Quantum criticality at the superconductor-insulator transition revealed by specific heat measurements

Abstract

The superconductor-insulator transition (SIT) is considered an excellent example of a quantum phase transition that is driven by quantum fluctuations at zero temperature. The quantum critical point is characterized by a diverging correlation length and a vanishing energy scale. Low-energy fluctuations near quantum criticality may be experimentally detected by specific heat, c_p , measurements. Here we use a unique highly sensitive experiment to measure c_p of two-dimensional granular Pb films through the SIT. The specific heat shows the usual jump at the mean field superconducting transition temperature marking the onset of Cooper pairs formation. As the film thickness is tuned towards the SIT, is relatively unchanged, while the magnitude of the jump and low-temperature specific heat increase significantly. This behaviour is taken as the thermodynamic fingerprint of quantum criticality in the vicinity of a quantum phase transition.

ARTICLE

Received 7 Jun 2016 | Accepted 3 Jan 2017 | Published 22 Feb 2017

The superconductorinsulator transition (SIT) is considered an excellent example of a quantum phase transition that is driven by quantum uctuations at zero temperature. The quantum critical point is characterized by a diverging correlation length and a vanishing energy scale. Low-energy uctuations near quantum criticality may be experimentally detected by specic heat, cp, measurements. Here we use a unique highly sensitive experiment to measure cp of two-dimensional granular Pb lms through the SIT. The specic heat shows the usual jump at the mean eld superconducting transition temperature Tmfc marking the onset of Cooper pairs formation. As the lm thickness is tuned towards the SIT, Tmfc is

relatively unchanged, while the magnitude of the jump and low-temperature specic heat increase signicantly. This behaviour is taken as the thermodynamic ngerprint of quantum criticality in the vicinity of a quantum phase transition.

DOI: 10.1038/ncomms14464 OPEN

Quantum criticality at the superconductor-insulator transition revealed by specic heat measurements

S. Poran1,2, T. Nguyen-Duc2,3, A. Auerbach4,5, N. Dupuis5, A. Frydman1,2,3 & Olivier Bourgeois2,3

1 Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel. 2 Institut NEL, CNRS, 25 avenue des Martyrs, F-38042 Grenoble, France. 3 Univ. Grenoble Alpes, Inst NEEL, F-38042 Grenoble, France. 4 Department of Physics, Technion, 32000 Haifa, Israel. 5 Laboratoire de Physique Thorique de la Matire Condense, CNRS UMR 7600, UPMC-Sorbonne Universits, 4 Place Jussieu, 75252 Paris, France. Correspondence and requests for materials should be addressed to A.F. (email: mailto:[email protected]

Web End [email protected] ) or to O.B. (email: mailto:[email protected]

Web End [email protected] ).

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Quantum criticality is a central paradigm in physics. It unies the description of diverse systems in the vicinity of a second-order, zero-temperature quantum phase transi

tion, governed by a quantum critical point (QCP). QCPs have been discovered and extensively studied primarily in metallic and magnetic systems. At many of these QCPs, especially in two dimensions (2D), conventional mean-eld and Fermi-liquid theories fail in lack of well-dened quasiparticles. QCPs inspired innovative non-perturbative approaches14, including those relevant to eld theory/gravity duality5.

In 2D superconducting lms, the zero-temperature superconductorinsulator transition (SIT) has been viewed as a prototype of a quantum phase transition that is controlled by a non-thermal tuning parameter g6. Experimentally, the transition has been driven utilizing various g such as inverse thickness716, magnetic eld12,13,1726, disorder25,27,28 chemical composition29 and gate voltage30. For gogc the lm is a superconductor with well-dened quasiparticles and superconducting collective modes as well as a nite 2D superuid density. As g is increased, the system enters the critical regime in which excitations are strongly correlated, while the superuid density vanishes as g-gc. For

g4gc, the system becomes insulating with gapped charge excitations.

2D superconducting granular lms have been shown to exhibit signs for Cooper-pairing effects such as the presence of an energy gap, D well into the insulator phase31, and it has been argued that they may be modelled by a bosonic quantum eld theory with O(2) symmetry32. Similar ndings were found for disordered thin lms27,29,33 in which disorder is assumed to generate emergent granularity34. If one ignores the broadening of the SIT due to inhomogeneities, divergent correlation length and time are expected at the transition. Indeed, recent optical conductivity measurements have detected signatures of the critical amplitude (Higgs) mode, becoming soft at the SIT35,36.

One of the salient thermodynamic signatures of a QCP is the presence of excess entropy or specic heat1,2. In heavy electron metals, this has been observed as the divergence of linear specic heat coefcients, or electronic effective mass, which signals a deviation from conventional Fermi-liquid theory behaviour37. Such a signature would be very important to measure in the SIT system, to probe the critical thermodynamics near this non-mean-eld-type QCP.

Entropy S(T) is a fundamental physical quantity of signicant importance for the quantum phase transition; however, its absolute value cannot be directly measured. On the other hand, the specic heat cp(T) at constant pressure can have its absolute value determined experimentally. Apart from its interest regarding QCP, it is above all the most appropriate physical property to categorize the order of a phase transition or to probe signatures of uctuations. Despite its crucial interest, specic heat experiments have never been performed close to the QCP in the context of the SIT. The main reason is that the systems under study are 2D ultra-thin lms involving ultra-low mass. The substrate mass onto which thin lms are deposited is usually much larger than that of the lm itself rendering ultra-thin lm specic heat unmeasurable.

Here we report on the rst experimental demonstration of excess specic heat in 2D granular lms spanning the superconductorinsulator quantum phase transition. We employ a unique technique based on thin suspended silicon membrane used as a thermal sensor that enables ultra-sensitive measurements of the specic heat of superconducting lms close to the SIT. We nd that the mean-eld critical temperature, Tmfc, remains basically unchanged through the SIT. Nevertheless, the specic heat jump at Tmfc and the specic heat magnitude at temperatures lower than the critical temperature increase

progressively towards the transition. These results are interpreted as thermodynamic indications for quantum criticality close to the QCP.

ResultsSpecic heat experimental set-up. The samples used in this study were ultra-thin granular lms grown by the quench condensation technique. In this method, sequential layers are deposited directly on an insulating substrate held at cryogenic temperatures (T 8 K) under ultra-high-vacuum condi

tions7,3840. The rst stages of evaporation result in a discontinuous layer of isolated superconducting islands. As material is added, the inter-grain coupling increases and the system undergoes a transition from an insulator to a super-conductor.

The measurement set-up consists of a thermally sensitive thin membrane that is suspended by 10 silicon arms used for mechanical support, thermal isolation and for electrical wiring (Fig. 1). This results in a calorimetric cell that enables the simultaneous measurement of transport properties and heat capacity with energy sensitivity as low as an attoJoule around 1 K (ref. 41) so that temperature variation as low as few microkelvin can be detected on ultra-thin samples with masses down to few tens of nanograms. This set-up provides a unique opportunity to measure simultaneously the lm resistance, R, and heat capacity, Cp, of a single lm as a function of thickness through the entire

SIT without the need to warm up the sample or to expose it to atmosphere; both processes being harmful to ultra-thin lms. Further experimental details are specied elsewhere42, see also Methods.

Specic heat and resistance through the SIT. Panels a and b of Fig. 2 show concomitant R(T) and Cp(T) measurements performed on a series of 18 consecutive depositions of Pb on a single nano-calorimetric cell. The thinnest layer is an insulator with R441 GO, making it unmeasurable within the sensitivity of our set-up, and the thickest is a superconductor with a sharp transition corresponding to the bulk critical temperature of Pb. The R(T) curves are typical of granular Pb lms11,31, where the rst six depositions are on the insulating side of the SIT, the evaporations 7 and 8 show resistance re-entrance behaviour, 916 are superconducting with long exponential tails that become increasingly sharper until, in stages 17 and 18, the transition is sharp.

Such samples have been considered as prototype systems for the bosonic SIT in which the grains are believed to be large enough to sustain superconductivity with bulk properties11,31,39,43. However, for the thinnest layers, phase uctuations between the grains are strong enough to suppress global superconductivity and lead to an insulating state. The critical temperature measured by transport (temperature of zero resistance) is thus governed by order-parameter uctuations and not by actual pair breaking.

As opposed to electrical transport, thermodynamic measurements can be performed deep into the insulating regime (purple line in Fig. 2b). The measured heat capacity contains contributions from phononic, electronic and superconducting degrees of freedom. In the following, we will only focus on the specic heat cp (the heat capacity cp normalized to the mass). Above Tc, the normal specic heat, cn, should follow the well-known form:cnT gn bT2 1

with gn and b proportional to the electron and phonon specic heat (heat capacity divided by the mass of the Pb lm), respectively. In our case, the phonon contributions to cp

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a

b

VH

a.c.

IHa.c.

ISa.c.

c

VTh

a.c.

IThd.c.

Figure 1 | Sketch of the experimental set-up. (a) The suspended membrane acting as the thermal cell contains a copper meander, used as a heater, and a niobium nitride strip, used as a thermometer. These are lithographically fabricated close to the two edges of the active sensor. (b) The quench condensation set-up is constituted by an evaporation basket containing the Pb material that is thermally evaporated on the substrate held at cryogenic temperatures and in UHV conditions. The granular quench-condensed lm is evaporated through a shadow mask which, together with the measurement leads, denes its geometry. The biasing of the heater is done with a a.c. current IHa:c: (used for heat dissipation), IThd:c: is the d.c. current biasing the thermometer for the measurement of the temperature through the voltage Va.c. and the measurement of the resistance of the quench-condensed lms is done using the d.c.

current ISd:c:. The inset shows a low-temperature STM image of the quench-condensed granular Pb56. (c) The whole experimental set-up is immersed in a liquid helium bath.

a

108

b

107

106

105

104

103

102

101

4

1.5

7.0

6.5

6.0

5.5

2 )

R square()

C p/T(nJ K

1.0

0.5

10 20

T 2 (K2)

5 6 7 8 9 10

30 40 50

T (K)

c d

4

3

2

1

0 3 4 5 6 7 8

7.5

1 K1 )

mf (K)

c se(mJ g

T c

Cp

5.0 5 10 15 20t (nm)

25 30 35

T (K)

Figure 2 | Resistance and heat capacity versus temperature. (a) Resistance per square and (b) the heat capacity of ultra-thin lead versus temperature of the 18 sequential deposited lms. The same colour code is used for all the panels of the gure. The resistances of the two rst depositions are unmeasurable, unlike heat capacity that can be measured deeply in the insulating regime (in purple in b). (c) cse versus T for a number of layers clearly depicting the growth of specic heat as the sample is thinned. The thicknesses are 8.9, 10.2, 10.95, 12.7 and 29 nm from top to bottom, respectively.

(d) Quasi-constant mean-eld critical temperature, Tmfc of the granular Pb layers as extracted from the midpoint of the heat capaity jump as a function of lm thickness through the SIT.

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overwhelms the electronic contribution by at least one order of magnitude as can be seen from the linear behaviour of the heat capacity shown in Fig. 2b. This strong phonon contribution originates from the amorphous nature of the superconducting Pb materials obtained by quench condensation; the discussion of this point goes far beyond the scope of this paper (see Methods). Hence, the specic heat above Tc can be t using a simpler relation than equation (1), that is, cnT bT2 since gnoobT2.

In this study, we focus on cse, the specic heat of electrons in the superconducting state compared with the one in the normal state, as dened by the following relation: cse(T) cp(T) cn(T).

To do this, we subtract from each of the measured cp(T) of Fig. 2b the specic heat extrapolated from the normal-state regime cn(T) bT3 above Tc. The superconducting electronic specic

heat cse (J g 1 K 1) for a number of selected layers are shown in Fig. 2c. The remainder of this article will focus on cse of the lms

close to the quantum phase transition.

Two prominent facts are clearly observed in the specic heat data of Fig. 2c. The rst is the position of the specic heat jumps Dcp and the second is the amplitude of the specic heat below the critical temperature.

Specic heat jumps Dcp. The amplitude of these specic heat jumps at the transition temperature Tmfc is quantied by:

Dcp cp(T Tmfc) cn(T Tmfc) cse(T Tcmf). These jumps are

identied as the usual ngerprint of the superconducting second-order phase transition.

The temperature of the jumps Tmfc, (dened by the midpoint of the jumps) is close to the superconducting transition temperature of bulk Pb Tbulkc 7.2 K. However, Tmfc does not mark the onset

of macroscopic superconductivity since the thinnest lms remain resistive below it (Fig. 2a). It rather reects the onset of local, intra-grain Cooper pairing. Interestingly, while the jump becomes increasingly broadened for thinner layers, Tmfc[t] changes very little with thickness t (Fig. 2d). This is true in the whole regime between the thickest layer, which is a high-quality super-conductor, and the thinnest layer, which is insulating at low temperatures. In fact Tmfc has similar behaviour to the tunnelling gap 2Dtunn, which was found to remain unchanged through the SIT31. The fact that it does not decrease towards the insulating state means that the SIT is driven by inter-grain phase uctuations. A model for such a bosonic SIT transition is provided by a disordered network of weakly Josephson-coupled, low-capacitance superconducting grains44.

Excess specic heat below Tmfc. The key observation of Fig. 2c is that the electronic specic heat in the superconducting state increases for thinner lms. This increase is throughout the temperature range 3oToTmfc. The specic heat jump Dcp also increases as t decreases towards the SIT as seen in Fig. 2c. Since superconducting grains are involved, one may naively expect that the anomaly at T Tc would be spread over a broader

temperature range as the grains become smaller as calculated by Muhlschlegel45,46, and hence reduced in amplitude. Indeed, superconductivity may be suppressed in very small grains due to energy level splitting being of the order of the superconducting gap and the specic heat anomaly is expected to be less pronounced. Here the opposite is observed: Dcp is more pronounced as the lm is made thinner and pushed towards the QCP as illustrated in Fig. 3. We note that demonstrating a decrease of specic heat as the sample crosses the QCP is extremely difcult for two main reasons: rst, pushing the sample into the insulating regime requires increasingly thinner lms. This results in the signal-to-noise ratio becoming less and less favourable. Second, we cannot enter too deeply into the insulating regime because this

would require measurements of a very-low-mass sample beyond the sensitivity of our set-up (which is state-of-the-art for specic heat experiments).

DiscussionThe results presented show that the low-temperature specic heat is enhanced towards the QCP. Here we present a possible scenario for the effect of quantum criticality on the electronic specic in the superconducting state. We recall that for a weakly interacting metal, such as Pb, the normal-state specic heat follows equation (1), where the electronic contribution coefcient is given by:

gn

p2

3 k2Bg EF

2

here g EF

is the single-particle density of states at the Fermi

energy. Hence, for free electrons, gn mk

2BkF

3 2 is proportional to

the electron mass.

a

1 K1 )

C se

[afii9828]*

c p(mJ g

3.75

3.50

3.25

3.00

2.75

2.50

2.25

10 15 20

t (nm)

25 30

b

1.0

[afii9828] 0 (a.u.)

3.3

3.0

2.7

0.8

0.6

0.4

0.2

0.5 0.6 0.7 0.8 0.9 1.0

2.4

5 10 20

15 25 30

t (nm)

1.51.0

[afii9825][notdef]1.5

0.05 10 15 20 25 30

t (nm)

T Tc

Figure 3 | Excess specic heat. (a) The specic heat jump, Dcp, versus the thickness of the layer. The hatched region marks the position of the QCP.

The error bars are estimated from the noise of the heat capacitance measurements which becomes larger the thinner the lm. (b) The superconducting electronic specic heat, cse for the layers of Fig. 2c scaled according to equations (6 and 7). The colour code is similar to that of

Fig. 2c. The insets show that both a(t), (the power of equation (7)) and gn increase towards the QCP. The dashed black line is the BCS prediction of equation (3).

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Excess specic heat is not expected near an ordinary disorder driven metal-to-insulator transition via Anderson localization. In such a transition, the linear coefcient persists into the insulator phase as measured in, for example, silicon-doped phosphorous system47. Therefore the enhancement observed here at low temperatures, ToTmfc, indicates that it is related to pairing and superconductivity.

The BCS prediction for the electronic specic heat in the superconducting state is given by:48

csT 10gnTc exp 1:76

Tc

T

3

hence, it depends on the same gn as that of the normal state. In addition, the BCS specic heat jump also scales with gn since48,

Dcpcn 1:43 4

Since, as we demonstrated experimentally, Tmfc remains constant through the SIT, we interpret the cs enhancement as a signature of the renormalization of the electron mass appearing in the coefcient gn. This is described through the presence of a self-energy in the Green function formalism, which is an interaction driven effect in the vicinity of the QCP. Indeed close to a QCP, following quantum eld theory, the electron effective mass is renormalized by the self-energy S by49,50:

g gn

1 @Ek EkF; o

1 @o EkF; o

5

where S depends on the many-body interactions. These interactions can be with phonons or with other electrons. However, the main contribution to the self-energy arises from the interaction of the fermions with low-energy collective super-conducting quantum uctuations. These quantum uctuations can be either gapless phase-density uctuations (called Gold-stone/plasmon modes) or amplitude uctuations (called Higgs modes), both lead to an infrared singularity in the self-energy51. This can be phenomenologically modelled by a divergent g*(T) by replacing equation (3) by:

csT; t 10Tmfcg T; texp 1:76

Tmfc

T

6

The specic heat curves depicted in Fig. 2c imply that such an interpretation would require a g*(T) that would signicantly increase at low temperatures. We note that a divergence of linear specic heat coefcient gn has been widely observed at the

QCP separating a magnetic and paramagnetic Fermi liquids2 in good agreement with what is currently observed in granular superconducting Pb. In those systems, the temperature dependence of gn was predicted to be2: gnT g0 log1=T. In our

limited temperature interval, we nd that a better t is given by assuming a power-law dependence of g* on temperature (Fig. 3):

g g0 Tmfc=T a;

4pf dTa:c: 8

Pa.c. being the joule heating power dissipated in the heater, allowing highly sensitive

1.5

1st depositon

18th depositon

1.0

C 0.5

7

where both g0 and a increase as the thickness t decreases towards the QCP. The appropriateness of such scaling is illustrated in Fig. 3, where all curves can be collapsed by the same equation 7.

Finally, the bosonic degrees of freedom due to order-parameter uctuations of phase and amplitude should also contribute to the specic heat. Indeed, such specic heat, cboson, was computed for the 2D O(2) relativistic GinzburgLandau theory, by the nonperturbative renormalization group4. The results are shown in the Methods section. cboson/T2 exhibits a peak that reects a relative enhancement of order 2 at the QCP. This peak is associated with the excess of low-temperature entropy of the softening amplitude uctuations (Higgs mode). As for the granular superconducting

lms, the overall magnitude of the bosonic specic heat is controlled by the density of grains so that cboson n

grainskB, which is at least two orders of magnitude smaller than the measured excess specic heat scale. Hence, the effects of the bosonic collective modes close to the QCP are measurable only indirectly through their effects on the electronic effective mass.

To conclude, we experimentally demonstrated the enhancement of specic heat towards the QCP in a granular super-conductor. This is interpreted as the thermodynamic indication for quantum criticality in the quantum critical regime of the SIT. A possible mechanism for this specic heat enhancement is the increase of the electronic effective mass in the vicinity of the quantum phase transition. The effective mass increase is correlated to the self-energy emerging from the interactions between the fermions and bosonic collective modes that become pronounced close to the quantum critical point of the SIT. From our results, it is not possible to tell whether the specic heat decreases as the sample is pushed deep into the insulating phase. The data become too noisy to draw any denite conclusion since the lm becomes too thin. The direct detection of the collective modes by specic heat require either more sensitive instrumentation or more adapted systems.

Methods

Experimental methods. The samples used in this study are ultra-thin granular lms quench condensed directly on an insulating substrate. In these systems, superconducting layers are sequentially evaporated on a cryogenic substrate under ultra high vacuum (UHV) conditions38. The rst stages of evaporation result in a discontinuous layer of isolated superconducting islands. As material is added the grains become larger and thereby the inter-grain coupling increases.

The experimental set-up is composed of a thin silicon membrane on which the evaporation of Pb is performed. A copper meander to be used as a heater and a niobium nitride strip to be used as a thermometer, both close to two edges of the active sensor, are structured by photolithography and lift-off processes. The sensitive part of the membrane is suspended by 10 silicon arms holding the electrical wiring. This results in a calorimetric cell into which one can supply heating power and measure its temperature while being effectively separated from the heat bath. Transport measurements of the thin Pb lms was enabled by depositing 5 nm titanium and 25 nm gold on two additional leads through a mechanical mask. The quench condensation apparatus consists of a high-vacuum chamber containing tungsten thermal evaporation boat. The membrane is wire-bonded to the sample holder that is mounted in the quench condensation chamber. Quench condensation evaporations are made through a mechanical shadow mask dening a window of 1.3 mm 3.2 mm on the membrane. The

chamber is then immersed in liquid helium, and by pumping on a 1-K pot the system is capable of reaching 1.5 K.

Measurement of the membrane heat capacity is conducted by a.c. calorimetry, in which a current at frequency f is driven through the heater. This oscillates the cell temperature at the second harmonic 2f with amplitude dTa.c.. This amplitude is

picked up by the thermometer, which allows the calculation of heat capacity through:

Cp

Pa:c:

2 )

p/T(nJ K

0.010 20 30 40 50 T 2 (K2)

Figure 4 | Heat capacity of the granular Pb versus T2. The linear cubic t in temperature is used to extract the electronic contribution to the specic heat in the superconducting state cse without the phononic contribution.

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Table 1 | Experimental data extracted from the heat capacity measurement of the 18 evaporations (refereed to by sample no).

Deposition no. Mass (lg) Cp (nJ K 1) b (lJ K 4) t (nm) DCp (nJ K 1) Dcp (mJ g 1K 1) Rsq (X) Tc (K) 1 0.42 16 97 8.9 1.44 3.5 NA 7.02 2 0.44 17 96 9.4 1.39 3.1 6.6 108 7.02

3 0.46 18 98 9.7 1.41 3.05 7.8 107 7.08

4 0.47 18.3 98.6 10 1.62 3.45 1.5 107 7.2

5 0.48 18.8 98.5 10.2 1.60 3.3 1.5 106 7.2

6 0.49 19.2 98.5 10.4 1.68 3.4 0.5 106 7.13

7 0.497 19.3 97.4 10.5 1.46 2.95 1.5 105 7.12

8 0.498 19.4 99 10.6 1.58 3.2 4.8 104 7.15

9 0.514 19.9 97 10.85 1.35 2.6 2.4 104 7.12

10 0.517 20 97 10.95 1.35 2.6 1.1 104 7.1

11 0.518 20.1 98 11 1.76 3.4 7,160 7.2 12 0.526 20.4 97.7 11.1 1.69 3.2 3,080 7.2 13 0.54 21 98 11.4 1.67 3.1 1,380 7.02 14 0.56 22.4 99.5 11.9 1.71 3.05 647 7.17 15 0.6 23 98 12.7 1.65 2.75 330 7.08 16 0.67 26 98.5 14.1 1.85 2.75 168 7.2 17 0.8 31 99.5 17 2.15 2.7 75 7.12 18 1.37 53 100 29 3.25 2.4 27 7.18

For each evaporation of Pb, we give the mass, the heat capacity C at 7.5 K, the b used in the equation C (T) b T

3 for the t, the thickness t, the heat capacity jump DC at T , the specic heat jump Dc at

T , the resistance per square R and the T extracted from the heat capacity measurements.

heat capacity measurement41. To extract only the heat capacity of the evaporated metallic layer, the heat capacity of the raw membrane (containing the heater and the thermometer) is measured from 1.5 to 10 K. This background is subtracted from all the measurements we report in this letter. Further details on the experimental system can be found elsewhere42.

Specic heat components. The specic heat (the heat capacity normalized to the mass of the sample) of the granular lm has two contributions

cnTgnT bT3; 9 one linear in temperature that comes from the electronic part of the degrees of freedom, and the second part cubic in temperature, coming from the phonons of the lattice. In our samples, as an experimental fact, the phonon contribution is far bigger than its electronic counterpart: gnToobiT3; this is illustrated in Fig. 4 by the dominating cubic variation of the heat capacity versus temperature. Such behaviour has been already observed in many granular systems such as granular Al or granular Al-Ge. The large phonon contribution to specic heat may come from a lower Debye temperature in the thin lm than in the bulk, additional degrees of freedom from surface phonons (soft surface modes) or amorphous structure of the quench-condensed Pb grains52,53.

To extract information on the electronic specic heat in the superconducting state, one has to remove the contribution of phonons from the overall specic heat. In regular superconductors (crystals or thick lms), the electronic specic heat dominates the total signal. Traditionally, to single out the superconducting electronic contribution to the specic heat, one can apply a magnetic eld larger than the critical magnetic eld, thus suppressing the superconducting state. In this limit, only the normal-state electrons contribute to the specic heat; the electronic contribution in the superconducting state cse is then obtained by subtracting the normal-state specic heat cn from the superconducting specic heat cs: that is,cse cs cn. Here this protocol cannot be applied since the expected critical

magnetic eld for small superconducting grains is far larger than the eld that can be applied in our experiment (Bc4410 T)54. If the critical magnetic eld happens to be too high, there is a second traditional way to extract the electronic contribution, that is, by tting the specic heat in the normal state above Tc using equation (9), and then extrapolating to temperatures lower than Tc to nd gn. Again, this cannot be applied in our case since the electronic contribution in the normal state is overwhelmed by the phononic contribution. The consequence of this is twofold:(i) one cannot have access to gn from the normal-state specic heat and (ii) one can only t the normal state cn by a cubic power law.

Hence, to extract the signicant information contained in the superconducting specic heat in granular materials, the following protocol has been used: the subtraction of the phononic contribution to the specic heat in the normal state is done by tting the part of the specic heat curves above Tc only by a cubic term with temperature such as ci(T) biT3, i being the ith evaporation. All the relevant

numerical data extracted from the heat capacity measurements are gathered in the Table 1.

Analysis of the bosonic contribution. In this section, we present the analysis of the contribution to specic heat of quantum uctuations within the bosonic model. Since the SIT is bosonic in nature, its universal properties can be described by a 2D

effective bosonic theory. Standard scaling arguments imply that the singular part of the specic heat reads:

cs kB

kBT

vc

1.0

0.5

3 K3.5 K4 K G

2 (mJ gK3 )

c s/T

0.0

0 2 4 6 8 10 12

[afii9829]/T (1/K)

Figure 5 | Scaling of the electronic specic heat. The specic heat cs normalized to T2 is scaled as a function of the characteristic energy scale D related to disorder normalized to temperature. The brown curve corresponds to the best adjustment obtained from the bosonic model developed by Ranon et al.3 (see text).

2G

D kBT

; 10

where G is a dimensionless scaling function. vc is the velocity of critical uctuations at the QCP and D denotes a characteristic energy scale that vanishes at the QCP: Dp|d dc|nz with d the non-thermal parameter that controls the transition

(here the inverse of the lm thickness), and n and z being the correlation length and the dynamical critical exponents, respectively. D corresponds to the excitation gap in the insulating phase and is dened by the superuid stiffness in the super-conducting phase.

The universal scaling function F dening the pressure P(T) P(0) (kBT)3/

(vc)2F(D/kBT) near the QCP has recently been calculated in the framework of the

relativistic quantum O(2) model (quantum j4 theory for a complex eld j)3. The

scaling function G(x) in equation (10) is simply G 6F 4xF0 x2F00 (Fig. 5).

A striking observation is that both the entropy and the specic heat are non-monotonic when d is varied at xed temperature, with a pronounced maximum near the QCP d dc. Although this result was obtained for a clean system, we

expect it to hold also for a disordered system similar to the calculations of the collective amplitude modes in this region55. In any case, this scaling cannot explain the full physics of the data since at least two orders of magnitude differs between the expected specic heat variation in the bosonic picture and what has been observed on the granular superconducting lms.

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Data availability. The data that support the ndings of this study are available from the corresponding author on reasonable request.

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Acknowledgements

We are grateful for useful discussions with M. Molina Ruiz, P. Gandit, N. Trivedi,E. Shimshoni and A. Kapitulnik, and for technical support from E. Andr, P. Brosse-Marron and A. Grardin. We acknowledge support from the Laboratoire dexcellence LANEF in Grenoble (ANR-10-LABX-51-01). A.F. acknowledges support from theEU project MicroKelvin, and by the US-Israel Binational Science Foundation grant no. 2014325. A.A. acknowledges support from the US-Israel Binational Science Foundation grant number 2012233 and from the Israel Science Foundation, grant number 1111/16.

Author contributions

S.P., T.N.-D, A.F. and O.B. carried out the experiments. A.F and O.B. initiated and supervised the research. A.A. and N.D. carried out the theoretical analysis. All the authors discussed the results and wrote the manuscript.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Poran, S. et al. Quantum criticality at the superconductor-insulator transition revealed by specic heat measurements. Nat. Commun. 8, 14464 doi: 10.1038/ncomms14464 (2017).

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