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Reflections on the superconductor-insulator transition
1. Phys. Chem So/i& Vol57, No. 4, pp. I-III. 1996 Cowright Q 1996 Ekvier Scima Ltd Printed& &at Britain. All rights remwd 0022-36911% 115.00 + 0.00
Pergamon
00223697(9S)Ml74-3
ERRATUM Corrected version of paper from ‘Proceedings of the Conference on Spectroscopies in Novel Superconductors’, published in J: Whys.Chem Solids, 56 (No. 12) (1995).
REFLECTIONS ON THE SUPERCONDUCTOR-INSULATOR
TRANSITION
S. DONIACH Depts of Applied Physics and Physics Stanford University, Stanford, CA 94305, U.S.A.
Abstract-The
anomalously long superconducting coherence lengths observedin YBCOIPBCOmultilayers are interpreted in terms of a boson localization picture for the superconductor-insulator transition in YBCO as Pr is substituted for Y. It is suggested that the combination of tinite vortex mass and reduced supertluid density for the proximity-induced order parameter in PBCO should lead to enhanced quantum creep in these multilayer samples:
1.
INTRODUCHON
Some years ago Doniach and Inui [l] suggested that the superconductor-insulator transition observed in the cuprates as doping is lowered from that needed for optimal superconductivity is a “boson localization” or “Cooper crystal” transition. The purpose of this note is to suggest that the insulating state induced in YBCO by Pr substitution for Y is also in this class of localized boson states. This hypothesis is motivated by the extraordinarily long superconductivity proximity tunneling lengths observed in b-axis YBCOlPBCO multilayers by Suzuki et al. [2] which extends to some 480 A. If the PBCO contains bound Cooper pairs which are weakly localized, then proximity tunneling from the adjacent YBCO interface will have a proximity coherence length I$ oi 1/rprtr-tdtion and hence can become very long as the superconductor-insulator boundary is approached in the PBCO. This weak localization could also account for variable range hopping conductivity observed in pure PBCO by Suzuki et al. [2].
2. PSEUDO-SPIN MODEL Because the T = 0 Ginzburg Landau coherence length 5s is so short (- 20 A) in the cuprates_ a convenient cOarse grained phenomenological model for the superconductorinsulator transition is that of Robasczkiewia ef al. 13)generalized to include long range Coulomb interactions. Here, pseudospins T defined on a lattice with unit cell of order 50 have x, y components representing the superconducting order parameter and z- components representing the number operator of the Cooper pairs.
In the simplest such model T will be a spin 1 object for which the m = 0 eigenstate represents the average Cooper pair density, while M = +l eigenstates represent charging fluctuations of k2e. The Hamiltonian becomes H = &$(T;,~
-J
c(Ti+T,:
t
hc)
, (TilTf) +gx
(1) i,j
I&
-
RjI
where $ is the on-site charging energy, J is the Cooper pair tunneling matrix element and em is a phenomenological dielectric constant. In the classical limit J > > 4$/e@a, one has < TV >= 1 and the superconducting condensate energy becomes Es = -dd$
= -4J
(2)
where we consider superconductivity in two-dimensional planes of thickness d and square lattice spacing a but treat the long range Coulomb interactions in three dimensions the expansion about A Ginzburg-Landau superconductor-insulator transition was given by Doniach [4]. A generalized form which includes the effects of long range Coulomb interactions was proposed by by Doniach and Inui [l]. This phenomenological Ginzburg Landau free energy functional of the complex order parameter w(x) = (I,v(x)]@‘(“)is given by P
3TIwl= l&l jd2rj&W12 0
- (1-~u/or,)l~~I~
S.
II
DONIACH
where Em= 4e’lc,a, a = E&J, a, = 2.50 z a and the on-site charging term has (somewhat arbitrarily) been neglected. A similar expression was given by Eckem and Schmid [S]for the Josephson junction array model.
3. PROXlMlTY TUNNELING IN PBCO If we assume that a is greater than, but close to, the critical value in PBCO, then the 1~1’ term becomes positive and PBCO is an insulator when not coupled to YBCO. However,assuming a boundary condition for the superconducting order parameter of w = 1 at the PBCO-YBCO interface, then 9 will decay exponentially into the PBCO as wPBCO(X)
=
v/YBcO
e-“”
(4)
where g = g2ac/(a - a,). For &,/z z (480/20) = 24, the Cooper pair localization energy will be given approximately by the q = 0 plasmon energy of eqn (3): Qp = [4Q(a/a,
5. FIELD-INDUCED QUANTUM MELTING OF THE VORTEX LATTICE
Use of the phenomenologictl functional (3) allows an estimate of the vortex mass at the mean field level. Following Eckem and Schmid [S]wecan write the vorttex contribution to the phase 4(x, T) of the order parameter as q(x) = I~,le”‘“) with
Y [ 1
4(x) = Cq”arctan
5
where qv = 2 1 are the vortex charges and (x,, y,) are the vortex positions Substituting in eqn (3) this leads to a vortex mass Af, = 21T2h2/E&?.
(10)
At sufficiently large applied magnetic field, the vortex lattice will then be driven through a quantum melting transition, at T = 0, at a value of H which may be estimated using the Lindemrnn criterion as
- I)]“‘. Using
a = O(KT,), where T, is the BCS transition temerature one then finds
(11)
where 0” is the frequency of the zone boundary phonon modes of the vortex lattice [8]. When the uniform system is close to the insulaHopping of the Cooper pairs with this weak pairtor/superconductor phase boundary (but on the superconlocalization energy might account for the variable range ducting side), the effective mass M, of the vortices starts hopping in the CuOz planes of films of pure PBCO obto become very small due to renormalization by the charg served by Suzuki et al. [2]. ing fluctuations of the order parameter as shown by van Otterloo, Fazio & Schiin [I. So as the system approaches the superconductor-insulator phase boundary, the melting 4. VORTEX ACIIVATION ENERGlES JN THE field HM will be driven to smaller values both due to the YBCO/PBCO COMPOSITE LAYERS softening of the vortex lattice as p, - 0 and due to the renormaliztion of the vortex mass. Thus HM will tend to Suzuki er a[. [2] measure a vortex activation energy & as zero at the superconductor-insulator phase boundary (see a function of PBCO spacer thickness in their a-axis films. Fisher \9]). They fit it to the theory of Feigel’man et al. [6] for the This field-induced quantum melting provides a nice exactivated resistivity of a pinned vortex lattice planation for the field-induced superconductor-insulator transition seen in oxygen-depleted YBCO by Seidler er gd = &ln(&,/H) (6) al. [lo]. In fact, their measurements provide experimental evidence for the boson-localization mechanism for the where rd = &d/64~‘A2. For the composite films, A-* oc superconductor-insulator phase boundary in underdoped A;‘$’ where we take b to be the direction of the cuprate samples. If the application of the field had lead to YBCO/PBCO multilayers. Then pair breaking then a transition to a metallic state would be expected, which was not found in these experiments LYBCO i Ep(l - emLPBm’Ep) The YBCO/PBCO multilayers provide an interesting case Ai2a pssps{ 1 (7) LYBCO + LPBCO where the proximity-induced superconducting order parameter remains finite even though the PBCO segments have provided LPBCO z &,leading to crossed into the insulating regime. So in the multilayer case we may expect that the mass reduction to zero caused by the charge fluctuations is inhibited by the proximity effect and M, will remain finite. The effect of the charge fluctuations on the mean field for intermediate Valuesof LPBCO/&. estimate of M, in the presence of the proximity-induced
Reflectionson the superconductor-insulatortransition order parameter is hard to estimate. However, there will be a red&ion of the melting field HH due to softening of the vortex lattice as ps is reduced. Thus using eqn (1 I) we may estimate
where we have set Iz, CCpi’“. At fields of order H ” H~mxim’y, any vortices induced in the PBCO regions will then be driven through the quantum melting transition, thus leading to a field-indu~d decoupling of the YBCO regions at a critical field whose value will depend on the magnitude of the proximity-induced order parameter. It would also be interesting to observe if the reduced HM in the PBCO regions leads to enhanced quantum creep at very low temperature as the PBCO iayer thickness is increased towards I$. Acknowledgements-The
author wishesto thank Ted Geballe and Yuri Suzuki for many interesting discussions and the NSF for support through grant number DMR-9302882.
REFERENCES 1. Doniach S. and Inui M.. P&X.Rev. B 41, 6668-78(1990).
2. Suzuki Y., T&cone J.-M., Eom C. B., Beasley M. R. Geballe T. H., WhysRev.Larts73, 328-31(199b). 3. RobaszckiewiczS., Micnas R. and Chao K. A., Phys. R 23. 1447(1981):26. 3915(19821 4. D&iach s., P/&s. kex B k, 5663(1981) 5. &kern U. &d ~SchmidA., P&s. RI&. B j9* 6441 (1989) 6. Fei8el’manM. V.,GashkenbeinV.B. and Larkin A. I., PI; c 167, 177 (1990) 7. van Otterloo A., Fazio R. and Schiin G.,Physica B 194 1153-4(1994). 8. Onogi T. and Doniach S., submitted for publication. 9. Fisher M. I? A., Phys. Rev. Left. 65,923 (1990). 10. Scidler C. T, RosenbaumT. E and VealB. W., Phys. I 45, 10162-4(1992).
Pergamon
00223697(9S)Ml74-3
ERRATUM Corrected version of paper from ‘Proceedings of the Conference on Spectroscopies in Novel Superconductors’, published in J: Whys.Chem Solids, 56 (No. 12) (1995).
REFLECTIONS ON THE SUPERCONDUCTOR-INSULATOR
TRANSITION
S. DONIACH Depts of Applied Physics and Physics Stanford University, Stanford, CA 94305, U.S.A.
Abstract-The
anomalously long superconducting coherence lengths observedin YBCOIPBCOmultilayers are interpreted in terms of a boson localization picture for the superconductor-insulator transition in YBCO as Pr is substituted for Y. It is suggested that the combination of tinite vortex mass and reduced supertluid density for the proximity-induced order parameter in PBCO should lead to enhanced quantum creep in these multilayer samples:
1.
INTRODUCHON
Some years ago Doniach and Inui [l] suggested that the superconductor-insulator transition observed in the cuprates as doping is lowered from that needed for optimal superconductivity is a “boson localization” or “Cooper crystal” transition. The purpose of this note is to suggest that the insulating state induced in YBCO by Pr substitution for Y is also in this class of localized boson states. This hypothesis is motivated by the extraordinarily long superconductivity proximity tunneling lengths observed in b-axis YBCOlPBCO multilayers by Suzuki et al. [2] which extends to some 480 A. If the PBCO contains bound Cooper pairs which are weakly localized, then proximity tunneling from the adjacent YBCO interface will have a proximity coherence length I$ oi 1/rprtr-tdtion and hence can become very long as the superconductor-insulator boundary is approached in the PBCO. This weak localization could also account for variable range hopping conductivity observed in pure PBCO by Suzuki et al. [2].
2. PSEUDO-SPIN MODEL Because the T = 0 Ginzburg Landau coherence length 5s is so short (- 20 A) in the cuprates_ a convenient cOarse grained phenomenological model for the superconductorinsulator transition is that of Robasczkiewia ef al. 13)generalized to include long range Coulomb interactions. Here, pseudospins T defined on a lattice with unit cell of order 50 have x, y components representing the superconducting order parameter and z- components representing the number operator of the Cooper pairs.
In the simplest such model T will be a spin 1 object for which the m = 0 eigenstate represents the average Cooper pair density, while M = +l eigenstates represent charging fluctuations of k2e. The Hamiltonian becomes H = &$(T;,~
-J
c(Ti+T,:
t
hc)
, (TilTf) +gx
(1) i,j
I&
-
RjI
where $ is the on-site charging energy, J is the Cooper pair tunneling matrix element and em is a phenomenological dielectric constant. In the classical limit J > > 4$/e@a, one has < TV >= 1 and the superconducting condensate energy becomes Es = -dd$
= -4J
(2)
where we consider superconductivity in two-dimensional planes of thickness d and square lattice spacing a but treat the long range Coulomb interactions in three dimensions the expansion about A Ginzburg-Landau superconductor-insulator transition was given by Doniach [4]. A generalized form which includes the effects of long range Coulomb interactions was proposed by by Doniach and Inui [l]. This phenomenological Ginzburg Landau free energy functional of the complex order parameter w(x) = (I,v(x)]@‘(“)is given by P
3TIwl= l&l jd2rj&W12 0
- (1-~u/or,)l~~I~
S.
II
DONIACH
where Em= 4e’lc,a, a = E&J, a, = 2.50 z a and the on-site charging term has (somewhat arbitrarily) been neglected. A similar expression was given by Eckem and Schmid [S]for the Josephson junction array model.
3. PROXlMlTY TUNNELING IN PBCO If we assume that a is greater than, but close to, the critical value in PBCO, then the 1~1’ term becomes positive and PBCO is an insulator when not coupled to YBCO. However,assuming a boundary condition for the superconducting order parameter of w = 1 at the PBCO-YBCO interface, then 9 will decay exponentially into the PBCO as wPBCO(X)
=
v/YBcO
e-“”
(4)
where g = g2ac/(a - a,). For &,/z z (480/20) = 24, the Cooper pair localization energy will be given approximately by the q = 0 plasmon energy of eqn (3): Qp = [4Q(a/a,
5. FIELD-INDUCED QUANTUM MELTING OF THE VORTEX LATTICE
Use of the phenomenologictl functional (3) allows an estimate of the vortex mass at the mean field level. Following Eckem and Schmid [S]wecan write the vorttex contribution to the phase 4(x, T) of the order parameter as q(x) = I~,le”‘“) with
Y [ 1
4(x) = Cq”arctan
5
where qv = 2 1 are the vortex charges and (x,, y,) are the vortex positions Substituting in eqn (3) this leads to a vortex mass Af, = 21T2h2/E&?.
(10)
At sufficiently large applied magnetic field, the vortex lattice will then be driven through a quantum melting transition, at T = 0, at a value of H which may be estimated using the Lindemrnn criterion as
- I)]“‘. Using
a = O(KT,), where T, is the BCS transition temerature one then finds
(11)
where 0” is the frequency of the zone boundary phonon modes of the vortex lattice [8]. When the uniform system is close to the insulaHopping of the Cooper pairs with this weak pairtor/superconductor phase boundary (but on the superconlocalization energy might account for the variable range ducting side), the effective mass M, of the vortices starts hopping in the CuOz planes of films of pure PBCO obto become very small due to renormalization by the charg served by Suzuki et al. [2]. ing fluctuations of the order parameter as shown by van Otterloo, Fazio & Schiin [I. So as the system approaches the superconductor-insulator phase boundary, the melting 4. VORTEX ACIIVATION ENERGlES JN THE field HM will be driven to smaller values both due to the YBCO/PBCO COMPOSITE LAYERS softening of the vortex lattice as p, - 0 and due to the renormaliztion of the vortex mass. Thus HM will tend to Suzuki er a[. [2] measure a vortex activation energy & as zero at the superconductor-insulator phase boundary (see a function of PBCO spacer thickness in their a-axis films. Fisher \9]). They fit it to the theory of Feigel’man et al. [6] for the This field-induced quantum melting provides a nice exactivated resistivity of a pinned vortex lattice planation for the field-induced superconductor-insulator transition seen in oxygen-depleted YBCO by Seidler er gd = &ln(&,/H) (6) al. [lo]. In fact, their measurements provide experimental evidence for the boson-localization mechanism for the where rd = &d/64~‘A2. For the composite films, A-* oc superconductor-insulator phase boundary in underdoped A;‘$’ where we take b to be the direction of the cuprate samples. If the application of the field had lead to YBCO/PBCO multilayers. Then pair breaking then a transition to a metallic state would be expected, which was not found in these experiments LYBCO i Ep(l - emLPBm’Ep) The YBCO/PBCO multilayers provide an interesting case Ai2a pssps{ 1 (7) LYBCO + LPBCO where the proximity-induced superconducting order parameter remains finite even though the PBCO segments have provided LPBCO z &,leading to crossed into the insulating regime. So in the multilayer case we may expect that the mass reduction to zero caused by the charge fluctuations is inhibited by the proximity effect and M, will remain finite. The effect of the charge fluctuations on the mean field for intermediate Valuesof LPBCO/&. estimate of M, in the presence of the proximity-induced
Reflectionson the superconductor-insulatortransition order parameter is hard to estimate. However, there will be a red&ion of the melting field HH due to softening of the vortex lattice as ps is reduced. Thus using eqn (1 I) we may estimate
where we have set Iz, CCpi’“. At fields of order H ” H~mxim’y, any vortices induced in the PBCO regions will then be driven through the quantum melting transition, thus leading to a field-indu~d decoupling of the YBCO regions at a critical field whose value will depend on the magnitude of the proximity-induced order parameter. It would also be interesting to observe if the reduced HM in the PBCO regions leads to enhanced quantum creep at very low temperature as the PBCO iayer thickness is increased towards I$. Acknowledgements-The
author wishesto thank Ted Geballe and Yuri Suzuki for many interesting discussions and the NSF for support through grant number DMR-9302882.
REFERENCES 1. Doniach S. and Inui M.. P&X.Rev. B 41, 6668-78(1990).
2. Suzuki Y., T&cone J.-M., Eom C. B., Beasley M. R. Geballe T. H., WhysRev.Larts73, 328-31(199b). 3. RobaszckiewiczS., Micnas R. and Chao K. A., Phys. R 23. 1447(1981):26. 3915(19821 4. D&iach s., P/&s. kex B k, 5663(1981) 5. &kern U. &d ~SchmidA., P&s. RI&. B j9* 6441 (1989) 6. Fei8el’manM. V.,GashkenbeinV.B. and Larkin A. I., PI; c 167, 177 (1990) 7. van Otterloo A., Fazio R. and Schiin G.,Physica B 194 1153-4(1994). 8. Onogi T. and Doniach S., submitted for publication. 9. Fisher M. I? A., Phys. Rev. Left. 65,923 (1990). 10. Scidler C. T, RosenbaumT. E and VealB. W., Phys. I 45, 10162-4(1992).