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Energy gap in a superconductor — normal proximity — effect sandwich
Solid State Communications,
Vol. 22, pp. 29-31,
Pergamon Press.
1977.
ENERGY GAP IN A SUPERCONDUCTOR
- NORMAL PROXIMITY
Printed in Great Britain
- EFFECT SANDWICH
J. Bar-S@ and Ora Entin-Wohlman Dept. of Physics and Astronomy,
Tel Aviv University,
Tel Aviv, Israel
(Received 3 November 1976 by C. W. McCombie)
The energy gap in the tunneling density of states at the free surface of a normal metal backed by a superconductor is considered. The energy gap dependence upon the normal metal thickness and the barrier at the interface between the two metals are analysed. We also find the effect of the barrier on the Rowell-McMillan oscillations.
WE CONSIDER a clean normal metal (N) deposited on a superconductor (S), and backed by an insulating layer Q through which electrons can tunnel into another metal. A possible oxide layer, vacancies, etc. at the S-N interface are described by a potential barrier (B). For the sake of simplicity, it is assumed%? both metals are indentical, except for the interaction leading to superconductivity in S. (Most effects arising from differences in Fermi momenta and velocities are, actually, the same as those of the Barrier (B) [ 1] ). The normal metal occupies the region - dEI< x < 0, while S occupies x > 0, with a delta-function barrier in between, u&(x). The excitation spectrum for energies below the gap of S is found as follows [2]. The twocomponent wave function inside N, corresponding to energy E is ik+x
+ eik+Qd
527 =E2-A2
(4)
Here A is the pair potential inside S, which vanishes in N, i.e. has a step function form as a function of x. At the interface, the wavefunctions are continuous Gh@) = Il/.s@)
(5)
However, owing to the potential barrier u&(x), there is a jump in their first derivatives [4]. km
- &to)
= 4k@)
(6)
where 4 = (2m/h2)u. Inserting (1), (3), into (5) and (6) we find that a non-trivial solution of the coefficients a,, a2, bl, b2, for energies less than A exists when (1 +cr)sin(f$--xXNcos@)+O1sin(#+x,cos@) = - 2 sin 4 [o
px)
COS
(2kFdN) - &
sin (2kFxdN)]
(7)
1
Here
+a2 with
x,
k
(2)
where i,, = hvF/nA, x-direction and
Here kF, /cl are the Fermi momentum and the momenta components parallel to the barrier, respectively. The derivative of It/N(x) vanishes at x = - dN. The other possible boundary condition at the surface is the vanishing of the wavefunction itself; however, both boundary conditions lead to the same final result [3]. The wave function inside S is
&(x)
with
= 61(-i
j3eik+-
2 & = - -71
kF
to kF,
kFx is the Fermi momentum
(8) in the
/ \” (9)
To obtain the result (7) we have approximated k+_-k’ -k in the coefficients of G’(x), but not in the phase factors. This involves errors of the order A/eF [3,4]. The result of Saint James [2] is reproduced for (Y= 0, i.e. for a transparent barrier. The excitations important for tunneling are those with momentum normal to the tunneling junction (Z) (i.e. kF, = kF). The expression in square brackets in the right-hand side of (7) is a rapid oscillatory function
+b2(F)emikex
(3)
29
ENERGY GAP IN A SUPERCONDUCTOR
30
Vol. 22, No. 1
F
1.11
C--
___.-.--
__-.C-
QCKS
Fig. 1. The apparent gap divided by that of a bulk superconductor vis. normalized normal layer thickness for dif. ferent barrier reflectivities. R = 0, --R = 0.3, -._. R = 0.6, and - .. - .. R = 0.9. between the two extreme values + d-). Hence the first solution, for kFX 21 IcF, occurs when the left-hand side equals + 2 sin @&m The energy gap is therefore given by the solution corresponding to the smallest cos 4 of the equation sin (@- XN cos $) + R sin (@ + XN cos 4) = 24
sin fp (10)
where R is the reflection
coefficient
R=cw IfCY
of the barrier [4].
(11)
The apparent energy gap, Eo, as a function ofXN, is given in Fig. 1. According the McMillan’s tunneling model [S] for the proximity effect
f OJF)
I
____------
---_I I, I 0.744 I.Il3
__-__-.__
I I 1483
I I852
Fig. 2. Check of the approximation EodN S constant. Normalized apparent gap E. times normal layer thickness 4 vis. normalized normal layer thickness, for different barrier reflectivities. -R=O,---R=0.3, -.-. R = 0.6, and - . . - . . R = 0.9. It is well known that the density of states at x = -dN has an oscillatory structure, i.e. the RowellMcMillan oscillations, [6]. As was noted by Rowe11 and McMillan [6], the potential barrier at the interface affects the amplitude of these oscillations. Here we fmd this effect. The tunneling density of states is calculated from the 11 matrix element of the Green’s functions. These can be constructed from the wavefunctions solutions, [7], [see (1) and (3)]. Their detailed calculation will be given elsewhere; here we give only the result;
NC-GE)
= ---
Re E,/dE’
-Ai
(13)
FX
where AaE)
E--R = 4~~E’r7~
(14) where u is the penetration probability of the barrier, u = 1 -R, and B = 2 for a clean normal metal [5]. There is an apparent disagreement between our result and McMillan’s result, especially for moderate and low values of RI This disagreement can be cured by adjusting the parameter B for each value R(a). We find that B - 1 + R for R > 0.3, but B - 0.5 for R = 0. The approximate relation proposed by McMillan [4] (.!?,-,/A) (2d,&te) z constant is practically valid for all XN when R = 0.9 but becomes a poor approximation for lower values of R. It is not surprising that the disagreement is serious for low values of R, since the tunneling model is justified for high R. The theory presented here is valid for intermediate and low values of R as well, thus is more suitable for comparison with experiments.
For E >>A this yields N(-dN,E)
=
2
i7’f-l VFx
(15)
This expression has to be integrated over 8, where VFx = VF cos 0, to obtain the total density of states [6]. Note that for R = 0 (transparent barrier), we recover the
Vol. 22, No. 1
31
ENERGY GAP IN A SUPERCONDUCTOR square of the transmission coefficient to the remark made in reference 6.
result of reference 6. However, for a ftite potential barrier, the amplitude is not simply proportional to the
REFERENCES 1.
BAR-SAG1 J., Solid State Commun.
15,729
(1974).
2.
SAINT-JAMES D.,J. Phys. 25,899
3.
WOLFRAM T., Phys. Rev. 170,481(1968).
4.
GRIFFIN
5.
McMILLAN W.L., Phys. Rev. 175,537
6.
ROWELL J.M. & McMILLAN W.L., Phys. Rev. Lett. 16,453
7.
McMILLAN W.L., Phys. Rev. 175,559
(1964).
A. & DEMERS J., Phys. Rev. B4,2202
(1971).
(1968).
(1968).
(1966).
1 - R, contrary
Vol. 22, pp. 29-31,
Pergamon Press.
1977.
ENERGY GAP IN A SUPERCONDUCTOR
- NORMAL PROXIMITY
Printed in Great Britain
- EFFECT SANDWICH
J. Bar-S@ and Ora Entin-Wohlman Dept. of Physics and Astronomy,
Tel Aviv University,
Tel Aviv, Israel
(Received 3 November 1976 by C. W. McCombie)
The energy gap in the tunneling density of states at the free surface of a normal metal backed by a superconductor is considered. The energy gap dependence upon the normal metal thickness and the barrier at the interface between the two metals are analysed. We also find the effect of the barrier on the Rowell-McMillan oscillations.
WE CONSIDER a clean normal metal (N) deposited on a superconductor (S), and backed by an insulating layer Q through which electrons can tunnel into another metal. A possible oxide layer, vacancies, etc. at the S-N interface are described by a potential barrier (B). For the sake of simplicity, it is assumed%? both metals are indentical, except for the interaction leading to superconductivity in S. (Most effects arising from differences in Fermi momenta and velocities are, actually, the same as those of the Barrier (B) [ 1] ). The normal metal occupies the region - dEI< x < 0, while S occupies x > 0, with a delta-function barrier in between, u&(x). The excitation spectrum for energies below the gap of S is found as follows [2]. The twocomponent wave function inside N, corresponding to energy E is ik+x
+ eik+Qd
527 =E2-A2
(4)
Here A is the pair potential inside S, which vanishes in N, i.e. has a step function form as a function of x. At the interface, the wavefunctions are continuous Gh@) = Il/.s@)
(5)
However, owing to the potential barrier u&(x), there is a jump in their first derivatives [4]. km
- &to)
= 4k@)
(6)
where 4 = (2m/h2)u. Inserting (1), (3), into (5) and (6) we find that a non-trivial solution of the coefficients a,, a2, bl, b2, for energies less than A exists when (1 +cr)sin(f$--xXNcos@)+O1sin(#+x,cos@) = - 2 sin 4 [o
px)
COS
(2kFdN) - &
sin (2kFxdN)]
(7)
1
Here
+a2 with
x,
k
(2)
where i,, = hvF/nA, x-direction and
Here kF, /cl are the Fermi momentum and the momenta components parallel to the barrier, respectively. The derivative of It/N(x) vanishes at x = - dN. The other possible boundary condition at the surface is the vanishing of the wavefunction itself; however, both boundary conditions lead to the same final result [3]. The wave function inside S is
&(x)
with
= 61(-i
j3eik+-
2 & = - -71
kF
to kF,
kFx is the Fermi momentum
(8) in the
/ \” (9)
To obtain the result (7) we have approximated k+_-k’ -k in the coefficients of G’(x), but not in the phase factors. This involves errors of the order A/eF [3,4]. The result of Saint James [2] is reproduced for (Y= 0, i.e. for a transparent barrier. The excitations important for tunneling are those with momentum normal to the tunneling junction (Z) (i.e. kF, = kF). The expression in square brackets in the right-hand side of (7) is a rapid oscillatory function
+b2(F)emikex
(3)
29
ENERGY GAP IN A SUPERCONDUCTOR
30
Vol. 22, No. 1
F
1.11
C--
___.-.--
__-.C-
QCKS
Fig. 1. The apparent gap divided by that of a bulk superconductor vis. normalized normal layer thickness for dif. ferent barrier reflectivities. R = 0, --R = 0.3, -._. R = 0.6, and - .. - .. R = 0.9. between the two extreme values + d-). Hence the first solution, for kFX 21 IcF, occurs when the left-hand side equals + 2 sin @&m The energy gap is therefore given by the solution corresponding to the smallest cos 4 of the equation sin (@- XN cos $) + R sin (@ + XN cos 4) = 24
sin fp (10)
where R is the reflection
coefficient
R=cw IfCY
of the barrier [4].
(11)
The apparent energy gap, Eo, as a function ofXN, is given in Fig. 1. According the McMillan’s tunneling model [S] for the proximity effect
f OJF)
I
____------
---_I I, I 0.744 I.Il3
__-__-.__
I I 1483
I I852
Fig. 2. Check of the approximation EodN S constant. Normalized apparent gap E. times normal layer thickness 4 vis. normalized normal layer thickness, for different barrier reflectivities. -R=O,---R=0.3, -.-. R = 0.6, and - . . - . . R = 0.9. It is well known that the density of states at x = -dN has an oscillatory structure, i.e. the RowellMcMillan oscillations, [6]. As was noted by Rowe11 and McMillan [6], the potential barrier at the interface affects the amplitude of these oscillations. Here we fmd this effect. The tunneling density of states is calculated from the 11 matrix element of the Green’s functions. These can be constructed from the wavefunctions solutions, [7], [see (1) and (3)]. Their detailed calculation will be given elsewhere; here we give only the result;
NC-GE)
= ---
Re E,/dE’
-Ai
(13)
FX
where AaE)
E--R = 4~~E’r7~
(14) where u is the penetration probability of the barrier, u = 1 -R, and B = 2 for a clean normal metal [5]. There is an apparent disagreement between our result and McMillan’s result, especially for moderate and low values of RI This disagreement can be cured by adjusting the parameter B for each value R(a). We find that B - 1 + R for R > 0.3, but B - 0.5 for R = 0. The approximate relation proposed by McMillan [4] (.!?,-,/A) (2d,&te) z constant is practically valid for all XN when R = 0.9 but becomes a poor approximation for lower values of R. It is not surprising that the disagreement is serious for low values of R, since the tunneling model is justified for high R. The theory presented here is valid for intermediate and low values of R as well, thus is more suitable for comparison with experiments.
For E >>A this yields N(-dN,E)
=
2
i7’f-l VFx
(15)
This expression has to be integrated over 8, where VFx = VF cos 0, to obtain the total density of states [6]. Note that for R = 0 (transparent barrier), we recover the
Vol. 22, No. 1
31
ENERGY GAP IN A SUPERCONDUCTOR square of the transmission coefficient to the remark made in reference 6.
result of reference 6. However, for a ftite potential barrier, the amplitude is not simply proportional to the
REFERENCES 1.
BAR-SAG1 J., Solid State Commun.
15,729
(1974).
2.
SAINT-JAMES D.,J. Phys. 25,899
3.
WOLFRAM T., Phys. Rev. 170,481(1968).
4.
GRIFFIN
5.
McMILLAN W.L., Phys. Rev. 175,537
6.
ROWELL J.M. & McMILLAN W.L., Phys. Rev. Lett. 16,453
7.
McMILLAN W.L., Phys. Rev. 175,559
(1964).
A. & DEMERS J., Phys. Rev. B4,2202
(1971).
(1968).
(1968).
(1966).
1 - R, contrary