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Monte Carlo study of a two-dimensional array of Josephson junctions
Physica C 153-155 (1988) 312-313 North-Holland, Amsterdam
MONTE CARLO STUDY OF A TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
ARRAY
OF
J. M. ttoulrik and M. A. Novotny
Bergen Scientific Centre, Alldgaten 36, N-5007 Bergen, Norway A q u a n t u m Monte Carlo study of a two-dimensional array of Josephson junctions including the effects of Josephson tunnelling and long-range Coulomb interactions has been performed. The inclusion of long-range interactions does not measurably shift, the location of the Berezinskii-Kosterlitz-Thouless transition. 1. I N T R O D U C T I O N A realistic description of a granular thin film, well below the single grain bulk critical temperature, is expected to result from the llamiltonian 7-( = " H c + ~ j + 7fR + Tip, where the individual components represent respectively the charging energy 7/c, the Josephson tunnelling effect 7f j, the dissipation "/in associated with normal conduction of electrons, and the effect due to disorder in the array "HD. This model, with an applied magnetic field and only the terms 7fj and 7-/D, has been applied to the superconducting glassy state of high-temperature superconducting oxides (Miiller et al 1987, Morgenstern 19881. These materials are characterized by a n u m b e r of weak links occuring either at grain boundaries or at twin boundaries inside the superconducting domains themselves (Deutscher and Miiller 1987). In this report we consider only the effect of the first two terms, 7-[ = "He + 7~j. The Josephson Itamiltonian is taken to be "Ha = ~i,j Ej{1 - cos(¢i - Cj)}. The phase angle ¢i is, as usual, associated with the phase of the superconducting (Ginzburg-Landau) order parameter on each grain. The charging energy Hamiltonian is defined in terms of the charge opera~ ' which are canonically conjugate to tots Pi = - z °0¢i the phase variables (Anderson 1964), [p~, q~j] = -,Sij. The energy 7/c is given by
7~c -
( 2 e ) ~ - ~ # pipj, 2 C-1
(1.l)
i,j where C - I is a real, symmetric matrix. The diagonal elements of the matrix C are known as the capacity coefficients, while the off diagonal elements are the electrostatic induction coefficients (Landau and Lifshitz 1960). This form can be simply rewritten using
0921-4534/88/$03.50 © ElsevierSciencePublishersB.V. (North-HollandPhysicsPublishingDivision)
tleisenberg's equation of motion, - , h a°~t = [7fc,¢], and the commutator [pipj, ¢~] = -,(pi6jk + pj6ik), giving (Anderson 1964, Kampf and Schfn 1987)
i,j
Ot Ot
This is the expression we USe in the path integral representation. If we assume the distance between grains is large relative to their typical size, i. e. Cola << 1, where a is the lattice constant of a square array and Co is the capacity of a single grain, the elements of C -1 are respectively C~ 1 = l/Co and C;~ 1 = l / I r i - rj[. Thus C is completely specified by the ratio Co/a. 2. Q U A N T U M
MONTE
CARLO METHOD
The partition function can be formally written as a path integral Z ~_ f D[¢(r)]e-S[¢(~)]/~, where S[¢(r)] is the action in imaginary time r = zt:
s =
d~
~
+Ej ~{1 (ij)
\T~]
+
] - cos(C, - ¢ ~ ) ) / • J
(2.11
Temperature enters in the integration limits in the time-direction and the paths ¢(r) satisfy the periodic boundary condition ¢(0) = ¢(~h). We study a discrete-time version given by s L -
-
CiiLr
~c,,o~j
[flo:El Ei(l--cOS[(gi(Tk+l)--(Oi(Tk)]) -}-
E i c j sin[¢i(~k+~)
- ¢;(~)]sin[¢i(~+l)
-
where ~ = ~ , and e, the spacing between different time-slices, is given by /3h = ~L~. We start
J.M. Houlrik and M.A. Novotny / Two-dimensional array of Josephson junctions
with a physical L × L array of grains with periodic boundaries. Then using the expression given above, we simulate the model on a. on a L × L × L~ lattice using a standard importance sampling Monte Carlo technique. A transition to a superconducting phase can be followed by measuring the helicity modulus, which is proportional to the superfluid density. The helicity modulus is given by* T = -~z-[(~::~y k cos(¢~+1 ( T k ) hL~
L~ (Y:zyk sin(¢~+l(rk) -¢z(wk)))~] " In the picture of Berezinskii, Kosterlitz and Thouless, superconductivity is a result of long range phase coherence in the phase-variables ¢i (Berezinskii 1979, Kosterlitz and Thouless 1973). The BKT-temperature may be estimated as the point, where the helicity modulus intersects the line T(fl) = 2/rrfl.
313
7~K7.(c~) as estimated above for Co/a = 0.1. For this value of Cola, TBKT does not change mesurably from its value in the self-charging limit. Our present results on the normal to superconducting transition provide a small amount of new information compared to the behavior already observed when grain-grain interactions are neglected. We expect the additional quantum fluctuations caused by intergrain capacitances to be important for the firstorder quantum induced transition (QU1T) found at, low temperatures (Jacobs et al 1987, 1988). Simulations in this part of the phase diagram, as well as calculations including different C matrices and applied magnetic field are in progress.
1,00-
• ....................... II .................................................. •
ICOl" = 0.II
/'" rJl
i _ o.ao
i__
]COla= 0.1
]
,,F
1. o Alpha
. e . . o aLo)j
' -3o
./o
Temperature
1. o
F i g u r e 2. TBK T
as
a flmction of c~ for Co/a = 0.1.
REFERENCES
F i g u r e 1. 11elicity modulus as a function of temperature for Co/a = 0.1 and c~ = 1.0 and a = 2.0. Lattice sizes a r e 4 × 4 × 10 and 8 × 8 × 10. 3. R E S U L T S
Earlier MC-studies considering only the self-charging limit Cij = Co6q (Jacobs et a] 1984, 1987, 1988) have shown that the inclusion of this charging energy renormalizes the BKT-transition observed in the planar model ('H = "]~J) downwards. We have examined this behavior, when the off-diagonal charging energy is also included in the ttamiltonian. Figure 1 shows the helicity modulus measured as a. function of temperature for Cola = 0.1 and different values of c~. From this figure a transition temperature separating the phase-ordered state from the hightemperature disordered state can be obtained. As in the the self-charging limit, the critical temperature is found to be a decreasing function ofc~. Figure 2 shows
P. W. Anderson, in Lectures on the Many Body Problem, edited by E. R. Caianello (Academic, New York, 1964), Vol. 2, p 113 V. L. Berezhinskii, Zh. Theor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 34, 610 (1971)] G. Deutscher and K. A. Mfiller, Phys. Rev. Lett. 59, 1745 (1987) L. Jacobs, J. V. los6, and M. A. Novotny, Phys. Rev. Lett. 53, 2177 (1984) L. Jacobs, J. V. Josd, M. A. Novotny, and A. M. Goldman, Europhys. Lett. 3, 1295 (1987); Phys. Rev. B, to be published (1988). A. Kampf and G. SchSn, Phys. Rev. B36~ 3651 (1987) J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973) L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media, Chapter I, Pergamon (1960) I. Morgenstern, Z. Phys. B, to be published (1988) K. A. Miiller, M. Takashige, and J. G. Bednorz, Phys. Rev. Lett. 58, 1143 (1987)
MONTE CARLO STUDY OF A TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
ARRAY
OF
J. M. ttoulrik and M. A. Novotny
Bergen Scientific Centre, Alldgaten 36, N-5007 Bergen, Norway A q u a n t u m Monte Carlo study of a two-dimensional array of Josephson junctions including the effects of Josephson tunnelling and long-range Coulomb interactions has been performed. The inclusion of long-range interactions does not measurably shift, the location of the Berezinskii-Kosterlitz-Thouless transition. 1. I N T R O D U C T I O N A realistic description of a granular thin film, well below the single grain bulk critical temperature, is expected to result from the llamiltonian 7-( = " H c + ~ j + 7fR + Tip, where the individual components represent respectively the charging energy 7/c, the Josephson tunnelling effect 7f j, the dissipation "/in associated with normal conduction of electrons, and the effect due to disorder in the array "HD. This model, with an applied magnetic field and only the terms 7fj and 7-/D, has been applied to the superconducting glassy state of high-temperature superconducting oxides (Miiller et al 1987, Morgenstern 19881. These materials are characterized by a n u m b e r of weak links occuring either at grain boundaries or at twin boundaries inside the superconducting domains themselves (Deutscher and Miiller 1987). In this report we consider only the effect of the first two terms, 7-[ = "He + 7~j. The Josephson Itamiltonian is taken to be "Ha = ~i,j Ej{1 - cos(¢i - Cj)}. The phase angle ¢i is, as usual, associated with the phase of the superconducting (Ginzburg-Landau) order parameter on each grain. The charging energy Hamiltonian is defined in terms of the charge opera~ ' which are canonically conjugate to tots Pi = - z °0¢i the phase variables (Anderson 1964), [p~, q~j] = -,Sij. The energy 7/c is given by
7~c -
( 2 e ) ~ - ~ # pipj, 2 C-1
(1.l)
i,j where C - I is a real, symmetric matrix. The diagonal elements of the matrix C are known as the capacity coefficients, while the off diagonal elements are the electrostatic induction coefficients (Landau and Lifshitz 1960). This form can be simply rewritten using
0921-4534/88/$03.50 © ElsevierSciencePublishersB.V. (North-HollandPhysicsPublishingDivision)
tleisenberg's equation of motion, - , h a°~t = [7fc,¢], and the commutator [pipj, ¢~] = -,(pi6jk + pj6ik), giving (Anderson 1964, Kampf and Schfn 1987)
i,j
Ot Ot
This is the expression we USe in the path integral representation. If we assume the distance between grains is large relative to their typical size, i. e. Cola << 1, where a is the lattice constant of a square array and Co is the capacity of a single grain, the elements of C -1 are respectively C~ 1 = l/Co and C;~ 1 = l / I r i - rj[. Thus C is completely specified by the ratio Co/a. 2. Q U A N T U M
MONTE
CARLO METHOD
The partition function can be formally written as a path integral Z ~_ f D[¢(r)]e-S[¢(~)]/~, where S[¢(r)] is the action in imaginary time r = zt:
s =
d~
~
+Ej ~{1 (ij)
\T~]
+
] - cos(C, - ¢ ~ ) ) / • J
(2.11
Temperature enters in the integration limits in the time-direction and the paths ¢(r) satisfy the periodic boundary condition ¢(0) = ¢(~h). We study a discrete-time version given by s L -
-
CiiLr
~c,,o~j
[flo:El Ei(l--cOS[(gi(Tk+l)--(Oi(Tk)]) -}-
E i c j sin[¢i(~k+~)
- ¢;(~)]sin[¢i(~+l)
-
where ~ = ~ , and e, the spacing between different time-slices, is given by /3h = ~L~. We start
J.M. Houlrik and M.A. Novotny / Two-dimensional array of Josephson junctions
with a physical L × L array of grains with periodic boundaries. Then using the expression given above, we simulate the model on a. on a L × L × L~ lattice using a standard importance sampling Monte Carlo technique. A transition to a superconducting phase can be followed by measuring the helicity modulus, which is proportional to the superfluid density. The helicity modulus is given by* T = -~z-[(~::~y k cos(¢~+1 ( T k ) hL~
L~ (Y:zyk sin(¢~+l(rk) -¢z(wk)))~] " In the picture of Berezinskii, Kosterlitz and Thouless, superconductivity is a result of long range phase coherence in the phase-variables ¢i (Berezinskii 1979, Kosterlitz and Thouless 1973). The BKT-temperature may be estimated as the point, where the helicity modulus intersects the line T(fl) = 2/rrfl.
313
7~K7.(c~) as estimated above for Co/a = 0.1. For this value of Cola, TBKT does not change mesurably from its value in the self-charging limit. Our present results on the normal to superconducting transition provide a small amount of new information compared to the behavior already observed when grain-grain interactions are neglected. We expect the additional quantum fluctuations caused by intergrain capacitances to be important for the firstorder quantum induced transition (QU1T) found at, low temperatures (Jacobs et al 1987, 1988). Simulations in this part of the phase diagram, as well as calculations including different C matrices and applied magnetic field are in progress.
1,00-
• ....................... II .................................................. •
ICOl" = 0.II
/'" rJl
i _ o.ao
i__
]COla= 0.1
]
,,F
1. o Alpha
. e . . o aLo)j
' -3o
./o
Temperature
1. o
F i g u r e 2. TBK T
as
a flmction of c~ for Co/a = 0.1.
REFERENCES
F i g u r e 1. 11elicity modulus as a function of temperature for Co/a = 0.1 and c~ = 1.0 and a = 2.0. Lattice sizes a r e 4 × 4 × 10 and 8 × 8 × 10. 3. R E S U L T S
Earlier MC-studies considering only the self-charging limit Cij = Co6q (Jacobs et a] 1984, 1987, 1988) have shown that the inclusion of this charging energy renormalizes the BKT-transition observed in the planar model ('H = "]~J) downwards. We have examined this behavior, when the off-diagonal charging energy is also included in the ttamiltonian. Figure 1 shows the helicity modulus measured as a. function of temperature for Cola = 0.1 and different values of c~. From this figure a transition temperature separating the phase-ordered state from the hightemperature disordered state can be obtained. As in the the self-charging limit, the critical temperature is found to be a decreasing function ofc~. Figure 2 shows
P. W. Anderson, in Lectures on the Many Body Problem, edited by E. R. Caianello (Academic, New York, 1964), Vol. 2, p 113 V. L. Berezhinskii, Zh. Theor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 34, 610 (1971)] G. Deutscher and K. A. Mfiller, Phys. Rev. Lett. 59, 1745 (1987) L. Jacobs, J. V. los6, and M. A. Novotny, Phys. Rev. Lett. 53, 2177 (1984) L. Jacobs, J. V. Josd, M. A. Novotny, and A. M. Goldman, Europhys. Lett. 3, 1295 (1987); Phys. Rev. B, to be published (1988). A. Kampf and G. SchSn, Phys. Rev. B36~ 3651 (1987) J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973) L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media, Chapter I, Pergamon (1960) I. Morgenstern, Z. Phys. B, to be published (1988) K. A. Miiller, M. Takashige, and J. G. Bednorz, Phys. Rev. Lett. 58, 1143 (1987)