-field radiation efficiency is low for the reasons outlined
above, it can be significantly enhanced in a resonant geometry and provides
significant size advantages in the design of very low frequency antennas [9].
In this work, we focus on the near-field (surface) component of this
oscillating magnetic field ($B_{1}$), and ask if these $B_{1}$ fields can be
exploited for improving the spin detection sensitivity of nanoscale electron
spin resonance (ESR) experiments [10, 11]. Our aim is to apply ideas from
cavity quantum electrodynamics (cQED) [12] to nanoscale spin systems [13, 14],
with the key sensitivity enhancement being provided by the vastly reduced mode
volume ($V_{m}$) of piezoelectric microresonators in comparison to their
electromagnetic counterparts.

##  II Surface current density scaling in piezoelectric devices

In a piezoelectric material, the propagating acoustic displacement field is
accompanied by a surface polarization ($\rho$). The evanescent electric fields
($\vec{E}$) generated at the material-air boundary curl as depicted in
Fig.1(a). The plot shows a finite element method (FEM) simulation of a surface
acoustic wave (SAW) mode propagating on a Scandium Aluminum Nitride (ScAlN) on
Si substrate. The accompanying electric fields are shown by an arrow plot.
This curling of surface fields is one instance of the universal phenomenon of
spin-momentum locking [15] that applies to all evanescent fields (cf. Appendix
A for a discussion of the analogy between the surface fields in surface
plasmons and surface acoustic waves). The accompanying surface magnetic field
($B_{1}$) orientation can be directly obtained from the ($\vec{E}$) fields,
and in this case would form loops that come periodically into and out of the
plane (shown schematically in Appendix A). One can verify the $\vec{B_{1}}$
orientation by noting the direction of the surface polarization currents
($J=\frac{\partial\rho}{{\partial}t}$), plotted in Fig.1(b) and invoking
Ampere’s law. For focusing acoustic field geometries as would be necessary for
increasing the local field strength, the field orientation is shown in
Fig.1(c), with a zoomed-in plot of the focus region shown in Fig.1(d). We
would like to note here that while the $\vec{E}$ field lines can be computed
using an FEM solver like COMSOL, due to the quasistatic approximation being
imposed on the solver, the magnetic field is strictly zero. Therefore, current
FEM solvers cannot be used to visualize this ($B_{1}$) field directly from
taking the curl of the $\vec{E}$ field. The results presented in Figs.1(a-d)
make certain simplifications. We model (Sc)AlN films using aluminum nitride’s
material parameters, and in Fig.1(c,d), we simulate focusing on a thin AlN
film to reduce the simulation’s memory constraints.

Figure 1: (a) FEM simulation showing the surface displacement of a surface
acoustic wave propagating on a ScAlN on Si substrate. The accompanying
$\approx$ 2.9GHz electric field is shown using an overlaid arrow plot. As can
be seen there exists an evanescent field in the air whose orientation
(helicity) is determined by spin momentum locking [15], (b) FEM simulation,
same as (a) but showing the overlaid oscillating polarization current density
using an arrow plot. This oscillating current density which exists at both the
interfaces (air-ScAlN and ScAlN-Si) is responsible for the surface
$\vec{B_{1}}$ fields we investigate in this work. (c) Focusing the SAW using
curved electrodes in AlN (d) At the focus, the displacement, and corresponding
evanescent field and current densities, are enhanced by the focusing ratio.

To estimate the scaling of this surface $B_{1}$ field, we therefore start with
the oscillating surface current density ($J$, [□]) instead [3] following prior
work on SAW based sensors [16]:

$J^{2}=2K^{2}{\omega}k^{2}(\epsilon_{c}+\epsilon_{s})P$ (1)

where $K^{2}$ is the material’s piezoelectric coefficient and represents the
fraction of the acoustic wave energy that is stored in the electric field.
$\omega$ is the wave frequency, $k=2\pi/\lambda$ the wave vector,
$\epsilon_{c,s}$ [] represent the dielectric constants of the cladding (air)
and substrate respectively, and $P$ is the power density of the acoustic wave
expressed in power per unit beam width []. We can see that for a tightly
focused acoustic beam with focus width $\approx\lambda$,
$P{\propto}1/{\lambda}$, and hence $B_{1}^{2}{\sim}J^{2}{\propto}f^{4}$, or
the surface field $B_{1}{\propto}f^{2}$. With $K^{2}$ = 0.05, $\lambda_{a}$=
1, $f$ = 3GHz, $\epsilon_{c}+\epsilon_{s}\approx$ 10, beam width at focus
$\approx\lambda_{a}$ and $P_{a}$=1mW, we expect a surface current density at
the focus of $\approx$ 81.15□. The scaling of $J$ with $\epsilon$ can be
visualized directly in Fig.1(b), where the current density at the (Sc)AlN-Si
interface is stronger than at the (Sc)AlN-air interface.

Assuming the current flows uniformly in a (semi-circular) loop of size
$\approx\lambda_{a}$, one can roughly estimate the $B_{1}$ field as
$B_{1}\approx{\mu_{0}}J\lambda_{a}/4$ = 25.5, with $\mu_{0}$ the vacuum
permeability. By trapping the acoustic field in wavelength scale microcavities
[17], the surface current density and therefore the accompanying $B_{1}$ field
can be enhanced by the cavity mechanical quality factor $Q_{m}$, with
$Q_{m}\approx 10^{4}$ feasible in crystalline media at ambient conditions
[18]. Alternately, one can also estimate the spin-resonator single photon
coupling strength $g$ in a cavity-QED framework [12, 10] with:

$g=\frac{\mu_{B}}{\hbar}\sqrt{\frac{2\mu_{0}\hbar\omega_{0}}{V_{m}}}$ (2)

where $\mu_{B}$ is the Bohr magneton, $\hbar$ the reduced Planck constant,
$\mu_{0}$ vacuum permeability and $V_{m}$ is the cavity mode volume. Assuming
an acoustic cavity mode with dimensions of $5\lambda_{a}^{3}$ at 3GHz with
$\lambda_{a}$=1, this gives us a coupling strength $g\approx$ $2\pi*$14kHz. We
would like to emphasize that we are interested in the near-field $B_{1}$ here.
Due to the alternating signs of the surface current density, the far-field
radiation is relatively weak.

Such piezoelectric approaches to enhancing spin detection sensitivity are in
many ways complementary to superconducting cavity approaches which have
recently achieved single spin sensitivity [11, 19] at temperatures. Both
approaches rely on a Purcell-like [12] enhancement of the spin detection
sensitivity which scales $\propto\sqrt{Q_{c}/V_{c}}$, where $Q_{c}$ is the
quality factor and $V_{c}$ is the mode volume of the cavity. The
superconducting approaches support high $Q$ factors ($>10^{5}$) and small
magnetic mode volume ($\approx 10^{-12}\lambda_{RF}^{3}$) [20] by exploiting
lumped element inductor-capacitor ($LC$) cavity designs, where the $B_{1}$
field is strongly localized in close proximity to the inductor. Piezoelectric
resonators, on the other hand provide moderate $Q_{m}$ ($\approx 10^{4}$), but
vastly reduced cavity mode volumes ($\approx 10^{-15}\lambda_{RF}^{3}$) with
the additional advantage of enabling experiments in ambient conditions, which
makes it feasible to use this technique with chemical and biological samples
which might deteriorate appreciably in cryogenic environments.

We would like to distinguish our work from previous results that have studied
the interaction between acoustic fields and spin systems (both nanomagnets and
magnetic thin films) that primarily exploit the magnetostrictive effect [21,
22, 23] or the strain field to perturb spin systems [24, 25, 26]. In this
work, we instead focus on using piezoelectric devices for generating the
oscillating magnetic field ($B_{1}$) directly and the methods developed here
can be applied in principle to any spin resonance experiment. Given the field
generated is on the surface and evanescent, physical contact between the
sample and the spin system can be completely avoided, as discussed in the
experiments below.

##  III Using spin resonance absorption to probe the evanescent $B_{1}$ field

Figure 2: (a) Schematic of the experimental setup used to probe the $B_{1}$.
Curved IDTs are used to launch and detect SAWs on a ScAlN on Si substrate. A
polished YIG sphere is brought in proximity (non-contact) of the evanescent
field at the focus of the SAW and the IDT transmission is monitored as the
magnon mode is tuned across the SAW transmission resonance. The magnitude and
phase of the transmission is monitored using a vector network analyzer and the
magnon mode is tuned by a $B_{z}$ field applied using an electromagnet
underneath the sample. (b) and (c) show images from the side and top view
cameras of the experiment during operation.

We perform spin resonance measurements by positioning a polished Yttrium-Iron-
Garnet (YIG) sphere (diameter $\approx$ 150) in the evanescent field
($<\lambda_{a}\approx$ 1) of the SAW devices. YIG spheres support high-Q
($\approx 10^{4}$ at 10GHz) collective spin wave excitations (magnons), which
have been used for a wide range of applications, ranging from tunable filters
and low noise oscillators for wireless communication [27] to hybrid quantum
transduction, wherein quantum states from superconducting qubits are mapped
back and forth from the magnon modes [28]. By mounting the YIG sphere on a
movable three-axis translation stage and positioning it in the evanescent
field of the acoustic wave, we can spatially probe the surface $B_{1}$ field
by monitoring the SAW transmission and looking for changes in the amplitude
and phase response of the received SAW signal as the magnon mode frequency is
tuned across the SAW resonance.

The schematic of our experimental setup is illustrated in Fig.2(a). Fig.2(b,c)
show the side and top view of the YIG sphere positioned at the centre of the
SAW transmit-receive circuits as captured by the two zoom lenses indicated in
Fig.2(a). SAWs are launched and detected using a vector network analyzer (VNA)
by patterning a set of interdigitated transducers (IDT) [2] on a piezoelectric
(c-axis oriented) Sc0.06Al0.94N film deposited on a [100] oriented silicon
substrate. We pattern IDTs with both straight fingers to launch quasi plane
waves of sound, and curved fingers [29] to focus the acoustic field down to a
beam width of 1-5 $\lambda_{a}$. The curved IDT devices are designed as
confocal transmit-receive pairs [30, 17]. To achieve the highest spin
detection sensitivity, focusing devices are necessary as they significantly
enhance the local power density ($P$ in equation 1) by the focusing ratio
($\approx 20-50x$) and the strongest $B_{1}$ fields are therefore always
generated at the focus. The size of the YIG sphere in our experiments presents
two constraints. It requires us to separate the focusing IDTs by $\approx
100-125\lambda_{a}$, which magnifies the effect of wave diffraction and the
anisotropy of the underlying silicon substrate with the net result being a
relatively low acoustic throughput.The second issue, which can be seen from
Fig.2(c) is the difficulty of determining the precise acoustic focus location
while aligning the YIG sphere for maximum SAW extinction.

Figure 3: The SAW-magnon interaction is probed by monitoring the VNA
transmission ($S_{21}$) as a function of $B_{z}$. The 2D colorplots in (a,c)
plot $\lvert{S_{21}}\rvert$ as a function of frequency and $B_{z}$ for two
different curved IDT devices with the same period, but different curvature.
The SAW mode frequency is constant with $B_{z}$, whereas the magnon mode
frequency shifts linearly with $B_{z}$. When the magnon mode frequency crosses
the SAW transmission resonance, one expects an excess attenuation in the SAW
transmission due to magnon mode excitation and dissipation. (b,d) show
representative linecuts from (a,c) respectively for $f_{m}<f_{s}$ (red),
$f_{m}{\approx}f_{s}$ (black), and $f_{m}>f_{s}$ (blue). The excess
attenuation when $f_{m}{\approx}f_{s}$ is especially clear in (b), and the
same trend can also be observed, albeit with lower magnitude in (d). The
plotted data are time gated with a 40ns. Uncorrected datasets are available in
Appendix B, and the bare IDT reflection and transmission spectra are plotted
in Appendix D.

The magnon frequency can be tuned by applying a static DC magnetic field
($B_{z}$) using an electromagnet, as shown in Fig.2(a). Due to the frequency
of operation and the orientation (shown in Fig.2(b)), the YIG sphere is not
saturated, and the magnon mode relaxes to lower frequencies with time.
Therefore, there is a finite time offset between the setting of the voltage on
the electromagnet and the data acquisition on the VNA, which would result in a
lower effective $B_{z}$ due to relaxation. In principle, one can park the
magnon mode on the higher frequency side of the SAW transmission and let the
mode relax and down-shift in frequency across the SAW resonance, while
recording VNA traces as a function of time. In practice, we found this method
did not give us the resolution needed to capture the exact crossing between
the magnon mode and the SAW. Therefore, we rely on incrementing the voltage in
small steps and acquiring the data quickly to minimize the effects of mode
relaxation.

As the $B_{z}$ field is tuned, ferromagnetic resonance (FMR) in the YIG sphere
induces an excess absorption which shows up as a reduced transmission
(attenuation) at the SAW frequency. This excess absorption is maximized when
the frequencies of the SAW and the magnon mode are identical. This is shown in
Fig.3. Fig.3(a,c) correspond to datasets taken from two distinct curved IDT
devices with the same period but different curvature. The colorbar on the 2D
plot indicates the magnitude of the (normalized) transmitted signal ($S_{21}$)
as a function of both $B_{z}$ and frequency. As the plot shows, we observe a
variety of magnon modes [31] in our experiment, with varying coupling
strengths and quality factors. Without mode imaging [31], it is hard to
ascertain the mode symmetry from a pure microwave transmission experiment.
Here, we focus on the mode that gives us the strongest signal and generically
label it as magnon in Fig.3(a,c). As the 2D color plots show, the SAW
frequency stays relatively independent of $B_{z}$, whereas the magnon mode
linearly tunes to higher frequency with increasing $B_{z}$. When the magnon
mode frequency ($f_{m}$) crosses the SAW mode ($f_{s}$), one can clearly see
an excess absorption which is stronger in Fig.3(a) compared to the dataset in
Fig.3(c). One can see this excess absorption effect more clearly by taking 1D
cuts through the 2D data corresponding to the cases when $f_{m}<f_{s}$,
$f_{m}{\approx}f_{s}$ and $f_{m}>f_{s}$. These 1D cuts are shown respectively
in Fig.3(b) and (d), with the cases colored red, black and blue respectively.
The background SAW transmission when the magnon mode is way off resonance
($f_{m}{\ll}f_{s}$, labelled $f_{s}$) is indicated in magenta. Especially, in
Fig.3(b) the excess attenuation as the magnon mode passes through the SAW
frequency is clear. While we do observe a similar effect in Fig.3(d), the
magnitude is much weaker which we believe is due to a combination of the mode
drifting with time and the difficulty of positioning the YIG sphere at the
focus, given the geometry shown in Fig.2(c).

Our analysis here is complicated by the fact that there is significant
electromagnetic crosstalk between the two ports and one needs to distinguish
between the local magnon-SAW interaction occurring on the sample and possible
interference effects occurring at the VNA. In particular, the electromagnetic
radiation from the probes and the IDT, which act as inefficient antennas [32],
can excite the magnon mode and the scattered signal picked up by the receiving
port will interfere with the acoustic transmission. We refer to this second
pathway as a nonlocal interaction due to the phase sensitive detection
employed by the VNA. In our experiments, the crosstalk signal is larger than
the acoustic transmission because of the diffraction effects mentioned above.
One can see this in action, by noting that in the case of $f_{m}>f_{s}$ and
$f_{m}<f_{s}$, the magnon signal appears as a transmission peak rather than a
dip, which is clear signature of a non-local interference pathway [33]. One
way to reduce this background is to exploit the significantly lower speed of
sound compared to light and time-gate the VNA transmission [34]. The datasets
shown in Fig.3 have been time gated with a notch of 40 ns. While this helps to
improve the signal to noise ratio (cf. the raw datasets in Appendix B), the
high quality factor of the YIG sphere makes the cross talk persist for
significantly longer than the electromagnetic transit time. This residual
cross-talk makes it challenging for us to extract the SAW-magnon interaction
strength from the experiments, although the excess local interaction with the
SAW through the surface $B_{1}$ field is clear from Fig.3(a,b). As a control,
we repeat the experiment with a straight IDT device and do not observe an
excess attenuation at the magnon-SAW crossing (cf. Appendix C).

Given that the SAW-magnon mode interaction is observed with the YIG sphere not
touching the sample, this experiment provides strong preliminary evidence that
GHz frequency, localized $B_{1}$ fields can be generated on the surface of
piezoelectric devices and can be used to interface with spin systems. While
this is encouraging, we would like to emphasize that the experimental modality
has a few limitations obvious with hindsight and the results should be
interpreted within these constraints. In particular, the size of commercially
available YIG spheres limits the sensitivity of the experiment by physically
requiring the IDTs be separated by $>100\lambda_{a}$, and making it
challenging to determine the focus in real-time. The size of the YIG sphere is
also responsible for the magnitude of the crosstalk, which makes it
challenging to infer the coupling strength and the coupling dynamics from our
experiments. In particular, one of the key effects we were hoping to confirm
was the frequency dependence of the interaction, which should scale $\propto
f^{2}$. Although we made devices with varying SAW frequency, we were unable to
quantify this effect. Moving forward, by impedance matching the transducers
(to reduce electromagnetic radiation and reflections) and working with high Q
YIG samples with dimensions $<$ 5, one can potentially scan the surface and
verify the spatial extent of the $B_{1}$ field both in-plane ($x,y$) and in
$z$. Mapping the spatial confinement of the $B_{1}$ field is critical to the
spin sensitivity enhancement experiments and this is something we are unable
to do with our current setup. Finally, moving to higher acoustic frequencies
(and higher $B_{z}$) would enable us to saturate the YIG sphere and avoid the
relaxation drift with time, which is another source of error in our
experiments. In passing, we would like to note that the $B_{1}$ field
description provides a complimentary route towards understanding the spin
rotation effects in nanoparticles interacting with SAWs [35], and interpret
the switching of magnetization observed in previous experiments [36].

##  IV Prospects for single spin electrical readout

As noted in a recent review [10], the problem of improving spin detection
sensitivity in electron spin resonance experiments boils down to focusing the
magnetic field to deeply sub-wavelength geometries while maintaining high-Q.
In effect, piezoelectric microresonators are ideal in that they naturally
provide both strong confinement and high Q, and therefore provide a natural
complement to traditional electromagnetic approaches [37]. The key issue is
whether the magnetic field strengths can be substantial in these devices. As
we have shown above using both scaling arguments and proof-of-principle
experiments, the surface current density has a ${\propto}f^{2}$ and can be
further enhanced by working with stronger piezoelectric materials /
orientations ($K_{eff}^{2}>$ 0.2) and designing small mode volume acoustic
cavities [17] to exploit the power scaling. With advances in materials and
device geometries pushing acoustic device operation to ever-higher frequencies
($>$ 50GHz) [38], the prospect of acoustics enabled X-Band ESR is within
reach. There is still the open question of how one can efficiently load the
near field of these devices efficiently to separate the pure field induced
effects from strain effects. One possible route could be to employ suspended
membranes (similar to [39], but made with an insulator like alumina) in close
proximity ($<$ 50nm) to the piezoelectric substrate.

We can estimate the minimum spin detection sensitivity, following [40]:

$N_{min}=\frac{\kappa}{2gp}\sqrt{\frac{nw}{\kappa_{2}}}$ (3)

where $N_{min}$ is the single-shot spin detection sensitivity (per echo),
$\kappa$ is the total cavity decay rate given by $\kappa=\omega_{0}/Q_{L}$
with $\omega_{0}=2{\pi}f$ being the operating frequency and $Q_{L}$ the loaded
Q factor of the cavity. We assume the cavity is operated at critical coupling
with external coupling rate $\kappa_{2}\approx\kappa/2$, $n$ is the average
number of noise photons, given by $n=k_{B}T/\hbar\omega_{0}$, $p$ is the spin
polarization
$p=\frac{1-e^{-\frac{\hbar\omega_{0}}{k_{B}T}}}{1+e^{-\frac{\hbar\omega_{0}}{k_{B}T}}}$,
and the spin resonance width $w=2/T_{2}$ with $T_{2}$ the average spin
dephasing time. $g$ is the spin cavity coupling rate defined above in eqn.2.
For operating frequencies of 10GHz, temperature 4, mode volume 5^3 and
$T_{2}\approx$ 50ms, achieving $N_{min}\approx$ 1 requires $Q_{L}\approx
10^{5}$, which is challenging for mechanical systems, but feasible given
recent results [18] and the favourable scaling of acoustic dissipation with
temperature. In any case, this shows that provided the $B_{1}$ field in
piezoelectric microresonators has a spatial extent comparable to that of the
acoustic field which requires experimental confirmation, then detecting
individual magnetic nanoparticles at room temperature
($N_{spins}{\geq}10^{5}$) is well-within reach using current devices.

The exquisite spin detection potential of these resonators can be understood
from a different perspective by treating them as the high frequency analogs of
mechanical cantilever based spin sensors [41], where the piezoelectric effect
enables inductive detection. It was noted [42] that the signal to noise ratio
for both inductive and mechanical detection of spin resonance scales
$\propto\sqrt{\omega_{0}Q/k_{m}}$, with $\omega_{0}$ is the operating
frequency, $Q$ the cavity quality factor and $k_{m}$ an effective magnetic
spring constant which scales with the cavity mode volume $V_{m}$ for inductive
detection. The sensitivity enhancement therefore derives from achieving high
quality factors in deeply sub-wavelength mode volumes, which gives an
effective Purcell enhancement $\propto\sqrt{Q/V_{m}}$ to the sensitivity [12].

We would like to conclude by noting that while in this work, we have primarily
focused on using piezoelectric devices for improving spin detection
sensitivity in ESR, the strong field confinement is also of interest in
scenarios involving large-scale closely packed efficient spin addressing [43]
without deleterious crosstalk effects, as would be necessary in future spin-
based quantum computing.

##  V Acknowledgements

We would like to thank John Rarity, Joe Smith, Vivek Tiwari, Hao-Cheng Weng,
Alex Clark, Rowan Hoggarth and Edmund Harbord for valuable discussions and
suggestions. We acknowledge funding support from the UK’s Engineering and
Physical Sciences Research Council (EP/N015126/1, EP/V048856/1) and the
European Research Council (ERC-StG SBS 3-5, 758843).

##  Appendix A Spin Momentum Locking of Evanescent fields in surface acoustic
waves and surface plasmons

Figure 4: (a) Schematic illustration of surface plasmon propagating on a
metal-air interface. The evanescent electric field and the magnetic field
orientation are shown, alongwith the surface charges that terminat the
electric field on the metal surface. (b) Illustration of a surface acoustic
wave propagating on a piezoelectric material. The evanescent electric field
orientation is similar to that in (a), but terminated by bound polarization
charges induced in the piezoelectric. By analogy, the orientation of the
$B_{1}$ field can be determined, as shown.

One can see the universality of spin-momentum locking [15] as applied to all
evanescent fields by looking at two very different surface waves: a surface
plasmon propagating at a metal air interface (at $>$100THz) and a surface
acoustic wave ($<$ 50GHz) propagating at a piezoelectric-air interface. The
respective cases are shown in Fig.4(a,b). As can be seen in both cases, the
evanescent electric fields curl with an orientation determined by spin-
momentum locking. The main difference between the two scenarios is the
termination of the fields on free charges in the metal and on bound
polarization charges in the piezoelectric case. Given that the surface plasmon
dispersion relation is traditionally derived by solving for the magnetic
field, the $B_{1}$ orientation can be derived by analogy as shown in Fig.4(b).

##  Appendix B Uncorrected datasets without time-gating

Figure 5: (a) Uncorrected data sets for the data corresponding to Fig.3(a,b).
Without the time-gating, the background electromagnetic crosstalk makes it
impossible to observe the excess attenuation during the SAW-magnon mode
crossing, although even with the raw dataset, a net attenuation (cf. black
curve) is clearly visible.

Figure 5 plots the raw data sets corresponding to the time-gated datasets
plotted in Fig.3(a,b). As discussed in the main text, the presence of the
background electromagnetic crosstalk makes it challenging to infer the SAW-
magnon interaction, which can be best seen by comparing the black curves
($f_{m}{\approx}f_{s}$) in Fig.3(b) and Fig.5(b). On the other hand, even with
the raw datasets, we can clearly observe an overall attenuation as the magnon
mode frequency comes close to the SAW frequency. This is in contrast to what
we observe with the straight IDT devices, as discussed in the next section.

We would like to note that the choice of the notch gate time (40ns) was not
optimized for this analysis and neither was the gate shape. Given the
separation between IDTs was 200, and a speed of sound of $\approx$ 3750sec,
the acoustic wave takes $\approx$ 53ns to reach the receiving IDT. The choice
of gate time was made as a rough trade-off between minimizing the background
crosstalk signal and ensuring minimal attenuation of the acoustic signal in
the transmitted spectrum. We would like to note again that time gating does
not fully eliminate the background crosstalk because of the Q factor of the
YIG sphere.

##  Appendix C Straight IDT control results

Figure 6: (a) 2D colorplot of the time-gated $\lvert{S_{21}}\rvert$ for a
straight IDT transmit-receive device with the YIG sphere positioned in
between. The YIG was mounted vertically in this experiment to align the 110
axis with $z$, which shifts the modes to higher frequencies in the 3.4GHz
range (b) 1D datasets from (a) corresponding to the three different cases
($f_{m}<f_{s}$, $f_{m}{\approx}f_{s}$, and $f_{m}>f_{s}$), along with the
background SAW transmission (magenta). The excess attenuation at the magnon-
SAW crossing is not observable here.

The power dependence of $J^{2}$ in equation 1 can be tested by measuring the
relative performance of IDT with curved and straight fingers in inducing
magnon absorption. Given the local field intensity in curved devices at the
focus is increased by the focusing ratio, the straight IDT devices here serve
as a control experiment to separate the local SAW-magnon interaction from the
nonlocal EM crosstalk-SAW interaction occuring in the VNA, discussed in the
main text.

Experiments identical to that reported in Fig.3 were done with a straight IDT
device. The YIG sphere in these experiments was mounted vertically with a view
towards saturating the sphere and avoiding the time drift in the magnon modes.
This moves the magnon modes to higher frequencies in the 3.4 GHz range, and
the IDT period was reduced correspondingly to shift the SAW response to higher
frequencies. Fig.6(a) plots the 2D colorplot of the transmitted
${\lvert}S_{21}{\rvert}$ as a function of frequency and $B_{z}$. To achieve
the higher $B_{z}$, we use a permanent magnet in combination with our
electromagnet. Fig.6(b) shows linecuts from 6(a) that correspond to the three
different cases $f_{m}<f_{s}$, $f_{m}{\approx}f_{s}$ and $f_{m}>f_{s}$. Here,
we don’t observe an excess attenuation as the magnon mode crosses through the
SAW resonance, which can be interpreted as a signature of the dependence of
the local current density on the local power density.

While mounting the YIG sphere vertically makes the magnon mode frequency
stable due to field saturation, it makes it very challenging to determine the
positioning of the sphere with respect to the beam focus using imaging
cameras. In particular, the top view camera, shown in Fig.2(a) can not be used
anymore and one has to rely more on the side view camera with associated
parallax errors. We tried to repeat the experiments in Fig.3 with focusing
IDTs and the sphere mounted vertically, but we were not able to observe a
clear signature of excess attenuation as in Fig.3(b,d) which we currently
attribute to the difficulty of positioning the sphere at the beam focus in
this configuration. As noted in the main text, many of the issues detailed
here can be addressed by moving to high Q YIG samples with dimensions $<$5.

##  Appendix D Reference IDT spectra

Figure 7: Bare IDT (without the YIG sphere) reflection ($S_{11}$, $S_{21}$)
and transmission ($S_{12}$, $S_{21}$) spectra for the device in Fig.3(a,b)
with VNA power 1mW). The spectra correspond to the raw VNA measurements
without time-gating.

.

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