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Novel boundary effect in high temperature superconductors: superconductivity induced by pseudogap proximity effect
Physica C 392–396 (2003) 48–52 www.elsevier.com/locate/physc
Novel boundary effect in high temperature superconductors: superconductivity induced by pseudogap proximity effect M. Hayashi *, H. Ebisawa Graduate School of Information Sciences, Tohoku University, Aramaki Aoba-ku, Sendai 980-8579, Japan Received 13 November 2002; accepted 3 February 2003
Abstract The boundary between highly under-doped and over-doped cuprate superconductors is studied within the framework of the slave-boson mean-field theory of the t–J model. We assume that the under-doped region is in the pseudogap state and the singlet resonating valence bond (s-RVB) order is formed there. The over-doped region, on the other hand, is in a hole-rich state and the pseudogap is destroyed by the holes. Because of the proximity effect, the s-RVB order and high-density holes can coexist, leading to the superconducting behavior. Remarkably, the transition temperature of this ‘‘boundary superconductivity’’ exceeds the maximum superconducting transition temperature of the bulk region, thus possessing a new possibility of application for superconducting devices. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.76.Bz; 74.80.)g; 74.20.Mn Keywords: Boundary; High-Tc compound; Pseudogap; Proximity effect
1. Introduction In the slave-boson mean-field theory of the t–J model, the singlet resonating valence bond (sRVB) order of spinons and the bose condensation of holons are considered to be essential ingredients of the superconductivity [1–4]. Until now, it has been clarified that this picture explains the most physical properties of the cuprate superconductors
*
Corresponding author. Tel.: +81-22-217-5847; fax: +81-22217-5851. E-mail address: [email protected] (M. Hayashi).
very well [5,6]. Although there still remain some subtle questions about the validity of the spin– charge separation, the t–J model may be considered as an effective model, which captures the essence of the cuprate superconductors. In this paper, we consider the situation where the doping rate is spatially varying. Here we especially study the boundary between highly overdoped and under-doped regions, existing in the same CuO2 plane. We consider that, in the overdoped region, holons are bose-condensed and, in the under-doped region, the s-RVB order exists. In general, when an ordered phase touches a disordered phase at a boundary, the ordered state
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01070-0
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
penetrates into the disordered region with a characteristic penetration length, that is a phenomenon known as the ‘‘proximity effect’’ [7]. Therefore, in our case, it is expected that the holon condensate and the s-RVB order coexist at the boundary. From the basic structure of the mean field theory of the t–J model we immediately result in a possibility of finding superconductivity at the boundary. Let us call this the ‘‘boundary superconductivity’’. The remarkable nature of this superconductivity is that the transition temperature can be as high as the transition temperature of the s-RVB order, namely the pseudogap temperature, which largely exceeds the maximum (optimum) superconducting transition temperature of the base material. This implies that we can actually raise the transition temperature by artificially making the situation stated above. Several ideas to realize this is also discussed.
2. Model First we introduce the t–J model which describes the correlated electrons in a CuO2 plane. The Hamiltonian is given by X X y ~ H ¼ t fir fjr byj bi þ h:c: þ J Sj Si ~ hi;jir
X i
ki
hi;ji
X
!
firy fir þ byi bi 1 ;
ð1Þ
r
where i and j denote lattice points in the CuO2 plane, hi; ji means the nearest neighbors, fir (r ¼"; #) and bi are the annihilation operators of the and the holon, respectively, ~ Si denotes Pspinon y 1 ~ ~ f f with r being Pauli matrices, and r ab ib ab ab ia 2 k is the Lagrange multiplier for the constraint, i P y y r fir fir þ bi bi ¼ 1. The spinon fir and holon bi are related to the electron operator cir by cir ¼ byi fir . In this paper the lattice spacing is taken to be unity. In addition to Eq. (1), we later introduce the coupling to the electromagnetic field. Eq. (1) is treated by the slave-boson mean-field theory. The local constraint represented by ki is replaced by the global one, which is described by the constant chemical potential for spinons lF and that for holons lB . Several mean fields are intro-
49
duced to describe the ordering in the P system; the hopping order parameters, n h r firy fjr i and g hbyi bj i, are treated as constants in the following, since their critical temperatures are higher ð0Þ than the temperatures of our interest, T < TRVB , ð0Þ where TRVB is the transition temperature of the sRVB order. The s-RVB order parameter is denoted by Dij ¼ 3J8 hfi" fj# fi# fj" i. To incorporate the anisotropic pairing of spinons, we set Dij ¼ Dx ð~ ri Þ if j ¼ i þ x^ and Dij ¼ Dy ð~ ri Þ if j ¼ i þ y^ where x^ and y^ indicate the unit vector in x- and y-directions, respectively. Including also the scalar and vector potential, u and ~ A, we obtain the following Lagrangian: X y L¼ fir fh@s lF gfir ir
þ
X
o n byi h@s lB þ ieuð~ r i Þ bi
i
X ihji y 3 tg þ J n e fir fjr þ h:c: 8 hi;jir X y tn bi bj þ h:c:
hi;ji
X
Dx ð~ ri Þviiþ^x þ Dy ð~ ri Þviiþy^
i
þ h:c:
8 2 2 jDx ð~ ri Þj þ jDy ð~ ri Þj ; 3J
where vij ¼ fi" fj# fi# fj" and Z ~ri e ~ hji ¼ A d~ l: hc ~rj
ð2Þ
ð3Þ
The condensation of the holons is described by introducing the wave function of the condensate, usually defined by hbi i, which is not written explicitly here. We assume that at the doping rate, which we are interested in, the vector potential and the scalar potential couple only to the spinons and the holons, respectively [5,8,9]. The chemical potentials lF and lB are determined from N 1 X hf y fir i ¼ 1 dðxi Þ; N i¼1 ir N 1 X hby bi i ¼ dðxi Þ; N i¼1 i
ð4Þ
50
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
where dðxi Þ is the average holon density. Here xi denotes the x-coordinate of the lattice point i and N is the total number of sites. Although dðxi Þ should be determined in principle from self-consistent treatment of electrostatic screening, we simply define dðxi Þ as 0 ðx < 0Þ; dðxi Þ ¼ ð5Þ d0 ðx > 0Þ; and treat it as a given quantity. Now d0 is assumed to be large enough to cause bose condensation of holons in the region xi > 0. Now the problem is reduced to the one of determining Dx ð~ ri Þ and Dy ð~ ri Þ.
3. Ginzburg–Landau theory of t–J model
2
4
2
ð6Þ
where P ¼ i» þ ð2p=/0 ÞA with /0 ¼ hc=ð2eÞ. Að~ rÞ is the vector potential. Here the CuO2 plane is set parallel to x–y plane. The GL coefficients are given by 2
ad ¼ 2N ð0Þð1 dÞ ln b¼
T ; Td
21fð3ÞN ð0Þ c1 4 ð1 dÞ 2 ; 2 2 2p T T D
7fð3ÞN ð0ÞD2 c2 D cd ¼ 2 ; T 16p2 T 2
Hereafter we simply assume that Td , Td0 , ad and a0d are parameters and do not go into the details of their dependence on dðxÞ.
4. The boundary superconductivity
Our next purpose is to derive the GL theory which describe the spatial variation of Dij ð~ rÞ. We assume that Dij ð~ rÞ favors d-wave symmetry and it is described by a single complex order parameter (OP), Dd ð~ rÞ ¼ 12 fDx ð~ rÞ Dy ð~ rÞg, in the continuum limit. The GL free energy of Dd ð~ rÞ in the bulk is obtained using a conventional perturbation technique as [10] Fs-RVB ¼ ad jDd j þ bjDd j þ cd jPDd j ;
GL coefficients comes from Td , ad and b. Among them, the dependence arising from Td is dominant. Therefore hereafter we neglect the dðxÞ dependence other than that in Td . Spatial variation of the doping rate is incorporated by setting ðx < 0Þ; Td Td ðxÞ ¼ Td0 Td ðx > 0Þ; ð10Þ ad < 0 ðx < 0Þ; ad ðxÞ ¼ a0d > 0 ðx > 0Þ:
ð7Þ ð8Þ ð9Þ
where N ð0Þ, d and D are the density of states at Fermi energy, the doping rate (assumed to be constant here) and the band width, respectively. fðxÞ is the zeta function. The transition tempera2 ture is given by Td ’ T exp½1=f2VN ð0Þð1 dÞ g c with T ¼ 2De =p where c is the EulerÕs constant and V denotes 3J =8. The doping dependence of the
Now we solve the Ginzburg–Landau equation obtained from the free energy above. After a simple calculation, the solution for Dd ð~ rÞ leads to ( ) x=nF d 1 nF e 0 Dd ¼ D for x < 0; ð11Þ nF þ nF 0
x=n0F
d nF e 0 Dd ¼ D nF þ nF
for x > 0;
ð12Þ
d is the equilibrium value of the gap at where D x ! 1. We note that Dd and its derivative is continuous at x ¼ 0. This boundary condition is rather strange if one compares it with the boundary condition of the conventional superconducting proximity effect. In the conventional case, the energy gap Dð~ ri Þ is determined from the anomalous amplitude hci" ci# i as ri Þhci" ci# i; Dð~ ri Þ ¼ V ð~
ð13Þ
where V ð~ ri Þ is the BCS interaction at site i. Therefore at the superconductor-normal metal boundary, where V ð~ ri Þ usually changes discontinuously, the gap Dð~ ri Þ also shows a discontinuity, since the continuous quantity at the boundary is not the gap but the anomalous amplitude. However, in the present case, the interaction parameter corresponding to V ð~ ri Þ is J , which is continuous at the boundary, and, then, the gap also changes continuously. This difference of the boundary
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
condition makes the penetration of the s-RVB gap to the hole-rich region large. The coherence lengths, nF and n0F , characterize the strength of the proximity effect. These are given by rffiffiffiffi pffiffiffiffi c2 c2 D 1 D 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ; nF ’ ; nF ¼ 0 T 2 Td 1 T =Td ln T =Td c1 and c2 are constants defined in Eqs. (8) and (9). Since the holon condensate is assumed in x > 0, we conclude that the superconductivity appears in the region 0 < x < n0F . In Fig. 1 we have depicted the behavior of the solution. In order to obtain a strong superconductivity, two conditions are needed: (a) The proximity length n0F should be long. (b) The amplitude of the gap at the boundary should be large. The condition (a) is rather easily satisfied since n0F is pffiffiffiffi c2 D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 T ln Td =Td d at T Td and it can be the same order as the sRVB coherence length nF at T ¼ 0. On the other hand, the condition (b) is not satisfied if T is too close to Td where nF diverges and the gap at the boundary is much suppressed. To avoid this situation the temperature must be rather lower than
51
Td . However, since nF rapidly decreases below Td , this condition is also satisfied at temperatures a little lower than Td . These points indicates that we have a good chance to increase the critical temperature using the ‘‘boundary superconductivity’’ mechanism. 5. Discussion We have demonstrated that the superconductivity appears at the boundary due to the proximity effect of s-RVB order. Except when nF is divergently large (namely T Td ), the energy gap is of d and the width of the superconthe order of D ducting region is of the order of the bulk superconducting coherence length of the base material at T ¼ 0. Therefore it may be experimentally observable. For example, the field effect transistor may be an appropriate experimental setup to realize our proposal [11,12], since it can control the density of the holes by applied gate electrode. If we can make a sharp edge of the electrode, the spatially varying doping rate can be managed and the situation studied in this paper is realized. If the doping concentration is controlled at the length of the lattice spacing, a careful crystal growth may also realize this system. However in both cases much effort is required to realize the condition needed for the boundary superconductivity to be significant. The extensive theoretical studies on the various situations in which the doping rate is changing is also required to find more suitable experimental setup to give rise to the boundary superconductivity. 6. Conclusions
γ
Fig. 1. The solution for Dd ðxÞ (bold line). The dotted line stands for the spatial variation of the hole concentration dðxÞ, which is approximated by a step function.
We have studied the effect of highly underdoped and over-doped region of high-Tc cuprate superconductor. It is predicted that at the boundary the superconductivity appears at much higher temperature than the bulk transition temperature. By employing the Ginzburg–Landau theory of the t–J model, we have presented quantitative analysis of this phenomenon and clarified the condition for its appearance. Some experimental setups are also proposed to observe this phenomenon.
52
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
Acknowledgement This research is supported by a Grant-in-Aid for Scientific Research (1246207) from the Ministry of Education, Science, Sports and Culture, Japan. References [1] P.W. Anderson, Science 235 (1987) 1196. [2] G. Baskaran, Z. Zou, P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] Y. Suzumura, Y. Hasegawa, H. Fukuyama, J. Phys. Soc. Jpn. 57 (1988) 2768. [4] G. Kotlier, J. Liu, Phys. Rev. B 38 (1988) 5142. [5] N. Nagaosa, P.A. Lee, Phys. Rev. Lett. 64 (1990) 2450; N. Nagaosa, P.A. Lee, Phys. Rev. B 46 (1992) 5621.
[6] T. Tanamoto, K. Kuboki, H. Fukuyama, J. Phys. Soc. Jpn. 60 (1991) 3072; T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn. 62 (1993) 717; T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn. 62 (1993) 2793. [7] A similar idea was previously introduced related to the stripe order and its superconducting properties by V.J. Emery, S.A. Kivelson, O. Zacher, Phys. Rev. B 56 (1997) 6120; V.J. Emery, S.A. Kivelson, O. Zacher, Phys. Rev. B 59 (1999) 15641; S.A. Kivelson, cond-mat/0109151. [8] N. Nagaosa, P.A. Lee, Phys. Rev. B 45 (1992) 966. [9] S. Sachdev, Phys. Rev. B 45 (1992) 389. [10] D.L. Feder, C. Kallin, Phys. Rev. B 55 (1997) 559. [11] R.E. Grover, M.D. Sherrill, Pays. Rev. Lett. 5 (1960) 248. [12] J. Mannhart, J.G. Bednorz, K.A. M€ uller, D.G. Schlom, Z. Pays. B 83 (1991) 307.
Novel boundary effect in high temperature superconductors: superconductivity induced by pseudogap proximity effect M. Hayashi *, H. Ebisawa Graduate School of Information Sciences, Tohoku University, Aramaki Aoba-ku, Sendai 980-8579, Japan Received 13 November 2002; accepted 3 February 2003
Abstract The boundary between highly under-doped and over-doped cuprate superconductors is studied within the framework of the slave-boson mean-field theory of the t–J model. We assume that the under-doped region is in the pseudogap state and the singlet resonating valence bond (s-RVB) order is formed there. The over-doped region, on the other hand, is in a hole-rich state and the pseudogap is destroyed by the holes. Because of the proximity effect, the s-RVB order and high-density holes can coexist, leading to the superconducting behavior. Remarkably, the transition temperature of this ‘‘boundary superconductivity’’ exceeds the maximum superconducting transition temperature of the bulk region, thus possessing a new possibility of application for superconducting devices. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.76.Bz; 74.80.)g; 74.20.Mn Keywords: Boundary; High-Tc compound; Pseudogap; Proximity effect
1. Introduction In the slave-boson mean-field theory of the t–J model, the singlet resonating valence bond (sRVB) order of spinons and the bose condensation of holons are considered to be essential ingredients of the superconductivity [1–4]. Until now, it has been clarified that this picture explains the most physical properties of the cuprate superconductors
*
Corresponding author. Tel.: +81-22-217-5847; fax: +81-22217-5851. E-mail address: [email protected] (M. Hayashi).
very well [5,6]. Although there still remain some subtle questions about the validity of the spin– charge separation, the t–J model may be considered as an effective model, which captures the essence of the cuprate superconductors. In this paper, we consider the situation where the doping rate is spatially varying. Here we especially study the boundary between highly overdoped and under-doped regions, existing in the same CuO2 plane. We consider that, in the overdoped region, holons are bose-condensed and, in the under-doped region, the s-RVB order exists. In general, when an ordered phase touches a disordered phase at a boundary, the ordered state
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01070-0
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
penetrates into the disordered region with a characteristic penetration length, that is a phenomenon known as the ‘‘proximity effect’’ [7]. Therefore, in our case, it is expected that the holon condensate and the s-RVB order coexist at the boundary. From the basic structure of the mean field theory of the t–J model we immediately result in a possibility of finding superconductivity at the boundary. Let us call this the ‘‘boundary superconductivity’’. The remarkable nature of this superconductivity is that the transition temperature can be as high as the transition temperature of the s-RVB order, namely the pseudogap temperature, which largely exceeds the maximum (optimum) superconducting transition temperature of the base material. This implies that we can actually raise the transition temperature by artificially making the situation stated above. Several ideas to realize this is also discussed.
2. Model First we introduce the t–J model which describes the correlated electrons in a CuO2 plane. The Hamiltonian is given by X X y ~ H ¼ t fir fjr byj bi þ h:c: þ J Sj Si ~ hi;jir
X i
ki
hi;ji
X
!
firy fir þ byi bi 1 ;
ð1Þ
r
where i and j denote lattice points in the CuO2 plane, hi; ji means the nearest neighbors, fir (r ¼"; #) and bi are the annihilation operators of the and the holon, respectively, ~ Si denotes Pspinon y 1 ~ ~ f f with r being Pauli matrices, and r ab ib ab ab ia 2 k is the Lagrange multiplier for the constraint, i P y y r fir fir þ bi bi ¼ 1. The spinon fir and holon bi are related to the electron operator cir by cir ¼ byi fir . In this paper the lattice spacing is taken to be unity. In addition to Eq. (1), we later introduce the coupling to the electromagnetic field. Eq. (1) is treated by the slave-boson mean-field theory. The local constraint represented by ki is replaced by the global one, which is described by the constant chemical potential for spinons lF and that for holons lB . Several mean fields are intro-
49
duced to describe the ordering in the P system; the hopping order parameters, n h r firy fjr i and g hbyi bj i, are treated as constants in the following, since their critical temperatures are higher ð0Þ than the temperatures of our interest, T < TRVB , ð0Þ where TRVB is the transition temperature of the sRVB order. The s-RVB order parameter is denoted by Dij ¼ 3J8 hfi" fj# fi# fj" i. To incorporate the anisotropic pairing of spinons, we set Dij ¼ Dx ð~ ri Þ if j ¼ i þ x^ and Dij ¼ Dy ð~ ri Þ if j ¼ i þ y^ where x^ and y^ indicate the unit vector in x- and y-directions, respectively. Including also the scalar and vector potential, u and ~ A, we obtain the following Lagrangian: X y L¼ fir fh@s lF gfir ir
þ
X
o n byi h@s lB þ ieuð~ r i Þ bi
i
X ihji y 3 tg þ J n e fir fjr þ h:c: 8 hi;jir X y tn bi bj þ h:c:
hi;ji
X
Dx ð~ ri Þviiþ^x þ Dy ð~ ri Þviiþy^
i
þ h:c:
8 2 2 jDx ð~ ri Þj þ jDy ð~ ri Þj ; 3J
where vij ¼ fi" fj# fi# fj" and Z ~ri e ~ hji ¼ A d~ l: hc ~rj
ð2Þ
ð3Þ
The condensation of the holons is described by introducing the wave function of the condensate, usually defined by hbi i, which is not written explicitly here. We assume that at the doping rate, which we are interested in, the vector potential and the scalar potential couple only to the spinons and the holons, respectively [5,8,9]. The chemical potentials lF and lB are determined from N 1 X hf y fir i ¼ 1 dðxi Þ; N i¼1 ir N 1 X hby bi i ¼ dðxi Þ; N i¼1 i
ð4Þ
50
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
where dðxi Þ is the average holon density. Here xi denotes the x-coordinate of the lattice point i and N is the total number of sites. Although dðxi Þ should be determined in principle from self-consistent treatment of electrostatic screening, we simply define dðxi Þ as 0 ðx < 0Þ; dðxi Þ ¼ ð5Þ d0 ðx > 0Þ; and treat it as a given quantity. Now d0 is assumed to be large enough to cause bose condensation of holons in the region xi > 0. Now the problem is reduced to the one of determining Dx ð~ ri Þ and Dy ð~ ri Þ.
3. Ginzburg–Landau theory of t–J model
2
4
2
ð6Þ
where P ¼ i» þ ð2p=/0 ÞA with /0 ¼ hc=ð2eÞ. Að~ rÞ is the vector potential. Here the CuO2 plane is set parallel to x–y plane. The GL coefficients are given by 2
ad ¼ 2N ð0Þð1 dÞ ln b¼
T ; Td
21fð3ÞN ð0Þ c1 4 ð1 dÞ 2 ; 2 2 2p T T D
7fð3ÞN ð0ÞD2 c2 D cd ¼ 2 ; T 16p2 T 2
Hereafter we simply assume that Td , Td0 , ad and a0d are parameters and do not go into the details of their dependence on dðxÞ.
4. The boundary superconductivity
Our next purpose is to derive the GL theory which describe the spatial variation of Dij ð~ rÞ. We assume that Dij ð~ rÞ favors d-wave symmetry and it is described by a single complex order parameter (OP), Dd ð~ rÞ ¼ 12 fDx ð~ rÞ Dy ð~ rÞg, in the continuum limit. The GL free energy of Dd ð~ rÞ in the bulk is obtained using a conventional perturbation technique as [10] Fs-RVB ¼ ad jDd j þ bjDd j þ cd jPDd j ;
GL coefficients comes from Td , ad and b. Among them, the dependence arising from Td is dominant. Therefore hereafter we neglect the dðxÞ dependence other than that in Td . Spatial variation of the doping rate is incorporated by setting ðx < 0Þ; Td Td ðxÞ ¼ Td0 Td ðx > 0Þ; ð10Þ ad < 0 ðx < 0Þ; ad ðxÞ ¼ a0d > 0 ðx > 0Þ:
ð7Þ ð8Þ ð9Þ
where N ð0Þ, d and D are the density of states at Fermi energy, the doping rate (assumed to be constant here) and the band width, respectively. fðxÞ is the zeta function. The transition tempera2 ture is given by Td ’ T exp½1=f2VN ð0Þð1 dÞ g c with T ¼ 2De =p where c is the EulerÕs constant and V denotes 3J =8. The doping dependence of the
Now we solve the Ginzburg–Landau equation obtained from the free energy above. After a simple calculation, the solution for Dd ð~ rÞ leads to ( ) x=nF d 1 nF e 0 Dd ¼ D for x < 0; ð11Þ nF þ nF 0
x=n0F
d nF e 0 Dd ¼ D nF þ nF
for x > 0;
ð12Þ
d is the equilibrium value of the gap at where D x ! 1. We note that Dd and its derivative is continuous at x ¼ 0. This boundary condition is rather strange if one compares it with the boundary condition of the conventional superconducting proximity effect. In the conventional case, the energy gap Dð~ ri Þ is determined from the anomalous amplitude hci" ci# i as ri Þhci" ci# i; Dð~ ri Þ ¼ V ð~
ð13Þ
where V ð~ ri Þ is the BCS interaction at site i. Therefore at the superconductor-normal metal boundary, where V ð~ ri Þ usually changes discontinuously, the gap Dð~ ri Þ also shows a discontinuity, since the continuous quantity at the boundary is not the gap but the anomalous amplitude. However, in the present case, the interaction parameter corresponding to V ð~ ri Þ is J , which is continuous at the boundary, and, then, the gap also changes continuously. This difference of the boundary
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
condition makes the penetration of the s-RVB gap to the hole-rich region large. The coherence lengths, nF and n0F , characterize the strength of the proximity effect. These are given by rffiffiffiffi pffiffiffiffi c2 c2 D 1 D 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ; nF ’ ; nF ¼ 0 T 2 Td 1 T =Td ln T =Td c1 and c2 are constants defined in Eqs. (8) and (9). Since the holon condensate is assumed in x > 0, we conclude that the superconductivity appears in the region 0 < x < n0F . In Fig. 1 we have depicted the behavior of the solution. In order to obtain a strong superconductivity, two conditions are needed: (a) The proximity length n0F should be long. (b) The amplitude of the gap at the boundary should be large. The condition (a) is rather easily satisfied since n0F is pffiffiffiffi c2 D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 T ln Td =Td d at T Td and it can be the same order as the sRVB coherence length nF at T ¼ 0. On the other hand, the condition (b) is not satisfied if T is too close to Td where nF diverges and the gap at the boundary is much suppressed. To avoid this situation the temperature must be rather lower than
51
Td . However, since nF rapidly decreases below Td , this condition is also satisfied at temperatures a little lower than Td . These points indicates that we have a good chance to increase the critical temperature using the ‘‘boundary superconductivity’’ mechanism. 5. Discussion We have demonstrated that the superconductivity appears at the boundary due to the proximity effect of s-RVB order. Except when nF is divergently large (namely T Td ), the energy gap is of d and the width of the superconthe order of D ducting region is of the order of the bulk superconducting coherence length of the base material at T ¼ 0. Therefore it may be experimentally observable. For example, the field effect transistor may be an appropriate experimental setup to realize our proposal [11,12], since it can control the density of the holes by applied gate electrode. If we can make a sharp edge of the electrode, the spatially varying doping rate can be managed and the situation studied in this paper is realized. If the doping concentration is controlled at the length of the lattice spacing, a careful crystal growth may also realize this system. However in both cases much effort is required to realize the condition needed for the boundary superconductivity to be significant. The extensive theoretical studies on the various situations in which the doping rate is changing is also required to find more suitable experimental setup to give rise to the boundary superconductivity. 6. Conclusions
γ
Fig. 1. The solution for Dd ðxÞ (bold line). The dotted line stands for the spatial variation of the hole concentration dðxÞ, which is approximated by a step function.
We have studied the effect of highly underdoped and over-doped region of high-Tc cuprate superconductor. It is predicted that at the boundary the superconductivity appears at much higher temperature than the bulk transition temperature. By employing the Ginzburg–Landau theory of the t–J model, we have presented quantitative analysis of this phenomenon and clarified the condition for its appearance. Some experimental setups are also proposed to observe this phenomenon.
52
M. Hayashi, H. Ebisawa / Physica C 392–396 (2003) 48–52
Acknowledgement This research is supported by a Grant-in-Aid for Scientific Research (1246207) from the Ministry of Education, Science, Sports and Culture, Japan. References [1] P.W. Anderson, Science 235 (1987) 1196. [2] G. Baskaran, Z. Zou, P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] Y. Suzumura, Y. Hasegawa, H. Fukuyama, J. Phys. Soc. Jpn. 57 (1988) 2768. [4] G. Kotlier, J. Liu, Phys. Rev. B 38 (1988) 5142. [5] N. Nagaosa, P.A. Lee, Phys. Rev. Lett. 64 (1990) 2450; N. Nagaosa, P.A. Lee, Phys. Rev. B 46 (1992) 5621.
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