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A numerical study of dc Josephson effect in SNS junctions of d-wave superconductors
Physica C 367 (2002) 157–160 www.elsevier.com/locate/physc
A numerical study of dc Josephson effect in SNS junctions of d-wave superconductors Yasuhiro Asano * Department of Applied Physics, Hokkaido University, Kita-13, Nishi-8 Kitaku, Sapporo 060-8628, Japan
Abstract We numerically study the dc Josephson current in d-wave superconductor/normal metal/d-wave superconductor junctions by using the recursive Green function method. The Josephson current is sensitive to the orientation angle between the crystalline axis of the high-Tc superconductors and the normal of the junction interface. It was pointed out that the ensemble average of the Josephson current, hJ i, vanishes when the orientation angle is p=4 and the normal metal is in the diffusive transport regime. We show in this paper that hJ i disappears even when the normal metal is in the quasi-ballistic transport regime. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 74.80.Fp; 74.25.Fy; 74.50.þr Keywords: Josephson current; Mesoscopic fluctuations; Zero-energy states
1. Introduction The transport phenomena in junctions of d-wave superconductors have been studied intensively in recent years [1,2] because the high-Tc superconductors might have the d-wave pairing symmetry [3,4]. The transport properties in the d-wave junctions are sensitive to the orientation angle a between the crystalline axis of the high-Tc superconductors and the normal of the junction interface. This is because the zero-energy states (ZES) [5] are formed at the normal metal/superconductor (NS) interface when the potential barrier at the interface is large enough. In N/d-wave superconductor junctions, the ZES are observed in the conductance spectra [6] and the zero-bias conductance takes its maximum at a ¼ p=4 [7]. It is known that the ZES are responsible for the low-temperature anomaly of the *
Tel.: +81-11-706-6792; fax: +81-11-716-6157. E-mail address: [email protected] (Y. Asano).
Josephson current in SIS junctions of the d-wave superconductor [8,9]. In a previous paper [10], we studied the dc Josephson current in superconductor/dirty normal metal/superconductors (SNS) junctions. It was shown that the ensemble average of the Josephson current hJ i vanishes for a ¼ p=4, whereas the Josephson current in a single sample has a large amplitude because of the ZES at the NS interface. The applicability of the analytical results [10], however, is limited to the SNS junctions where the normal metal is in the diffusive transport regime. In this paper, we numerically study the Josephson effect in SNS junctions as a function of degree of disorder in the normal metal. The results show that hJ i vanishes even when the normal metal is in the quasi-ballistic transport regime. 2. Model Let us consider the SNS junction on the twodimensional tight-binding model as shown in Fig. 1,
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 9 9 9 - 6
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Y. Asano / Physica C 367 (2002) 157–160
Fig. 1. The system under consideration consists of three region: two superconductors (j 6 0 and j P LN þ 1) and the normal conductor (1 6 j 6 LN ). The phase of the left (right) superconductor is uL (uR ). The orientation angle between the crystalline axis and junction interface normal is a in the superconductors on both sides.
where the length of the normal segment is LN a0 and with of the junctions is WJ a0 , where a0 is the lattice constant. The pair potential in the momentum space is schematically illustrated in the figure. In the high-Tc superconductors, a-axis points the direction in which the amplitude of the pair potential take its maximum. The orientation angle between the a-axis and the junction interface normal (x-direction) is a. In this paper, we assume that the orientation angle of the two superconductors are equal to each other. The phase of the pair potential in the superconductor on the left (right) hand side is uL (uR ). The Hamiltonian of the system reads, H ¼ t t
WJ X h 1 X X j¼1 m¼1 WJ 1 X X
cyjþ1;m;r cj;m;r þ h:c:
i
r
i Xh y cj;mþ1;r cj;m;r þ h:c: r
j¼1 m¼1
WJ X 1 X X ðj;m þ 4t lÞcyj;m;r cj;m;r þ j¼1 m¼1
X Xh j0 ;m0
r
i D ðj0 ; m0 ; j; mÞcj0 ;m0 ;" cj;m;# þ h:c: ;
j;m
ð1Þ where r ¼ ðj; mÞ is the lattice index, cyj;m;r (cj;m;r ) is the creation (annihilation) operator of an electron
at (j; m) with spin rð¼" or #Þ, t is the nearest neighbor hopping-integral, and l is the Fermi energy, respectively. Throughout this paper, we take units of h ¼ kB ¼ 1, where kB is the Boltzmann constant. The energy and the length are measured in units of t and a0 . In the y-direction, the periodic boundary condition is applied. We assume that the on-site potential j;m is zero in the two superconductors, (i.e., j;m ¼ 0 for j 6 0 and LN þ 1 6 j). In the d-wave superconductor the pair ðr; r0 Þ has the site-off-diagonal elements potential D depending on the orientation angle [11]. In the normal segment (2 6 j 6 LN 1), we introduce the impurities which is described by the on-site potential j;m given randomly in the range of VI =2 6 j;m 6 VI =2. At the NS interface (j ¼ 1 and j ¼ LN ), we consider the potential barrier which is represented by j;m ¼ VB . The Hamiltonian is diagonalized by the Bogoliubov transformation and the wave functions of a Bogoliubov quasi-particle obeys the Bogoliubov–de Gennes (BdG) equation [13]. The dc Josephson current can be calculated from the Matsubara Green function of BdG equation by using the recursive Green function method [12]. Details of the numerical method was discussed in the previous paper [11]. In this method, it is possible to calculate the Josephson current of a single sample with specific random potential con-
Y. Asano / Physica C 367 (2002) 157–160
figuration. After the numerical simulation over a number of samples with different random configuration, it is also possible to compute the ensemble average of the Josephson current. These are the advantages of this method.
3. Results In Fig. 2, we show the conductance of the normal segment at zero temperature as a function of LN by using the usual recursive Green func-
Fig. 2. The conductance in the normal metal at T ¼ 0 is shown, where WJ ¼ 20, l ¼ 2t and VI ¼ 2t, respectively. The ensemble average of the conductance is GN ¼ ð2e2 =hÞgN . We plot gN LN =WJ as a function of LN .
159
tion method [12], where GN ¼ ð2e2 =hÞgN is the ensemble average of the conductance in the normal metal. Throughout this paper, we fix the parameters such as l ¼ 2t, WJ ¼ 20 and VI ¼ 2t, respectively. When the normal metal is in the quasiballistic regime, gN LN =WJ is proportional to LN since gN Nc , where Nc ¼ WJ kF =p is the number of propagation channels. When the normal metal is in the diffusive transport regime, gN L=WJ becomes kF ‘=2 independent of LN , where kF is the Fermi wave number and ‘ is the elastic mean free path. When the normal metal is in the localization regime, gN LN =WJ behaves like expðLN =nL Þ and decreases with increasing LN , where nL is the localization length. Thus in Fig. 2, the quasi-ballistic, the diffusive and the localization regimes correspond to LN < 60, 60 < LN < 110 and LN > 110, respectively. From the numerical results in the diffusive regime, we estimate the mean free path to be ‘ 6:7a0 and the ratio of LN =‘ is shown in the lower horizontal axis of Fig. 2(a). The diffusive regime is roughly characterized by a relation LN =‘ P 10. The ensemble average of the Josephson current hJ ðaÞi in units of eD0 for a ¼ 0 and p=4 are shown as a function of LN =‘ in Fig. 2(b), where the amplitude of the pair potential at zero-temperature D0 is 0.01t, the temperature is fixed at T =Tc ¼ 0:1, and uL –uR is taken to be p=2. We describe the dependence of the amplitude of the pair potential on T by the BCS theory and Tc 0:57D0 is the critical temperature. The numerical data for a ¼ p=4 and a ¼ 0 are shown with the solid circles and the solid diamonds, respectively. The results of hJ ð0Þi decreases with increasing LN since hJ ð0Þi / gN expðLN =nD Þ for LN =‘ 1, where nD is the coherence length. The Josephson current hJ ðp=4Þi rapidly decreases with increasing LN in comparison with hJ ð0Þi and almost vanishes for LN =‘ P 5. When we compare hJ ðp=4Þi with the conductance in Fig. 2(a), the hJ ðp=4Þi vanishes even when the normal metal is in the quasi-ballistic regime. At least for LN =‘ > 3, hJ ðp=4Þi becomes smaller than 2 1=2 its fluctuations, dJ ¼ ðhJ 2 i hJ i Þ , which are shown with the open circles. The fluctuations increase with decrease LN because the Josephson current itself increases. In addition to this, in the quasi-ballistic regime, the amplitudes of the
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Y. Asano / Physica C 367 (2002) 157–160
conductance fluctuations become larger [14] than the universal values [15] in the diffusive regime. This also explains the increase of dJ for small LN . The amplitude of dJ for a ¼ p=4 depends on T and VB because of the ZES at the interface [10] and increases with decreasing T, which indicates an importance of the mesoscopic fluctuations in the Josephson current.
4. Conclusion In a previous paper, we analytically show that the ensemble average of the Josephson current vanishes in SNS junctions of the d-wave superconductors when the orientation angle is p=4 and the degree of disorder in normal metal is sufficiently large. Within the analytic calculation, the applicability of the conclusion is limited to the SNS junctions where the normal metal is in the diffusive transport regime. In this paper, we numerically study the Josephson current as a function of degree of disorder in the normal metal. The numerical results show the averaged Josephson current vanishes even when the normal metal is in the quasi-ballistic transport regime. In experiments [16], the junctions are not free from disorder near the interfaces. In such junctions, we should pay attention to the mesoscopic fluctuations in the Josephson current.
Acknowledgements The author is indebted to N. Tokuda, H. Akera and Y. Tanaka for useful discussion. References [1] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2001) 1641. [2] T. L€ ofwander, V.S. Shumeiko, G. Wendin, Supercond. Sci. Technol. 14 (2001) R53. [3] M. Sigrist, T.M. Rice, J. Phys. Soc. Jpn. 61 (1992) 4283; M. Sigrist, T.M. Rice, Rev. Mod. Phys. 67 (1995) 503. [4] D.A. Wollman, D.J. van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [5] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [6] J.Y. Wei, N.-C. Yeh, D.F. Garrigus, M. Strasik, Phys. Rev. Lett. 81 (1998) 2542. [7] I. Iguchi, W. Wang, M. Yamazaki, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 62 (2000) R6131; Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [8] Y. Tanaka, S. Kashiwaya, Phys. Rev. B 53 (1996) R11957. [9] Y.S. Barash, H. Burkhardt, D. Rainer, Phys. Rev. Lett. 77 (1996) 4070. [10] Y. Asano, Phys. Rev. B 64 (2001) 014511. [11] Y. Asano, Phys. Rev. B 63 (2001) 052512. [12] P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. [13] P.G. de Gennes, Superconductivity of Metals and Alloys, Benjamin, New York, 1966. [14] Y. Asano, G.E.W. Bauer, Phys. Rev. B 54 (1996) 11602; Y. Asano, G.E.W. Bauer, Physica B 249–251 (1998) 523. [15] P.A. Lee, A.D. Stone, H. Fukuyama, Phys. Rev. B 35 (1987) 1039. [16] E. Ilichev et al., Phys. Rev. Lett. 81 (1998) 894; E. Ilichev et al., Phys. Rev. Lett. 86 (2001) 5369.
A numerical study of dc Josephson effect in SNS junctions of d-wave superconductors Yasuhiro Asano * Department of Applied Physics, Hokkaido University, Kita-13, Nishi-8 Kitaku, Sapporo 060-8628, Japan
Abstract We numerically study the dc Josephson current in d-wave superconductor/normal metal/d-wave superconductor junctions by using the recursive Green function method. The Josephson current is sensitive to the orientation angle between the crystalline axis of the high-Tc superconductors and the normal of the junction interface. It was pointed out that the ensemble average of the Josephson current, hJ i, vanishes when the orientation angle is p=4 and the normal metal is in the diffusive transport regime. We show in this paper that hJ i disappears even when the normal metal is in the quasi-ballistic transport regime. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 74.80.Fp; 74.25.Fy; 74.50.þr Keywords: Josephson current; Mesoscopic fluctuations; Zero-energy states
1. Introduction The transport phenomena in junctions of d-wave superconductors have been studied intensively in recent years [1,2] because the high-Tc superconductors might have the d-wave pairing symmetry [3,4]. The transport properties in the d-wave junctions are sensitive to the orientation angle a between the crystalline axis of the high-Tc superconductors and the normal of the junction interface. This is because the zero-energy states (ZES) [5] are formed at the normal metal/superconductor (NS) interface when the potential barrier at the interface is large enough. In N/d-wave superconductor junctions, the ZES are observed in the conductance spectra [6] and the zero-bias conductance takes its maximum at a ¼ p=4 [7]. It is known that the ZES are responsible for the low-temperature anomaly of the *
Tel.: +81-11-706-6792; fax: +81-11-716-6157. E-mail address: [email protected] (Y. Asano).
Josephson current in SIS junctions of the d-wave superconductor [8,9]. In a previous paper [10], we studied the dc Josephson current in superconductor/dirty normal metal/superconductors (SNS) junctions. It was shown that the ensemble average of the Josephson current hJ i vanishes for a ¼ p=4, whereas the Josephson current in a single sample has a large amplitude because of the ZES at the NS interface. The applicability of the analytical results [10], however, is limited to the SNS junctions where the normal metal is in the diffusive transport regime. In this paper, we numerically study the Josephson effect in SNS junctions as a function of degree of disorder in the normal metal. The results show that hJ i vanishes even when the normal metal is in the quasi-ballistic transport regime. 2. Model Let us consider the SNS junction on the twodimensional tight-binding model as shown in Fig. 1,
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 9 9 9 - 6
158
Y. Asano / Physica C 367 (2002) 157–160
Fig. 1. The system under consideration consists of three region: two superconductors (j 6 0 and j P LN þ 1) and the normal conductor (1 6 j 6 LN ). The phase of the left (right) superconductor is uL (uR ). The orientation angle between the crystalline axis and junction interface normal is a in the superconductors on both sides.
where the length of the normal segment is LN a0 and with of the junctions is WJ a0 , where a0 is the lattice constant. The pair potential in the momentum space is schematically illustrated in the figure. In the high-Tc superconductors, a-axis points the direction in which the amplitude of the pair potential take its maximum. The orientation angle between the a-axis and the junction interface normal (x-direction) is a. In this paper, we assume that the orientation angle of the two superconductors are equal to each other. The phase of the pair potential in the superconductor on the left (right) hand side is uL (uR ). The Hamiltonian of the system reads, H ¼ t t
WJ X h 1 X X j¼1 m¼1 WJ 1 X X
cyjþ1;m;r cj;m;r þ h:c:
i
r
i Xh y cj;mþ1;r cj;m;r þ h:c: r
j¼1 m¼1
WJ X 1 X X ðj;m þ 4t lÞcyj;m;r cj;m;r þ j¼1 m¼1
X Xh j0 ;m0
r
i D ðj0 ; m0 ; j; mÞcj0 ;m0 ;" cj;m;# þ h:c: ;
j;m
ð1Þ where r ¼ ðj; mÞ is the lattice index, cyj;m;r (cj;m;r ) is the creation (annihilation) operator of an electron
at (j; m) with spin rð¼" or #Þ, t is the nearest neighbor hopping-integral, and l is the Fermi energy, respectively. Throughout this paper, we take units of h ¼ kB ¼ 1, where kB is the Boltzmann constant. The energy and the length are measured in units of t and a0 . In the y-direction, the periodic boundary condition is applied. We assume that the on-site potential j;m is zero in the two superconductors, (i.e., j;m ¼ 0 for j 6 0 and LN þ 1 6 j). In the d-wave superconductor the pair ðr; r0 Þ has the site-off-diagonal elements potential D depending on the orientation angle [11]. In the normal segment (2 6 j 6 LN 1), we introduce the impurities which is described by the on-site potential j;m given randomly in the range of VI =2 6 j;m 6 VI =2. At the NS interface (j ¼ 1 and j ¼ LN ), we consider the potential barrier which is represented by j;m ¼ VB . The Hamiltonian is diagonalized by the Bogoliubov transformation and the wave functions of a Bogoliubov quasi-particle obeys the Bogoliubov–de Gennes (BdG) equation [13]. The dc Josephson current can be calculated from the Matsubara Green function of BdG equation by using the recursive Green function method [12]. Details of the numerical method was discussed in the previous paper [11]. In this method, it is possible to calculate the Josephson current of a single sample with specific random potential con-
Y. Asano / Physica C 367 (2002) 157–160
figuration. After the numerical simulation over a number of samples with different random configuration, it is also possible to compute the ensemble average of the Josephson current. These are the advantages of this method.
3. Results In Fig. 2, we show the conductance of the normal segment at zero temperature as a function of LN by using the usual recursive Green func-
Fig. 2. The conductance in the normal metal at T ¼ 0 is shown, where WJ ¼ 20, l ¼ 2t and VI ¼ 2t, respectively. The ensemble average of the conductance is GN ¼ ð2e2 =hÞgN . We plot gN LN =WJ as a function of LN .
159
tion method [12], where GN ¼ ð2e2 =hÞgN is the ensemble average of the conductance in the normal metal. Throughout this paper, we fix the parameters such as l ¼ 2t, WJ ¼ 20 and VI ¼ 2t, respectively. When the normal metal is in the quasiballistic regime, gN LN =WJ is proportional to LN since gN Nc , where Nc ¼ WJ kF =p is the number of propagation channels. When the normal metal is in the diffusive transport regime, gN L=WJ becomes kF ‘=2 independent of LN , where kF is the Fermi wave number and ‘ is the elastic mean free path. When the normal metal is in the localization regime, gN LN =WJ behaves like expðLN =nL Þ and decreases with increasing LN , where nL is the localization length. Thus in Fig. 2, the quasi-ballistic, the diffusive and the localization regimes correspond to LN < 60, 60 < LN < 110 and LN > 110, respectively. From the numerical results in the diffusive regime, we estimate the mean free path to be ‘ 6:7a0 and the ratio of LN =‘ is shown in the lower horizontal axis of Fig. 2(a). The diffusive regime is roughly characterized by a relation LN =‘ P 10. The ensemble average of the Josephson current hJ ðaÞi in units of eD0 for a ¼ 0 and p=4 are shown as a function of LN =‘ in Fig. 2(b), where the amplitude of the pair potential at zero-temperature D0 is 0.01t, the temperature is fixed at T =Tc ¼ 0:1, and uL –uR is taken to be p=2. We describe the dependence of the amplitude of the pair potential on T by the BCS theory and Tc 0:57D0 is the critical temperature. The numerical data for a ¼ p=4 and a ¼ 0 are shown with the solid circles and the solid diamonds, respectively. The results of hJ ð0Þi decreases with increasing LN since hJ ð0Þi / gN expðLN =nD Þ for LN =‘ 1, where nD is the coherence length. The Josephson current hJ ðp=4Þi rapidly decreases with increasing LN in comparison with hJ ð0Þi and almost vanishes for LN =‘ P 5. When we compare hJ ðp=4Þi with the conductance in Fig. 2(a), the hJ ðp=4Þi vanishes even when the normal metal is in the quasi-ballistic regime. At least for LN =‘ > 3, hJ ðp=4Þi becomes smaller than 2 1=2 its fluctuations, dJ ¼ ðhJ 2 i hJ i Þ , which are shown with the open circles. The fluctuations increase with decrease LN because the Josephson current itself increases. In addition to this, in the quasi-ballistic regime, the amplitudes of the
160
Y. Asano / Physica C 367 (2002) 157–160
conductance fluctuations become larger [14] than the universal values [15] in the diffusive regime. This also explains the increase of dJ for small LN . The amplitude of dJ for a ¼ p=4 depends on T and VB because of the ZES at the interface [10] and increases with decreasing T, which indicates an importance of the mesoscopic fluctuations in the Josephson current.
4. Conclusion In a previous paper, we analytically show that the ensemble average of the Josephson current vanishes in SNS junctions of the d-wave superconductors when the orientation angle is p=4 and the degree of disorder in normal metal is sufficiently large. Within the analytic calculation, the applicability of the conclusion is limited to the SNS junctions where the normal metal is in the diffusive transport regime. In this paper, we numerically study the Josephson current as a function of degree of disorder in the normal metal. The numerical results show the averaged Josephson current vanishes even when the normal metal is in the quasi-ballistic transport regime. In experiments [16], the junctions are not free from disorder near the interfaces. In such junctions, we should pay attention to the mesoscopic fluctuations in the Josephson current.
Acknowledgements The author is indebted to N. Tokuda, H. Akera and Y. Tanaka for useful discussion. References [1] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2001) 1641. [2] T. L€ ofwander, V.S. Shumeiko, G. Wendin, Supercond. Sci. Technol. 14 (2001) R53. [3] M. Sigrist, T.M. Rice, J. Phys. Soc. Jpn. 61 (1992) 4283; M. Sigrist, T.M. Rice, Rev. Mod. Phys. 67 (1995) 503. [4] D.A. Wollman, D.J. van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [5] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [6] J.Y. Wei, N.-C. Yeh, D.F. Garrigus, M. Strasik, Phys. Rev. Lett. 81 (1998) 2542. [7] I. Iguchi, W. Wang, M. Yamazaki, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 62 (2000) R6131; Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [8] Y. Tanaka, S. Kashiwaya, Phys. Rev. B 53 (1996) R11957. [9] Y.S. Barash, H. Burkhardt, D. Rainer, Phys. Rev. Lett. 77 (1996) 4070. [10] Y. Asano, Phys. Rev. B 64 (2001) 014511. [11] Y. Asano, Phys. Rev. B 63 (2001) 052512. [12] P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. [13] P.G. de Gennes, Superconductivity of Metals and Alloys, Benjamin, New York, 1966. [14] Y. Asano, G.E.W. Bauer, Phys. Rev. B 54 (1996) 11602; Y. Asano, G.E.W. Bauer, Physica B 249–251 (1998) 523. [15] P.A. Lee, A.D. Stone, H. Fukuyama, Phys. Rev. B 35 (1987) 1039. [16] E. Ilichev et al., Phys. Rev. Lett. 81 (1998) 894; E. Ilichev et al., Phys. Rev. Lett. 86 (2001) 5369.