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Proximity effect in superconducting tunnel junctions with bilayer electrodes
PHYSICA
Physica B 194-196 (1994) 1675-1676 North-Holland
P r o x i m i t y e f f e c t in s u p e r c o n d u c t i n g tunnel j u n c t i o n s w i t h b i l a y e r e l e c t r o d e s M.G. Blamire, Z.H. Barber, and P.A. Warburton Department of Materials Science, University of Cambridge, Pembroke St., Cambridge CB2 3QZ, U.K. A wide variety of theories exists for modelling the proximity effect in superconductor/normal metal bilayers. However, their predictions of the resultant energy gap, its temperature dependence and the tunneling density of states are generally only qualitatively correct. In order to test these theories we have fabricated tunnel junctions with bilayer electrodes from a variety of different materials and have found that the relatively simple McMillan model agrees quantitatively in most cases. In addition, we have shown theoretically that the major undetermined parameter of this model, the transmission factor cy, may be determined exactly from fundamental quantities. 1. I N T R O D U C T I O N Despite the complexity of recent theories of the proximity e f f e c t 1,2, it has been found experimentally that in many cases the simple McMillan tunneling model (MTM) 3 predicts phenomena such as the temperature dependence of the induced gap 4,5 and the induced tunneling density of states at least as accurately. 6,7 However, the MTM has serious limitations; most importantly the model relies on the spatial invariance of the proximity-induced parameters, and this is ensured by a low transmission coefficient, ~, between the normal and superconducting layers. Since the proximity effect is the only measure of cy, this is a completely adjustable parameter in any fit to experimental data. The original justification of a low transmission barrier was that bilayers deposited in the low quality vacuum systems then current were likely to be degraded at the interface. Here we demonstrate that: (1) results from epitaxially grown heterostructures can still be fitted to the model and (2) by extending the MTM for structures with a good structural interface, ~ can be replaced by a function of the Fermi velocities of the normal and superconducting layers, removing the major undetermined parameter from the MTM. 2.
EXPERIMENTAL
accurately measured by anodisation spectroscopyl0; the Mo thickness was obtained by subtracting the Nb layer thickness from the total base layer thickness. These results are plotted in Fig. 1 as AS/If2 N vs 2 d N / ~ where kS is the coherence length (~SNb=40nm q 1, ~SNbN(epitaxial)=7nm 12), and ASNbN=3. lmeV. Also plotted on this figure are points from a variety of systems in the literature. With the exception of the Pb/Cu and Pb/Sn systems, all the data lie close to a straight line of the form AS/K~N = I+dN/~S plotted on the same figure. Fig. 2 shows experimental tunneling densities of states for the NbN/AI electrode from a series of device structures with different A1 thicknesses. This data was obtained at 4.2K. Also plotted on this graph are theoretical curves corresponding to the same AI thickness using the extended MTM model discussed below. 10 . . . . I ' ' ' ' I .... I .... I ' F '@_
x
Z 6
"
-
RESULTS
The aim of our experiments was to provide data on the variation with thickness of the gap and tunneling density of states induced in a normal metal from high quality bilayers with a superconductor for comparison with theory. We investigated three systems: Nb/A1, NbN/AI and Nb/Mo. In each case the barrier was A1Ox and the upper electrode was Nb. All of these systems were prepared by UHV dc magnetron sputtering and low-leakage tunnel junctions were fabricated by a whole-wafer route. In all cases the superconducting base layer was grown epitaxially, with the normal metal layer showing a good epitaxial relationship to the base layer. 8,9. Values of tile induced tunneling energy gap, f~N, were obtained from each system at 4.2K for a variety of normal metal layer thicknesses dR. The thickness of the A1 proximity layers the depth could be
2
~
0
ii
0
O© ii
Ii
r i i[i
iii
+-I i k III
III
I
1
2 3 4 5 2dN/rC~S Fig. 1. Inverse normalised tunneling energy gap vs normalised normal metal thickness, ANbZr/A16, ONb/Mo, ~¢Pb/Cu 5, - F P b / S n 5, oRbS/A1 oRb/A1. Lines from Eqn. (2) for normal metals (straight line) and Sn/Pb (curved line) 3. E X T E N D E D M A C M I L L A N
MODEL
This model differs from the original MTM3 in that an exchange interaction occurs between the normal and superconducting layers (of thickness dN, ds respectively) in which the bulk pair potentials A~hand A~hare coupled to produce derived pair
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93) 1419-M
1676 potentials f~S and ~ N in the superconducting and normal layers. This coupling is mediated by the coherence length of the material ¢ or by the correlation length ]3 CN= h VfN/2rckT and by a barrier transmission coefficient T i at the interface. 5 J i ] i t i i i i - l - T ~ t: r i i i o4
~ :
/'i/
~3
Z2 l
0
.
.
0.25
.
"
:-'.:i-...
.
li,
0.5 E/A 0.75
,
" "--
,
I
.25
2)
Fig. 2. Experimental normalised density of states for NbN/A1 (solid lines): dN(Al)=9nm (lowest energy), 7, 6, 2nm. Broken lines are derived from our extension of the MTM 3 for the same A1 thicknesses. Thus the induced pair potential at the tunnel barrier is obtained by integrating the equilibrium pair potential over space, weighted by an exponential with decay equal to the relevant coherence length ~: oo n~-). N = T i J A ~ h e - - d N / ~ a 0
e-Z/~s
dz + d~NA ~qh e - z / ~ , N dz 1) 0
where n is a normalising factor obtained by evaluating the integrals with A~hand A ~ s e t to (. In the standard experimental limit dN<<~N this gives: A ~ h + A~ ~dN / Ti CS (2) ~N(dN) = I+dN/T~s This model reproduces the qualitative features of other proximity models, the main difference lying in the continuity of ~(x) at the NS interface for Ti= 1. With A~qh=0, the original MTM 3 produces a result which~s identical in form to (2): A~h hvlNO (3) ~2N= where F N l+A~ h / F N 2BdN B is a function of the ratio of the mean free path and the film thickness. Using this relationship wc can equate FN/A S with T i t s . Replacing h v f s / g A S by kS allows us to relate (:5 to fundamental parameters cY= T i (2B / rc)(Vfs / VfN ) (4) Within the MTM B=I; to this accuracy the first bracketed term may be neglected and we are left with one term, Ti, which we take as a structural transmission coefficient, and the ratio of the Fermi velocities in the two materials. In the dirty limit this result is multiplied by an additional term(1s/~s) where IS is the electron mean free path in the superconductor. In heteroepitaxial structures Ti=l
and so our extension to the MTM can relate directly to fundamental quantities. 4.
DISCUSSION
Fig. 1 shows the theoretical line from Eqn (2) with Ti=l plotted with the experimental data. With the exception of the Pb/Cu data, all the normal metal bilayer results lie around this line. The Pb/Cu line shows a different slope, but remains straight; in this case Ti
CONCLUSIONS
We have demonstrated that the undetermined transmission coefficient (:5 of the MTM may be replaced by a simple function of the Fermi velocities of the two materials. Using this model we have obtained an excellent fit to experimental results without thc use of any adjustable parameters. REFERENCES
1. G.B. Arnold Phys. Rev. B 23, 1171 (1981) 2. A.A. Golubov & M.Yu. Kuprianov, J. Low Temp. Phys. 70, 83 (1988) 3. W.L. McMillan, Phys. Rev. 175, 537 (1968) 4. D.J. Goldie, N.E. Booth, C. Patel, & G.L. Sahnon, Phys. Rev. Lett. 64,954 (1990) 5. A. Gilabert, Ann. Phys. 2, 203 (1977) 6. R. Delesclefs & O. Fischer, J. Low Temp. Phys. 53, 339 (1983) 7. E.L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford University Press, 1985 8. Z.H. Barber, M.G. Blamire, R.E. Somekh and J.E. Evctts, IEEE Trans Supercon. (in press) 9. P.A.Warburton & M. G. Blamire, ESA SP-356, 413 (1993) 10.M.G. Blamire, R.E. Somekh, Z.H. Barber, G.W. Morris & J.E. Evetts, J. Appl. Phys. 64, 6396 (1988) I1.T.P. Orlando & K.A. Delin, Foundations of Appl. Superconductivity, Addison Wesley, 1991 12.A. Sho]i, S. Kiryu, S. Kohjiro, Appl. Phys. Lett. 60, 1624 (1992) 13.V.G. Kogan, Phys. Rev. B 26, 88 (1982) 14.G.B. Arnold, Phys. Rev. B 18, 1076 (1978)
Physica B 194-196 (1994) 1675-1676 North-Holland
P r o x i m i t y e f f e c t in s u p e r c o n d u c t i n g tunnel j u n c t i o n s w i t h b i l a y e r e l e c t r o d e s M.G. Blamire, Z.H. Barber, and P.A. Warburton Department of Materials Science, University of Cambridge, Pembroke St., Cambridge CB2 3QZ, U.K. A wide variety of theories exists for modelling the proximity effect in superconductor/normal metal bilayers. However, their predictions of the resultant energy gap, its temperature dependence and the tunneling density of states are generally only qualitatively correct. In order to test these theories we have fabricated tunnel junctions with bilayer electrodes from a variety of different materials and have found that the relatively simple McMillan model agrees quantitatively in most cases. In addition, we have shown theoretically that the major undetermined parameter of this model, the transmission factor cy, may be determined exactly from fundamental quantities. 1. I N T R O D U C T I O N Despite the complexity of recent theories of the proximity e f f e c t 1,2, it has been found experimentally that in many cases the simple McMillan tunneling model (MTM) 3 predicts phenomena such as the temperature dependence of the induced gap 4,5 and the induced tunneling density of states at least as accurately. 6,7 However, the MTM has serious limitations; most importantly the model relies on the spatial invariance of the proximity-induced parameters, and this is ensured by a low transmission coefficient, ~, between the normal and superconducting layers. Since the proximity effect is the only measure of cy, this is a completely adjustable parameter in any fit to experimental data. The original justification of a low transmission barrier was that bilayers deposited in the low quality vacuum systems then current were likely to be degraded at the interface. Here we demonstrate that: (1) results from epitaxially grown heterostructures can still be fitted to the model and (2) by extending the MTM for structures with a good structural interface, ~ can be replaced by a function of the Fermi velocities of the normal and superconducting layers, removing the major undetermined parameter from the MTM. 2.
EXPERIMENTAL
accurately measured by anodisation spectroscopyl0; the Mo thickness was obtained by subtracting the Nb layer thickness from the total base layer thickness. These results are plotted in Fig. 1 as AS/If2 N vs 2 d N / ~ where kS is the coherence length (~SNb=40nm q 1, ~SNbN(epitaxial)=7nm 12), and ASNbN=3. lmeV. Also plotted on this figure are points from a variety of systems in the literature. With the exception of the Pb/Cu and Pb/Sn systems, all the data lie close to a straight line of the form AS/K~N = I+dN/~S plotted on the same figure. Fig. 2 shows experimental tunneling densities of states for the NbN/AI electrode from a series of device structures with different A1 thicknesses. This data was obtained at 4.2K. Also plotted on this graph are theoretical curves corresponding to the same AI thickness using the extended MTM model discussed below. 10 . . . . I ' ' ' ' I .... I .... I ' F '@_
x
Z 6
"
-
RESULTS
The aim of our experiments was to provide data on the variation with thickness of the gap and tunneling density of states induced in a normal metal from high quality bilayers with a superconductor for comparison with theory. We investigated three systems: Nb/A1, NbN/AI and Nb/Mo. In each case the barrier was A1Ox and the upper electrode was Nb. All of these systems were prepared by UHV dc magnetron sputtering and low-leakage tunnel junctions were fabricated by a whole-wafer route. In all cases the superconducting base layer was grown epitaxially, with the normal metal layer showing a good epitaxial relationship to the base layer. 8,9. Values of tile induced tunneling energy gap, f~N, were obtained from each system at 4.2K for a variety of normal metal layer thicknesses dR. The thickness of the A1 proximity layers the depth could be
2
~
0
ii
0
O© ii
Ii
r i i[i
iii
+-I i k III
III
I
1
2 3 4 5 2dN/rC~S Fig. 1. Inverse normalised tunneling energy gap vs normalised normal metal thickness, ANbZr/A16, ONb/Mo, ~¢Pb/Cu 5, - F P b / S n 5, oRbS/A1 oRb/A1. Lines from Eqn. (2) for normal metals (straight line) and Sn/Pb (curved line) 3. E X T E N D E D M A C M I L L A N
MODEL
This model differs from the original MTM3 in that an exchange interaction occurs between the normal and superconducting layers (of thickness dN, ds respectively) in which the bulk pair potentials A~hand A~hare coupled to produce derived pair
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93) 1419-M
1676 potentials f~S and ~ N in the superconducting and normal layers. This coupling is mediated by the coherence length of the material ¢ or by the correlation length ]3 CN= h VfN/2rckT and by a barrier transmission coefficient T i at the interface. 5 J i ] i t i i i i - l - T ~ t: r i i i o4
~ :
/'i/
~3
Z2 l
0
.
.
0.25
.
"
:-'.:i-...
.
li,
0.5 E/A 0.75
,
" "--
,
I
.25
2)
Fig. 2. Experimental normalised density of states for NbN/A1 (solid lines): dN(Al)=9nm (lowest energy), 7, 6, 2nm. Broken lines are derived from our extension of the MTM 3 for the same A1 thicknesses. Thus the induced pair potential at the tunnel barrier is obtained by integrating the equilibrium pair potential over space, weighted by an exponential with decay equal to the relevant coherence length ~: oo n~-). N = T i J A ~ h e - - d N / ~ a 0
e-Z/~s
dz + d~NA ~qh e - z / ~ , N dz 1) 0
where n is a normalising factor obtained by evaluating the integrals with A~hand A ~ s e t to (. In the standard experimental limit dN<<~N this gives: A ~ h + A~ ~dN / Ti CS (2) ~N(dN) = I+dN/T~s This model reproduces the qualitative features of other proximity models, the main difference lying in the continuity of ~(x) at the NS interface for Ti= 1. With A~qh=0, the original MTM 3 produces a result which~s identical in form to (2): A~h hvlNO (3) ~2N= where F N l+A~ h / F N 2BdN B is a function of the ratio of the mean free path and the film thickness. Using this relationship wc can equate FN/A S with T i t s . Replacing h v f s / g A S by kS allows us to relate (:5 to fundamental parameters cY= T i (2B / rc)(Vfs / VfN ) (4) Within the MTM B=I; to this accuracy the first bracketed term may be neglected and we are left with one term, Ti, which we take as a structural transmission coefficient, and the ratio of the Fermi velocities in the two materials. In the dirty limit this result is multiplied by an additional term(1s/~s) where IS is the electron mean free path in the superconductor. In heteroepitaxial structures Ti=l
and so our extension to the MTM can relate directly to fundamental quantities. 4.
DISCUSSION
Fig. 1 shows the theoretical line from Eqn (2) with Ti=l plotted with the experimental data. With the exception of the Pb/Cu data, all the normal metal bilayer results lie around this line. The Pb/Cu line shows a different slope, but remains straight; in this case Ti
CONCLUSIONS
We have demonstrated that the undetermined transmission coefficient (:5 of the MTM may be replaced by a simple function of the Fermi velocities of the two materials. Using this model we have obtained an excellent fit to experimental results without thc use of any adjustable parameters. REFERENCES
1. G.B. Arnold Phys. Rev. B 23, 1171 (1981) 2. A.A. Golubov & M.Yu. Kuprianov, J. Low Temp. Phys. 70, 83 (1988) 3. W.L. McMillan, Phys. Rev. 175, 537 (1968) 4. D.J. Goldie, N.E. Booth, C. Patel, & G.L. Sahnon, Phys. Rev. Lett. 64,954 (1990) 5. A. Gilabert, Ann. Phys. 2, 203 (1977) 6. R. Delesclefs & O. Fischer, J. Low Temp. Phys. 53, 339 (1983) 7. E.L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford University Press, 1985 8. Z.H. Barber, M.G. Blamire, R.E. Somekh and J.E. Evctts, IEEE Trans Supercon. (in press) 9. P.A.Warburton & M. G. Blamire, ESA SP-356, 413 (1993) 10.M.G. Blamire, R.E. Somekh, Z.H. Barber, G.W. Morris & J.E. Evetts, J. Appl. Phys. 64, 6396 (1988) I1.T.P. Orlando & K.A. Delin, Foundations of Appl. Superconductivity, Addison Wesley, 1991 12.A. Sho]i, S. Kiryu, S. Kohjiro, Appl. Phys. Lett. 60, 1624 (1992) 13.V.G. Kogan, Phys. Rev. B 26, 88 (1982) 14.G.B. Arnold, Phys. Rev. B 18, 1076 (1978)