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conventional superconductor junctions
Physica C 392–396 (2003) 249–253 www.elsevier.com/locate/physc
Effect of the randomness on the electron transport in normal metal/insulator/conventional superconductor junctions Y. Tanaka
a,*
, N. Kitaura a, H. Itoh a, J. Inoue a, A. Golubov b, N. Yoshida c, S. Kashiwaya d
a CREST, Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Department of Applied Physics, University of Twente, 7500 AE Enschede, The Netherlands Department of Microelectronics and Nanoscience, School of Physics and Engineering Physics, Chalmers University of Technology and Goteborg University, S-412 96 Goteborg, Sweden d National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568, Japan b
c
Received 13 November 2002; accepted 31 January 2003
Abstract Tunneling conductance at zero voltage in diffusive normal metal (DN)/insulator/s-wave superconductor junctions is calculated for various situations based on the quasiclassical (QC) theory. In QC theory, the generalized boundary condition introduced by Nazarov [Superlattices and Microstructures 25 1221 (1999)] is applied, where ballistic theory by Blonder Tinkham and Klapwijk and diffusive theory by Volkov Zaitsev and Klapwijk based on the boundary condition by Kupriyanov and Lukichev are naturally reproduced. It is shown that with the increase of the magnitude of resistance of DN ðRD Þ, normalized tunneling conductance is enhanced (suppressed) for low (high) transparent barrier. When the length of DN is larger than that of localization length, tunneling conductance obtained by numerical simulation deviates from that by QC theory. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.40.+k; 74.50.+r
1. Introduction The electron coherence in mesoscopic superconducting systems has been one of the important topics of solid state physics. The phase coherence between incoming electrons and Andreev reflected holes persists in the normal metal at mesoscopic
*
Corresponding author. Tel.: +81-52-789-3701; fax: +81-52789-3298. E-mail address: [email protected] (Y. Tanaka).
length scale [1]. When the physical dimensions of the sample are shorter than the phase breaking length due to inelastic scattering, we can expect interesting interference effects between electrons and holes [2]. Early theory of Blonder et al. [3] derived a compact formula of tunneling conductance using the corresponding transmission coefficients by solving the Andreev equations in the presence of the potential barrier at the interface. The method in Blonder et al. [3] is confined to ballistic systems. The generalization of this method to systems with
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01096-7
250
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
impurities has been attempted by several authors [4–7]. Quasiclassical (QC) GreenÕs function calculation based on nonequilibrium superconductivity theories [8] is powerful and convenient for the actual calculations [9]. In this approach, the impurity scattering is only included in the selfconsistent Born approximation and the weak localization effects are neglected. Volkov, Zaitsev and Klapwijk (VZK) developed a theory of conductance in diffusive normal metal(DN)/superconductor junctions [9] using Kupriyanov and Lukichev (KL) boundary condition for the Keldysh–Nambu GreenÕs function [10] for the diffusive NS interface. After that much more general boundary condition has been presented by Nazarov [11]. Actually, this boundary condition is very general since in the ballistic limit BTK theory is reproduced while in the diffusive limit with low transmissivity at the interface, KL boundary condition applied to VZK theory is reproduced. Although there are several works about charge transport in mesoscopic junctions, almost all of them are based on the KL boundary conditions or simply using BTK theory neglecting impurity scattering. However in the actual junctions, transparency of the junction is not necessarily to be quite small and impurity scattering effect in DN is important. To address this issue, recently, Tanaka Golubov and Kashiwaya (TGK) performed a systematic calculation of tunneling conductance based on a NazarovÕs general boundary condition and clarified many important features [12]. In the present paper, we look at zero voltage conductance using TGK theory in detail and compare with that obtained numerical simulation where localization effect can be taken into account.
free path in DN. The position of boundary between normal metal and DN is denoted as x ¼ 0 and that between DN and S is denoted as x ¼ L, respectively. The insulating barrier at the interface between DN and S is expressed as Tm ¼ 4 cos2 h= ð4 cos2 h þ Z 2 Þ, where Z is a dimensionless constant and h is the injection angle measured from the interface normal to the junction. In VZK and TGK theories, spatial dependence of distribution function ft ðxÞ and a certain parameter cðxÞ which indicates the measure of the proximity effect are determined by solving the following equations o2 cðxÞ þ 2i sin½cðxÞ ¼ 0 ox2 o oft ðxÞ cosh2 cimag ðxÞ ¼ 0 D ox ox
In this section, we explain a model and formulation of QC theory by TGK. Let us consider a two dimensional normal electrode/normal diffusive conductor (DN)/superconductor (DN/S) junction where the length of DN is L and its resistance is RD , respectively where DN is connected to normal electrode at the boundary. The magnitude of L is assumed to be sufficiently larger than that of mean
ð2Þ
where cimag ðxÞ denotes the imaginary part of cðxÞ with diffusion constant D. At x ¼ 0, since DN is attached to normal electrode cð0Þ ¼ 0 and ft ð0Þ ¼ ft0 is satisfied with 1 ð þ eV Þ ð eV Þ tanh ft0 ¼ tanh ; 2 ð2kB T Þ ð2kB T Þ where is a quasiparticle energy and V is a bias voltage, respectively. At x ¼ L, i.e. DN/S interface, L ocðxÞ hF i ¼ ; ð3Þ RD ox x¼L RB F ¼
2ðf cos cL g sin cL ÞTm ð2 Tm Þ þ Tm ½g cos cL þ f sin cL
with cL ¼ cðL Þ and RB ¼
2. Formulation of quasiclassical theory
ð1Þ
D
2e2 2 : R p=2 h dhT ðhÞ cos h p=2
ð4Þ
In the above RD and RB are the resistance in DN and the insulating barrier at the DN/S interface, respectively. The average over the various angles of injected particles at the interface is defined as ,Z Z p=2
hBðhÞi ¼
p=2
dhT ðhÞ cos h
dh cos hBðhÞ p=2
p=2
ð5Þ
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
with BðhÞ ¼ B and T ðhÞ ¼ Tm . While ft ðxÞ satisfies L oft hIb0 i 2 ¼ ð6Þ cosh cimag ðxÞ RD ox RB x¼L
251
eV=0
2 a
Ib0 ¼
σ T (eV)
with Tm2 K1 þ 2Tm ð2 Tm ÞK2 2jð2 Tm Þ þ Tm ½g cos cL þ f sin cL j2
1 b
K1 ¼ ð1 þ j cos cL j2 þ j sin cL j2 Þ 2
2
þ 4Imag½fg
c
ðjgj þ jf j þ 1Þ
0
ð7Þ
K2 ¼ Real gðcos cL þ cos cL Þ þ f ðsin cL þ sin cL Þ ð8Þ
1
2
3 RD /R B
Fig. 1. Normalized tunneling conductance at zero voltage is plotted as a function of RD =RB using QC theory. a: Z ¼ 0, b: Z ¼ 1, and c: Z ¼ 5.
0.4
L
After a simple manipulation, we can obtain total resistance R at zero temperature as Z RB RD L dx þ R¼ : ð9Þ hIb0 i L 0 cosh2 cimag ðxÞ
0
Real(γ )/π
Imag½cos cL sin cL
In the following section, we will look at normalized tunneling conductance at zero voltage rT with rT ¼ rS =rN where rSðNÞ is the tunneling conductance in superconducting (normal) state and is given by rS ¼ 1=R and rN ðeV Þ ¼ rN ¼ 1=ðRD þ RB Þ at zero voltage, respectively.
0.2
a b
3. Results The RD =RB dependence of tunneling conductance rT is plotted in Fig. 1 for various Z. For small magnitude of Z, rT decreases with the increase of RD =RB (curve a). While for larger Z, it is enhanced with the increase of RD =RB . As shown in Fig. 2, the measure of the proximity effect cL is enhanced with the increase of RD =RB . Although the magnitude of cL is enhanced independent of the magnitude of Z, its influence on the tunneling conductance is quite different depending on the magnitude of Z. It is also interesting to compare these results with those obtained by different theoretical approach. We have developed a numerical method to calculate tunneling conductance in DN/S junction [13]. In this approach, localization of wave functions due to impurity scattering in DN can be
0 0
1
2
3 RD /RB
Fig. 2. Measure of the proximity effect cL is plotted as a function of RD =RB using QC theory. a: Z ¼ 0 and b: Z ¼ 5.
taken into account which cannot be treated in QC theory. We consider a DN/S junction on the twodimensional square lattice, where mean-field (BCS) Hamiltonian H on the single-orbital tight-binding model is given by X H ¼ t ðcyl0 ;m0 ;r cyl;m;r þ h:c:Þ l;m;l0 ;m0 ;r
þ
X
vl;m cyl;m;r cl;m;r l^ n
l;m;r
X l;m
½Dl;m cyl;m;# cyl0 ;m0 ;" þ h:c:
ð10Þ
252
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
In the above, ðl; mÞ are lattice indices, cl;m;r (cyl;m;r ) is the annihilation (creation) operator of an electron at ðl; mÞ with spin r ( ¼ " or #), n^ is the number operator and l is the chemical potential of a junction,Prespectively. In the first term, the summation l;m;l0 ;m0 runs over nearest-neighbor sites, and t is the nearest-neighbor hopping integral. In the above, Dl;m ¼ D0 in the S and Dl;m ¼ 0 in DN where D0 is the amplitude of the pair potential in S. The impurity potential in DN is considered through on-site potential vl;m which takes random values uniformly distributed within a range of Vdis =2 6 vl;m 6 Vdis =2 in a disordered region DN. In an insulator, vl;m is set to be Vins independent of ðl; mÞ. Far from the interface, vl;m is taken to be zero. By applying the Bogoliubov transformation, we can calculate conductance using Kubo formula. The obtained results are plotted in Fig. 3. Through the magnitude of RB , we estimate the corresponding magnitude of Z using Eq. (6). For Z ¼ 0 or Z ¼ 1, line shape of rT is similar to that obtained by QC theory. However, for Z ¼ 5, rT does not approach to unity with the increase of RD =RB . For Z ¼ 5 with RD =RB ¼ 1, the localization length in DN estimated by numerical simulation is the same order as that of the length of DN. We think the deviation from QC theory is due to localization effect in DN which cannot be taken into account in the QC theory.
eV=0
2
σT (eV)
a
1 b c 0 0
1
2
R D /RB
3
Fig. 3. Normalized tunneling conductance obtained by numerical simulation at zero voltage is plotted as a function of RD =RB using lattice model. a: Z ¼ 0, b: Z ¼ 1 and c: Z ¼ 5.
4. Conclusions In the present paper, we have calculated tunneling conductance between diffusive normal metal/conventional superconductor junctions. We concentrate on the tunneling conductance at zero voltage. It is shown that with the increase of the magnitude of resistance of DN (RD Þ, normalized tunneling conductance is enhanced (suppressed) for low (high) transparent barrier. When the length of DN is larger than that of localization length, tunneling conductance obtained by numerical simulation in the lattice model deviates from that by QC theory. In the present paper, superconductor is restricted to be conventional s-wave one. Recently, we have developed a theory of transport of unconventional junctions in the diffusive transport regime [14] where ZBCP from Andreev bound state can be taken into account [15,16]. It is actually quite interesting to apply this novel theory to the actual calculation of rT as in the present paper.
References [1] A.F. Andreev, Sov. Phys. JETP 19 (1996) 1228. [2] A. Kastalsky et al., Phys. Rev. Lett. 67 (1991) 3026; C. Nguyen, H. Kroemer, E.L. Hu, Phys. Rev. Lett. 69 (1992) 2847; P. Xiong, G. Xiao, R.B. Laibowitz, Phys. Rev. Lett. 71 (1993) 1907; W. Poirier, D. Mailly, M. Sanquer, Phys. Rev. Lett. 79 (1997) 2105. [3] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25 (1985) 4515. [4] C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731. [5] C.J. Lambert, J. Phys. Condens. Matter 3 (1991) 6579. [6] Y. Takane, H. Ebisawa, J. Phys. Soc. Jpn. 61 (1992) 2858. [7] Yu.V. Nazarov, Phys. Rev. Lett. 73 (1994) 1420. [8] A.I. Larkin, Yu.V. Ovchinnikov, Sov. Phys. JETP 41 (1975) 960. [9] A.F. Volkov, A.V. Zaitsev, T.M. Klapwijk, Physica C 210 (1993) 21. [10] M.Yu. Kupriyanov, V.F. Lukichev, Zh. Exp. Teor. Fiz. 94 (1988) 139, Sov. Phys. JETP 67 1163 (1988). [11] Yu.V. Nazarov, Superlatt. Microstruct. 25 (1999) 1221, cond-mat/9811155. [12] Y. Tanaka, A. Golubov, S. Kashiwaya, cond-mat/0212019. [13] H. Itoh et al., Physica C 367 (2002) 133.
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253 [14] Y. Tanaka, Yu.V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003. [15] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451; S. Kashiwaya et al., Phys. Rev. B 53 (1996) 2667; S. Kashiwaya et al., Phys. Rev. B 51 (1995) 1350; J.Y.T. Wei et al., Phys. Rev. Lett. 81 (1998) 2542;
253
I. Iguchi et al., Phys. Rev. B 62 (2000) R6131; L. Alff et al., Phys. Rev. B 55 (1997) 14757. [16] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641; Y. Tanaka, S. Kashiwaya, Phys. Rev. B 53 (1996) 11957; Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 892.
Effect of the randomness on the electron transport in normal metal/insulator/conventional superconductor junctions Y. Tanaka
a,*
, N. Kitaura a, H. Itoh a, J. Inoue a, A. Golubov b, N. Yoshida c, S. Kashiwaya d
a CREST, Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Department of Applied Physics, University of Twente, 7500 AE Enschede, The Netherlands Department of Microelectronics and Nanoscience, School of Physics and Engineering Physics, Chalmers University of Technology and Goteborg University, S-412 96 Goteborg, Sweden d National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568, Japan b
c
Received 13 November 2002; accepted 31 January 2003
Abstract Tunneling conductance at zero voltage in diffusive normal metal (DN)/insulator/s-wave superconductor junctions is calculated for various situations based on the quasiclassical (QC) theory. In QC theory, the generalized boundary condition introduced by Nazarov [Superlattices and Microstructures 25 1221 (1999)] is applied, where ballistic theory by Blonder Tinkham and Klapwijk and diffusive theory by Volkov Zaitsev and Klapwijk based on the boundary condition by Kupriyanov and Lukichev are naturally reproduced. It is shown that with the increase of the magnitude of resistance of DN ðRD Þ, normalized tunneling conductance is enhanced (suppressed) for low (high) transparent barrier. When the length of DN is larger than that of localization length, tunneling conductance obtained by numerical simulation deviates from that by QC theory. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.40.+k; 74.50.+r
1. Introduction The electron coherence in mesoscopic superconducting systems has been one of the important topics of solid state physics. The phase coherence between incoming electrons and Andreev reflected holes persists in the normal metal at mesoscopic
*
Corresponding author. Tel.: +81-52-789-3701; fax: +81-52789-3298. E-mail address: [email protected] (Y. Tanaka).
length scale [1]. When the physical dimensions of the sample are shorter than the phase breaking length due to inelastic scattering, we can expect interesting interference effects between electrons and holes [2]. Early theory of Blonder et al. [3] derived a compact formula of tunneling conductance using the corresponding transmission coefficients by solving the Andreev equations in the presence of the potential barrier at the interface. The method in Blonder et al. [3] is confined to ballistic systems. The generalization of this method to systems with
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01096-7
250
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
impurities has been attempted by several authors [4–7]. Quasiclassical (QC) GreenÕs function calculation based on nonequilibrium superconductivity theories [8] is powerful and convenient for the actual calculations [9]. In this approach, the impurity scattering is only included in the selfconsistent Born approximation and the weak localization effects are neglected. Volkov, Zaitsev and Klapwijk (VZK) developed a theory of conductance in diffusive normal metal(DN)/superconductor junctions [9] using Kupriyanov and Lukichev (KL) boundary condition for the Keldysh–Nambu GreenÕs function [10] for the diffusive NS interface. After that much more general boundary condition has been presented by Nazarov [11]. Actually, this boundary condition is very general since in the ballistic limit BTK theory is reproduced while in the diffusive limit with low transmissivity at the interface, KL boundary condition applied to VZK theory is reproduced. Although there are several works about charge transport in mesoscopic junctions, almost all of them are based on the KL boundary conditions or simply using BTK theory neglecting impurity scattering. However in the actual junctions, transparency of the junction is not necessarily to be quite small and impurity scattering effect in DN is important. To address this issue, recently, Tanaka Golubov and Kashiwaya (TGK) performed a systematic calculation of tunneling conductance based on a NazarovÕs general boundary condition and clarified many important features [12]. In the present paper, we look at zero voltage conductance using TGK theory in detail and compare with that obtained numerical simulation where localization effect can be taken into account.
free path in DN. The position of boundary between normal metal and DN is denoted as x ¼ 0 and that between DN and S is denoted as x ¼ L, respectively. The insulating barrier at the interface between DN and S is expressed as Tm ¼ 4 cos2 h= ð4 cos2 h þ Z 2 Þ, where Z is a dimensionless constant and h is the injection angle measured from the interface normal to the junction. In VZK and TGK theories, spatial dependence of distribution function ft ðxÞ and a certain parameter cðxÞ which indicates the measure of the proximity effect are determined by solving the following equations o2 cðxÞ þ 2i sin½cðxÞ ¼ 0 ox2 o oft ðxÞ cosh2 cimag ðxÞ ¼ 0 D ox ox
In this section, we explain a model and formulation of QC theory by TGK. Let us consider a two dimensional normal electrode/normal diffusive conductor (DN)/superconductor (DN/S) junction where the length of DN is L and its resistance is RD , respectively where DN is connected to normal electrode at the boundary. The magnitude of L is assumed to be sufficiently larger than that of mean
ð2Þ
where cimag ðxÞ denotes the imaginary part of cðxÞ with diffusion constant D. At x ¼ 0, since DN is attached to normal electrode cð0Þ ¼ 0 and ft ð0Þ ¼ ft0 is satisfied with 1 ð þ eV Þ ð eV Þ tanh ft0 ¼ tanh ; 2 ð2kB T Þ ð2kB T Þ where is a quasiparticle energy and V is a bias voltage, respectively. At x ¼ L, i.e. DN/S interface, L ocðxÞ hF i ¼ ; ð3Þ RD ox x¼L RB F ¼
2ðf cos cL g sin cL ÞTm ð2 Tm Þ þ Tm ½g cos cL þ f sin cL
with cL ¼ cðL Þ and RB ¼
2. Formulation of quasiclassical theory
ð1Þ
D
2e2 2 : R p=2 h dhT ðhÞ cos h p=2
ð4Þ
In the above RD and RB are the resistance in DN and the insulating barrier at the DN/S interface, respectively. The average over the various angles of injected particles at the interface is defined as ,Z Z p=2
hBðhÞi ¼
p=2
dhT ðhÞ cos h
dh cos hBðhÞ p=2
p=2
ð5Þ
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
with BðhÞ ¼ B and T ðhÞ ¼ Tm . While ft ðxÞ satisfies L oft hIb0 i 2 ¼ ð6Þ cosh cimag ðxÞ RD ox RB x¼L
251
eV=0
2 a
Ib0 ¼
σ T (eV)
with Tm2 K1 þ 2Tm ð2 Tm ÞK2 2jð2 Tm Þ þ Tm ½g cos cL þ f sin cL j2
1 b
K1 ¼ ð1 þ j cos cL j2 þ j sin cL j2 Þ 2
2
þ 4Imag½fg
c
ðjgj þ jf j þ 1Þ
0
ð7Þ
K2 ¼ Real gðcos cL þ cos cL Þ þ f ðsin cL þ sin cL Þ ð8Þ
1
2
3 RD /R B
Fig. 1. Normalized tunneling conductance at zero voltage is plotted as a function of RD =RB using QC theory. a: Z ¼ 0, b: Z ¼ 1, and c: Z ¼ 5.
0.4
L
After a simple manipulation, we can obtain total resistance R at zero temperature as Z RB RD L dx þ R¼ : ð9Þ hIb0 i L 0 cosh2 cimag ðxÞ
0
Real(γ )/π
Imag½cos cL sin cL
In the following section, we will look at normalized tunneling conductance at zero voltage rT with rT ¼ rS =rN where rSðNÞ is the tunneling conductance in superconducting (normal) state and is given by rS ¼ 1=R and rN ðeV Þ ¼ rN ¼ 1=ðRD þ RB Þ at zero voltage, respectively.
0.2
a b
3. Results The RD =RB dependence of tunneling conductance rT is plotted in Fig. 1 for various Z. For small magnitude of Z, rT decreases with the increase of RD =RB (curve a). While for larger Z, it is enhanced with the increase of RD =RB . As shown in Fig. 2, the measure of the proximity effect cL is enhanced with the increase of RD =RB . Although the magnitude of cL is enhanced independent of the magnitude of Z, its influence on the tunneling conductance is quite different depending on the magnitude of Z. It is also interesting to compare these results with those obtained by different theoretical approach. We have developed a numerical method to calculate tunneling conductance in DN/S junction [13]. In this approach, localization of wave functions due to impurity scattering in DN can be
0 0
1
2
3 RD /RB
Fig. 2. Measure of the proximity effect cL is plotted as a function of RD =RB using QC theory. a: Z ¼ 0 and b: Z ¼ 5.
taken into account which cannot be treated in QC theory. We consider a DN/S junction on the twodimensional square lattice, where mean-field (BCS) Hamiltonian H on the single-orbital tight-binding model is given by X H ¼ t ðcyl0 ;m0 ;r cyl;m;r þ h:c:Þ l;m;l0 ;m0 ;r
þ
X
vl;m cyl;m;r cl;m;r l^ n
l;m;r
X l;m
½Dl;m cyl;m;# cyl0 ;m0 ;" þ h:c:
ð10Þ
252
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253
In the above, ðl; mÞ are lattice indices, cl;m;r (cyl;m;r ) is the annihilation (creation) operator of an electron at ðl; mÞ with spin r ( ¼ " or #), n^ is the number operator and l is the chemical potential of a junction,Prespectively. In the first term, the summation l;m;l0 ;m0 runs over nearest-neighbor sites, and t is the nearest-neighbor hopping integral. In the above, Dl;m ¼ D0 in the S and Dl;m ¼ 0 in DN where D0 is the amplitude of the pair potential in S. The impurity potential in DN is considered through on-site potential vl;m which takes random values uniformly distributed within a range of Vdis =2 6 vl;m 6 Vdis =2 in a disordered region DN. In an insulator, vl;m is set to be Vins independent of ðl; mÞ. Far from the interface, vl;m is taken to be zero. By applying the Bogoliubov transformation, we can calculate conductance using Kubo formula. The obtained results are plotted in Fig. 3. Through the magnitude of RB , we estimate the corresponding magnitude of Z using Eq. (6). For Z ¼ 0 or Z ¼ 1, line shape of rT is similar to that obtained by QC theory. However, for Z ¼ 5, rT does not approach to unity with the increase of RD =RB . For Z ¼ 5 with RD =RB ¼ 1, the localization length in DN estimated by numerical simulation is the same order as that of the length of DN. We think the deviation from QC theory is due to localization effect in DN which cannot be taken into account in the QC theory.
eV=0
2
σT (eV)
a
1 b c 0 0
1
2
R D /RB
3
Fig. 3. Normalized tunneling conductance obtained by numerical simulation at zero voltage is plotted as a function of RD =RB using lattice model. a: Z ¼ 0, b: Z ¼ 1 and c: Z ¼ 5.
4. Conclusions In the present paper, we have calculated tunneling conductance between diffusive normal metal/conventional superconductor junctions. We concentrate on the tunneling conductance at zero voltage. It is shown that with the increase of the magnitude of resistance of DN (RD Þ, normalized tunneling conductance is enhanced (suppressed) for low (high) transparent barrier. When the length of DN is larger than that of localization length, tunneling conductance obtained by numerical simulation in the lattice model deviates from that by QC theory. In the present paper, superconductor is restricted to be conventional s-wave one. Recently, we have developed a theory of transport of unconventional junctions in the diffusive transport regime [14] where ZBCP from Andreev bound state can be taken into account [15,16]. It is actually quite interesting to apply this novel theory to the actual calculation of rT as in the present paper.
References [1] A.F. Andreev, Sov. Phys. JETP 19 (1996) 1228. [2] A. Kastalsky et al., Phys. Rev. Lett. 67 (1991) 3026; C. Nguyen, H. Kroemer, E.L. Hu, Phys. Rev. Lett. 69 (1992) 2847; P. Xiong, G. Xiao, R.B. Laibowitz, Phys. Rev. Lett. 71 (1993) 1907; W. Poirier, D. Mailly, M. Sanquer, Phys. Rev. Lett. 79 (1997) 2105. [3] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25 (1985) 4515. [4] C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731. [5] C.J. Lambert, J. Phys. Condens. Matter 3 (1991) 6579. [6] Y. Takane, H. Ebisawa, J. Phys. Soc. Jpn. 61 (1992) 2858. [7] Yu.V. Nazarov, Phys. Rev. Lett. 73 (1994) 1420. [8] A.I. Larkin, Yu.V. Ovchinnikov, Sov. Phys. JETP 41 (1975) 960. [9] A.F. Volkov, A.V. Zaitsev, T.M. Klapwijk, Physica C 210 (1993) 21. [10] M.Yu. Kupriyanov, V.F. Lukichev, Zh. Exp. Teor. Fiz. 94 (1988) 139, Sov. Phys. JETP 67 1163 (1988). [11] Yu.V. Nazarov, Superlatt. Microstruct. 25 (1999) 1221, cond-mat/9811155. [12] Y. Tanaka, A. Golubov, S. Kashiwaya, cond-mat/0212019. [13] H. Itoh et al., Physica C 367 (2002) 133.
Y. Tanaka et al. / Physica C 392–396 (2003) 249–253 [14] Y. Tanaka, Yu.V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003. [15] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451; S. Kashiwaya et al., Phys. Rev. B 53 (1996) 2667; S. Kashiwaya et al., Phys. Rev. B 51 (1995) 1350; J.Y.T. Wei et al., Phys. Rev. Lett. 81 (1998) 2542;
253
I. Iguchi et al., Phys. Rev. B 62 (2000) R6131; L. Alff et al., Phys. Rev. B 55 (1997) 14757. [16] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641; Y. Tanaka, S. Kashiwaya, Phys. Rev. B 53 (1996) 11957; Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 892.