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. Lowenergy 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 twodimensional 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 lowtemperature specific heat increase significantly. This behaviour is taken as the thermodynamic fingerprint of quantum criticality in the vicinity of a quantum phase transition.
Introduction
Quantum criticality is a central paradigm in physics. It unifies the description of diverse systems in the vicinity of a secondorder, zerotemperature quantum phase transition, 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 meanfield and Fermiliquid theories fail in lack of welldefined quasiparticles. QCPs inspired innovative nonperturbative approaches^{1,2,3,4}, including those relevant to field theory/gravity duality^{5}.
In 2D superconducting films, the zerotemperature superconductor–insulator transition (SIT) has been viewed as a prototype of a quantum phase transition that is controlled by a nonthermal tuning parameter g^{6}. Experimentally, the transition has been driven utilizing various g such as inverse thickness^{7,8,9,10,11,12,13,14,15,16}, magnetic field^{12,13,17,18,19,20,21,22,23,24,25,26}, disorder^{25,27,28} chemical composition^{29} and gate voltage^{30}. For g<g_{c} the film is a superconductor with welldefined quasiparticles and superconducting collective modes as well as a finite 2D superfluid density. As g is increased, the system enters the critical regime in which excitations are strongly correlated, while the superfluid density vanishes as g→g_{c}. For g>g_{c}, the system becomes insulating with gapped charge excitations.
2D superconducting granular films have been shown to exhibit signs for Cooperpairing effects such as the presence of an energy gap, Δ well into the insulator phase^{31}, and it has been argued that they may be modelled by a bosonic quantum field theory with O(2) symmetry^{32}. Similar findings were found for disordered thin films^{27,29,33} in which disorder is assumed to generate emergent granularity^{34}. 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 SIT^{35,36}.
One of the salient thermodynamic signatures of a QCP is the presence of excess entropy or specific heat^{1,2}. In heavy electron metals, this has been observed as the divergence of linear specific heat coefficients, or electronic effective mass, which signals a deviation from conventional Fermiliquid theory behaviour^{37}. Such a signature would be very important to measure in the SIT system, to probe the critical thermodynamics near this nonmeanfieldtype QCP.
Entropy S(T) is a fundamental physical quantity of significant importance for the quantum phase transition; however, its absolute value cannot be directly measured. On the other hand, the specific heat c_{p}(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 fluctuations. Despite its crucial interest, specific 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 ultrathin films involving ultralow mass. The substrate mass onto which thin films are deposited is usually much larger than that of the film itself rendering ultrathin film specific heat unmeasurable.
Here we report on the first experimental demonstration of excess specific heat in 2D granular films spanning the superconductor–insulator quantum phase transition. We employ a unique technique based on thin suspended silicon membrane used as a thermal sensor that enables ultrasensitive measurements of the specific heat of superconducting films close to the SIT. We find that the meanfield critical temperature, , remains basically unchanged through the SIT. Nevertheless, the specific heat jump at and the specific 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.
Results
Specific heat experimental setup
The samples used in this study were ultrathin granular films 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 ultrahighvacuum conditions^{7,38,39,40}. The first stages of evaporation result in a discontinuous layer of isolated superconducting islands. As material is added, the intergrain coupling increases and the system undergoes a transition from an insulator to a superconductor.
The measurement setup 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 ultrathin samples with masses down to few tens of nanograms. This setup provides a unique opportunity to measure simultaneously the film resistance, R, and heat capacity, C_{p}, of a single film 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 ultrathin films. Further experimental details are specified elsewhere^{42}, see also Methods.
Specific heat and resistance through the SIT
Panels a and b of Fig. 2 show concomitant R(T) and C_{p}(T) measurements performed on a series of 18 consecutive depositions of Pb on a single nanocalorimetric cell. The thinnest layer is an insulator with R>>1 GΩ, making it unmeasurable within the sensitivity of our setup, 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 films^{11,31}, where the first six depositions are on the insulating side of the SIT, the evaporations 7 and 8 show resistance reentrance behaviour, 9–16 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 properties^{11,31,39,43}. However, for the thinnest layers, phase fluctuations 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 orderparameter fluctuations 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 specific heat c_{p} (the heat capacity c_{p} normalized to the mass). Above T_{c}, the normal specific heat, c_{n}, should follow the wellknown form:
with γ_{n} and β proportional to the electron and phonon specific heat (heat capacity divided by the mass of the Pb film), respectively. In our case, the phonon contributions to c_{p} 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 specific heat above T_{c} can be fit using a simpler relation than equation (1), that is, =βT^{2} since γ_{n}<<βT^{2}.
In this study, we focus on c_{se}, the specific heat of electrons in the superconducting state compared with the one in the normal state, as defined by the following relation: c_{se}(T)=c_{p}(T)−c_{n}(T). To do this, we subtract from each of the measured c_{p}(T) of Fig. 2b the specific heat extrapolated from the normalstate regime c_{n}(T)=βT^{3} above T_{c}. The superconducting electronic specific heat c_{se} (J g^{−1} K^{−1}) for a number of selected layers are shown in Fig. 2c. The remainder of this article will focus on c_{se} of the films close to the quantum phase transition.
Two prominent facts are clearly observed in the specific heat data of Fig. 2c. The first is the position of the specific heat jumps Δc_{p} and the second is the amplitude of the specific heat below the critical temperature.
Specific heat jumps Δc _{p}
The amplitude of these specific heat jumps at the transition temperature is quantified by: Δc_{p}=c_{p}(T=)−c_{n}(T=)=c_{se}(T=T_{c}^{mf}). These jumps are identified as the usual fingerprint of the superconducting secondorder phase transition.
The temperature of the jumps , (defined by the midpoint of the jumps) is close to the superconducting transition temperature of bulk Pb =7.2 K. However, does not mark the onset of macroscopic superconductivity since the thinnest films remain resistive below it (Fig. 2a). It rather reflects the onset of local, intragrain Cooper pairing. Interestingly, while the jump becomes increasingly broadened for thinner layers, [t] changes very little with thickness t (Fig. 2d). This is true in the whole regime between the thickest layer, which is a highquality superconductor, and the thinnest layer, which is insulating at low temperatures. In fact has similar behaviour to the tunnelling gap 2Δ^{tunn}, which was found to remain unchanged through the SIT^{31}. The fact that it does not decrease towards the insulating state means that the SIT is driven by intergrain phase fluctuations. A model for such a bosonic SIT transition is provided by a disordered network of weakly Josephsoncoupled, lowcapacitance superconducting grains^{44}.
Excess specific heat below
The key observation of Fig. 2c is that the electronic specific heat in the superconducting state increases for thinner films. This increase is throughout the temperature range 3<T<. The specific heat jump Δc_{p} 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=T_{c} would be spread over a broader temperature range as the grains become smaller as calculated by Muhlschlegel^{45,46}, 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 specific heat anomaly is expected to be less pronounced. Here the opposite is observed: Δc_{p} is more pronounced as the film is made thinner and pushed towards the QCP as illustrated in Fig. 3. We note that demonstrating a decrease of specific heat as the sample crosses the QCP is extremely difficult for two main reasons: first, pushing the sample into the insulating regime requires increasingly thinner films. This results in the signaltonoise ratio becoming less and less favourable. Second, we cannot enter too deeply into the insulating regime because this would require measurements of a verylowmass sample beyond the sensitivity of our setup (which is stateoftheart for specific heat experiments).
Discussion
The results presented show that the lowtemperature specific heat is enhanced towards the QCP. Here we present a possible scenario for the effect of quantum criticality on the electronic specific in the superconducting state. We recall that for a weakly interacting metal, such as Pb, the normalstate specific heat follows equation (1), where the electronic contribution coefficient is given by:
here g is the singleparticle density of states at the Fermi energy. Hence, for free electrons, γ_{n}=m is proportional to the electron mass.
Excess specific heat is not expected near an ordinary disorder driven metaltoinsulator transition via Anderson localization. In such a transition, the linear coefficient persists into the insulator phase as measured in, for example, silicondoped phosphorous system^{47}. Therefore the enhancement observed here at low temperatures, T<, indicates that it is related to pairing and superconductivity.
The BCS prediction for the electronic specific heat in the superconducting state is given by:^{48}
hence, it depends on the same γ_{n} as that of the normal state. In addition, the BCS specific heat jump also scales with γ_{n} since^{48},
Since, as we demonstrated experimentally, remains constant through the SIT, we interpret the c_{s} enhancement as a signature of the renormalization of the electron mass appearing in the coefficient γ_{n}. This is described through the presence of a selfenergy in the Green function formalism, which is an interaction driven effect in the vicinity of the QCP. Indeed close to a QCP, following quantum field theory, the electron effective mass is renormalized by the selfenergy Σ by^{49,50}:
where Σ depends on the manybody interactions. These interactions can be with phonons or with other electrons. However, the main contribution to the selfenergy arises from the interaction of the fermions with lowenergy collective superconducting quantum fluctuations. These quantum fluctuations can be either gapless phasedensity fluctuations (called Goldstone/plasmon modes) or amplitude fluctuations (called Higgs modes), both lead to an infrared singularity in the selfenergy^{51}. This can be phenomenologically modelled by a divergent γ*(T) by replacing equation (3) by:
The specific heat curves depicted in Fig. 2c imply that such an interpretation would require a γ*(T) that would significantly increase at low temperatures. We note that a divergence of linear specific heat coefficient γ_{n} has been widely observed at the QCP separating a magnetic and paramagnetic Fermi liquids^{2} in good agreement with what is currently observed in granular superconducting Pb. In those systems, the temperature dependence of γ_{n} was predicted to be^{2}: . In our limited temperature interval, we find that a better fit is given by assuming a powerlaw dependence of γ* on temperature (Fig. 3):
where both γ_{0} and α 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 orderparameter fluctuations of phase and amplitude should also contribute to the specific heat. Indeed, such specific heat, c^{boson}, was computed for the 2D O(2) relativistic Ginzburg–Landau theory, by the nonperturbative renormalization group^{4}. The results are shown in the Methods section. c^{boson}/T^{2} exhibits a peak that reflects a relative enhancement of order 2 at the QCP. This peak is associated with the excess of lowtemperature entropy of the softening amplitude fluctuations (Higgs mode). As for the granular superconducting films, the overall magnitude of the bosonic specific heat is controlled by the density of grains so that , which is at least two orders of magnitude smaller than the measured excess specific 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 specific heat towards the QCP in a granular superconductor. This is interpreted as the thermodynamic indication for quantum criticality in the quantum critical regime of the SIT. A possible mechanism for this specific 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 selfenergy 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 specific heat decreases as the sample is pushed deep into the insulating phase. The data become too noisy to draw any definite conclusion since the film becomes too thin. The direct detection of the collective modes by specific heat require either more sensitive instrumentation or more adapted systems.
Methods
Experimental methods
The samples used in this study are ultrathin granular films quench condensed directly on an insulating substrate. In these systems, superconducting layers are sequentially evaporated on a cryogenic substrate under ultra high vacuum (UHV) conditions^{38}. The first stages of evaporation result in a discontinuous layer of isolated superconducting islands. As material is added the grains become larger and thereby the intergrain coupling increases.
The experimental setup 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 liftoff 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 films 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 highvacuum chamber containing tungsten thermal evaporation boat. The membrane is wirebonded to the sample holder that is mounted in the quench condensation chamber. Quench condensation evaporations are made through a mechanical shadow mask defining 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 1K 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 δT_{a.c.}. This amplitude is picked up by the thermometer, which allows the calculation of heat capacity through:
P_{a.c.} being the joule heating power dissipated in the heater, allowing highly sensitive heat capacity measurement^{41}. 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 elsewhere^{42}.
Specific heat components
The specific heat (the heat capacity normalized to the mass of the sample) of the granular film has two contributions
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: γ_{n}T<<β_{i}T^{3}; 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 AlGe. The large phonon contribution to specific heat may come from a lower Debye temperature in the thin film than in the bulk, additional degrees of freedom from surface phonons (soft surface modes) or amorphous structure of the quenchcondensed Pb grains^{52,53}.
To extract information on the electronic specific heat in the superconducting state, one has to remove the contribution of phonons from the overall specific heat. In regular superconductors (crystals or thick films), the electronic specific heat dominates the total signal. Traditionally, to single out the superconducting electronic contribution to the specific heat, one can apply a magnetic field larger than the critical magnetic field, thus suppressing the superconducting state. In this limit, only the normalstate electrons contribute to the specific heat; the electronic contribution in the superconducting state c_{se} is then obtained by subtracting the normalstate specific heat c_{n} from the superconducting specific heat c_{s}: that is, c_{se}=c_{s}−c_{n}. Here this protocol cannot be applied since the expected critical magnetic field for small superconducting grains is far larger than the field that can be applied in our experiment (B_{c}>>10 T)^{54}. If the critical magnetic field happens to be too high, there is a second traditional way to extract the electronic contribution, that is, by fitting the specific heat in the normal state above T_{c} using equation (9), and then extrapolating to temperatures lower than T_{c} to find γ_{n}. 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 γ_{n} from the normalstate specific heat and (ii) one can only fit the normal state c_{n} by a cubic power law.
Hence, to extract the significant information contained in the superconducting specific heat in granular materials, the following protocol has been used: the subtraction of the phononic contribution to the specific heat in the normal state is done by fitting the part of the specific heat curves above T_{c} only by a cubic term with temperature such as c_{i}(T)=β_{i}T^{3}, 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 specific heat of quantum fluctuations 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 specific heat reads:
where G is a dimensionless scaling function. v_{c} is the velocity of critical fluctuations at the QCP and Δ denotes a characteristic energy scale that vanishes at the QCP: Δ∝δ−δ_{c}^{νz} with δ the nonthermal parameter that controls the transition (here the inverse of the film thickness), and ν and z being the correlation length and the dynamical critical exponents, respectively. Δ corresponds to the excitation gap in the insulating phase and is defined by the superfluid stiffness in the superconducting phase.
The universal scaling function defining the pressure P(T)=P(0)+(k_{B}T)^{3}/(ℏv_{c})^{2}(Δ/k_{B}T) near the QCP has recently been calculated in the framework of the relativistic quantum O(2) model (quantum theory for a complex field )^{3}. The scaling function (x) in equation (10) is simply =6−4x+x^{2} (Fig. 5). A striking observation is that both the entropy and the specific heat are nonmonotonic when δ is varied at fixed temperature, with a pronounced maximum near the QCP δ=δ_{c}. 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 region^{55}. In any case, this scaling cannot explain the full physics of the data since at least two orders of magnitude differs between the expected specific heat variation in the bosonic picture and what has been observed on the granular superconducting films.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Additional information
How to cite this article: Poran, S. et al. Quantum criticality at the superconductorinsulator transition revealed by specific heat measurements. Nat. Commun. 8, 14464 doi: 10.1038/ncomms14464 (2017).
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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. André, P. BrosseMarron and A. Gérardin. We acknowledge support from the Laboratoire dexcellence LANEF in Grenoble (ANR10LABX5101). A.F. acknowledges support from the EU project MicroKelvin, and by the USIsrael Binational Science Foundation grant no. 2014325. A.A. acknowledges support from the USIsrael Binational Science Foundation grant number 2012233 and from the Israel Science Foundation, grant number 1111/16.
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Affiliations
Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
 S. Poran
 & A. Frydman
Institut NÉEL, CNRS, 25 avenue des Martyrs, F38042 Grenoble, France
 S. Poran
 , T. NguyenDuc
 , A. Frydman
 & Olivier Bourgeois
Univ. Grenoble Alpes, Inst NEEL, F38042 Grenoble, France
 T. NguyenDuc
 , A. Frydman
 & Olivier Bourgeois
Department of Physics, Technion, 32000 Haifa, Israel
 A. Auerbach
Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, UPMCSorbonne Universités, 4 Place Jussieu, 75252 Paris, France
 A. Auerbach
 & N. Dupuis
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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.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to A. Frydman or Olivier Bourgeois.
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