- Email: [email protected]
The charge unbinding transition in arrays of tunnel junctions
Physica B 165&166 (1990) 1127-1128 North-Holland
THE CHARGE UNBINDING TRANSITION IN ARRAYS OF TUNNEL JUNCTIONS Rosario Fazio and Gerd ScMn Department of Applied Physics, Delft University of Technology, Lorentzweg 1 - CJ 2628 Delft, The Netherlands. Two dimensional arrays of normal and superconducting low capacitance tunnel junctions provide a physical realization of a 2-d Coulomb gas. A Kosterlitz-Thouless-Berezinskii (KTB) phase transition can occur where charge anticharge pairs dissociate. Recent experiments suggest such a transition. It differs from the more familiar vortex unbinding transition. We discuss the influence of single electron tunneling and of the Josephson coupling on the transition temperature. 1. INTRODUCTION Phase transitions in junction arrays have attracted much attention. Most of the work concentrated on the phase coherence and the vortex unbinding transition (l) but little attention has been devoted to the properties of the charge gas. Charging effects have been demonstrated to be important in low capacitance junctions (2). In two dimensional arrays with dominant capacitances between the islands the interaction between the charges on the islands depends logarithmically on the distance, at least up to reasonably large values. This implies that a KTB transition can occur where charge anticharge pairs dissociate (3). Recent experiments on 2-d arrays of normal and superconducting low capacitance junctions hint at such a transition (3). The possibility of a charge KTB transition in superconducting granular fIlms in the limit where the charging energy is much larger than the Josephson coupling energy was indicated in Ref. 4. In the following we will discuss normal and superconducting arrays and study the influence of single electron tunneling and of the Josephson coupling on the transition.
2. ARRAYS OF NORMAL JUNCTIONS An array of normal tunnel junctions is described by the action (5)
t «it)
_ ~ _1_ A[q>]-Jm [ 16Eo
dq>i 2 _1_ ~ ~2 + 16Ec <~>( dt - dt) ]
- L fcttJ~t'a(t - t') cos[Pij(t);Pij(t')]
[1]
d The first two terms represent the charging energy due to the self-capacitance Co (Eo=e2/2Co) to the ground and the nearestneighbors capacitance C (Ec=e 2/2C), respectively. In the following we assume that the former is small (CO « C), which is well realized in the experiments of Ref. 3. The third term describes the dissipation due to quasi-particle tunnelling. The subscript i labels the islands of the array, and q>jj=q>i-q>j refers to nearest neighbors. The 'phase' q>i introduced here formally is related to the voltage on the island by the relation dq>Jdt = 2eVj. The dissipative kernel is a(t) = a ~2/sin2(1t't/~), the di 0921-4526/90/$03.50
© 1990 -
mensionless parameter a is defined as [2]
and RT is the junction resistance (except in [2] we choose units where 11 = 1). The partition function can be expressed as a path integral over the field q> with limits which depend on the allowed charge states of the system. Since the charge on each island is integer we have to choose q>i(~) = q>i(D) + 41tl1i, i.e. the partition function includes a summation over the winding numbers nj = D,±1,.... If we decompose q>j(t) = q>iO + t}i(t) + 41tnit/~, where t}j(D) = t}i(~) = D, the charging part of the action decouples into two terms, one Ac[t}] depending on t}i(t), the other depending on the winding numbers 2 1t2 1t2 Ac[n] = REo ~ n i + REc ....
I
....
L (ni - nj) 2
[3]
This action describes the Discrete Gaussian model (DGM) which has been studied in connection with the roughening transition (6). It shows a KTB transition at a temperature (0)
Tc = Ec!41t
[4]
which separates a low temperature phase where the charges form dipoles from the high temperature phase where free charges are present. The result [4] is obtained in the limit of vanishing self-capacitance; a finite Co rounds off the transition. The result [4] ignores the effect of electron tunneling, except that it is needed to establish a thermal equilibrium charge distribution. However, the tunneling itself influences the transition. In the limit of weak dissipation a « 1 we expand the partition function in a and obtain in lowest order the following effective action for the winding number part Aeff=Ac[n]-
~ ~ 21t JdtJdt'a(t-t')g(t-t') COS['Rnij(t-t)] [5] ....
L
Here g(t-t')=exp{ -<[t}ij(t)-t}ij(t')]2>c} is a phase correlation function evaluated with Ac[t}]. For small frequencies the correction can be expressed as a renormalization of the nearest
Elsevier Science Publishers B.V. (North-Holland)
R. Fazio, G. Schon
1128 neighbor capacitance. This means it leads transition temperature (0)
Tcrrc
to
a shift of the KTB
= 1-0.11 a
[6]
In the limit of strong dissipation it is more appropriate to expand the action in the ~ fields. In lowest order, the winding number contribution leads to the effective action Aeftln] = Adn] + 2a
L
Inj - njl
17]
where the particular form of the second term in [7] arises as Fourier transform of the dissipative kernel. At very low temperatures the effective action approaches the so called ASOS model (7) and we find KTB transition at a critical value of ac = 0.45 for l' = O. Finite temperatures lower this critical value to
ac = 0.45 [I -
(0)
(7r/4) Tffc ]
(0)
for l' « T c
[8]
3. SUPERCONDUCTING ARRAYS In the superconducting state, disregarding for the moment the effect of quasi-particles, the unit of charge is 2e and the charging energy is 4Ec. If the system is dominated by the charging energy we can proceed as described above. However, the difference in the allowed charge states leads to a different requirement for the limits of the path integrals, CPi(~) = CPi(O) + 21tl1i. It is clear that the corresponding KTB transition temperature is increased (4) - Ed 1'(0) cs 1t
[9]
The experimental data of Ref. 3 show a shift in the critical temperature supporting this picture. The Josephson tunneling and the quasi-particle tunneling modify this picture and lead to a shift of the critical temperature. The action of a Josephson junction is As[cp]= A[cp]- EJ
L
~
Jdt COSCPij(t)
[10]
If we treat the Josephson coupling perturbatively EJ « Ec we can determine the corrections to the bare KTB transition (similarly as we evaluated the corrections due to finite a) (0)
TcsITcs = I - 0.98 (EJ/Ec)
2
[11]
The effect of quasi-particle tunneling depends strongly of the relative values of the superconducting gap 6 and the charging energy Ec. If the gap is large 6 » Ec the tunneling processes are frozen out. Nevertheless, virtual transitions occur which renormalize the nearest-neighbor capacitance (5)
E~n/Ec= [I +~; arl
[12]
As a result the transition temperature [9] or [11] is lowered in an obvious way. On the other hand, in the limit where the superconducting gap is small 6 :s; Ec the quasi-particle transitions dominate the picture. The charges on the islands are then integer multiples of the single electron charge e, and the transition temperature is given by the normal array values. A crossover between the single electron and the Cooper pair charge unbinding transition should occur for 6 = Ec. Finally we consider the limit EJ » Ee. In this case the phase of the superconducting islands provides a more natural description. The elementary excitations of the system are spin waves and vortices. The latter are bound in pairs at low temperatures, but become free at a KTB transition temperature T~O) = EJ. In small capacitance junctions (decreasing values of EJ/Ec) the quantum fluctuations lower the transition temperature l'y. This effect has been studied in the literature (8), but mostly for the case where the self capacitance dominates. Due to the duality between the charge and the vortex picture we expeet that the charge and the vortex ordered states exclude one another. Hence with increasing values of EJ/Ec the system undergoes a transition from a charge ordered to a disordered to a phase ordered state ('ordered' in the sense of the KTB transition). 4. CONCLUSIONS We presented new results for the charge KTB transition in normal and superconducting arrays. In the low temperature phase the charges of the islands are bound in dipoles. This has strong effects on the transport properties of the system. In the case of Josephson arrays we find a phase diagram which is much more complex than anticipated before (4). In the limits where Ec or EJ are dominant the charge-KTB or the phaseKTB are found. When both are nonzero quantum fluctuations decrease the transition temperatures. REFERENCES (1) Proc. of the NATO Advanced Research Workshop on Coherence in Superconducting Networks, J.E. Mooij and G. Schtln eds., Physica B 152 (1988) (2) D.V. Averin and K.K. Likharev, in: 'Quantum effects in small disordered systems', edited by B.L. Altshuler, PA Lee and R.A. Webb, to be published. (3) J.E. Mooij, B.J. van Wees, L.J. Gcerligs, M. Peters, R. Fazio and G. SchOn, submitted to Phys. Rev. Lett (4) A. Widom and S. Badjou, Phys. Rev. B37, 7915 (1988) (5) V. Ambegaokar, U. Eckem and G. Schon, Phys. Rev. Len. 48,1745 (1982); Phys. Rev. B 30, 6419 (1984) (6) S.T. Chui and J.D. Weeks, Phys. Rev. B14, 4978 (1976) (7) R.H. Swendsen, Phys. Rev. B15, 5421 (1977) (8) L. Jacobs, J.V. Jose, M.A. Novotny and A.M. Goldman, Phys. Rev. 838,4562 (1988)
THE CHARGE UNBINDING TRANSITION IN ARRAYS OF TUNNEL JUNCTIONS Rosario Fazio and Gerd ScMn Department of Applied Physics, Delft University of Technology, Lorentzweg 1 - CJ 2628 Delft, The Netherlands. Two dimensional arrays of normal and superconducting low capacitance tunnel junctions provide a physical realization of a 2-d Coulomb gas. A Kosterlitz-Thouless-Berezinskii (KTB) phase transition can occur where charge anticharge pairs dissociate. Recent experiments suggest such a transition. It differs from the more familiar vortex unbinding transition. We discuss the influence of single electron tunneling and of the Josephson coupling on the transition temperature. 1. INTRODUCTION Phase transitions in junction arrays have attracted much attention. Most of the work concentrated on the phase coherence and the vortex unbinding transition (l) but little attention has been devoted to the properties of the charge gas. Charging effects have been demonstrated to be important in low capacitance junctions (2). In two dimensional arrays with dominant capacitances between the islands the interaction between the charges on the islands depends logarithmically on the distance, at least up to reasonably large values. This implies that a KTB transition can occur where charge anticharge pairs dissociate (3). Recent experiments on 2-d arrays of normal and superconducting low capacitance junctions hint at such a transition (3). The possibility of a charge KTB transition in superconducting granular fIlms in the limit where the charging energy is much larger than the Josephson coupling energy was indicated in Ref. 4. In the following we will discuss normal and superconducting arrays and study the influence of single electron tunneling and of the Josephson coupling on the transition.
2. ARRAYS OF NORMAL JUNCTIONS An array of normal tunnel junctions is described by the action (5)
t «it)
_ ~ _1_ A[q>]-Jm [ 16Eo
dq>i 2 _1_ ~ ~2 + 16Ec <~>( dt - dt) ]
- L fcttJ~t'a(t - t') cos[Pij(t);Pij(t')]
[1]
© 1990 -
mensionless parameter a is defined as [2]
and RT is the junction resistance (except in [2] we choose units where 11 = 1). The partition function can be expressed as a path integral over the field q> with limits which depend on the allowed charge states of the system. Since the charge on each island is integer we have to choose q>i(~) = q>i(D) + 41tl1i, i.e. the partition function includes a summation over the winding numbers nj = D,±1,.... If we decompose q>j(t) = q>iO + t}i(t) + 41tnit/~, where t}j(D) = t}i(~) = D, the charging part of the action decouples into two terms, one Ac[t}] depending on t}i(t), the other depending on the winding numbers 2 1t2 1t2 Ac[n] = REo ~ n i + REc ....
I
....
L (ni - nj) 2
[3]
This action describes the Discrete Gaussian model (DGM) which has been studied in connection with the roughening transition (6). It shows a KTB transition at a temperature (0)
Tc = Ec!41t
[4]
which separates a low temperature phase where the charges form dipoles from the high temperature phase where free charges are present. The result [4] is obtained in the limit of vanishing self-capacitance; a finite Co rounds off the transition. The result [4] ignores the effect of electron tunneling, except that it is needed to establish a thermal equilibrium charge distribution. However, the tunneling itself influences the transition. In the limit of weak dissipation a « 1 we expand the partition function in a and obtain in lowest order the following effective action for the winding number part Aeff=Ac[n]-
~ ~ 21t JdtJdt'a(t-t')g(t-t') COS['Rnij(t-t)] [5]
L
Here g(t-t')=exp{ -<[t}ij(t)-t}ij(t')]2>c} is a phase correlation function evaluated with Ac[t}]. For small frequencies the correction can be expressed as a renormalization of the nearest
Elsevier Science Publishers B.V. (North-Holland)
R. Fazio, G. Schon
1128 neighbor capacitance. This means it leads transition temperature (0)
Tcrrc
to
a shift of the KTB
= 1-0.11 a
[6]
In the limit of strong dissipation it is more appropriate to expand the action in the ~ fields. In lowest order, the winding number contribution leads to the effective action Aeftln] = Adn] + 2a
L
Inj - njl
17]
where the particular form of the second term in [7] arises as Fourier transform of the dissipative kernel. At very low temperatures the effective action approaches the so called ASOS model (7) and we find KTB transition at a critical value of ac = 0.45 for l' = O. Finite temperatures lower this critical value to
ac = 0.45 [I -
(0)
(7r/4) Tffc ]
(0)
for l' « T c
[8]
3. SUPERCONDUCTING ARRAYS In the superconducting state, disregarding for the moment the effect of quasi-particles, the unit of charge is 2e and the charging energy is 4Ec. If the system is dominated by the charging energy we can proceed as described above. However, the difference in the allowed charge states leads to a different requirement for the limits of the path integrals, CPi(~) = CPi(O) + 21tl1i. It is clear that the corresponding KTB transition temperature is increased (4) - Ed 1'(0) cs 1t
[9]
The experimental data of Ref. 3 show a shift in the critical temperature supporting this picture. The Josephson tunneling and the quasi-particle tunneling modify this picture and lead to a shift of the critical temperature. The action of a Josephson junction is As[cp]= A[cp]- EJ
L
~
Jdt COSCPij(t)
[10]
TcsITcs = I - 0.98 (EJ/Ec)
2
[11]
The effect of quasi-particle tunneling depends strongly of the relative values of the superconducting gap 6 and the charging energy Ec. If the gap is large 6 » Ec the tunneling processes are frozen out. Nevertheless, virtual transitions occur which renormalize the nearest-neighbor capacitance (5)
E~n/Ec= [I +~; arl
[12]
As a result the transition temperature [9] or [11] is lowered in an obvious way. On the other hand, in the limit where the superconducting gap is small 6 :s; Ec the quasi-particle transitions dominate the picture. The charges on the islands are then integer multiples of the single electron charge e, and the transition temperature is given by the normal array values. A crossover between the single electron and the Cooper pair charge unbinding transition should occur for 6 = Ec. Finally we consider the limit EJ » Ee. In this case the phase of the superconducting islands provides a more natural description. The elementary excitations of the system are spin waves and vortices. The latter are bound in pairs at low temperatures, but become free at a KTB transition temperature T~O) = EJ. In small capacitance junctions (decreasing values of EJ/Ec) the quantum fluctuations lower the transition temperature l'y. This effect has been studied in the literature (8), but mostly for the case where the self capacitance dominates. Due to the duality between the charge and the vortex picture we expeet that the charge and the vortex ordered states exclude one another. Hence with increasing values of EJ/Ec the system undergoes a transition from a charge ordered to a disordered to a phase ordered state ('ordered' in the sense of the KTB transition). 4. CONCLUSIONS We presented new results for the charge KTB transition in normal and superconducting arrays. In the low temperature phase the charges of the islands are bound in dipoles. This has strong effects on the transport properties of the system. In the case of Josephson arrays we find a phase diagram which is much more complex than anticipated before (4). In the limits where Ec or EJ are dominant the charge-KTB or the phaseKTB are found. When both are nonzero quantum fluctuations decrease the transition temperatures. REFERENCES (1) Proc. of the NATO Advanced Research Workshop on Coherence in Superconducting Networks, J.E. Mooij and G. Schtln eds., Physica B 152 (1988) (2) D.V. Averin and K.K. Likharev, in: 'Quantum effects in small disordered systems', edited by B.L. Altshuler, PA Lee and R.A. Webb, to be published. (3) J.E. Mooij, B.J. van Wees, L.J. Gcerligs, M. Peters, R. Fazio and G. SchOn, submitted to Phys. Rev. Lett (4) A. Widom and S. Badjou, Phys. Rev. B37, 7915 (1988) (5) V. Ambegaokar, U. Eckem and G. Schon, Phys. Rev. Len. 48,1745 (1982); Phys. Rev. B 30, 6419 (1984) (6) S.T. Chui and J.D. Weeks, Phys. Rev. B14, 4978 (1976) (7) R.H. Swendsen, Phys. Rev. B15, 5421 (1977) (8) L. Jacobs, J.V. Jose, M.A. Novotny and A.M. Goldman, Phys. Rev. 838,4562 (1988)