A S I M P L E M O D E L O F T H I N - F I L M P O I N T C O N T A C T I N T W O A N D T H R E E D I M E N S I O N S
P. Exner*), P. Seba**)
Laboratory o f Theoretical Physics, JINR, 141 980 Dubna, USSR
We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two- parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admis- sible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.
1. INTRODUCTION
The theory o f self-adjoint extensions is a part of functional analysis which is commonly regarded as classical. Recently it has attracted a new interest connected with applications in quantum physics which include, e.g., rigorous treatment of point interaction [ 1 - 8 ] , a model of three-particle resonances [9], the so-called metallic models of molecules [10], some properties o f singular potentials such as tunneling effect in one dimension [11] or various regularisation procedures [ 1 2 - 1 4 ] etc.
The theory o f self-adjoint extensions gives us also a possibility to construct models describing experiments in the quantum point-contact spectroscopy, where one studies deviations from the O h m ' s law on metallic contacts whose diameter is small compar- ing to the mean free path of the electrons [15]. In a recent series o f papers [ 1 6 - 18], we have analyzed very simple models, in which the electrons are supposed to be free and spinless, for two typical situations usually dubbed spear-and-anvil contact and thin-film contact. Despite their simplicity, the models are able to reproduce the basic non-linear shape o f the observed current-voltage characteristics of the " p o i n t "
contact as we have illustrated on examples in ref. [18]. On the other hand, one cannot obtain in this way the fine structure of the current-voltage characteristics, which is not surprising, because we have completely neglected the structure o f the metal (which might be modelled by adding a suitable periodic potential). Another open question concerns the choice of the right self-adjoint extension which should
*) On leave of absence from Nuclear Physics Institute, Czechosl. Acad. Sci., 250 68 l~e~ near Prague, Czechoslovakia.
**) On leave of absence from Nuclear Centre, Charles University, V Holegovi~kdeh 2, 180 O0 Prague 8, Czechoslovakia.
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play the role of Hamiltonian; one would invite a choice guided by some physical consideration rather than a fit to experimental data.
Before dealing with these problems, however, one must analyse basic features o f this class of models. The present paper is a part of this program: we concentrate our attention here to the problem how the current-voltage characteristics of a thin- film contact is influenced by thickness of the metallic films. The typical experiment is sketched schematically in fig. 1. Two metallic films are produced on a substrate
oxide layer
//~/~zmetaUic him _~_
d
Fig. 1. S c h e m e o f t h e p o i n t - c o n t a c t e x p e r i m e n t with thin films.
by an evaporation technique, separated by a very thin oxide layer in which a crack is produced by an electric breakdown or mechanically. It is well known from the pioneering experiments [19] that the current-voltage characteristics at low energies exhibits oscillations whose frequency is inversely proportional to the thickness of the film.
We are going to discuss two extreme situations. In the first of them, the film is supposed to be very thick, i.e., d = oo; and we have to analyze motion of a particle in two halfspaces separated by a plane with one " p o i n t " hole. One must choose, o f course, suitable boundary conditions on this plane. Since we have in mind motion of electrons in metal, the Neumann condition is an appropriate choice. Other pos- sibilities are discussed in section 3; in particular, we show there that one cannot construct a model o f this type, in which the electrons are allowed to pass from one halfspace to the other, with Dirichlet condition.
Starting from the Hamiltonian in halfspace, we construct a four-parameter family of self-adjoint operators H u which may be used as Hamiltonians for a (spinless) electron moving on the described configuration manifold; each of them is characteri- zed by suitable singular boundary conditions at the point connecting the halfspaces.
Among the admissible Hamiltonians, there is a physically interesting two-parameter family containing the operators commuting with the modified parity operator which exchanges the halfspaces. In section 6, we calculate the transmission coefficient for each of the operators H r .
The second extreme is represented by the situation when the film is very thin so that one can set d -- 0. The configuration manifold can be modelled in this case by two planes connected at one point. This problem has been discussed in ref. [17];
in section 7, we rewrite the boundary conditions obtained there in a more convenient way and calculate the transmission coefficient. Since this yields easily the current- voltage characteristics, we are able to compare results for the two situations. This is
1096 Czech. J. Phys. B 38 (1988)
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done in the concluding section; the result is that for thin films, the observed deviations from the Ohms law are (at least, in part) a global quantum (or geometrical)effect rather than a consequence of the electron-phonon interaction.
2. THE OPERATOR h o
Consider first the Laplacian in a halfspace specified by the Neumann boundary condition, i.e., the operator h on L2(~ 2 • R+) defined by
' h f = - A f (2.1a)
with
(2.1b) D(h) = { f ~ L2(n z x r 2 • n + ) in the sense of distributions,
Czech. J. Phys. B 38 (1988) 1097
S(x)l.,:o -- o}.
~x3
Points o f R 2 x R+ are denoted as x = ( x l , x2, xs) w i t h x l ; x2 ~ R2 and x 3 ~ R+.
The operator h is self-adjoint. Our construction w i l l start from its non-seffadjoint restriction
(2.2) h o = h ~" Do,
D o = { f ~ D ( h ) : f ( x ) = 0 for x of some neighborhood of the point 0} . Let us look for the deficiency indices of ho. It is useful to work in the spherical co- ordinates with the center at 0,
xx = r s i n 0 c o s ~ o ,
(2.3) x2 = r sin ,9 sin ~o,
X 3 = r C O S ,9
where r ~ R+, ~o ~ [0, 2n) and ,9 ~ [0, 89 The Hilbert space decomposes con- ventionally as
(2.4) L2(n 2 • t~+) = L2(t~+, r 2 dr) | Lz(S~ ) , dO)
where S(+ z) is the halfsphere or unit radius and dO the rotationally invariant measure on it. The operator h can be rewritten in the coordinates (2.3) as
(2.5a) h f = - ( 1 / r 2 ) ~ r ,.2 ~ f + (l/r 2) Af
where A is the modified squared angular momentum operator defined by the differen- tial expression
(2.5b) A = sinl ,9 ~,9~3 (sin `9 a ~ ) - (l/sin 2 0) ~q 9za---~2
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together with the b o u n d a r y condition
(2.5c) = o
which follows f r o m the N e u m a n n condition in (2.1b).
Proposition 2.1: A has a purely point spectrum,
Af, m = l ( l + l)f~m, l = 0 , 1 , 2 . . . m = - l , - l + 2 . . . 1 where fl,, = Ylm ~ S(+ 2) with l + m even.
Proof: h i t easy to see that
fln
a r e eigenfunctions o f A. It remains to check that they span L2(S(+2)). Suppose that a function 9 ~ L2(S(+ 2)) fulfils['-/2 ['2. d
(2.6a) Jo sinSdOJo o
for all 1 -- O, 1 . . . . and m = - l , - l + ' 2 . . . . , I. We extend g to the sphere S (2) by 0 0 , ~o) = )'gO, e ) . . . 0 = O = 89
to( - o, < o =<
so 0 is an even function with respect to 0 -~ ~ - 0 (with a possible exception o f a zero- measure set). It holds
Ytm(TZ -- ~9, (p) = (-- 1) t+m Ytm(O, qO, and therefore (2.6a) gives
(2.6b) O dO dq~ 0(0, 4o) Y,,(O, q~) = 0
for I + m even; the same relation holds trivially for l + m odd. Since {Ylm} forms an o r t h o n o r m a l basis in L2(S(2)), it follows that 0 = 0, i.e., g = 0. 9
We denote further by Kt the eigenspaces lin {fin: m = -- l, -- l + 2, ..., l~, then the Hilbert space (2.4) decomposes as
oo
(2.7a) where (2.7b)
and for h we obtain (2.8)
where (2.9a)
L2(~ 2 x N + ) = • L t
1 = 0
Lt = Lz(~+, r 2 dr) | K z
oo
h = E) h (~) | I
/ = 0
d ( r 2 d ) l ( l + 1) h ( ' ) f = - ( 1 / r 2 ) d r \ dr f + r 2 f
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with (2.9b) In the (2.10 / where (2.111
D(h CO) = { f e LZ(n+, r 2 dr): f , f ' ~ A C ( n + ) and ht')f e L2(R+, r 2 dr)}
same way, we get the expansion o f ho:
co
h o = @ h(o ~ | I
1=0
h(o0fis given by the rhs. o f (2.9a) and
D(h(o O) = { f e D(hC~ = 0 on some neighborhood o f 0}.
Hence the self-adjointness problem for h o is reduced to analysis o f the operators h(o 0. They are e.s.a for l > 1 (cf. Theorem X.10 o f [20]), while h(o ~ can be easily seen to have deficiency indices equal to (1, 1) - solution to the deficiency equations will be presented below. Consequently we have
Proposition 2.2: The deficiency indices o f ho are (1, 1). Before using this information we are going to make a short digression concerning a more general type o f boundary conditions.
3. T H E M I X E D B O U N D A R Y C O N D I T I O N S
If we replace the Neumann boundary condition in (2.1b) by the Dirichlet one, the corresponding operator h o will be e.s.a. In order to see this, one has to realize that the modified squared angular momentum operator is specified in this case by the boundary condition f(72/2, rp) = 0 instead o f (2.5c). An argument similar to the p r o o f o f the Proposition 2.1 then shows that it has a pure point spectrum
Aft m = l(l q- 1)ftm,
where now ftm = Y z m ~ S ~ ) with l + m odd. It means that l = 1, 2 . . . . and all terms in the decomposition analogous to (2.10) are e.s.a.
These boundary conditions represent particular cases o f a more general condition o f the mixed type, namely
(3.1a) _ O f(X)lx3= ~ _ = c O)
(~X 3 F
where c is a real number. In spherical coordinates, this condition is independent o f r and looks as follows
(3.1b) a__ f(O, rp)[o=,/z = - c f(n/2, go).
O0
The condition (3.11 can be interpreted as adding a surface interaction, term which is repulsive if c > 0 and attractive for c < 0. The Dirichlet condition corresponds to the case of infinitely strong repulsion, while the Neumann one means that the surface term is absent. Somewhere between them there is a critical point Cn+~ in which
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the repulsion becomes weak enough that the operator h 0 ceases to be e.s.a., i.e., the deficiency indices jump from (0, 0) to (1, 1) - cf. fig. 2. There is another critical point in which the attraction becomes so strong that the particle can collapse into the singularity, i.e., the spectrum of h 0 is unbounded from below.
h o below Dirichtet
. unbounded N e u m o n n h o is e.s.o; /
1 / f ( .
o o I o I
ccott 0 1 Cns a 2
Fig. 2. T h e c r i t i c a l v a l u e s o f t h e b o u n d a r y c o n d i t i o n s (3.1).
The aim of this section is to find the above named critical points. In the same way as above, one can decompose h0 into an orthogonal sum, and reduce therefore the problem to analysis of the operators h~o v)
(3.2a) with (3.2b)
h ( o ~ ) f = - ( 1 / r 2 ) d ( r z d f ) + - - v(v +
1 ) f/.2
= ~ !
D(hr v)) ( f e L2(n+, r z dr). f , f e AC(n+),f(r)
= 0 in some neighborhood of 0 andh~oV)f ~
L2(~ +, r 2 dr)}.The numbers
v(v +
1) are obtained by solving the eigenvalue problem(3.3)
Aft, = v(v +
1)f~,,for the modified squared angular momentum operator, which is now defined by the expression (2.5b) together with the boundary condition (3.1b) (at the same time one must demand, of course, the functions f~,, to be regular at the pole (0 = 0)).
Conventionally, we substitute
f~m(0, ~0) = g(cos 8) e imp~
for m = 0, _ 1, + 2 . . . . obtaining in this way the eigenvalue problem for the ordinary differential operator L,,:
B,,/2
(3.4a) L,,g = (z 2 - 1 ) g " +
2zg'+ 1 - z -~ g'
i.e., the Legendre equation, with the boundary conditions
(3.4) 9 regular at z = 1,
g'(0) = e g ( 0 ) .
In distinction to the particular cases c = O, ~ , the solution to (3.3) need not generally exhibit degeneracy with respect to m; only the degeneracy with respect to the sign
1100 Czech. J. Phys. B 38 (1988)
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of m persists. Hence for each rn = 0, -+ 1, _+2 . . . . we have a sequence of values v(v + 1) which obey (3.3). The equation
L,,g = v(v + 1) g
is solved by the first-order Legendre function g(z) = P~; the solution Q~ is excluded by the first of the conditions (3.4b) - el. [21], 8.711.4. The second condition leads to the equation
(3.5) 2 tan [89 + m)] = c F(89 + 89 + 89 F(89 - 89 + 89 F(89 + 89 + 1) F(89 -- 89 + 1) (cf. [22], 8.6); solving it for v one obtains the eigenvalues o f (3.3).
Fortunately, it is not necessary to perform the task in the general setting. As we have mentioned, the operator h(o ") is e.s.a, iff v(v + 1) > 88 It is also well-known (see, e.g., [23]) that h(o ") is bounded from below if v(v + 1) > - 88 Hence the lowest eigenvalue in (3.3) is decisive for the both of our problems. It is an easy consequence of the minimax principle ([20], Theorem XIII.1) that the ground state of L,, is not lower than that of L0, i.e., inf a(Lo) < inf a(Lm) for each m. It is therefore sufficient, to solve the equation (3.5) for m = 0 when it acquires the form
(3.6a) 2 tan (89 = c F(89 + 89
r(89 + 1) 2.
The eigenvalues v(v + 1) have to be real, so v is either real belonging to [ - 8 9 ~ ) or v = - 8 9 + ifl with f l e •. The r.h.s, of (3.6a) is a monotonous function of v in [ - 8 9 oo) (decreasing for c > 0); we denote it as c V(v). It means that a solution in ( - 8 9 89 which corresponds to v(v + 1) e ( - 8 8 88 exists if c F(v) ~ ( - 1 , 1). On the other hand, for v = - 8 9 + ifl the eq. (3.6a) can be after simple manipulations re- written in the form
(3.6b) ch
-c 21r(88 + i/3)1-"
It is clearly sufficient to consider 13 => 0. The r.h.s, of (3.6b) which we denote as - c G(fl) is increasing for c < 0, but not so fast as ch (nil); its asymptotic for large/3 is
G(/3)
- 88
e + o ( / 3 - I ) ] .Hence (3.6b) has a solution if - c G(0) > 1; one can check that ch (nil) > G(/3)/G(O) for all/3 # 0 so there is no solution if c G(0) < 1.
The above considerations allow us to calculate the sought critical values: we obtain
(F(S)~Z = 1"09422 ...
(3.7a) Cn~a = 2 \ F ~ ]
and
(3.7b) Coon = - 2 ( F ( 8 8 z = -0"22847...
\r( 88
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4. T H E A D M I S S I B L E H A M I L T O N I A N S
As mentioned in the introduction, our model consists of a free spinless particle moving in two halfspaces (with Neumann boundary conditions) which are connected at one point. The state Hilbert space of such a system is
(4.1) ~ = LZ(~ 2 x ~+) | Lz(R 2 x R+).
Since the particle is assumed to be free everywhere except at the point singularity.
a suitable starting operator for constructing the Hamiltonian is
(4.2) Ho = Ho,1 0 Ho,2,
where Ho, j = h o for j = 1,2. According to Proposition 2.2, the deficiency indices of H o are (2, 2) so there is a four-parameter family of self-adjoint extensions H v specified by 2 • 2 unitary matrices U. The decomposition (2.10) shows that each extension is of the form
cO o 0
(4.3) Hv = Av @ ( 9 h~o')x | I) 9 ( 9 h~o'~2 | I)
l = l l = l
where h(o~)y = h~o ~) for j = 1, 2 (it makes no difference how we number the halfspaces) and Av is some self-adjoint extension of the operator A0
(4.4a) Ao =,(h(~174 I) @ v,o,2tt'(~ | I) which acts on the Hilbert space
(4.4b) ~r o = (L2(~+, r 2 dr) | Ko) @ (LZ(R+, r z dr) | Ko).
Here K o is the one-dimensional subspace in L2(S~ )) spanned by the constant function fo,o; the angular part is therefore trivial and we shall drop it out in the following.
The adjoint operator A* is given by the same differential expression as Ao (cf.
[20], appendix to sec. X.1). so the deficiency subspaces K (+) = Ker (A* -T- iI) can be found easily: they are spanned by the functions {~0~ +), ~0~z +)} defined as
( 0o) ,
(4.5a) where (4.5b) and
(4"5c)
fo(r) = ( 1 / r ) e x p ( - ~ r ) , e = exp(i88
~o}-)= ~0J +), j = 1 , 2 . According to the yon Neumann theory, we have
(4.6a) A v c A*
and every extension is specified by its domain
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(4.6b)
D(Au) = Du = { f = ~p +
Cl((~(1 +) -~- u11~(1 -) -~- Ul2(P (-)) -~+ c2(e + u . e l - ) + c . c2 c , where ujk are the matrix elements of U.
N o t every extension is interesting, however. It is reasonable to suppose that the two halfspaces are physically equivalent, i.e., to require the extensions to commute with the modified parity operator P which is defined on ~ by
(4.7) p f f l ~ = ( f ~ )
\f2,#
9In the following, we restrict therefore our attention to such extensions
Av
which fulfil(4.8a)
PoAv c A•P o
where Po = P ~" ~ o , or more explicitely, PoDv c
Du
andPoAd = AuPof
forf e D U.
An inspection of the relations (4.6) shows that it is equivalent to (4.85)
ull
= / ' / 2 2 ' 1/12 = U 2 1 'since Pogo~ +) = go<2 +) and Po.4o c .4oP o. The family of extensions selected in this way is therefore two-parametric and its elements can be specified by the matrices (4.9)
U = e ' ~ ( cOsfl'i
sin fl, i sin f l ) c o swith ~, fl s [0, 2~z).
5. C H A R A C T E R I Z A T I O N O F T H E E X T E N S I O N S T H R O U G H B O U N D A R Y C O N D I T I O N S
According to (4.6a), every extension acts as the differential operator (2.9a) with l = 0 on each component of the wave function. The expression (4.6b) of the domain
Dr,
however, is not very suitable for practical calculations; one would prefer rather to have some boundary conditions in the connection point. Since the deficiency functions (4.5) are singular there, one must introduce regularized boundary values similarly as in [4, 16, 17]:(5.1a)
Lo(f)
= lim r f ( r ) ,r'} 0
(5.1b)
Ll(f)
= lira I f ( r ) - (l/r) L0(f)] 9r"~ 0
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Before proceeding further, we shall make one more restriction in the class of conside- red extensions. If the matrix U is diagonal, then one can see directly from (4.3) and (4.6b) together with the von Neumann formula for A v f that H v is reduced by the projections Ej on the subspaces of ~ referring to the two halfspaces; the situation is completely analogous to what happens for a diagonal U in the models treated in [16, 17]. In such a case, motion in the two parts of the configuration manifold is separated; the particle cannot pass through the singularity to the other halfspace.
Since this situation is not interesting from the viewpoint of modelling a thin-film contact, we restrict our attention in the following to the operators H v corresponding to non-diagonal matrices U only.
Proposition 5.1: Each operator H v which is not reduced by the projections Ej is of the form (4.3), where the operator A v c A* is characterized uniquely by its domain specified by the following requirements:
(fa) withfx,fj~A C(R+) and he~ dr), j = 1,2,
(i)
f = S2
(ii) the function Sj fulfil the following boundary conditions at r = 0:
(5.2) Lo(f,) =
aLo(f2 ) + b Li(f2),
L i ( f , ) = c L0(f2 ) + d Ll(f2)
where the coefficients a, b, c, d are related to the elements of a non-diagonal 2 x 2 unitary matrix U by
(5.3a) a =
2-1/2~u121(i
-t- iult -- //22 -- det U), (5.3b) b = 2 -1/2 iu721(1 + tr U + det U).(5.3c) c = 2-'/2u~-2~( - 1 - i tr U + det U), (5.3d) d = 2-1/2gu~-21(-i + u l , - iu22 + det U).
In particular, for the extensions commuting with the operator (4.7) we have (5.4a) a = - d = (cos/7 + cos r - sin r
(5.4b) b = 2i/2(cos/7 + cos r (5.4c) c = 21/2(sin r -- cos/7)/sin fl
where the parameters/7 + 0 and ~ refer to the matrix (4.9).
Proof: Each ~0 e D(,4o) fulfils the condition (i), and the same is true for any linear combination of the deficiency functions. In view of (4.6b) and (5.1), the conditions (5.2) yield the following equations for the coefficients a, b, c, d:
(5.5) 1 + ual = aul2 - b e u l 2 ,
u21 = a(1 + u 2 2 ) - b(g + CUe2),
] ] 0 4 Czech. J. Phys. B 38 (1988)
P. E x n e r , P. ~eba: A m o d e l o f p o i n t c o n t a c t . . .
- - ~ - - f U l l = c u 1 2 - - d e u 1 2 ,
- - ~U21 : C(I --~ /'/22) - - d ( ~ + Eu22 ) .
Solving it we obtain (5.3), and substituting for U from (4.9), we arrive at the relations (5.4). It remains to check that the map U ~-, {a, b, c, d} is injective. Suppose that two matrices U, U' lead to the same values of the coefficients. The relations (5.3) then give
(5.6) i + iu~l - u22 - det U' = ~(i + i u l , - u22 - Oet U ) , l + t r U ' + d e t U ' = o ~ ( 1 + t r U + d e t U ) ,
- 1 - i t r U ' + d e t U ' = ~ ( - 1 - i t r U + d e t U ) ,
- i + u ~ - iu~2 + det U' = o~(-i + u l l - iu22 ~- det U ) ,
! .
where ~ -- u12/u12, this number is non-zero by assumption. Taking suitable linear combinations of these relations, we get
t ?
(5.7a) ul~ - u22 = ~(u11 - u22),
(5.7a) 22/2 + e tr U' = ~(21/2 + e tr U ) , (5.7c) e + g det U' = ~(e + g det U ) ,
(5.7d) 1 + u~t = ct(1 + u~l).
The first two of them give the relation 1 - i + 2u~1 = ~(1 - i + 2ull ), which is compatible with (5.7b) only if ~ = 1. Hence we have u~2 = u12, and using (5.7a, b) again, we get u~.j = ujj. Finally combining these results with (5.7c), we obtain
I
/ ' / 2 2 = U 2 1 " 9
6. T H E T R A N S M I S S I O N C O E F F I C I E N T
Let us now examine the situation when the particle passes from one halfspace to the other one through the point singularity; our aim is to find probability per unit time o f the transmission. Similarly as in [ 1 6 - 1 8 ] , we use the time-independent approach since it makes the problem more manageable. We start with the function
(;:) w ore
(6.1) f l ( r ) = 1_ e-ikr + A 1_ eikr '
r r
f2(r) = ! elk,
r
and demand it to belong locally to D(Av). Clearly f j, f j ~ AC(R+ ) and h(~ i = k2f j belongs to L2oc(R+, r 2 dr). The boundary conditions (5.2) applied to (6.1) yield simple equations which are solved by
Czech. J. Phys. a 38 (W88) 1105
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(6.2a) A ( k ) = ik(a + d) + c - bk 2 ik(a - d ) - c - b k 2 and
(6.2b) B ( k ) = 2ik
ik(a - d ) - c - bk 2
We are interested in the extensions c o m m u t i n g with the o p e r a t o r (4.7), then the last relations reduce in the parametrization (4.9) to the f o r m
sin 4 - c o s / / - k2(cos 4 + cos fl)
(6.3a) A ( k ) = i 2 1 / Z k ( c o s / / + cos 4 - sin 4) - sin 4 + cos fl - k2(cos r + cos f l ) ' (6.3b) B ( k ) = i 2 ' / 2 k sin fl
i21/2k(cos fl + cos 4 - sin 4) - sin ~ + cos fl - k2(cos 4 + c o s f l ) "
It is easy to see that these coefficients fulfil
(6.4) Ia(k)l ~ + IB(k)]2 - - 1.
Let us ask n o w how the transmission coefficient can be derived f r o m here 9 The S- matrix a p p r o a c h is not applicable, at least without a m o r e sophisticated formulation, since we have no free H a m i l t o n i a n to c o m p a r e (it is w o r t h mentioning, however, that a formal calculation using the " p a r t i a l wave d e c o m p o s i t i o n " (4.3) yields So(k) =
= - A ( k ) leading to 1 - [ S o ( k ) l 2 = In(k)l ). Instead we shall calculate the prob- ability current t h r o u g h the halfspheres o f radius R in the two halfspaces. One obtains (6.5a) jx(R) = - r c R 2
Im I.fl(r ) ~0 r
f l ( r ) l , = R = r e k ( 1 - l A ( k ) l 2)where we set h = 2m = 1 and the sign is chosen to correspond to the incoming probability current, and similarly
(6.5b) j 2 ( R ) = ~klB(k)l 2 9
These quantities do not depend on R, so it is natural to interpret them as the prob- ability current t h r o u g h the connection point. T h e transmission coefficient at the energy E = k 2 is therefore equal to
(6.6)
T(E)
= IB(k)? =2k 2 sin 2 fl
2k2(cos fl + cos 4 - sin r + (sin 4 - cos fl + kZ(cos 4 + cos fl))2 It is easy to see that there are two extensions (in the " p a r i t y preserving" class), namely those corresponding to 4 = 88 and fl = 88 7 ~rc, which yield full transmission l a ( k ) l 2 = 1. These are also the only cases in which the transmission coefficient is energy independent. Notice that the second named extension (with fl = 88 corre-
] ].06 Czech. J. Phys. B 38 (1988)
P. Exner, P. ~eba: A model o f point contact ...
sponds to the situation when the wavefunctions are "glued together" in such a way that the regularized function r ~ r f ( r ) is continuous at r = 0 together with its first derivative.
7. T H E C A S E O F T W O P L A N E S
In this section we are going to discuss the second extreme case mentioned in the introduction, a free particle moving on two planes connected in one point. This situation has been analyzed in [17], and our aim here is to rephrase the results in a more suitable way. Recall that the boundary conditions can be written in terms o f the regularized boundary values
(7.1) Lo(f) = lira f(r)
r-~0 In r
L l ( f ) = lim I f ( r ) -- Lo(f)In r ] .
r'-*0
The admissible Hamiltonians are o f the form H u = Av E) h, where h is a self- adjoint operator analogous to the second and third term in (4.3) (cf. Proposition 2 of [17]) and A v a selfadjoint operator on L2(•+, r dr) 9 L2(R+, r dr) which acts
a s
(7.2) Av(fl~ =(-(llr)~r(r~rfl) I.
f2] t_O/r) ~ (r ~ f2)]
The domain o f Av consists o f the functions f with fs, fJ e AC(R+) and A v f s e e L2(R+, r dr), which are (for a given U) specified by suitable boundary conditions.
Proposition 7.1: The boundary conditions formulated in proposition 3 o f [17] can be for a non-diagonal U replaced by
(7.3) Lo(fl) = a Lo(f2) + b Ll(f2)
Ll(fl)
= cLo(f2 )
+ dLi(f2)
where the coefficients a, b, c, d are related to the matrix elements of U as follows (7.4a) a = u?21[Z(ual - 1) + ~(det U - u22)]
(7.4b) b = 2 i . u~_21[1 _ tr U + det U]
7~
ni - 1 2 X-2
(7.4c) c = - - . U l z [ Z + z ~ t r U + d e t U ] 2
Czech. J. Phys. B 38 (1988) ] 107
P. Exner, P. Seba: A m o d e l o f p o i n t contact . . .
(7.4d) d = u~-~[X(1 - uz2) + ~(Ull - det U)]
where Z = 89 + (2i/zc)(y - In 2) and 7 = 0.577216 ... is the Euler's constant.
Proof can be performed in a complete analogy with that of Proposition 5.I or of Proposition 3 of [17]. Alternatively, one can write down and solve the linear relations between the boundary conditions (7.3) and those of [17]. I i
Like above, we are interested primarily in the extensions which commute with the modified parity operator which is defined on the state space of the present problem again by the relation (4.7). The admissible matrices U are then of the form (4.9}
and the coefficients (7.4) can be rewritten as
(7.5a) a = - d = (1/sin fl) [sin r + (4/~) (~ - In 2)(cos fl - cos r (7.5b) b = (4/~ sin fl) (cos r - cos fl),
(7.5c)
c = (~/2 sin fl) [(a/n) (7 - In 2) sin { + (89 - (8/re 2) (~ - In 2) 2 (cos { + cos fl)].
(,,)
Now we want to calculate the transmission coefficient. We start with f = f2 ' 9 where
(7.6a) Sl(r) = Hg (kr) +
(7.6b) f2(r) = B(k) HCot)(kr)
and demand it to belong locally to D(Av). A simple calculation yields expressions for A, B which reduce in the "parity preserving" case to the form
c - 2 a 7 + l n - b + y + l n
(7.7a) A(k) =
c - 2 a r c + y + l n - b 7z + ? + l n --ire
(7.7b) B(k) =
c - - 2 a = + 7 + l n - b r~ + 7 + l n
It can be seen easily that they fulfil the relation (6.4). Then one can calculate the incoming and outgoing probability currents through the circle of radius R in the first (second) plane; it holds
(7.8)
j,(R)
= 4(1 - I A ( k ) ] 2 ) ,A(R)
-- 4 l ~ ( k ) l = .Notice that if we use instead the ansatz of [17], i.e., if the first term on the r.h.s, o f (7.6a) is replaced by Jo(kr), then j r ( R ) = 1 - I 1 +
2A(k)l =
=1 -Is0(~)l =,
where So(k) is the s-wave scattering matrix. The relations (7.8) show that the transmission1108 Czech. J. Phys. B 38 (1988)
P. Exner, P. ~eba: A model o f point contact ...
coefficient is equal to
(7.9) T(k 2) = IB(k)[ 2 ,
In distinction to the case o f two halfspaces, no extension provides us with an energy- independent transmission coefficient. There is again one extension, namely the one corresponding to the matrix (4.9) with ~ = fl = - a r c c t g ( ( 4 / ~ ) ( Y - In 2b), such that the wave functions are joined "continuously" at r = 0, as well as their "deriva- tives", but even in this case the transmission probability depends on energy,
T(k 2) = [1 + (4/~2) (y + In (k/2))2] - ' .
8. CONCLUSIONS
With the knowledge o f the transmission coefficient
T(E)
as a function o f energy, we are able to calculate the current-voltage characteristics. I f the metals involved have the same Fermi energy, then the current is given by [24](8.1a)
I = - 2__eeh f ~176
[fT(E) - - f T ( E-- eU)] dE,
where e is the electron charge, U is the applied voltage, and
is the electron-gas density at the temperature T and Fermi energy E v. The results are particularly simple in the zero-temperature limit. The differential resistance, for instance, is then given by
dU h
(8.2)
- [T(E v +
e U ) ] - ' ;dI 2e
this formula is of interest because
dU/dI
is a measured quantity, and the measure- ments are usually performed at temperature o f few K. To be just, we must mention that the formula (8.1) has been challenged, however, the alternative proposed in [25]differs by an additive constant, which is unimportant for the arguments presented below.
Let us turn now to the two extreme situations mentioned in the introduction.
In the case, when the films are very thick, which can be modelled by two haVspaces, we found two admissible Hamiltonians which yielded a transmission coefficient which is energy-independent, i.e., which gives a linear relation between the current and the applied voltage. As we have remarked, the problem of choosing the right self-adjoint extension remains open. What is important, however, is that there are extensions leading to the Ohm's law in this simple model.
Czech. J. Phys, B 38 (1988) ] ] 0 9
P. Exner, P. Seba: A model o f point contact ...
T h e s i t u a t i o n is e n t i r e l y different i f the films a r e very thin. T h e a n a l y s i s o f section 7 shows t h a t the t r a n s m i s s i o n coefficient is e n e r g y - d e p e n d e n t in this case for every extension. T h i s is in c o n t r a s t w i t h t h e u s u a l e x p l a n a t i o n [-15] o f the n o n l i n e a r s h a p e o f c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s f o r p o i n t c o n t a c t s as a result o f e l e c t r o n - p h o n o n i n t e r a c t i o n . O u r m o d e l shows t h a t O h m ' s l a w s h o u l d be e x p e c t e d t o b e v i o l a t e d for sufficiently t h i n films i n d e p e n d e n t l y o f a n y i n t e r a c t i o n .
Received 25 August 1987
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