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Interface density of states in d-wave superconducting proximity effect through Andreev reflections
Journal of Physics and Chemistry of Solids 67 (2006) 132–135 www.elsevier.com/locate/jpcs
Interface density of states in d-wave superconducting proximity effect through Andreev reflections Y. Tanuma a,*, Y. Tanaka b, S. Kashiwaya c a Institute of Physics, Kanagawa University, Yokohama, 221-8686, Japan Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan c National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8568, Japan b
Abstract Tunneling spectra via Andreev bound states between a normal metal ðNÞ=dx2Ky2 -wave superconductor (S) (in the presence of a subdominant s-wave pair potential) junction are investigated. In the present work, we employ quasiclassical Green’s function methods in order to study the role of the proximity effect in detail. In the case of a high transparent contact to the (100) interface of the d-wave superconductor, the pair potential penetrates into the inside of the N due to the proximity effect, where the is-wave is not indeed at all. Then, the tunneling spectra has a very sharp zero-energy peak (ZEP). This ZEP originates from the fact that quasiparticles feel different sign of the pair potentials between normal metals and d-wave superconductors through Andreev reflections. On the other hand, in the N/S junction with the (110) interface, the induction of is-wave component in the S side leads to an induced is-wavepair potential also in the N side due to the proximity effect. In this case, the interface density of states has a dip structure. We show that the spatial dependence of pair potentials is significantly sensitive to the transparency of the junction. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.50.Cr; 74.20.Rp; 74.72.-h
1. Introduction Nowadays, most promising symmetry of superconducting state of high-Tc cuprates is d-wave. One of the important features of the d-wave symmetry is the so-called zero-bias conductance peak (ZBCP) [1,2] due to the formation of Andreev bound states (ABS’s) in normal metal/d-wave superconductor (N/S) junctions. The ABS’s originate from the interference effect in the predominant dx2Ky2 -wave symmetry through reflection at a surface or an interface [3]. Up to now, the consistency between the theories and experiments has been checked in details [1,4–7]. On the other hand, the reduction of the dx2Ky2 -wave state at the surface or interface allows the coexistence of different symmetry of pair potentials. The subdominant interaction induces the broken time reversal symmetry state (BTRSS), i.e. dCis-wave state [8,9]. The splitting of ZBCP in tunneling spectra at low temperatures may be one of the evidence for the BTRSS. However, as regards this point, tunneling experiments are still controversial. Some experiments report the splitting of ZBCP, * Correspondence author: Tel.: C81-45-491-1701; fax: C81-45-413-7288. E-mail address: [email protected] (Y. Tanuma).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.034
others do not show the splitting even in low temperatures [10,11]. More recently, it is proposed that the induced s-wave component of the pair potential by proximity effect in the N side may enhance the magnitude of the subdominant s-wave component in the S side, which forms the BTRSS based on the analysis of the tunneling experiments [12]. The proximity effect in N/S junction without the BTRSS was theoretically studied by Ohashi [13]. In order to understand the actual tunneling spectroscopy quantitatively, we must study the proximity effect in detail. Motivated by this point, we extend our previous formula [6] to take into account the induced pair potential in the N side. In the present paper, we study the local density of states at the interface of N/S junctions based on the self-consistently determined pair potentials by changing the transmission probability of the junctions. 2. Formulation We consider the normal metal (N)/d-wave superconductor (S) junctions separated by an insulating interface at xZ0, where the normal metal is located at x!0 and the dx2Ky2 -wave superconductor extends elsewhere In order to study the proximity effect in the N/S junction, we determine the spatial variation of the pair potentials self-consistently. For this purpose, we make use of the quasi-classical Green’s function
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
procedure [14] developed by Nagai and co-workers [15,16]. Here a cylindrical Fermi surface is assumed and the magnitude of the Fermi momentum and the effective mass are chosen to be equal both in the N and S sides. The pair potential in S [N] side will tend to the bulk value [zero] Ds(fa,N) [DN(fa,-N)] at sufficiently large x. We introduce the quasiclassical Green’s functions in N and S regions given by S ðfC ;xÞZ g^ CC
N g^ KK ðfK ;xÞZ
the Matsubara frequency. Initial conditions of these equations are DS ðfC ;NÞ* N ; DK ðKNÞZ0; (5) S um CUC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Ula Z u2m CjDl ðfa ;NÞj2 . Moreover, the boundary conditions of the Gla ð0Þ and Fal ð0Þ at the interface xZ0 are
DSC ðNÞZ
! S S 1CDSC ðxÞFC ðxÞ2iFC ðxÞ i ; S ðxÞ S S 1KDSC ðxÞFC ðxÞ K1KDSC ðxÞFC ðxÞð2Þ 2iDC (1) i N ðxÞF N ðxÞ 1KDK K
!
N N 1CDK ðxÞFK ðxÞ
N 2iFK ðxÞ
N ðxÞ 2iDK
N N K1KDK ðxÞFK ðxÞ
:
S FC Z
N S RGN C KGK Cð1KRÞGK ; S N N N GN GK ½GC KRGK Kð1KRÞGC K
(6)
N Z FK
S N RGK KGSC Cð1KRÞGC : S S S S GN C ½GK KRGC Kð1KRÞGK GC
(7)
The pair potentials are given by [6,13,16] X 1 Dl ðf;xÞZ 2p 0%m!u =2pT
(2)
In the above, Dla ðxÞ and Fal ðxÞðlZN;S;aZGÞ obey the following Riccati-type equations ZvFx
133
c
p=2 ð
v l D ðxÞZa½2um Dla ðxÞCDl ðfa ;xÞDla ðxÞ2 KDl ðfa ;xÞ* ; vx a
! (3)
df0
X
V l ðf;f0a Þ½^glaa ðfa0 ;xÞ12 ;
Kp=2
where uc is the cutoff energy and ½^glaa ðfa0 ;xÞ12 denotes the 12 element of g^ laa ðf0a ;xÞ. Here V l ðf;fa0 Þ is the effective interelectron potential of the Copper pair. In our numerical calculations, a new Dl(f,x) is calculated using Eq. (8) and g^ laa ðfa ;xÞ is obtained again from Eqs. (1)–(7). We reiterate this process until the convergence is sufficiently obtained.
v ZvFx Fal ðxÞZKa½2um Fal ðxÞCDl ðfa ;xÞ* Fal ðxÞ2 KDl ðfa ;xÞ; vx (4) where nFx is the x component of the Fermi velocity and t^ 3 is the Pauli matrix. Here umZpT(2mC1) with integer m is
(100) (c) 1
(a)
∆N,S(x)/ ∆d0
1
Re[∆d]
Re[∆ ] d
R = 1 (solid lines) R = 0 (dot lines) 0.5
R = 0.25 (solid lines) R = 0.5 (dot lines) R = 0.75 (dash lines)
0.5
Re[∆N]
Re[∆N]
Re[∆s]
Re[∆s]
0
0 –20
–10
0
10
20
–10
0
10
(110) (d) 1
(b)
∆N,S(x)/ ∆d0
1
Re[∆ ] d
Re[∆ ] d
R = 1 (solid lines) R = 0 (dot lines)
0.5
0.5
Im[∆N]
R = 0.25 (solid lines) R = 0.5 (dot lines) R = 0.75 (dash lines)
Im[∆N]
Im[∆s]
Im[∆s]
0
0 –10
0 x / ξd
10
(8)
a
–10
0 x / ξd
10
Fig. 1. Spatial dependence of the pair potentials in the N/D junctions for various R. (a) (100) interface [qZ0], (b) (110) interface [qZp/4].
134
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
(100) (c) 4
NS(E,0) / N0
(a) 3
R = 0.75 R=1
R=0 R = 0.25 R = 0.5
2
2 1
0 –2
0
2
0 –2
0
2
(110)
NS(E,0) / N0
(b) 3
(d) 4 R = 0.75 R=1
R=0 R = 0.25 R = 0.5
2
2 1
0 –2
0 E / ∆d0
2
0 –2
0 E / ∆d0
2
Fig. 2. The local density of states at the interface on the S side. (a) (100) interface [qZ0], (b) (110) interface [qZp/4].
Next, the local density of states based on the selfconsistently determined pair potentials is obtained as follows,
respectively. The attractive potentials Vl(f,f 0 ) are given by VN Z
Nl ðE; xÞ Z Im
1 2i
p=2 ð
ln TTN C
Vd;s Z df Tr½g^laa ðfa ; xÞt^3 ium/ECid :
(9)
Kp=2
In this paper, almost calculations are performed on the temperature T/TdZ0.02, where Td is the critical temperature of the bulk dx2Ky2 -wave superconductor.
3. Results In this section, the spatial variation of the self-consistently determined pair potentials in the N/S junction and local density of states at the interface are calculated by varying reflection probabilities R. In the bulk region of the dx2Ky2 -wave superconductor, the pair potential is given by Ds(fa,N)ZDd0 cos 2(f-q), where q is the angle between the x-axis and the crystal a-axis of the dx2Ky2 -wave. The spatial dependencies of the pair potentials in the N and S regions are expressed as DN(f,x)ZDN(x), DS ðf; xÞZ Dd ðxÞcos 2ðfKqÞC Ds ðxÞ, where DN(x), Dd(x), and Ds(x) correspond to the amplitude of s-wave in the N side, dx2Ky2 -wave, and s-wave superconducting states,
2pk T P B
ln TTd;s C
1 0%m!uc =2pT mC1=2
2pk T P B
;
1 0%m!uc =2pT mC1=2
(10) :
Here, we take Ts/TdZ0.3. In Fig. 1(a) and (b), we show the spatial variations of the pair potentials near (100) and (110) interfaces of the N/S junctions, respectively. Here, Re[DN,d(x)] and Im[DN,s(x)] stand for real components of DN,d(x) and imaginary components of DN,s(x), respectively. The x-axis of Fig. 1 is normalized by xd Z ZvF =Dd0 , which is the coherence length of the superconductor. Near the (100) surface/interface, Re[Dd(x)] is suppressed near the interface, while Re[Ds(x)] is slightly mixed with the decrease of R [Fig. 1(a)]. Since Im[Ds(x)]Z0 is satisfied, the time reversal symmetry (TRS) is not broken. Turning to the N side, we can readily see that the pair potential Re[DN(x)] is induced toward the inside. On the other hands, near the (110) surface/interface, both the Im[DN(x)] and Im[Ds(x)] are enhanced nearly in RZ0.75. This proximity effect is recently found by Lo¨fwander [18], and our results are consistent with his work. When Re[Dd(x)] is suppressed at the interface in the S side, the quasiparticle forms the ABS with zero-energy at the interface [3]. With R close to unity, the ABS becomes unstable with the introduction of the subdominant s-wave attractive potential. And then Im[Ds(x)] is induced in the vicinity of the interface in the S side [17]. Next, let us show
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
4 (100)
135
d, while the height of it increases monotonically. For the dZ0 limit, the ZEP is reduced to be expressed by the d-function.
NS (E,0) / N0
4. Summary δ = 0.001 δ = 0.005 δ = 0.01 δ = 0.02 δ = 0.05
2
0
0
0.04
In this paper, the spatial dependence of the pair potential in the N/S junction is determined based on the quasiclassical Green’s function method. We have shown that the local density of states at the interface is quite sensitive to the transparency of the junction. For lowtransparency cases, our results are consistent with previous theoretical works [6,8,13,17]. In order to compare with recent experiments, we need to calculate not only the local density of states but also tunneling conductance by changing transparency and the orientation angle of the junction much more in detail.
0.08 E / ∆d0
Fig. 3. The local density of states at the interface near zero energy with RZ0 for d.
the corresponding local density of states at the interface. Fig. 2 is plotted for the (100) and (110) interfaces where the same parameters of Fig. 1 are used, respectively. As shown in Fig. 1(a), the pair potential Re[DN(x)] in the N side remains. In this case, Re[DN(x)] can induce the ZEP. The quasiparticles feel the different sign between Re[DN(x)] and Re[Dd(x)] for jfjOp/4 through Andreev reflections. As for the low transparent cases [the right panel of Fig. 1(a)], the LDOS on the S side has a V-shaped structure similar to the bulk d-wave density of states [6,8,17]. On the other hands, for (110) junctions, as shown in Fig. 1(b), not only Im[DN(x)] but also Im[Ds(x)] remain to be non-zero due to the proximity effect for high transparency. In this case, the LDOS has a dip structure reflecting the induced Im[DN,s(x)] around the interface. With the increase of R, the magnitude of the induced s-wave component which breaks TRS enhances, and then the resulting LDOS has the ZEP splitting. As for low transparent cases, these features are consistent with previous results where the proximity effect in the N side is neglected [6,8,17]. Finally, we concentrate on the width of ZEP in the (100) interface. We see that the relevance of the peak width and the infinitesimal number d introduced in Eq. (9) to avoid divergence in the actual calculation where the inverse of d can be regarded as a life time of the quasiparticle. As shown in Fig. 3, the width of the ZEP becomes narrow with the decrease of the magnitude of
Acknowledgements The authors would like to thank Dr Lofwander for critical and valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18]
Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641. C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. S. Kashiwaya, Y. Tanaka, H. Takashima, M. Koyanagi, K. Kajimura, Phys. Rev. B 51 (1995) 1350. J.Y.T. Wei, N.-C. Yeh, D.F. Garrigus, M. Strasik, Phys. Rev. Lett. 81 (1998) 2542. Y. Tanuma, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 64 (2001) 214519. I. Iguchi, W. Wang, M. Yamazaki, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 62 (2000) R6131. M. Fogelstro¨m, D. Rainer, J.A. Sauls, Phys. Rev. Lett. 79 (1997) 281. M. Covington, M. Aprili, E. Paraoanu, L.H. Greene, F. Xu, J. Zhu, C.A. Mirkin, Phys. Rev. Lett. 79 (1997) 277. S. Kashiwaya, Y. Tanaka, N. Terada, M. Koyanagi, S. Ueno, L. Alff, H. Takashima, Y. Tanuma, K. Kajimura, J. Phys. Chem. Solids 59 (1998) 2034. G. Koren, L. Shkedy, E. Polturak, cond-mat/0306594 (2003). A. Kohen, G. Leibovitch, G. Deutscher, Phys. Rev. Lett. 90 (2003) 207005. Y. Ohashi, J. Phys. Soc. 65 (1996) 823. G. Eilenberger, Z. Phys. 214 (1968) 195. K. Nagai, in: D. Rainer, J.A. Sauls (Eds.), Quasiclassical Methods in Superconductivity and Superfluidity (unpublished). M. Ashida, S. Aoyama, J. Hara, K. Nagai, Phys. Rev. B 40 (1989) 8673. M. Matsumoto, H. Shiba, J. Phys. Soc. Jpn 64 (1995) 4867. T. Lo¨fwander, Phys. Rev. B 70 (2004) 094518.
Interface density of states in d-wave superconducting proximity effect through Andreev reflections Y. Tanuma a,*, Y. Tanaka b, S. Kashiwaya c a Institute of Physics, Kanagawa University, Yokohama, 221-8686, Japan Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan c National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8568, Japan b
Abstract Tunneling spectra via Andreev bound states between a normal metal ðNÞ=dx2Ky2 -wave superconductor (S) (in the presence of a subdominant s-wave pair potential) junction are investigated. In the present work, we employ quasiclassical Green’s function methods in order to study the role of the proximity effect in detail. In the case of a high transparent contact to the (100) interface of the d-wave superconductor, the pair potential penetrates into the inside of the N due to the proximity effect, where the is-wave is not indeed at all. Then, the tunneling spectra has a very sharp zero-energy peak (ZEP). This ZEP originates from the fact that quasiparticles feel different sign of the pair potentials between normal metals and d-wave superconductors through Andreev reflections. On the other hand, in the N/S junction with the (110) interface, the induction of is-wave component in the S side leads to an induced is-wavepair potential also in the N side due to the proximity effect. In this case, the interface density of states has a dip structure. We show that the spatial dependence of pair potentials is significantly sensitive to the transparency of the junction. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.50.Cr; 74.20.Rp; 74.72.-h
1. Introduction Nowadays, most promising symmetry of superconducting state of high-Tc cuprates is d-wave. One of the important features of the d-wave symmetry is the so-called zero-bias conductance peak (ZBCP) [1,2] due to the formation of Andreev bound states (ABS’s) in normal metal/d-wave superconductor (N/S) junctions. The ABS’s originate from the interference effect in the predominant dx2Ky2 -wave symmetry through reflection at a surface or an interface [3]. Up to now, the consistency between the theories and experiments has been checked in details [1,4–7]. On the other hand, the reduction of the dx2Ky2 -wave state at the surface or interface allows the coexistence of different symmetry of pair potentials. The subdominant interaction induces the broken time reversal symmetry state (BTRSS), i.e. dCis-wave state [8,9]. The splitting of ZBCP in tunneling spectra at low temperatures may be one of the evidence for the BTRSS. However, as regards this point, tunneling experiments are still controversial. Some experiments report the splitting of ZBCP, * Correspondence author: Tel.: C81-45-491-1701; fax: C81-45-413-7288. E-mail address: [email protected] (Y. Tanuma).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.034
others do not show the splitting even in low temperatures [10,11]. More recently, it is proposed that the induced s-wave component of the pair potential by proximity effect in the N side may enhance the magnitude of the subdominant s-wave component in the S side, which forms the BTRSS based on the analysis of the tunneling experiments [12]. The proximity effect in N/S junction without the BTRSS was theoretically studied by Ohashi [13]. In order to understand the actual tunneling spectroscopy quantitatively, we must study the proximity effect in detail. Motivated by this point, we extend our previous formula [6] to take into account the induced pair potential in the N side. In the present paper, we study the local density of states at the interface of N/S junctions based on the self-consistently determined pair potentials by changing the transmission probability of the junctions. 2. Formulation We consider the normal metal (N)/d-wave superconductor (S) junctions separated by an insulating interface at xZ0, where the normal metal is located at x!0 and the dx2Ky2 -wave superconductor extends elsewhere In order to study the proximity effect in the N/S junction, we determine the spatial variation of the pair potentials self-consistently. For this purpose, we make use of the quasi-classical Green’s function
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
procedure [14] developed by Nagai and co-workers [15,16]. Here a cylindrical Fermi surface is assumed and the magnitude of the Fermi momentum and the effective mass are chosen to be equal both in the N and S sides. The pair potential in S [N] side will tend to the bulk value [zero] Ds(fa,N) [DN(fa,-N)] at sufficiently large x. We introduce the quasiclassical Green’s functions in N and S regions given by S ðfC ;xÞZ g^ CC
N g^ KK ðfK ;xÞZ
the Matsubara frequency. Initial conditions of these equations are DS ðfC ;NÞ* N ; DK ðKNÞZ0; (5) S um CUC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Ula Z u2m CjDl ðfa ;NÞj2 . Moreover, the boundary conditions of the Gla ð0Þ and Fal ð0Þ at the interface xZ0 are
DSC ðNÞZ
! S S 1CDSC ðxÞFC ðxÞ2iFC ðxÞ i ; S ðxÞ S S 1KDSC ðxÞFC ðxÞ K1KDSC ðxÞFC ðxÞð2Þ 2iDC (1) i N ðxÞF N ðxÞ 1KDK K
!
N N 1CDK ðxÞFK ðxÞ
N 2iFK ðxÞ
N ðxÞ 2iDK
N N K1KDK ðxÞFK ðxÞ
:
S FC Z
N S RGN C KGK Cð1KRÞGK ; S N N N GN GK ½GC KRGK Kð1KRÞGC K
(6)
N Z FK
S N RGK KGSC Cð1KRÞGC : S S S S GN C ½GK KRGC Kð1KRÞGK GC
(7)
The pair potentials are given by [6,13,16] X 1 Dl ðf;xÞZ 2p 0%m!u =2pT
(2)
In the above, Dla ðxÞ and Fal ðxÞðlZN;S;aZGÞ obey the following Riccati-type equations ZvFx
133
c
p=2 ð
v l D ðxÞZa½2um Dla ðxÞCDl ðfa ;xÞDla ðxÞ2 KDl ðfa ;xÞ* ; vx a
! (3)
df0
X
V l ðf;f0a Þ½^glaa ðfa0 ;xÞ12 ;
Kp=2
where uc is the cutoff energy and ½^glaa ðfa0 ;xÞ12 denotes the 12 element of g^ laa ðf0a ;xÞ. Here V l ðf;fa0 Þ is the effective interelectron potential of the Copper pair. In our numerical calculations, a new Dl(f,x) is calculated using Eq. (8) and g^ laa ðfa ;xÞ is obtained again from Eqs. (1)–(7). We reiterate this process until the convergence is sufficiently obtained.
v ZvFx Fal ðxÞZKa½2um Fal ðxÞCDl ðfa ;xÞ* Fal ðxÞ2 KDl ðfa ;xÞ; vx (4) where nFx is the x component of the Fermi velocity and t^ 3 is the Pauli matrix. Here umZpT(2mC1) with integer m is
(100) (c) 1
(a)
∆N,S(x)/ ∆d0
1
Re[∆d]
Re[∆ ] d
R = 1 (solid lines) R = 0 (dot lines) 0.5
R = 0.25 (solid lines) R = 0.5 (dot lines) R = 0.75 (dash lines)
0.5
Re[∆N]
Re[∆N]
Re[∆s]
Re[∆s]
0
0 –20
–10
0
10
20
–10
0
10
(110) (d) 1
(b)
∆N,S(x)/ ∆d0
1
Re[∆ ] d
Re[∆ ] d
R = 1 (solid lines) R = 0 (dot lines)
0.5
0.5
Im[∆N]
R = 0.25 (solid lines) R = 0.5 (dot lines) R = 0.75 (dash lines)
Im[∆N]
Im[∆s]
Im[∆s]
0
0 –10
0 x / ξd
10
(8)
a
–10
0 x / ξd
10
Fig. 1. Spatial dependence of the pair potentials in the N/D junctions for various R. (a) (100) interface [qZ0], (b) (110) interface [qZp/4].
134
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
(100) (c) 4
NS(E,0) / N0
(a) 3
R = 0.75 R=1
R=0 R = 0.25 R = 0.5
2
2 1
0 –2
0
2
0 –2
0
2
(110)
NS(E,0) / N0
(b) 3
(d) 4 R = 0.75 R=1
R=0 R = 0.25 R = 0.5
2
2 1
0 –2
0 E / ∆d0
2
0 –2
0 E / ∆d0
2
Fig. 2. The local density of states at the interface on the S side. (a) (100) interface [qZ0], (b) (110) interface [qZp/4].
Next, the local density of states based on the selfconsistently determined pair potentials is obtained as follows,
respectively. The attractive potentials Vl(f,f 0 ) are given by VN Z
Nl ðE; xÞ Z Im
1 2i
p=2 ð
ln TTN C
Vd;s Z df Tr½g^laa ðfa ; xÞt^3 ium/ECid :
(9)
Kp=2
In this paper, almost calculations are performed on the temperature T/TdZ0.02, where Td is the critical temperature of the bulk dx2Ky2 -wave superconductor.
3. Results In this section, the spatial variation of the self-consistently determined pair potentials in the N/S junction and local density of states at the interface are calculated by varying reflection probabilities R. In the bulk region of the dx2Ky2 -wave superconductor, the pair potential is given by Ds(fa,N)ZDd0 cos 2(f-q), where q is the angle between the x-axis and the crystal a-axis of the dx2Ky2 -wave. The spatial dependencies of the pair potentials in the N and S regions are expressed as DN(f,x)ZDN(x), DS ðf; xÞZ Dd ðxÞcos 2ðfKqÞC Ds ðxÞ, where DN(x), Dd(x), and Ds(x) correspond to the amplitude of s-wave in the N side, dx2Ky2 -wave, and s-wave superconducting states,
2pk T P B
ln TTd;s C
1 0%m!uc =2pT mC1=2
2pk T P B
;
1 0%m!uc =2pT mC1=2
(10) :
Here, we take Ts/TdZ0.3. In Fig. 1(a) and (b), we show the spatial variations of the pair potentials near (100) and (110) interfaces of the N/S junctions, respectively. Here, Re[DN,d(x)] and Im[DN,s(x)] stand for real components of DN,d(x) and imaginary components of DN,s(x), respectively. The x-axis of Fig. 1 is normalized by xd Z ZvF =Dd0 , which is the coherence length of the superconductor. Near the (100) surface/interface, Re[Dd(x)] is suppressed near the interface, while Re[Ds(x)] is slightly mixed with the decrease of R [Fig. 1(a)]. Since Im[Ds(x)]Z0 is satisfied, the time reversal symmetry (TRS) is not broken. Turning to the N side, we can readily see that the pair potential Re[DN(x)] is induced toward the inside. On the other hands, near the (110) surface/interface, both the Im[DN(x)] and Im[Ds(x)] are enhanced nearly in RZ0.75. This proximity effect is recently found by Lo¨fwander [18], and our results are consistent with his work. When Re[Dd(x)] is suppressed at the interface in the S side, the quasiparticle forms the ABS with zero-energy at the interface [3]. With R close to unity, the ABS becomes unstable with the introduction of the subdominant s-wave attractive potential. And then Im[Ds(x)] is induced in the vicinity of the interface in the S side [17]. Next, let us show
Y. Tanuma et al. / Journal of Physics and Chemistry of Solids 67 (2006) 132–135
4 (100)
135
d, while the height of it increases monotonically. For the dZ0 limit, the ZEP is reduced to be expressed by the d-function.
NS (E,0) / N0
4. Summary δ = 0.001 δ = 0.005 δ = 0.01 δ = 0.02 δ = 0.05
2
0
0
0.04
In this paper, the spatial dependence of the pair potential in the N/S junction is determined based on the quasiclassical Green’s function method. We have shown that the local density of states at the interface is quite sensitive to the transparency of the junction. For lowtransparency cases, our results are consistent with previous theoretical works [6,8,13,17]. In order to compare with recent experiments, we need to calculate not only the local density of states but also tunneling conductance by changing transparency and the orientation angle of the junction much more in detail.
0.08 E / ∆d0
Fig. 3. The local density of states at the interface near zero energy with RZ0 for d.
the corresponding local density of states at the interface. Fig. 2 is plotted for the (100) and (110) interfaces where the same parameters of Fig. 1 are used, respectively. As shown in Fig. 1(a), the pair potential Re[DN(x)] in the N side remains. In this case, Re[DN(x)] can induce the ZEP. The quasiparticles feel the different sign between Re[DN(x)] and Re[Dd(x)] for jfjOp/4 through Andreev reflections. As for the low transparent cases [the right panel of Fig. 1(a)], the LDOS on the S side has a V-shaped structure similar to the bulk d-wave density of states [6,8,17]. On the other hands, for (110) junctions, as shown in Fig. 1(b), not only Im[DN(x)] but also Im[Ds(x)] remain to be non-zero due to the proximity effect for high transparency. In this case, the LDOS has a dip structure reflecting the induced Im[DN,s(x)] around the interface. With the increase of R, the magnitude of the induced s-wave component which breaks TRS enhances, and then the resulting LDOS has the ZEP splitting. As for low transparent cases, these features are consistent with previous results where the proximity effect in the N side is neglected [6,8,17]. Finally, we concentrate on the width of ZEP in the (100) interface. We see that the relevance of the peak width and the infinitesimal number d introduced in Eq. (9) to avoid divergence in the actual calculation where the inverse of d can be regarded as a life time of the quasiparticle. As shown in Fig. 3, the width of the ZEP becomes narrow with the decrease of the magnitude of
Acknowledgements The authors would like to thank Dr Lofwander for critical and valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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