# Single and two-particle energy gaps across the disorder-driven superconductor-insulator transition

###### Abstract

The competition between superconductivity and localization raises profound questions in condensed matter physics. In spite of decades of research, the mechanism of the superconductor-insulator transition (SIT) and the nature of the insulator are not understood. We use quantum Monte Carlo simulations that treat, on an equal footing, inhomogeneous amplitude variations and phase fluctuations, a major advance over previous theories. We gain new microscopic insights and make testable predictions for local spectroscopic probes. The energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT, despite a robust single-particle gap.

###### pacs:

74.40.Kb,74.62.En,74.78.-w,74.81.-gAttractive interactions between electrons lead to superconductivity, a spectacular example of long range order in physics, while disorder leads to localization of electronic states. One of the most fascinating examples of the interplay between the effects of interactions and localization is the destruction of superconductivity in thin films with increasing disorder and the resulting superconductor-to-insulator transition (SIT) goldman-markovic_physicstoday1998 (); gantmakher2010 (); haviland1989 (); hebard1990 (); shahar1992 (); adams2004 (); steiner2005 (); valles_nanohc_science2007 (); sachdev_qpt ().

It was recognized several decades ago that -wave superconductivity (SC) is remarkably robust against weak disorder anderson1959 (); abrikosovgorkov1959 (). It was later argued ma1985 () that SC can survive even when disorder localizes the single-particle states. Thus the superconductor-to-insulator transition must occur in a strong disorder regime that is difficult to treat theoretically in an interacting system. Critical phenomena at the SIT have been described in terms of disordered bosons fisher-grinstein-girvin_prl1990 (), which model fermion pairs and describe phase fluctuations of the SC order parameter. A more microscopic description must necessarily start with the fermionic degrees of freedom. A Bogoliubov-de Gennes (BdG) treatment of attractive electrons in a random potential shows that the SC pairing amplitude becomes spatially inhomogeneous with strong disorder ghosal1998 (); ghosal2001 (); feigelman_prl2007 (). This leads to a robust energy gap and a large suppression of the superfluid density ghosal1998 (); ghosal2001 (). However, the phase fluctuations ultimately responsible for the SIT are beyond the BdG approach and are treated in an approximate manner ghosal1998 (); ghosal2001 (); dubi_nature2007 ().

In this paper we make a major advance using quantum Monte Carlo (QMC) simulations on a fermionic model, which include thermal and quantum fluctuations of the SC phase and the spatially inhomogeneous amplitude on an equal footing. While confirming the bosonic mechanism for the SIT, our work also gives new insights into the experimentally observable electronic spectral functions. Our results provide us with a detailed description of the phases, the transition, and the quantum critical region at finite temperature.

Our main results are as follows:

(1) Single-particle gap:
At the gap in the single-particle density of states (DOS) survives through the SIT,
so that one goes from a gapped superconductor to a gapped insulator.
Although the local gap extracted from the local density of states (LDOS)
is highly inhomogeneous, it is nevertheless finite at every site.

(2) Coherence peaks:
These characteristic pile-ups in the DOS at the gap edges
are directly correlated with superconducting order and vanish
as the temperature is raised above ,
or as the disorder is increased across the SIT.

(3) Pseudogap:
Near the SIT, a pseudogap – a suppression in the low-energy DOS –
persists well above the superconducting up
to a crossover temperature scale , in marked deviation from BCS theory.
This disorder-driven pseudogap also exists at finite temperatures in the insulating state
and grows with disorder.

(4) Two-particle gap:
There is a characteristic energy scale to insert a pair in the insulator
that collapses upon approaching the SIT from the insulating side.
In addition the two-particle spectral function may also have
very small spectral weight at low energies coming from
rare regions.

Our predictions for the local tunneling density of states and the dynamical pair susceptibility as a function of temperature and disorder have the potential to guide future experiments using scanning tunneling spectroscopy (STS) sacepe2008 (); cren2000 (); lang2002 (); yazdani07 () and other dynamical probes crane_fluctuations_2007 ().

Model and methods: To model the competition between superconductivity and localization that leads to the SIT in quench-condensed films with thicknesses less than the coherence length, we take the simplest lattice Hamiltonian that has an -wave superconducting ground state in the absence of disorder () and exhibits Anderson localization when the attractive interaction is turned off (). Thus, we study the two-dimensional attractive Hubbard model in a random potential:

(1) |

with lattice sites and , spin indices or , fermion creation and annihilation operators and , number operators , hopping between neighboring sites , and a chemical potential chosen such that the average density is . is a random potential at each site drawn from the uniform distribution on , and is the on-site attraction leading to -wave SC. We will measure all energies in units of .

We use the determinantal QMC method blankenbecler (), which is free of the fermion sign problem for the Hamiltonian (1). We choose , so that the coherence length is within the system size, and . We have made extensive comparisons of the QMC results with self-consistent BdG calculations, which take into account only the spatial amplitude variations; see supplementary material. These comparisons permit us to separate the effects of amplitude inhomogeneity and phase fluctuations.

We compute frequency-dependent observables across the SIT for the first time. The single-particle DOS, LDOS and the pair susceptibility are obtained using the maximum entropy method (MEM) for analytic continuation jarrell (); sandvik1998 (). We have verified that these results obey various sum rules to high precision, and that the MEM correctly reproduces the low-energy structure of test spectra as shown in the supplementary material. What gives us confidence is that our central results on the single- and two-particle gaps can be equally well estimated directly from the exponential decay of the imaginary-time QMC data, without recourse to MEM.

Phase diagram: In Fig. 1 we summarize our key results for the disorder dependence of various temperature and energy scales. Since the finite temperature transition is expected to be in the Berezinskii-Kosterlitz-Thouless universality class, we estimate the critical temperature from the superfluid density , calculated from the transverse current correlator scalapinowhitezhang1993 (); trivedi1996 (). We note that this procedure on finite systems provide an upper bound on the actual in the thermodynamic limit. As disorder strength increases, falls and finally vanishes at the critical disorder , which defines the SIT. The single-particle energy gap remains non-zero across the SIT, whereas the two-particle energy scale is finite in the insulator and goes to zero at the transition. These gap scales are extracted from the DOS and the dynamical pair susceptibility discussed in detail below. Figure 1 can be interpreted as a phase diagram: is the superconducting transition temperature, is a crossover scale between the insulator and the quantum critical region, and is related to the pseudogap crossover scale described below.

Single-particle spectra: We show in Fig. 2 the disorder and temperature dependence of the DOS . Panels (A,B) show the evolution with disorder at a very low temperature . The gap clearly remains finite in both superconducting and insulating states, a counterintuitive observation that agrees qualitatively with BdG results ghosal1998 (); ghosal2001 (). In contrast, the coherence peaks diminish with increasing and disappear near the SIT at .

Figures 2(C,D) show the temperature evolution of at weak disorder . Unlike in BCS theory, the hard SC gap does not close with increasing . Instead, the coherence peaks gradually disappear as the temperature increases across . Above , the gap gradually fills up, with a pseudogap persisting well above .

The temperature evolution of at strong disorder is shown in Fig. 2(E,F). Here the ground state is an insulator with a hard gap and little evidence for coherence peaks, and the pseudogap persists up to an even higher temperature.

Two-particle spectra: Given that we find an insulator with a single-particle gap, what is the energy scale that vanishes upon approaching the quantum critical point from the insulating side? We propose that it is the typical energy for a two-particle excitation in the insulator. To access this scale, we examine the pair susceptibility obtained by analytical continuation of the correlation function where . Thus is the amplitude for a pair created at a site at to be found at the same site at a later time . We find that in the insulating phase decays exponentially, which allows us to define , the characteristic energy scale for two-particle excitations.

In Fig. 3 we show the imaginary part of the pair susceptibility for three disorder strengths. At weak disorder is very large at low , whereas at strong disorder it has a clear two-peak structure with a characteristic energy scale . This dominant scale represents the typical energy required to insert a pair into the system. We find that collapses to zero at the SIT because there is no cost for inserting a pair into a condensate.

At sufficiently small energies our insulating state is similar to a Bose glass, in which rare regions fisher1989 () give rise to a very small but non-zero spectral weight in at low energies. Such Griffiths-McCoy-Wu singularities can be very difficult to pin down in numerical simulations and even in experiments. Nevertheless, we do indeed see some signs of low-energy spectral weight in, e.g., Fig. 3B. In this paper, however, we focus on the most salient features in . These are the peaks at , which imply that the typical energy cost to insert a pair is finite.

Local probes: In Fig. 4 we track the behavior of various local quantities with increasing disorder strength . We show the LDOS at representative points, maps of the spatial variation of the density , and the BdG pairing amplitude (which cannot be computed in QMC). We see that the system becomes increasingly inhomogeneous with increasing disorder, as we move from left to right in Fig. 4. The SIT occurs due to loss of phase coherence between superconducting islands, seen as blue patches in the map of the pairing amplitude .

We predict experimentally measurable signatures of the local density and pairing amplitude in the LDOS . Let us focus on three representative sites , , and . At moderate and strong disorder, is located on a potential hill, with a low density and a negligible pairing amplitude . Thus the LDOS at is highly asymmetric, with most of the spectral weight at , for adding an electron. In contrast, is in a potential well, with a high density and a negligible pairing amplitude . Thus also has a highly asymmetric LDOS, but most of the spectral weight is at , for removing an electron. We believe that MEM correctly captures the gap, coherence peaks, and integrated spectral asymmetry (tested by sum rules); it is much less reliable for high-energy spectral features, which are in any case irrelevant for our purposes.

Finally, lies in a superconducting island close to half-filling, , which permits particle-hole mixing, and therefore a large pairing amplitude . The LDOS at is much more symmetrical, with large coherence peaks that persist across the SIT and even in the insulating state. Note that all the LDOS curves have a clear gap. We thus find that symmetrical coherence peaks in the LDOS, and not the local energy gap, are a clear experimental signature of a local pairing amplitude, which is difficult to probe by other means.

Discussion: We now discuss our results in light of existing theories. We have ignored the renormalization of the effective interaction between electrons arising from changes in screening with increasing disorder finkelstein1994 (). Our point of view is that electronic inhomogeneity (that we focus on) is much more important in the vicinity of the SIT than the disorder dependence of the effective (that we neglect), so long as the latter is not driven to zero. This assumption is validated by experiments that find a non-zero gap across the SIT sacepe2008 ().

Our results are consistent with the absence of a fermionic or bosonic metal phase in between the superconductor and the insulator. Although we have not computed transport here (see ref. trivedi1996 () for an approximate calculation of the resistivity in the same model), we do not find any extended low-energy excitations characteristic of a metallic phase.

The existence of gapped fermions implies a phase-fluctuation-dominated “bosonic” picture for the superconductor-insulator transition fisher-grinstein-girvin_prl1990 (); fisher1989 (). However, we must emphasize that we did not assume such a bosonic picture from the outset. A nontrivial aspect of our results is that even though we started with a model of interacting fermions in a random potential and could have, in principle, obtained (localized) gapless fermions in the insulator, we did not find such excitations. The reason all fermionic excitations are gapped is intimately related to the structure of the inhomogeneous local pairing amplitude generated in the presence of large disorder, as we now explain.

We show in Fig. 5 that even for “homogeneous” disorder, i.e., an uncorrelated random potential (see panel A), the pairing amplitude exhibits an emergent “granular” structure (shown in panel B). The system self-organizes into superconducting islands, on the scale of the coherence length, with finite , interspersed with insulating regions where is negligible. The spatial variations of spectral features (asymmetry and coherence peaks) in this inhomogeneous state were already discussed above in connection with Fig. 4.

The close connection between inhomogeneity and energy gaps is made clear in Figs. 5 B and C which demonstrate two striking facts. We see that (i) there is an energy gap in the LDOS at every site, and (ii) small gaps in the LDOS are spatially correlated with large SC islands.

A simple way to understand these results is to use the pairing-of-exact-eigenstates approach generalized to highly disordered systems ghosal2001 (). In the limit of weak attraction, pairing gaps out the low-energy density of states in the underlying Anderson insulator and leads to the islands with non-zero and a small energy gap. On the other hand, the insulating sea corresponds to the higher-energy strongly localized states in the system.

From this perspective one can see that the gap , observed in the spatially average DOS, initially decreases with increasing disorder due to a reduction in the density of states near the chemical potential in our model. (In a real material, the coupling will also decrease finkelstein1994 () with disorder). However, at high disorder, the gap grows (consistent with Fig. 1) like where is the single-particle localization length ghosal2001 (). This is due to the enhanced effective attraction between fermions confined to a smaller localization volume .

The phase stiffness (or superfluid density) , on the other hand, decreases monotonically with disorder as the SC islands become smaller and the Josephson coupling between islands becomes weaker. Thus, even if one starts with a weak-coupling BCS superconductor with , disorder will necessarily drive it into the regime. Eventually quantum phase fluctuations destroy long-range order at , leading to an insulator whose low-energy excitations are pairs localized on SC islands.

The low- regime on the SC side of the SIT leads to a finite-temperature transition driven by thermal phase fluctuations emery1995 () with . The large energy gap then leads to a marked deviation from conventional BCS theory, with a pairing pseudogap in the the temperature range . This pseudogap exists even in the weak-coupling regime, provided one is close enough to the SIT so that .

Comparison with experiments: We describe the connection between our predictions and experiments on the disorder-tuned SIT in systems such as indium oxide, titanium nitride, and niobium nitride films, for which our theory seems to be the most appropriate. First let us discuss the insulating side of the SIT. The existence of a gap in the insulator implies activated transport, consistent with early measurements on amorphous InO films shahar1992 (). In addition, there is evidence for pairs on the insulating side of the transition valles_nanohc_science2007 () in specially patterned amorphous bismuth films.

Recent STM experiments are directly relevant to our predictions on the superconducting side of the SIT. Experiments on homogeneously disordered TiN films sacepe2008 () have shown that, while goes to zero at the SIT, the STM gap remains finite, in agreement with Fig. 1. In addition, the gap in the LDOS shows significant inhomogeneity, which supports our picture of emergent granularity (see Figs. 4 and 5). After our paper was written, we became aware of new experiments that corroborate our predictions. STM experiments on InO sacepe2011 (), TiN sacepe_natcomm (), and NbN films mondal2011 () have all found a pseudogap persisting up to many times . In particular, they observe a marked suppression of the low-energy DOS together with a destruction of coherence peaks above , in complete agreement with our predictions.

We hope that future STM experiments will study in detail the anticorrelation that we predict between the height of the coherence peaks (associated with large pairing amplitude) and the small energy gaps in the local DOS. The obvious quantum critical scaling between and at the SIT, well studied in rather different systems hetel2007 (), also remains to be tested experimentally in -wave superconducting films.

Conclusion: In conclusion, we have obtained detailed insights and predictions for observable properties of the highly disordered superconducting and insulating states in 2D films, and of the transition between these states. Although we focused on -wave SC films, it has not escaped our attention that aspects of our results bear a striking resemblance to the completely different – and much less understood – problem of the pseudogap in the -wave high superconductors. Features like the loss of low-energy spectral weight persisting across thermal or quantum phase transitions, even as coherence peaks are destroyed, may well be common to all systems where the small superfluid stiffness drives the loss of phase coherence. The pseudogap in underdoped cuprates is driven by the proximity to the Mott insulator and further complicated by competing order parameters, with disorder probably playing a secondary role, unlike the disorder-induced pseudogap near the SIT discussed in this paper.

Correspondence and request for materials shoud be addressed to: N. Trivedi
(trivedi.15@osu.edu).

We gratefully acknowledge support from
NSF DMR-0907275 (KB),
US Department of Energy, Office of Basic Energy Sciences grant DOE DE-FG02-07ER46423 (NT,YLL),
NSF DMR-0706203 and NSF DMR-1006532 (MR),
and computational support from the Ohio Supercomputing Center.
KB and YLL performed the numerical calculations;
MR and NT were responsible for the project planning;
KB, YLL, MR and NT contributed to the data analysis, discussions and writing.

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