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Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 505702 (http://iopscience.iop.org/0953-8984/26/50/505702) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 137.207.120.173 This content was downloaded on 16/06/2017 at 21:03 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 505702 (7pp)
doi:10.1088/0953-8984/26/50/505702
Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions Yen Lee Loh, Rajesh Dhakal, John F Neis and Evan M Moen Department of Physics and Astrophysics, University of North Dakota, Grand Forks, ND 58201, USA E-mail: [email protected] Received 11 June 2014, revised 25 October 2014 Accepted for publication 28 October 2014 Published 24 November 2014 Abstract
Random inductor–capacitor (LC) networks can exhibit percolative superconductor-insulator transitions (SITs). We use a simple and efficient algorithm to compute the dynamical conductivity σ (ω, p) of one type of LC network on large (4000 × 4000) square lattices, where δ = p − pc is the tuning parameter for the SIT. We confirm that the conductivity obeys a scaling form, so that the characteristic frequency scales as ∝ |δ|νz with νz ≈ 1.91, the superfluid stiffness scales as ϒ ∝ |δ|t with t ≈ 1.3, and the electric susceptibility scales as χE ∝ |δ|−s with s = 2νz − t ≈ 2.52. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of Im σ (ω) at small ω changes across the SIT. Most importantly, we find that right at the SIT Re σ (ω) ∝ ωt/νz−1 ∝ ω−0.32 , so that the conductivity diverges in the DC limit, in contrast with most other classical and quantum models of SITs. Keywords: superconductor-insulator transition, dynamical conductivity, percolation, disorder, circuit, network, quantum phase transition (Some figures may appear in colour only in the online journal)
One of the most important questions concerns the AC conductivity in the ‘collisionless DC’ limit [22, 23]1 , σ ∗ = limω→0 limT →0 σ (ω, T ). It is often claimed that at the SIT σ ∗ is finite and takes a universal value of the order of σQ = 4e2 / h, but there are large discrepancies between the ‘universal’ values from various studies, and it is not clear whether there really is a universal value [1, 4, 16, 23, 24–31]. In this paper we consider a coarse-grained superconductor-insulator composite (e.g. millimeter-sized superconducting particles deposited on an insulating substrate) where the SIT is governed by classical percolation. Then the kinetic inductances and mutual capacitances of the superconducting grains can be represented by inductors and capacitors (see appendix A for justification). We find that for one of the simplest inductorcapacitor network models, σ (ω) diverges as ω → 0, so that σ ∗ is infinite (figure 1)! This is surprising and important, especially in light of prior work on classical models [32]. Furthermore, it ties in well with recent studies [33–36] to
Materials are classified as superconductors, metals, or insulators based on the way they respond to an electric field. A thin film of a superconducting material can be turned into an insulator by increasing the disorder, decreasing the thickness [1], applying a parallel [2] or perpendicular magnetic field [3], or changing the gate voltage [4, 5]. As a quantum phase transition occurring at zero temperature, this superconductorinsulator transition (SIT) has attracted much interest. Early work focused on the most easily measurable quantity, the DC conductivity [1–3, 6–10]. Recently, due to the availability of local scanning probes, attention has turned to the tunneling behavior [11–15]. In the case of the disorder-tuned SIT [15] the SIT is ultimately due to a bosonic mechanism [16] rather than a fermionic one [17]. The last step towards a full characterization of the SIT is to develop an understanding of the behavior of the AC conductivity. This is a powerful probe of fluctuations on both long and short length and time scales, and it is of great interest especially as recent technological developments begin to open up more windows of the electromagnetic spectrum for measurement [18–21]. 0953-8984/14/505702+07$33.00
In general one must be careful to distinguish between σ ∗ and the true DC conductivity σ DC = limT →0 limω→0 σ (ω, T ).
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© 2014 IOP Publishing Ltd Printed in the UK
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Figure 2. (Left) In the Lij Ci model, every site is connected to ground via a capacitance. (Right) Such self-capacitances are absent from the Lij Cij model.
inductances Lij = L0 along bonds ij with probability p. This can be thought of as a model of spherical superconducting grains each of capacitance C0 = 4π 0 R, connected by superconducting links obeying the London equation dj = dt ns e 2 E. The latter are represented by kinetic inductances L0 m (since L0 dI = V for an inductor). The Lij Ci model is almost dt the same as a Josephson junction array, except that it lacks Coulomb blockade effects resulting from charge quantization. The dynamical electromagnetic response ϒ = −dj/dA, conductivity σ = dj/dE, and electric susceptibility χE = dP /dE can be defined in the usual way for a 2D system.
Figure 1. Dissipative conductivity Re σ (ω) for the Lij Ci model. (Top) Right at the SIT (p = pc = 0.5), Re σ diverges as ω−0.32 at small ω, indicated by a downward-sloping straight line on the log–log plot. In the superconducting state (p > pc ), Re σ obeys a ω1 power law. The delta function δ(ω) is not shown. (Bottom) In the insulating state (p < pc ), Re σ is exponentially small (e−(p)/ω ), such that a log plot of Re σ (ω) versus 1/ω shows straight lines.
2. Methods
Previous authors have attacked similar problems numerically using a matrix formalism [38], transfer matrix methods [39–41], and the Frank–Lobb bond propagation algorithm [42, 43]. In this study we employ a variant of the equationof-motion method [44], which is much simpler, more general, and more efficient. Our approach is the theoretical analogue of Fourier transform nuclear magnetic resonance (FT-NMR) spectroscopy, where the frequency response is inferred from the free induction decay signal—the impulse response in the time domain. We apply a transient uniform electric field Ex (t) = δ(t), evolve currents and voltages according to the dynamical Kirchhoff equations, record the uniform component of the current Ix (t), and extract the dissipative conductivity Re σ (ω) using a fast Fourier transform. The discretization error in the time evolution enters entirely in the form of systematic phase error, which we eliminate by a suitable transformation of the frequency variable. This procedure can be shown to be formally equivalent to Chebyshev methods such as the kernel polynomial method [45–47]. The only source of error is the finite duration of the simulation, which leads to a finite frequency resolution. We use a Kaiser window function that gives Re σ (ω) with sidelobe amplitude below σ ≈ 10−8 and main lobe width ω ≈ 15ωMmax (where M is the number of timesteps). This prevents exponentially small tails in the spectrum from being contaminated by spectral leakage. We compute Im σ (ω) using a Kramers–Kronig transformation. We estimate the superfluid stiffness ϒ(ω) from the weight in the lowest-frequency bin, and the electric susceptibility from ∞ χE = 0 dω 2Re σ (ω)/ω2 . See appendices B and C for details. We simulated 4000 × 4000 lattices for 40 000 timesteps (i.e. 40 000 Chebyshev moments), giving a resolution of
complete a bigger picture. For disordered Josephson junction arrays (JJAs) that include Coulomb blockade effects, those studies found that σ ∗ ≈ 0.4(4e2 / h) for zero disorder (p = 0) whereas σ ∗ ≈ 0.5(4e2 / h) for moderate disorder (p = 0.337). According to our work, σ ∗ = ∞ at a percolative SIT, corresponding to an extremely disordered JJA (p = 0.5). This suggests a monotonic trend in σ ∗ (p). There have been several high-profile studies [19, 21, 37] giving detailed measurements of σ (ω) across a finitetemperature superconductor-normal transition. Thin NbN films had a large fluctuation regime: for a film with Tc = 3.14 K, finite-frequency superfluid response persisted up to T = 6 K. A careful scaling analysis indicated a ‘fundamental breakdown of the standard Cooper-pair fluctuation scenario’ due to phase fluctuations. The ultimate goal is to measure σ (ω) across the superconductor-insulator transition. This has not yet been achieved, partly because it is difficult to characterize insulating films by tunneling. Thus, theoretical predictions are crucial in guiding experiments. In this paper we point out that Im σ has a sign change across the SIT, which would be hidden in traditional plots of log |Im σ (ω)| [19, 37]. We elucidate the general structure of σ (ω), addressing the static limit, characteristic frequency, reactive response, and lowfrequency contributions from rare regions in the insulator and from Goldstone modes in the superconductor. 1. Model
We focus on the Lij Ci model (see figure 2), which contains capacitances-to-ground Ci = C0 on every site i and 2
J. Phys.: Condens. Matter 26 (2014) 505702
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Figure 3. Dynamical conductivity for the Lij Ci model for p = 0.02, 0.04, . . . , 0.40, 0.41, . . . , 0.60, 0.62, . . . , 0.98 for single realizations on a 4000 × 4000 lattice. Red, green, and blue indicate insulating (capacitive), metallic (resistive), and superconducting (inductive) behavior respectively. The SIT occurs at the percolation threshold (pc = 0.5). The dissipative conductivity Re σ (ω, p) has a delta function in the superconductor (p > pc ), visible as a thin green line; most of the remaining weight occurs above a characteristic frequency ∝ (p − pc )νz , forming a fan shape reminiscent of quantum criticality. The reactive conductivity Imσ (ω, p) changes sign from capacitive to inductive as p increases.
ω ≈ 0.0015 after windowing and rebinning. We found that larger (6144 × 6144) lattices gave no discernible differences. This is because the finite-frequency (ω 0.005) dynamical response is dominated by clusters considerably smaller than the simulation size, and is thus relatively insensitive to finitesize effects. See appendix D. 3. Results
√ We quote angular frequency ω in units of 1/√ L0 C0 and 2D conductivity (sheet conductance) σ in units of C0 /L0 . Color plots of Re σ (ω, p) and Im σ (ω, p) are shown in figure 3. The data contain a wealth of interesting information that we list below. Figure 4. Static response functions of LC networks as a function of superconducting bond fraction p. The insulating state is characterized by a finite electric susceptibility χE , which diverges at the percolation transition. The superconducting state is characterized by a finite superfluid stiffness ϒ, which vanishes at the transition. Near the percolation threshold p, the superfluid stiffness scales as ϒ ∼ (p − pc )1.30 . For the Lij Cij model near pc , χE = ϒ −1 ∼ (pc − p)−1.30 due to duality. For the Lij Ci model near pc , χE ∼ (pc − p)−2.52 .
(a) Superfluid stiffness: On the superconducting side of the SIT (p > pc ), the superfluid stiffness scales as ϒ(p) ∼ δ t where δ = p − pc and t ≈ 1.30, as shown in figure 4. This agrees with results for resistor networks [48, 49]. (b) Characteristic frequency: For 2D percolation the correlation length diverges as ξ ∼ |δ|−ν with ν = 4/3 [50]. This suggests that there is a characteristic (angular) frequency ∼ ξ −z ∼ |δ|νz , where z is a dynamical critical exponent. Indeed, figure 3 suggests that most of the spectral weight in Re σ (ω) occurs above a frequency ∼ δ νz , forming a shape analogous to a ‘quantum critical’ fan. (c) Divergent conductivity at SIT: At the SIT (p = pc ), Re σ (ω) does not tend to a finite limit as in many other systems, but instead it diverges! This is illustrated in figure 1. We find a very good fit to a power law of the form Re σ (ω) ≈ ω−a where a ≈ 0.32(1). (d) Scaling collapse: Based on observations (a) and (b), we postulate the scaling form σ (ω, δ) = ωt/νz−1 f (ω−1/νz δ). This mandates that σ (ω) ∝ ωt/νz−1 at the SIT. Comparing with (c), we see that we must have t/νz − 1 = −a, so that νz = t/(1 − a) ≈ 1.91. Indeed, we find that both Re σ and Im σ collapse onto single curves for νz ≈ 1.91, as shown in figure 5. The details of f (x) will be reported elsewhere.
(e) Electric susceptibility: On the insulating side of the SIT (p < pc ), the scaling form dictates that the electric susceptibility must scale as χE ∼ |δ|−s with s = 2νz−t ≈ 2.52. Indeed, figure 4 shows a good fit to this power law. (f) Low-frequency dissipation: The characteristic frequency does not correspond to a hard gap in the spectrum. In the insulating state, large rare regions contribute exponentially small weight to Re σ all the way down to zero frequency, as shown in the bottom panel of figure 1. This is a ramification of Griffiths–McCoy– Wu physics [51–53] in systems with quenched disorder. The superconducting state has linear low-frequency dissipation Re σ ∼ ω, as shown in the top panel of figure 1. We believe this is due to the excitation of acoustic ‘transmission-line’ modes that is permitted in the presence of disorder. (It is also reminiscent of [54].) 3
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particular, the Lij Ci model. We used an efficient algorithm to compute σ (ω, p) on large lattices (4000 × 4000 sites). We find the critical exponents t ≈ 1.30 (in agreement with results on resistor networks), νz ≈ 1.91, and s = 2νz − t ≈ 2.52. We have extracted the complex-valued scaling function. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of Im σ (ω) at small ω changes across the SIT. Most surprisingly, right at the SIT, Re σ (ω) diverges as ω → 0. Our results form an important baseline to which to compare simulations of more complicated models such as XY, Bose–Hubbard, and Fermi–Hubbard models, thus allowing one to separate the effects of quenched disorder, quantum phase fluctuations, and pairbreaking physics.
Figure 5. Real and imaginary parts of the scaling function f (x) extracted using scaling collapse of σ (p, ω) for 0.3 p 0.7 and 0 < ω < 0.2. This confirms the scaling form σ (ω, δ) = ωt/νz−1 f (ω−1/νz δ) where δ = p − pc , t = 1.30, and νz = 1.91. Colors of data points indicate ω values. We have verified that scaling collapse is good for 10−4 < f (x) < 101 , beyond the dynamic range shown above.
Acknowledgment
We gratefully acknowledge M Swanson, M Randeria and P Karki for useful discussions. Appendix A. Modeling superconductors and inductors by lumped circuit elements
(g) Reactive conductivity: The reactive part of the dynamical conductivity, Im σ (ω), is shown in figure 3 and implicitly in figure 5. In the insulating state, the low-frequency response is capacitive (Im σ (ω) < 0). In the superconducting state, it is inductive (Im σ (ω) > 0). This suggests that the sign of Im σ (ω) may be used in experiments as a criterion to distinguish between insulating and superconducting states.
We first review the relationship between Ohmic conductance and conductivity. First, consider a solid metal tube of length l, cross-sectional area A, and DC conductivity σ . The potential difference and electric field are related by V = El, whereas the current and current density are related by I = j A. Thus the DC conductance of the tube is jA A I = =σ . (A1) G= V El l Second, consider a solid superconducting tube of length l and cross-sectional area A. Suppose the superconductor carries a small uniform current density, much smaller than the critical value. In this regime the superconductor satisfies the London equation ∂t j = ϒE, where ϒ is the superfluid stiffness (conventionally expressed as a superfluid density ns 2 se via ϒ = nm ). Then, the superconducting tube behaves as an inductor of ‘kinetic’ inductance L such that ∂t I (∂t j )A A = =ϒ . (A2) L−1 = V El l Indeed, the low-frequency dynamical conductivity of a superconductor (Im σ ∼ ω−1 ) is the same as the lowfrequency admittance of an inductor (Im G ∼ ω−1 ). Thus a superconductor can be represented by an inductor provided that nonlinear effects are negligible (i.e. phase fluctuations are small compared to π ). (In this paper we are not concerned with high frequencies, where σ (ω) in a real superconductor is dominated by the Mattis–Bardeen contribution coming from two-quasiparticle excitations.) Finally, consider a solid insulating tube of the same geometry containing a uniform polarization P and electric field E. The surface polarization charge is σpol = 0 P = χe E, where χe is the electric susceptibility of the material. This gives a na¨ıve estimate of the capacitance,
4. Importance of on-site capacitances
In previous studies of the dynamical conductivity of classical systems near percolation, the insulating (capacitive) elements were placed along bonds, in series with the superconducting (inductive) elements [32, 38]. We have studied an LC model of this type, which we call the Lij Cij model (right panel in figure 2). The dynamical conductivities of the Lij Ci and Lij Cij models are compared in figure 6. We find that for the latter model at percolation (p = 0.5), σ (ω) tends to a finite value σ ∗ as ω → 0. This is a consequence of self-duality [55], which implies that νz = t ≈ 1.3. Also, Re σ indicates exponentially small low-frequency dissipation in the superconducting state (as opposed to Re σ ∝ ω for the Lij Ci model). The Lij Ci model includes on-site capacitances, which are absent from the Lij Cij model. In general, one expects that any real system of superconducting grains will have finite on-site capacitances, literally or in a ‘renormalization group’ sense. (All previous studies of bosonic models of SITs [29, 30] included finite on-site capacitances; in fact, the Lij Ci model is the limiting case of a JJA when the charging energy is negligible.) Thus our results suggest that classical percolative SITs should generically have a divergent conductivity. 5. Conclusions
We have studied the percolative superconductor-insulator transition in two-dimensional classical LC networks, in
C= 4
σpol A A Q = = χe . V El l
(A3)
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Figure 6. Stacked plots of Re σ (ω) for Lij Ci model (left column) and Lij Cij model (right column). At the percolation threshold (p = 0.5) at low frequency (ω → 0), σ (ω) diverges as ω−0.32 for the Lij Ci model, whereas it tends to a constant σ ∗ for for the Lij Cij model.
In reality the surface charges produce an induced electric field that partially opposes the applied electric field, so that the capacitance of a parallel-plate capacitor is C = (1 + χe ) Al . In fact, two electrodes separated by vacuum still have a finite capacitance. Even a single electrode can have a finite capacitance to ground; for example, a sphere of radius a has C = 4π 0 a to ground. Also note that real metals and insulators do not respond instantaneously to an applied voltage, because of the inertia of the electrons; this can be modeled by inserting a kinetic inductance in series with the resistor or capacitor. In principle, superconducting grains deposited on an insulator can be treated by dividing the system into small cubes (or other finite elements) and going from there to a lumped element model. However, for elucidating universal critical behavior, it is better to study simple models with well-defined disorder distributions, as we do here.
(x, y)–(x + 1, y), and LYxy is the inductance on the link (x, y)– (x, y + 1). Cxy = 1 ∞ X,Y Lxy = 1
(B2)
Identify the highest possible eigenfrequency according to √ For Gerschgorin’s theorem (which is 8 for this model). √ safety, choose ωmax to be slightly greater than 8, and set the timestep to τ = 2/ωmax . Initialize node voltages and link currents: X Ixy = 1/LX xy 0 Y Ixy =0 0 τ X X Y Y Vxy τ = (I − I + I − I ) . (B3) x−1,y xy x,y−1 xy Y 0 2Cxy 2
Appendix B. Algorithm description for Lij Ci model
Evolve system using leapfrog algorithm: τ X X Ixy = Ixy + X (Vxy − Vx+1,y ) τ t t−τ t− 2 Lxy τ Y Y Ixy = Ixy + Y (Vxy − Vx,y+1 ) τ t t−τ t− 2 Lxy τ X X Y Y Vxy τ = Vxy τ + (Ix−1,y − Ixy + Ix,y−1 − Ixy ) t+ 2 t− 2 t Cxy
The dynamical Kirchhoff equations for the Lij Ci model are simply I˙ij = L−1 ij (Vi − Vj ) Ij i V˙i = Ci−1
with probability 1 − p with probability p.
(B1)
j
(B4)
where Vi is the current on node i and Ij i is the current along link j i. A detailed simulation algorithm is as follows. Set up a square lattice with periodic boundary conditions with N = Nx Ny nodes, where x = 0, 1, 2, . . . , Nx − 1 and y = 0, 1, 2, . . . , Ny − 1. Initialize the node capacitances Cxy and the link inductances. LX xy is the inductance on the link
Record the total x-current at each timestep, which is an estimate of the real-time conductivity: X I (t) = Ixy (t). (B5) xy
5
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Appendix C. Spectral estimation
Given the time series Im ≡ I (t = mτ ) for m = 0, 1, 2, . . . , M − 1, multiply it by a Kaiser window function Wb involving modified Bessel functions I0 (x) with a suitable width parameter β = 24, Wm = I0 β 1 − (
m 2 ) M −1
I0 (β).
(C1)
Perform a type-III discrete cosine transform to obtain the weights σb for b = 0, 1, 2, . . . , M − 1:
M−1 π m(b + 21 ) 1 cos (C2) I0 + 2 Wm Im . σb = N M m=1 In a naive implementation of the equation-of-motion method, {σb } represent the integrals of Re σ ( ω) over bins delimited by b the frequencies ωb = 2π . As explained in the text, the phase τM ω error can be eliminated by writing ω = τ2 sin τ2 . Therefore {σb } can be interpreted as weights in M unequal bins covering the interval ω ∈ [0, ωmax ]: ωb+1 dω Re σ (ω), σb ≈
Figure 7. Test of the windowing procedure. We define a test spectrum containing two delta functions and a box function δ(ω) + δ(ω − 0.017) + (0.03 − ω) − (0.02 − ω). Taking ωmax = 1, we compute M = 25 000 Chebyshev moments Im analytically. We then estimate the spectrum Re σ (ω) using a Kaiser window. The top panel shows the raw estimate (ωb , σb ). The bottom panel shows the estimated spectrum resampled in M/15 equal bins. Spectral leakage is clearly negligible ( 10−8 ) for |ω − ωexact | 10−3 .
ωb
2 πb ωb = sin τ M
(m = 0, 1, 2, . . . , M).
(C3)
For the Kaiser window with a width parameter β = 24 as defined above, the spectral leakage function has a central lobe of width β/π ≈ 15 bins, and the ratio of centrallobe to side-lobe amplitude is about 108 . Thus, if the true Re σ (ω) contains a delta function at some frequency, the numerically calculated weights σb will have the delta function spread over about 15 consecutive bins. To save memory, and for later convenience, it makes sense to resample the spectrum into approximately M/15 equal bins. This should be done in a way that preserves the sum rule. Thus, we calculate the cumulative distribution function C(ωb ) for b = 0, 1, 2, . . . , M; interpolate this and evaluate C(ωj ) at equally spaced frequencies ωj , j = 0, 1, 2, . . . , M/15 ; and take differences to give the new weights [56]. The final result can be quoted, roughly speaking, as an estimate of Re σ (ω) with frequency resolution (horizontal error bars) ω ≈ 15ωMmax and ‘vertical error bars’ σ ≈ 10−8 . Figure 7 illustrates the result of this procedure for a test spectrum. Other than the Kaiser window, there are other possible window functions such as the Dolph–Chebyshev window (which is optimal but slightly more complicated to set up), and the window function derived by Storck and Wolff [57].
Figure 8. Dynamical conductivity Re σ (ω); L of square lattice Lij Ci model for various system sizes L, on a log–log plot. The dynamical conductivity is almost independent of L. The only discernible finite-size effect is that the lowest-frequency data point lies slightly above the curve. Thus, the data for L = 6144 have no significant finite-size effects down to ω ≈ 0.0015. Error bars were estimated by comparing 2–32 disorder realizations.
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As mentioned in the text, the dynamical response σ (ω) is mainly determined by finite-sized clusters that are smaller than the system sizes in this study. Thus, finite-size errors are practically negligible, as shown in figure 8. 6
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Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 505702 (http://iopscience.iop.org/0953-8984/26/50/505702) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 137.207.120.173 This content was downloaded on 16/06/2017 at 21:03 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 505702 (7pp)
doi:10.1088/0953-8984/26/50/505702
Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions Yen Lee Loh, Rajesh Dhakal, John F Neis and Evan M Moen Department of Physics and Astrophysics, University of North Dakota, Grand Forks, ND 58201, USA E-mail: [email protected] Received 11 June 2014, revised 25 October 2014 Accepted for publication 28 October 2014 Published 24 November 2014 Abstract
Random inductor–capacitor (LC) networks can exhibit percolative superconductor-insulator transitions (SITs). We use a simple and efficient algorithm to compute the dynamical conductivity σ (ω, p) of one type of LC network on large (4000 × 4000) square lattices, where δ = p − pc is the tuning parameter for the SIT. We confirm that the conductivity obeys a scaling form, so that the characteristic frequency scales as ∝ |δ|νz with νz ≈ 1.91, the superfluid stiffness scales as ϒ ∝ |δ|t with t ≈ 1.3, and the electric susceptibility scales as χE ∝ |δ|−s with s = 2νz − t ≈ 2.52. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of Im σ (ω) at small ω changes across the SIT. Most importantly, we find that right at the SIT Re σ (ω) ∝ ωt/νz−1 ∝ ω−0.32 , so that the conductivity diverges in the DC limit, in contrast with most other classical and quantum models of SITs. Keywords: superconductor-insulator transition, dynamical conductivity, percolation, disorder, circuit, network, quantum phase transition (Some figures may appear in colour only in the online journal)
One of the most important questions concerns the AC conductivity in the ‘collisionless DC’ limit [22, 23]1 , σ ∗ = limω→0 limT →0 σ (ω, T ). It is often claimed that at the SIT σ ∗ is finite and takes a universal value of the order of σQ = 4e2 / h, but there are large discrepancies between the ‘universal’ values from various studies, and it is not clear whether there really is a universal value [1, 4, 16, 23, 24–31]. In this paper we consider a coarse-grained superconductor-insulator composite (e.g. millimeter-sized superconducting particles deposited on an insulating substrate) where the SIT is governed by classical percolation. Then the kinetic inductances and mutual capacitances of the superconducting grains can be represented by inductors and capacitors (see appendix A for justification). We find that for one of the simplest inductorcapacitor network models, σ (ω) diverges as ω → 0, so that σ ∗ is infinite (figure 1)! This is surprising and important, especially in light of prior work on classical models [32]. Furthermore, it ties in well with recent studies [33–36] to
Materials are classified as superconductors, metals, or insulators based on the way they respond to an electric field. A thin film of a superconducting material can be turned into an insulator by increasing the disorder, decreasing the thickness [1], applying a parallel [2] or perpendicular magnetic field [3], or changing the gate voltage [4, 5]. As a quantum phase transition occurring at zero temperature, this superconductorinsulator transition (SIT) has attracted much interest. Early work focused on the most easily measurable quantity, the DC conductivity [1–3, 6–10]. Recently, due to the availability of local scanning probes, attention has turned to the tunneling behavior [11–15]. In the case of the disorder-tuned SIT [15] the SIT is ultimately due to a bosonic mechanism [16] rather than a fermionic one [17]. The last step towards a full characterization of the SIT is to develop an understanding of the behavior of the AC conductivity. This is a powerful probe of fluctuations on both long and short length and time scales, and it is of great interest especially as recent technological developments begin to open up more windows of the electromagnetic spectrum for measurement [18–21]. 0953-8984/14/505702+07$33.00
In general one must be careful to distinguish between σ ∗ and the true DC conductivity σ DC = limT →0 limω→0 σ (ω, T ).
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Figure 2. (Left) In the Lij Ci model, every site is connected to ground via a capacitance. (Right) Such self-capacitances are absent from the Lij Cij model.
inductances Lij = L0 along bonds ij with probability p. This can be thought of as a model of spherical superconducting grains each of capacitance C0 = 4π 0 R, connected by superconducting links obeying the London equation dj = dt ns e 2 E. The latter are represented by kinetic inductances L0 m (since L0 dI = V for an inductor). The Lij Ci model is almost dt the same as a Josephson junction array, except that it lacks Coulomb blockade effects resulting from charge quantization. The dynamical electromagnetic response ϒ = −dj/dA, conductivity σ = dj/dE, and electric susceptibility χE = dP /dE can be defined in the usual way for a 2D system.
Figure 1. Dissipative conductivity Re σ (ω) for the Lij Ci model. (Top) Right at the SIT (p = pc = 0.5), Re σ diverges as ω−0.32 at small ω, indicated by a downward-sloping straight line on the log–log plot. In the superconducting state (p > pc ), Re σ obeys a ω1 power law. The delta function δ(ω) is not shown. (Bottom) In the insulating state (p < pc ), Re σ is exponentially small (e−(p)/ω ), such that a log plot of Re σ (ω) versus 1/ω shows straight lines.
2. Methods
Previous authors have attacked similar problems numerically using a matrix formalism [38], transfer matrix methods [39–41], and the Frank–Lobb bond propagation algorithm [42, 43]. In this study we employ a variant of the equationof-motion method [44], which is much simpler, more general, and more efficient. Our approach is the theoretical analogue of Fourier transform nuclear magnetic resonance (FT-NMR) spectroscopy, where the frequency response is inferred from the free induction decay signal—the impulse response in the time domain. We apply a transient uniform electric field Ex (t) = δ(t), evolve currents and voltages according to the dynamical Kirchhoff equations, record the uniform component of the current Ix (t), and extract the dissipative conductivity Re σ (ω) using a fast Fourier transform. The discretization error in the time evolution enters entirely in the form of systematic phase error, which we eliminate by a suitable transformation of the frequency variable. This procedure can be shown to be formally equivalent to Chebyshev methods such as the kernel polynomial method [45–47]. The only source of error is the finite duration of the simulation, which leads to a finite frequency resolution. We use a Kaiser window function that gives Re σ (ω) with sidelobe amplitude below σ ≈ 10−8 and main lobe width ω ≈ 15ωMmax (where M is the number of timesteps). This prevents exponentially small tails in the spectrum from being contaminated by spectral leakage. We compute Im σ (ω) using a Kramers–Kronig transformation. We estimate the superfluid stiffness ϒ(ω) from the weight in the lowest-frequency bin, and the electric susceptibility from ∞ χE = 0 dω 2Re σ (ω)/ω2 . See appendices B and C for details. We simulated 4000 × 4000 lattices for 40 000 timesteps (i.e. 40 000 Chebyshev moments), giving a resolution of
complete a bigger picture. For disordered Josephson junction arrays (JJAs) that include Coulomb blockade effects, those studies found that σ ∗ ≈ 0.4(4e2 / h) for zero disorder (p = 0) whereas σ ∗ ≈ 0.5(4e2 / h) for moderate disorder (p = 0.337). According to our work, σ ∗ = ∞ at a percolative SIT, corresponding to an extremely disordered JJA (p = 0.5). This suggests a monotonic trend in σ ∗ (p). There have been several high-profile studies [19, 21, 37] giving detailed measurements of σ (ω) across a finitetemperature superconductor-normal transition. Thin NbN films had a large fluctuation regime: for a film with Tc = 3.14 K, finite-frequency superfluid response persisted up to T = 6 K. A careful scaling analysis indicated a ‘fundamental breakdown of the standard Cooper-pair fluctuation scenario’ due to phase fluctuations. The ultimate goal is to measure σ (ω) across the superconductor-insulator transition. This has not yet been achieved, partly because it is difficult to characterize insulating films by tunneling. Thus, theoretical predictions are crucial in guiding experiments. In this paper we point out that Im σ has a sign change across the SIT, which would be hidden in traditional plots of log |Im σ (ω)| [19, 37]. We elucidate the general structure of σ (ω), addressing the static limit, characteristic frequency, reactive response, and lowfrequency contributions from rare regions in the insulator and from Goldstone modes in the superconductor. 1. Model
We focus on the Lij Ci model (see figure 2), which contains capacitances-to-ground Ci = C0 on every site i and 2
J. Phys.: Condens. Matter 26 (2014) 505702
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Figure 3. Dynamical conductivity for the Lij Ci model for p = 0.02, 0.04, . . . , 0.40, 0.41, . . . , 0.60, 0.62, . . . , 0.98 for single realizations on a 4000 × 4000 lattice. Red, green, and blue indicate insulating (capacitive), metallic (resistive), and superconducting (inductive) behavior respectively. The SIT occurs at the percolation threshold (pc = 0.5). The dissipative conductivity Re σ (ω, p) has a delta function in the superconductor (p > pc ), visible as a thin green line; most of the remaining weight occurs above a characteristic frequency ∝ (p − pc )νz , forming a fan shape reminiscent of quantum criticality. The reactive conductivity Imσ (ω, p) changes sign from capacitive to inductive as p increases.
ω ≈ 0.0015 after windowing and rebinning. We found that larger (6144 × 6144) lattices gave no discernible differences. This is because the finite-frequency (ω 0.005) dynamical response is dominated by clusters considerably smaller than the simulation size, and is thus relatively insensitive to finitesize effects. See appendix D. 3. Results
√ We quote angular frequency ω in units of 1/√ L0 C0 and 2D conductivity (sheet conductance) σ in units of C0 /L0 . Color plots of Re σ (ω, p) and Im σ (ω, p) are shown in figure 3. The data contain a wealth of interesting information that we list below. Figure 4. Static response functions of LC networks as a function of superconducting bond fraction p. The insulating state is characterized by a finite electric susceptibility χE , which diverges at the percolation transition. The superconducting state is characterized by a finite superfluid stiffness ϒ, which vanishes at the transition. Near the percolation threshold p, the superfluid stiffness scales as ϒ ∼ (p − pc )1.30 . For the Lij Cij model near pc , χE = ϒ −1 ∼ (pc − p)−1.30 due to duality. For the Lij Ci model near pc , χE ∼ (pc − p)−2.52 .
(a) Superfluid stiffness: On the superconducting side of the SIT (p > pc ), the superfluid stiffness scales as ϒ(p) ∼ δ t where δ = p − pc and t ≈ 1.30, as shown in figure 4. This agrees with results for resistor networks [48, 49]. (b) Characteristic frequency: For 2D percolation the correlation length diverges as ξ ∼ |δ|−ν with ν = 4/3 [50]. This suggests that there is a characteristic (angular) frequency ∼ ξ −z ∼ |δ|νz , where z is a dynamical critical exponent. Indeed, figure 3 suggests that most of the spectral weight in Re σ (ω) occurs above a frequency ∼ δ νz , forming a shape analogous to a ‘quantum critical’ fan. (c) Divergent conductivity at SIT: At the SIT (p = pc ), Re σ (ω) does not tend to a finite limit as in many other systems, but instead it diverges! This is illustrated in figure 1. We find a very good fit to a power law of the form Re σ (ω) ≈ ω−a where a ≈ 0.32(1). (d) Scaling collapse: Based on observations (a) and (b), we postulate the scaling form σ (ω, δ) = ωt/νz−1 f (ω−1/νz δ). This mandates that σ (ω) ∝ ωt/νz−1 at the SIT. Comparing with (c), we see that we must have t/νz − 1 = −a, so that νz = t/(1 − a) ≈ 1.91. Indeed, we find that both Re σ and Im σ collapse onto single curves for νz ≈ 1.91, as shown in figure 5. The details of f (x) will be reported elsewhere.
(e) Electric susceptibility: On the insulating side of the SIT (p < pc ), the scaling form dictates that the electric susceptibility must scale as χE ∼ |δ|−s with s = 2νz−t ≈ 2.52. Indeed, figure 4 shows a good fit to this power law. (f) Low-frequency dissipation: The characteristic frequency does not correspond to a hard gap in the spectrum. In the insulating state, large rare regions contribute exponentially small weight to Re σ all the way down to zero frequency, as shown in the bottom panel of figure 1. This is a ramification of Griffiths–McCoy– Wu physics [51–53] in systems with quenched disorder. The superconducting state has linear low-frequency dissipation Re σ ∼ ω, as shown in the top panel of figure 1. We believe this is due to the excitation of acoustic ‘transmission-line’ modes that is permitted in the presence of disorder. (It is also reminiscent of [54].) 3
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particular, the Lij Ci model. We used an efficient algorithm to compute σ (ω, p) on large lattices (4000 × 4000 sites). We find the critical exponents t ≈ 1.30 (in agreement with results on resistor networks), νz ≈ 1.91, and s = 2νz − t ≈ 2.52. We have extracted the complex-valued scaling function. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of Im σ (ω) at small ω changes across the SIT. Most surprisingly, right at the SIT, Re σ (ω) diverges as ω → 0. Our results form an important baseline to which to compare simulations of more complicated models such as XY, Bose–Hubbard, and Fermi–Hubbard models, thus allowing one to separate the effects of quenched disorder, quantum phase fluctuations, and pairbreaking physics.
Figure 5. Real and imaginary parts of the scaling function f (x) extracted using scaling collapse of σ (p, ω) for 0.3 p 0.7 and 0 < ω < 0.2. This confirms the scaling form σ (ω, δ) = ωt/νz−1 f (ω−1/νz δ) where δ = p − pc , t = 1.30, and νz = 1.91. Colors of data points indicate ω values. We have verified that scaling collapse is good for 10−4 < f (x) < 101 , beyond the dynamic range shown above.
Acknowledgment
We gratefully acknowledge M Swanson, M Randeria and P Karki for useful discussions. Appendix A. Modeling superconductors and inductors by lumped circuit elements
(g) Reactive conductivity: The reactive part of the dynamical conductivity, Im σ (ω), is shown in figure 3 and implicitly in figure 5. In the insulating state, the low-frequency response is capacitive (Im σ (ω) < 0). In the superconducting state, it is inductive (Im σ (ω) > 0). This suggests that the sign of Im σ (ω) may be used in experiments as a criterion to distinguish between insulating and superconducting states.
We first review the relationship between Ohmic conductance and conductivity. First, consider a solid metal tube of length l, cross-sectional area A, and DC conductivity σ . The potential difference and electric field are related by V = El, whereas the current and current density are related by I = j A. Thus the DC conductance of the tube is jA A I = =σ . (A1) G= V El l Second, consider a solid superconducting tube of length l and cross-sectional area A. Suppose the superconductor carries a small uniform current density, much smaller than the critical value. In this regime the superconductor satisfies the London equation ∂t j = ϒE, where ϒ is the superfluid stiffness (conventionally expressed as a superfluid density ns 2 se via ϒ = nm ). Then, the superconducting tube behaves as an inductor of ‘kinetic’ inductance L such that ∂t I (∂t j )A A = =ϒ . (A2) L−1 = V El l Indeed, the low-frequency dynamical conductivity of a superconductor (Im σ ∼ ω−1 ) is the same as the lowfrequency admittance of an inductor (Im G ∼ ω−1 ). Thus a superconductor can be represented by an inductor provided that nonlinear effects are negligible (i.e. phase fluctuations are small compared to π ). (In this paper we are not concerned with high frequencies, where σ (ω) in a real superconductor is dominated by the Mattis–Bardeen contribution coming from two-quasiparticle excitations.) Finally, consider a solid insulating tube of the same geometry containing a uniform polarization P and electric field E. The surface polarization charge is σpol = 0 P = χe E, where χe is the electric susceptibility of the material. This gives a na¨ıve estimate of the capacitance,
4. Importance of on-site capacitances
In previous studies of the dynamical conductivity of classical systems near percolation, the insulating (capacitive) elements were placed along bonds, in series with the superconducting (inductive) elements [32, 38]. We have studied an LC model of this type, which we call the Lij Cij model (right panel in figure 2). The dynamical conductivities of the Lij Ci and Lij Cij models are compared in figure 6. We find that for the latter model at percolation (p = 0.5), σ (ω) tends to a finite value σ ∗ as ω → 0. This is a consequence of self-duality [55], which implies that νz = t ≈ 1.3. Also, Re σ indicates exponentially small low-frequency dissipation in the superconducting state (as opposed to Re σ ∝ ω for the Lij Ci model). The Lij Ci model includes on-site capacitances, which are absent from the Lij Cij model. In general, one expects that any real system of superconducting grains will have finite on-site capacitances, literally or in a ‘renormalization group’ sense. (All previous studies of bosonic models of SITs [29, 30] included finite on-site capacitances; in fact, the Lij Ci model is the limiting case of a JJA when the charging energy is negligible.) Thus our results suggest that classical percolative SITs should generically have a divergent conductivity. 5. Conclusions
We have studied the percolative superconductor-insulator transition in two-dimensional classical LC networks, in
C= 4
σpol A A Q = = χe . V El l
(A3)
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Figure 6. Stacked plots of Re σ (ω) for Lij Ci model (left column) and Lij Cij model (right column). At the percolation threshold (p = 0.5) at low frequency (ω → 0), σ (ω) diverges as ω−0.32 for the Lij Ci model, whereas it tends to a constant σ ∗ for for the Lij Cij model.
In reality the surface charges produce an induced electric field that partially opposes the applied electric field, so that the capacitance of a parallel-plate capacitor is C = (1 + χe ) Al . In fact, two electrodes separated by vacuum still have a finite capacitance. Even a single electrode can have a finite capacitance to ground; for example, a sphere of radius a has C = 4π 0 a to ground. Also note that real metals and insulators do not respond instantaneously to an applied voltage, because of the inertia of the electrons; this can be modeled by inserting a kinetic inductance in series with the resistor or capacitor. In principle, superconducting grains deposited on an insulator can be treated by dividing the system into small cubes (or other finite elements) and going from there to a lumped element model. However, for elucidating universal critical behavior, it is better to study simple models with well-defined disorder distributions, as we do here.
(x, y)–(x + 1, y), and LYxy is the inductance on the link (x, y)– (x, y + 1). Cxy = 1 ∞ X,Y Lxy = 1
(B2)
Identify the highest possible eigenfrequency according to √ For Gerschgorin’s theorem (which is 8 for this model). √ safety, choose ωmax to be slightly greater than 8, and set the timestep to τ = 2/ωmax . Initialize node voltages and link currents: X Ixy = 1/LX xy 0 Y Ixy =0 0 τ X X Y Y Vxy τ = (I − I + I − I ) . (B3) x−1,y xy x,y−1 xy Y 0 2Cxy 2
Appendix B. Algorithm description for Lij Ci model
Evolve system using leapfrog algorithm: τ X X Ixy = Ixy + X (Vxy − Vx+1,y ) τ t t−τ t− 2 Lxy τ Y Y Ixy = Ixy + Y (Vxy − Vx,y+1 ) τ t t−τ t− 2 Lxy τ X X Y Y Vxy τ = Vxy τ + (Ix−1,y − Ixy + Ix,y−1 − Ixy ) t+ 2 t− 2 t Cxy
The dynamical Kirchhoff equations for the Lij Ci model are simply I˙ij = L−1 ij (Vi − Vj ) Ij i V˙i = Ci−1
with probability 1 − p with probability p.
(B1)
j
(B4)
where Vi is the current on node i and Ij i is the current along link j i. A detailed simulation algorithm is as follows. Set up a square lattice with periodic boundary conditions with N = Nx Ny nodes, where x = 0, 1, 2, . . . , Nx − 1 and y = 0, 1, 2, . . . , Ny − 1. Initialize the node capacitances Cxy and the link inductances. LX xy is the inductance on the link
Record the total x-current at each timestep, which is an estimate of the real-time conductivity: X I (t) = Ixy (t). (B5) xy
5
J. Phys.: Condens. Matter 26 (2014) 505702
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Appendix C. Spectral estimation
Given the time series Im ≡ I (t = mτ ) for m = 0, 1, 2, . . . , M − 1, multiply it by a Kaiser window function Wb involving modified Bessel functions I0 (x) with a suitable width parameter β = 24, Wm = I0 β 1 − (
m 2 ) M −1
I0 (β).
(C1)
Perform a type-III discrete cosine transform to obtain the weights σb for b = 0, 1, 2, . . . , M − 1:
M−1 π m(b + 21 ) 1 cos (C2) I0 + 2 Wm Im . σb = N M m=1 In a naive implementation of the equation-of-motion method, {σb } represent the integrals of Re σ ( ω) over bins delimited by b the frequencies ωb = 2π . As explained in the text, the phase τM ω error can be eliminated by writing ω = τ2 sin τ2 . Therefore {σb } can be interpreted as weights in M unequal bins covering the interval ω ∈ [0, ωmax ]: ωb+1 dω Re σ (ω), σb ≈
Figure 7. Test of the windowing procedure. We define a test spectrum containing two delta functions and a box function δ(ω) + δ(ω − 0.017) + (0.03 − ω) − (0.02 − ω). Taking ωmax = 1, we compute M = 25 000 Chebyshev moments Im analytically. We then estimate the spectrum Re σ (ω) using a Kaiser window. The top panel shows the raw estimate (ωb , σb ). The bottom panel shows the estimated spectrum resampled in M/15 equal bins. Spectral leakage is clearly negligible ( 10−8 ) for |ω − ωexact | 10−3 .
ωb
2 πb ωb = sin τ M
(m = 0, 1, 2, . . . , M).
(C3)
For the Kaiser window with a width parameter β = 24 as defined above, the spectral leakage function has a central lobe of width β/π ≈ 15 bins, and the ratio of centrallobe to side-lobe amplitude is about 108 . Thus, if the true Re σ (ω) contains a delta function at some frequency, the numerically calculated weights σb will have the delta function spread over about 15 consecutive bins. To save memory, and for later convenience, it makes sense to resample the spectrum into approximately M/15 equal bins. This should be done in a way that preserves the sum rule. Thus, we calculate the cumulative distribution function C(ωb ) for b = 0, 1, 2, . . . , M; interpolate this and evaluate C(ωj ) at equally spaced frequencies ωj , j = 0, 1, 2, . . . , M/15 ; and take differences to give the new weights [56]. The final result can be quoted, roughly speaking, as an estimate of Re σ (ω) with frequency resolution (horizontal error bars) ω ≈ 15ωMmax and ‘vertical error bars’ σ ≈ 10−8 . Figure 7 illustrates the result of this procedure for a test spectrum. Other than the Kaiser window, there are other possible window functions such as the Dolph–Chebyshev window (which is optimal but slightly more complicated to set up), and the window function derived by Storck and Wolff [57].
Figure 8. Dynamical conductivity Re σ (ω); L of square lattice Lij Ci model for various system sizes L, on a log–log plot. The dynamical conductivity is almost independent of L. The only discernible finite-size effect is that the lowest-frequency data point lies slightly above the curve. Thus, the data for L = 6144 have no significant finite-size effects down to ω ≈ 0.0015. Error bars were estimated by comparing 2–32 disorder realizations.
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As mentioned in the text, the dynamical response σ (ω) is mainly determined by finite-sized clusters that are smaller than the system sizes in this study. Thus, finite-size errors are practically negligible, as shown in figure 8. 6
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