Unconventional rf photoresponse from a superconducting spiral resonator
ABSTRACT Superconducting thin film resonators employing strip geometries show great
promise in rf/microwave applications due to their low loss and compact nature.
However, their functionality is limited by nonlinear effects at elevated
rf/microwave powers. Here, we show that by using a planar spiral geometry
carrying parallel currents in adjacent turns, this limitation can be minimized.
We investigate the rf current distributions in spiral resonators implemented
with Nb thin films via laser scanning microscopy. The rf current density
profile along the width of the individual turns of the resonators reveals an
unconventional trend: maximum current in the middle of the structure and
decaying toward its edges. This unusual behavior is associated with the
circular nature of the geometry and the cancellation of magnetic field between
the turns, which is favorable for handling high powers since it allows the
linear characteristics to persist at high rf current densities.
- SourceAvailable from: Behnood GhamsariApplied Physics Letters 10/2012; 102(1). · 3.79 Impact Factor
- Physical Review X 12/2013; 3(4):041029. · 8.39 Impact Factor
- Physical Review Letters 12/2012; 110(15). · 7.73 Impact Factor
Unconventional RF Photo-Response from a Superconducting Spiral Resonator
Alexander P. Zhuravel,1Cihan Kurter,2Alexey V. Ustinov,3and Steven M. Anlage2,3
1B. Verkin Institute for Low Temperature Physics and Engineering,
National Academy of Sciences of Ukraine, 61103 Kharkov, Ukraine
2Center for Nanophysics and Advanced Materials, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111 USA
3Physikalisches Institut and DFG-Center for Functional Nanostructures (CFN),
Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
(Dated: May 2, 2012)
Superconducting thin film resonators employing strip geometries show great promise in
RF/microwave applications due to their low-loss and compact nature. However, their function-
ality is limited by non-linear effects at elevated RF/microwave powers. Here, we show that by
using a planar spiral geometry carrying parallel currents in adjacent turns, this limitation can be
minimized. We investigate the RF current distributions in spiral resonators implemented with Nb
thin films via Laser Scanning Microscopy. The RF current density profile along the width of the
individual turns of the resonators reveals an unconventional trend: maximum current in the mid-
dle of the structure and decaying towards its edges. This unusual behavior is associated with the
circular nature of the geometry and the cancellation of magnetic field between the turns, which is
favorable for handling high powers since it allows the linear characteristics to persist at high RF
PACS numbers: 74.25.N-, 74.81.-g, 74.62.Dh, 74.25.nn, 74.70.-b
Superconducting thin film RF/microwave resonators
play a prominent role in many applications including
quantum computing1,2, single photon detection3, bifur-
cation amplifiers4along with the quest to develop novel
devices5,6and media such as metamaterials7–9. However,
superconductors show nonlinear response when driven
strongly by RF signals/microwaves10–13, that manifests
itself with a significant dependence of the surface resis-
tance and reactance on the input power14–17, PRF. It is
important to find an effective way to keep the resonant
characteristics linear for a long range of PRF to maxi-
mize the power handling capability of the resonators and
expand their range of applicability.
Many superconducting resonators generally employ
planar geometries made up of finite-width thin strips to
carry a longitudinal high frequency current. The mag-
netic fields generated by flowing currents along the strips
have a common characteristic of being perpendicular to
the edges of the strip. Such a field configuration poses a
challenge to the superconductor. In order to remain in
the Meissner state, the strip must generate strong dia-
magnetic shielding currents to screen the perpendicular
magnetic field from its interior. This gives rise to a large
current build-up at the edges of a superconducting film
shaped into a strip-geometry resonator18,19. Screening
currents can approach or exceed the critical current at
the edges leading to a local breakdown of superconduc-
tivity and the onset of nonlinear behavior20. Therefore,
the microwave properties of superconducting resonators
are strongly dependent on the geometry of the design17.
Apart from simple single strip-lines, co-planar waveg-
uides19,21, hairpin22and meander-line resonators23are
FIG. 1: (Color online) Schematic sketch of a coplanar wave
guide (a) and a spiral resonator (b). The RF currents flow as
shown with the red arrows when the resonators are excited.
other planar designs based on strip geometries. Many of
these designs include parallel conductors where the cur-
rents in neighboring strips flow in opposite directions, see
Fig. 1(a). This causes the induced normal oriented mag-
netic fields to be enhanced between the strips, and in turn
results in an accumulation of RF screening currents at
the edges. Such an inhomogeneous RF current density,
JRF can create changes in the superconducting proper-
ties of the film, therefore limiting the functionality of the
superconducting resonator by leading to non-linearity in
its response even at low stimulus.
Here we consider a unique resonator in the form of a
continuous planar spiral designed to generate a strong
electromagnetic response below 100 MHz. The spirals
are intended to be deep sub-wavelength meta-atoms of a
metamaterial which could be utilized, e.g., in magnetic
resonance imaging applications as compact and low loss
flux guides24,25. The resonators have a superior geome-
try in which the currents flowing in neighboring strips
are in the same direction and approximately equal in
magnitude, at least for the first few resonant modes, see
Fig. 1(b). The perpendicular components of the induced
arXiv:1203.3998v2 [cond-mat.supr-con] 1 May 2012
magnetic fields largely cancel in the region between the
windings, leading to a magnetic field pattern mainly par-
allel to the plane of the strips. This renders the distri-
bution of total current density to be relatively uniform
within the sample compared to the anti-parallel current
case discussed above, eliminating RF current build-up
at the edges of the windings. This kind of configuration
maintains the linear characteristics at elevated excitation
power, and can be a better candidate in applications re-
quiring linear RF/microwave response.
We have applied the spatially-resolved technique of
low-temperature Laser Scanning Microscopy (LSM) to
map RF current distributions globally (on the entire
sample) and locally (in an individual winding) on spi-
ral resonators made of Nb thin films. From the two di-
mensional (2D) LSM images of the spirals excited at the
fundamental resonance we have observed an unconven-
tional RF current pattern with the absence of a build-up
at the edges of the turns until a critical power value is
reached. The evolution of the RF current distribution
with increasing PRF has been examined to investigate
the power handling capability of these resonators.
The LSM technique has various contrast modes for
imaging26. Here, we have utilized only two of them:
optical reflectivity and ordinary high-frequency pho-
toresponse modes. As was demonstrated in previous
works19,20, the response of the ordinary high-frequency
photoresponse mode is a superposition of two com-
ponents; inductive and dissipative/resistive responses.
Both generally require the superconducting sample to
show a nonlinear response under laser irradiation. At
low PRFvalues, only laser heating plays a significant role
in the nonlinearity, however once the power is elevated,
extra dissipation mechanisms will be added due to RF
heating. Such a response in superconductors well below
their critical temperature, Tcis mainly attributed to the
formation of local dissipative (non-superconducting) do-
mains where JRF may exceed the local critical current
density, Jc. The superconducting state is extremely sen-
sitive to variations in the superfluid density that changes
either with temperature or magnetic field, hence nonlin-
earity is inevitable17. The effect manifests itself globally
as distortion and/or bistable switching in the resonant
transmission as a function of frequency, |S21(f)|, at some
microwave powers12,13,17,27due to increased absorption
of microwave radiation by quasi-particles.
The LSM measurements presented in this paper use
planar spiral resonators fabricated with 200 nm Nb thin
films sputtered onto 350 µm thick single crystal quartz
substrates. Photolithography and reactive ion etching
(CF4:O2, 90%:10%) are applied to give a spiral shape to
the thin film. The Tcof the Nb film (9.2 K) is obtained
from resistance vs temperature measurements25. Below
the Tc of Nb, the microwave surface resistance, Rs of
the film will be very small (about 20 µΩ at 10 GHz and
Each spiral is made up of 40 turns, has an outer di-
ameter of 6 mm and an inner diameter of 4.4 mm. The
windings in the spirals and the spacing between them are
of 10 µm width. Prior results show that the spirals act
as very compact self-resonant strips, supporting up to
10 half-wavelength standing waves of current along their
A single spiral resonator is placed on a sapphire disk
plate (50 mm in diameter, 2 mm in thickness) where a
thermometer is attached nearby, in a cryogenic environ-
ment. The sample is stimulated with RF power applied
via two coaxial cables terminated by shorted loops at the
end with a diameter slightly larger than the outer diam-
eter of the spiral as shown in Fig. 2. The planes contain-
ing the excitation (RF in) and the pickup loops (RF out)
are parallel and the two loops are placed sandwiching the
sample between them7. The sample temperature is con-
trolled with a heater located on the Cu cylinder on the
cold head supporting the sapphire plate. The global reso-
nant response was characterized with transmission mea-
surements at different RF power levels between PRF=
-30 dBm and +30 dBm and at a bath temperature of
TB= 4.5 K using a Microwave Vector Network Analyzer
(Anritsu MS4640A). From these measurements, the fun-
damental resonant frequency is found to be ∼74 MHz,
followed by higher harmonics.
Cooling the spiral samples in the range Tc≥TB≥4.5 K
takes place inside the vacuum cavity of a variable tem-
perature optical cryostat. The temperature of the cold
Cu cylinder below the sample (50 mm in outer diam-
eter with a 5 mm thick wall), see Fig. 2, is stabilized
with an accuracy of 1 mK. The cylinder temperature
is controlled with a bifilar coil heater connected to the
temperature controller and wound around the cold Cu
plate having the same temperature as the cylinder. This
Cu cylinder also cools both coaxial cables to eliminate
a possible temperature gradient with the sample. The
top surface of the sample faces the laser probe while the
bottom surface is temperature stabilized by gluing it to
the sapphire disk with cryogenic vacuum grease, assuring
a reliable thermal heat sink. The same grease is used on
the thermally conducting interface between the sapphire
and Cu cylinder.
objective lens objective lens
FIG. 2: (Color online) Simplified schematic representation of
the LSM setup used for 2D visualization of microwave pho-
toresponse of the tested resonator structure. Drawing is not
V. LASER SCANNING MICROSCOPY (LSM)
For LSM imaging, the spirals are excited by RF sig-
nals, (while being kept well below the Tcof Nb) and illu-
minated by a focused laser beam acting as a non-contact
optical probe. The LSM photoresponse (PR) dominantly
comes from thermally-induced changes in the RF trans-
mission characteristics of the spiral due to absorption of
the laser light with a wavelength of 670 nm. The small-
est diameter of the laser probe spot is 1.5 µm when a
20x magnification (NA=0.42) objective lens is used for
detailed LSM imaging (scan area up to 250x250 µm2).
Large scale (up to an area of 50x50 mm2) LSM images
are acquired with an f-theta objective lens creating a 20
µm diameter laser illuminated spot. The intensity of the
laser is modulated at a frequency of 100 kHz creating
an oscillating thermal and/or optical probe. Changes in
|S21(f)| due to the laser heating are synchronously de-
tected with a lock-in amplifier.
PR∼(∂|S21(f)|/∂T)δT due to local temperature change
δT, can be uniquely decomposed into inductive and
resistive components.19The inductive photoresponse,
PRI is proportional to Aλ2(x,y)J2
is the area heated by the laser spot and λ(x,y) is the
local value of the penetration depth at position (x,y)
and can be interpreted as arising from the changes in
penetration depth, δλ induced by the laser heating.
When λ(x,y) and δλ have uniform values, the PRI has
a profile proportional to the local value of RF current
density squared, J2
RF(x,y). The resistive photoresponse,
PRR arises from thermally caused changes in the local
resistance of the sample Rsand is a convolution of the
laser modulated surface resistance, δRsweighted by the
local value of J2
RF(x,y)δλ , where A
In non-equilibrium (non-thermal) mode, the main
mechanism of the LSM PR contrast is the following.
Below Tc, the absorbed portion of laser power, δPL
causes nonequilibrium changes in the quasiparticle pop-
ulation, NQP, resulting from the high-energy excita-
tion of the superconducting film by individual optical
photons with an energy of hfL= 1.85 eV ? 2∆Nb(0)
where ∆Nbis the superconducting energy gap of Nb, h
is Planck’s constant, and fL is the laser (irradiation)
frequency.Because of electron-electron and electron-
phonon scattering as well as direct Stokes-like depairing
(with continuous frequency spectrum hf≤hfL− 2∆Nb),
every high-energy quasiparticle is capable of producing
an extra population of low-energy excitations, NQP=
γhfL/2∆Nb, where γ is the quantum efficiency and
smaller than 1. The excess quasiparticles create a non-
equilibrium superconducting state due to the reduced
superfluid density beneath the laser probe.
sult, local changes in NQP(ρ,δPL) cause modifications
in the surface impedance δZs(ρ,δPL)= δRs(ρ,δPL) +
iωδLk(ρ,δPL) due to δRs as well as photoinduced
changes in local kinetic inductance, δLk .
?(x − x0)2+ (y − y0)2is the radial coordinate on the
We observe that the Nb samples do not show any sig-
nificant inductive photoresponse, PRI at temperatures
well below Tc. While increasing both RF and/or laser
power, it has been found that resistive photoresponse,
PRRis produced at a lower critical RF power, Pc1cor-
responding to the first local switching of the sample into
the nonlinear regime. The first detectable resistive com-
ponent of PR can be written as
As a re-
sample surface relative to the position (x0,y0) of the laser
PRR∝ |S21(f,PRF)|2− |S21(f,PRF+ δPL)|2
for a condition of JRF≥Jc(x0,y0,PRF) − δJc(x0,y0,PL)
combining the effects of the local microwave field (first
term) and suppression of the critical current by the laser
beam (second term).
It has been shown in the literature (see, for instance,
Refs.17,29) that the first nonlinear distortion of |S21(f)|
appears as a deviation where the |S21(f)| curves fall on
to curves with smaller quality factor, Q, in a narrow-
band near the resonant frequency f0(with PRFexceeding
Pc1). In the case of a small optical probe perturbation
δPL?Pc1−δPL, the resistive component of LSM PR may
be undetectable outside this narrow band, while strong
PR signals are generated inside the band.
The LSM work presented here follows a modified pro-
cedure originally developed in Ref.20which is based
on the insertion loss component of the photoresponse,
PRIL, rather than PRI and PRR measured at a fre-
quency in the vicinity of f0. At a fixed RF frequency and
spatially independent laser probe perturbation, the LSM
PR is proportional to the laser-beam-induced changes in
resonator transmission, δ|S21(f)|2that can be expressed
in a form close to that introduced in Ref.30.
PR ∝ δ|S21(f)|2=1
where the transmission coefficient, |S21(f)|2[ratio of
the transmitted power, POUT
PRF(f)] as a function of driving frequency f ∼ f0 is
given in the limit of weak coupling by31
(f), to the input power,
1 + 4Q2(f/f0− 1)2
at the peak of the resonance. By substitution of Eq. 3 in
Eq. 2, one finds that the inductive
2is the maximum of the transmission coefficient
[1 + 4Q2(f
and the resistive
[1 + 4Q2(f
components of total LSM PR are nulled at f = f0, while
the insertion loss component
1 + 4Q2(f
is peaked at f = f0.
In terms of local photo induced changes, PRILis di-
rectly linked with Ohmic dissipation generated by the
laser probe at position (x0,y0),32,33
PRIL(x0,y0) ∝ J2
In the frame of the paradigm described in Ref.20(in the
case of a linear response function and a small probe per-
turbation) for a strip geometry oriented along the path
L in the ? direction, the change in surface resistance due
to a change in local critical current [JRF≥Jc(?0,PRF) −
δJc(?0,PL)] at a specific laser probe position ?0may be
for large scale imaging mode (Λ≥W) where ? L is the path
along the entire spiral with total length of L, W is the
width of the film, PCIRC
is the circulating RF power in
the resonator and Λ is the characteristic healing length
describing spatial decay of PRIL(?0) ∝ e−|?−?0|/Λat a
distance ? outside the intense beam focus.
As was postulated in Ref34, one can assume that both
quantities ∂Rs(x0)/∂Jc(x0) and ∂Jc(x0)/∂PLare invari-
able in the probed sample area if d, δPL and Λ are
spatially uniform through the whole resonator structure.
Combining Eq. 7 with the integral value of Eq. 8 over the
laser probe profile ΛδPLleads to
at location x0in the one dimensional strip geometry.
Note that Eq. 9 demonstrates a threshold mecha-
nism of PRILgeneration relative to excitations by both
and δPL. In the undercritical state of the su-
perconducting structure at P = PCIRC
the value of ∂Rs(x0)/∂Jc(x0) is zero at any position of
the laser probe. In this case there is no PRIL(x0) de-
tectable by the LSM technique at f0in microwave imag-
ing mode. In addition, very weak response is observed in
purely normal regions of the sample. A detectable PRIL
signal is generated only in the narrow range of power
between Pc1 and Pc2 (upper critical RF power). Note
that Pc1(see Fig. 4b) denotes the total (PCIRC
power initiating the first local dissipative source that de-
stroys superconductivity. By Pc2 we denote the power
of this source giving rise to normal state switching. As
seen from Eq. 9, PRIL(x0) is proportional to J2
in this range and spatial variations of LSM PR ampli-
tude directly show the distribution of J2
part of the standing wave that generates an overcritical
state in the superconducting strips. Any deviation of
PRIL(x0) from the shape of a sinusoidal standing wave
pattern then gives evidence for an inhomogeneous distri-
bution of Jc(x0) due to the term ∂Jc(x0)/∂PLin Eq. 9.
Also, it is clear that manipulations by both PRFand δPL
may be used to probe local values of Jc(x0) as either PRF
or δPLis increased.
In the case of 2D LSM probing (characteristic length of
the laser-probe induced non-equilibrium state, Λ ≤ strip
width, W), the main LSM PRIL imaging mode results
from laser probe induced redistribution of the microwave
current around the illuminated area. This effect leads to
additional Ohmic dissipation in the nearby unilluminated
areas of the superconducting strip generating
+ δPL ≤ Pc1,
RF(x0) along that
The effect is linked with the laser-induced modulation of
the local critical current,
underneath the laser probe allowing direct measurement
of Ic. Here, ∂Jc(x0,y0)/∂PL∝Jc(x0,y0) if Λ and δPL
are independent of the beam position. Larger critical
current densities produce larger LSM PRILas a result
of redistribution of JRF through the cross-section of the
undercritical currents of width W − Wc, thus increasing
the averaged JRF flowing there. Here Wc denotes the
width of the critical region.
As one can see from Eq. 10, the highest microwave cur-
rent densities produce the largest PRIL(x0,y0) resulting
in quantitative profiles of J2
RF(x0,y0) in the area of the
laser beam raster on the superconductor surface.
VI.GLOBAL PHOTORESPONSE RESULTS
To characterize the resilience of the superconducting
spiral resonators at high PRF, it is important to exam-
ine how current is distributed in the entire sample when
driven by strong RF signals. Fig. 3(a) is a 2D LSM image
showing the global photoresponse of a Nb spiral excited
at its fundamental resonant mode of 74 MHz. The laser
is scanned over a 7.6 x 7.6 mm2area at TB= 4.5 K, PRF=
14.8 dBm and 1 mW laser power. The contrast in the im-
age is mainly produced by PRILwhere the bright areas
can be interpreted as J2
RF(x,y) to first approximation,
and illustrates a mode in which a single half-wavelength
of standing wave current spans the length of the spiral.
As seen, RF current mainly flows in the middle windings
in a quite uniform way, and diminishes towards the inner
and outer edges of the spiral. Fig. 3(b) is a 2D LSM
reflectivity image of the same spiral and shows the turns
in an area on the spiral shown with the green box in (a).
The evolution of the PRILcoming from the individual
windings along the cross section of the spiral [marked as
S in (a)] is shown in Fig. 3(c) for four different PRFvalues
and reveals the JRF distribution in greater detail; note
that the maximum PR corresponds to the center of the S
cut line, and the ends show no response, confirming what
is seen in (a). The asymmetric shape of the standing wave
profile in the fundamental mode is understood from the
fact that the spiral turns at larger radius have a greater
circumference. The dots show the estimated J2
the cut S for the case of a half sinusoid wave wrapped into
a spiral. These dots describe well the observed PRIL,
indicating that the measured PR distribution is quite
similar to the naive interpretation of imaging J2
In Fig. 4(a), the individual line-scans of PRILat dif-
ferent incident power levels are shown in a three dimen-
sional (3D) image.Fig. 4(b) shows power-dependent
evolution of LSM PR at three fixed positions of the
laser probe coinciding with the centers of three neigh-
boring Nb strips (strips A, B and C) exposed to maxi-
LSM PR (a.u.)
LSM PR (mV)
FIG. 3: (Color online) (a) 2D LSM image showing current
distributions in a Nb spiral with an outer diameter of 6 mm
and 40 turns, at the fundamental resonant mode of 74 MHz,
TB= 4.5 K, PRF= 14.8 dBm. (b) 2D LSM reflectivity im-
age showing the individual turns within an area on the spiral
marked with a green box in (a). (c) The power dependent
PRR along the cross section of the spiral shown with S-line;
maximum at the center, minimum at the edges. The dots are
the estimated J2
pattern at each PRF.
RFprofile for a simple standing wave current
LSM PR (a.u.)
i Bstrip B
SM PR (mV)
RF power (mW)
FIG. 4: (Color online) (a) 3D LSM image showing the power
dependence of PRIL over the S-line scan shown in Fig. 3(a).
(b) Experimental LSM PR vs. PRF on a linear scale, taken
at three neighboring strips (strip B is located at the center of
the S-line scan between strip A and strip C). Both data are
obtained at a temperature well below Tc, 4.5 K.
mum JRFnear the peak of the microwave standing wave.
Note that a linear power scale is used. In the purely
superconducting or normal states, LSM PR is not ob-
servable [notice the zero PR at the low and high lim-
its of PRF values in (a)].
PRIL(SA,B,C,PRF) ∝ J2
ear trend for a long range of nonequilibrium states of
the Nb film starting from an PRF corresponding to the
first observable LSM PR at Pc1 up to a switching to
the normal state at Pc2= 14.8 dBm where the PR drops.
These observations validate Eq. 9 in explaining our re-
sults. Also, one can see that based on the values of Pc1,
the Jcof all three strips is practically the same, indicat-
ing a spatial uniformity in Nb film microstructure.
As evident from Fig. 4(b),
RF(∼ PRF) shows an almost lin-
VII.LOCAL PHOTORESPONSE RESULTS
Upon more detailed examination of the Nb resonators,
one can see that the LSM PR is also the strongest at
the center of an individual turn forming the spiral, fol-
lowing the same trend of the global behavior shown in
Fig. 3. Fig. 5(a) shows 2D LSM PRILof a 40x40 µm2
area on the same resonator scanned with a 1.5 µm di-
ameter laser probe in the fundamental resonant mode of
the spiral, a TBof 4.5 K, and PRF of 7 dBm, and laser
power of 1 µW, while Fig. 5(b) is a LSM reflectivity
image obtained from the same area. By comparing the
simultaneously measured LSM PR and reflectivity one
finds that the PR is centered in the windings and does
not extend to the edge of the strip. This observation is
verified by studying the RF PR images as a function of
increasing temperature or PRF, and noting that the PR
spreads out laterally in both directions from the center
of the strip as the critical temperature and power are
approached (see the power evolution of PR coming from
Nb turns in Fig. 5(c); beyond +10 dBm the entire strip
starts to show strong resistive response).
The accumulation of PR in the center of the turns is in
contrast with previously published PR profiles of strip-
geometries that show substantial concentration of the
LSM PR at the edges of current-carrying strips26,33,35,36.
Qualitatively, this fact can be easily understood. Here,
the vertical components of magnetic field between the
strips are partially cancelled, as discussed above, since
adjacent strips have nearly equal and parallel currents
(at least in the fundamental mode). The spiral effectively
acts like a disk carrying an approximately homogeneous
current distribution, in which the current density goes to
zero at the inner and outer radii of the disk.
As well as PRF, laser power has an impact on the
RF PR profile in the spirals. Fig. 5(d) shows the ini-
tial depression of Jc(x0,y0) by modulated laser power,
δPL(x0,y0)= 1 µW in detail (blue curve). The main fea-
ture of PRIL(x0,y0) induced by δPL is generated only
inside a very narrow resistive strip, directed along the
center of the strip-line. Moreover, no spatial modula-
tion in LSM PR is visible in the scanned area along
the direction of RF current flow, indicating that the Nb
film is quite homogenous, which rules out a structure-
related mechanism of hot-spot formation. Taking Eqs. 1
and 10 into account, as well as considering the fact
that the laser beam illumination is spatially uniform,
one can deduce that RF current is peaked half way be-
tween the Nb strip-line edges reaching local maxima of
JRF(x0,y0)≤Jc(x0,y0) there. Larger laser power (see the
red curve in Fig. 5(d) corresponding to δPL(x0,y0)= 10
µW) increases the area of the strip in the critical state
and consequently JRF(x0,y0) adjusts itself accordingly,
since in the superconducting state, JRF(x0,y0) cannot
exceed Jc(x0,y0). Thus, the distribution of PRIL(x0,y0)
spreads all over the strip occupying the dissipative re-
gions of the still superconducting strip.
Line-scan profiles across two strips of such spatial evo-
RF PRRF PR
R (a. u.)
FIG. 5: (Color online) 2D LSM (a) PRIL and (b) reflectivity
images taken from 40x40 µm2area on the Nb spiral resonator
at a laser power of about 1 µW. (c) 2D LSM PRIL image at
10 µW. Inset shows RF power dependence of LSM PRIL on
the same area, showing the JRF profiles at low and high RF
stimulus. The x-line cut is at the same location in the figure
and inset. (d) LSM PR coming from 2 neighboring Nb turns
at two different laser powers; 1 µW and 10 µW. The data
are taken at PRF= 7 dBm.
lution of Jc(x) are shown in Fig. 6 as a function of PRF.
Small laser probe perturbation (PL= 1 µW <<Pc1) gen-
erates the first observable LSM PRILexactly at the cen-
ters of the superconducting strips carrying a current den-
sity JRF(x0,y0)= Jc(x0,y0) at PRF = Pc1= 12.8 dBm,
described by Eq. 10. Emergence of this signal is linked
with the creation of sub-micron critical-state nonequilib-
rium domains at the centers of the strips, much smaller
than the size of the laser probe.
maximum (FWHM) of the position dependent LSM PR
is about 2Λ (see Fig. 5d and Fig. 6). Further increase
in PRF leads to a broadening of the critical state area
which results in an increase of the FWHM of the dis-
sipative LSM PRIL profiles. With reference to Eq. 10
we see that as JRFincreases, the width of the film in the
critical state (Wc) will increase, forcing more current into
the under-critical region (W − Wc) and thus increasing
Further, we estimate Jcby using the measured LSM
PRIL. In the inset of Fig. 5(c), there is a sharp transi-
tion from a center concentrated strip-like resistive state
to an almost uniform resistive state where PR covers the
whole width of the strip. For a wide range of PRF below
Pc1∼ 10 dBm, the LSM PR is almost independent of RF
stimulus. We associate this effect with the auto-adjusting
of instantaneous circulating JRF to a value which could
be accommodated in the superconducting resonator with
varying local values of Jc. Considering the absence of vis-
ible imperfections in the reflectivity data taken from the
The full width half
LSM PR (a.u.)
FIG. 6: (Color online) Power dependence of dissipative LSM
PRIL profiles showing the broadening of the critical state
along the same 40 µm line x-scan through the width of two
Nb turns at a laser power of about 1 µW.
same area of this spatial power dependence we can con-
firm that this transition results from heating effects gen-
erated by hot-spot formation at a location far from the
scanned line. Thus, this defect-free section of the scan is
chosen for rough estimation of the Jc(x0,y0) limit for our
resonator. Estimation is done based on measurements of
Pc1generating the first detectable LSM resistive image
similar to that shown in Fig. 5(a).
Since the use of smaller laser power is favorable in these
measurements in terms of eliminating extra dissipation,
we have scanned the spiral surface at δPL= 500 nW and
found that Pc1≈9.5 dBm. There is no LSM PR observed
below this laser power except only at very high PRF.
Using measured data [Pc1= 8.91 mW (9.5 dBm), W= 10
µm, d= 0.2 µm, Q= 545.4, |S21(f0)|= 0.1553, harmonic
number n = 3 and Z0= 96 Ω which is the characteristic
impedance estimated for a co-planar waveguide of similar
geometry] and a simple model for homogenous current
distribution in the stripline we estimate an upper limit
S21(f0)(1 − S21(f0))4QPc1
and obtain Jc∼ 2.7×1010A/m2from Eq. 12. This value
is more than an order of magnitude smaller than the
theoretical estimation for the depairing current density
A/m2at 4.5 K (using JGL
Nb at TB= 0 K38). This result implies that the mea-
sured critical current is limited by factors other than the
dp(0)(1 − T/Tc)3/2, which is 0.44×1012
dp(0) = 1.26×1012A/m2for
R (a. u.)
120 140 160 180 200 220 240
x-scan (m)( )
FIG. 7: (Color online) Large-scale 7x7 mm2LSM PR images
showing RF current induced dissipation in a Nb spiral at 4.5
K, 10 dBm and at (a) the second harmonic (219 MHz), (b) the
third harmonic (355 MHz), and (c) the forth harmonic (498
MHz). The area marked A in (b) indicates the position of
a detailed 2D image (d) that shows the same RF dissipation
in a 125x125 µm2region localized at the center of the 3λ/2
standing wave pattern. (e) LSM PR profile along a 125 µm
x-scan corresponding to the bottom line-scan in (d).
We observe that higher order harmonic modes of the
Nb spiral resonator have more inhomogeneous current
distributions in the windings due to the larger spatial
gradients of the current. Moreover, the resonant charac-
teristics of the Nb spiral are more power dependent in
those higher modes than the fundamental mode.
Fig. 7(a)-(c) show LSM PR in the 2ndto 4thharmonic
standing wave patterns over a 7x7 mm2area. If we focus
on a small area (125x125 µm2) in the 3rd harmonic LSM
image shown in (b), we see a different scenario from that
discussed above, Fig. 7(d). In locations where there is
a gradient in the current in the radial direction, a more
traditional current profile is observed (notice the 4thturn
from left in d). Fig. 7(e) shows the profile of this PR as
a function of position in the radial direction. The PR is
uniformly distributed across the strip at the peak of the
standing wave pattern. However in regions where there
is a significant change in the amount of current flowing
in neighboring strips, the PR tends to be concentrated
along one edge of the strip. For example on the right
hand side of Fig. 7(d) there are decreasing magnitudes of
current flowing through the strips to the right. This leads
to an asymmetry of the perpendicular magnetic field so
that there is a larger field on the left side of each gap com-
pared to the right. This in turn leads to an asymmetric
buildup of current on the right side of each strip to screen
out the field. The analogous phenomenon occurs on the
other side of the peak in the current distribution as the
current carried by windings further to the left decrease
We have seen similar effects in the third harmonic
standing wave pattern of similar spirals implemented
with Y Ba2Cu3O7.Two distinct photoresponse peaks
are seen on the edges of the Y Ba2Cu3O7 strip.
maximum of the standing wave pattern is visible in the
middle strip, and the current decreases to either side.
In conclusion, we have mapped the global and local
current profile in planar spiral resonators implemented
with superconducting Nb thin films via LSM imaging.
The PR analyses reveal that the RF current in the fun-
damental mode mainly flows at the center of the turns
of the spiral, which is contrary to the profile that is tra-
ditionally seen in stripline and coplanar waveguide res-
onators. The continuous spiral geometry plays an im-
portant role in this unusual current profile contrasting
with the conventional strip resonator case having anti-
parallel currents in adjacent elements where RF current
accumulates at the edges.
We gratefully acknowledge the contributions of John
Abrahams, Tian Lan, Liza Sarytchev, Brian Straughn,
and Frederic Sirois.The work at Maryland was sup-
ported by ONR Grants No.
20101144225000, the US DOE DESC 0004950, the
ONR/University of Maryland AppEl Center, Task D10
(N000140911190), and Center for Nanophysics and Ad-
vanced Materials (CNAM). The work in Karlsruhe is
supported by the Fundamental Researches State Fund
of Ukraine and the German Federal Ministry of Ed-
ucation and Research under Grant No.
the Deutsche Forschungsgemeinschaft (DFG) and the
State of Baden-Wurttemberg through the DFG Center
for Functional Nanostructures (CFN), and a National
Academy of Sciences of Ukraine program on Nanostruc-
tures, Materials and Technologies. S.M.A. acknowledges
sabbatical support from the CFN at Karlsruhe Institute
N000140811058 and No.
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