Microwave properties of superconductors close to SIT

TL;DR

This paper develops a theoretical framework to understand microwave dissipation in strongly disordered superconductors near the superconductor-insulator transition, highlighting the role of collective modes as a source of noise.

Contribution

It introduces a new theory of collective modes in disordered superconductors near SIT and analyzes their impact on microwave dissipation and noise.

Findings

Collective modes become significant sources of dissipation near SIT.

Disorder level influences the spectrum of collective modes.

High kinetic inductance is associated with proximity to SIT.

Abstract

Strongly disordered pseudogapped superconductors are expected to display arbitrary high values of kinetic inductance close to superconductor-insulator transition (SIT) that makes them attractive for the implementation of dissipationless superinductance. We develop the theory of the collective modes in these superconductors and discuss associated dissipation at microwave frequencies. We obtain the collective mode spectra dependence on the disorder level and conclude that collective modes become relevant source of dissipation and noise in the far vicinity of SIT.

Equations

$$H=\sum _{i}2{\xi }_i{s}_i^z-\sum _{(ij)}(J_{ij}{s}_i^{+}{s}_j^{-}+h.c.)$$

$$K_c=g{e}^{1/(eg)};K_1=g{e}^{1/2g};K_2=\frac{g}{4}{e}^{1/g}$$

$$\mathcal{A}=-\sum _{i,j}{b}_i(\omega ){\hat{J}}_{ij}^{-1}{b}_j(\omega )+\sum _i\frac{b_i^2(\omega )\sqrt{{\xi }_i^2+{\Delta }_i^2}}{{\xi }_i^2+{\Delta }_i^2-{\bar{\omega }}^2}$$

$$b_i(\omega )=\sum _j{J}_{ij}{b}_j(\omega ){\eta }_j(\omega )\text{ where}\eta (\omega )\equiv \frac{\sqrt{{\xi }_j^2+{\Delta }^2}}{{\xi }_j^2+{\Delta }^2-{\bar{\omega }}^2}$$

$$e^{f(x)}\equiv K{\int }_0^{1}d\xi {\left[\frac{g}{K}\frac{\sqrt{{\xi }^2+{\Delta }^2}}{{\xi }^2+{\Delta }^2-{\bar{\omega }}^2}\right]}^{2x}=1\text{, }\frac{\partial f}{\partial x}=0$$

$$E^2=ϵ\frac{\delta K}{K_1}-\frac{{ϵ}^2}{24g^2}\text{, }ϵ=12g^2\frac{\delta K}{K_1}$$

$$\frac{{\omega }_1}{2\Delta }\equiv E(K)=\sqrt{6}g\frac{K-K_1}{K_1}$$

$$K_2(\omega )=K_2\frac{\Delta }{\sqrt{{\Delta }^2-{\bar{\omega }}^2}}\approx K_2\left[1+\frac{1}{2}{\left(\frac{\omega }{2\Delta }\right)}^2\right]$$

Full text

A piece of superconductor is characterized by the phase of the order parameter, $\varphi$. Because the order parameter $\Psi=\left|\Psi\right|e^{i\varphi}$, the state of the superconductor does not change $\varphi\rightarrow\varphi+2\pi$. This periodicity is due to the charge quantization for the isolated pieces of superconductors. If it is violated, a plethora of new physical effects becomes possible such as formation of Bloch states in the Josephson potential, phase slip quantization and current Shapiro steps, etc. All these effects require that the phase change by $2\pi$ leads to the state of the same energy but distinguishable from the original one. This can be achieved if the superconductor is connected to the ground by a very large inductor characterized by the energy $E=(1/2)E_{L}\varphi^{2}$ where $\varphi$ is the phase differennce and $E_{L}=e^{2}/\hbar^{2}L\rightarrow 0$. Physically the limit $E_{L}\rightarrow 0$ means that $E_{L}$ is much less than all relevant energy scales, for a typical problem this translates into L\gtrsim 1\text{\muH}. The *superinductor *should be dissipationless, and as such it should contain no low energy modes, in particular it should not form a low frequency resonator. This limits the geometrical size of the superinductor to a few $\mu m$ and therefore $L_{\square}\gtrsim 10\text{ nH}$. The question is if such superinductors are physically possible?

An attractive candidate for superinductors is the superconductor close to the superconductor-insulator transition (quantum critical point). One expects that at the transition the superfluid stiffness $\rho_{S}^{\square}=0$ ($\rho_{S}^{\square}=e^{2}/\hbar^{2}L_{\square}$), so if this transition leads to the insulating state with a large gap, in the vicinity of it the superfluid stiffness can be arbitrary small corresponding to arbitrary large superinductances. Generally, there are two mechanisms for the destruction of the superconductivity by disorder that lead to quantum critical point where $\rho_{S}$ is exactly zero (for recent reviews see Gantmakher and Dolgopolov (2010); Feigel’man et al. (2010a)). The first (fermionic) mechanism attributes the suppression of the superconductivity to the increase of the Coulomb interaction that results in the decrease of the attraction between electrons and their eventual depairing.Finkelstein (1994) In this mechanism the state formed upon the destruction of the superconductor is essentially a poor conductor. This mechanism clearly does not lead to the formation of the superinductance. The alternative (bosonic) mechanism attributes superconductivity suppression to the localization of Cooper pairs that remain intact even when superconductivity is completely suppressed. The theory of the bosonic mechanism has a long history: this scenario of the superconductor-insulator transition was suggested long agoMa and Lee (1985); Kapitulnik and Kotliar (1985); Bulaevskii and Sadovskii (1985); Kotliar and Kapitulnik (1986) but was not developed further until recently Feigei’man et al. (2007); Feigel’man et al. (2010a) when experimental data indicated it might indeed occur in InO. Shahar and Ovadyahu (1992); Sambandamurthy et al. (2004); Ovadia et al. (2009); Sacepe et al. (2011)

In this Letter we show that as the bosonic SIT is approached the collective modes are pushed down to low energies. In BCS theory the critical temperature of the superconductor or its low energy gap does not depend on the disorder. In the simplest model of the bosonic SIT the critical temperature does not depend on the disorder until the latter exceeds some critical value. At larger values of the disorder the transition temperature decreases quickly and eventually becomes zero while single electon gap, $\Delta_{P}$ remains constant.Feigel’man et al. (2010a) It is natural to associate the regime where the transition temperature depends on the disorder with the critical regime of the SIT in the bosonic model. As we show below, the collective modes are pushed to low energies even outside the critical regime. This severely limits the possible values of the kinetic inductances that can be achieved in the strongly disordered superconductors close to SIT.

Before we give the details of the model of the bosonic SIT and its low energy properties we discuss its main physical assumptions and materials in which such physics might be realized. The main physical assumption of the bosonic model is that Coulomb repulsion does not lead to the electron depairing at large disorder. This might occur if it is screened by the electrons far from the Fermi surface. In other words, the Coulomb interaction between superconducting electrons is small due to a large effective dielectric constant of the material. Empirically, in this case one expects that superconductivity occurs against the background of the insulating $R(T)$. This is the situation in InO that displays strong insulating temperature behavior that is followed by superconductivity at very low $T$.Shahar and Ovadyahu (1992); Sambandamurthy et al. (2004); Ovadia et al. (2009) Large dielectric constants, $\kappa\gtrsim 10^{3}$ are expected in superconductors derived from high-$\kappa$ hostMuller and Burkard (1979), $\text{SrTiO}_{3}$, such as \text{SrTiO{}{3}-LaAlO}_{3} interfacesReyren et al. (2007); Caviglia et al. (2008) or Nb-doped SrTi1-xNbxO3Lin et al. (2015). In the material where Coulomb interaction is completely suppressed by large $\kappa$ one expects that $T_{c}$ and $\Delta_{P}$ initially increase with disorder due to the electron wave function localization before the effects of the suppression of Cooper pair tunneling suppress $T_{c}$ and $\rho_{S}$ leading to SIT while the single electron gap $\Delta_{P}$ remains large everywhere. Such unusual behavior (with the maximum of $T_{c}$) was indeed observed in \text{SrTiO{}{3}-LaAlO}_{3} system.Caviglia et al. (2008); Richter et al. (2013); Fillis-Tsirakis et al. (2016) The increase of $T_{c}$ followed by abrupt transition to the insulating state was also observed in $\text{Li}_{x}\text{ZrNCl}$ crystalsTaguchi et al. (2006) as well as in slightly oxidized aluminum wires (aka granular aluminum), in the latter the suppression of the superfluid density is not accompanied by a significant dissipation at high frequenciesRotzinger et al. (2017), pointing towards the bosonic mechanism. Finally, a likely candidate for this physics are superconducting semiconductors with low density of carriers, such as In-doped $\text{Pb}_{z}\text{Sn}_{1-z}\text{Te}$Parfeniev et al. (2015); Nemov and Ravich (1998)

Wires made from thin films with large $\Delta_{P}$ and small $\rho_{S}$ are expected to exhibit coherent phase slips. This phenomena was indeed observedAstafiev et al. (2012) in InO wires and other strongly disordered superconductors that retain significant single electron gap: NbN and $\text{TiN}.$Peltonen et al. (2013) However, in all these materials the quality factor of the phase slip transition remains low indicating a significant intrinsic dissipation. While expected for fermionic suppression mechanism in NbNChockalingam et al. (2009); Chand et al. (2012); Kamlapure et al. (2013) and TiNSacepe et al. (2008, 2010) that leads to the formation of the subgap states, the reason for the dissipation in InO remains unclear.

*Model. *We consider a simplified model of a pseudogapped superconductor where single-particle excitation are totally absent so that all electronic degrees of freedom can be represented in terms of Anderson pseudospins Anderson (1959) that describe population and hopping of localized electron pairs. In other words, we assume that $\Delta_{p}$ is larger than all relevant energy scales of the problem. The low energy physics is described by

[TABLE]

where indices $i,j$ enumerate localized single-electron states, notation $(i,j)$ indicates a pair of connected sites, $\xi_{i}$ represent their energies, and spin-$\frac{1}{2}$ operators $\mathbf{s}_{i}$ are related with electron creation/annihilation operators $a_{i,\sigma}^{+},a_{i,\sigma}^{-}$ by $2s_{i}^{z}=a_{i,\uparrow}^{+}a_{i,\uparrow}+a_{i,\downarrow}^{+}a_{i,\downarrow}-1$, $s_{i}^{+}=a_{i,\uparrow}^{+}a_{i,\uparrow}^{+}$ and $s_{i}^{-}=a_{i,\downarrow}a_{i,\uparrow}$. Matrix elements $J_{ij}$ that describe hopping of localized Cooper pairs are determined by single-electron wavefunctions $\psi_{i}^{2}(\mathbf{r})$ which are supposed to be localized at relatively long spatial scale: $J_{ij}=\tilde{g}\int d^{3}\mathbf{r}\,\psi_{i}^{2}(\mathbf{r})\,\psi_{j}^{2}(\mathbf{r})$. In a 3D pseudogapped superconductor typical value of matrix element $J_{ij}$ depends in a non-trivial way on the energy difference between the participating states: $\epsilon_{ij}=|\xi_{i}-\xi_{j}|$, see Ref. Feigel’man et al. (2010a); this dependence is due to fractal nature of nearly-critical (in terms of Anderson localization) electron eigenfunctions. An effective number $Z$ of localized electron states $j(i)$ coupled to a given state $i$ by hopping matrix elements $J_{ij}$ depend on the difference between Fermi energy $E_{F}$ and localization threshold $E_{c}$; increase of disorder moves $E_{F}$ further into the localized part of the spectrum, decreasing $Z$.

Solution. In order to obtain the analytical solution we simplify further the model (1). Namely, we assume that all the sites $i,j$ where spins $\mathbf{s}_{i}$ are located, belong to a Bethe lattice with coordination number $Z=K+1$ and all nonzero couplings $J_{ij}$ are equal and connect each spin with its $Z$ nearest neighbors: $J_{ij}=2g/K$, such normalization is used to allow for a well-defined limit of $K\to\infty$. Random variables $\xi_{i}$ are distributed independently over sites $i$ with the flat density $P(\xi)=\frac{1}{2}\theta(1-|\xi|)$. Within this model, increase of disorder corresponds to the decrease of $K$. We have shown previously Feigel’man et al. (2010b) that within such a model a standard BCS-type phase transition takes place at very large $K\geq g\exp(1/g)$, while at lower (but still large) values of $K$ spatial fluctuations of superconducting order parameter become large and eventually lead to an unusual kind of a quantum $T=0$ phase transition from superconducting to insulating state.

In the present Letter we concentrate upon the low-temperature properties of a superconducting state at moderately large values of $K$ in the range $K_{c}<K_{1}<K\leq K_{2}$, where $g\ll 1$ and

[TABLE]

The region $K>K_{1}$ is known Feigel’man et al. (2010b) to possess a usual BCS-like temperature-controlled superconducting transition with $T_{c}=T_{c0}(g)=\frac{4e^{\mathrm{C}}}{\pi}e^{-1/g}$ and low-temperature amplitude of the order parameter $\Delta(T=0,g)=2e^{-1/g}$. At smaller $K$ superconducting transition temperature $T_{c}(g,K)$ is suppressed with respect to $T_{c0}(g)$ and eventually vanishes at $K=K_{c}$. In the range $K_{c}<K<K_{1}$ local values $\Delta_{i}$ of the order parameter fluctuate strongly Feigel’man et al. (2010b), with a "fat tail"   extending to the range of $\Delta_{i}$ much larger than its typical value $\Delta_{{\rm typ}}=\exp(\langle\ln|\Delta_{i}|\rangle)$ that also vanishes at $K\to K_{c}+0$. At larger $K>K_{1}$ the order parameter follows BCS relation and its spatial fluctuations are relatively weak.

Contrary to expectations at $K>K_{1}$ there is a whole band of delocalized low-lying collective excitation modes with a lower cutoff of their energies $\omega_{{\rm 1}}(K)$ growing upon the increase of $K$. Moreover, we find a band of localized collective modes with $\omega<\omega_{{\rm 1}}(K)$ which extends down to zero energy as long as $K\leq K_{2}$.

We start the derivation of our results by writing the action for low-$\omega$ transverse fluctuations $b_{i}(\omega)$ of the order parameter. These fluctuations are parametrized via phase rotation of the mean-field solution: $\Delta_{i}(\omega)=\Delta_{i}e^{i\varphi(\omega)}\equiv\Delta_{i}+b_{i}(\omega)$ with action

[TABLE]

where $\bar{\omega}\equiv\omega/2$. At $K>K_{1}$, $\Delta_{i}\approx\Delta=2e^{1/g}$. The action (3) is directly applicable for $\omega\ll\Delta$; at energies comparable to $\Delta$ antisymmetric coupling (neglected in (3) between transverse mode and longitudinal (gapful) mode might become relevant. Equation for the collective mode can be obtained as an extremum of the action (3) with respect to $b_{i}(\omega)$:

[TABLE]

At $\omega=0$ it is satisfied automatically for $b_{i}=\mathrm{const}\cdot\Delta$ due to self-consistency equations for local order parameters $\Delta$.

Eqs. (3,4) are general, below we study eigenmodes of (4) defined on the Bethe lattice and employ the method developed in the seminal paper Abou-Chacra et al. (1973). To use this method we need to introduce the self-ajoint linear operator $\hat{L}$ related to (4), its matrix elements are $C_{ij}=J_{ij}\left[\eta_{i}(\omega)\eta_{j}(\omega)\right]^{1/2}$. Eqs.(4) possess delocalized solutions if the expansion for the imaginary part of the Green function $\hat{G}=\left(\hat{1}-\hat{C}+i\delta\right)^{-1}$ in powers of $\hat{C}$ is singular. This singularity is indicated by the nonzero value of typical imaginary part $(\Im G_{ii})_{{\rm typ}}$ of the local Green function in the limit of $\delta\to 0$. We look for the singularity threshold within the "forward path" approximation Ioffe and Mezard (2010); Feigel’man et al. (2010b) equivalent to the "Anderson upper limit" conditionAbou-Chacra et al. (1973), i.e. we neglect self-energy corrections for the Green function $G_{ii}(\omega)$. Each path over Bethe lattice that contribute to $(\Im G_{ii})$ is traversed twice (forward and backward). Therefore summation over the paths is equivalent to calculation of partition function $Z_{DP}(N)$ for the $N$-links directed polymer (DP) model with weights $w_{ij}=J_{ij}^{2}\eta_{i}(\omega)\eta_{j}(\omega)$ defined on nearest-neighbour links: $Z_{DP}(N)=\sum_{P}\prod_{\{l(P)\}}w_{ij}$.

We need to find an extensive part of the DP free energy $F_{DP}(N)=\ln Z_{DP}(N)\approx Nf$ at $N\to\infty$; localization threshold is determined by the condition $\langle f\rangle=0$ where averaging is over distribution of random $\xi_{i}$. An equivalent way to calculate $f$ is to use modified weights $\tilde{w}_{ij}=J_{ij}^{2}\eta_{j}^{2}$; the difference between corresponding partition functions $Z_{DP}$ and $\tilde{Z}_{DP}$ is concentrated at the end points of each contributing path and thus does not contribute to $f=\lim_{N\to\infty}\frac{1}{N}F_{DP}(N)$.

The shortest method to calculate $f$ is to use replica trick as described in Ioffe and Mezard (2010); Feigel’man et al. (2010b). It gives:

[TABLE]

Here $0<x<1$ is an anomalous exponent that measures the degree of Replica Symmetry Breaking (RSB) for the DP problem (within usual mean-field theory $x=1$ and second equation in (5) is absent). Extremal condition $\partial f/\partial x|_{x_{0}}=0$ selects typical Green functions of the operator $\hat{C}$ introduced above; the first equation in (5) then leads to $f(x_{0})=0$ which indicates a critical point between localized domain for $f(x_{0})<0$ where typical Green function decays upon iterations, and extended domain, which corresponds then to $f(x_{0})>0$, where linear iterations diverge and nonlinear terms should be taken into account to get stable distribution.

At $K=K_{1}=ge^{1/2g}$ and $\omega=0$ the system of equations (5) can be solved exactly (up to relative corrections $\sim e^{-1/g}\ll 1$), with $x=1/2$. At slightly large $K>K_{1}$ and low energies $\bar{\omega}=E\Delta$ we look for the solution assuming $2x-1\equiv\epsilon\ll 1$ and $E\ll 1$. Expanding the integral in (5) up to the 2nd order in $\epsilon$ and up to the 1st order in $\delta K=K-K_{1}$, we find (the term $\propto E^{2}$ can be omitted in the second of Eqs.(5)):

[TABLE]

leading to the result for the threshold energy in the main order expansion over $\delta K/K_{1}\ll 1$:

[TABLE]

Eigenmodes with $\omega>\omega_{1}$ are extended, while those with lower energies are localized. Numerically obtained delocalization line for $\omega_{\mathrm{1}}(K)$ is shown in green in Fig. 1 for specific choice of $\Delta=10^{-3}$, that corresponds to $g=0.129$ and $K_{1}=5.85$.

To find the domain of existence of localized eigenmodes with low energies $\omega\ll\Delta$, we use another criterion based upon (5). Namely, we look for solutions of the equation $\partial f/\partial x|_{x_{0}}=0$ such that $x_{0}<1$ and $f(x_{0})<0$. The condition $x_{0}<1$ guarantees RSB that implies the different behavior of typical and average of Green functions. Namely, in the limit in the limit of $\delta\to 0$ the average imaginary part of the Green function has a finite value which implies that the density of states is non-zero in this regime. The condition $f(x_{0})<0$ implies that the wave function decreases, so this regime corresponds to the localized states. This band of localized state ends when $x_{0}$ coincides with unity: at this point typical average of the imaginary part of the Green function $\langle\Im G(\omega)\rangle_{typ}$ becomes equal to the simple average, $\langle\Im G(\omega)\rangle=\pi\rho(\omega)$. Because at the same time $f(x_{0}=1)<0$, $\left\langle\Im G(\omega)\right\rangle$ decays upon interations over the Bethe lattice, and $\rho(\omega)=0$ at the stationary point of these iterations. Therefore, the boundary of the parameter region with $\rho(\omega)>0$ is given by the solution of the equation $\partial f/\partial x|_{x_{0}=1}=0$, where $f(x)\equiv f(x,\omega,K)$ is defined in (5). At $\omega=0$ a straightforward calculation leads to the result (2); in deriving it we used the equality $\int_{0}^{\infty}\frac{dt}{\cosh t}\ln\cosh t=\frac{\pi}{2}\ln 2$. At $\omega>0$ the same procedure provides the dependence of the spectrum boundary $\omega_{2}$ on $K$ in the region $\omega\ll\Delta$:

[TABLE]

Numerical solution of the equation $\partial f/\partial x|_{x_{0}=1}=0$ gives the red line in Fig. 1b. Qualitatively, the appearance of $K_{2}$ as one of characteristic value for coordination number $Z=K+1$ in our model can be understood by noticing that at $K\gg K_{2}$ the total number of neighbors in which local energies $\xi_{i}\sim\Delta$ becomes large, so at these $K$ the system becomes similar to conventional Ginzburg-Landau superconductor.

We now justify the neglect of Coulomb interaction between electron pairs. The presence of preformed pairs implies that Coulomb repulsion is strongly suppressed at the spatial scale of pair size $\xi_{0}$, which may be due to large dielectric constant $\epsilon\gtrsim 100$, as discussed in the introductory part. However, this is not sufficient to completely suppress a plasmon gap to frequencies below $2\Delta$. For a usual dirty superconductor $\omega_{0}^{2}=4\pi\rho_{s}/\epsilon$ where superfluid density $\rho_{S}=\pi\Delta\sigma/\hbar$ is defined by $\mathbf{j}=-\rho_{s}\mathbf{A}/c$ and $\sigma$ is a bulk conductivity. For instance, for moderately disordered amorphous InOx far from SIT: $e\Delta\approx 0.5meV$ and $\sigma^{-1}\sim 2m\Omega\cdot cm$, we get $(\hbar\omega_{0}/2\Delta)^{2}=\pi^{2}\hbar\sigma/\Delta\approx 300/\epsilon$. In the pseudogapped regime two important factors suppress further the effects of long-range Coulomb. First, the superfluid density in the pseudogapped state is much smaller than the semiclassical equation predicts. This suppression was observed in Astafiev et al. (2012); Peltonen et al. (2013) and predicted to be a general property of the pseudogapped superconductors inFeigel’man and Ioffe (2015): $\rho_{S}\approx\nu(eR_{0}\Delta/\hbar)^{2}\ll\rho_{BCS}$, where $R_{0}$ is a typical overlap length between localized electron eigenstates and $\nu$ is the electron DoS. For films studied inAstafiev et al. (2012)** **we get $(\hbar\omega_{0}/2\Delta)^{2}\sim\pi\nu e^{2}R_{0}^{2}/\epsilon\approx 100/\epsilon$. Second, the low-lying eigenstates are strongly inhomogeneous in real space as evidenced the Replica-Symmetry-Breaking solution, i.e. the eigenfunctions of the self-ajoint operator $\hat{L}$ which enter spectral expansion of the Green function $G_{ij}(\omega)$, are fractal. As a result, the matrix elements of long-range Coulomb interaction $V(r)\propto 1/r$ between these eigenstates are strongly suppressed with respect to their usual magnitude. Finally, in the case of really thin films Coulomb interaction do not produce a gap anyway (plasmon frequency $\omega(q)\propto q^{1/2}$). For all these reasons we believe that Coulomb interaction does not modify our qualitative conclusions summarized in Fig. 1b.

Experimentally observable properties. The spectrum shown in Fig. 1b translates into microwave properties of the superconductors. In the vicinity of the transition the spectrum of delocalized collective modes extends to zero frequency. Even for $K>K_{1},$ at which the critical temperature of the superconductor does not experience the suppression due to the quantum critical point, the low energy modes are delocalized at relatively low frequencies $\Delta>\omega>\omega_{1}(K)$ resulting in a relatively large intrinsic dissipation of the superconductors at these frequencies. The resonators made from such superconductors exhibits low quality factors. As the disorder is decreased the delocalized modes are shifted to higher frequencies. At $\omega<\omega_{1}(K)$ the oscillation with frequency $\omega$ excite only long-living localized states, so that the dissipation in the superconductor is suppressed. However, the localized modes extend down to zero frequencies for $K<K_{2}.$ At any non-zero temperatures these low frequency bosonic modes are excited. Because the relaxation of these modes is slow, their occupation numbers fluctuate slowly with time. This, together, with the mode-mode interaction implies that the frequency of the high energy modes experience significant jitter in the range $K_{1}<K<K_{2}.$ The microwave properties described above can be compared with the other predictions of the model (1). Namely, one expects broadening of the distribution function at $K<K_{1}$ sketched in Fig. 1a that was observed inSacepe et al. (2011**). Another experimentally measurable characteristic is the behavior of superfluid stiffness that is proportional to the $\Delta^{2}$ in the whole range of $K$ considered here.Feigel’man and Ioffe (2015) Finally, we note that fluctuational conductivity is given by a slightly modifiedPoboiko and Feigel’man (2017) Aslamazov-Larkin formula above $T_{c}$ for $K>K_{2}$ ** which can serve as yet another verification of the applicability of the theory; similarly one can estimate the value of $K_{2}$ from ultrasound attenuation measurements that are expectedShtyk and Feigel’man (2016) to become exponentially low only at $K>K_{2}$. Notice that these different regimes happens *within *the pseudogapped regime where localization of single electron function leads to the formation of preformed Cooper pairs.Feigel’man et al. (2010a). Such materials are expected to have normal-state resistivity $R_{n}$ only several times below the critical value $R_{c}$. Experimentally, for moderately thin films the value of R_{c}^{\square}\sim 10\text{ k\Omega}. Assuming that $\rho_{s}$ for the film is suppressed by a factor of $2-5$ compared to BCS formula $\rho_{BCS}=\pi\Delta/R_{\square}$Astafiev et al. (2012); Peltonen et al. (2013); Feigel’man and Ioffe (2015) we conclude that for the films with $\Delta\sim 1-2K$ and R_{\square}\sim 1-2\text{ k\Omega} one should be able to reach $L_{\square}\sim 10\text{ nH}$ as required for superinductor. However to achieve this goal the material should be tuned into the regime where resistance is already large but not too large so that effective $K>K_{2}.$

This research was supported by the Russian Science Foundation grant 14-42-00044 and ARO W911NF-13-1-0431.