Superconductivity from a melted insulator

2025-04-24 0 0 5.43MB 24 页 10玖币

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S. Mukhopadhyay,1, ∗J. Senior,1, ∗J. Saez-Mollejo,1D. Puglia,1M. Zemlicka,1J. Fink,1and A.P. Higginbotham1, †

1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria

Quantum phase transitions typically result

in a broadened critical or crossover region at

nonzero temperature [1]. Josephson arrays are

a model of this phenomenon [2], exhibiting a

superconductor-insulator transition at a critical

wave impedance [3–13], and a well-understood

insulating phase [14,15]. Yet high-impedance

arrays used in quantum computing [16–19] and

metrology [20] apparently evade this transition,

displaying superconducting behavior deep into

the nominally insulating regime [21]. The ab-

sence of critical behavior in such devices is not

well understood. Here we show that, unlike the

typical quantum-critical broadening scenario, in

Josephson arrays temperature dramatically shifts

the critical region. This shift leads to a regime

of superconductivity at high temperature, aris-

ing from the melted zero-temperature insula-

tor. Our results quantitatively explain the low-

temperature onset of superconductivity in nomi-

nally insulating regimes, and the transition to the

strongly insulating phase. We further present,

to our knowledge, the ﬁrst understanding of the

onset of anomalous-metallic resistance saturation

[22]. This work demonstrates a non-trivial inter-

play between thermal eﬀects and quantum criti-

cality. A practical consequence is that, counterin-

tuitively, the coherence of high-impedance quan-

tum circuits is expected to be stabilized by ther-

mal ﬂuctuations.

Josephson-array superinductors are characterized by a

Josephson energy EJ, junction charging energy EC, and

ground charging energy Eg[17]. A common experimen-

tal strategy for avoiding insulating behavior is to make

the fugacity for quantum phase slips y∝e−4√2EJ/EC

small. However, for high-impedance arrays the fugacity

is always renormalized towards inﬁnity as temperature

goes to zero [13,23], resulting in insulating behavior.

Our key insight is that long superinductors avoid this

fate by operating above the melting point of the insu-

lating phase, where the low-temperature renormalization

has yet to occur, and that this results in apparent super-

conducting behavior. This eﬀect quantitatively explains

the presence of superconducting behavior, resistance sat-

uration, and transition to strongly insulating regimes in

superinductors.

Two nearly identical devices are studied: one galvani-

cally coupled to electrical leads permitting the measure-

ment of resistance, and one capacitively coupled to mi-

crowave transmission lines permitting the measurement

of plasma modes [17,21]. Both devices consist of an ar-

ray of approximately 1220 Josephson junctions fabricated

using electron-beam lithography and a standard shadow

evaporation process on high-resistivity silicon substrates

(Fig. 1a) [24]. For nanofabrication reasons the array

islands have alternating thickness, which, in the pres-

ence of magnetic ﬁeld, should give rise to an alternat-

ing gap structure while maintaining a uniform Josephson

energy throughout the chain. At zero magnetic ﬁeld,

each junction has nominally identical EJ/h ≈76 GHz,

Eg/h ≈1400 GHz, and EC/h ≈5 GHz. These parame-

ters are determined from analyzing microwave (EJ, Eg)

and transport (EC) measurements with several consis-

tency checks, as described below and in the Supplement

[24].

The working principle of the experiment is to leverage

the complementary strengths of low-frequency electrical

transport and microwave-domain circuit quantum elec-

trodynamics. These techniques diﬀer by nine orders of

magnitude in characteristic frequency, and combine to

give access to both the scaling behavior, associated with

low energies (transport), and the microscopic system pa-

rameters, associated with high energies (microwave).

In the transport device, a linear current-voltage char-

acteristic at large applied voltage bias gives way to a high

resistance region at low bias, whose extent is approxi-

mately given by the number of junctions Ntimes twice

the superconducting gap ∆ (Fig. 1b). Over a smaller

range of applied voltage a series of evenly spaced cur-

rent peaks are observed with an apparent supercurrent

at zero bias (Fig. 1b inset). The successive current peaks

can be qualitatively understood within a picture of suc-

cessive voltage drops across Nvoltage-biased Josephson

junctions, with low current on the quasiparticle branches

and high current when bias is a multiple of 2∆/e [11].

Increasing magnetic ﬁeld Bparallel to the chip plane

suppresses supercurrent, suggesting a ﬁeld-driven tran-

sition from a superconducting to an insulating state

(Fig. 1c). The spacing between current peaks also de-

creases with B, indicating a reduction in the supercon-

ducting gap with magnetic ﬁeld. In the strongly super-

conducting regime (B= 0), zero-bias diﬀerential resis-

tance per junction (speciﬁc resistance) associated with

the superconducting branch decreases dramatically with

cryostat temperature (Fig. 1d), dropping over more than

three decades before saturating to a low value of <1 Ω

per junction. Due to the long length of the array, we

rule out ﬁnite-size eﬀects as a possible origin of the low-

temperature saturation [25]. The precipitous drop in re-

arXiv:2210.06508v1 [cond-mat.mes-hall] 12 Oct 2022

0 12

f(GHz)

-85

-60

S(dB)

-1 1

V(V)

-0.4

0.4

I( A)

V(mV)

B(mT)

T(K)

I(nA)

MICROWAVE

TRANSPORT

10 10

0 12

fP(GHz)

-35

B(mT)

-2 2

-20

-5

( )

B-25 B(mT)

0.1

INS ULATOR

SUPERCONDUC TOR

ac d

e f g

Figure 1: Device, transport, and microwave measurement techniques. a, Scanning electron micrograph

of a small segment of the Josephson array. Left scale bar indicates 1.5µm. Arrow indicates direction of magnetic

ﬁeld B.b, Current Iversus source-drain bias voltage V. Inset shows small-scale current peaks over a narrow range

(−3,3) mV. c, Current Iversus bias Vand magnetic ﬁeld Bover a bias range similar to Fig. 1b inset. d, Diﬀer-

ential resistance per junction (speciﬁc resistance) ρversus cryostat temperature Tmeasured at V= 0 and B= 0.

Blue line shows power-law ﬁt. ρreﬂects the resistance associated with the zero-bias superconducting branch, found

by measuring the two-probe resistance, subtracting oﬀ four-probe-measured line resistance, and then dividing by

number of junctions. e, Two-tone microwave spectroscopy. Probe tone transmission Sversus pump-tone frequency

f, with probe tone frequency ﬁxed to resonance at approximately 6.11 GHz. Extracted plasma-mode resonant fre-

quencies fPindicated by colored markers. f, Evolution of measured plasma-mode frequencies fPwith applied mag-

netic ﬁeld B.g, Superﬂuid stiﬀness Kg=pEJ/(2Eg), experimentally inferred from plasma modes in f, versus

B(black line). Theoretically expected superconducting and insulating regimes labeled, and demarcated by a band

covering the clean [2] and dirty [23] limits.

sistance at low temperature and supercurrent features in

nonlinear transport give a preliminary indication of the

dominance of superconducting behavior. We will develop

a framework for understanding the behavior of speciﬁc

resistance in detail, but ﬁrst turn to the complementary

use of microwave techniques to independently determine

system parameters.

Microwave spectroscopy is performed by monitoring

the transmission of a weak probe signal while the fre-

quency of a strong pump tone is varied [17]. A series

of sharp dips are observed in probe-tone transmission S

(Fig. 1e), corresponding to plasma modes of the array.

The plasma modes are evenly spaced at low frequency,

reﬂecting the speed of light and length of the array, and

are clustered at high frequency due to proximity with the

single-junction plasma frequency. A simple ﬁtting proce-

dure allows extraction of the array parameters from the

microwave data [24]. Performing two-tone spectroscopy

as a function of ﬁeld (Fig. 1f), the array parameters Eg,

EC, and EJ(B) are fully characterized as a function of

magnetic ﬁeld. With these values ﬁxed experimentally, it

is straightforward to perform parameter-free comparisons

with the theory of the superconductor-insulator transi-

tion in one dimension.

Performing this comparison (Fig. 1g) reveals that the

array’s superﬂuid stiﬀness Kg=pEJ/(2Eg) is as much

as an order of magnitude below the critical value for

insulating behavior [2,23], in contrast to the observed

superconducting behavior in transport. Thus, com-

bining the transport and microwave measurements re-

veals an apparent conﬂict with basic expectations for

the superconductor-insulator phase transition. Resolv-

ing this conﬂict is the central subject of this work.

The theoretical picture for understanding our ob-

servations was developed in Ref. [13]. Near the

superconductor-insulator transition, thermal ﬂuctuations

are controlled by the timescale τ=h/kBTand the asso-

ciated thermal length lth =vτ, where vis a characteristic

velocity with dimensions of unit cells per time. lth must

be compared with the electrostatic screening length in

units of unit cells, Λ = pEg/EC. At high temperature

(lth <Λ) the system is governed by the local superﬂuid

EJZ²

,1/

ϖKc~ 1 ϖKg~ 3/2

Tins

LOCAL

SUPERCONDUCTOR

SUPER-

CONDUCTOR

INSULATOR

SUPER-

INDUCTOR

(LSC)

Figure 2: Proposed phase diagram. Map of su-

perconducting and insulating states as a function of

Josephson energy EJand temperature T. Dashed

line marks the boundary between long-range and

short-range behavior, Tins, given by Eq. 1. Below

Tins, physics is governed by the long-range superﬂuid

stiﬀness Kgwith a superconductor-insulator tran-

sition at πKg∼3/2. Above Tins, physics is gov-

erned by the short-range superﬂuid stiﬀness KCwith a

superconductor-insulator transition as πKC∼1. Solid

black curve traces the crossover from local to global

superconductor-insulator transition. Outlined box indi-

cates superinductance region probed in this experiment.

stiﬀness, KC=pEJ/(2EC). In contrast, at low tem-

perature (lth >Λ) the system is governed by the long-

range superﬂuid stiﬀness Kg, as assumed by standard

theories of the superconductor-insulator transition. In

the superinductor limit superconductivity is locally stiﬀ,

KCKg, which results in a curious regime of local

superconductivity that arises from a melted T= 0 in-

sulator (Fig. 2). The “melting point” of the insulator,

above which local superconductivity dominates, is

Tins ∼p2EJEC/Λ.(1)

In the locally superconducting regime, we ﬁnd that the

high-temperature behavior of the speciﬁc resistance fol-

lows a power law

ρ=ρ0(T/Tp)πKC−1,(2)

where Tp=√2EJEC/kBis the plasma temperature [24].

Local superconductivity gives way to insulating behavior

when πKC∼1. In contrast, in the low-temperature

limit the power law is 2πKg−3, which yields the typical

superconductor-insulator prediction πKg∼3/2.

0.1 1

T(K)

104

105

( )

64 mT

40 mT

35 mT

30 mT

25 mT

20 mT

15 mT

0 mT 0 2

-1

25 75

EJ(GHz)

103

105

A(Kp)

a b

Figure 3: Power law nature of local supercon-

ductivity. a, Zero-bias speciﬁc diﬀerential resistance

ρas a function of temperature T, at various mag-

netic ﬁelds. Solid lines are ﬁts to power law expression

ρ=AT p.T∗is the crossover temperature from power

law to saturation behavior, extracted from the point

where speciﬁc resistance goes 20% above its minimum

value. b, Exponent pfrom power-law ﬁts to trans-

port data in aversus the local superﬂuid stiﬀness KC

from microwave measurements. Solid line is a linear ﬁt.

Shaded blue region in bdepicts the systematic error

resulting from the choice of lower resistance cutoﬀ in

the power law ﬁts [24]. c, Amplitude Afrom power-law

ﬁts to transport data versus Josephson energy EJfrom

microwave measurements.

The experimentally studied devices have, at B = 0,

πKg<1< πKC, and Tins ∼70 mK, giving an initial

suggestion that they are governed by local superconduc-

tivity even at low temperatures. This hypothesis can

be tested by comparing experimental measurements of

temperature dependent speciﬁc resistance, ρ(T), with the

predicted power law in Eq. 2. As shown in Fig. 3a, in-

creasing magnetic ﬁeld weakens the temperature depen-

dence of the speciﬁc resistance, eventually giving way to

a superconductor insulator transition at high magnetic

ﬁeld (B∼44 mT). Fitting each speciﬁc resistance curve

to a power law ρ=AT pindicates that, on the supercon-

ducting side, the exponent psteadily decreases with ﬁeld.

Comparing pfrom the transport measurements with the

local superﬂuid stiﬀness KCinferred from microwave

measurements reveals a linear behavior (Fig. 3b) with

slope 2.7±0.50 and intercept of −1.3±1.0, in agree-

ment with the predicted slope πand intercept −1 for

local superconductivity from Eq. 2, p=πKC−1 [26]. We

0 50

B(mT)

0 - 41 mT

43 mT

43.5 mT

44 mT

44.5 mT

45 mT

104

106

( )

-100 100

B(mT)

0.2

T(K)

-25

B(mT)

100

105

( )

a b

c d

EXPERIMENT THEORY

50 150

T(mK)

INS

LSC

INS

T*(K)

105x

Figure 4: Crossover physics and phase diagram.

a, Crossover temperature T∗versus magnetic ﬁeld B.

Black line is T∗∝Tins with a proportionality constant

of 2.2. Colored markers indicate crossover temperature

at higher ﬁelds from dataset in b. Red vertical line in-

dicates πKC= 1, where local superconductor-insulator

transition is expected. b, Zero-bias diﬀerential speciﬁc

resistance ρversus temperature Tat higher magnetic

ﬁelds, measured with higher excitation voltage and

more averaging than in Fig. 3. c, ρversus temperature

Tand magnetic ﬁeld B. Dome of local superconduc-

tivity (LSC), and wings of insulating behavior (INS)

labeled. Red vertical lines indicate πKC= 1, where lo-

cal superconductor-insulator transition is expected. d,

Calculated speciﬁc resistance ρas a function of temper-

ature Tand magnetic ﬁeld B.

note that, near the superconductor-insulator transition,

power-law behavior is interrupted by a shoulder-like fea-

ture at high temperature, which is not understood. Am-

plitude dependence on EJ(Fig. 3c) is also in reasonable

agreement with the prediction of Eq. 2, A=ρ0/T πKC−1

with a single free parameter, ρ0= 4.8±0.30 kΩ, which

is of the order of single-junction normal-state resistance.

Experimental agreement with Eq. 2 gives strong evidence

that superconductivity is local, and resolves the appar-

ent paradox of superconductivity at low Kgsuggested by

Fig. 1g.

The boundaries of local superconductivity can also be

understood within the picture of Fig. 2. At low tem-

peratures, the experimentally observed power-law behav-

ior in resistance saturates at a crossover temperature T∗

(indicated in Fig. 3a). The crossover temperature de-

creases with magnetic ﬁeld, as shown in Fig. 4a, agreeing

with the expected square-root dependence for T∗∝Tins,

which supports the view that the low-temperature sat-

uration is in fact a crossover into the insulating state.

At high magnetic ﬁelds corresponding to πKC>1, T∗

increases with magnetic ﬁeld (Fig. 4b), consistent with

a superconductor-insulator transition entering into the

non-perturbative insulating regime of Ref. [13], where

the phase-slip fugacity, ∝e−8√EJ/EC, is no longer small.

We caution that the experimental interpretation of T∗is

complicated for two reasons. First, although we have

performed normal-state electron thermometry and ra-

diation thermometry and found that all characteristic

temperatures are below T∗, thermalization at the actual

superconductor-insulator transition is diﬃcult to verify

directly. Second, diﬀerent metrics for T∗give quantita-

tively diﬀerent scaling with B, although the decreasing

trend predicted by Eq. 1 is a robust feature.

The complete behavior of the Josephson array can be

summarized by measuring a resistance “phase diagram.”

Mapping zero-bias diﬀerential resistance as a function of

magnetic ﬁeld and temperature reveals a characteristic

dome at low ﬁeld, already identiﬁed from the power-law

analysis as a local superconductor, giving way to a high-

resistance insulating phase as magnetic ﬁeld is increased

(Fig. 4c). The low-temperature boundary between su-

perconducting and insulating states occurs at πKC∼1,

as expected. We speculate that the high-ﬁeld boundary

of the high-resistance regime corresponds to the upper

critical ﬁeld of the thinnest islands of the array.

The local superconducting dome and its boundaries

can be quantitatively modeled as follows. The ther-

mal boundary of the dome is T=Tp, the upper cut-

oﬀ scale of our renormalization-group approach [13]. For

αTins < T < TpEq. 2 applies, with ρ0from Fig. 3c. For

T < αTins resistance saturates due to a crossover into

the insulating regime, and would presumably increase

at lower, experimentally inaccessible temperatures. The

constant α= 5, which tunes the crossover to insulating

behavior in the model, is ﬁxed from the experimentally

observed saturation resistance at B= 0 and in reason-

able agreement with the constant found in Fig. 4a. For

suﬃciently large Bone approaches πKC= 1, which sets

the magnetic ﬁeld boundaries of the dome. Calculating

ρaccording to this procedure results in a local super-

conducting dome in satisfactory agreement to the exper-

iment (Fig. 4d). This gives evidence that the presence of

local superconductivity, and its proximity to insulating

phases, is well understood.

Summarizing, by combining transport and microwave

measurements, we have uncovered strong evidence for a

locally superconducting state in Josephson arrays aris-

ing from a T= 0 insulator. This resolves the prob-

lem of apparent superconductivity in nominally insulat-

ing regimes, and clariﬁes where superconductor-insulator

transitions are actually observed in experiment. Our

work sheds new light on the observation of high-quality

microwave response in the nominally insulating regime

of superinductors [21], suggesting eﬀects in addition to

high-frequency mechanisms that have been previously

discussed [27–29]. Such devices operate near the “sweet

spot” T≈Tins where temperature is low enough for well-

developed local superconductivity, yet high enough to

melt insulating behavior. As a consequence, we suggest

that the performance of some high-impedance quantum

devices [18,19,30] is actually improved by thermal ﬂuc-

tuations. It is also interesting to consider if experimen-

tal studies of insulating behavior in resistively shunted

Josephson junctions [31–34] could be understood by care-

fully considering the role of non-zero temperature, ﬁnite-

size, or non-perturbative eﬀects [35].

Viewed from the broader perspective of response func-

tions near quantum criticality, we have demonstrated

a rare example where the thermal ﬂuctuations with

timescale τ=h/kBTcan be quantitatively traced

through to experimentally measured resistance [13]. This

does not result in an eﬀectively Planckian scattering [24],

as was recently observed in a diﬀerent superconductor-

insulator system [36]. It is also interesting to note that

our saturating speciﬁc resistance curves empirically bear

a strong resemblance to the anomalous-metallic phase in

two-dimensional systems [22]. In our case, saturation is

understood as a crossover eﬀect towards insulating be-

havior. It would be interesting to perform a similar ex-

perimental program on a known anomalous-metallic sys-

tem to test if saturation can be understood as a similar

crossover eﬀect.

ACKNOWLEDGMENTS

We thank David Haviland, Jukka Pekola, Anton Bu-

bis and Alexander Shnirman for helpful feedback on the

manuscript. This research was supported by the Scien-

tiﬁc Service Units of IST Austria through resources pro-

vided by the MIBA Machine Shop and the Nanofabri-

cation Facility. Work funded by Austrian FWF grant

P33692-N. J.S. acknowledges funding from the European

Union’s Horizon 2020 research and innovation program

under the Marie Sk lodowska-Curie Grant Agreement No.

754411.

SUPPLEMENTARY MATERIALS

Raw data for all ﬁgures to be uploaded before ﬁnal

publication

Materials and Methods

Supplementary Text

Figs. S1 to S15

References [2,13,15,23,36–45]

∗Equal contribution

†andrew.higginbotham@ist.ac.at

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SuperconductivityfromameltedinsulatorS.Mukhopadhyay,1,J.Senior,1,J.Saez-Mollejo,1D.Puglia,1M.Zemlicka,1J.Fink,1andA.P.Higginbotham1,y1ISTAustria,AmCampus1,3400Klosterneuburg,AustriaQuantumphasetransitionstypicallyresultinabroadenedcriticalorcrossoverregionatnonzerotemperature[1].Josephsonarraysare...

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