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The resistance of point contacts between normal metals
Physica 109 & I10B (1982) 1955-1963 North-Holland Publishing Company
1955
THE RESISTANCE OF POINT CONTACTS BETWEEN NORMAL METALS A.P. V A N G E L D E R ,
A . G . M . J A N S E N and P. W Y D E R
Research Institute for Materials, University of Nijmegen, Toernooiveld, 6525 E D Nijmegen, The Netherlands
Point contacts between normal metals at low temperatures show very interesting non-linear phenomena in the current-voltage characteristics. The observed deviations from Ohm's law in metallic constrictions can be used for the determination of the energy dependence of the electron-phonon interaction. An analysis of the relevant theory is given, which allows us to understand the detailed nature of the characteristic signal (the Eliashberg function ~2F, modified with a mean free path dependent efficiency factor), and also of the smooth superimposed background signal which is due to the presence of non-equilibrium phonons.
1. Problem analysis Electrical contacts between metals have received a g o o d deal of interest during the past decade, for instance with respect to applications in microelectronics. Usually, such contacts m a y very well be described by a linear c u r r e n t - v o l tage relationship as expressed by O h m ' s law. F r o m a theoretical point of view, it might seem reasonable to conjecture that "linear" transport theory may be used to describe the physics of this transport problem. By the term "linear" we refer to the same concept as is customarily int r o d u c e d for describing the ohmic resistance of a c o n d u c t o r (metal or semiconductor, etc.); " l i n e a r " implies for these theories that the distribution function of the electrons (and that of the p h o n o n s ) deviates slightly from its equilibrium value, and that the difference is proportional to the driving electrical field or to the applied voltage. The conjecture that such a linearized transport theory may suffice to describe electronic transport for cases where the c u r r e n t - v o l t a g e relation is linear is not necessarily correct, however, as is k n o w n from the theory of electron tunneling between (normal) metals for instance. In this case, the current transport should be described by a theory similar to that of field emission, which is not at all linear 0378-4363/82/0000-0000/$02.75 © 1982 N o r t h - H o l l a n d
in the sense m e n t i o n e d above. T h e fact that such theories are not linear in this sense b e c o m e s obvious if the density of electronic states is not constant near the Fermi surfaces of the metals in contact, or if one looks very carefully at the structure of the c u r r e n t - v o l t a g e ( I - V ) characteristics of the contacted metals. A very important discovery was m a d e by Y a n s o n [1] some time ago for m e t a l - i n s u l a t o r metal tunnel junctions with a short circuit in the oxide layer between the metal films. A l t h o u g h the c u r r e n t - v o l t a g e characteristics are nearly linear (such that a "resistance" may be defined), these characteristics exhibit a structure which could not be identified with that of an ordinary tunnel contact between two metal films. Moreover, the second derivative d2//d V 2 of such junctions appears to have a shape which is very similar to the Eliashberg function o~2F(~), with = eV/h, where c~2 is the coupling constant between electrons and p h o n o n s and F(oo) is the spectral function of the lattice frequencies. A perhaps less spectacular, yet equally startling, observation was the o r d e r of magnitude of the resistance of the contact, in m a n y cases larger than 1 ohm. If the junctions were not shorted and simple electron tunneling were the relevant process, it would be easy to account for resistances of this order, but then the structure would
105¢'~
A . P . pan (Selder et al. / Point contacts h e t w e e n n o r m a l metals
have to c o r r e s p o n d with that of a tunnel diode: on the other hand, if the metals did have a small region with a direct contact without an insulating layer, assuming the validity of O h m ' s (linear) law, the estimated size of such a region would have to be smaller than the distance between two atoms. It was Sharvin [2] w h o long ago clearly realized that the resistance of a point contact could be much in excess of the values predicted by a linearized transport theory (Maxwell resistance RM-~ p/b, where p is the resistivity of the metals, and b is the (linear) dimension of the contact area). T h e physical picture for the nonequilibrium system in the current-carrying state introduced by Sharvin is similar to that of a tunnel diode. T o a g o o d approximation, the Fermi levels of the contacted metals may be considered to be shifted relative to each other by an a m o u n t equal to eV. Electrons with energies lying between the two Fermi levels are permitted to penetrate through the orifice between the contacted metals, as if one were dealing with a case of field emission. T o a g o o d approximation, collisions between the electrons and the lattice or with impurities may be ignored, so that the Maxwell resitance RM is increased by a factor of order ((/b), where t' is the effective mean free path of the electrons which appears to be larger than the size of the orifice (b) in Y a n s o n ' s experiments. Thus, the value of the resistance of a point contact, as calculated by Sharvin, equals to RSh - ~mvF/(rrben°ee), where n" is the electron density and VF the velocity of an electron at the Fermi surface. T h e problem of how to explain the aeF(m) like structure of the deI/dV 2 characteristics as o b s e r v e d by Yanson, and later by Jansen et al. [3] with different techniques, concerns the question how to solve the transport equations for electrons and p h o n o n s in such a way that Sharvin's solution is obtained as a "'lowest o r d e r " result. Considering that Sharvin's solution is characterized by the absence of collisions and that the observed structure (first order) is related to a single collision between an
electron and a p h o n o n , it seems reasonable t~, conjecture that an expansion may be made in terms of the n u m b e r (t~) of collisions which has to be taken into account. Hence, S h a r v i n s solution corresponds with n : t!, Y a n s o n ' s structure corresponds with n 1. and possible b a c k g r o u n d or other contributions should c o r r e s p o n d with n = 2, n - 3, etc. Without going into the details of the transport equations, it is thcreforc obvious from a dimensional point of view that the expansion p a r a m e t e r of such a multiple-scattering expansion is (h/t) since the characteristic uni! of length of the problem may be chosen as equal to the size of the orilice (h). F u r t h e r m o r e , i| might be fair to cnnjecture that the o r d e r ~,f magnitude of the nth term of the expansion for the distribution function or for the current is (b/l)" relative to the Sharvin contribution. Although a rigorous analysis shows that this coniecture is not quite correct, it can hc used as a qualitative explanation that in the "'local" Maxwell limit (,~ <{h) the resistance of the point contact is given by the Sharvin value multiplied with a factor equal to (h/l), For this purpose we first sum the contributions to the current (t~ (t. 1,2, etc.) if h .--~, leading to a factor of order {1 + (b/c')} ' relative to the Sharvin contribution. From this, a factor of ( t / h i is then found b} analytic continuation to the local limit. An elaborate and less qualitative discussion of thc expansion of the distribution functions (or the current) in terms of the p a r a m e t e r (b/;) has becll given [4], indicating thc existence of logarithmic terms in (h/#) which ¢1c~ not fit into a simple Taylor-series expansion. ()riginally, Kulik et al. [5, 6] introduced an explicit form for the singlecollision contribution to the current. From ~ physical point of view this contribution corresponds to a reduction of the Sharvin resistance Rsh(n -- ()) by electrons which return through thc orifice after having collided with a phonon. This contribution (n = 1) can he u n d e r s t o o d in terms of a simple kinetic concept without explicitly solving the transport equations. H o w e v e r , if one were to p r o c e e d to calculate the next contribu-
A.P. van Gelder et al. / Point contacts betweennormal metals tion (n = 2) using the same kinetic considerations, the resulting term for the current would diverge. This divergence can be understood with the geometrical concepts used in ref. 4. The solution of this problem is to "renormalize" the kinetic theory by adding a damping factor for each particle trajectory with a range equal to the mean free part for the corresponding energy. The result is a series expansion for the distribution functions (or the current) for which the leading terms are of order (b/F), and (b/() 2 ln(b/Q. The logarithm may be considered to indicate that the kinetic series expansion has a divergent coefficient for the term n = 2. Having realized this, it is not difficult to see how the hitherto presented theories for the structure of the d21/dV 2 characteristics, based on kinetic reasoning only, have to be improved. Recently, two elaborate and rather complete review papers on point contacts in metals have appeared [7, 8] which give full reference to most of the published work on these problems. Therefore, in this paper we limit ourselves to the discussion of some essential theoretical questions which still seem to be open.
2. Transport-efficiency factor The single-collision backflow contribution to the current which passes through the orifice may be expressed as an integral over the coordinates of the position at the orifice (r~), and the velocities (v) of the returning electrons with directions opposite to that of the "'Sharvin-like" emitted current (n = 0). Then (n = 1) contribution to the current (jl) may hence be written as
J'=-e f dr, f
dvv f'(r,,v),
(1)
where fJ(r~, v) is the contribution to the distribution function, resulting from a single scattering event which has taken place at a time t prior to arrival at the orifice (at r~), and v~ is the velocity c o m p o n e n t perpendicular to the plane of
1957
the orifice. Since this collision has occurred at r~ = r ~ - vt, the distribution function f~(r~, v) of eq. (1) can be expressed as an integral over the elapsed time (t), and a weighted integration over the velocities (v') of the Sharvin-emitted electrons (n = 0) at r = r ~ - vt. T h e time integration is the result of pure kinetic considerations and should be supplemented with a damping factor of the form exp(-vFt/F) in order to ensure the proper renormalization for convergence of the multiple-scattering expansion. Therefore, the v' integration contains the distribution function f s h ( r l - vt, v') of the "Sharvin-emitted" electrons at the position where the electron has scattered. If n and n ' are unit vectors, parallel with v and v', and the corresponding energies are e and e', the distribution function f~ of eq. (1) may be expressed as eV
f l ( r l , lo)= f dt f dg' f d2ntp(n[n ') o xa{l
xD(r,,
-
h rl -
(nln')}
vt)fsh(r, - vt, V') .
(2)
Here, P ( n l n ' ) is the scattering rate of an electron from a direction n' to n. The energy loss of the electron during this collision is equal to the energy hw(nln') of a spontaneously emitted phonon with wave-vector equal to k v ( n ' - n ) , modulo a reciprocal lattice vector. Finally, D(rl, r x - v t ) is equal to the exponential damping factor along the trajectory ( r l - v t ~ r~), as mentioned above. The kinetic expression for fsh(r~vt, V') is 2(m/27rh) 3 if v' is parallel to the vector ( r l - - v t - - r 2 ) , where r2 is an arbitrary point at the orifice; this vector r2 corresponds to the position where the electron has initially passed through the orifice, before it has scattered into the direction of v. If this condition for v' is not met, the electron does not pass through the orifice and fsh vanishes. In this way, the properly renormalized expression for fsh differs from the kinetic one by a similar exponential damping factor D ( r ~ vt, r2) for the corresponding trajectory.
A . P . v a n Gelder el al.
195S
Point contacts betw'ee#l nt~rt#lal metals
For convenience, in eq. (2) we have limited ourselves to zero t e m p e r a t u r e s so that ~" > , and only s p o n t a n e o u s emission of a p h o n o n is possible. Generalization to f n i t e t e m p e r a t u r e s with thermal distribution functions for electrons and p h o n o n s is straightforward and gives rise only to a "'smearing" of the structure function [c)]. In order to account for non-equilibrium cc)rrecnons of the distribution functions, terms of the order n = 2, 3, etc. have to be considered. The leading correction appears to be due to the non-equilibrium phonon distribution function (n =-2, order (b/()(bl,~ph...... )). A discussion will be given in the final section of this paper. In view of the nine integrations involved, thc single collision back-flow current as given by eqs. (I) and (2) looks rather c u m b e r s o m e . Since the integrations over the energies s' and e (corresponding to v) are i n d e p e n d e n t of the other variables, we may write:
!1
(3) where the integrand S on the right-hand side of this expression still contains seven integrations. i.e. the directions n and n' of v and t~'. over r~ (orifice) and over t. Explicitly:
S(,ol=(Srr ~) ' f d:n f d'~n'P(nln') ×,~{w - to(nln')}~)(nln').
(4)
where r/(nln' ) is a factor which characterizes the efficiency of the scattering of the transport process under consideration. 'Fhis efficiency function 7/(nln' ) may be considered to be the result of the three remaining integrations, i.e. ovcr r~ and t. Note that m this n o m e n c l a t u r e both S(w) and ~/(nln' ) are chosen to be dimensionless, which is achieved by the factorization of the dimensional factor h 3 of the (rbt) integrations from the integrand in eq. (3). The
wHtage d e p e n d e n c e of the logarithmic derivative of the resistance of the point contact from cq, (3) and is given by [ 1()]. I
- 1,, e 3
khz:~/
I S
, i[
"
(5 t
where b is the size of the orifice (i.e. its radius, il circular). If the d a m p i n g factors D of eq. (2) are ignored, then the transport-efficiency function r/(n[n') has a simplc geometric meaning: apart from the extracted dimensional factor 1~", o(njn't is equal to the c o m m o n volume which is enclosed by two (intersection) cylinders whose axes are parallel with n and with n'. respectively. Fhe position of both cylinders is uniquely defined by the requirement that their intersections with the plane of the orilice coincide with the circumference of this o r i f c c . (:orrespondingly, apart from a factor tl'. thc transport efficiency ("structure factor") as given by Kulik ct al. [5 i is precisely equal to this c o m n l o n wHumc of these two cylinders. It is clear l:ronl this geometrical consideration that 71(nln') diverges if n and n are anti-parallel, i.e. for back-scattering over an ailglc 0 ,7. 1 o tnldcrstand this geometrical picture <51 the trallsporl efficiency, cHIc has Io realize thai ,~(,,I,,') is the resull <51 integrating the function v , . t~l,(r~ vl. v *) ~5\.'er the variables r~ and I. I1 we neglect t e m p o r a r i l y the' l~'-clependence of lsh (which imposes all upper bond for t). the integrations would give ~i \ a l u e for ~1 equal to the ~oitinle enclosed by the cylinder through ttle circumference
1959
A . P . van Gelder et al. / Point contacts between n o r m a l metals
damping factors D are ignored. This neglect is consistent with a simple kinetic picture which leads to divergencies if higher order corrections, such as contributions from double-collision backflow processes [11], would be included. Note that several efforts have already beerl made to take into account the damping into the efficiency calculation [4, 12]. (ii) The geometric picture is correct for an orifice with arbitrary shape, which is not always valid for other expressions [5]. (iii) The angular integrations over n and n' in eq. (4) is restricted in such a way that n' and n correspond to directions of incoming and backscattered electrons through the orifice; therefore, they are sensitive to the relative orientation of the orifice with respect to crystalline directions. If the scattering rate P(nln') 6{]e e']-hw(nln')} has an angular dependence of n and n', which depends on the value of the scattering angle 0 only (with cos(0)= n - n ' ) , the integrations can be performed in a different order, making the transport efficiency rl(nln') of eq. (4) a function of 0 only. The result obtained in this way may be considered to correspond with an average over the possible directions of the crystallographic axes relative to the plane of the orifice. The angular integrations of eq. (4) reduce to a single integration over the scattering angle 0 = O(n]n'). Denoting rt(0) by r/0(0) if damping is ignored, it can be shown that [10]
the rl and r2 integrations decouple from the one over/3, which finally leads to eq. (6). If the damping factors are taken into account, these last integrations are coupled and one gets for ~ the following expression: 0
r/(0, K) = {2 sin(0)} -I
f duf{x(u, O, K)} 0
x {cos(u)- cos(0)}
(7)
with, if the shape of the orifice is circular, I
f(x)
(8)
= 23 x1 f dy(1 - y2)1/2( 1 - e -2xy) 0
and
x(u,
0, K) -- l COS(U/2) COS(0/2) "
(9)
Here K represents the Knudsen number defined by K = g/b, where g is the mean free path of the damping factors D of eq. (2). The integration variable u corresponds to /3 and is given by u = 0 - 2 / 3 . Note that eq. (6) is found in the limit g ~ . For small values of x,f(x) in eq. (8) can be expanded in a Taylor series and one obtains:
fix) = ~
c.x ° ,
(10)
n=0
n0(0) = ½{l- tg-~0~}
(6)
which, as expected, diverges at 0 = 7r. This result can be obtained by a transformation of the n and n' and t-integrations of eqs. (1) and (2) into integrations over 0 (scattering angle), r2 (defined as before as the position of the initial passage through orifice), /3 (angle between n (or v) with ( r ~ - r2)) and ~ (angle between the plane of the trajectories and that of the orifice). The integration over ~ then decouples from the other integrations. If the damping terms are neglected,
with -z
n
3
-2
leading to the following expansion of r/(0) in powers of (b/g):
rl(O) = ~, c,~7,(O)(b/g)",
(12)
n=0
with co = 1, c~ =-37r/16, c2 = 4/15, etc. r/0 is the
A . P . van Gelder et a L /
1961)
Point contacts b e t w e e n n o r m a l metals
-n(o,K)
expression for rl(0) of eq. (6), and
c~
rl ,(0) = { tge(0/2). 3
0
]g L.
e
0
+1[
0
1}
30
7k
20
16b
(13) Obviously, the expansion of eq. (12) does not converge near the divergence of r1(0) of eq. (6): i.e. at 0 ~ rr. The behaviour of the efficiency factor for large-angle scattering is therefore found from the expansion of f(x) for large values of x, and leads to
rl(0) ~ ( ~ - ~ ) ( b ) { 1 +
cos(O~2)
tg(O/2)In t g ( ~ - ~ 0 ) }
12~ 10
99
which bears a resemblance to the well-known transport efficiency for d.c. conductivity (rta~-(3rr/16) (g/b) { 1 - cos(0)}). Eq. (14) may be considered to be the characteristic efficiency function for m e t a l - m e t a l point contacts in the "'dirty" limit ( L ~ b ) , where t' is the impuritydominated mean free path. Eq. (14) also indicates how the singularity near 0 7r is removed by the introduction of the damping terms. In fig. 1 the effciency function rj is shown for different values of the Knudsen number ~,-. In order to predict the dependence of ( I / R ) × (dR/dV) of eq. (5) on the impurity concentration (or its mean free path), knowledge is required of the function
(15)
which characterizes the distribution of scattering angles for a given frequency, or vice versa. If P(nln')=P{w(ntn')}, eqs. (4) and (15) can be combined:
S(w) = P(w) f dOrl(O)p(m, 0).
0b~
, (14) 0
p(,o,O)=(Srr2)-' f d'-n f den'6{oo "~0(nin')} ×,8{0 O(nln')},
/, ,5 Z
(i6)
~ I
F i g . 1. P o i n t c o n l a c l
¢llicicnc~
values of the K n u d s c l l free path: h
0 nincuon
~710, ~," 1o¢ d i l ' / c r c n i
/ / h [t r a d i u s o[' t h e c i r c u l a r o r i f i c c l , llUlllbt2r h
~.l\t2r~.tgc [llei|l]
The leading contributions to S(co) tire from Umklapp scattering. Since U m k l a p p scattering is characterized by large angles, there exists a lower bond for the 0-integration which, for monovalent simple cubic metals, decreases from 0 = rr (for very low frequcnccs) to 0 - 7r/3 (near the Debye frequency). Therefore, in this case il can be concluded that due to impurity scattering the relative signal reduction is expected to decrease with increasing frequency (w)ltage). Another possibility may be realized for metals with such complicated band structure that the function p(o), 0) may be considered to he independent of ~o. In this case cq. (15) gives p(¢o, 0) = ~ol>1sin(0). It then follows fronl eq. (lf~) that
S(w) = {P(~o)/~OD} i dO sin(0)r/(0).
(17)
Physically, we hereby introduce a model for which all kinematic constraints for the collision
A . P . van Gelder et al. / P o i n t contacts between n o r m a l metals
processes are completely ignored [4]. In view of the lower bond for Umklapp scattering angles, as discussed above, this model does not apply to the monovalent simple cubic metals. In the limit of small values of (b/te), eq. (17) may be approximated by
In this case, if the mean free path is due to impurity scattering, the shape of the signal does not change, but its amplitude simply drops with the increased impurity concentration. If the relevant mean free path in eq. (18) is due to electron-phonon interaction, in this particular model the relative signal reduction increases with the frequency (or with the applied voltage).
3. Background structure Characteristic of most of the observed (voltage-dependent) structures of function ( 1 / R ) x (dR/dV) of point contacts is an extra, more or less structureless, contribution which differs distinctly from the predicted theoretical result of eq. (5). One of the most striking features of this "background signal" is a non-vanishing voltageindependent contribution at V > VDebye, where S(eV/h) vanishes. Moreover, this background contribution has an amplitude which decreases as R -~/2, relative to that of the main signal S, if R is larger than - 1 0 o h m [1, 3, 7, 8]. As is obvious from our general theoretical discussion in the section on the problem analysis, in order to understand this background signal one has to take into account the next correction term in the series expansion in powers of (b/g). In particular, we have to deal with an " n = 2" process, involving two collisions and hence two Knudsen numbers (g/b). One factor corresponds to the singlescattering event considered above; the second factor may either correspond with another, similar scattering process, or with a contribution
1961
from the phonon-distribution function. (It should be emphasized that the transport problem involves coupled equations for the electron- and for the phonon-distribution functions.) If the second factor corresponds with the phonon-distribution function, it means that we also have to take into account the effects of the non-equilibrium phonons on the single-collision backflow processes. Note that it is not difficult to demonstrate that the effects of the thermal distribution functions simply introduce a "thermal smearing" of the zero-temperature structure function, which we shall ignore in what follows [9]. Before going into the details of the non-equilibrium phonon distributions, it is possible to understand in a simple way why a saturating background signal may be caused by non-equilibrium phonons, and also why this contribution should be expected to be of order R -1/2 relative to the main signal. For this purpose, let us assume that the applied voltage is much in excess of the Debye value, VD. Then in processes which are similar to the ones which give rise to the single-collision backflow current Jl of eq. (3) non-equilibrium phonons are produced. Since the integrations in eq. (3) may also be written in the form of f~Vde{eV-e}S(e/h), it can be concluded that the production rate of spontaneously emitted phonons with a given frequency w = e/h is directly proportional to the factor ( e V - e ) . For sufficiently large values of V, this rate will be proportional to V and hence the non-equilibrium value of the phonon-distribution function as well. In order to relate the phonon-production rate with the value of the distribution function at the orifice, the phonon=transport equation has to be solved. In analogy with the corresponding solution of eq. (3) for the electrons, there is a time (spatial) integration which gives rise to a factor b, since the distribution function should vanish if b ~ 0. For dimensional reasons, a factor (1/taph . . . . ) enters into the expression for the phonon-distribution function. It is therefore plausible to expect that the single-collision backflow current j1 of eq. (3) has to be multiplied with a factor of the type.
A.P. t:an Gelder el al.
19(~2
j t __~j l { 1 +
oz(kR/R)D~hy~(V/VD~.by~)},
Point contacts between normal metals
19)
where ( A R / R ) D e b y e is the relative resistance increment at V = VD = Vl~bv~, which is proportional to b and inversely proportional to thc (electron) mean free path. Of course, the electron mean free path is related to the mean free path of the phonons, as the same processes are responsible for both. The factor o~ is dimensionless and independent of b or V and is characteristic of a point contact of a given geometry (shape). As j i of eq. (3) is equal to j l D e b y e ( g / V D ) " if V ~ VD, it is now obvious why the background signal saturates for values of V larger than VD. Multiplication with the factor on the right of eq. (19) gives a term proportional to (V/VD) ~ which corresponds with a saturating (voltage-independent) contribution to the logarithmic derivative of the resistance on the right of eq. (5), obtained from deJ~/dV 2. Roughly, the relative order of magnitude of this saturating background contribution may be estimated by assuming that j~ l = JDebye ( V / V D ) 4, if V --< Vt), leading to S(eV/ti)~ V 2. It is then easy to see that the ratio of the background signal to the peak of the main signal is equal to
(20) In view of eq. (5), this ratio is indeed proportional to R v2 A simple theoretical estimate gives a value of cr ~ I1 which leads to an expected background-to-signal ratio of the order of 9% for a relative resistance ratio of 5% at the D e b y e energy. It is possible to solve explicitly the transport equation for the non-equilibrium phonons [10], and one gets for the occupation number ~,(w, V) of a phonon with frequency ~o at the orifice a value of 0.32 ( e V - hoJ) •p(CO) (hw)
43
-F
d ,
K,,'(~o) : -[q(~o)/ ~0 d-~0~K, '(~01}.
(22)
(q(o0) frequency-dependent wave-vector of the phonon). This cquation expresses the relation between the average value of the life-time of a phonon and that of an electron, both being determined by the same scattering processes. If the non-equilibrium phonons are taken into account, the b a c k f l o w current c a n n o w b c calc u l a t e d from cq. (3) for the single collision backflow contribution by adding a factor {I f p(oo, V)} to account for stimulated emission and also adding extra terms (energy integrations) to a c c o u n t for absorption processes of thc nol/equilibrium p h o m m s ~,(~o, V), The energy integrations of cq. (3) then transform to ,%
,I'
- ] ds{eV
F}S(F/~ )
t, ,t
j dsp(~, V)S(~'/h), !,
background/signal-peak = (a/6)(dR/R)t,~h,~..
v(w, V ) -
phonons, defined by K :::("ph ......./h), and is intimately related to the corresponding Knudscn number for the electrons through the relation:
(21)
The factor Kp(W) is a Knudsen n u m b e r for the
Note that the lirst term on the right corresponds to eq. (3). Our simple relation, given by eq. (19), follows from eq. (23) if Kp~(~O) (0.(}4) ~(~o/¢o,))(*(JR/R)tx.b,~,.
(24)
In view of eq. (22), this relation can be true for all frequencies only if S(o)) is proportional to ~o' (as was assumed for the estimate of eq. (2(t)). Eq. (22) then immediately leads to o~ 4.5 (Sph ....... kF/WD)e 11, if we substitute for the sound-velocity Sph........ =(rr/2)(wD/kv), valid for monovalent simple cubic metals only. Here. ,*,c have used the relation K~'(~o) = 1.18{AR(oo)/R}.
(25)
which follows from eq. (5) after integration with
A . P . van Gelder et al. / P o i n t contacts between n o r m a l metals
respect to V and approximating S(eV/h) by c~2F in order to determine the average electronic life-time. Although it is clear that our discussion is formally limited to the case where S ( w ) - w 2, it is immediately obvious how the method can be generalized. For this purpose, we have to start from eq. (23) in conjunction with eqs. (21) and (22) to obtain the precise background structure, which involves both S(w) and a2F(to). The details have been given elsewhere [10]. Other possible contributions to the background signal have been discussed as well [10], for instance the role of direct electron-electron interactions, leading to a possible contribution to the background signal which is linear as a function of V. Here, we would like to mention another simple mechanism which may be directly understood from eq. (19). The linear term ( - V) corresponds to the effect of a single stimulated emission or an absorption process, because the distribution function for the phonons is roughly proportional to V. If we now also consider the backflow contribution from double-scattering processes, this contribution may be considered to be proportional to the single-collision contribution, multiplied by an extra factor due to the Knudsen number (-AR/RD) and an extra factor (V/VD), accounting for two factors of order v V. It is then obvious that this double-scattering contribution, again involving the non-equilibrium phonon distribution, will give rise to a linear contribution ( - V) to the background signal, as is
1963
indeed frequently observed for low-ohmic point contacts or for high voltages.
Acknowledgements Part of this work has been supported by the "Stichting voor Fundamenteel Ondersoek der Materie" (FOM) with financial support of the "Nederlandse Organisatie voor Zuiver Wetenschappelijk O n d e r z o e k " (ZWO).
References [1] I.K. Yanson, Zh, Eksp. Teor. Fiz. 66 (1974) 1035 [Sov. P h y s . - JETP 39 (1974) 51)6]. [2] Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48 11965) 984 [Soy. P h y s . - J E T P 21 (1965) 655]. [3] A.G.M. Jansen, F.M. Mueller and P. Wyder, Science 199 (1978) 1037. 14] A.P. van Gelder, Solid State Commun. 25 (1978) 1097. [5] 1.O. Kulik, R.I. Shekhter and A,N. Omelyanchuk, Solid State Commun. 23 11977) 301. [6] A.N. Omelyanchuk, I.O. Kulik and R.I. Shekhter, Pis'ma Zh. Eksp, Teor. Fiz. 25 (1977) 465 [JETP Lett. 25 (1977) 437]. [7] I.K. Yanson, I.O. Kulik and A.G. Batrak, J. Low Temp. Phys. 42 (1981) 527. [8] A.G.M. Jansen, A.P. van Gelder and P. Wyder, J. Phys. C: Solid State Phys. 13 (1980) 6073. [9] A.P. van Gelder, A.G.M. Jansen and P. Wyder, Phys. Rev. B22 11980) 1515. [10] A.P. van Gelder, Solid State Commun. 35 (1980) 19, [11] A.P. van Gelder, A.G.M. Jansen, S. Str~issler and P. Wyder, J. Phys. (Paris) 39 (1978) C6-326. [12] M. Askraf and J.C. Swihart, J. Phys. F: Metal Phys,, to be published. We are most grateful to Professor Swihart for sending us a preprint of his work.
1955
THE RESISTANCE OF POINT CONTACTS BETWEEN NORMAL METALS A.P. V A N G E L D E R ,
A . G . M . J A N S E N and P. W Y D E R
Research Institute for Materials, University of Nijmegen, Toernooiveld, 6525 E D Nijmegen, The Netherlands
Point contacts between normal metals at low temperatures show very interesting non-linear phenomena in the current-voltage characteristics. The observed deviations from Ohm's law in metallic constrictions can be used for the determination of the energy dependence of the electron-phonon interaction. An analysis of the relevant theory is given, which allows us to understand the detailed nature of the characteristic signal (the Eliashberg function ~2F, modified with a mean free path dependent efficiency factor), and also of the smooth superimposed background signal which is due to the presence of non-equilibrium phonons.
1. Problem analysis Electrical contacts between metals have received a g o o d deal of interest during the past decade, for instance with respect to applications in microelectronics. Usually, such contacts m a y very well be described by a linear c u r r e n t - v o l tage relationship as expressed by O h m ' s law. F r o m a theoretical point of view, it might seem reasonable to conjecture that "linear" transport theory may be used to describe the physics of this transport problem. By the term "linear" we refer to the same concept as is customarily int r o d u c e d for describing the ohmic resistance of a c o n d u c t o r (metal or semiconductor, etc.); " l i n e a r " implies for these theories that the distribution function of the electrons (and that of the p h o n o n s ) deviates slightly from its equilibrium value, and that the difference is proportional to the driving electrical field or to the applied voltage. The conjecture that such a linearized transport theory may suffice to describe electronic transport for cases where the c u r r e n t - v o l t a g e relation is linear is not necessarily correct, however, as is k n o w n from the theory of electron tunneling between (normal) metals for instance. In this case, the current transport should be described by a theory similar to that of field emission, which is not at all linear 0378-4363/82/0000-0000/$02.75 © 1982 N o r t h - H o l l a n d
in the sense m e n t i o n e d above. T h e fact that such theories are not linear in this sense b e c o m e s obvious if the density of electronic states is not constant near the Fermi surfaces of the metals in contact, or if one looks very carefully at the structure of the c u r r e n t - v o l t a g e ( I - V ) characteristics of the contacted metals. A very important discovery was m a d e by Y a n s o n [1] some time ago for m e t a l - i n s u l a t o r metal tunnel junctions with a short circuit in the oxide layer between the metal films. A l t h o u g h the c u r r e n t - v o l t a g e characteristics are nearly linear (such that a "resistance" may be defined), these characteristics exhibit a structure which could not be identified with that of an ordinary tunnel contact between two metal films. Moreover, the second derivative d2//d V 2 of such junctions appears to have a shape which is very similar to the Eliashberg function o~2F(~), with = eV/h, where c~2 is the coupling constant between electrons and p h o n o n s and F(oo) is the spectral function of the lattice frequencies. A perhaps less spectacular, yet equally startling, observation was the o r d e r of magnitude of the resistance of the contact, in m a n y cases larger than 1 ohm. If the junctions were not shorted and simple electron tunneling were the relevant process, it would be easy to account for resistances of this order, but then the structure would
105¢'~
A . P . pan (Selder et al. / Point contacts h e t w e e n n o r m a l metals
have to c o r r e s p o n d with that of a tunnel diode: on the other hand, if the metals did have a small region with a direct contact without an insulating layer, assuming the validity of O h m ' s (linear) law, the estimated size of such a region would have to be smaller than the distance between two atoms. It was Sharvin [2] w h o long ago clearly realized that the resistance of a point contact could be much in excess of the values predicted by a linearized transport theory (Maxwell resistance RM-~ p/b, where p is the resistivity of the metals, and b is the (linear) dimension of the contact area). T h e physical picture for the nonequilibrium system in the current-carrying state introduced by Sharvin is similar to that of a tunnel diode. T o a g o o d approximation, the Fermi levels of the contacted metals may be considered to be shifted relative to each other by an a m o u n t equal to eV. Electrons with energies lying between the two Fermi levels are permitted to penetrate through the orifice between the contacted metals, as if one were dealing with a case of field emission. T o a g o o d approximation, collisions between the electrons and the lattice or with impurities may be ignored, so that the Maxwell resitance RM is increased by a factor of order ((/b), where t' is the effective mean free path of the electrons which appears to be larger than the size of the orifice (b) in Y a n s o n ' s experiments. Thus, the value of the resistance of a point contact, as calculated by Sharvin, equals to RSh - ~mvF/(rrben°ee), where n" is the electron density and VF the velocity of an electron at the Fermi surface. T h e problem of how to explain the aeF(m) like structure of the deI/dV 2 characteristics as o b s e r v e d by Yanson, and later by Jansen et al. [3] with different techniques, concerns the question how to solve the transport equations for electrons and p h o n o n s in such a way that Sharvin's solution is obtained as a "'lowest o r d e r " result. Considering that Sharvin's solution is characterized by the absence of collisions and that the observed structure (first order) is related to a single collision between an
electron and a p h o n o n , it seems reasonable t~, conjecture that an expansion may be made in terms of the n u m b e r (t~) of collisions which has to be taken into account. Hence, S h a r v i n s solution corresponds with n : t!, Y a n s o n ' s structure corresponds with n 1. and possible b a c k g r o u n d or other contributions should c o r r e s p o n d with n = 2, n - 3, etc. Without going into the details of the transport equations, it is thcreforc obvious from a dimensional point of view that the expansion p a r a m e t e r of such a multiple-scattering expansion is (h/t) since the characteristic uni! of length of the problem may be chosen as equal to the size of the orilice (h). F u r t h e r m o r e , i| might be fair to cnnjecture that the o r d e r ~,f magnitude of the nth term of the expansion for the distribution function or for the current is (b/l)" relative to the Sharvin contribution. Although a rigorous analysis shows that this coniecture is not quite correct, it can hc used as a qualitative explanation that in the "'local" Maxwell limit (,~ <{h) the resistance of the point contact is given by the Sharvin value multiplied with a factor equal to (h/l), For this purpose we first sum the contributions to the current (t~ (t. 1,2, etc.) if h .--~, leading to a factor of order {1 + (b/c')} ' relative to the Sharvin contribution. From this, a factor of ( t / h i is then found b} analytic continuation to the local limit. An elaborate and less qualitative discussion of thc expansion of the distribution functions (or the current) in terms of the p a r a m e t e r (b/;) has becll given [4], indicating thc existence of logarithmic terms in (h/#) which ¢1c~ not fit into a simple Taylor-series expansion. ()riginally, Kulik et al. [5, 6] introduced an explicit form for the singlecollision contribution to the current. From ~ physical point of view this contribution corresponds to a reduction of the Sharvin resistance Rsh(n -- ()) by electrons which return through thc orifice after having collided with a phonon. This contribution (n = 1) can he u n d e r s t o o d in terms of a simple kinetic concept without explicitly solving the transport equations. H o w e v e r , if one were to p r o c e e d to calculate the next contribu-
A.P. van Gelder et al. / Point contacts betweennormal metals tion (n = 2) using the same kinetic considerations, the resulting term for the current would diverge. This divergence can be understood with the geometrical concepts used in ref. 4. The solution of this problem is to "renormalize" the kinetic theory by adding a damping factor for each particle trajectory with a range equal to the mean free part for the corresponding energy. The result is a series expansion for the distribution functions (or the current) for which the leading terms are of order (b/F), and (b/() 2 ln(b/Q. The logarithm may be considered to indicate that the kinetic series expansion has a divergent coefficient for the term n = 2. Having realized this, it is not difficult to see how the hitherto presented theories for the structure of the d21/dV 2 characteristics, based on kinetic reasoning only, have to be improved. Recently, two elaborate and rather complete review papers on point contacts in metals have appeared [7, 8] which give full reference to most of the published work on these problems. Therefore, in this paper we limit ourselves to the discussion of some essential theoretical questions which still seem to be open.
2. Transport-efficiency factor The single-collision backflow contribution to the current which passes through the orifice may be expressed as an integral over the coordinates of the position at the orifice (r~), and the velocities (v) of the returning electrons with directions opposite to that of the "'Sharvin-like" emitted current (n = 0). Then (n = 1) contribution to the current (jl) may hence be written as
J'=-e f dr, f
dvv f'(r,,v),
(1)
where fJ(r~, v) is the contribution to the distribution function, resulting from a single scattering event which has taken place at a time t prior to arrival at the orifice (at r~), and v~ is the velocity c o m p o n e n t perpendicular to the plane of
1957
the orifice. Since this collision has occurred at r~ = r ~ - vt, the distribution function f~(r~, v) of eq. (1) can be expressed as an integral over the elapsed time (t), and a weighted integration over the velocities (v') of the Sharvin-emitted electrons (n = 0) at r = r ~ - vt. T h e time integration is the result of pure kinetic considerations and should be supplemented with a damping factor of the form exp(-vFt/F) in order to ensure the proper renormalization for convergence of the multiple-scattering expansion. Therefore, the v' integration contains the distribution function f s h ( r l - vt, v') of the "Sharvin-emitted" electrons at the position where the electron has scattered. If n and n ' are unit vectors, parallel with v and v', and the corresponding energies are e and e', the distribution function f~ of eq. (1) may be expressed as eV
f l ( r l , lo)= f dt f dg' f d2ntp(n[n ') o xa{l
xD(r,,
-
h rl -
(nln')}
vt)fsh(r, - vt, V') .
(2)
Here, P ( n l n ' ) is the scattering rate of an electron from a direction n' to n. The energy loss of the electron during this collision is equal to the energy hw(nln') of a spontaneously emitted phonon with wave-vector equal to k v ( n ' - n ) , modulo a reciprocal lattice vector. Finally, D(rl, r x - v t ) is equal to the exponential damping factor along the trajectory ( r l - v t ~ r~), as mentioned above. The kinetic expression for fsh(r~vt, V') is 2(m/27rh) 3 if v' is parallel to the vector ( r l - - v t - - r 2 ) , where r2 is an arbitrary point at the orifice; this vector r2 corresponds to the position where the electron has initially passed through the orifice, before it has scattered into the direction of v. If this condition for v' is not met, the electron does not pass through the orifice and fsh vanishes. In this way, the properly renormalized expression for fsh differs from the kinetic one by a similar exponential damping factor D ( r ~ vt, r2) for the corresponding trajectory.
A . P . v a n Gelder el al.
195S
Point contacts betw'ee#l nt~rt#lal metals
For convenience, in eq. (2) we have limited ourselves to zero t e m p e r a t u r e s so that ~" > , and only s p o n t a n e o u s emission of a p h o n o n is possible. Generalization to f n i t e t e m p e r a t u r e s with thermal distribution functions for electrons and p h o n o n s is straightforward and gives rise only to a "'smearing" of the structure function [c)]. In order to account for non-equilibrium cc)rrecnons of the distribution functions, terms of the order n = 2, 3, etc. have to be considered. The leading correction appears to be due to the non-equilibrium phonon distribution function (n =-2, order (b/()(bl,~ph...... )). A discussion will be given in the final section of this paper. In view of the nine integrations involved, thc single collision back-flow current as given by eqs. (I) and (2) looks rather c u m b e r s o m e . Since the integrations over the energies s' and e (corresponding to v) are i n d e p e n d e n t of the other variables, we may write:
!1
(3) where the integrand S on the right-hand side of this expression still contains seven integrations. i.e. the directions n and n' of v and t~'. over r~ (orifice) and over t. Explicitly:
S(,ol=(Srr ~) ' f d:n f d'~n'P(nln') ×,~{w - to(nln')}~)(nln').
(4)
where r/(nln' ) is a factor which characterizes the efficiency of the scattering of the transport process under consideration. 'Fhis efficiency function 7/(nln' ) may be considered to be the result of the three remaining integrations, i.e. ovcr r~ and t. Note that m this n o m e n c l a t u r e both S(w) and ~/(nln' ) are chosen to be dimensionless, which is achieved by the factorization of the dimensional factor h 3 of the (rbt) integrations from the integrand in eq. (3). The
wHtage d e p e n d e n c e of the logarithmic derivative of the resistance of the point contact from cq, (3) and is given by [ 1()]. I
- 1,, e 3
khz:~/
I S
, i[
"
(5 t
where b is the size of the orifice (i.e. its radius, il circular). If the d a m p i n g factors D of eq. (2) are ignored, then the transport-efficiency function r/(n[n') has a simplc geometric meaning: apart from the extracted dimensional factor 1~", o(njn't is equal to the c o m m o n volume which is enclosed by two (intersection) cylinders whose axes are parallel with n and with n'. respectively. Fhe position of both cylinders is uniquely defined by the requirement that their intersections with the plane of the orilice coincide with the circumference of this o r i f c c . (:orrespondingly, apart from a factor tl'. thc transport efficiency ("structure factor") as given by Kulik ct al. [5 i is precisely equal to this c o m n l o n wHumc of these two cylinders. It is clear l:ronl this geometrical consideration that 71(nln') diverges if n and n are anti-parallel, i.e. for back-scattering over an ailglc 0 ,7. 1 o tnldcrstand this geometrical picture <51 the trallsporl efficiency, cHIc has Io realize thai ,~(,,I,,') is the resull <51 integrating the function v , . t~l,(r~ vl. v *) ~5\.'er the variables r~ and I. I1 we neglect t e m p o r a r i l y the' l~'-clependence of lsh (which imposes all upper bond for t). the integrations would give ~i \ a l u e for ~1 equal to the ~oitinle enclosed by the cylinder through ttle circumference
1959
A . P . van Gelder et al. / Point contacts between n o r m a l metals
damping factors D are ignored. This neglect is consistent with a simple kinetic picture which leads to divergencies if higher order corrections, such as contributions from double-collision backflow processes [11], would be included. Note that several efforts have already beerl made to take into account the damping into the efficiency calculation [4, 12]. (ii) The geometric picture is correct for an orifice with arbitrary shape, which is not always valid for other expressions [5]. (iii) The angular integrations over n and n' in eq. (4) is restricted in such a way that n' and n correspond to directions of incoming and backscattered electrons through the orifice; therefore, they are sensitive to the relative orientation of the orifice with respect to crystalline directions. If the scattering rate P(nln') 6{]e e']-hw(nln')} has an angular dependence of n and n', which depends on the value of the scattering angle 0 only (with cos(0)= n - n ' ) , the integrations can be performed in a different order, making the transport efficiency rl(nln') of eq. (4) a function of 0 only. The result obtained in this way may be considered to correspond with an average over the possible directions of the crystallographic axes relative to the plane of the orifice. The angular integrations of eq. (4) reduce to a single integration over the scattering angle 0 = O(n]n'). Denoting rt(0) by r/0(0) if damping is ignored, it can be shown that [10]
the rl and r2 integrations decouple from the one over/3, which finally leads to eq. (6). If the damping factors are taken into account, these last integrations are coupled and one gets for ~ the following expression: 0
r/(0, K) = {2 sin(0)} -I
f duf{x(u, O, K)} 0
x {cos(u)- cos(0)}
(7)
with, if the shape of the orifice is circular, I
f(x)
(8)
= 23 x1 f dy(1 - y2)1/2( 1 - e -2xy) 0
and
x(u,
0, K) -- l COS(U/2) COS(0/2) "
(9)
Here K represents the Knudsen number defined by K = g/b, where g is the mean free path of the damping factors D of eq. (2). The integration variable u corresponds to /3 and is given by u = 0 - 2 / 3 . Note that eq. (6) is found in the limit g ~ . For small values of x,f(x) in eq. (8) can be expanded in a Taylor series and one obtains:
fix) = ~
c.x ° ,
(10)
n=0
n0(0) = ½{l- tg-~0~}
(6)
which, as expected, diverges at 0 = 7r. This result can be obtained by a transformation of the n and n' and t-integrations of eqs. (1) and (2) into integrations over 0 (scattering angle), r2 (defined as before as the position of the initial passage through orifice), /3 (angle between n (or v) with ( r ~ - r2)) and ~ (angle between the plane of the trajectories and that of the orifice). The integration over ~ then decouples from the other integrations. If the damping terms are neglected,
with -z
n
3
-2
leading to the following expansion of r/(0) in powers of (b/g):
rl(O) = ~, c,~7,(O)(b/g)",
(12)
n=0
with co = 1, c~ =-37r/16, c2 = 4/15, etc. r/0 is the
A . P . van Gelder et a L /
1961)
Point contacts b e t w e e n n o r m a l metals
-n(o,K)
expression for rl(0) of eq. (6), and
c~
rl ,(0) = { tge(0/2). 3
0
]g L.
e
0
+1[
0
1}
30
7k
20
16b
(13) Obviously, the expansion of eq. (12) does not converge near the divergence of r1(0) of eq. (6): i.e. at 0 ~ rr. The behaviour of the efficiency factor for large-angle scattering is therefore found from the expansion of f(x) for large values of x, and leads to
rl(0) ~ ( ~ - ~ ) ( b ) { 1 +
cos(O~2)
tg(O/2)In t g ( ~ - ~ 0 ) }
12~ 10
99
which bears a resemblance to the well-known transport efficiency for d.c. conductivity (rta~-(3rr/16) (g/b) { 1 - cos(0)}). Eq. (14) may be considered to be the characteristic efficiency function for m e t a l - m e t a l point contacts in the "'dirty" limit ( L ~ b ) , where t' is the impuritydominated mean free path. Eq. (14) also indicates how the singularity near 0 7r is removed by the introduction of the damping terms. In fig. 1 the effciency function rj is shown for different values of the Knudsen number ~,-. In order to predict the dependence of ( I / R ) × (dR/dV) of eq. (5) on the impurity concentration (or its mean free path), knowledge is required of the function
(15)
which characterizes the distribution of scattering angles for a given frequency, or vice versa. If P(nln')=P{w(ntn')}, eqs. (4) and (15) can be combined:
S(w) = P(w) f dOrl(O)p(m, 0).
0b~
, (14) 0
p(,o,O)=(Srr2)-' f d'-n f den'6{oo "~0(nin')} ×,8{0 O(nln')},
/, ,5 Z
(i6)
~ I
F i g . 1. P o i n t c o n l a c l
¢llicicnc~
values of the K n u d s c l l free path: h
0 nincuon
~710, ~," 1o¢ d i l ' / c r c n i
/ / h [t r a d i u s o[' t h e c i r c u l a r o r i f i c c l , llUlllbt2r h
~.l\t2r~.tgc [llei|l]
The leading contributions to S(co) tire from Umklapp scattering. Since U m k l a p p scattering is characterized by large angles, there exists a lower bond for the 0-integration which, for monovalent simple cubic metals, decreases from 0 = rr (for very low frequcnccs) to 0 - 7r/3 (near the Debye frequency). Therefore, in this case il can be concluded that due to impurity scattering the relative signal reduction is expected to decrease with increasing frequency (w)ltage). Another possibility may be realized for metals with such complicated band structure that the function p(o), 0) may be considered to he independent of ~o. In this case cq. (15) gives p(¢o, 0) = ~ol>1sin(0). It then follows fronl eq. (lf~) that
S(w) = {P(~o)/~OD} i dO sin(0)r/(0).
(17)
Physically, we hereby introduce a model for which all kinematic constraints for the collision
A . P . van Gelder et al. / P o i n t contacts between n o r m a l metals
processes are completely ignored [4]. In view of the lower bond for Umklapp scattering angles, as discussed above, this model does not apply to the monovalent simple cubic metals. In the limit of small values of (b/te), eq. (17) may be approximated by
In this case, if the mean free path is due to impurity scattering, the shape of the signal does not change, but its amplitude simply drops with the increased impurity concentration. If the relevant mean free path in eq. (18) is due to electron-phonon interaction, in this particular model the relative signal reduction increases with the frequency (or with the applied voltage).
3. Background structure Characteristic of most of the observed (voltage-dependent) structures of function ( 1 / R ) x (dR/dV) of point contacts is an extra, more or less structureless, contribution which differs distinctly from the predicted theoretical result of eq. (5). One of the most striking features of this "background signal" is a non-vanishing voltageindependent contribution at V > VDebye, where S(eV/h) vanishes. Moreover, this background contribution has an amplitude which decreases as R -~/2, relative to that of the main signal S, if R is larger than - 1 0 o h m [1, 3, 7, 8]. As is obvious from our general theoretical discussion in the section on the problem analysis, in order to understand this background signal one has to take into account the next correction term in the series expansion in powers of (b/g). In particular, we have to deal with an " n = 2" process, involving two collisions and hence two Knudsen numbers (g/b). One factor corresponds to the singlescattering event considered above; the second factor may either correspond with another, similar scattering process, or with a contribution
1961
from the phonon-distribution function. (It should be emphasized that the transport problem involves coupled equations for the electron- and for the phonon-distribution functions.) If the second factor corresponds with the phonon-distribution function, it means that we also have to take into account the effects of the non-equilibrium phonons on the single-collision backflow processes. Note that it is not difficult to demonstrate that the effects of the thermal distribution functions simply introduce a "thermal smearing" of the zero-temperature structure function, which we shall ignore in what follows [9]. Before going into the details of the non-equilibrium phonon distributions, it is possible to understand in a simple way why a saturating background signal may be caused by non-equilibrium phonons, and also why this contribution should be expected to be of order R -1/2 relative to the main signal. For this purpose, let us assume that the applied voltage is much in excess of the Debye value, VD. Then in processes which are similar to the ones which give rise to the single-collision backflow current Jl of eq. (3) non-equilibrium phonons are produced. Since the integrations in eq. (3) may also be written in the form of f~Vde{eV-e}S(e/h), it can be concluded that the production rate of spontaneously emitted phonons with a given frequency w = e/h is directly proportional to the factor ( e V - e ) . For sufficiently large values of V, this rate will be proportional to V and hence the non-equilibrium value of the phonon-distribution function as well. In order to relate the phonon-production rate with the value of the distribution function at the orifice, the phonon=transport equation has to be solved. In analogy with the corresponding solution of eq. (3) for the electrons, there is a time (spatial) integration which gives rise to a factor b, since the distribution function should vanish if b ~ 0. For dimensional reasons, a factor (1/taph . . . . ) enters into the expression for the phonon-distribution function. It is therefore plausible to expect that the single-collision backflow current j1 of eq. (3) has to be multiplied with a factor of the type.
A.P. t:an Gelder el al.
19(~2
j t __~j l { 1 +
oz(kR/R)D~hy~(V/VD~.by~)},
Point contacts between normal metals
19)
where ( A R / R ) D e b y e is the relative resistance increment at V = VD = Vl~bv~, which is proportional to b and inversely proportional to thc (electron) mean free path. Of course, the electron mean free path is related to the mean free path of the phonons, as the same processes are responsible for both. The factor o~ is dimensionless and independent of b or V and is characteristic of a point contact of a given geometry (shape). As j i of eq. (3) is equal to j l D e b y e ( g / V D ) " if V ~ VD, it is now obvious why the background signal saturates for values of V larger than VD. Multiplication with the factor on the right of eq. (19) gives a term proportional to (V/VD) ~ which corresponds with a saturating (voltage-independent) contribution to the logarithmic derivative of the resistance on the right of eq. (5), obtained from deJ~/dV 2. Roughly, the relative order of magnitude of this saturating background contribution may be estimated by assuming that j~ l = JDebye ( V / V D ) 4, if V --< Vt), leading to S(eV/ti)~ V 2. It is then easy to see that the ratio of the background signal to the peak of the main signal is equal to
(20) In view of eq. (5), this ratio is indeed proportional to R v2 A simple theoretical estimate gives a value of cr ~ I1 which leads to an expected background-to-signal ratio of the order of 9% for a relative resistance ratio of 5% at the D e b y e energy. It is possible to solve explicitly the transport equation for the non-equilibrium phonons [10], and one gets for the occupation number ~,(w, V) of a phonon with frequency ~o at the orifice a value of 0.32 ( e V - hoJ) •p(CO) (hw)
43
-F
d ,
K,,'(~o) : -[q(~o)/ ~0 d-~0~K, '(~01}.
(22)
(q(o0) frequency-dependent wave-vector of the phonon). This cquation expresses the relation between the average value of the life-time of a phonon and that of an electron, both being determined by the same scattering processes. If the non-equilibrium phonons are taken into account, the b a c k f l o w current c a n n o w b c calc u l a t e d from cq. (3) for the single collision backflow contribution by adding a factor {I f p(oo, V)} to account for stimulated emission and also adding extra terms (energy integrations) to a c c o u n t for absorption processes of thc nol/equilibrium p h o m m s ~,(~o, V), The energy integrations of cq. (3) then transform to ,%
,I'
- ] ds{eV
F}S(F/~ )
t, ,t
j dsp(~, V)S(~'/h), !,
background/signal-peak = (a/6)(dR/R)t,~h,~..
v(w, V ) -
phonons, defined by K :::("ph ......./h), and is intimately related to the corresponding Knudscn number for the electrons through the relation:
(21)
The factor Kp(W) is a Knudsen n u m b e r for the
Note that the lirst term on the right corresponds to eq. (3). Our simple relation, given by eq. (19), follows from eq. (23) if Kp~(~O) (0.(}4) ~(~o/¢o,))(*(JR/R)tx.b,~,.
(24)
In view of eq. (22), this relation can be true for all frequencies only if S(o)) is proportional to ~o' (as was assumed for the estimate of eq. (2(t)). Eq. (22) then immediately leads to o~ 4.5 (Sph ....... kF/WD)e 11, if we substitute for the sound-velocity Sph........ =(rr/2)(wD/kv), valid for monovalent simple cubic metals only. Here. ,*,c have used the relation K~'(~o) = 1.18{AR(oo)/R}.
(25)
which follows from eq. (5) after integration with
A . P . van Gelder et al. / P o i n t contacts between n o r m a l metals
respect to V and approximating S(eV/h) by c~2F in order to determine the average electronic life-time. Although it is clear that our discussion is formally limited to the case where S ( w ) - w 2, it is immediately obvious how the method can be generalized. For this purpose, we have to start from eq. (23) in conjunction with eqs. (21) and (22) to obtain the precise background structure, which involves both S(w) and a2F(to). The details have been given elsewhere [10]. Other possible contributions to the background signal have been discussed as well [10], for instance the role of direct electron-electron interactions, leading to a possible contribution to the background signal which is linear as a function of V. Here, we would like to mention another simple mechanism which may be directly understood from eq. (19). The linear term ( - V) corresponds to the effect of a single stimulated emission or an absorption process, because the distribution function for the phonons is roughly proportional to V. If we now also consider the backflow contribution from double-scattering processes, this contribution may be considered to be proportional to the single-collision contribution, multiplied by an extra factor due to the Knudsen number (-AR/RD) and an extra factor (V/VD), accounting for two factors of order v V. It is then obvious that this double-scattering contribution, again involving the non-equilibrium phonon distribution, will give rise to a linear contribution ( - V) to the background signal, as is
1963
indeed frequently observed for low-ohmic point contacts or for high voltages.
Acknowledgements Part of this work has been supported by the "Stichting voor Fundamenteel Ondersoek der Materie" (FOM) with financial support of the "Nederlandse Organisatie voor Zuiver Wetenschappelijk O n d e r z o e k " (ZWO).
References [1] I.K. Yanson, Zh, Eksp. Teor. Fiz. 66 (1974) 1035 [Sov. P h y s . - JETP 39 (1974) 51)6]. [2] Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48 11965) 984 [Soy. P h y s . - J E T P 21 (1965) 655]. [3] A.G.M. Jansen, F.M. Mueller and P. Wyder, Science 199 (1978) 1037. 14] A.P. van Gelder, Solid State Commun. 25 (1978) 1097. [5] 1.O. Kulik, R.I. Shekhter and A,N. Omelyanchuk, Solid State Commun. 23 11977) 301. [6] A.N. Omelyanchuk, I.O. Kulik and R.I. Shekhter, Pis'ma Zh. Eksp, Teor. Fiz. 25 (1977) 465 [JETP Lett. 25 (1977) 437]. [7] I.K. Yanson, I.O. Kulik and A.G. Batrak, J. Low Temp. Phys. 42 (1981) 527. [8] A.G.M. Jansen, A.P. van Gelder and P. Wyder, J. Phys. C: Solid State Phys. 13 (1980) 6073. [9] A.P. van Gelder, A.G.M. Jansen and P. Wyder, Phys. Rev. B22 11980) 1515. [10] A.P. van Gelder, Solid State Commun. 35 (1980) 19, [11] A.P. van Gelder, A.G.M. Jansen, S. Str~issler and P. Wyder, J. Phys. (Paris) 39 (1978) C6-326. [12] M. Askraf and J.C. Swihart, J. Phys. F: Metal Phys,, to be published. We are most grateful to Professor Swihart for sending us a preprint of his work.