Physica 109 & ll0B (1982) 1657-1664 North-Holland Publishing Company
EXPLANATION OF SUBHARMONIC ENERGY GAP STRUCTURE IN SUPERCONDUCTING CONTACTS T.M. K L A P W I J K * , G.E. B L O N D E R and M. T I N K H A M Department of Physics and Division of Applied Sciences, Harvard University, Cambridge, Ma. 02138, USA Using a simple physical model, we show that subharmonic gap structure and "excess current" observed in the I - V curves of superconducting weak links can be explained by multiple Andreev reflections between two superconductors. Our model allows one to treat the cross over between metallic weak links and tunnel junctions.
2. Historical review
The occurrence of subharmonic energy gap structure has been one of the tantalizing and unsolved problems in nonequilibrium superconductivity. Despite various attempts to explain these observations, and an abundance of experimental work, a consistent explanation has not emerged. Recent progress by Artemenko, Volkov and Zaitsev [1,2] in understanding excess currents has led us to a careful reexamination of energy-dependent Andreev  reflections between weakly linked superconductors. We find that the structure at submultiples of 2A can be attributed to multiple Andreev reflections. Our paper is divided into sections. First, a short summary of the relevant experiments which characterize the phenomena, as well as a description of previous theories in the field, is presented. In section 3 we introduce the Bogoliubov equations, and explain their role in calculating transmission coefficients at a n o r m a l : s u p e r c o n d u c t o r (N:S) boundary. In section 4 our method for utilizing the transmission coefficients in the computation of I - V curves is explained. Lastly, in section 5, we compare our computer-generated results with examples from the available experimental literature.
Superconducting weak links and their currentvoltage ( I - V ) curves have been an area of substantial experimental interest for the past 20 years, and remain so today. It is surprising, therefore, to find a well substantiated theory only for the case of tunnel junctions, while only a limited understanding exists for microbridges, point contacts and dielectric breakdown sandwiches. Although many features of the I - V curve remain to be explained, we will focus our attention on two features: the "excess current" and the "subharmonic gap structure". One might expect, for an applied voltage, eV, much greater than the energy gap A, that any weak link would carry the same d.c. current as if it were in the normal state, as is the case with tunnel junctions. However, observations by many workers, including Divin and Nad'  in point contacts and Octavio et al.  in microbridges, show that the current in the high voltage limit, while linear, does not extrapolate back to zero current at zero voltage. The V = 0 intercept on the current axis is referred to as "excess current" (EC). At intermediate voltages, e V ~< A, there is rich structure superimposed on a smooth background. Distinct peaks are easily seen in the differential resistance at integral submultiples of the gap. This p h e n o m e n o n is referred to as the "subharmonic gap structure" (SGS), and has
* Permanent Address: Laboratorium voor Technische Natuurkunde, Technische Hogeschool Delft, Delft, The Netherlands.
0378-4363/82/0000-0000/$02.75 O 1982 North-Holland
T.M. Klapwijk et al. / Subharmonic gap structure in superconducting contacts
been reported by a variety of workers [6-131, We have examined much of the experimental literature with the goal of correlating c o m m o n experimental p a r a m e t e r s with the two features, EC and SGS. This is hardly a straightforward task, given the confusing and often conflicting observations in the literature. Heating effects (Octavio et al. ) can distort and move the position of structure, or add curvature in the high voltage limit. Also, the constrictions, which are usually < 1 0 0 0 4 in diameter, are poorly characterized. It is rarely known whether there is oxide at the constriction, as in point contacts, or if there is extra scattering in the neck region of microbridges due to fabrication techniques. In addition, there does not seem to be a unique weak-link I - V curve but, rather, an entire family of different shapes. However, we believe one can make, with hindsight, the following five generalizations: (1) There is no subgap structure seen when one bank is normal, but excess current is still observed. (2) SGS and E C only occur in metallic constrictions. Observations in tunnel junctions are due to pinholes or shorts in the oxide. (3) The position of the peaks, and the magnitude of the excess current, scale with the gap. (4) The presence of scattering in the constriction weakens the structure. (5) The structure weakens monotonically with increasing n, Considerable effort has been spent, by a variety of workers, in trying to understand the theoretical basis of this p h e n o m e n o n . Schrieffer and Wilkins  used multiparticle tunneling to explain SGS. Structure at 2_4/n involves n particle tunneling, and so must proceed with a probability proportional to I TI 2", where T is the tunneling matrix element. However, since a typical value of ITt 2 is 10-1°, it is hard to reconcile this model with Rowell's  observation of a structure up to n = 12. W e r t h a m e r  proposed coupling of the single particle tunneling process to Josephson radi-
ation. If a single particle tunneling event occurs while simultaneously a photon is absorbed from the Josephson radiation, an increase in tunnel probability will occur for e V + fi~o = 2_4, i.e. for 2A/3, since ho~ = 2 e V for Josephson radiation. Inclusion of higher order photons gives structure at 2_4/2n + 1, i.e. for odd values of the denominator. One can obtain the even series by including absorption of the Josephson radiation by the electrodes, thus generating quasiparticles at nhoJ : 2_4. The main problem with this selfcoupling explanation is that two different explanations have to be invoked for the even and odd series, while the observed steps occur with the same appearance and strength in both series (McDonald et al. ). The low voltage limit of the I - V curve has received much attention by various authors . A r t e m e n k o , Volkov and Zaitsev  have extended their e V ~ O treatment to all voltages, and have derived expressions for the excess current. Their prediction has been verified by G u b a n k o v and Margolin . Unfortunately, this G r e e n function formalism is not easily understood by most workers in the field. O u r technique, based on a simple physical model, yields the same result and also provides an explanation for the SGS. The next two sections outline this method.
3. The Bogoliubov equations As a starting point, we illustrate our method of analysis for an S : c : N (superconductor:constriction :normal) contact. We assume that the energy gap rises steeply at the S : N interface, and can be treated as a step function. If the orifice in the constriction is chosen small c o m p a r e d to the electronic mean free path, scattering can be neglected and detailed calculations are easily p e r f o r m e d by matching solutions of the Bogoliubov equations  at the interface. The Bogoliubov equations are a quantum
T.M. Klapwijk et al. I Subharmonic gap structure in superconducting contacts
mechanical description of the excitations in a superconductor. As extended by Kfimmel , the equations may also be used to describe the condensate. We will defer this consideration to a later paper, and for now work with a conventional version of the equations, namely
fh2V2 ] ih ~t = - [ - ~ + ~ f(x, t ) - a (x)g(x, t), Og
ih-aT= [-~-+u g(x,t)-a(x)f(x,t).
f(x, t) = u e ik-+xe -iEtl* ,
g(x, t)= v e ik-*x e -iE'/* ,
where, generalizing from the normal state [ A ( x ) = 0], we identify f with electron wave functions, and g with hole wave functions. Here, E, u, and v are the usual BCS quantities:
the _+ refers to hk +-= ~ / ~ ( I z +" x / ~ - A 2 ) m and the group velocity is given by h v g = 3ElOk.
There is a wealth of information about the behavior of quasiparticles contained within (1), but for now we will concentrate on the current produced by a normal electron incident on an N : S boundary. The incident electron wave is rll
0i,c= [ 0 ] e
~ref = a ( ~ ] e iq-x ,
and the transmitted wave is
It is easy to show that steady state solutions to these equations take the form
E~= \[~:k 2 m ~- / * f 7t- A 2 ' u2 ~ ':12]
Vg< 0) hole wave is
where hq +-= X/2-m~/~ +_E. The reflected (i.e.
Matching amplitude boundary we find
A ( E ) =- aa * = reflection coeff. =
IEI -> a
I~7 = JEIIEI+~ 7 V'E - :- ,a:
T ( E ) = 1 - A ( E ) = bb * = transmission coefficient. Thus, for [ E l < A , an incident electron is completely reflected as a hole. This is Andreev reflection. For IEI>zl, A ( E ) < I and there is partial Andreev reflection. The total probability current carried, on either side of the constriction, is given by J•S(E) = VF[1-- A(E)]. There is a similar process involving an incident quasiparticle from the superconducting side. Direct calculation shows that J ~ " ( E ) N ( E ) = J~S(E), where N ( E ) is the normalized BCS density of states, as required by detailed balance.
4. Currents We calculate currents within the context of a conventional semiconductor model. As our first example, we consider the N : S case. Since Andreev reflection interchanges electrons with holes, it is natural to treat both on an equal basis. Thus, we consider both the electrons and holes incident on the barrier, which produce reflected holes and electrons, respectively. To avoid double counting, however, we sum only electron contributions to the current. Similarly, in the superconductor we treat electron- and
T.M. Klapwijk et al. / Subharmonic gap structure in superconducting contacts
hole-like incident quasiparticles. Although we might calculate the current at any point in the metal, the most convenient place is just inside the normal region. With that choice, there are three contributions to the electron current: electrons incident from the normal metal, electrons resulting from Andreev reflection of holes, and electrons injected into the normal metal via incident quasiparticles in the superconductor. Consistent with the assumption of clean metal banks is the use of equilibrium Fermi functions to describe the distribution of particles that originate deep inside the banks and travel to the interface. Using the reflection and transmission coefficients derived above, we find for our three currents:
fo(E - eV) d E ,
A ( E ) [ I - f.(E - eV)] d E ,
After a little manipulation using the property that A ( E ) = A ( - E ) , we find the total current is given by
the limit of small voltages and low temperatures we find a conductance of 2/Ro, i.e. an excess conductance of 1/R~,. For d ~ kT, (6) reduces to
For sufficiently high voltages ( e V - ,3 >>k T) one has an excess current 4/.1/3eRo, at all temperatures, One can extend these calculations to the experimentally relevant case of a semitransparent hole by including a &function potential between the normal metal and the superconductor. In this more general case, an incident electron may produce a reflected electron, rather than a reflected hole. We have included such a term in our expression for the total current, and have calculated the I - V curve as a function of the 8-function strength H. Fig. 1 shows the effect of H on the differential conductance dI/dV, at T = I). For high barrier potentials (fig. l(d)), one approaches the usual normal-superconductor tunnel result, d I / d V ~ N(E), giving further support for our methodology. dl dV
V 1 f I : -~,,+7~,~ I dE[fo(E--
I = ~ + ~7~,, - e~,l tanh 2k T "
where R0 is the normal state resistance. Although obtained by an entirely different procedure, this result agrees in detail with that found by Zaitsev [2, 22]. As is clear from (6), the integral expression is a positive addition to the current in the normal state at the same voltage. Basically, this "excess current" results from the extra charge transmitted in the Andreev reflection process. In
- - - ~
Fig. 1. Differential c o n d u c t a n c e v e r s u s v o l t a g e , at T - II. H is t h e s t r e n g t h of t h e a - f u n c t i o n b a r r i e r , Z is a d i m e n s i o n l e s s b a r r i e r s t r e n g t h d e f i n e d as Z = H/(2ev/kF), a n d d V / d l is in units of t h e n o r m a l state r e s i s t a n c e for n o b a r r i e r , i.e. Z = 0. d l / d V r e a c h e s a limit of 2 at e V = A i n d e p e n d e n t of Z.
T.M. Klapwijk et al. / Subharmonic gap structure in superconducting contacts
If both metals are superconducting, excess current is again generated by the Andreev reflection process. However, a much more complicated situation occurs because multiple Andreev reflections are now possible. Consider a superconductor : normal metal construction : superconductor device, with an applied voltage eV and with no &function barrier ( H = 0). Let us concentrate on an electron injected from SL into the normal metal at an energy E relative to /~L. Passing through the normal metal, it gains energy eV, and reflects at the N:SR interface. By energy conservation, this electron reflects around the pair chemical potential, tZR, of the superconductor forcing the reflection, and returns as a hole; in response, current is induced to flow in SR. Because our particle has switched the sign of its charge, the hole gains another eV by retraversing the field. A similar reflection process will occur at the SL:N interface. Clearly, then, by iteration of this process, Andreev reflection provides a way to gain arbitrarily large integral multiples of eV from the field. To keep track of these various trajectories, we
A+eV r ...... :.....i~z
~v-Ai .... .LIN
have found it useful to construct diagrams such as fig. 2. This picture is patterned after a semiconductor-type model, where we plot the Andreev reflection coefficient, instead of the density of states, as a function of energy. Ignoring for a m o m e n t the partial reflections outside the gap region, one can easily see that electrons incident from SL into SR, with A - e V < E < e V - d are totally reflected as holes in the same symmetrical band of energies. On the other hand, electrons incident in the shaded band with - A < E < A - eV from S L onto SR are totally reflected as holes in the band e V - A < E < A . These holes incident on SL are again totally reflected and emerge into SR as electrons in the band of energies from 2 e V - A to e V + A . (All energies are relative to the chemical potential of SR.) In making these constructions, we have used the fact that Andreev reflection of an electron incident at E causes it to reemerge as a hole at - E , and vice versa, E being relative to the chemical potential of the superconductor which forces the reflection to occur. Increasing the voltage allows the particle to gain 2A in energy with fewer reflections; whenever eV passes through 2A/n, the" (n + 1)-fold reflection process disappears, and the (n - D-fold process opens up. In order to deal with a computationally tractable problem, we assume the charge transported by each incident particle is given by the sum of squared amplitudes, ignoring possible interferences between multiple reflections. Using the same method as in the N : S case above, but summing over the multiple reflections, the current generated by particles incident from SL is given by
ILR-Fig. 2. Semiconductor picture for trajectory following in the symmetric case, when e V >~A. The filled circles are electrons, the open circles are holes, and the arrows point in the direction of the group velocity. T h e reflection coefficient, A ( E ) , is shown in the place of the density of states.
1 f dEfo(E-eV)[1-A eRo x [1 + A 0 + AoA1 + • • "],
where A m ( E ) = A ( E + m e V ) . The reverse cur-
T . M . K l a p w i j k et al.
S u b h a r m o n i c g a p structure in s u p e r c o n d u c t i n g contacts
rent is given by
Each term in the integral has a simple meaning. For example, JI~(E-eV)[1 A t]A,~AI can be interpreted as: an electron-like quasiparticle in SL, with a distribution function f , ( E - e V ) approaches the SL interface, and a fraction [ t A_~] is transmitted into the normal region as an electron. The probability of reflecting twice is A~A~, while the increasing subscripts indicate the gain in energy with each traversal. Since we count only electron current, terms with an odd n u m b e r of subscripts are the result of incoming holes, and those with an even n u m b e r from incoming electrons. For A ~ k T and A ~ eV, (8) and (9) reduce to a total current given by V
I = ~ + 3-~0 tanh 2 - ~ ,
two trajectories. The particles in these two paths will reflect off the second interface along four possible trajectories, and so on, the numbers increasing geometrically. Needless to say. the computational difficulties increase proportionately, but we expect to generate I - V curves in the near future.
5. Comparison with experiment We have integrated (8) and (9) numerically, and have plotted d V / d I versus e V in fig. 3. One observes that the I - V characteristic has sharp changes in differential resistance at 2A/n. Comparison with experiments from the available literature shows good qualitative agreement , as indicated by the data from ref. 12, reproduced in fig. 3. Quantitative agreement, however, will be difficult to achieve for several reasons. First, as noted above, we have ignored all phasedependent effects that link the two banks,
or an excess current twice as large  as that found for an S - c - N contact in (7). A potential difficulty with our "trajectory foilowing" method is a lack of self-consistency; that is, we specify only the incoming particle distribution function, with no constraint on the occupancy of the final states. However, we have also analyzed this problem within the context of a Boltzmann equation approach. Using our reflection and transmission coefficients as boundary conditions at both N : S interfaces, we find an expression for the total current identical to the difference of (8) and (9), and feel that this gives added confirmation to our approach. A natural extension of the S : c : S problem would be to include a barrier at the interface, but this task has proven to be quite difficult. For a clean interface ( H = 0), one incident particle produces one reflected particle, but for a dirty interface ( H # 0) the reflected particle can follow
RN ~ n =1
8 7 ~
Fig, 3. Computer-generated d V l d l versus e V , using (S) and (9). T/T~ - 0.989 and A (T) is given by the usual BCS expression. T h e data shown, taken from ref. 12, fig. 1, T/T,, - (I.989, exhibit the shape exaggeration and voltage shifts characteristic of heating (see text). An offset zero and arbitrary resistance units have been used for presenting the experimental data.
T.M. Klapwijk et al. / Subharmonic gap structure in superconducting contacts
thereby eliminating any influence of the Josephson effect. Second, as Octavio et al  showed, heating effects are known to exaggerate or distort the shape and placement of the observed structure. Third, any realizable contact should be modelled in three dimensions with a finite 6function potential, representing the inevitable dirtiness of the constriction. These considerations should not effect the qualitative picture, however. Striking evidence for the qualitative correctness of this picture is provided by application of the model to contacts between two different superconductors. As Rowell and Feldman  observed in Sn-Pb devices, structure appears in the derivative at e V = AR, A R + A L , and 2AL/2n, when AL'~ AR. This unusual pattern is a simple consequence of our model, and is easily understood when the two gaps are very different. For example, fig. 4 illustrates the case of eV--~AL. As the voltage drops through AL, the minimum number of transits changes from two to four, and similarly from four to six if pictures are drawn for eV~AL/2. Thus, the 2AL/2n structure results. The structure at e V = AR+AL and at e V = AR can likewise be explained by the drawing of pictures for voltages just above and just below the specified value. In both cases the number of trajectories changes by two as the voltage is swept by these points.
. . . . . . ] AR
.aj.-.~ L'.'Y.".' i
- ..... -~-A R
eV > A L ;'
Fig. 5. Computer-generated d V / d I versus e V for the case T/TcL = 0.9, Aa = 2.3A L. Ar is given by the usual BCS expression. The absence of structure at 2AL/n, for n odd, is evident. For comparison, data from ref.  are also shown, with an offset zero and with arbitrary resistance units (see also ).
Fig. 5 shows the result of a computer calculation for the I - V curve derivative in an asymmetric SL:c:SR contact, using suitably generalized expressions similar to (8) and (9). We also show, in fig. 5, experimental data taken from Rowell and Feldmann . The locations of the peaks, and the pattern of subharmonic features predicted by our simulation, match those seen experimentally. However, our model, as explained above for the symmetric case, is not expected to provide a quantitative fit .
6. C o n c l u s i o n ~SR'"
AL+eV r - " eV ', eV- A LL,~..\............
Fig. 4. Semiconductor picture for trajectory following in the asymmetric case, A L "~ A R and e V ~ A L.
We believe subharmonic gap structure is the result of multiple Andreev reflections. Although some details of our model may be in error, the qualitative picture is well-founded, and provides an attractive explanation for a complex problem. Additional work, using the 8-function approximation, should lead to a better understanding of point contact I - V curves, and further unify our understanding of both tunnel junctions and micro-constrictions.
et al. 1 Suhharn~onic
National Research, Program. the
Science and One
of us (TMK)
the Office of Naval Electronics like to thank voor
for a grant.
References [I1 S.N. Artemenko.
A.F. Volkov and A.V. Zaitscv. Sov. Phys. JETP 49 (1070) Y24. A.V. Zaitsev. Sov. Phyx. JETP 51 (IYXO) I I I. i:,’ A.F. Andreev, Sov. Phys. JETP IO (1064) 122X. [4 Yu. Ya. Divin and F. Ya. Nad’, Sov. J. I-ow Temp Phy\. 1(107X) 520: JETP Lett. 20 (107’)) 517. [-‘I M. Octavia. W.J. Skocpol and M. Tinkham. f’hys. KC\. B17 (lY78) 1%. IhI B.N. Taylor and E Burstein. Phys. Re\. lxtt. IO (IYhi) 14. 171 C.J. Adkins. Phil. Msg. X (1Yh.i) 1051: Rev. Mod. Phys. 36 (1064) ?I I. Soerensen, B. Kofoed, N.F. Pedersen and IX1 0. Hoffmann S. Shapiro. Rev. de Phys. Appl. 0 (1074) 153. W.J. Skocpol and M. Tinkham. IEEE PI M. Octavia. I-rans. MAC-13 (lY77) 73Y. [I()1J.M. Rowell. Rev. Mod. Phys. 36 (1063) 715.
gap structure in .superconducting
D.G. McDonald, E.G. Johnson and R.E. Harris. Phy\. Rev. HI3 (1076) 102X. P.E. Gregers-Hansen. E. Hendricks, M.1. Lcvinsen and G.R. Pick&t. Phys. Rev. Lett. 31 (1073) 524. J.M. Rowell and W.L. Feldmann. Phys. Re\ 177 (IYhX) 393. J.R. Schrieffer and J.W. Wilkins. Phy\. Rev. I.ett. IO (1063) 17. N.R. Werthamer, Phys. Re\. 137 (1966) 255. A.A. Golub. Sov Phyq. JETP 44 (IY76) 17X. 1..G. Aslsmaso\ and AI. Larkin. Sov. Phys. JETP .li (lY7h) 61)X. A. Schmid. G. Schbn. and M. Tinkham. Phys. KC\. 1~21 (IYXO) 507h. V.M. Gubankov and Nhl. Margolin. JETP I.&t. 3 (I Y7Y) 673. P.G. De Gennes. Superconductivity of Metal\ and Alloys (W.A. Benjamin. New York. IYhh). R. Kiimmel, Z. Phys. 21X (IYhY) 472. Eq. (55) of ref. 7 can he simplified to agree with OUI-(6). We differ only by a factor of two in the excess current. It ih our understanding that %aitxx now believes his result to he in error by this factor. For % ~ 0 and with our assumption of an infinite mean free path. the I-\’ curves exhibit an effective “critical current” that scales like A tanh(A/2kT). However. this i\ not a true supercurrent. Gnce it\ presence depends on the absence of any impurity \catterinp. The Rowe11 and Feldmann experimental system [ 12) consists of a tunnel junction In parallel with a metallic short. This parallel contribution will modify the \tructure at the usual tunnel junction value\ of A, t JR and JR _I,. Also note their data is at ‘UT,(%) = I).36