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Quasiparticle bound states near the interface of high-Tc superconductors
Physica C 367 (2002) 73–78 www.elsevier.com/locate/physc
Quasiparticle bound states near the interface of high-Tc superconductors Y. Tanaka a,b,*, H. Itoh c, H. Tsuchiura b, Y. Tanuma d, J. Inoue a,b, S. Kashiwaya e,b a
e
Department of Applied Physics, Nagoya University, Chikusaku, Nagoya 464-8603, Japan b CREST, Japan Science and Technology Corporation (JST), Japan c Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan d Department of Physics, Okayama University, Okayama 700-8530, Okayama, Japan National Institute of Advanced Industrial Science and Technology, Tsukuba 305-0045, Japan
Abstract In high-Tc superconductor junctions, due to the sign changing behavior of the pair potential, the zero bias conductance peak (ZBCP) appears in the quasiparticle tunneling spectroscopy. In this paper, we will discuss unresolved problems about ZBCP, i.e., effect of magnetic field and Fermi surface. In the former part of the paper, it is shown that ZBCP does not split into two by the magnetic field, when the transparency of the junction is sufficiently large. In the latter part of the paper, we will extend our theory to include the effect of Fermi surface based on the Green’s function theory and linear response theory. We will clarify ZBCP always appears for [1 1 0] interface even if we take into account of the Fermi surface mismatch at the interface. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: N/I/D junction; Tunnel conductance; Zero bias conductance peak
1. Introduction A great deal of experimental and theoretical studies have revealed that the zero bias conductance peak (ZBCP) seen in the tunneling spectroscopy in high-Tc superconductor junctions is a direct consequence of a dx2 y 2 symmetry of the pair potential [1–12]. The origin of this ZBCP is thought to be the formation of Andreev bound state (ABS) at the Fermi energy (zero energy) near a specularly reflecting surface [3] or interface [13], * Corresponding author. Tel.: +81-52-789-4447; fax: +81-52789-3298. E-mail address: [email protected] (Y. Tanaka).
when the angle between the lobe direction of the dx2 y 2 -wave pair potential and the normal to the interface is nonzero. This state is originated from the interference effect between the injected and reflected quasiparticles at the surface or interface and the sign change of the dx2 y 2 -wave pair potential. There are large number of studies about ZBCP [14–22] and its related problems [23–30]. However, the effect of the magnetic field dependence and the Fermi surface on the tunneling conductance in the normal metal (N)/insulator (I)/ d-wave superconductor (D) junctions is not clarified yet. It was pointed out that in a magnetic field H, screening currents shift the ABS spectrum and lead
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Y. Tanaka et al. / Physica C 367 (2002) 73–78
to a splitting of ZBCP that is linear in H at low fields [20]. This splitting of the ZBCP in a magnetic field has been seen in some experiments [16–18] while others cannot reproduce it [14,15,31]. At this stage, we must clarify the effect of the magnetic field on the ZBCP. It was revealed through the various studies that the degree of the splitting of the ZBCP is affected by transparency of the junctions [32] seriously. In the former part of the paper, we will clarify the effect of the transparency of the junction on the tunneling conductance in the presence of magnetic field. We will clarify that there is a characteristic value of the magnetic field Hc , which crucially determines the line shape of the tunneling conductance. For H > Hc , ZBCP splits into two by magnetic field as pointed out by Fogelstr€ om et al. [20]. While for H < Hc , ZBCP splitting does not occur by magnetic field. In the latter part of this paper, we extend our theory of tunneling spectroscopy to tight-binding model [33] so as to take into account the band mismatch effect. Although there are many theoretical works about tunneling effect in d-wave superconductors, no explicit calculations are performed taking into account the finite transmissivity of the junction and Fermi surface of high-Tc cuprates. We calculate the electrical conductance of N/I/D junctions using the tight-binding model in the normal metal and t–J model in d-wave superconductor in order to include the realistic electronic structure of the high-Tc superconductor [33–35]. We will show that ZBCP always appears even if we take into account of the band mismatch effect as long as the interface is flat.
2. Magnetic field dependence of the tunneling conductance In the following, we calculate the tunneling conductance of N/I/D junctions in the presence of the magnetic field. Since the detailed formulation is written in another paper, we will briefly show the obtained results. We start with the Bogoliubov–de Gennes (BdG) equation for unconventional spinsinglet superconductors and apply the quasiclassical approximation [36–39]. Now, we consider the case where a specularly reflecting surface or in-
terface runs along the y-direction. The insulator located at the interface between normal metal and d-wave superconductor is expressed by H dðxÞ. The magnetic field is applied parallel to the z-axis and the resulting vector potential can be chosen as AðrÞ ¼ ð0; Ay ðxÞ; 0Þ. In this case, the pair potential depends only on x since the system is homogeneous along the y-direction. Since we are now considering the situation where the coherence length of the pair potential n is much smaller than the penetration depth of the magnetic field k, we can ignore the spatial dependence of Ay ðxÞ in the actual calculations. We assume Ay ðxÞ ¼ A0 ¼ H k, where H is the applied magnetic field. We first determine the spatial dependence of the pair potential and then using this pair potential normalized tunneling conductance is determined as a function of bias voltage V. In the following, the applied magnetic field is normalized by /0 =8n0 k with /0 ¼ h=2e. The resulting rT ðeV Þ for a ¼ p=4 with Z ¼ 5 and Z ¼ 2 is plotted in Fig. 1, where ZBCP is obtained without magnetic field. In the case of Z ¼ 5, with the increase of H, the height of rT ðeV Þ around zero voltage is reduced and rT ðeV Þ has a ZBCP splitting (see Fig. 1(a)) for H P 0:2H0 . These features are consistent with the previous theory where the tunneling conductance is obtained in the tunneling limit [20]. On the other hand, for Z ¼ 2, where the transparency of the junction is enhanced, only the magnitude of rT ðeV Þ is reduced with the increase of H, no ZBCP splitting appears. Apparently, H dependence of the line shape of the tunneling conductance strongly depends on the transparency of the junctions. In the presence of magnetic field, the energy of the quasiparticle E is substituted with E evFy Ay ðxÞ ¼ E þ
H D0 sin h p2 : H0 8
In the present case, since positive H is chosen, quasiparticles energy E increases (decreases) by H for h > 0 (h < 0). The order of this increase (decrease) is about H D0 =H0 . We can find the critical value of the magnetic field Hc . For H > Hc , splitting of ZBCP appears by magnetic field as pointed out by Fogelstr€ om et al. [20], while for H < Hc , ZBCP splitting does not occur. We can estimate
Y. Tanaka et al. / Physica C 367 (2002) 73–78
75
where C is a constant with order one. The roughly estimated C is 0:6D0 and 1:2D0 for Z ¼ 5 and Z ¼ 2, respectively, and the resulting Hc is 0:3CH0 and 0:6CH0 for Z ¼ 5 and Z ¼ 2, respectively. By choosing C 0:6, we can understand the overall features of the line shape of rT ðeV Þ obtained in Fig. 1. It is known [2,40] that the width of ZBCP C is almost proportional to the transmissivity of the junctions and the resulting Hc is enhanced with the increase of the transparency junction. We must need large magnitude of H to observe ZBCP splitting in high-transparent junctions.
3. Band mismatch effect In this section, we discuss the effect of band mismatch between normal metal and d-wave superconductor on the conductance in N/I/D junctions. We use tight-binding formulation for the conductance [33] based on Kubo formula and real space Green’s function method [41–45]. We consider a N/I/D junction on two-dimensional lattice system, where N and I are stacked along (1 0 0) axis of square lattice with lattice spacing a and I is connected to the (1 1 0) surface of D. In order to describe the d-wave superconductor, we use t–J model which is one of the promising models for high-Tc superconductors. By using the Gutzwiller approximation and mean-field approximation, the model can be mapped into a BCS like mean-filed Hamiltonian [46]. Obtained effective Hamiltonian is H ¼ t1
X
cyir cjr t2
D¼ the order of Hc by comparing the width of ZBCP C without H. The amplitude of energy shift by magnetic field is given by H D0 =H0 . We can define Hc as Hc ¼
CCH0 ; 2D0
ð1Þ
X
cyir cjr l^ n;
ð2Þ
ði;jÞ0 ;r
ði;jÞ;r
Fig. 1. The normalized tunneling conductance rT ðeV Þ with a ¼ p=4, T =Tc ¼ 0:05: (a) Z ¼ 5 and (b) Z ¼ 2.
X
ð3Þ
Dij cir cjr ;
ði;jÞ;r 0
where ð Þ and ð Þ denote the nearest-neighbor and next-nearest-neighbor pairs, respectively. The hopping parameters t1 and t2 include so-called Gutzwiller factor and the pair potential Dij has dx2 y 2 symmetry.
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Y. Tanaka et al. / Physica C 367 (2002) 73–78
As for the normal region (N and I), we use the single orbital tight-binding model. The Hamiltonian is X y ðclþ1;m;r cl;m;r þ cyl;mþ1;r cl;m;r þ h:c:Þ H ¼ t l;m;r
þ
X
vl;m cyl;m;r cl;m;r l^ n;
ð4Þ
l;m;r
where l is the index of layer along y-direction and m denotes the y position in the layer. Since the system considered has the translational invariance in y-direction, the conductance CS of the junction is given by Kubo formula using the Green’s function G as [33] e2 t2 X ky ky ky G ky CS ðeV Þ ¼ 2 ½G þG G lþ1;l lþ1;l h 2 ky l;lþ1 l;lþ1 ky G ky ky ky G l;l lþ1;lþ1 Glþ1;lþ1 Gl;l 11 ; ¼ GðE i0Þ GðE þ i0Þ; G GðzÞ ¼
z1 H Dy
D z1 þ H y
ð5Þ
Fig. 2. The normalized tunneling conductance rT ðeV Þ for lattice model, t ¼ 10 for both N and I, v ¼ 20 for N, v ¼ 35 (solid line) and 25 (dot-dashed line) for I. The thickness of I is a. In the inset, the Fermi surface in N and normal state of D is plotted with dashed line and solid line, respectively.
ð6Þ
1 ;
ð7Þ
P where the relations P cl;" ðky Þ ¼ m cl;m;" expðiky maÞ and cl;# ðky Þ ¼ m cl;m;# expðiky maÞ were used ky 0 is a 2 2 matrix. and G l;l In Figs. 2 and 3, we plot the normalized tunneling conductance rT ðeV Þ ¼ CS ðeV Þ=CN ðeV Þ, where CN ðeV Þ denotes the corresponding quantity in the normal state where Dij is chosen to be zero. We use the tight-binding parameters of D such as t1 ¼ 0:2765t0 , t2 ¼ 0:0727t0 , l ¼ 0:2075t0 [46], where t0 denotes the next nearest transfer without taking account of the Gutzwiller approximations. In Fig. 2, as for the parameters of N and I, we use t ¼ 10t0 for both N and I, v ¼ 20t0 for N, and v ¼ 25t0 (dot-dashed line) and 35t0 (solid line) for I. The corresponding Fermi surface is plotted in the inset. Although the topology of the Fermi surface is very different in N and D, the ZBCP appears. The height of the ZBCP is enhanced (reduced) for solid line (dot-dashed line), where the magnitude of the transmissivity of the junction is small (large). In Fig. 3, rT ðeV Þ is plotted for different material parameters in N, where we use t ¼ 10t0 for both N and I, v ¼ 0 for N, v ¼ 10t0
Fig. 3. The normalized tunneling conductance rT ðeV Þ for lattice model, t ¼ 10 for both N and I, v ¼ 0 for N, v ¼ 20 (solid line) and 10 (dot-dashed line) for I. The thickness of I is a. In the inset, the Fermi surface in N and normal state of D is plotted with dashed line and solid line, respectively.
(dot-dashed line) and 20t0 (solid line) for I. The corresponding Fermi surface in N plotted in the inset is different from that in Fig. 2. As in the case of Fig. 3, the height of the ZBCP is enhanced (reduced) for solid line (dot-dashed line), where the magnitude of the transmissivity of the junction is
Y. Tanaka et al. / Physica C 367 (2002) 73–78
small (large). In the low transmissive junction with the larger magnitude of barrier, the line shape of rT ðeV Þ is insensitive to the detailed shape of the Fermi surface in the normal region.
4. Summary In this paper, the influence of the magnetic field on the tunneling conductance in N/I/D junctions is studied. The splitting of ZBCP due to the energy shift of the ABS does not always happen when the transparency of the junction is not low. In such a case, only the height of the zero bias conductance peak is suppressed. To understand these features, we introduced Hc CH0 =D0 . For H > Hc , ZBCP splitting occurs by magnetic field as pointed by Fogelstr€ om et al. [20]. While for H < Hc , ZBCP splitting does not occur by magnetic field. Since the magnitude of C is almost proportional to the transparency of the junction, we can see the splitting of ZBCP easily for low transparent junction without applying large magnitude of H. These results serve as a guide to analyze the actual tunneling spectroscopy data in the presence of magnetic field. At this stage, some experiments reported the splitting of the ZBCP [16,18] and others not [14,15,31], by magnetic field. In the light of the present theory, the transparency of the junction is an important key factor to determine the observability of ZBCP splitting. In the latter part of the present paper, we have used tight-binding model in order to treat Fermi surface effect such as band mismatch between normal metal and d-wave superconductor. We have calculated the conductance for N/I/D junctions including the realistic electronic structure for d-wave superconductor. It has been clarified that ZBCP is a robust property for the N/I/D junctions with [1 1 0] interface for d-wave superconductors independent of the topology of the Fermi surface in N and D. Our formulation can be extended to include the magnetic field. It is also revealed recently that the Fermi surface effect influences significantly on the magnetic field splitting [47,48]. In d-wave superconductor junctions, Josephson effect is quite different from conventional superconductor junctions and two of the authors Y.T. and S.K.
77
predicted anomalous temperature dependence of the Josephson current [25,26]. Recently, the anomalous temperature dependence of the Josephson current is actually observed by Il’ichev et al. [49]. It is actually an interesting problem to calculate Josephson current based on the lattice model to understand the temperature dependence in detail.
Acknowledgements This work is supported in part by the Core Research for Evolutional Science and Technology (CREST). Numerical computation in this work was partially carried out at the Yukawa Institute Computer Facility, and the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. J.I. is supported by the NEDO project NAME.
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Quasiparticle bound states near the interface of high-Tc superconductors Y. Tanaka a,b,*, H. Itoh c, H. Tsuchiura b, Y. Tanuma d, J. Inoue a,b, S. Kashiwaya e,b a
e
Department of Applied Physics, Nagoya University, Chikusaku, Nagoya 464-8603, Japan b CREST, Japan Science and Technology Corporation (JST), Japan c Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan d Department of Physics, Okayama University, Okayama 700-8530, Okayama, Japan National Institute of Advanced Industrial Science and Technology, Tsukuba 305-0045, Japan
Abstract In high-Tc superconductor junctions, due to the sign changing behavior of the pair potential, the zero bias conductance peak (ZBCP) appears in the quasiparticle tunneling spectroscopy. In this paper, we will discuss unresolved problems about ZBCP, i.e., effect of magnetic field and Fermi surface. In the former part of the paper, it is shown that ZBCP does not split into two by the magnetic field, when the transparency of the junction is sufficiently large. In the latter part of the paper, we will extend our theory to include the effect of Fermi surface based on the Green’s function theory and linear response theory. We will clarify ZBCP always appears for [1 1 0] interface even if we take into account of the Fermi surface mismatch at the interface. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: N/I/D junction; Tunnel conductance; Zero bias conductance peak
1. Introduction A great deal of experimental and theoretical studies have revealed that the zero bias conductance peak (ZBCP) seen in the tunneling spectroscopy in high-Tc superconductor junctions is a direct consequence of a dx2 y 2 symmetry of the pair potential [1–12]. The origin of this ZBCP is thought to be the formation of Andreev bound state (ABS) at the Fermi energy (zero energy) near a specularly reflecting surface [3] or interface [13], * Corresponding author. Tel.: +81-52-789-4447; fax: +81-52789-3298. E-mail address: [email protected] (Y. Tanaka).
when the angle between the lobe direction of the dx2 y 2 -wave pair potential and the normal to the interface is nonzero. This state is originated from the interference effect between the injected and reflected quasiparticles at the surface or interface and the sign change of the dx2 y 2 -wave pair potential. There are large number of studies about ZBCP [14–22] and its related problems [23–30]. However, the effect of the magnetic field dependence and the Fermi surface on the tunneling conductance in the normal metal (N)/insulator (I)/ d-wave superconductor (D) junctions is not clarified yet. It was pointed out that in a magnetic field H, screening currents shift the ABS spectrum and lead
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 5 - 9
74
Y. Tanaka et al. / Physica C 367 (2002) 73–78
to a splitting of ZBCP that is linear in H at low fields [20]. This splitting of the ZBCP in a magnetic field has been seen in some experiments [16–18] while others cannot reproduce it [14,15,31]. At this stage, we must clarify the effect of the magnetic field on the ZBCP. It was revealed through the various studies that the degree of the splitting of the ZBCP is affected by transparency of the junctions [32] seriously. In the former part of the paper, we will clarify the effect of the transparency of the junction on the tunneling conductance in the presence of magnetic field. We will clarify that there is a characteristic value of the magnetic field Hc , which crucially determines the line shape of the tunneling conductance. For H > Hc , ZBCP splits into two by magnetic field as pointed out by Fogelstr€ om et al. [20]. While for H < Hc , ZBCP splitting does not occur by magnetic field. In the latter part of this paper, we extend our theory of tunneling spectroscopy to tight-binding model [33] so as to take into account the band mismatch effect. Although there are many theoretical works about tunneling effect in d-wave superconductors, no explicit calculations are performed taking into account the finite transmissivity of the junction and Fermi surface of high-Tc cuprates. We calculate the electrical conductance of N/I/D junctions using the tight-binding model in the normal metal and t–J model in d-wave superconductor in order to include the realistic electronic structure of the high-Tc superconductor [33–35]. We will show that ZBCP always appears even if we take into account of the band mismatch effect as long as the interface is flat.
2. Magnetic field dependence of the tunneling conductance In the following, we calculate the tunneling conductance of N/I/D junctions in the presence of the magnetic field. Since the detailed formulation is written in another paper, we will briefly show the obtained results. We start with the Bogoliubov–de Gennes (BdG) equation for unconventional spinsinglet superconductors and apply the quasiclassical approximation [36–39]. Now, we consider the case where a specularly reflecting surface or in-
terface runs along the y-direction. The insulator located at the interface between normal metal and d-wave superconductor is expressed by H dðxÞ. The magnetic field is applied parallel to the z-axis and the resulting vector potential can be chosen as AðrÞ ¼ ð0; Ay ðxÞ; 0Þ. In this case, the pair potential depends only on x since the system is homogeneous along the y-direction. Since we are now considering the situation where the coherence length of the pair potential n is much smaller than the penetration depth of the magnetic field k, we can ignore the spatial dependence of Ay ðxÞ in the actual calculations. We assume Ay ðxÞ ¼ A0 ¼ H k, where H is the applied magnetic field. We first determine the spatial dependence of the pair potential and then using this pair potential normalized tunneling conductance is determined as a function of bias voltage V. In the following, the applied magnetic field is normalized by /0 =8n0 k with /0 ¼ h=2e. The resulting rT ðeV Þ for a ¼ p=4 with Z ¼ 5 and Z ¼ 2 is plotted in Fig. 1, where ZBCP is obtained without magnetic field. In the case of Z ¼ 5, with the increase of H, the height of rT ðeV Þ around zero voltage is reduced and rT ðeV Þ has a ZBCP splitting (see Fig. 1(a)) for H P 0:2H0 . These features are consistent with the previous theory where the tunneling conductance is obtained in the tunneling limit [20]. On the other hand, for Z ¼ 2, where the transparency of the junction is enhanced, only the magnitude of rT ðeV Þ is reduced with the increase of H, no ZBCP splitting appears. Apparently, H dependence of the line shape of the tunneling conductance strongly depends on the transparency of the junctions. In the presence of magnetic field, the energy of the quasiparticle E is substituted with E evFy Ay ðxÞ ¼ E þ
H D0 sin h p2 : H0 8
In the present case, since positive H is chosen, quasiparticles energy E increases (decreases) by H for h > 0 (h < 0). The order of this increase (decrease) is about H D0 =H0 . We can find the critical value of the magnetic field Hc . For H > Hc , splitting of ZBCP appears by magnetic field as pointed out by Fogelstr€ om et al. [20], while for H < Hc , ZBCP splitting does not occur. We can estimate
Y. Tanaka et al. / Physica C 367 (2002) 73–78
75
where C is a constant with order one. The roughly estimated C is 0:6D0 and 1:2D0 for Z ¼ 5 and Z ¼ 2, respectively, and the resulting Hc is 0:3CH0 and 0:6CH0 for Z ¼ 5 and Z ¼ 2, respectively. By choosing C 0:6, we can understand the overall features of the line shape of rT ðeV Þ obtained in Fig. 1. It is known [2,40] that the width of ZBCP C is almost proportional to the transmissivity of the junctions and the resulting Hc is enhanced with the increase of the transparency junction. We must need large magnitude of H to observe ZBCP splitting in high-transparent junctions.
3. Band mismatch effect In this section, we discuss the effect of band mismatch between normal metal and d-wave superconductor on the conductance in N/I/D junctions. We use tight-binding formulation for the conductance [33] based on Kubo formula and real space Green’s function method [41–45]. We consider a N/I/D junction on two-dimensional lattice system, where N and I are stacked along (1 0 0) axis of square lattice with lattice spacing a and I is connected to the (1 1 0) surface of D. In order to describe the d-wave superconductor, we use t–J model which is one of the promising models for high-Tc superconductors. By using the Gutzwiller approximation and mean-field approximation, the model can be mapped into a BCS like mean-filed Hamiltonian [46]. Obtained effective Hamiltonian is H ¼ t1
X
cyir cjr t2
D¼ the order of Hc by comparing the width of ZBCP C without H. The amplitude of energy shift by magnetic field is given by H D0 =H0 . We can define Hc as Hc ¼
CCH0 ; 2D0
ð1Þ
X
cyir cjr l^ n;
ð2Þ
ði;jÞ0 ;r
ði;jÞ;r
Fig. 1. The normalized tunneling conductance rT ðeV Þ with a ¼ p=4, T =Tc ¼ 0:05: (a) Z ¼ 5 and (b) Z ¼ 2.
X
ð3Þ
Dij cir cjr ;
ði;jÞ;r 0
where ð Þ and ð Þ denote the nearest-neighbor and next-nearest-neighbor pairs, respectively. The hopping parameters t1 and t2 include so-called Gutzwiller factor and the pair potential Dij has dx2 y 2 symmetry.
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Y. Tanaka et al. / Physica C 367 (2002) 73–78
As for the normal region (N and I), we use the single orbital tight-binding model. The Hamiltonian is X y ðclþ1;m;r cl;m;r þ cyl;mþ1;r cl;m;r þ h:c:Þ H ¼ t l;m;r
þ
X
vl;m cyl;m;r cl;m;r l^ n;
ð4Þ
l;m;r
where l is the index of layer along y-direction and m denotes the y position in the layer. Since the system considered has the translational invariance in y-direction, the conductance CS of the junction is given by Kubo formula using the Green’s function G as [33] e2 t2 X ky ky ky G ky CS ðeV Þ ¼ 2 ½G þG G lþ1;l lþ1;l h 2 ky l;lþ1 l;lþ1 ky G ky ky ky G l;l lþ1;lþ1 Glþ1;lþ1 Gl;l 11 ; ¼ GðE i0Þ GðE þ i0Þ; G GðzÞ ¼
z1 H Dy
D z1 þ H y
ð5Þ
Fig. 2. The normalized tunneling conductance rT ðeV Þ for lattice model, t ¼ 10 for both N and I, v ¼ 20 for N, v ¼ 35 (solid line) and 25 (dot-dashed line) for I. The thickness of I is a. In the inset, the Fermi surface in N and normal state of D is plotted with dashed line and solid line, respectively.
ð6Þ
1 ;
ð7Þ
P where the relations P cl;" ðky Þ ¼ m cl;m;" expðiky maÞ and cl;# ðky Þ ¼ m cl;m;# expðiky maÞ were used ky 0 is a 2 2 matrix. and G l;l In Figs. 2 and 3, we plot the normalized tunneling conductance rT ðeV Þ ¼ CS ðeV Þ=CN ðeV Þ, where CN ðeV Þ denotes the corresponding quantity in the normal state where Dij is chosen to be zero. We use the tight-binding parameters of D such as t1 ¼ 0:2765t0 , t2 ¼ 0:0727t0 , l ¼ 0:2075t0 [46], where t0 denotes the next nearest transfer without taking account of the Gutzwiller approximations. In Fig. 2, as for the parameters of N and I, we use t ¼ 10t0 for both N and I, v ¼ 20t0 for N, and v ¼ 25t0 (dot-dashed line) and 35t0 (solid line) for I. The corresponding Fermi surface is plotted in the inset. Although the topology of the Fermi surface is very different in N and D, the ZBCP appears. The height of the ZBCP is enhanced (reduced) for solid line (dot-dashed line), where the magnitude of the transmissivity of the junction is small (large). In Fig. 3, rT ðeV Þ is plotted for different material parameters in N, where we use t ¼ 10t0 for both N and I, v ¼ 0 for N, v ¼ 10t0
Fig. 3. The normalized tunneling conductance rT ðeV Þ for lattice model, t ¼ 10 for both N and I, v ¼ 0 for N, v ¼ 20 (solid line) and 10 (dot-dashed line) for I. The thickness of I is a. In the inset, the Fermi surface in N and normal state of D is plotted with dashed line and solid line, respectively.
(dot-dashed line) and 20t0 (solid line) for I. The corresponding Fermi surface in N plotted in the inset is different from that in Fig. 2. As in the case of Fig. 3, the height of the ZBCP is enhanced (reduced) for solid line (dot-dashed line), where the magnitude of the transmissivity of the junction is
Y. Tanaka et al. / Physica C 367 (2002) 73–78
small (large). In the low transmissive junction with the larger magnitude of barrier, the line shape of rT ðeV Þ is insensitive to the detailed shape of the Fermi surface in the normal region.
4. Summary In this paper, the influence of the magnetic field on the tunneling conductance in N/I/D junctions is studied. The splitting of ZBCP due to the energy shift of the ABS does not always happen when the transparency of the junction is not low. In such a case, only the height of the zero bias conductance peak is suppressed. To understand these features, we introduced Hc CH0 =D0 . For H > Hc , ZBCP splitting occurs by magnetic field as pointed by Fogelstr€ om et al. [20]. While for H < Hc , ZBCP splitting does not occur by magnetic field. Since the magnitude of C is almost proportional to the transparency of the junction, we can see the splitting of ZBCP easily for low transparent junction without applying large magnitude of H. These results serve as a guide to analyze the actual tunneling spectroscopy data in the presence of magnetic field. At this stage, some experiments reported the splitting of the ZBCP [16,18] and others not [14,15,31], by magnetic field. In the light of the present theory, the transparency of the junction is an important key factor to determine the observability of ZBCP splitting. In the latter part of the present paper, we have used tight-binding model in order to treat Fermi surface effect such as band mismatch between normal metal and d-wave superconductor. We have calculated the conductance for N/I/D junctions including the realistic electronic structure for d-wave superconductor. It has been clarified that ZBCP is a robust property for the N/I/D junctions with [1 1 0] interface for d-wave superconductors independent of the topology of the Fermi surface in N and D. Our formulation can be extended to include the magnetic field. It is also revealed recently that the Fermi surface effect influences significantly on the magnetic field splitting [47,48]. In d-wave superconductor junctions, Josephson effect is quite different from conventional superconductor junctions and two of the authors Y.T. and S.K.
77
predicted anomalous temperature dependence of the Josephson current [25,26]. Recently, the anomalous temperature dependence of the Josephson current is actually observed by Il’ichev et al. [49]. It is actually an interesting problem to calculate Josephson current based on the lattice model to understand the temperature dependence in detail.
Acknowledgements This work is supported in part by the Core Research for Evolutional Science and Technology (CREST). Numerical computation in this work was partially carried out at the Yukawa Institute Computer Facility, and the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. J.I. is supported by the NEDO project NAME.
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