# 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.

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- Applied Physics Letters 10/2012; 102(1). · 3.79 Impact Factor
- SourceAvailable from: Daimeng Zhang
##### Article: Realization and Modeling of Metamaterials Made of rf Superconducting Quantum-Interference Devices

Physical Review X 12/2013; 3(4):041029. · 8.39 Impact Factor - SourceAvailable from: Stefan Heinz WuenschPhysical Review Letters 12/2012; 110(15). · 7.73 Impact Factor

Page 1

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

current densities.

PACS numbers: 74.25.N-, 74.81.-g, 74.62.Dh, 74.25.nn, 74.70.-b

I. INTRODUCTION

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

(a)(b)

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

Page 2

2

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.

II. SAMPLE

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

4 K)28.

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

length25.

III.RF EXCITATION

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.

IV.CRYOGENICS

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.

Page 3

3

L

Lock-in

k i

diode laser

x-y scanner

objective lens objective lens

Laser Intensity

Modulation

crystal diode

hi

RFin

RFout

Nb spiral

V

microwave amplifier

sapphire

t0sensor

Cu

cylinder

PR

Data acquisition

computercomputer

network analyzer

temperature

controller controller

Cold plate

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

to scale.

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.

Inthebolometric

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).

(thermal)regime, the

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

probe focus.

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-

Here, ρ=

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.

(1)

Page 4

4

PR ∝ δ|S21(f)|2=1

2(∂|S21(f)|2

∂f0

∂f0

∂PL

2

+

∂|S21(f)|2

∂(1/2Q)

∂(1/2Q)

∂PL

+∂|S21(f)|2

∂ˆ

S21

2

∂ˆ

S21

∂PL

)δPL

(2)

where the transmission coefficient, |S21(f)|2[ratio of

the transmitted power, POUT

RF

PRF(f)] as a function of driving frequency f ∼ f0 is

given in the limit of weak coupling by31

(f), to the input power,

|S21(f)|2=

ˆ

S21

2

1 + 4Q2(f/f0− 1)2

(3)

and ˆ

at the peak of the resonance. By substitution of Eq. 3 in

Eq. 2, one finds that the inductive

S21

2is the maximum of the transmission coefficient

PRI∝∂|S21(f)|2

∂f0

=

8ˆS21

[1 + 4Q2(f

2Q2(f

f0− 1)

f0− 1)2]2

f

f2

0

(4)

and the resistive

PRR∝∂|S21(f)|2

∂(1/2Q)

=

16ˆ

[1 + 4Q2(f

S21

2Q3(f

f0− 1)2

f0− 1)2]2

(5)

components of total LSM PR are nulled at f = f0, while

the insertion loss component

PRIL∝|S21(f)|2

δ(ˆS21)

=

2ˆS21

1 + 4Q2(f

f0− 1)2

(6)

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

described as

RF(x0,y0)δRs(x0,y0)(7)

δRs(?0) ∝π

4

Λ

WL

?

? L

d?∂Rs(?0)

∂Jc(?0)|J=JRF

∂Jc(?0)

∂PL

|P=PCIRC

RF

+δPLδPL(?0)(8)

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

RF

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

PRIL(x0) ∝

1

WdJ2

RF(x0)∂Rs(x0)

∂Jc(x0)|J=JRF

∂Jc(x0)

∂PL

|P=PCIRC

RF

+δPLdΛδPL(x0)(9)

at location x0in the one dimensional strip geometry.

Note that Eq. 9 demonstrates a threshold mecha-

nism of PRILgeneration relative to excitations by both

PCIRC

RF

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

RF

+ δPL ≤ Pc1,

RF

+ δPL)

RF(x0)

RF(x0) along that

PRIL(x0,y0) ∝

Λ2

WL< J2

RF>W−Wc

∂Jc(x0,y0)

∂PL

∂Rs(x0,y0)

∂Jc(x0,y0)

δPL

(10)

Page 5

5

The effect is linked with the laser-induced modulation of

the local critical current,

δIc(x0,y0) =πΛ2

4

∂Jc(x0,y0)

∂PL

δPL

(11)

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-

RFalong

RF(x,y).

14.8 dBm

14.3 dBm

13.8 dBm

13.3 dBm

0.00.20.40.60.81.0

0.0

0.3

0.5

0.8

1.0

1.3

(c)

(b)

(a)

S

0

max

LSM PR (a.u.)

x-distance (mm)

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.)

max

1 21.2

Pc1

(a)

strip A

i Bstrip B

strip C

(b)

Pc2

16

0

0

0.6

SM PR (mV)

L

12

0.5

1

152025303540

0.0

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-

Page 6

6

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-

(a)

RF PRRF PR

(c)

10 dBm

xx

PRF

10 m

7 dBm

Reflectivity Reflectivity

0max

LSM PR

(b)

(d)

2 22.2

1 11.1

R (a. u.)

LSM PR

NbNbquartz

reflectivity

10 W

1 W

001010202030304040

0.0

x-scan (m)

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

< J2

RF>W−Wc.

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

Page 7

7

14.8 dBm

14.7 dBm

14.5 dBm

14.2 dBm

13.6 dBm

13.2 dBm

12.8 dBm

0 11223344

0.0

0.5

1.0

1.5

LSM PR (a.u.)

x-scan (m)

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

of Jcas37

Jc(f0) =

1

Wd

?

S21(f0)(1 − S21(f0))4QPc1

nπZ0

(12)

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

JGL

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

depairing limit.

dp(T/Tc)∼JGL

dp(0)(1 − T/Tc)3/2, which is 0.44×1012

dp(0) = 1.26×1012A/m2for

(a)

(d)

(b)

1 mm

x

y

A

(c)

2

3

R (a. u.)

LSM PR

(e)

120 140 160 180 200 220 240

x-scan (m)( )

0

1

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).

VIII.HIGHER HARMONICS

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

Page 8

8

current carried by windings further to the left decrease

in magnitude.

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.

The

IX.CONCLUSIONS

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.

ACKNOWLEDGEMENTS

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

of Technology.

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