# Superconducting vortex-charge measurement using cavity electromechanics

Sudhir Kumar Sahu  Department of Physics, Indian Institute of Science,
Bangalore-560012 (India)  Supriya Mandal  Department of condensed matter
physics and material sciences, Tata Institute of Fundamental Research, Mumbai
- 400005 (India)  Sanat Ghosh  Department of condensed matter physics and
material sciences, Tata Institute of Fundamental Research, Mumbai - 400005
(India)  Mandar M. Deshmukh  Department of condensed matter physics and
material sciences, Tata Institute of Fundamental Research, Mumbai - 400005
(India)  Vibhor Singh<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science, Bangalore-560012 (India)

###### Abstract

As the magnetic field penetrates the surface of a superconductor, it results
in the formation of flux-vortices. It has been predicted that the flux-
vortices will have a charged vortex core and create a dipolelike electric
field. Such a charge trapping in vortices is particularly enhanced in high-Tc
superconductors (HTS). Here, we integrate a mechanical resonator made of a
thin flake of HTS Bi2Sr2CaCu2O8+δ into a microwave circuit to realize a
cavity-electromechanical device. Due to the exquisite sensitivity of cavity-
based devices to the external forces, we directly detect the charges in the
flux-vortices by measuring the electromechanical response of the mechanical
resonator. Our measurements reveal the strength of surface electric dipole
moment due to a single vortex core to be approximately 30 $|e|a_{B}$,
equivalent to a vortex charge per CuO2 layer of $3.7\times 10^{-2}|e|$, where
$a_{B}$ is the Bohr radius and $e$ is the electronic charge.

The penetration of a magnetic field in type-II superconductors in the form of
Abrikosov vortices is well-known [1, 2]. Such vortices have a normal-core of
size of the coherence length and each vortex is surrounded by a circulating
supercurrent that decays over a characteristic magnetic length scale. It has
been predicted that such vortices can trap charges [3, 4, 5, 6]. The origin of
the vortex charge can be understood from the difference in the chemical
potentials in the superconducting-state and the normal region due to the
particle-hole asymmetry. It leads to a redistribution of electrons to maintain
the same electrochemical potential across a vortex [7], as shown schematically
in Fig. 1(a). Intuitively, the idea of charged vortex core can also be
captured by considering the inertial and Lorentz forces acting on the Cooper
pairs encircling the normal core, resulting in the depletion of charges from
the core [6].

To probe the trapped charge in the vortex core, the high-$T_{c}$
superconductors (HTS) are particularly attractive. The magnitude of the vortex
charge is approximately given by $|e|\left(\Delta/\epsilon_{F}\right)^{2}$,
where $\Delta$ is the superconducting gap and $\epsilon_{F}$ is the Fermi
energy. The ratio $\Delta/\epsilon_{F}$ is usually high in HTS
$(\Delta/\epsilon_{F}\sim 0.1)$. Indeed, in the past, the inference of the
charged vortex core has been made on bulk crystals of cuprate superconductors
based on techniques probing the sign reversal of Hall coefficient [8], the
nuclear quadrupole resonance [9, 10], and instability of the vortex lattice
[11]. However, a measurement of the charged vortex core directly measuring its
electrostatic field is a challenging task due to its small magnitude and
screening by the surrounding charges.

In recent years, nanoelectromechanical systems based approach to probe phase
transitions and the thermodynamical properties such as heat capacity, thermal-
conductivity, and magnetization have drawn a lot of attention [12, 13, 14, 15,
16]. Particularly, the integration of exfoliated thin flakes into cavity
optomechanical and hybrid devices have resulted into enhanced sensitivity to
external forces [17, 18, 19]. With these motivations in mind, we apply cavity
optomechanical techniques to detect the charges associated with vortices in a
high-$T_{c}$ superconductor by directly probing their electrostatic effect.

Figure 1: Cavity optomechanical architecture for detecting the vortex charge:
(a) Schematic showing the redistribution of the charge density at vortex core
due to the chemical potential mismatch in the normal and in the
superconducting state. The cyan curve shows the radial profile of the charge
density redistribution around the vortex core and indicates the normal core of
size of the coherence length $\xi$. (b) Electromechanical system with a
single-port reflection cavity. The device is configured such that both direct
current (dc) and microwave signals can be added to the drive port. The
equivalent lumped inductance and capacitance are 1.42 nH and 350 fF,
respectively. (c) Optical microscope image (false color) of the complete
device. It shows a quarter-wavelength reflection-cavity in coplanar waveguide
geometry patterned on an intrinsic Si substrate. (d) Scanning electron
micrograph (false color) of the device near the coupling capacitor showing the
suspended part of BSCCO flake which is 5 UC thick ($\sim$15 nm). The circular
suspended part has a diameter of 7 $\mu$m.

In this work, we use a mechanical resonator of 5 unit cell (UC) thick
exfoliated flake of Bi2Sr2CaCu2O8+δ (BSCCO) and integrate it with a microwave
cavity to realize a cavity electromechanical device. Due to their low mass,
such mechanical resonators have a large coupling with the microwave field of
the cavity, which enhances their force sensitivity. Fig. 1(b) shows the
electrical equivalent schematic of the device. Such a device is analogous to a
Fabry-Perot cavity with a mechanically compliant mirror, where vibrations of
the mirror parametrically modulate the resonant frequency of the cavity [20].
We use a superconducting coplanar waveguide-based $\lambda/4$ microwave cavity
as shown in Fig. 1(c). The cavity is fabricated by patterning a 200 nm thick
sputtered molybdenum and rhenium (MoRe) alloy film on top of intrinsic silicon
wafer [21, 22]. The MoRe film has a $T_{c}$ of 11 K. Near the coupler end of
the microwave cavity, a feedline is selectively etched to reduce the MoRe film
thickness by 120 nm. The difference in MoRe film thickness provides the
clearance to form a suspended mechanical drumhead resonator by the transfer of
an exfoliated BSCCO flake. A scanning electron microscopy (SEM) image of the
suspended BSCCO flake after the transfer is shown in Fig. 1(d). In this
configuration, the BSCCO drumhead-shaped mechanical resonator provides
capacitive coupling between the feedline and microwave cavity. In addition,
the device design allows application of both alternating current (ac) and
direct current (dc) signals through the feedline.

Figure 2: Optomechanically induced absorption (OMIA) measurements:  (a)
Normalized reflection $|S_{11}(\omega)|$ measurement of cavity at 20 mK. The
blue line shows the fitted curve. (b) Measurement of $|S_{11}(\omega)|$ in the
presence of a red-detuned pump signal $\omega_{d}=\omega_{c}-\omega_{m}$. The
sharp feature at the center of the cavity shows the optomechanically induced
absorption (OMIA), resulting from the coherent oscillation of the mechanical
resonator. (c) Zoomed-in view of the OMIA feature showing the detailed
mechanical response. (d) Colorplot of the demodulated signal $V_{mix}$ showing
the quadratic tuning of the mechanical resonant frequency with $V_{dc}$
applied at the feedline.

The device is cooled down to 20 mK in a dilution refrigerator with
sufficiently attenuated microwave input lines. Additional details of the low-
temperature setup are included in the Supporting Information. The reflection
from the cavity $|S_{11}(\omega)|$ is measured using a vector network
analyzer. Fig. 2(a) shows the measurement of $|S_{11}(\omega)|$ with
fundamental resonant mode at $\omega_{c}/2\pi=$ 7.124 GHz. The solid line is
the fit to the cavity response, yielding the internal and external linewidths
of $\kappa_{i}/2\pi=$ 0.48 MHz and $\kappa_{e}/2\pi=$ 4.92 MHz, respectively.

For mechanical mode characterization, we drive the BSCCO resonator at its
fundamental resonant frequency $\omega_{m}$ using the radiation-pressure
force. To achieve this, we add a pump signal at
$\omega_{d}=\omega_{c}-\omega_{m}$ along with a weak probe signal $\omega_{p}$
near $\omega_{c}$. In the presence of the pump and the probe signals, the
mechanical resonator experiences a beating radiation-pressure force near its
resonant frequency which drives it coherently [23, 24, 17]. The mechanical
motion of the resonator, in turn, modulates the intracavity pump field,
resulting in an upconverted signal exactly at the probe frequency. As a
coherent process, the upconverted signal interferes with the original probe
signal and therefore results in an interference feature with linewidth set by
the mechanical losses. Such interference in the cavity reflection is referred
to as optomechanically induced absorption (OMIA) [24, 17].

The OMIA interference appears as a sharp absorption feature in the cavity
response as shown in Fig. 2(b). A narrow span measurement of the absorption
feature is shown in Fig. 2(c). The absorption dip directly manifests the
coherently driven mechanical response, and thus allows the complete
characterization of the mechanical resonator. From this measurement, we
determine the fundamental mechanical mode frequency of $\omega_{m}/2\pi$
$\sim$15.383 MHz and linewidth of $\gamma_{m}/2\pi$ $\sim$165 Hz,
corresponding to a quality factor of $Q_{m}\sim 93000$. The high quality
factor is a direct indication of the low contact resistance between the MoRe
film and BSCCO [22]. For the cavity-electromechanical device studied here, we
estimated the single photon coupling rate
$g_{0}=\left(\partial\omega_{c}/\partial x\right)x_{zp}$ to be $2\pi\times$30
Hz, where $x_{zp}$ is the quantum zero-point motion of the BSCCO resonator.

Due to the novel electrical design of the device, we can add a dc voltage
$V_{dc}$ across the capacitor formed by the BSCCO resonator and the MoRe
feedline underneath. While such a scheme has a significant advantage for
probing the vortex charge [7], it allows characterization of the BSCCO
resonator by an independent technique. A dc signal, $V_{dc}$, and an ac
signal, $V_{ac}$, can be used to actuate the resonator by a force
$C_{g}V_{dc}V_{ac}/z_{0}$, where $z_{0}$ is the separation between BSCCO and
the bottom electrode. By using the cavity as an interferometer, the mechanical
mode is detected by demodulating the reflected microwave signal from the
cavity. Fig. 2(d) shows the plot of demodulated signal $(V_{mix})$ as $V_{dc}$
is varied. We observe a quadratic tuning of the mechanical resonant frequency
due to capacitive spring softening [25].

Figure 3: Modification of the mechanical response in the magnetic field: (a)
Shift in the mechanical frequency with magnetic field at $V_{dc}=0$ V, as
magnetic field is swept in forward and reverse direction. The solid lines show
the results obtained from the critical state model. (b) Plot of calculated
local magnetic field and normalized critical current computed in the BSCCO
plane. Dotted lines show the spatial profile after a forward sweep of
$B=~{}$0-10 mT. Solid lines show the profiles for a subsequent field descent
to $B=~{}$6 mT. The side panel shows a side view of the normalized current
distribution for the two cases stressing upon the change in the direction of
current as field is lowered after reaching a maximum value.

Having described the experimental technique, we measure the electromechanical
response when a small magnetic field, perpendicular to the CuO2 plane of the
BSCCO crystal is applied. It is represented in Fig. 1(d). We perform OMIA
measurements while sweeping the magnetic field in the forward and reverse
directions. In the presence of the magnetic field, both the cavity frequency
and the mechanical resonant frequency can change. Therefore, we first record
the cavity response and readjust the pump frequency based on the measurement
of $\omega_{c}(B)$. Fig. 3(a) shows the plot of the mechanical resonant
frequency shift $\Delta f_{m}(B)=f_{m}(B)-f_{m}(0)$ at $V_{dc}=0$ V. We limit
the measurements in the range of $\pm$32.6 mT due to the reduction in the
cavity quality factor at higher fields, which obscure the OMIA feature.

The hysteresis in $\Delta f_{m}(B)$ suggests the role of flux-pinning, the
Lorentz force of vortex transferring to the lattice and hence the
irreversibility of the frequency shift, similar to the magnetostriction
measurements on bulk single crystal of BSCCO [26]. For quantitative analysis,
we use the critical state model, which is widely used to explain the
magnetization of HTS [27, 28]. Here the maximum sheet current is capped at a
critical current which in general could be magnetic field dependent [29, 30].
Due to this sheet current and the local magnetic field $B_{z}(r)$, the flake
experiences a Lorentz force in the lateral direction. It results in a magnetic
field dependent tension in the flake and hence a change in the mechanical
resonant frequency as observed in Fig. 3(a).

We use a local critical sheet current as
$J_{c}(r)=J_{c0}\exp(-|B_{z}(r)|/B_{0})$, where $B_{z}(r)$ is the local
magnetic field in the BSCCO plane, and the maximum critical current $J_{c0}$
and characteristic magnetic field $B_{0}$ are two model parameters [29, 30].
We note that here BSCCO resonator thickness (5 UC) is much smaller than the
London penetration length [31]. Therefore, a small field can push the flux-
front, the boundary between the Meissner phase and the mixed phase, to nearly
the center of the BSCCO flake. We assume that the steady-state critical
currents remain confined to the boundaries of the BSCCO flake. To keep the
calculations simple, we treat BSCCO as a thin circular plate. Calculation for
the realistic geometry would only results in a scaling factor. Our model
neglects geometrical [32] or Bean-Livingston barrier [33]. Furthermore, we
assume no interaction between the vortex lines passing through the MoRe film
(bottom capacitor plate) and BSCCO resonator (top capacitor plate).

To calculate the force experienced by the suspended part of the BSCCO flake,
we first calculate the radial profile of the sheet current density $J(r)$ and
$B_{z}(r)$ [34]. The Lorentz force on the suspended part of the BSCCO flake is
then estimated by $\int_{0}^{R}J(r)B_{z}(r)2\pi rdr$, resulting in a net
stress in the flake. The stress can be compressive or tensile depending on the
sign of the integral. The Lorentz-force induced stress can be used to compute
the frequency shift of mechanical resonator, as shown by the solid lines in
Fig. 3(a). The radial profiles of the local magnetic field and the normalized
critical current are shown in Fig. 3(b). For these calculations, we have used
the critical current density of $J_{c0}=3680~{}\text{A}/\mu\text{m}$ [35, 36]
and $B_{0}=100$ mT [37]. Additional details of the model are included in the
Supporting Information..

Figure 4: Evidence for the vortex-charge:  (a) A schematic showing the
geometry of the capacitor formed by the BSCCO resonator (cyan) and MoRe plate
(grey). The red part of BSCCO resonator represents a single vortex, showing
two surface dipoles. (b-h) Measurement of shift in the mechanical frequency
$\Delta f_{m}=f_{m}(B,V_{dc})-f_{m}(B=0,V_{dc})$ as the magnetic field is
swept in forward and reverse direction in the presence of a dc voltage
$V_{dc}$. The solid lines are the calculated curves considering formation of
the vortex-dipole.

To probe the charge associated with vortices, their interaction with the
electrostatic field setup by a dc voltage plays a crucial role. Therefore, we
now turn our focus on the mechanical response when $V_{dc}$ is applied as
shown schematically in Fig. 4(a). The plots of $\Delta f_{m}$ at different
$V_{dc}$ applied across the BSCCO resonator are shown in Fig. 4(b-h). For
these measurements, we sweep the magnetic field in $\pm$16.3 mT range due to
the reduction in the mechanical quality factor at higher $|V_{dc}|$, which
limits the measurement of mechanical response using OMIA technique. The
measurement of $\Delta f_{m}$ shows a clear asymmetry depending on the sign of
$V_{dc}$. In addition, the overall dispersion with the magnetic field is
enhanced at large negative dc voltages as compared to the case when $V_{dc}$
is positive. The asymmetry in the electromechanical response with respect to
the sign of $V_{dc}$ (left vs right panels in Fig. 4(b-h) clearly suggests an
electrostatic origin. This could be explained by considering the charge
trapped in the vortex core. The charge per copper-oxide layer inside the
vortex core is given by $Q_{c}\sim|e|\left(\Delta/\epsilon_{F}\right)^{2}$ [4,
7]. Using $\Delta/\epsilon_{F}\sim 0.1$, and a CuO2-plane spacing of 0.75 nm
for BSCCO [38], a typical estimate of $Q_{c}=0.01~{}|e|$ can be made. It is
equivalent to a line charge density $Q_{\xi}$ of 0.013 $|e|$/nm.

In the limit $t\gg\xi$, where $t$ is the thickness of BSCCO, the net effect of
the vortex line charge can effectively be captured by considering an
equivalent surface electric dipole $\vec{p}_{0}$ [7]. To understand our
observations, we consider the interaction of surface dipoles $\vec{p}_{0}$
with the electric field set up by $V_{dc}$. A dc voltage across the capacitor
modifies the interaction energy by $U_{dip}(z)=-\vec{p}_{0}\cdot\vec{E}(z)$,
where $\vec{E}$ is the electric field between the capacitor plates, and hence
leads to an electromechanical coupling. As magnetic field is increased, the
number of vortices $n_{v}(B)$ penetrating the suspended part of the BSCCO
increases, as $n_{v}(B)=(\pi R^{2}|B|)/\Phi_{0}$, where $\Phi_{0}$ is the
superconducting flux quantum and $R$ is the radius of the suspended part of
BSCCO. By considering a uniform distribution of the vortices on a triangular
lattice across the BSCCO resonator, their contribution to the
electromechanical energy can be calculated by summing over all the vortices
present in the suspended part.

We include the electric dipole contribution along with the elastic, and the
capacitive contribution to the total electromechanical energy $U(z)$.
Therefore, the effective spring constant $\partial^{2}U(z)/\partial z^{2}$ and
hence the mechanical resonant frequency can be determined. The solid lines in
Fig. 4(b-h) show the calculated $\Delta f_{m}(B)$ by including a dc voltage
dependent energy in the electromechanical response. Here the curves are
plotted taking a dipole magnitude of $p_{0}=30~{}|e|a_{B}$ with a positively
charged core, where $a_{B}=5.29\times 10^{-11}$ m is the Bohr radius. It is
important to point out here that the sign of the vortex core charge is
determined by the oxygen doping level in the BSCCO crystal. For an overdoped
(oxygen-rich) crystal, it is expected to be positive [39]. For a positively
charged vortex core, the surface dipole moment points in an outward direction
from the BSCCO surface regardless to the direction of the applied magnetic
field. Therefore, the electromechanical response is expected show an asymmetry
with respect to the electric field created by $V_{dc}$. For a positively
charged vortex core, the mechanical resonant frequency is expected to show a
softening (hardening) behavior for negative (positive) $V_{dc}$ voltages as
observed in the measurement. Details of the electromechanical spring constant
calculation are provided in the Supporting Information. In next paragraph, we
calculate the line charge density equivalent to this surface dipole moment and
make a comparison with the values reported in the literature.

Following ref. [7], the charge density profile across a vortex core can be
written as
$\rho(r)\sim\left(ea_{B}\zeta/\pi^{3}\right)\left(\xi^{2}-r^{2}\right)/\left(\xi^{2}+r^{2}\right)^{3}$,
where $a_{B}$ is the Bohr radius, and $\zeta$ is the particle-hole asymmetry
parameter. The line charge density in the vortex core, therefore, is given by
$Q_{\xi}~{}=\int_{0}^{\xi}\rho(r)2\pi r\,dr=\zeta
ea_{B}/\left(2\pi\xi\right)^{2}$. Such a charge redistribution is equivalent
to a surface dipole pointing normal to the superconducting surface and having
a magnitude of $p_{0}=\left(\zeta
ea_{B}/\pi^{2}\right)\left(m/m_{eff}\right)\ln\left(z/\xi\right)$. Therefore,
the line charge can be related to the dipole moment as
$Q_{\xi}=p_{0}\left(m_{eff}/m\right)\left(4\xi^{2}\ln\left(z/\xi\right)\right)^{-1}$,
where $m_{eff}/m$ is the effective mass ratio of the charge carrier. Using
$\xi=3.2$ nm [40], $m_{eff}/m=4.6$ [41], and $z=120$ nm, we obtain
$Q_{\xi}=+4.9\times 10^{-2}~{}|e|$/nm, which is equivalent to a charge of
$+3.7\times 10^{-2}~{}|e|$ per CuO2 layer.

Our estimation of the charge per copper oxide layer is consistent with the
earlier measurements made on YBCO using the nuclear quadrupole resonance
technique [9]. Both the experiments reveal a larger value of the vortex charge
than that obtained via the theoretical predictions. The difference between the
vortex charge magnitude from the experiments and the theory can have several
interesting origins. First, it suggests that the gap anisotropy and Fermi
surface curvature might be playing an important role [42]. Second, given the
ultralow temperature in this experiment, the quantum effects in the vortex
could become relevant. Finally, the nature of the vortex core could be
different from the metallic phase [43, 44].

To summarize, we show clear evidence for the trapped-charges in the vortex
core probed using a cavity optomechanical device. The magnitude of the
equivalent dipole is higher than the estimates from the BCS-theory while
considering Thomas-Fermi screening. Such a difference opens up new
possibilities to revisit the vortex-charge problem in atomically thin
superconductors. Furthermore, our experiment and novel device approach show
the advantages of integrating exfoliated thin flake into the cavity
optomechanical devices for sensitive measurement of the thermodynamical
properties. With minor modifications in the design, such technique could
further be extended to other systems involving topological charge [45] or
bosonic Landau levels [46]. This work thus opens up a new avenue to study
quantum phase transitions as external parameters are varied.

##  .1 Acknowledgment

The authors thank Eli Zeldov, Subir Sachdev and Vijay Shenoy for insightful
discussions. The authors also thank D. Jangade and A. Thamizhavel for their
help during the crystal growth. M.M.D. acknowledges the Department of Atomic
Energy of the Government of India under Grant No. 12-R$\&$D-TFR-5.10-0100, DST
Nanomission under Grant No. SR/NM/NS-45/2016 and SERB CORE grant
CRG/2020/003836. V.S. acknowledges the research support under the Core
Research Grant CRG/2018/001132 by DST and ISTC-0395 by STC-ISRO. S.K.S.
acknowledges device fabrication facilities at the Department of Physics, and
CeNSE, IISc-Bangalore funded by Department of Science and Technology (DST),
Government of India.

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