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Effects of quasi-particle dissipation in small metallic junctions
Physica B 165&166 (1990) 975-976 North-Holland
EFFECTS OF QUASI.PARTICLE DISSIPATION IN SMALL METALLIC JUNCTIONS V.Scalia (*), G. Falci(*), R.Fazio(§) and G.Giaquinta(*) (*)Istituto di Fisica, Facolta' di Ingegneria, Universita' di Catania, viale A.Doria 6 - 95129Catania,ITALY. (§)Faculty of Applied Physics, Delft University of Technology, LorenlZweg I - CJ 2628 Delft, The Netherlands. We present a Monte Carlo simulation of a small metallic junction with quasi-particle dissipation. In the present contribution we report results concerning the phase correlation functions and the specific heat of the system. We observe the possibility of a phase transition as the strength of dissipation is varied.
1. INTRODUCI10N
Quantum dissipative systems at low temperatures are subject of current theoretical and experimental interest. In these systems , dissipation can arise from the tunnelling of quasi-particle or from leakage currents through the substrate. Depending on the nature of dissipation, different outcomes stem out (1). Starting from the Caldeira-Leggett model (2), Schmid has shown that a Josephson junction with Otunic dissipation undergoes a phase transition from a resistive to a superconducting state depending on the value of the shunt resistance. In the case of quasi-particle dissipation, the proper Euclidean effective action was obtained in reference (3). The partition function can be expressed as a path integral (in imaginary time) over a macroscopic field lp('t) (4). Guinea and ScMn (5) studied the phase transition induced by quasi-particle dissipation; in this case the critical value of the dissipation strength is not universal but depends on the Josephson energy EJ. In the limit in which the Josephson energy vanishes, the T=O is equivalent to the Hamiltonian of the one dimensional XY model with 1/n2 interaction (n being the distance between lattice sites in time direction). Although long range attractive interactions can induce critical behavior even in low-dimensional spin systems (6), the 1/n2 decay represents a marginal case for which the possibility of a phase transition is still an open question. Simanek and Brown (7), using a self-consistent harmonic approximation, showed that for T ~O the conductance of the metal junction exhibits a sudden change from activated to an Ohmic behavior when the nominal junction conductance exceeds a certain threshold. A subsequent Monte Carlo simulation (8) indicates the possibility of a phase transition with infinite susceptibility in the strong dissipation limit The obtained results strongly resemble the well known properties of the two dimensional XY model with short range interaction where a KosterlilZ-Thouless-Berezinskii (KTB) transition occurs. In the present contribution we will analyze the phase-phase correlation function and the specific heat. 2. lHEMODEL The euclidean effective action obtained in (4) involves two
0921·4526/90/$03.50
© 1990 -
contributions relative to the electrostatic effects and to the presence of the quasi-panicle tunneling:
P
d
P
S[cp]= (1/8EQ)Jd't(Cft)2 -Jd'td't'a.('t--'t')coS[(lp('t)-CP('t'»!2] where Eo is the charging energy and a.('t) is the dissipative kernel (to be specified below).The phase CPi is related to the voltage difference across the junction by the relation dcp/d't = 2eV. All the quantities are expressed in units of II. The action dermed above realizes a one-dirnensional XY model and the range of the interaction is dictated by the functional form of the dissipative kernel, the strength playing the role of the inverse temperature. The charging contribution to the action is not relevant as far a the critical properties are concerned providing only a high frequency cut-off. In the discretized time version of the action the charging energy introduces a nearest-neighbor interaction (9), which can be neglected and the quasi-particle damping reads: S[cp]= ~ a(k - k')cos[(CPk -CPk')!2]
[1]
kk'
where , _ 2 (?t/N)2 a.(k - k) - (1/4n; RN)sin 2 (n;(k-k')1N)
[2]
In eq. (2) RN is the nominal resistance of the junction, N is the number of lattice sites (labelled by k and k') and it goes to infinity when T=O. The variables CPk are dermed in [0,211] and the boundary CP()=CPN holds. Visualizing (1) as the XY hamiltonian RN plays the role of the temperature and adopting the usual notation we define PQJ = 1/4n;2RN ; the subscript Q means that PQ controls quantum fluctuations.
It seems to us that a further improvement of the understanding of the transition envisaged in (8) requires the analysis of the the various correlation functions. We calculated the phase-phase correlator defined as: g(kl - k2 ) =
Elsevier Science Publishers B.V. (North-Holland)
-'Pkil/2l>
[3]
V. Scalia, G. Falci, R. Fazio, G. Giaquinta
976
where the average is calculated over the ensemble dermed by [2]. Simanek found (10) that in the hannonic approximation the correlation function exihbits an algebraic decay in the strong coupling limit. 3. RESULTS AND CONCLUSIONS We used the Metropolis Monte Carlo procedure to simulate the model dermed above. The ensemble averages were calculated over 7000 passes after 5000 thermalization steps. Many independent runs, both heating and cooling, were performed and no hysteresis was detected. Starting from low temperatures, the initial configuration with aligned phases was chosen. In figure 1 we plot the phase correlator [3] in a log-log scale; the data clearly show the change from algebraic to exponential decay when (PQJ)-l exceeds a critical value consistent with that obtained in (8).From a preliminar analysis the characteristic exponent 1'\ has a non linear increase with (PQJ)-l , tending to a value = .5 when the critical point is approached.
10-1
,..... JJ,
~
.
......... . 10-3
-.,.:
FIGURE 1 The phase correlator is plotted against the distance between the lattice sites. 0 (PQJ)-l =0.9; D(PQJ)-l =1.0; .(PQJ)-l = 1.2
A more detailed analysis of the behavior of the exponent 1'\ will be presented elsewhere (9). For the sake of completeness
we present in figure 2 the results obtained for the specific heat that fully confirm the previous simulation of (8). A further understanding of the picture emerging from this analysis comes from the behavior of the helicity modulus which proved to be useful in characterizing the transition (11).
5rr-----,r---.--::----.------, 4
3
Cv 2
o LQ..odlt:::....---L_ _-L._-...,L-_...J o 0.5 1 1.5 2 (,Ba J )"1
FIGURE 2 It is shown the behavior of the specific heat in the critical region.
ACKNOWLEDGMENTS We are particularly grateful to Hilda Cerdeira for the constant support and for making us available the computer facilities at ICTP (Trieste) REFERENCES (1) Proc. of the NATO Advanced Workshop on Coherence in Superconducting Networks, J.E.Mooij and G.ScMn Eds. , Physica 8152 (1988) (2) A.O.Caldeira and A.J.Leggett, Ann.Phys. (N.Y.) 149, 374 (1983) (3) A.Schmid, Phys.Rev.Lett. 51, 1506 (1983) (4) V.Ambegaokar, U.Eckem and G.SchBn, Phys.Rev. Lett. 48, 1745 (1982); E.Ben-Jacob E.Mottola and G.ScMn, Phys.Rev.Lett. 51,2064 (1983) (5) F.Guinea and G.SchOn, J.Low temp. Phys. 69, 219 (1987) (6) M.E.Fisher, S. Ma and B.Nickel, Phys.Rev.Lett. 29, 917 (1972); J. Banacharjee et al., Phys.Rev. B24, 3862(1980) (7) RBrown and E.Simanek, Phys.Rev. B34, 2957 (1986) (8) RBrown and E.Simanek, Phys.Rev. B38, 9264 (1988) (9) V.Scalia, G.Falci,RFazio and G.Giaquinta, to be published (10) E. Simanek, Phys.Lett. AU9, 477 (1987) (11) L.Jacobs et al.,Phys. Rev. B38, 4562 (1988)
EFFECTS OF QUASI.PARTICLE DISSIPATION IN SMALL METALLIC JUNCTIONS V.Scalia (*), G. Falci(*), R.Fazio(§) and G.Giaquinta(*) (*)Istituto di Fisica, Facolta' di Ingegneria, Universita' di Catania, viale A.Doria 6 - 95129Catania,ITALY. (§)Faculty of Applied Physics, Delft University of Technology, LorenlZweg I - CJ 2628 Delft, The Netherlands. We present a Monte Carlo simulation of a small metallic junction with quasi-particle dissipation. In the present contribution we report results concerning the phase correlation functions and the specific heat of the system. We observe the possibility of a phase transition as the strength of dissipation is varied.
1. INTRODUCI10N
Quantum dissipative systems at low temperatures are subject of current theoretical and experimental interest. In these systems , dissipation can arise from the tunnelling of quasi-particle or from leakage currents through the substrate. Depending on the nature of dissipation, different outcomes stem out (1). Starting from the Caldeira-Leggett model (2), Schmid has shown that a Josephson junction with Otunic dissipation undergoes a phase transition from a resistive to a superconducting state depending on the value of the shunt resistance. In the case of quasi-particle dissipation, the proper Euclidean effective action was obtained in reference (3). The partition function can be expressed as a path integral (in imaginary time) over a macroscopic field lp('t) (4). Guinea and ScMn (5) studied the phase transition induced by quasi-particle dissipation; in this case the critical value of the dissipation strength is not universal but depends on the Josephson energy EJ. In the limit in which the Josephson energy vanishes, the T=O is equivalent to the Hamiltonian of the one dimensional XY model with 1/n2 interaction (n being the distance between lattice sites in time direction). Although long range attractive interactions can induce critical behavior even in low-dimensional spin systems (6), the 1/n2 decay represents a marginal case for which the possibility of a phase transition is still an open question. Simanek and Brown (7), using a self-consistent harmonic approximation, showed that for T ~O the conductance of the metal junction exhibits a sudden change from activated to an Ohmic behavior when the nominal junction conductance exceeds a certain threshold. A subsequent Monte Carlo simulation (8) indicates the possibility of a phase transition with infinite susceptibility in the strong dissipation limit The obtained results strongly resemble the well known properties of the two dimensional XY model with short range interaction where a KosterlilZ-Thouless-Berezinskii (KTB) transition occurs. In the present contribution we will analyze the phase-phase correlation function and the specific heat. 2. lHEMODEL The euclidean effective action obtained in (4) involves two
0921·4526/90/$03.50
© 1990 -
contributions relative to the electrostatic effects and to the presence of the quasi-panicle tunneling:
P
d
P
S[cp]= (1/8EQ)Jd't(Cft)2 -Jd'td't'a.('t--'t')coS[(lp('t)-CP('t'»!2] where Eo is the charging energy and a.('t) is the dissipative kernel (to be specified below).The phase CPi is related to the voltage difference across the junction by the relation dcp/d't = 2eV. All the quantities are expressed in units of II. The action dermed above realizes a one-dirnensional XY model and the range of the interaction is dictated by the functional form of the dissipative kernel, the strength playing the role of the inverse temperature. The charging contribution to the action is not relevant as far a the critical properties are concerned providing only a high frequency cut-off. In the discretized time version of the action the charging energy introduces a nearest-neighbor interaction (9), which can be neglected and the quasi-particle damping reads: S[cp]= ~ a(k - k')cos[(CPk -CPk')!2]
[1]
kk'
where , _ 2 (?t/N)2 a.(k - k) - (1/4n; RN)sin 2 (n;(k-k')1N)
[2]
In eq. (2) RN is the nominal resistance of the junction, N is the number of lattice sites (labelled by k and k') and it goes to infinity when T=O. The variables CPk are dermed in [0,211] and the boundary CP()=CPN holds. Visualizing (1) as the XY hamiltonian RN plays the role of the temperature and adopting the usual notation we define PQJ = 1/4n;2RN ; the subscript Q means that PQ controls quantum fluctuations.
It seems to us that a further improvement of the understanding of the transition envisaged in (8) requires the analysis of the the various correlation functions. We calculated the phase-phase correlator defined as: g(kl - k2 ) =
Elsevier Science Publishers B.V. (North-Holland)
-'Pkil/2l>
[3]
V. Scalia, G. Falci, R. Fazio, G. Giaquinta
976
where the average is calculated over the ensemble dermed by [2]. Simanek found (10) that in the hannonic approximation the correlation function exihbits an algebraic decay in the strong coupling limit. 3. RESULTS AND CONCLUSIONS We used the Metropolis Monte Carlo procedure to simulate the model dermed above. The ensemble averages were calculated over 7000 passes after 5000 thermalization steps. Many independent runs, both heating and cooling, were performed and no hysteresis was detected. Starting from low temperatures, the initial configuration with aligned phases was chosen. In figure 1 we plot the phase correlator [3] in a log-log scale; the data clearly show the change from algebraic to exponential decay when (PQJ)-l exceeds a critical value consistent with that obtained in (8).From a preliminar analysis the characteristic exponent 1'\ has a non linear increase with (PQJ)-l , tending to a value = .5 when the critical point is approached.
10-1
,..... JJ,
~
.
......... . 10-3
-.,.:
FIGURE 1 The phase correlator is plotted against the distance between the lattice sites. 0 (PQJ)-l =0.9; D(PQJ)-l =1.0; .(PQJ)-l = 1.2
A more detailed analysis of the behavior of the exponent 1'\ will be presented elsewhere (9). For the sake of completeness
we present in figure 2 the results obtained for the specific heat that fully confirm the previous simulation of (8). A further understanding of the picture emerging from this analysis comes from the behavior of the helicity modulus which proved to be useful in characterizing the transition (11).
5rr-----,r---.--::----.------, 4
3
Cv 2
o LQ..odlt:::....---L_ _-L._-...,L-_...J o 0.5 1 1.5 2 (,Ba J )"1
FIGURE 2 It is shown the behavior of the specific heat in the critical region.
ACKNOWLEDGMENTS We are particularly grateful to Hilda Cerdeira for the constant support and for making us available the computer facilities at ICTP (Trieste) REFERENCES (1) Proc. of the NATO Advanced Workshop on Coherence in Superconducting Networks, J.E.Mooij and G.ScMn Eds. , Physica 8152 (1988) (2) A.O.Caldeira and A.J.Leggett, Ann.Phys. (N.Y.) 149, 374 (1983) (3) A.Schmid, Phys.Rev.Lett. 51, 1506 (1983) (4) V.Ambegaokar, U.Eckem and G.SchBn, Phys.Rev. Lett. 48, 1745 (1982); E.Ben-Jacob E.Mottola and G.ScMn, Phys.Rev.Lett. 51,2064 (1983) (5) F.Guinea and G.SchOn, J.Low temp. Phys. 69, 219 (1987) (6) M.E.Fisher, S. Ma and B.Nickel, Phys.Rev.Lett. 29, 917 (1972); J. Banacharjee et al., Phys.Rev. B24, 3862(1980) (7) RBrown and E.Simanek, Phys.Rev. B34, 2957 (1986) (8) RBrown and E.Simanek, Phys.Rev. B38, 9264 (1988) (9) V.Scalia, G.Falci,RFazio and G.Giaquinta, to be published (10) E. Simanek, Phys.Lett. AU9, 477 (1987) (11) L.Jacobs et al.,Phys. Rev. B38, 4562 (1988)