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Bond and flux-disorder effects on the superconductor-insulator transition of a honeycomb array of Josephson junctions
Physica B xxx (xxxx) xxx–xxx
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Physica B journal homepage: www.elsevier.com/locate/physb
Bond and flux-disorder effects on the superconductor-insulator transition of a honeycomb array of Josephson junctions Enzo Granato
⁎
Laboratório Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, SP, Brazil
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconductor-insulator transition Josephson-junction arrays Granular superconductors
We study the effects of disorder on the zero-temperature quantum phase transition of a honeycomb array of Josephson junctions in a magnetic field with an average of fo flux quantum per plaquette. Bond disorder due to spatial variations in the Josephson couplings and magnetic flux disorder due to variations in the plaquette areas are considered. The model can describe the superconductor-insulator transition in ultra-thin films with a triangular pattern of nanoholes. Path integral Monte Carlo simulations of the equivalent (2 + 1)-dimensional classical model are used to study the critical behavior and estimate the universal resistivity at the transition. The results show that bond disorder leads to a rounding of the first-order phase transition for fo = 1/3 to a continuous transition. For integer fo , the decrease of the critical coupling parameter with flux disorder is significantly different from that of the same model defined on a square lattice. The results are compared with recent experimental observations on nanohole thin films with geometrical disorder and external magnetic field.
1. Introduction As a theoretical model, a Josephson-junction array consisting of a lattice of superconducting grains coupled by the Josephson or proximity effect, has a very wide applicability. In addition to describing inhomogeneous superconductors [1,2], it is also closely related to the Bose-Hubbard model [3–5] and ultracold atoms on optical lattices [6,7]. The superconductor-insulator transition in Josephson-junction arrays is particularly interesting as a physical realizations of a quantum phase transition [2,3,8–12]. This transition can occur when quantum phase fluctuations due to charging effects of the grains are sufficiently large compared to the Josephson coupling to drive the system into an insulating state. Under an applied magnetic field, frustration effects lead to phase transitions with distinct universality classes, characterized by different critical exponents and universal conductivities at the transition, depending on the geometry of the array and the value of the frustration parameter f. The parameter f corresponds to the number of flux quantum per plaquette. There has been a growing interest on the superconductor-insulator transition of ultra-thin superconducting films with a triangular lattice of nanoholes [13–17]. A simple model for this system consists of a Josephson-junction array on a honeycomb lattice, with the triangular lattice of nanoholes corresponding to the dual lattice [12,18,19]. In the insulating phase near the transition, the magnetoresistance oscillates with the applied magnetic field at low temperatures [13], displaying
⁎
minima at integer values of f = n and maxima at f = n + 1/2 . This is analogous to the expected behavior for a Josephson-junction array when the ratio of the charging energy to the Josephson coupling is above the critical coupling, at different frustration parameters. However, a secondary minima on the magnetoresistance at f = n + 1/3 is not observed in the nanhole film as would be expected for a periodic Josephson-junction array on a honeycomb lattice [12,18]. Recently, the effect of controlled amount of positional disorder of the nanoholes near the superconductor-insulator transition was also investigated experimentally [16,17]. For Josepshon-junction arrays, positional disorder of the grains or disorder in the plaquette areas [20,21], leads to radomness in the magnetic flux per plaquette which increases with the applied field. The experimental data on nanohole ultrathin films shows that the amplitude of the magnetoresistance oscillations decreases with disorder and disappears above a critical value. At average integer frustration fo , the critical coupling for the transition also decreases with disorder. Numerical results obtained with a honeycomb array of Josesphon-junctions for the critical disorder and the behavior of the resistivity for integer and noninteger fo appear consistent with the experimental data [19]. However, a quantitative comparison of the decrease of the critical coupling with disorder [17] for integer fo was also found to be consistent with a similar model on a square lattice [11]. In this work, we study the superconductor-insulator transition at zero temperature using a self-charging model of Josephson-junction
Corresponding author. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.physb.2017.09.086 Received 15 May 2017; Received in revised form 19 September 2017; Accepted 21 September 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: granato, e., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.09.086
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arrays on a honeycomb lattice with an average of fo flux quanta per plaquette. Bond disorder due to spatial variations in the Josephson coupling and magnetic flux disorder due to variations in the plaquette areas are considered. The path integral representation of the model corresponds to a (2+1)-dimensional classical model, which is used to investigate the critical behavior by extensive Monte Carlo (MC) simulations. The conductivity at the transition is also estimated. The results show that bond disorder leads to a rounding of the first-order phase transition for fo = 1/3 to a continuous transition. For average integer fo , the decrease of the critical coupling parameter with flux disorder is significantly different from that of the same model defined on a square lattice. The results are compared with recent experimental observations on nanohole ultrathin films [13,15–17].
The 3D classical Hamiltonian in Eq. (2) is used for the MC simulations, regarding g as a temperature-like parameter. The honeycomb lattice is defined on a rectangular geometry with linear size given by a dimensionless length L. In terms of L, the linear size in the x and y 3 directions correspond to L x = L 3 and L y = 2 L , respectively. The parallel tempering method [22] is used for equilibrium MC simulations with periodic boundary conditions. Since the correlation lengths in the spatial and imaginary-time directions are related by dynamical scaling as ξτ ∝ ξ z , the finite-size scaling analysis is performed for different sizes L with L τ = aLz , where a is a constant aspect ratio. This requires to know the value of the dynamic exponent z in advance. In presence of disorder, it is difficult to determine the critical behavior with the above MC method since z ≠ 1 and long computer times are required for averaging over many realizations of disorder. To overcome this problem, simulations are performed with a driven MC dynamics method [19,23]. In this method, the layered honeycomb model of Eq. (2) is viewed as 3D superconductor and the corresponding ”current-voltage” scaling near the transition is used to determine the critical coupling and critical exponent [24]. In the presence of an external driving perturbation Jx (”current density”) which couples to the phase difference θτ , i + x − θτ , i along the x direction, the classical Hamiltonian of Eq. (2) is modified to
2. Model and monte carlo simulation We consider a Josephson-junction array model, described by the Hamiltonian [1,2,11,19,20]
=−
Ec 2
⎛ d ⎞2 ⎟ − dθi ⎠
∑ ⎜⎝ i
∑ Eij cos (θi − θj − Aijo ij
− tij ). (1)
The first term in Eq. (1) describes quantum fluctuations induced by the charging energy, Ec ni2 /2 , of a non-neutral superconducting grain located at site i of a honeycomb lattice, where Ec = 4e 2 / C , e is the electronic charge, and ni is the operator representing the deviation of the number of Cooper pairs from a constant integer value. C is an effective capacitance to the ground for each grain, which is assumed to be spatially uniform for simplicity. The second term in (1) is the Josephson-junction coupling between nearest-neighbor grains described by phase variables θi . The effect of the magnetic field B applied in the perpendicular (z -direction) appears through the link variables Aijo and tij , which satisfy the constraints ∑ij Aijo = 2πfo and ∑ij tij = 2πδfp , where the gauge-invariant sums ∑ij are over the links ij surrounding the site p of the plaquette centers. fo is a uniform constant parameter and δfp is a spatially varying random variable with zero average. The effects of the positional disorder of the nanoholes, which corresponds to random plaquette areas Sp of the array, can be incorporated in this model by identifying fo as the average number of flux quanta per plaquette BSo / Φo , where Φo = hc /2e is the flux quantum, and So as the uniform plaquette area of a reference lattice. δfp then represents the additional random flux fo δSp / So , where δSp = Sp − So . Experimentally, the flux disorder δfp can be varied by changing fo via the external field or the geometrical disorder δSp using different samples [16,17]. To study the quantum phase transition at zero temperature, we employ the imaginary-time path-integral formulation of the model [3]. In this representation, the 2D quantum model of Eq. (1) maps into a (2+1)D classical statistical mechanics problem. The extra dimension corresponds to the imaginary-time direction. Dividing the time axis τ into slices Δτ , the ground state energy corresponds to the reduced free energy F of the classical model per time slice. The classical reduced Hamiltonian can be written as
H=
HJ = H −
∑ < ij >, τ
∑ (θτ,i +x − θτ,i ). i, τ
(3)
The MC simulations are carried out using the Metropolis algorithm and the time dependence is obtained by identifying the time t as the MC time. The measurable quantity of interest is the phase slippage response (”nonlinear resistivity”) defined as Rx = Vx / Jx , where Vx = < d (θτ , i + x − θτ , i )/ dt >. Similarly, we define R τ as the phase slippage response in the layered (imaginary-time) direction. Above the phasecoherence transition, g > gc , Rx should approach a nonzero value when Jx → 0 while it should approach zero below the transition. From the nonlinear scaling behavior near the transition [19,23–25] of a sufficiently large system, one can extract the critical coupling gc , and the critical exponents ν and z. The conductivity at the transition is determined from the frequency and finite-size dependence of the phase stiffness γ (w ) in the spatial direction, following the scaling method described by Cha et al. [4,5]. The conductivity is given by the Kubo formula
σ = 2πσQ lim
wn →0
γ (iwn ) , wn
(4)
where σQ = (2e)2 / h is the quantum of conductance and γ (iwn ) is the frequency dependent phase stiffness [4,12] evaluated at the finite frequency wn = 2πn / L τ , with n an integer. At the transition, γ (iwn ) vanishes linearly with frequency and σ assumes a universal value σ *, which can be extracted from the scaling behavior [4]
σ (iwn ) σ* w 2π = − c( n − α )⋯ σQ σQ 2π wn L τ
1 − [ ∑ cos(θτ , i − θτ +1, i ) g τ,i +
Jx g
(5)
The parameter α is determined from the best data collapse of the frequency dependent curves for different systems sizes in a plot of σ (iwn) σQ
eij cos(θτ , i − θτ , j − Aijo − tij )],
w
2π
versus x = ( 2πn − α w L ). The universal conductivity is obtained from n τ the intercept of these curves with the line x = 0 .
(2)
where eij = Eij / EJ and τ labels the sites in the imaginary-time direction. The ratio g = (Ec / EJ )1/2 , which drives the superconductor to insulator transition for the model of Eq. (1), corresponds to an effective ”temperature” in the 3D classical model of Eq. (2). In general, a quantum phase transition shows intrinsic anisotropic scaling, with diverging correlation lengths ξ and ξτ in the spatial and imaginary-time directions [3], respectively. They are related by the dynamic critical exponent z as ξτ ∝ ξ z . Disorder in eij or tij are completely correlated in the imaginary-time direction.
3. Results 3.1. Bond disorder We first consider the effect of Josephson-coupling disorder on the nature of the transition at fo = 1/3. Disorder is introduced as Eij = EJ (1 ± D ) with equal probability and disorder parameter strength D. The nature of the transition can be determined from the behavior of 2
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Fig. 2. Phase slippage response R τ in the imaginary-time direction for fo = 1/2 near the Fig. 1. Free-energy barrier ΔF for fo = 1/3 as a function of bond disorder D and system
transition. Disorder strength D = 0.3 and L = 60 . The couplings g from top down are 1.2 , 1.18 , 1.16 , 1.14 , 1.12 , 1.1, 1.09 , 1.08, 1.07 and 1.06 . Inset: scaling plot for data near the transition with ξ = |g /gc − 1|−ν for gc = 1.095, z o = 2.6 , z = 1.22 and ν = 0.9 .
size L. Inset: restricted free energy A = −ln p (E ) as a function of energy density −E at the transition, for L = 48 .
the histogram p (E ) for the energy density near the transition. The structure and finite-size dependence of p (E ) provide information on the nature of the transition, if it is first order or continuous [26]. As found previously [12], in the pure case the transition is first order. Since such transition is characterized by phase coexistence, an interesting question is if the macroscopic coexistence is still possible in presence of weak disorder. The answer that comes from a recent theorem [27] is that phase coexistence is not possible in two dimensions for quantum systems at zero temperature in presence of disorder. In the present case, for sufficiently small values of the bond disorder parameter D, the restricted free energy A = −ln p (E ) near the transition still displays a double minima as shown in the Inset of Fig. 1. The two coexisting phases are separated by a free-energy barrier ΔF = AM − Am , where Am corresponds to one of the minima of A (E ) and AM corresponds to the maxima between them. Increasing bond disorder leads to a rounding of the first order transition, as the free energy barrier ΔF decreases, vanishing at some value of disorder parameter D c . For higher values of D, the transition becomes continuous. From Fig. 1, the apparent location of this tricritical point is D c ∼ 0.1 for a system size L = 48. However, since D c decreases significantly with increasing L, this finite value should be the result of the finite size effects and one expects that D c → 0 in the thermodynamic limit. For fo = 1/2 , on the other hand, the transition was found to be continuous in the pure case [12] and it remains continuous in presence of bond disorder. We determined the critical behavior for large bond disorder using the driven MC dynamics [23] described in Section 2. Figs. 2 and 3 show the behavior of the nonlinear phase slippage response Rx and R τ as a function of the applied perturbation Jx and Jτ coupling to the phase difference in the spatial and imaginary-time direction, respectively, for different couplings g and large system size. The behavior is consistent with a phase-coherence transition at an apparent critical coupling in the range gc ∼ 1.09 − 1.12 . For g > gc , both Rx and R τ tend to a finite value while for g < gc , they extrapolate to low values. Assuming the transition is continuous, the nonlinear response behavior sufficiently close the transition should satisfy a scaling form in terms of J and g. The critical coupling gc and critical exponents ν and z can then be obtained from the best data collapse. The scaling theory describing this behavior has been described in detail in ref. [25]. The nonlinear phase slippage response Rx and R τ should satisfy the scaling forms
gRx ξ z0 − z = F± (Jx ξ z +1/ g),
Fig. 3. Same as Fig. 2 but for the phase slippage response Rx in the x spatial direction. Inset: scaling plot for data near the transition with ξ = |g /gc − 1|−ν .
times, tx ∼ ξ zo and tτ ∼ ξτzo , in the spatial and imaginary-time directions, respectively, and ξ = |g / gc − 1|−ν . The + and - signs correspond to g > gc and g < gc , respectively. The two scaling forms are the same when z = 1, corresponding to isotropic scaling. The joint scaling plot according to Eqs. (6) and (7) are shown in the Insets of Figs. 2 and 3, obtained by adjusting the unknown parameters, providing the estimates gc = 1.095(5), z o = 2.6(1), z = 1.22(1) and ν = 0.9(1). We have also determined the conductivity at the critical point, as described in Section 2, for different values of fo . The calculations were performed using different system sizes with L τ = aLz , with the above estimates of z and gc . The scaling behavior for fo = 1/2 is shown in Fig. 4. The results for critical exponents and conductivity at different values of fo are summarized in Table 1. When compared with the pure σ* case [12], σ = 0.32(2) for fo = 0 and 1.29(2) for fo = 1/2 , bond disorder Q
changes significantly the value of the critical conductivity for fo = 1/2 while it remains essentially unchanged for fo = 0 . Finally, we show in Fig. 5 the behavior of the critical coupling gc for different values of frustration and increasing bond disorder. Disorder has the effect of washing out the secondary maximum at fo = 1/3 while still preserving the periodicity in fo with period 1. As will be discussed in Section 4, this implies that for large bond disorder the magnetoresistance should display minima at integer values of fo = n and maxima at fo = n + 1/2 but without secondary minima at fo = n + 1/3.
(6)
and
gRτ ξ z + z0 z −2 = H ± (Jτ ξ 2 / g),
3.2. Flux disorder
(7)
We now consider the effects of flux disorder, which introduces
where z o is an additional critical exponent describing the MC relaxation 3
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Fig. 6. Critical couplings gc for the superconductor-insulator transition at average
Fig. 4. Conductivity σ (iwn ) at the critical coupling gc = 1.095 for fo = 1/2 with bond w 2π , with α = 0.30 and z = 1.22 . disorder D = 0.3 as a function of the variable x = n − α 2π
integer frustration fo for increasing flux disorder Df . For the square lattice, they are
wn L τ
normalized to coincide with the honeycomb lattice at Df = 0 . Data for the honeycomb
The universal conductivity is given by the intercept with the x = 0 dashed line, leading to σ* = 0.94(3) .
lattice are replotted from ref. [19].
σQ
fo , the decrease of the critical coupling parameter for the honeycomb lattice with flux disorder is significantly different from that of the same model defined on a square lattice. This indicates that the lattice geometry of the array should be taken into account for a quantitative comparison with the experimental data.
Table 1 Critical exponents z, ν and critical conductivity σ* at different frustration parameters fo , with bond disorder D = 0.3.
fo
0
1/2
1/3
1/5
1/6
z ν σ */σQ
1.04(4) 0.9(1) 0.29(3)
1.22(4) 0.9(1) 0.94(3)
1.15(4) 0.95(9) 0.78(3)
1.20(4) 1.0(3) 0.77(3)
1.20(4) 1.0(3) 0.78(3)
4. Superconducting nanohole thin films We now compare the expected behavior of the magnetoresistance for the present model with experimental observations on ultrathin superconducting films with a triangular lattice of nanoholes [13–17] at low applied magnetic fields. The superconductor to insulator transition is tuned by decreasing the film thickness, which corresponds to decreasing the Josephson coupling and therefore increasing the coupling parameter g = Ec / EJ in the Josephson-junction array model. Measurements of the resistance in the insulating phase near the transition show magnetoresistance oscillations with the applied magnetic field, displaying minima at integer values of f = n and maxima at f = n + 1/2 , consistent with the results for the pure model [12]. The resistance is thermally activated with an activation energy increasing roughly linearly with the deviation from the critical value of the film thickness both for f = 0 and f = 1/2 , which corresponds to an exponent a ≳ 1 for the behavior of the energy gap in the insulating phase, Δ ∝ |g − gc ( f )|a . The secondary minimum of the magnetoresistance at f = 1/3, predicted by the pure model, is not observed. This behavior however is consistent with the effects of quenched disorder considered in present work. As shown in Section 3, bond disorder due to randomness in the Josephson coupling affects the critical coupling gc ( f ) as well as the critical exponents z and ν for the transition. The numerical results in Fig. 5 show that indeed increasing disorder in Eij washes out the secondary maximum of gc versus fo at fo = 1/3, which corresponds to a minimum in the energy gap. Moreover, from the results in Table 1, the exponent characterizing the dependence of the energy gap in the insulating phase [28], a = νz , is a ∼ 1 both for fo = 0 and fo = 1/2 , which is consistent with the experimental observation for the activation energy [13]. The universal conductivity at the field-induced transition determined from the experiments is also consistent with the numerical simulations. In the experiments [13,14], a field-induced superconductor to insulator transition is observed at fo ≈ 0.15, with the value of the conductivity at transition about 2.7 times its value in the absence of the magnetic field. From Table 1, the closest rational approximation to the experimental frustration parameter is fo = 1/6 . Indeed, the ratio of the conductivity for fo = 1/6 to its value at fo = 0 is very close to the experimental result.
Fig. 5. Critical coupling gc for the superconductor-insulator transition at different values of frustration fo for increasing bond disorder strength D.
random gauge fields tij in the model of Eq. (1), as described in Section 2. For convenience, we assume a uniform disorder distribution of fluxes, δfp = Df [−1, 1] at different plaquettes p. As found previously [19], increasing Df changes significantly the values of the critical couplings gc at different average frustrations fo and washes out the periodic oscillations as a function of fo above a critical value Dfc ∼ 0.5. For integer fo , the critical coupling decreases with Df and saturates above D fc . This decreasing behavior is consistent with experimental data on thin films with an initial triangular lattice of nanoholes [17]. On the other hand, previous results for a similar model [11] defined on a square lattice but with uncorrelated disorder in tij , instead of disorder in the flux δfp , also shows a decreasing behavior. It is therefore interesting to compare the behavior for the square and honeycomb lattices within the same flux disorder model considered here. We have performed calculations for the square lattice using the method described in the previous section. As show in Fig. 6, for integer 4
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Desenvolvimento Científico e Tecnológico)-Brazil (304197/2015-7), and computer facilities from CENAPAD-SP.
More recently [16,17], the effect of controlled amount of positional disorder of the nanoholes near the superconductor-insulator transition was also investigated experimentally. Weak positional disorder of the grains or weak disorder in the plaquette areas [20], leads to disorder in the magnetic flux per plaquette which increases with the applied field. Besides changing the universality class of the transition, it limits the number of oscillations in the magnetoresistance, with decreasing amplitude. The results obtained with the present model for the critical disorder above which the oscillations disappear and the behavior of the resistivity for integer and noninteger frustration [19] are consistent with the experimental data. Although the quantitative comparison of the decrease of the critical coupling with disorder [17] for integer fo was also found to be consistent with a similar model on a square lattice [11], we believe that the appropriate geometry should be a honeycomb lattice.
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5. Conclusions
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(b) E. Granato, J. Phys.: Conf. Ser. 568 (2014) 022017. [13] M.D. Stewart Jr., Aijun Yin, J.M. Xu, J.M. Valles Jr., Science 318 (2007) 1273. [14] M.D. Stewart Jr., Aijun Yin, J.M. Xu, J.M. Valles Jr., Phys. Rev. B 77 (2008) 140501. [15] G. Kopnov, O. Cohen, M. Ovadia, K. Hong Lee, C.C. Wong, D. Shahar, Phys. Rev. Lett. 109 (2012) 167002. [16] H.Q. Nguyen, S.M. Hollen, J.M. Valles Jr., J. Shainline, J.M. Xu, Phys. Rev. B 92 (2015) 140501. [17] H.Q. Nguyen, S.M. Hollen, J. Shainline, J.M. Xu, J.M. Valles Jr., Sci. Rep. 6 (2016) 38166. [18] (a) E. Granato, Phys. Rev. B 87 (2013) 094517;
We have studied the effects of bond and magnetic flux disorder on the superconductor to insulator transition in a self-charging model of Josephson-junction arrays on a honeycomb lattice with an average fo flux quantum per plaquette. From Monte Carlo simulations in the pathintegral representation of the model, we determined the critical behavior and the universal conductivity at the transition. The behavior of the magnetoresistance oscillations observed experimentally in superconducting films with a triangular lattice of nanoholes [13,15– 17] are qualitatively consistent with the predictions from the model. We showed that the absence of secondary minima at fo = n + 1/3 predicted by the pure model and the approximately linear behavior of the activation energy with film thickness are due to effects of Josephson-coupling disorder. Flux disorder due to variations in the plaquette areas leads to a decrease of the critical coupling parameter with flux disorder at integer fo . We found that this behavior is quantitatively different from that of the same model defined on a square lattice. This indicates that the lattice geometry should be taken into account when comparing theoretical results [11,19] with the experimental data [16,17].
(b) E. Granato, Phys. Rev. B 85 (2012) 054508. [19] E. Granato, Phys. Rev. B 94 (2016) 060504. [20] (a) E. Granato, J.M. Kosterlitz, Phys. Rev. B 33 (1986) 6533; (b) E. Granato, J.M. Kosterlitz, Phys. Rev. Lett. 62 (1989) 823. [21] (a) M.G. Forrester, Hu Jong Lee, M. Tinkham, C.J. Lobb, Phys. Rev. B 37 (1988) 5966;
[22] [23] [24] [25] [26] [27] [28]
Acknowledgements This work was supported by CNPq (Conselho Nacional de
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Physica B journal homepage: www.elsevier.com/locate/physb
Bond and flux-disorder effects on the superconductor-insulator transition of a honeycomb array of Josephson junctions Enzo Granato
⁎
Laboratório Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, SP, Brazil
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconductor-insulator transition Josephson-junction arrays Granular superconductors
We study the effects of disorder on the zero-temperature quantum phase transition of a honeycomb array of Josephson junctions in a magnetic field with an average of fo flux quantum per plaquette. Bond disorder due to spatial variations in the Josephson couplings and magnetic flux disorder due to variations in the plaquette areas are considered. The model can describe the superconductor-insulator transition in ultra-thin films with a triangular pattern of nanoholes. Path integral Monte Carlo simulations of the equivalent (2 + 1)-dimensional classical model are used to study the critical behavior and estimate the universal resistivity at the transition. The results show that bond disorder leads to a rounding of the first-order phase transition for fo = 1/3 to a continuous transition. For integer fo , the decrease of the critical coupling parameter with flux disorder is significantly different from that of the same model defined on a square lattice. The results are compared with recent experimental observations on nanohole thin films with geometrical disorder and external magnetic field.
1. Introduction As a theoretical model, a Josephson-junction array consisting of a lattice of superconducting grains coupled by the Josephson or proximity effect, has a very wide applicability. In addition to describing inhomogeneous superconductors [1,2], it is also closely related to the Bose-Hubbard model [3–5] and ultracold atoms on optical lattices [6,7]. The superconductor-insulator transition in Josephson-junction arrays is particularly interesting as a physical realizations of a quantum phase transition [2,3,8–12]. This transition can occur when quantum phase fluctuations due to charging effects of the grains are sufficiently large compared to the Josephson coupling to drive the system into an insulating state. Under an applied magnetic field, frustration effects lead to phase transitions with distinct universality classes, characterized by different critical exponents and universal conductivities at the transition, depending on the geometry of the array and the value of the frustration parameter f. The parameter f corresponds to the number of flux quantum per plaquette. There has been a growing interest on the superconductor-insulator transition of ultra-thin superconducting films with a triangular lattice of nanoholes [13–17]. A simple model for this system consists of a Josephson-junction array on a honeycomb lattice, with the triangular lattice of nanoholes corresponding to the dual lattice [12,18,19]. In the insulating phase near the transition, the magnetoresistance oscillates with the applied magnetic field at low temperatures [13], displaying
⁎
minima at integer values of f = n and maxima at f = n + 1/2 . This is analogous to the expected behavior for a Josephson-junction array when the ratio of the charging energy to the Josephson coupling is above the critical coupling, at different frustration parameters. However, a secondary minima on the magnetoresistance at f = n + 1/3 is not observed in the nanhole film as would be expected for a periodic Josephson-junction array on a honeycomb lattice [12,18]. Recently, the effect of controlled amount of positional disorder of the nanoholes near the superconductor-insulator transition was also investigated experimentally [16,17]. For Josepshon-junction arrays, positional disorder of the grains or disorder in the plaquette areas [20,21], leads to radomness in the magnetic flux per plaquette which increases with the applied field. The experimental data on nanohole ultrathin films shows that the amplitude of the magnetoresistance oscillations decreases with disorder and disappears above a critical value. At average integer frustration fo , the critical coupling for the transition also decreases with disorder. Numerical results obtained with a honeycomb array of Josesphon-junctions for the critical disorder and the behavior of the resistivity for integer and noninteger fo appear consistent with the experimental data [19]. However, a quantitative comparison of the decrease of the critical coupling with disorder [17] for integer fo was also found to be consistent with a similar model on a square lattice [11]. In this work, we study the superconductor-insulator transition at zero temperature using a self-charging model of Josephson-junction
Corresponding author. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.physb.2017.09.086 Received 15 May 2017; Received in revised form 19 September 2017; Accepted 21 September 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: granato, e., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.09.086
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arrays on a honeycomb lattice with an average of fo flux quanta per plaquette. Bond disorder due to spatial variations in the Josephson coupling and magnetic flux disorder due to variations in the plaquette areas are considered. The path integral representation of the model corresponds to a (2+1)-dimensional classical model, which is used to investigate the critical behavior by extensive Monte Carlo (MC) simulations. The conductivity at the transition is also estimated. The results show that bond disorder leads to a rounding of the first-order phase transition for fo = 1/3 to a continuous transition. For average integer fo , the decrease of the critical coupling parameter with flux disorder is significantly different from that of the same model defined on a square lattice. The results are compared with recent experimental observations on nanohole ultrathin films [13,15–17].
The 3D classical Hamiltonian in Eq. (2) is used for the MC simulations, regarding g as a temperature-like parameter. The honeycomb lattice is defined on a rectangular geometry with linear size given by a dimensionless length L. In terms of L, the linear size in the x and y 3 directions correspond to L x = L 3 and L y = 2 L , respectively. The parallel tempering method [22] is used for equilibrium MC simulations with periodic boundary conditions. Since the correlation lengths in the spatial and imaginary-time directions are related by dynamical scaling as ξτ ∝ ξ z , the finite-size scaling analysis is performed for different sizes L with L τ = aLz , where a is a constant aspect ratio. This requires to know the value of the dynamic exponent z in advance. In presence of disorder, it is difficult to determine the critical behavior with the above MC method since z ≠ 1 and long computer times are required for averaging over many realizations of disorder. To overcome this problem, simulations are performed with a driven MC dynamics method [19,23]. In this method, the layered honeycomb model of Eq. (2) is viewed as 3D superconductor and the corresponding ”current-voltage” scaling near the transition is used to determine the critical coupling and critical exponent [24]. In the presence of an external driving perturbation Jx (”current density”) which couples to the phase difference θτ , i + x − θτ , i along the x direction, the classical Hamiltonian of Eq. (2) is modified to
2. Model and monte carlo simulation We consider a Josephson-junction array model, described by the Hamiltonian [1,2,11,19,20]
=−
Ec 2
⎛ d ⎞2 ⎟ − dθi ⎠
∑ ⎜⎝ i
∑ Eij cos (θi − θj − Aijo ij
− tij ). (1)
The first term in Eq. (1) describes quantum fluctuations induced by the charging energy, Ec ni2 /2 , of a non-neutral superconducting grain located at site i of a honeycomb lattice, where Ec = 4e 2 / C , e is the electronic charge, and ni is the operator representing the deviation of the number of Cooper pairs from a constant integer value. C is an effective capacitance to the ground for each grain, which is assumed to be spatially uniform for simplicity. The second term in (1) is the Josephson-junction coupling between nearest-neighbor grains described by phase variables θi . The effect of the magnetic field B applied in the perpendicular (z -direction) appears through the link variables Aijo and tij , which satisfy the constraints ∑ij Aijo = 2πfo and ∑ij tij = 2πδfp , where the gauge-invariant sums ∑ij are over the links ij surrounding the site p of the plaquette centers. fo is a uniform constant parameter and δfp is a spatially varying random variable with zero average. The effects of the positional disorder of the nanoholes, which corresponds to random plaquette areas Sp of the array, can be incorporated in this model by identifying fo as the average number of flux quanta per plaquette BSo / Φo , where Φo = hc /2e is the flux quantum, and So as the uniform plaquette area of a reference lattice. δfp then represents the additional random flux fo δSp / So , where δSp = Sp − So . Experimentally, the flux disorder δfp can be varied by changing fo via the external field or the geometrical disorder δSp using different samples [16,17]. To study the quantum phase transition at zero temperature, we employ the imaginary-time path-integral formulation of the model [3]. In this representation, the 2D quantum model of Eq. (1) maps into a (2+1)D classical statistical mechanics problem. The extra dimension corresponds to the imaginary-time direction. Dividing the time axis τ into slices Δτ , the ground state energy corresponds to the reduced free energy F of the classical model per time slice. The classical reduced Hamiltonian can be written as
H=
HJ = H −
∑ < ij >, τ
∑ (θτ,i +x − θτ,i ). i, τ
(3)
The MC simulations are carried out using the Metropolis algorithm and the time dependence is obtained by identifying the time t as the MC time. The measurable quantity of interest is the phase slippage response (”nonlinear resistivity”) defined as Rx = Vx / Jx , where Vx = < d (θτ , i + x − θτ , i )/ dt >. Similarly, we define R τ as the phase slippage response in the layered (imaginary-time) direction. Above the phasecoherence transition, g > gc , Rx should approach a nonzero value when Jx → 0 while it should approach zero below the transition. From the nonlinear scaling behavior near the transition [19,23–25] of a sufficiently large system, one can extract the critical coupling gc , and the critical exponents ν and z. The conductivity at the transition is determined from the frequency and finite-size dependence of the phase stiffness γ (w ) in the spatial direction, following the scaling method described by Cha et al. [4,5]. The conductivity is given by the Kubo formula
σ = 2πσQ lim
wn →0
γ (iwn ) , wn
(4)
where σQ = (2e)2 / h is the quantum of conductance and γ (iwn ) is the frequency dependent phase stiffness [4,12] evaluated at the finite frequency wn = 2πn / L τ , with n an integer. At the transition, γ (iwn ) vanishes linearly with frequency and σ assumes a universal value σ *, which can be extracted from the scaling behavior [4]
σ (iwn ) σ* w 2π = − c( n − α )⋯ σQ σQ 2π wn L τ
1 − [ ∑ cos(θτ , i − θτ +1, i ) g τ,i +
Jx g
(5)
The parameter α is determined from the best data collapse of the frequency dependent curves for different systems sizes in a plot of σ (iwn) σQ
eij cos(θτ , i − θτ , j − Aijo − tij )],
w
2π
versus x = ( 2πn − α w L ). The universal conductivity is obtained from n τ the intercept of these curves with the line x = 0 .
(2)
where eij = Eij / EJ and τ labels the sites in the imaginary-time direction. The ratio g = (Ec / EJ )1/2 , which drives the superconductor to insulator transition for the model of Eq. (1), corresponds to an effective ”temperature” in the 3D classical model of Eq. (2). In general, a quantum phase transition shows intrinsic anisotropic scaling, with diverging correlation lengths ξ and ξτ in the spatial and imaginary-time directions [3], respectively. They are related by the dynamic critical exponent z as ξτ ∝ ξ z . Disorder in eij or tij are completely correlated in the imaginary-time direction.
3. Results 3.1. Bond disorder We first consider the effect of Josephson-coupling disorder on the nature of the transition at fo = 1/3. Disorder is introduced as Eij = EJ (1 ± D ) with equal probability and disorder parameter strength D. The nature of the transition can be determined from the behavior of 2
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Fig. 2. Phase slippage response R τ in the imaginary-time direction for fo = 1/2 near the Fig. 1. Free-energy barrier ΔF for fo = 1/3 as a function of bond disorder D and system
transition. Disorder strength D = 0.3 and L = 60 . The couplings g from top down are 1.2 , 1.18 , 1.16 , 1.14 , 1.12 , 1.1, 1.09 , 1.08, 1.07 and 1.06 . Inset: scaling plot for data near the transition with ξ = |g /gc − 1|−ν for gc = 1.095, z o = 2.6 , z = 1.22 and ν = 0.9 .
size L. Inset: restricted free energy A = −ln p (E ) as a function of energy density −E at the transition, for L = 48 .
the histogram p (E ) for the energy density near the transition. The structure and finite-size dependence of p (E ) provide information on the nature of the transition, if it is first order or continuous [26]. As found previously [12], in the pure case the transition is first order. Since such transition is characterized by phase coexistence, an interesting question is if the macroscopic coexistence is still possible in presence of weak disorder. The answer that comes from a recent theorem [27] is that phase coexistence is not possible in two dimensions for quantum systems at zero temperature in presence of disorder. In the present case, for sufficiently small values of the bond disorder parameter D, the restricted free energy A = −ln p (E ) near the transition still displays a double minima as shown in the Inset of Fig. 1. The two coexisting phases are separated by a free-energy barrier ΔF = AM − Am , where Am corresponds to one of the minima of A (E ) and AM corresponds to the maxima between them. Increasing bond disorder leads to a rounding of the first order transition, as the free energy barrier ΔF decreases, vanishing at some value of disorder parameter D c . For higher values of D, the transition becomes continuous. From Fig. 1, the apparent location of this tricritical point is D c ∼ 0.1 for a system size L = 48. However, since D c decreases significantly with increasing L, this finite value should be the result of the finite size effects and one expects that D c → 0 in the thermodynamic limit. For fo = 1/2 , on the other hand, the transition was found to be continuous in the pure case [12] and it remains continuous in presence of bond disorder. We determined the critical behavior for large bond disorder using the driven MC dynamics [23] described in Section 2. Figs. 2 and 3 show the behavior of the nonlinear phase slippage response Rx and R τ as a function of the applied perturbation Jx and Jτ coupling to the phase difference in the spatial and imaginary-time direction, respectively, for different couplings g and large system size. The behavior is consistent with a phase-coherence transition at an apparent critical coupling in the range gc ∼ 1.09 − 1.12 . For g > gc , both Rx and R τ tend to a finite value while for g < gc , they extrapolate to low values. Assuming the transition is continuous, the nonlinear response behavior sufficiently close the transition should satisfy a scaling form in terms of J and g. The critical coupling gc and critical exponents ν and z can then be obtained from the best data collapse. The scaling theory describing this behavior has been described in detail in ref. [25]. The nonlinear phase slippage response Rx and R τ should satisfy the scaling forms
gRx ξ z0 − z = F± (Jx ξ z +1/ g),
Fig. 3. Same as Fig. 2 but for the phase slippage response Rx in the x spatial direction. Inset: scaling plot for data near the transition with ξ = |g /gc − 1|−ν .
times, tx ∼ ξ zo and tτ ∼ ξτzo , in the spatial and imaginary-time directions, respectively, and ξ = |g / gc − 1|−ν . The + and - signs correspond to g > gc and g < gc , respectively. The two scaling forms are the same when z = 1, corresponding to isotropic scaling. The joint scaling plot according to Eqs. (6) and (7) are shown in the Insets of Figs. 2 and 3, obtained by adjusting the unknown parameters, providing the estimates gc = 1.095(5), z o = 2.6(1), z = 1.22(1) and ν = 0.9(1). We have also determined the conductivity at the critical point, as described in Section 2, for different values of fo . The calculations were performed using different system sizes with L τ = aLz , with the above estimates of z and gc . The scaling behavior for fo = 1/2 is shown in Fig. 4. The results for critical exponents and conductivity at different values of fo are summarized in Table 1. When compared with the pure σ* case [12], σ = 0.32(2) for fo = 0 and 1.29(2) for fo = 1/2 , bond disorder Q
changes significantly the value of the critical conductivity for fo = 1/2 while it remains essentially unchanged for fo = 0 . Finally, we show in Fig. 5 the behavior of the critical coupling gc for different values of frustration and increasing bond disorder. Disorder has the effect of washing out the secondary maximum at fo = 1/3 while still preserving the periodicity in fo with period 1. As will be discussed in Section 4, this implies that for large bond disorder the magnetoresistance should display minima at integer values of fo = n and maxima at fo = n + 1/2 but without secondary minima at fo = n + 1/3.
(6)
and
gRτ ξ z + z0 z −2 = H ± (Jτ ξ 2 / g),
3.2. Flux disorder
(7)
We now consider the effects of flux disorder, which introduces
where z o is an additional critical exponent describing the MC relaxation 3
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Fig. 6. Critical couplings gc for the superconductor-insulator transition at average
Fig. 4. Conductivity σ (iwn ) at the critical coupling gc = 1.095 for fo = 1/2 with bond w 2π , with α = 0.30 and z = 1.22 . disorder D = 0.3 as a function of the variable x = n − α 2π
integer frustration fo for increasing flux disorder Df . For the square lattice, they are
wn L τ
normalized to coincide with the honeycomb lattice at Df = 0 . Data for the honeycomb
The universal conductivity is given by the intercept with the x = 0 dashed line, leading to σ* = 0.94(3) .
lattice are replotted from ref. [19].
σQ
fo , the decrease of the critical coupling parameter for the honeycomb lattice with flux disorder is significantly different from that of the same model defined on a square lattice. This indicates that the lattice geometry of the array should be taken into account for a quantitative comparison with the experimental data.
Table 1 Critical exponents z, ν and critical conductivity σ* at different frustration parameters fo , with bond disorder D = 0.3.
fo
0
1/2
1/3
1/5
1/6
z ν σ */σQ
1.04(4) 0.9(1) 0.29(3)
1.22(4) 0.9(1) 0.94(3)
1.15(4) 0.95(9) 0.78(3)
1.20(4) 1.0(3) 0.77(3)
1.20(4) 1.0(3) 0.78(3)
4. Superconducting nanohole thin films We now compare the expected behavior of the magnetoresistance for the present model with experimental observations on ultrathin superconducting films with a triangular lattice of nanoholes [13–17] at low applied magnetic fields. The superconductor to insulator transition is tuned by decreasing the film thickness, which corresponds to decreasing the Josephson coupling and therefore increasing the coupling parameter g = Ec / EJ in the Josephson-junction array model. Measurements of the resistance in the insulating phase near the transition show magnetoresistance oscillations with the applied magnetic field, displaying minima at integer values of f = n and maxima at f = n + 1/2 , consistent with the results for the pure model [12]. The resistance is thermally activated with an activation energy increasing roughly linearly with the deviation from the critical value of the film thickness both for f = 0 and f = 1/2 , which corresponds to an exponent a ≳ 1 for the behavior of the energy gap in the insulating phase, Δ ∝ |g − gc ( f )|a . The secondary minimum of the magnetoresistance at f = 1/3, predicted by the pure model, is not observed. This behavior however is consistent with the effects of quenched disorder considered in present work. As shown in Section 3, bond disorder due to randomness in the Josephson coupling affects the critical coupling gc ( f ) as well as the critical exponents z and ν for the transition. The numerical results in Fig. 5 show that indeed increasing disorder in Eij washes out the secondary maximum of gc versus fo at fo = 1/3, which corresponds to a minimum in the energy gap. Moreover, from the results in Table 1, the exponent characterizing the dependence of the energy gap in the insulating phase [28], a = νz , is a ∼ 1 both for fo = 0 and fo = 1/2 , which is consistent with the experimental observation for the activation energy [13]. The universal conductivity at the field-induced transition determined from the experiments is also consistent with the numerical simulations. In the experiments [13,14], a field-induced superconductor to insulator transition is observed at fo ≈ 0.15, with the value of the conductivity at transition about 2.7 times its value in the absence of the magnetic field. From Table 1, the closest rational approximation to the experimental frustration parameter is fo = 1/6 . Indeed, the ratio of the conductivity for fo = 1/6 to its value at fo = 0 is very close to the experimental result.
Fig. 5. Critical coupling gc for the superconductor-insulator transition at different values of frustration fo for increasing bond disorder strength D.
random gauge fields tij in the model of Eq. (1), as described in Section 2. For convenience, we assume a uniform disorder distribution of fluxes, δfp = Df [−1, 1] at different plaquettes p. As found previously [19], increasing Df changes significantly the values of the critical couplings gc at different average frustrations fo and washes out the periodic oscillations as a function of fo above a critical value Dfc ∼ 0.5. For integer fo , the critical coupling decreases with Df and saturates above D fc . This decreasing behavior is consistent with experimental data on thin films with an initial triangular lattice of nanoholes [17]. On the other hand, previous results for a similar model [11] defined on a square lattice but with uncorrelated disorder in tij , instead of disorder in the flux δfp , also shows a decreasing behavior. It is therefore interesting to compare the behavior for the square and honeycomb lattices within the same flux disorder model considered here. We have performed calculations for the square lattice using the method described in the previous section. As show in Fig. 6, for integer 4
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Desenvolvimento Científico e Tecnológico)-Brazil (304197/2015-7), and computer facilities from CENAPAD-SP.
More recently [16,17], the effect of controlled amount of positional disorder of the nanoholes near the superconductor-insulator transition was also investigated experimentally. Weak positional disorder of the grains or weak disorder in the plaquette areas [20], leads to disorder in the magnetic flux per plaquette which increases with the applied field. Besides changing the universality class of the transition, it limits the number of oscillations in the magnetoresistance, with decreasing amplitude. The results obtained with the present model for the critical disorder above which the oscillations disappear and the behavior of the resistivity for integer and noninteger frustration [19] are consistent with the experimental data. Although the quantitative comparison of the decrease of the critical coupling with disorder [17] for integer fo was also found to be consistent with a similar model on a square lattice [11], we believe that the appropriate geometry should be a honeycomb lattice.
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We have studied the effects of bond and magnetic flux disorder on the superconductor to insulator transition in a self-charging model of Josephson-junction arrays on a honeycomb lattice with an average fo flux quantum per plaquette. From Monte Carlo simulations in the pathintegral representation of the model, we determined the critical behavior and the universal conductivity at the transition. The behavior of the magnetoresistance oscillations observed experimentally in superconducting films with a triangular lattice of nanoholes [13,15– 17] are qualitatively consistent with the predictions from the model. We showed that the absence of secondary minima at fo = n + 1/3 predicted by the pure model and the approximately linear behavior of the activation energy with film thickness are due to effects of Josephson-coupling disorder. Flux disorder due to variations in the plaquette areas leads to a decrease of the critical coupling parameter with flux disorder at integer fo . We found that this behavior is quantitatively different from that of the same model defined on a square lattice. This indicates that the lattice geometry should be taken into account when comparing theoretical results [11,19] with the experimental data [16,17].
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Acknowledgements This work was supported by CNPq (Conselho Nacional de
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