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Effect of the electromagnetic environment on the dynamics of charge and phase particles in
one-dimensional arrays of small Josephson junctions
View the table of contents for this issue, or go to the journal homepage for more 2011 EPL 96 47004
(http://iopscience.iop.org/0295-5075/96/4/47004)
EPL,96 (2011) 47004 www.epljournal.org doi:10.1209/0295-5075/96/47004
Effect of the electromagnetic environment on the dynamics
of charge and phase particles in one-dimensional arrays
of small Josephson junctions
I. L. Ho1, W. Kuo2, S. D. Lin3, C. P. Lee3, C. T. Liang4, C. S. Wu5(a) and C. D. Chen1,6
1Institute of Physics, Academia Sinica - Taipei 115, Taiwan
2Department of Physics, National Chung Hsing University - 250, Taichung, Taiwan
3Department of Electronic Engineering, National Chiao-Tung University - Hsinchu 300, Taiwan 4Department of Physics, National Taiwan University - Taipei 106, Taiwan
5Department of Physics, National Chang-Hua University of Education - ChangHua 500, Taiwan 6Department of Physics, National Chen-Kung University - Tainan 701, Taiwan
received 29 March 2011; accepted in final form 30 September 2011 published online 11 November 2011
PACS 74.40.Kb – Quantum critical phenomena
PACS 74.81.Fa – Josephson junction arrays and wire networks PACS 74.25.Dw – Superconductivity phase diagrams
Abstract – The effect of the electromagnetic environment on the dynamics of quasi-particles, Cooper pairs and phase particles in one-dimensional arrays of small Josephson junctions is investigated experimentally and theoretically. It is found that the environment enhances the phase ordering and thus suppresses quasi-particle tunneling at high temperature and localization of Cooper pairs at low temperature. The dynamics is studied in the context of phase-charge duality, and the experimental results are quantitatively analyzed in both charge-ordered and phase-ordered regimes. Based on these analyses, a low-temperature phase diagram as well as a finite-temperature crossover phase diagram are constructed and compared to the experimental diagrams.
Copyright c EPLA, 2011
The competition between phase-order and charge-order in superconducting systems containing small grains has been a long-standing yet fascinating subject of inter-est [1–6]. Strong inter-grain Josephson coupling locks the phase difference, and the system is in the phase-order regime. Conversely, strong Coulomb interaction suppresses inter-grain charge tunneling, and the system is in the charge-order regime. Upon cooling from high tempera-ture, the competition results in a continuous evolution from superconducting to quasi-reentrant [7] and to insulat-ing regimes provided that the strength of the Josephson coupling is comparable to that of the Coulomb interac-tion. However, the Josephson coupling and the Coulomb interaction are affected by the presence of external para-meters such as quasi-particles [8] and electromagnetic environment [3]. Recent theoretical [9–12] and experi-mental [13–15] advances revealed the important role of the electromagnetic environment on the fluctuations of
(a)E-mail: wucs@cc.ncue.edu.tw
phase and charge particles. In the phase-order regime, the electromagnetic environment can be considered to produce dissipation to the phase fluctuations and to support global superconductivity [16], which can eventu-ally lead to dissipative-phase transition [17]. In the charge-order regime, it provides electromagnetic energy needed for virtual tunneling in the Coulomb blockade regime and promotes charge transport [18]. A system consisting of lithographically made small Josephson junctions [19,20] provides a paradigmatic model for studying this competi-tion because here the charging energy can be designed precisely while the Josephson coupling energy can be controlled independently. The one-dimensional Josephson junction array (1D JJA) is an ideal system in which each junction is virtually decoupled from the measure-ment leads. The superconductor-insulator (SI) transition in 1D arrays of small Josephson junctions was previously explored [21] where the transition is controlled by tuning the Josephson coupling strength. Here, in addition to that, we show that the SI transition can also be tuned by
I. L. Ho et al.
changing the impedance of the electromagnetic environ-ment. This is of particular interest as it provides a direct test of the theory of dissipative-phase transition [17].
Using the 1D JJA as an example, in this work we study the effects of a two-dimensional electron gas (2DEG) environment [22,23] on the quasi-reentrant behavior, which reflects directly the dynamics of quasi-particles, phase particles and Cooper pairs. Experimentally, the Josephson coupling strength is varied by an applied magnetic field while the environment strength, which is inversely proportional to the impedance of the under-neath 2DEG sheet, is controlled by a pair of side-gates. Being able to tune both the Josephson coupling and the environment strength independently, we mapped out a low-temperature quantum phase diagram [24] in which the region for the quasi-reentrant behavior is identified. The quasi-reentrant behavior is characterized by two resistance turnover temperatures which divide the temperature dependence into three distinct regimes: from low temperature, they are charge-order regime, phase-order regime and quasi-particle dominating regime. Lowering the environment impedance will decrease the upturn temperature and increase the downturn temper-ature, suggesting enhancement of phase ordering. This, in turn, brings about the suppression of quasi-particle tunneling at high temperature and Cooper-pair blockade at low temperature. By modeling the environment as an ensemble of harmonic oscillators, the two turnover temperatures are calculated as a function of environ-ment and Josephson coupling strengths. Performing the calculations in both charge and phase presentations yields consistent results. Based on these calculations, the low-temperature phase diagram is extended to finite temperatures. This diagram agrees quantitatively with the diagram extracted from the experiment.
1D arrays comprising 100 aluminum SQUIDs (see inset of panel (A2) in fig. 1) are made on the top of a GaAs/AlGaAs hetero-structure, about 100 nm above the 2DEG sheet. Each SQUID consists of two parallel Josephson junctions with a junction area of 80× 180 nm2, corresponding to a sum junction capacitance C of about 1.5 fF [25] and a charging energyECP≡ 4e2/2C of about 212µeV. The arrays are fabricated by standard e-beam lithography and tilted-angle evaporation techniques as addressed in the previous works [19,21]. While the two arrays (denoted as A and B) present here have the same junction area, the junction resistances are different because of the difference in the tunnel barrier thickness. The Josephson coupling energy EJ0 can be
determined by using the Ambegoakar-Baratov relation-ship, EJ0≡ (∆/2)(RQ/RN). Here, RQ≡ h/4e2≈ 6.45 kΩ
is the quantum resistance,RN (6.75 kΩ for A and 7.7 kΩ
for B) is the measured normal state resistance of each SQUID (i.e. two parallel junctions) and ∆ = 200µeV is the superconducting energy gap. The normal state resistance can be controlled by the oxidation time of the bottom Al electrode before evaporation of the top
Vg +Vb/2 -Vb/2 R2deg -1.2 0.0 1.2 EJ/ECP=0.45 EJ/ECP=0.18 -0.50 -0.25 0.00 0.25 0.50 -0.5 0.0 0.5
I(nA)
Vb(mV)
α =70 α =0.01 -1.0 -0.5 0.0 0.5 1.0 -1 0 1I(nA)
EJ/ECP=0.39 EJ/ECP=0.15 -0.4 -0.2 0.0 0.2 0.4 -0.1 0.0 0.1Vb(mV)
α=70 α=5 0.0 0.4 0.8 1.2 1 2 R 0 ( M Ω ) T(K) (B1) (B2) (A1) (A2)Fig. 1: (Color online) Measured I-Vb characteristics of two arrays A and B. Panels (A1) and (B1) show the effects of suppressionEJ (from top traces) by the magnetic field when α was tuned to 70, whereas panels (A2) and (B2) illustrate the influence of decreasing α (from top traces) by the gate field when the EJ/ECP ratio was tuned to 0.26 and 0.29, respectively. The inset in panel (A2) is an optical microscope image of the measurement circuit. The inset in panel (B2) shows the quasi-reentrant behavior measured for array B with EJ/ECP= 0.29 and α = 5.
Al electrode. Accordingly, EJ0 of the SQUIDs in arrays A and B is 96.3 µeV and 83.8 µeV, respectively. The 2DEG sheet possesses a carrier concentration of about 5× 1011/cm2, yielding a sheet resistance R of about 80 Ω at 80 mK. The 1D arrays are placed at the center of a pair of side-gate electrodes which confines the under-neath 2DEG sheet into a long strip. The capacitance C0 between each superconducting island and 2DEG is estimated to be∼ 0.47 fF. The 2DEG structure is similar to that in ref. [23] except that in this work the backgate is replaced by a pair of metal side gates. The two ends of the strip are connected to Au pads via Ohmic contacts for measuring the 2DEG resistance. Through these Ohmic contacts, the zero-bias resistance of the 2DEG strip could be measured using a separated AC lock-in circuit. The gap between the two side-gates is about 5µm whereas the width of the 1D SQUID arrays is 1µm. The electrons in the strip were depleted by application of a negative voltage on the side-gates, causing an exponential increase in the 2DEG sheet resistance. The arrays were placed in a compartment in a dilution refrigerator equipped with a superconducting magnet, and the electric characteriza-tion was performed by using a symmetrical source-meter circuit to minimize any possible pick-up of common mode noises. The zero-bias resistance of the arrays was extracted from the current-voltage (I -Vb) traces taken at varying magnetic fields, side-gate voltages, and temperatures. A perpendicularly applied magnetic field B threading the SQUIDs with loop area A could reduce the Josephson coupling energy toEJ=EJ0cos(πB × A/Φ0);
here Φ0≡ h/2e is the flux quantum. The magnetic field corresponding to a flux quantum in the loop is about 42.5 Gs. The side-gate field has no measurable effect on the arrays themselves with similar EJ0 and ECP values;
this was confirmed separately on other 1D arrays made on bulk silicon chips. For a quantitative analysis, the strength of the environment is defined by a dimensionless conductance,α ≡ RQ/R2DEG.
Figure 1 illustrates the similarity between the effects of changingEJ and α on the I-Vb characteristics at low temperatures: decreasingEJ andα tends to suppress the critical current and to enhance the Coulomb blockade of Cooper-pair tunneling. However, it is noticed that the influence of α is prominent when the device EJ/ECP
value is tuned to be between 0.2 and 0.3 where both phase and charge fluctuations are significant. The small difference in the switching currents shown in panel (B1) is an indication of the uniformity of the junction parameters in the 1D arrays. The inset in fig. 2(a) shows a low-temperature phase diagram for array A with borders determined by the trend of R0(T ) at T = Tmin (Tmin≈
100 mK in the experiment). The phase diagram is largely divided into superconducting region I (dR0/dT |T =T min> 0) and insulating regions II and III (dR0/dT |T =T min< 0). However, here we are interested in region II in which the R0(T ) characteristics exhibit a downturn at Th and then an upturn at Tl upon cooling. As shown in the inset in panel (B2) of fig. 1, array B also exhibits a similar behavior. The upturning between Tl and Tmin
is a characteristic known as quasi-reentrant behavior, as shown in fig. 2(a). Figure 2(b) displays the I -Vb as
well as the differential conductance (Gd≡ dI/dVb) vs.Vb
curves at Th, Tl and Tmin. We note a clear evolution
from governing Josephson tunneling at Th to onset of
Coulomb blockade of Cooper-pair tunneling atTland then
to strong localization of Cooper pairs atTmin. Figure 2(a)
also shows the R0(T ) at different α. As α is increased,
R0 at all temperatures decreases; this is attributed to
the suppression of phase fluctuations (in the phase-order regime) as well as enhanced higher-order tunneling (in the charge-order regime). Moreover, we find thatThincreases andTl decreases with increasingα.
For a quantitative analysis of the quasi-reentrant behav-ior,ThandTlare calculated theoretically. The Lagrangian
of the system comprises items for an 1D JJA, a 2DEG sheet and the interaction between them and is given by [17,26]
Ltotal =LJJA+L2DEG+Linteraction=
1 2 ij Qi ˆ C−1 ij+ 1 4e2 3π 32∆Rt,ij ij Qj + i,j EJ[1− cos (ϕi− ϕj)] + 1 2 n (mnx˙2n −mnωn2x2n)− in Fin(Qi, ϕi, xn, λin). (1)
0.0
0.4
0.8
1.2
1.6
1
7
R
0(M
Ω
)
T (K)
(a) Tmin Tl Th α=0.1 α=67 α I II III 0.01 100 0 0.45 EJ / E CP Vb(mV) -0.5 0.0 0.5 0.0 0.4 0.8 -0.3 0.0 0.3 -0.15 0.15 G (µ S) I (nA ) Vb(mV) (b) Vb(mV) -0.5 0.0 0.5 0.0 0.4 0.8 -0.3 0.0 0.3 -0.15 0.15 G (µ S) I (nA ) Vb(mV) (b)Fig. 2: (Color online) (a) Quasi-reentrant R0(T ) traces at increasing α values for array A with EJ/ECP= 0.23. The arrows denote three temperature points,Th,TlandTmin, and the dashed curves mark α-dependences of Th (blue) and Tl
(orange). The error bars are large for low resistance values and small for high resistance values; shown in the plot are the maximum and minimum error bars. The correspondingα values are indicated in the inset in accordance with the color of theR0(T ) traces. The experimental data are presented as dots in theR0(T ) traces and as crosses in the inset. (b) Differential conductance as a function of bias voltage atT = Th(blue),Tl
(orange) andTmin(green). The inset shows I-Vb curves at the corresponding temperatures.
On the right-hand side, the first term describes the array charging energy for Cooper pairs including the renormal-ization correction to the Cooper-pair Coulomb interaction due to the presence of quasi-particles. The capacitance matrix [18] includes the junction capacitance and the capacitance to the 2DEG strip for diagonal elements and the junction capacitance for off-diagonal elements. In this way, the long-range Coulomb interaction enters automat-ically. The second term describes the Josephson coupling genergy [26]. The 2DEG environment is represented by an ensemble of harmonic oscillators with resonant (Matsub-ara) frequenciesωn= 2πnkBT [17]. In the last term, λin describes the coupling strength between superconducting island i and environment oscillator n. For simplicity, we model the 2DEG sheet as an Ohmic environment [27] by applying a constraint [26,28], n πλ2 in 2mnδ (ω − ωn) =R −1 2DEG. (2)
I. L. Ho et al.
This model is applicable for frequencies below the junction plasma frequency, which is about 300 GHz (or 14 K in temperature).
In the charge presentation, the EJ (i.e. the 2nd) term
is treated as a perturbation to the charge states and the environment (the 3rd and 4th) terms are taken into account by the P (E ) theory [18]. The tunneling rates for Cooper pairs (CPs) and quasi-particles (QPs) are given by the Fermi-Golden rule approximation [18]:
γCP=2πEJ2P (δ E˜ ch,CP), (3a) γQP = 1 e2Rt ∞ −∞dE ∞ −∞dE
N (E)N(E)f(E)[1 − f(E)]
×P (δEch,QP+E− E); (3b)
here P (E) is the probability function describing˜ the exchange of energy E between environment and Cooper-pairs tunneling. For quasi-particles, this prob-ability function is denoted as P (E ). N(E) = Θ (|E| − ∆) |E|/√E2− ∆2 is the BCS density of states with a
superconducting gap ∆ and f(E) is the Fermi-Dirac distribution.δEch,CP andδEch,QP are the energy changes
associated with the tunneling of the Cooper pairs and quasi-particles, respectively. Although γCP and γQP
are the tunneling rates for a single junction in the 1D array, the charge statuses of all islands in the array enter via the arguments δEch,CP and δEch,QP. In this
way, the net tunneling rate Γ for a junction is affected by the tunneling in the rest of the junctions and a correlation in the tunneling events is automatically established. Based on these two equations, the net tunneling rates at varying α and T can be calculated using the Monte Carlo technique and the result for ΓCP−←ΓCP, ΓQP−
←
ΓQP are displayed in figs. 3(a) and
(b), respectively. Details of the calculation technique are given in ref. [29]. ΓCP−
←
ΓCP and ΓQP−
←
ΓQP represent
the net tunneling rates for right-moving Cooper pairs and quasi-particles, which can be converted to current by simply multiplying the corresponding Coulomb charges. In panel (a), the temperatures corresponding to the maximum ΓCP−←ΓCP are marked by vertical arrows and are identified as Tl. Below Tl, ΓCP−
←
ΓCP decreases due
to the Coulomb blockade of the Cooper-pair tunneling. Above Tl, ΓCP−
←
ΓCP is suppressed due to the thermal
fluctuations of the island superconducting phase which follows a coth(1/T )-dependence [18] as addressed in the P (E ) theory. It is noted that the reentrant behavior exists even in the absence of quasi-particle tunneling (see the black dotted curve). The introduction of quasi-particle tunneling would affect the Cooper-pair tunneling in two ways (cf. the blue solid curve): Firstly, it would reduce the Cooper-pair tunneling rate because CP and QP tunneling are two competing processes as far as the
× ×108 (c) 0.2 0.4 0.6 2 4 6 8 10 R0 (M Ω ) T (K) α=0.2 α=0.7 α=1.2 Tl Th α=5.0 0.2 0.4 0.6 0 1 2 3 1 100 ΓQP -ΓQP T (K) 10 ΓCP -ΓCP (a) (b) ×105
Fig. 3: (Color online) Tunneling rates and zero-bias resistance calculated based on eqs. (3a) and (3b). Temperature depen-dence of a right-moving Cooper pair tunneling rate (a) and quasi-particle tunneling rate (b) calculated for a 20-junction array with the same junction parameter as array A. The blue and red curves are forα = 5 and α = 0.7, respectively. The black dotted curve in (a) is obtained by settingγQP= 0 andα = 5. Three downward arrows in (a) mark theTlvalues correspond-ing to the maximum Cooper-pair tunnelcorrespond-ing rate. The two verti-cal dashed lines in (b) corresponding to a sharp increase in the quasi-particle tunneling rate identify theThvalues. (c) Calcu-latedR0(T ) curves for different α. The two dashed lines mark
the Tl and Th values. To compare with the quasi-reentrant behavior shown in fig. 2(a), theEJ/ECP value is set to 0.23.
charging effect is concerned. Secondly, it gives rise to an additional dissipation to the phase fluctuations [30] and decreases Tl. On the other hand, since α represents
dissipation to the phase fluctuations, reducing α would raise Tl, as indicated by the red arrow. Regarding the
downturn dependence at high temperature, Th can be
identified as the temperature at which a sharp increase in the quasi-particle tunneling rate appears, as shown in fig. 3(b). Above Th, thermally assisted tunneling of quasi-particles gains importance and the transport is described by a simple activation behavior. The effect of the environment on Th can be understood through the Cooper-pair tunneling rate by comparing the blue and red curves in fig. 3(a). Fast Cooper-pair tunneling, as in the case of large α, would suppress quasi-particle tunneling and raise the Th value. Based on these calculations, the
array resistance at varying temperatures for different α is obtained and displayed in fig. 3(c), which exhibits a good agreement with the measurement results shown in fig. 2(a).
In the phase presentation, the Lagrangian is analyzed in the context of phase localization in the Josephson poten-tial well [31]. Similar to the Ginzburg-Landau mean-field theory for 2D and 3D junction arrays [32], the 1D JJAs are identified to be in the insulating phase when the phase correlation vanishes:EJcosϕij = 0. The presence of the environment (i.e.α = 0) introduces an effective reduction to ECP by a factor of 1 +αECP/2πωn. For T → 0, as
(a)
R
S
I
C
(b)
R
S
I
C
Fig. 4: (Color online) (a) Theoretical and (b) experimental crossover phase diagrams. The border surfaces are presented by different colors: blue forTl, red for Th and green forTm. The border surfaces divide the diagram into 4 regions: super-conducting (S), insulating (I), resistive (R) and conductive (C). Insets in (a) are schematic curves illustrating typicalI-Vb
characteristics in S, I and R/C regions. The projection of the intercept betweenTlandThborder surfaces onto the (EJ/ECP, α)-plane is identical to the blue dashed border shown in the inset of fig. 2(a).
α increases, the effective EJ/ECP value is increased and
can be brought across a critical value (EJ/ECP)∗. This gives rise to an insulator-to-superconductor transition, as illustrated by the red border in the low-temperature phase diagram shown in the inset of fig. 2(a). This is in quantitative agreement with the predication [28] that for T → 0 the border follows the relations (EJ/ECP)∗α = 2
for α 1 and (EJ/ECP)∗= 0.5 for α 1. In the T → 0
plane of the diagram, the SI phase transition is driven by changing EJ (along the EJ/ECP direction) and by
changing the dissipation (along theα direction). For T > 0, it is found that EJcosϕij = 0 has a double root for
(EJ/ECP,α) located in a region which coincides with the
quasi-reentrant region II defined in the low-temperature phase diagram. The two roots are identified asThandTl. When plotted as a function of (EJ/ECP, α), Th and Tl form a curled border surface, as displayed in fig. 4(a). This surface indicates the crossover between different
regions: a high-temperature resistive region, an inter-mediate superconducting region and a low-temperature insulating region. For a comparison, fig. 4(b) shows an experimental crossover phase diagram. This diagram can also be understood in charge presentation: For T > Th,
the increased quasi-particle tunneling suppresses the phase correlation and the array is pushed toward the resistive region. ForT < Tl, Cooper pairs are more localized, result-ing in strong fluctuations of phase and the array is thus moved toward the insulating region.
Outside the quasi-reentrant region, we show an addi-tional horizontal crossover surface (shown in green), which is referred to as Tm. This surface separates the diagram into a low-temperature “insulating” region and a high-temperature “conductive” region. The experimental crite-rion for Tm is the appearance of a dip structure in the
differential conductance (Gd≡ dI/dVb) vs. Vb curve at
the zero-bias point; see, e.g., orange and green traces in fig. 2(b). Within our measurement resolution,Tmseems to
overlap with theTl surface. TheTmsurface as a function
of EJ/ECP and α can also be calculated in the charge
presentation addressed above and the result is shown in fig. 4(a). The calculated Cooper-pair current shows a power-law dependence on Vb as I ∼ Va
b at low bias
volt-ages, and a border given by a = 1 separates the regions of bound charges (a > 1) and free charges (a 1) [31], which correspond to the insulating and conductive regions, respectively.
In summary, the effect of electromagnetic environment on the dynamics of charge and phase particles is studied by analyzing the quasi-reentrant behavior. By modeling the environment as an ensemble of harmonic oscillators, we calculated a finite-temperature crossover phase diagram, which agrees quantitatively with the experimental results. This study provides a leap toward understanding the effect of the electromagnetic environment on the phase-charge duality.
∗ ∗ ∗
Fruitful discussions with H.-J. Lee, S.-I. Lee and B. Altshulerare gratefully acknowledged. This research was funded by the National Science Council of Taiwan under contract Nos. NSC 98-2112-M-001-023-MY3 and NSC 99-2112-M-018-004-MY3. Technical support from NanoCore, the Core Facilities for Nanoscience and Nanotechnology at Academia Sinica, is acknowledged.
REFERENCES
[1] Dubi Y., Meir Y. and Avishai Y., Nature,449 (2007) 876.
[2] Chow E., Delsing P. and Haviland D. B., Phys. Rev. Lett., 81 (1998) 204.
I. L. Ho et al.
[4] Efetov K. B., Sov. Phys. JETP,51 (1980) 1015. [5] Feigel’man M. V., Korshunov S. E. and Pugachev
A. B., JETP Lett.,65 (1997) 566.
[6] Pop I. M., Protopopov I., Lecocq F., Peng Z., Pannetier B., Buisson O. and Guichard W., Nat. Phys., 6 (2010) 589.
[7] “Quasi-reentrant” is a term coined following the early works of Orr B. G., Jaeger H. M. and Goldman A. M., Phys. Rev. B, 32 (1985) 7586; Jaeger H. M., Haviland D. B., Orr B. G.and Goldman A. M., Phys. Rev. B, 40 (1989) 182. It should be noted that quasi-reentrance is not strictly a 1D effect.
[8] Kampf A. and Sch¨on G., Phys. Rev. B,36 (1987) 3651. [9] Lutchyn R. M., Galitski V., Refael G. and Sarma
S. D., Phys. Rev. Lett.,101 (2008) 106402.
[10] Wagenblast K. H., Otterlo A. V., Sch¨on G. and Zim´anyi G. T., Phys. Rev. Lett.,79 (1997) 2730. [11] Goswami P. and Chakravarty S., Phys. Rev. B, 73
(2006) 094516.
[12] Lobos A. M. and Giamarchi T., Phys. Rev. B,84 (2011) 024523.
[13] Corlevi S., Guichard W., Hekking F. W. J. and Haviland D. B., Phys. Rev. Lett.,97 (2006) 096802. [14] Miyazaki H., Takahide Y., Kanda A. and Ootuka
Y., Phys. Rev. Lett.,89 (2002) 197001.
[15] Penttil¨a J. S., Parts ¨U., Hakonen P. J., Paalanen M. A. and Sonin E. B., Phys. Rev. Lett., 82 (1999) 1004.
[16] Bobbert P. A., Fazio R., Sch¨on G.and Zaikin A. D., Phys. Rev. B, 45 (1992) 2294.
[17] Caldeira A. O. and Leggett A. J., Ann. Phys. (N.Y.), 149 (1983) 374.
[18] Ingold G. L. and Nazarov Yu V., in Single Charge Tunneling, edited by Grabert H. and Devoret M. H. (Plenum Press, New York and London) 1992, Chapt. 2, pp. 21–107.
[19] Lotkhov S. V., Bogoslovsky S. A., Zorin A. B. and Niemeyer J., Phys. Rev. Lett.,91 (2003) 197002. [20] Chen C. D., Delsing P., Haviland D. B., Harada Y.
and Claeson T., Phys. Rev. B,51 (1995) 15645. [21] Kuo W. and Chen C. D., Phys. Rev. Lett., 87 (2001)
186804.
[22] Kycia J. B., Chen J., Therrien R., Kurdak C., Campman K. L., Gossard A. C.and Clarke J., Phys. Rev. Lett., 87 (2001) 017002.
[23] Rimberg A. J., Ho T. R., Kurdak C. and Clarke J., Phys. Rev. Lett., 78 (1997) 2632.
[24] Refael G., Demler E., Oreg Y. and Fisher D. S., Phys. Rev. B, 75 (2007) 014522.
[25] This value was estimated assuming a specific capacitance of 45 fF/µm2, see Delsing P., Claeson T., Likharev K. K.and Kuzmin L. S., Phys. Rev. B,42 (1990) 7439. [26] Chakravarty S., Ingold G. L., Kivelson S. and
Luther A., Phys. Rev. Lett.,56 (1986) 2303.
[27] Wilhelm F. K., Sch¨on G. and Zim´any G. T., Phys. Rev. Lett., 87 (2001) 136802.
[28] Fisher M. P. A., Phys. Rev. Lett.,57 (1986) 885. [29] Ho I. L., Lin M. C., Aravind K., Wu C. S. and Chen
C. D., J. Appl. Phys.,108 (2010) 043907.
[30] Herrero C. P. and Zaikin A. D., Phys. Rev. B, 65 (2002) 104516.
[31] Panyukov S. V. and Zaikin A. D., J. Low Temp. Phys., 75 (1989) 361.