- Email: [email protected]
Tunneling and super-barrier transmission through a system of two real potential barriers
Nuclear Physics A147 (1970) 627- 649; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or mlcrofilmwithout written permissionfrom the publisher
T U N N E L I N G A N D SUPER-BARRIER T R A N S M I S S I O N T H R O U G H A S Y S T E M OF T W O REAL P O T E N T I A L BARRIERS NANNY FROMAN and ~3RJAN DAMMERT Department of Theoretical Physics, University of Uppsala, Uppsala, Sweden Received 18 December 1969
Abstract: The olle-dimensional quantal problem of tunneling and/or super-barrier transmission through a system of two potential barriers is treated rigorously by means of the higher-order phase-integral approximations introduced in a previous paper by one of the authors. These higher-order phase-integral approximations are closely related to the higher-order JWKB approximations but have great advantages compared to the latter approximations. General formulae for the transmission coefficient of a smooth double-barrier are given, pertinent to an arbitrary order of the approximations used. Upper bounds for the terms neglected in obtaining approximate formulae are derived. The case of resonant transparency is particularly considered. Approximate formulae for the wave function on tile real axis, except in the vicinity of the transition points, are also given. In the last section the final formula for the transmission coefficient in the first-order approximation is given with all the notations explained, for the benefit of those interested in this particular result only.
1. Introduction O n e - d i m e n s i o n a l p o t e n t i a l fields with two or m o r e p o t e n t i a l barriers s e p a r a t e d by p o t e n t i a l wells are met with in several currently used m o d e l s o f q u a n t a l systems. A t r e a t m e n t o f the p r o b l e m o f two barriers s e p a r a t e d by a well can be f o u n d in the textb o o k on q u a n t u m mechanics by B o h m 1), who uses the first-order J W K B a p p r o x i m a tion a n d o b t a i n s a n a p p r o x i m a t e expression for the t r a n s m i s s i o n coefficient for p a r t i cles with energy well b e l o w the p e a k s o f the barriers, which he assumes to be o f equal shape. The c o r r e s p o n d i n g f o r m u l a for the t r a n s m i s s i o n coefficient o f two u n e q u a l barriers * was given in a p a p e r by I o g a n s e n 2) a n d used b y h i m for treating the p r o b l e m o f resonance t r a n s m i s s i o n o f electrons t h r o u g h thin dielectric layers in a c o n d u c t o r 2, 3). The same f o r m u l a was also derived b y Child 4) a n d used for investigating the consequences o f a d i p in the activation b a r r i e r for certain chemical reactions. A s s u m i n g the m a x i m a o f the barriers to be r o u g h l y p a r a b o l i c , Child 4) also derived a f o r m u l a for the t r a n s m i s s i o n coefficient w h e n the energy a p p r o a c h e s o r exceeds the p e a k s o f the barriers. This case was e x a m i n e d in greater detail by C o n n o r 5), who i n c l u d e d certain p h a s e factors which were o m i t t e d by Child b u t are o f i m p o r t a n c e for energies n e a r to the b a r r i e r m a x i m a . P o n o m a r e v 6) used Z w a a n ' s 7) m e t h o d for h a n d l i n g the firsto r d e r J W K B a p p r o x i m a t i o n a n d o b t a i n e d a f o r m u l a for the t r a n s m i s s i o n coefficient which is correct for energies well below the p e a k s o f the barriers, b u t is n o t quite g o o d t According to a footnote in ref. 2) the formula was obtained by M. Ya. Azbel. 627
628
N. FRt~MAN AND O. DAMMERT
for energies near to either of the barrier maxima, since the above mentioned phase factors are lacking. Recently, through the work by Strutinsky s) and Nilsson et al. 9), the two-hump barrier has become of great interest in the theory of fission. In that context Ignatyuk et al. ~o) derived a formula for the penetrability, valid for energies far below the peaks of the barriers. This formula, as well as the corresponding formulae in refs. ~- 4), contains insignificant terms, originating from an inadequate use of the one-directional connection formulae of the conventional JWKB method. After such terms have been deleted the formulae in question agree with Ponomarev's 6) formula adapted to the case of particle energies far below the barrier maxima. In the present paper we shall give a rigorous treatment of the double-barrier problem based on the higher-order phase-integral approximations derived by one of us in a previous paper ~1), where also tile method for handling the connection problems for these approximations is outlined. The use of the higher-order phase-integral approximations improves the accuracy considerably, the first-order JWKB approximation being for many purposes rather unsatisfactory. An exact formula for the transmission coefficient of the double-barrier will be given. From this exact formula we shall obtain approximate formulae valid for any order of the phase-integral approximations and applicable to all possible situations of tunneling and/or superbarrier transmission. Upper bounds for the errors involved in obtaining the approximate formulae will also be given. The formulae given by Connor s) and Ponomarev 6) are obtainable as special cases of our results (see sect. 5 below).
2. The phase-integral approximations The higher-order phase-integral approximations introduced in reL 1~) have been used in a previous paper 12) by one of us for investigating the energy levels of doublewell potentials. Since the present problem will be treated along similar lines, the parts of ref. 12), which contain general theory and explanatory comments on the application of the methods developed in ref. 1~), form the basis for the present paper as well and will not be repeated here. Thus sects. 2 and 4 ofreL 12) are entirely pertinent to our present problem, and their content will be presupposed in the treatment below. We also refer to sect. 3 of ref. 12) and to refs. 13, 14) for the description of the Riemann surface and the contours of integration for obtaining the quantity w(x), which enters into our formulae. For details on the method for mastering the connection problems we refer to refs. 15, ~6). Thus we shall directly write down the expressions for the two linearly independent functions, being approximate solutions of the Schr6dinger equation and used on constructing the exact solution of the Schr6dinger equation [cf. sect. 2 of ref. ~2)], on which the treatment in the present paper is based. These functions have the following form
fl(z) = q-X~(z) exp { + iw(z)),
(la)
J2(z) = q-~(z) exp {-- iw(z)},
(lb)
T U N N E L I N G AND TRANSMISSION
629
where z
w(z)=
f q(z)dz,
(2)
N
q(z) =
Q(z) Z Yz,(z) •
(3)
n=0
Here
QZ(z) =
2m
~ - I-E- V(z)],
(4)
where E is the energy of a particle of mass m moving in the potential field V(x). The functions Yz,(z), n > 0, are considered to be small compared to Yo(z), which is equal to unity. Analytic expressions for the first few of the functions Yz,(z) can be found in ref. 14). Let Xl, x2, x3 and x4 be points on the real axis located as shown in figs. 1-4. It is convenient to introduce the following real quantities
Wl(Xl)
= Re
½frq(z)dz,
(negative)
(5a)
q(z)dz,
(negative)
(5b)
(positive)
(5c)
(positive)
(5d)
Wz(X3) = Re½~ '3/`3
q(z)dz,
w3(x3) = Re ½~ Fa
w4(xs) = Re ½~
- -
FL
q(z)dz,
J/'5
and, if classically forbidden regions exist, the following purely imaginary quantities
q(z)dz,
wl(x2) = ½f/"
(5e)
2
q(z)dz,
w2(x2) = ½ f r2 -
w3(x4)
=
(5f)
rK1
½fr q(z)dz,
(5g)
4
q(z)dz.
w4(x4) = ½ ( F4--r~
(5h)
2
The contours of integration to be used in (5a-h) lie on a two-sheet Riemann surface and are shown in figs. 1-4. In the notations introduced above a subscript on w is used to indicate that the integration contour encloses the turning point (tl, t2, t 3 or t4) with the same subscript.
630
N, FR()MAN AND O. DAMMERT
V(xl-E
(a)
x, :
F~
(b)
~
'-._.~
I
I
I
t2
t3
tz,
i
I ......
I
t2
t3
t4
tl
Ct I
X2 m ~ -i ~
(c)
X3 C :
•
' • ' i --:-"
(d} '
(e)
(f)
tl
q2 :jq]~
~KI
t2
//," q2:~ Z- qi~ / /
t3
q~:.jlql~
t4
/ q~:e-IT qi ~- -'// q~:.lqi~
(g)
Fig. I. Both barriers overdense, a Qualitative b e h a v i o u r o f V(x)--E. b-f C o n t o u r s o f integration for obtaining w(x). D a s h e d lines indicate parts o f the c o n t o u r s lying on the second R i e m a n n sheet. g Phase o f q~(z) on the real axis o f the first R i e m a n n sheet.
"
gg9
~.
~'~
~.o~ ~'~ S~
0
2
~J
/
x
=-n
_7 /
--o
X
~"
x
m
632
N. I~ROMANAND (). DAMMERT V(xI-E
XI
~I
I
t
~
X3 ~
X
X5
{a)
t~ (b)
"--'"
t2
tl
t3
t4 ,c)
C" .: tI
F'2
t2 t3
t4 (d)
,, r.,
?~
G
t3
{e)
tl
FKI
" .
.
.
.
, t3
K2
t4
Fig. 3. The left-hand barrier overdense, the right-hand barrier underdense, a Qualitative behaviour o f V(x)--E. b-e C o n t o u r s o f integration for obtaining w(x). Dashed lines indicate parts of the contours lying on the second Riemann sheet, f P h a s e o f q½(z)on the real axis o f the first Riemann sheet.
TUNNELING AND TRANSMISSION
633
V(x)-E (a}
.
.
.
.
.
.
:
=X
/ '(2 (b) t3
t4
I
,,,I
t3
t4
t! %2 (c)
~
3
7K I ~ tl
t2
~ ill
........
r K , ~
: .....
x4
(d) t~
"T; a
tt
t2 (e)
tl
t2 I
~
~
I
1
!
3it
1
y
J
Fig. 4. T h e left-hand u n d e r d e n s e , the r i g h t - h a n d barrier overdense, a Qualitative b e h a v i o u r o f V(x) --E. b-e C o n t o u r s o f integration for o b t a i n i n g w(x). D a s h e d lines indicate p a r t s o f the c o n t o u r s lying o n the s e c o n d R i e m a n n sheet, f P h a s e o f q~(z) o n t h e real axis o f t h e first R i e m a n n sheet.
634
N . F R O M A N A N D (). DAMiVIERT
It is also convenient to introduce the following notations
K 1 =- ½i~ q(z)dz,
(6a)
,g F K 1
K2 =--½i(
q(z)dz,
(6b)
•] F K 2
q(z)dz,
L----Re½~
(positive)
(7)
•] F L
where the paths of integration are the closed loops depicted in figs. 1-4. The quantities K1, K2 and L are all real. In fact, Kt and Kz are positive in the case of tunneling (subbarrier penetration) but negative in the case of super-barrier transmission, whereas L is always positive. In the case of sub-barrier penetration for both barriers the integral in (7) is real, and the symbol Re on the r.h.s, of (7) can be left out. In the case of superbarrier transmission for both barriers the closed-loop integral occurring in (7) can be written
-½ f rrq(z)dz = L +½i(Kt + K2).
(8)
It can be shown [cf. ref. 14)] that certain terms in the expressions for Y2,(z) do not contribute to the values of the quantities K1, Ka and L, defined by eqs. (6a, b) and (7) and entering into the final formula for the transmission coefficient. This fact considerably economizes the numerical work required for calculating the transmission coefficient by means of our final formula. Since q(z) is an analytic function on the Riemann surface under consideration, the contours of integration in eqs. (5), (6) and (7), may of course be deformed in a way that is convenient for performing numerical calculations. The contours of integration in figs. 1-4 are chosen so as to give a close analogy between all the possible cases of sub-bariier and/or super-barrier transmission. The formulae (5), (6) and (7) are the same in all cases. Only the contours of integration are different. If we imagine the vertical cuts in the super-barrier case to be rotated in the clockwise direction until they coincide with the real axis, we obtain the same scheme of integration contours as in the sub-barrier case. When super-barrier transmission occurs for one or both of the barriers, one can also use other paths of integration, as has been done in the above-mentioned work 12) on the double-oscillator. One should then let the cut between two complex conjugate transition points proceed along the Stokes' line t connecting these points. We then t For a symmetric barrier this Stokes' line coincides with the straight line connecting the two complex conjugate transition points.
TUNNELING AND TRANSMISSION
635
realize that the formulae (5a, b, c, d) and (7) can also be written as follows
wl(xl) = fs[1_oq(x)dx,
(9a)
q(x)dx,
(9b)
= f~[~_oq(x)dx,
(9c)
f[~ q(x)dx,
(9d)
wz(xa) =
wa(x3)
w4(xs) =
f f 1+0
2+0
/'s2 - o
L = -- Js~+o
q(x)dx,
(10)
tltz
where s a and s2 are the points on the real axis where the Stokes' lines and t3 t4, respectively, cross the real axis. Cf. fig. 2 in ref. 12). We shall now turn our attention to the function defined by (2). It is convenient to express where x is real, in terms of the quantities defined by (5a-h). With a convenient choice t of the lowex limit of integration in the integral (2) defining we have
w(z)
w(x),
w(z)
W(Xl) = wl(xl),
(lla)
w(x3) = w~(x3)-izq = w3(x~)-iX~-L,
(1tb)
w(xs) = w , ( x s ) - i(~q - X 2 ) - ~
01c)
and, if classically forbidden regions exist, w(x2) = wl(x2) = w2(x2)- iIq,
(lld)
w(x4) = w 3 ( x 4 ) - iKx - L
(lle)
= w4(x4) - i(K1 - K 2 ) - L .
3. E x a c t f o r m u l a e for the t r a n s m i s s i o n and r e f l e x i o n coefficients 3.1. G E N E R A L
CASE
The transmission problem for a single potential barrier has been treated by Fr6man and FrSman 15) in the first-order JWKB approximation and, in greater detail, with the aid o f the higher-order phase-integral approximations 14). Some of the results obtained in these investigations will be utilized in the present treatment of the twobarrier problem. t For the first-order JWKB approximation and the case of sub-barrier penetration through the left-hand barrier this choice amounts to choosing the lower limit of integration in (2) to be the transition point tl.
636
N. FRIDMAN AND (). DAMMERT
We shall make similar assumptions concerning the behaviourof V(x) for large values of Ixl as in chapter 9 of ref. 15). Thus we assume that Q2(x) and hence also q2(x) is positive for sufficiently large values of [xl and that the integral/~(xl, xs) defined by eq. (10) in ref. ~2) converges as xl tends to - c o and xs tends to + ~ . From this it follows similarly as on p. 92 in ref. 1 s) that, if z does not coincide with a singularity or a zero of q2(z), the limits F(-oo, z), F(z, +co), and F ( - c o , + o o ) exist and are finite, which in turn implies that the limits a~( ± oo) exist and are finite, and that f l (x) andf2(x) for increasing values of lxl approach two linearly independent solutions of the Scba'6dinger equation. These solutions may be interpreted as waves travelling in opposite directions. Far to the left as well as far to the right of the doublebarrier f , (x) represents a wave moving from the left to the right. Assuming that there is only an outgoing wave far to the right of the double-barrier, we obtain the following expressions for the transmission coefficient T and the reflexion coefficient R r =
a , ( + o o ) 2exp a,(-- oo) {2(K, - K2)},
(12a)
R=
az(-°°)
(12b)
<(-oo)
2.
Since there is only a transmitted wave to the right of the double-barrier, we have a2(+ co) = 0, and from formula (3.16) in ref. ,5) it follows that
a,(z) = F**(z, + co)at(+ co), az(z) = Fz,(z, + co)at(+ oo).
(13a) (13b)
From the symmetry relations (5.7a, b) in ref. ,5), which can be generalized to the higher-order phase-integral approximations, and eqs. (1 la, c) in the present paper, we obtain the symmetry relations
F2z(x,, xs) = exp {2(K2-K,)}F**(xl, Xs),
(14a)
F,2(x,, Xs) = exp {2(K2- Ka)}F**(Xl, xs).
(14b)
With the aid of (14a, b) the relation det F(x,, xs) = 1 [of. eq. (3.19) in ref. 15)] can be written
[Fzl(x,, Xs)l a = [FI,(x,, xs)l z-exp {2(K,-Kz)}.
(15)
Using (13a, b) and (15), we obtain from (12a, b) the following exact expressions for T and R T = exp {2(K, - Kz)} = exp { - 2 ( K , + K2) } I F , , ( - o o , +oo)1 z I F , l ( - o o , +oo) exp {-2K1}I 2'
(16a)
R = Fza(-°°' +oo) z = [ r l l ( _ o o ' +oo)12_exp {2(K _K2)} = 1 - T . (16b) F , l ( - o o , +oo)
IFl1(-~,
+oo)i z
TUNNELING AND TRANSMISSION
637
To be able to utilize the results obtained for a single barrier in refs. ,4,15), we shall express the denominator in (16a, b) in terms of quantities referring to the two separate barriers. To prepare this and for later reference we shall first consider the matrices F(xl, x3) and F(X3, x s ) . We have chosen the phase of q~(z) and the constant lower limit of integration in the integral (2) defining w(z) such that the matrix F(xl, xa) in the present paper corresponds for the first-order JWKB approximation to the matrix F(xl, x2) treated in subsects. 6.4 and 6.5 ofref. 15) and for the phase-integral approximations of an arbitrary order to the matrix F(xt, x2) in ref. ,4). In accordance with eqs. (6.17a, b) and (6.32a, b) in ref. ~5) the symmetry relations for the matrix F(xl, x3) in the present paper are t
F*2(x,, x3) exp {2K1},
(17a)
F21(x~, x3) = --F*a(Xl , x3) exp {2K1},
(17b)
F l l ( x , , x3) = -
where K, is defined by (6a) and is positive for the case of sub-barrier penetration and negative for super-barrier transmission.. Using (17a, b), we can write the relation det F(xl, x3) = 1 as follows
IF~2(xt, x3)[ 2 =
If22(xx, x3)12-t-exp {-2/£I}.
(18)
From this formula it follows that
F~z(xl, x3)
= [[Fz2(X,, x3)12+exp {-2K1}] ~ exp {i(½zc+2al)},
(19)
where cr~ is a real quantity defined by the formula o-, = ½[arg Fa2(x,, xa)-½7z].
(20)
This definition of at corresponds precisely to the definition (36c) in ref. i2). The symmetry relations for the matrix F(x3, xs) are F22(x3, Xs) = --F*l(x3, xs) exp {2K2},
(21a)
-F~,(xa,
(21b)
F12(x3, xs) =
xs) exp { 2 K z - 4 K , } ,
where K2 is defined by (6b). These symmetry relations can be obtained either directly as described in the previous footnote, or from the symmetry relations (17a, b) by means of the same arguments as will be used in sect. 4 to obtain the estimates of F(xa, xs) from those of F(x,, X3). By means of (21a, b) the relation det F(x3, xs) = 1 can be written as follows IF21(xa, xs) exp {-2/£1}12 -- ]F,~(x3, xs)lZ+exp {-2K2}.
(22)
t The symmetry relations (17a, b) and {21a, b) can be obtained by applying the symmetry relations (5.7a, b) in ref. ,5), generalized to apply to the phase-integral approximations of any order, and using the expressions ( l l a , b, c) for w(x).
638
?4. ]FR{}MAN AND O. DAMMERT
From this formula it follows that
Fz,(X3, xs) = [[Fll(x3, xs)]2+exp {-2K2}] ~ exp {2K1 +i(½~z-2L+2a2)}, (23) where 0"2 is a real quantity defined by 0-2 = ½[arg Fa 1(x3, xs) - ½n + 2L].
(24)
Although 0.1 and a2 are analogous quantities for the two barriers, the analytical expression for 0"2 is different from that for a 1 due to the fact that in the expression for w(x) the transition point tl has a preferred role compared to the transition point t3 (cf. the footnote on p. 635). The values of 0.1 and 0.2 depend only very slightly on the positions of the points xl, x3, and xs, provided no one of these points lies too close to a transition point. If, in particular, the two barriers are of approximately equal shape, we have 0-1 ~ 0-2. For a symmetric double-barrier we can achieve that 0-1 is exactly equal to 0-2 by choosing xl, x3 and x5 as described in the next subsection. According to the multiplication rule (3.27) in ref. 15) we have the relation r ( x l , xs) = F(xl, x3)r(x3, xs).
(25)
Using (17a, b), (19) and (23), we obtain from (25) the formulae
Fll(xl, xs) = - e x p {2K~}{F*2(x~, x3)F~(x3, xs) + [[F2a(xt, xa)]2 +exp {-2K~}]~[lFll(x3, xs)[ 2 + exp { - 2/£2} l~exp { - 2i(L- 0-1- 0-2)}} = - exp {2K1 - i[arg Fzz(Xl, x 3 ) - arg F l l (x3, xs)]} × {IF22(x1, x3)Flx(x3, Xs)[ q-[[Fz2(x 1, x3)12 +exp {-2321}] ~ x []Fl~(x3, xs)[ 2 +exp {-2/£2}] ~ exp
{-2i~}},
(26a)
Fzl(xl, xs) = i exp {2K1-2i0-1}{[[F22(xl, x3)12 + exp {-2K1}]@F11(x3, xs) +Fz2(xl, x3)[lFx1(x3, x5)12 +exp {-2Kz}] ~ x exp { - 2i(L- 0-1- a2)}}. (26b) The quantity c~in the last member of (26a) is defined by the formula c~ = L-(0.1 +0-2)-½[arg
F2z(x~, x a ) - a r g Fll(xa, xs)].
(27)
From (26a) we obtain
IFll(x~, xs) exp {-2K~}I 2 = {[]F22(x,, x3)lZ+exp
{-2K~}]-~[lFll(x3, Xs)[2+exp {-2K2}] @
-iF2 (xl, + 4[F22(xl,
xa)Fll (xa, xs)] [[F2z(xl, x3)] 2 + exp { - 2K1} ]~-
x [IFl~(xa, xs)12+exp {-2/£2}] -~ cos 2 c~.
(28)
TUNNELING AND TRANSMISSION
639
Inserting this expression with x~ = - o o and x 5 = + oo into the last member of (16a), we obtain an exact formula for T, which will be used in the following for analysing the transmission properties of the double-barrier. 3.2. T H E C A S E W H E N Q 2 ( - z ) = Q2(z)
As has been shown in ref. 14) it follows from the relation Q2(-z) = Q2(z) that qZ(-z) = q2(z). The results obtained in appendix A of ref. 12) can therefore be used for obtaining further symmetry relations between elements of the F-matrices. Using eqs. (A.5a) and (A.10) in ref. lz) and the symmetry relations (17a, b) and (21a, b) in the present paper, and noting that with the phase of q~(z) chosen as in figs. 1-4 we have q(xl) = q ( - x l ) , we obtain
Fll(xl, 0) = g2*z(0, xs),
(29a)
Flz(xt, O) = Fzl(O, xs) exp { - 2 K 1+2iL},
(29b)
F21(xl, O) = F12(0, x5) exp {2Kx -2iL},
(29c)
F22(xl, 0) = Fj*I(0, xs),
(29d)
provided xl = - x s . We remark that from (A.12) in ref. 12) and (14b) in the present paper it follows that F~z(Xl, x5) exp {-iL} and Fzt(xl, Xs) exp {iL} are purely imaginary quantities for the symmetric case considered in this subsection, provided xa = - x s . Furthermore, if xl = - x 5 and x 3 = 0 we have exactly al = 0"2 according to the definitions (20) and (24) and the symmetry relation (29b). We shall now obtain an exact formula for the transmission coefficient when Q2(-z) = QZ(z). Putting/£1 = Kz = K, xl = - x 5 = - o % x3 = 0, and 0"1 = 0"2 = a in (28), and using the symmetry relation (29d), we obtain from (16a) T -~ = 1 + 4 exp {4K}IFz2 ( - o0, 0)1211F22(-oo, 0)[Z+exp { - 2 K } ] cos 2 e.
(30)
This formula shows that the transmission coefficient is exactly equal to unity if the energy E satisfies the equation cc = (n+l)zr, (31) where n is an integer. Using the definition (27) of c~and the symmetry relation (29d), we can write (31) as follows L-[2a+arg
F22(--oo, 0)] = (n+½)~.
(32)
This is thus an exact equation for the determination of the resonance energies, i.e. the energies corresponding to complete transparency. From the formulae (7), (3) and (4) it is seen that L, which is positive, is a monotonically increasing function of the energy E when the first-order phase-integral approximation is used. Since for n > 0 the quantities Y2,(z) are considered to be small compared to unity, we expect L to be a monotonically increasing function of E also for the higher-order phase-integral
640
N. t'ROMAN AND O. DAMMERT
approximations. According to the estimates which will be given in sect. 4 the quantity 2o-+arg Fz2(-oe, 0) is small compared to unity for particle energies that are not near to the peaks of the barriers. Its change with E can then be neglected compared to the change of L, and hence the left-hand member of (32) is a monotonically increasing function of E. Thus we realize that eq. (32) has precisely one root for every non-negative value of n and that an approximate value of this root is obtained from the equation L = (n+½)r~, (33) except when the root of this equation turns out to lie close to the energy corresponding to a barrier peak.
4. Estimates of F(xl, xs) and F(xs, xs) In the following we shall assume that the well between the two barriers is not too narrow. More precisely, we shall assume that there is a region between the two barriers where the phase-integral approximations under consideration can be used. We can then utilize the results obtained for a single barrier 1s, t 4). According to eq. (43a, b) in ref. 14) we can write the estimates for F22(xl, x3) and Fi2(xi, x3) in the following unified way, which covers the case of sub-barrier penetration as well as the case of super-barrier transmission, Fz2(xl, x3) = 1 +exp {½([Kil-Ki)}O(pl),
(34a)
Flz(xl, xa) = i exp {2io'1} x [1 +exp {-2Ki}l~-[1 +exp {-½(IKll-K1)}O(pi)+O(Plz)].
(34b)
Here #t --/~(xl, x3) is a//-integral pertaining to the left-hand barrier and defined as in (44) in ref. 14). The integration is to be performed along the appropriate one of the paths depicted in figs. 6.4, 6.5 and 6.6 in ref. i 5). In the following we assume that /~l exp {½(]Kl]-K~)} is small compared to unity. Hence, the term O(#~) in (34b) can be omitted since this term is then at the most of the order of magnitude /A exp {-½(IKI[-K~)}. The matrices f(xl, x3) and F(x3, xs) are similar to each other in the sense that they connect two points lying on opposite sides of a potential barrier, but their elements do not directly correspond to each other. Evidently the phase of q(z) in the regions of the real axis pertaining to one barrier differs by ~z from the phase of q(z) in the corresponding regions pertaining to the other barrier. Furthermore, in the formulae (11) for w(x) the transition point tl has a preferred position compared to the transition point t a. Both these circumstances can, however, easily be taken account of, and the estimates of the elements of F(x 3, xs) can be obtained from those of F(xl, x3). Noting that, according to p. 20 in ref. is), the effect of adding ~ to the phase of q(z)is that F12(x ~, x3) and F22(x~, x3) should be compared with F21(x3, xs), and Fll(x3, xs), respectively, and utilizing the fact that, according to p. 21 in ref. 15) the quantities F12(xl, xa) exp {2iw(x3)} and F2t(x3, xa) exp {--2iw(x3)} as well as
TUNNELING AND TRANSMISSION
641
the diagonal elements of both of the F-matrices are independent of the choice of the lower limit of integration in the integral (2) defining w(z), we find that * F,t(x3, xs) corresponds to F22(x~, xa) and that F21(x 3, xs)exp {--2/(1 +2iL} corresponds to F12(Xl, x3). From (34a, b) we therefore obtain the following estimates F~I (x3, xs) = 1 + exp {21-(I/{21-K2)}O(p2), Fz,(X3, xs) exp { - 2 <
(35a)
+2/L} = i exp {2io-1}[1 +exp {-2/(2}1 ~ x [1 +exp {-½([K2I-K2)}O(/~2)+O(P2)],
05b)
where g2 =/~(x3, xs) is the #-integral from x3 to x s along an appropriate path similarly as for the estimates (34a, b). In the following we assume that #a exp {½(IKg]-K2)} is small compared to unity. Hence, the term O(# 2) in (35b) can be omitted for the reason explained in the comments to (34a, b). In ref. 1,v) it has been shown that if [Kt] is sufficiently large the quantity 0.1 is at the most of the order of magnitude of the integral ¢q, the integration being performed in the sub-barrier case along a path proceeding as depicted in fig. 6.4 in ref. ,s) and in the super-barrier case along the real axis. In the first-order approximation the quantity o-1 is rather small compared to ½re also for small values of [KI[, i.e. for particle energies in the neighbourhood of the top of tile barrier (see fig. 5), but when the higher-order phase-integral approximations are used, the quantity o-1 is very large compared to unity when [K,] is sufficiently small [cf. pp. 50 and 54 in reL 1s) and p. 89 in ref. 12)]. Similar remarks apply to the quantity 0.> An approximate analytic expression for 0.1 as a function of K1 and for o-2 as a function of 722 pertaining to the first-order approximation will be given in sect. 7. 5. Approximate formulae for the transmission coefficient
Using (28), (34a) and (35a), we obtain from (16a) the formula T -
1
A + B cos 2
,
(36)
where A = ([exp {2K~} + 11~ [exp {2K2} + 1 ]•- exp {K I + K2}) 2 x [1 + e x p {½(IX~l-K~)}O(/~z)+exp {½(]K2[-K2)}O(/~2)],
(37a)
B = 4 exp {K1 +K2} [exp {2K1}+ 1]~[exp {2/<2}+1] ~ x [I +exp {½([Kl[-K1)}O(/q)+exp {½([K2[--Kz)}O(/~2)].
(37b)
Since we are assuming that /~l exp {½(IKll-K1)}+p2 exp {½([KgI--Kz)} << 1,
(38)
* The symmetry relations (29a, b, c, d) although valid only in the special case when Q2(--z) = Q2(z), illuminate the correspondence between the matrices F(xl, xa) and B(x3, xs).
642
N. ]FI/OMAN AND O. DAMMERT
we obtain from (36) and (37a, b) the approximate formula Y - ' ~ ([exp {2K~}+ 1]~[exp {2K2} + 1 ] ~ - e x p {K~ +K2}) 2 + 4 exp {K1 +K2} [exp {2K~} + l li[exp {2/(2} + 1]÷ cos 2 ~,
(39)
which, introducing the notation S = [1 + exp { - 2K 1}1~ [1 + exp { - 2K2 } l~,
(40)
exp {-- 2(K1 + K2)} r ~ ( S - 1) 2 + 4S cos ~-c~ "
(4t)
we may also write
The relative error of this formula is at the most of the order p~ exp {½(IKll-K~)} + gz exp {½([Kal-K2)}. Sometimes the relative error actually amounts to the order of magnitude of this upper bound, but often it is much smaller. Consequently we may sometimes expect good results when applying (39) even to cases where the condition (38) is not fulfilled.
O'(KI~) _10"I
~
110"05 ' 1
-.3
-2
2
a
/
K/rr
-1
0.~ Fig. 5. Graphical representation of the function a(K/a). The full-drawn curve is obtained from eq. (57). The dashed curve is obtained by inserting the approximation (58) in the iast member of (57). Specializing formula (39) to the first-order 3WKB approximation, replacing ~ by L - 0 - 1 - 0-a [cf. eq. (27)], and substituting for 0-1 and 0-2 the expressions which will be given in sect. 7, we arrive at the formula given by Connor 5), while, if we replace cz by L and consider positive values of K1 and K2, we obtain Ponomarev's formula 6).
TUNNELING
643
AND TRANSMISSION
However, the replacement of c~ by L is not a very good approximation in the neighbourhood of the top of a barrier, where the term 0-1 +0- 2 on the r.h.s, of (27) may be important (cf. fig. 5). Also, for large positive values of K1 and K2 the replacement of c~ by L - 0 - 1 -er2 or by L may be dangerous when one is close to a resonance, since the half-width A ~ of the resonance is then much smaller than 1~ - ( L - 0-1 - 0-2)[ or ]~ - L [ , unless one uses phase-integral approximations of rather high orders. In subsect. 3.2 we have shown that for a symmetric double barrier there are certain energies, the so called resonance energies, which correspond to complete transparency of the double barrier. We also gave an exact relation in dosed form, namely eq. (31) or eq. (32), for the location of these energies. The approximate formula (39), valid for any smooth double barrier, yields T ~ 1 if K1 = K2 when c~ = (n + ½)Tr. However, whereas the resonance value of T is exactly equal to unity in the case of a symmetric double barrier, it is only approximately equal to unity if the double barrier is not symmetric but, incidentally, KI = /(2 when e = (n + ½)n. In the non-symmetric case we cannot in a simple way obtain exact expressions for the location of the energies corresponding to the transmission maxima. When exp { - 2 K 1 } and exp { - 2 / ( 2 } are not both small compared to unity, the quantities A, B and ~ all change appreciably with energy when E changes by the half-width of a m a x i m u m peak. There is then no simple way of determining the position of the maxim u m analytically. However, for particle energies far below the peaks of both barriers the quantity A is very small compared to B, and we realize from (36) and (37a, b) that the transmission coefficient exhibits narrow maxima. In this case A, B and c~all change only very slightly when E changes by the half-width of the maximum, and the maxima therefore lie very close to the energies for which cos 2 ~ = 0, i.e. c~ = (n+½)Tr where n is an integer. Since A is very small compared to B in the case under consideration, the term B cos 2 ~ will be much larger than the term A in the denominator of (36), except when cos c~ is very close to zero. This occurs in a narrow energy interval, in which we may regard A and B as constant. The maxima of the transmission coefficient, which we refer to as resonances also in the non-symmetric case, will appear as high but very narrow peaks in a diagram displaying T as a function of E. Confining our discussion of the resonances for a non-symmetric double barrier to the case when exp { - 2 K 1 } and exp { - 2 K 2 } are both small compared to unity, we obtain from (39) the following approximate formula for the transmission coefficient r-1 ~
1+
cosh 2 ( K 1 - K2).
(42)
exp { - 2 K 1 } + ¼ exp {--2K2} The maxima occur when a = (n + ½)rc, where n is a non-negative integer. The transmission coefficient at resonance is equal to 1/cosh 2 ( K 1 - / ( 2 ) and thus depends only on IK1-K21. When I~-(n+½)rc[ << ½~r eq. (42) can be written
T -1
1+
¼exp{-2K1}+¼exp{-2K2
cosh 2 (KI - K2).
(43)
644
N. ][ZR()MAN AND O. DAMMERT
From this formula it is seen that the total half-width of the resonance is Ac~ ~ ½ exp {-2K1}+½ exp {-2K2}.
(44)
To obtain the half-width on the energy scale we use eqs. (27), (7), (3) and (4), getting AE~
AL Ae --~-=
Ac~
A~
rL
f~3 | / 2 r a [E-- V ( x ) ] d x OEJ,2 I/ h 2
o
~3E
~E
~E L.
has
f,'a21/ dx
[2 [e-v(x)] m
Inserting the expression (44) for A ~, we obtain hv hv AE ~ - - e x p { - 2 K 1 ) + exp ( - 2 K 2 } ,
(45)
where
_l = 2 ( t3
dx
1
(46)
V is the time for a whole classical oscillation within the region tzt 3, and v, therefore, is the classical frequency of oscillation for a particle with energy E in the region between the two barriers. Let us now replace c~ by L in formula (42). When assessing the accuracy of the formula thus obtained one must be aware of the fact that the approximation a ~ L shifts the resonance curve slightly fi'om the position of the resonance and that this shift, although small, may be much larger than the half-width of the resonance curve, unless one uses phase-integral approximations of rather high orders. This means that although (42) gives a very good description of T also in the neighbourhood of a resonance, the replacement of c~ by L in (42) introduces in the neighbourhood of the resonance peak an error which is expected to be more marked the sharper is the resonance peak. Since we are assuming the energy of the impinging particle to lie well below the peaks of both barriers (i.e. K1 and/(2 both to be positive and sufficiently large), eqs. (27), (34a) and (35a) in the present paper together with eq. (51) in ref. 1,~) yield = L + O(/~1) + O(p2),
(47)
where/~1 and/~2 are integrals along paths of integration such asin fig. 6.4 in ref. 15). If # l + # z << Icos LI
(48a)
exp { - 2 K 1 } + exp { - 2/£2} << lcos L],
(48b)
we also have
TUNNELING
AND TRANSMISSION
645
by virtue of eq. (46) in ref. 14), and from (41) we get T N~"exp { - 2 ( K 1 + K2)} 4 cos 2 L
(49)
[t should be noted that for the first-order JWKB approximation the expression (49) for T can also be derived with the aid of the connection formulae (8.19), (8.20) and (8.21) in ref. t 5), and for the higher-order phase-integral approximations the expression (49) can be derived by means of the corresponding connection formulae given in ref. 16). In the case under consideration, i.e. when both barriers are overdense and sufficiently thick and lcosL{ >>/zl +/z2, we may use these connection formulae for passing through everyone of the four classical turning points. Starting with an outgoing wave to the right of the double-barrier, we then obtain in a quite straightforward way the corresponding approximate solution to the left of the double-barrier. The condition IcosLI >> #1 +#2 has to be fulfilled in order that we shall be able to trace the solution from the well between the two barriers further towards the left across the turning points t 2 and t 1. Using the approximate solution obtained in this way, we obtain the approximate formula (49) for T. For certain symmetric double barriers Skorupski 17) solved the Schr6dinger equation numerically and obtained accurate values of the transmission coefficient for different values of the energy E. He also calculated numerical values of T from the approximate formula (41) with c~ replaced by L. Comparison between these results revealed an extremely good accuracy of formula (41) when phase-integral approximations of higher order were used. For values of the energy above the barrier peaks, where the transmission coefficient is close to unity and the conventional JWKB approximation does not give any reflection, formula (41) yielded very good values for the reflection coefficient when the higher-order approximations were used. Skorupski's calculations include cases of physical interest where the first-order JWKB approximation is not good, while the higher-order phase-integral approximations are very accurate. In his calculations Skorupski also investigated the accuracy of the approximate equation (33) for locating the positions of the resonances. It turned out that although the error in the position obtained is very small compared with the distance to neighbouring resonances, it is yet, in general, much larger than the extremely small halfwidth of the resonance unless one uses phase-integral approximations of rather high order (the ninth order in a case considered by Skorupski). 6. The wave function
According to (4.3a, c) in ref. 1~) the following estimates are valid for the matrix
F(xs, + oo)
Ftl(xs, + oo) = Fzl(Xs, + oo) =
1 + O(#3),
(50a)
exp {2(K1-Ka)} 0(#3),
(50b)
646
N. F R O M A N A N D O. DAMMERT
where/Z3 = g(xs, + co) denotes the/z-integral along the real axis from xs to + oo. Estimates for the matrix f(x3, + co) are obtained from (35a, b) by putting xs = + co. For the matrix F(xl, + co) we obtain from (26a, b), (27), (34a) and (35a), denoting/zt +/z2 by/z, the estimates
F~t(xl, + o~) = - e x p {2Kt}{1 + [1 +exp {-2K1}]3[1 + e x p {-2K2}] 3 x exp { - 2 i ( L - o - , - o ' 2 ) } + [exp {½(IKt[-K1)}+exp {½([K2I-K2)}IO(/Z)},
(51a)
Fa,(x,, +oo) = i exp {2K,-2io',}{[1 +exp {-2K~}]3+ [1 +exp {-2/£2}] 3 x exp { - 2 i ( L - o - , - o - / ) } + exp {½(IK,[-K,)+½([K2I-K2)}O(#)}.
(51b)
We shall assume that far to the right of the system of barriers there is only an outgoing wave, i.e. that a2(+ oo) = 0. From the relations (3.25a) and (3.26) in re£ ,s), i.e. 0(z) = f(z)a(z) and a(z) = F(z, zo)a(zo), and the formulae (la, b), (lla, b, c), (50a, b), (35a, b) with xs = + 0% and (51a, b) in the present paper one can obtain approximate expressions for the wave function in the classically allowed regions of the real axis, except in the vicinity of the transition points t,, t2, t3 and t 4. Choosing a t ( + 0~) = --exp { K z - K , +i(L+¼~)}, and noting that w4(x,)is positive, whereas we(x3) and wl(x,) are negative, we get (,'(xs) ~ lq-3(xs)] exp
O(x3)
{i[[w4(xs)l +¼=1},
(52a)
exp {K2 + to'2}[q-3 (x3)[ exp {--i[[wz(xa)l+¼Tc--(L--o'z)]}
+ exp
{Kz + io-z} [1 + exp {-2Ke}] 3 Iq - (x3)lexp {+i[]w2(xa)[+¼~z-(L-o'2)]}, (52b)
0(xt) ~ exp {K, + K2} {1 + [1 + exp { - 2K1 } ]3 [1 + exp { - 2K2} ]3 x exp {--2i(L--o'1--o'2)}}[q-3(xt)[exp
{-i[[w~(xt)[-¼~z-L]}
+exp {K, +K2}{[1 +exp { - 2 K , } ] 3 exp {i(L-2o-,)} + [1 +exp {-2/(2}] 3 • xexp {--l(L-2o-2)}}lq
--3
(x,)lexp
{+i[[w1(x,)l-¼7c]}.
(52c)
For particle energies far below the peak of the right-hand barrier one can utilize the connection formula (19) in ref. * 6) for tracing the wave function to the interior of the classically forbidden region between t3 and t 4. Thus, if exp {-2K2} << 1, we obtain from (52a) -3 O(x4) ~ [q (x4)[ exp {Iw4(x4)l}. (52d) Analogously, if the particle energy lies far below the top of the left-hand barrier and is not in the neighbourhood of a resonance, i.e. if exp { - 2 K t } << 1 and Icos L[ >> #1 +/zz [cf. (48a)], we obtain from (52b), after having made the appropriate approximations in this expression by neglecting exp { - 2 K , } and o'1 compared to
TUNNELING AND TRANSMISSION
647
unity [cf. (46) and (51) in ref. 14)] 0(x2) ~ {1 + [1 +exp {-2K2}1 ~ exp {--2i(L--o-2)}} x exp {K2 q-iL}lq-~(x2)l exp {Iw2(x2)I}.
(52e)
If, in particular, the particle energy lies well below the top of the right-hand barrier also, the quantities exp {-2Kz} and o-2 are small compared to unity [cf. (46) and (51) in ref. 14)], and eq. (52e) becomes O(x2) ~ 2 cos L exp { K z } l q - ~ ( x 2 ) l e x p {Iw2(x2)]}.
(52e')
7. Transmission coefficient in first-order approximation. Brief review
In some physical situations, e.g. in the current theory of fission, where only a rather rough description of the transmission properties of the double-barrier is needed, the first-order approximation is sufficient. For the benefit of those who would like to use the above results in the first-order approximation only, without studying the whole treatment given above, we shall now specialize the general formula for the transmission coefficient and give approximate expressions for 0-1 and o-z pertaining to this particular case. The transmission coefficient is given by (39) or (41). In the first-order approximation we have K t = +_
Q(z)dz
,
(53a)
Q(z)dz
,
(53b)
1
K z = +__ 3
where Q ( z ) is defined by (4). The plus sign refers to the sub-barrier case and the minus sign to the super-barrier case. We may replace the exact expression (27) for c~ by c~ ~ L-(~l~-~r2),
(54)
except when the energy is well below the lowest barrier maximum and is furthermore in the neighbourhood of a resonance. In that case the half-width A c~of the resonance is in general much smaller than the terms neglected in the approximate expression (54) for e. In other words, the use of (54) instead of the exact expression (27) for e means that the resonance peak will be displaced, the displacement being in general much larger than the half-width of the resonance. The quantity L is given by L=
l Re ftta Q ( z ) d z 2
.
(55)
An approximate expression for o-1 as a function of K1/~z can be obtained from the definition (20) by substituting for F12(xl, x3) the expression for F12(- m, + ce) per-
648
N. FR()MA.N AND 0. DAM M E R T
taining to a parabolic barrier 1s). The corresponding expression for a2 is quite analogous except that/£1 should be replaced by Kz. Thus,
al ~ a(Ka/rc),
(56a)
a z "~ ~r(K2/rc),
(56b)
with a(K/zc) given by a(K/n)=-argF(¼+i~n)+
2nKlnlK - 2 nI
2nK+½[arctge-K--¼n]
_12 I KlnI K-~z __Krc+ a r g F ( ½ - i K ) l .
(57)
The function a(K/rc) is depicted in fig. 5. Cf. also the work by Ford, Hill, Wakano and Wheeler 19) on quantum effects near a barrier maximum, where the approximate formula [their eq. (16a)] ( ~) K ½argF ½-i ~ ---ln 2~r
E(K) ~ 2~ + (1__] ] ~ \4y] _1
7 = 1.78107 ....
(58)
is given and checked numerically (their table 1). The last naember of (57) corresponds to the expression given by Connor 5) for the phase factor ½~o(e)in his notation. We remark that for a single parabolic barrier represented by the potential V(x) Vo--½mo)Zx z we have [refs. 5,19)] =
K _ Vo-E 7z
(59)
hco
We are much indebted to Professor P. O. Fr6man for many valuable suggestions concerning the manuscript. We would also like to thank Dr. A. Skorupski for his interest in carrying out numerical calculations in connection with this paper. His results greatly stimulated our further work on the problem. References 1) 2) 3) 4) 5) 6)
D. Bohm, Quantum theory (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961) L. V. Iogansen, JETP (Soy. Phys.) 18 (1964) 146 L. V. Iogansen, JETP (Soy. Phys.) 20 (1965) 180 M. S. Child, Molec. Phys. 12 (1967) 401 J. N. L. Connor, Molec. Phys. 15 (1968) 37 L. L Ponomarev, Lectures on quasiclassics ITF-67-53, Institute for Theoretical Physics, Acad. Sc. Ukr.SSR (Kiev 1968) 7) A. Zwaan, Intensit~iten im Ca-Funktenspektrum (Academisch Proefschrift, Joh. Ensched6 en Zonen, Haarlem, 1929) 8) V. M. Strutinsky, Nucl. Phys. A95 (1967) 420; V. M. Strutinsky and S. Bjornholm, Proc. Syrup. on nuclear structure, Dubna, i968
TUNNELING AND TRANSMISSION 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
649
S. G. Nilsson et al., Nucl. Phys. A l l 5 (1968) 545 A. V. Ignatyuk, N. S. Rabotnov and G. N. Smirenkin, Phys. Lett. 29B (1969) 209 N. Frtiman, Ark. Fys. 32 (1966) 541 N. Fr~Sman, Ark. Fys. 32 (1966) 79 N. FrOman and U. Myhrman, Ark. Fys., to be published N. Fr~Sman and P. O. Fr6man, Nucl. Phys. A147 (1970) 606 N. Fr~Sman and P. O. FfiSman, JWKB Approximation, contributions to the theory (NorthHolland, Amsterdam, 1965) N. Fr~man, to be published A. Skorupski, to be published M. Soop, Ark. Fys. 30 (1965) 217 K. W. Ford, D. L. Hill, M. Wakano and J. A. Wheeler, Ann. of Phys. 7 (1959) 239
T U N N E L I N G A N D SUPER-BARRIER T R A N S M I S S I O N T H R O U G H A S Y S T E M OF T W O REAL P O T E N T I A L BARRIERS NANNY FROMAN and ~3RJAN DAMMERT Department of Theoretical Physics, University of Uppsala, Uppsala, Sweden Received 18 December 1969
Abstract: The olle-dimensional quantal problem of tunneling and/or super-barrier transmission through a system of two potential barriers is treated rigorously by means of the higher-order phase-integral approximations introduced in a previous paper by one of the authors. These higher-order phase-integral approximations are closely related to the higher-order JWKB approximations but have great advantages compared to the latter approximations. General formulae for the transmission coefficient of a smooth double-barrier are given, pertinent to an arbitrary order of the approximations used. Upper bounds for the terms neglected in obtaining approximate formulae are derived. The case of resonant transparency is particularly considered. Approximate formulae for the wave function on tile real axis, except in the vicinity of the transition points, are also given. In the last section the final formula for the transmission coefficient in the first-order approximation is given with all the notations explained, for the benefit of those interested in this particular result only.
1. Introduction O n e - d i m e n s i o n a l p o t e n t i a l fields with two or m o r e p o t e n t i a l barriers s e p a r a t e d by p o t e n t i a l wells are met with in several currently used m o d e l s o f q u a n t a l systems. A t r e a t m e n t o f the p r o b l e m o f two barriers s e p a r a t e d by a well can be f o u n d in the textb o o k on q u a n t u m mechanics by B o h m 1), who uses the first-order J W K B a p p r o x i m a tion a n d o b t a i n s a n a p p r o x i m a t e expression for the t r a n s m i s s i o n coefficient for p a r t i cles with energy well b e l o w the p e a k s o f the barriers, which he assumes to be o f equal shape. The c o r r e s p o n d i n g f o r m u l a for the t r a n s m i s s i o n coefficient o f two u n e q u a l barriers * was given in a p a p e r by I o g a n s e n 2) a n d used b y h i m for treating the p r o b l e m o f resonance t r a n s m i s s i o n o f electrons t h r o u g h thin dielectric layers in a c o n d u c t o r 2, 3). The same f o r m u l a was also derived b y Child 4) a n d used for investigating the consequences o f a d i p in the activation b a r r i e r for certain chemical reactions. A s s u m i n g the m a x i m a o f the barriers to be r o u g h l y p a r a b o l i c , Child 4) also derived a f o r m u l a for the t r a n s m i s s i o n coefficient w h e n the energy a p p r o a c h e s o r exceeds the p e a k s o f the barriers. This case was e x a m i n e d in greater detail by C o n n o r 5), who i n c l u d e d certain p h a s e factors which were o m i t t e d by Child b u t are o f i m p o r t a n c e for energies n e a r to the b a r r i e r m a x i m a . P o n o m a r e v 6) used Z w a a n ' s 7) m e t h o d for h a n d l i n g the firsto r d e r J W K B a p p r o x i m a t i o n a n d o b t a i n e d a f o r m u l a for the t r a n s m i s s i o n coefficient which is correct for energies well below the p e a k s o f the barriers, b u t is n o t quite g o o d t According to a footnote in ref. 2) the formula was obtained by M. Ya. Azbel. 627
628
N. FRt~MAN AND O. DAMMERT
for energies near to either of the barrier maxima, since the above mentioned phase factors are lacking. Recently, through the work by Strutinsky s) and Nilsson et al. 9), the two-hump barrier has become of great interest in the theory of fission. In that context Ignatyuk et al. ~o) derived a formula for the penetrability, valid for energies far below the peaks of the barriers. This formula, as well as the corresponding formulae in refs. ~- 4), contains insignificant terms, originating from an inadequate use of the one-directional connection formulae of the conventional JWKB method. After such terms have been deleted the formulae in question agree with Ponomarev's 6) formula adapted to the case of particle energies far below the barrier maxima. In the present paper we shall give a rigorous treatment of the double-barrier problem based on the higher-order phase-integral approximations derived by one of us in a previous paper ~1), where also tile method for handling the connection problems for these approximations is outlined. The use of the higher-order phase-integral approximations improves the accuracy considerably, the first-order JWKB approximation being for many purposes rather unsatisfactory. An exact formula for the transmission coefficient of the double-barrier will be given. From this exact formula we shall obtain approximate formulae valid for any order of the phase-integral approximations and applicable to all possible situations of tunneling and/or superbarrier transmission. Upper bounds for the errors involved in obtaining the approximate formulae will also be given. The formulae given by Connor s) and Ponomarev 6) are obtainable as special cases of our results (see sect. 5 below).
2. The phase-integral approximations The higher-order phase-integral approximations introduced in reL 1~) have been used in a previous paper 12) by one of us for investigating the energy levels of doublewell potentials. Since the present problem will be treated along similar lines, the parts of ref. 12), which contain general theory and explanatory comments on the application of the methods developed in ref. 1~), form the basis for the present paper as well and will not be repeated here. Thus sects. 2 and 4 ofreL 12) are entirely pertinent to our present problem, and their content will be presupposed in the treatment below. We also refer to sect. 3 of ref. 12) and to refs. 13, 14) for the description of the Riemann surface and the contours of integration for obtaining the quantity w(x), which enters into our formulae. For details on the method for mastering the connection problems we refer to refs. 15, ~6). Thus we shall directly write down the expressions for the two linearly independent functions, being approximate solutions of the Schr6dinger equation and used on constructing the exact solution of the Schr6dinger equation [cf. sect. 2 of ref. ~2)], on which the treatment in the present paper is based. These functions have the following form
fl(z) = q-X~(z) exp { + iw(z)),
(la)
J2(z) = q-~(z) exp {-- iw(z)},
(lb)
T U N N E L I N G AND TRANSMISSION
629
where z
w(z)=
f q(z)dz,
(2)
N
q(z) =
Q(z) Z Yz,(z) •
(3)
n=0
Here
QZ(z) =
2m
~ - I-E- V(z)],
(4)
where E is the energy of a particle of mass m moving in the potential field V(x). The functions Yz,(z), n > 0, are considered to be small compared to Yo(z), which is equal to unity. Analytic expressions for the first few of the functions Yz,(z) can be found in ref. 14). Let Xl, x2, x3 and x4 be points on the real axis located as shown in figs. 1-4. It is convenient to introduce the following real quantities
Wl(Xl)
= Re
½frq(z)dz,
(negative)
(5a)
q(z)dz,
(negative)
(5b)
(positive)
(5c)
(positive)
(5d)
Wz(X3) = Re½~ '3/`3
q(z)dz,
w3(x3) = Re ½~ Fa
w4(xs) = Re ½~
- -
FL
q(z)dz,
J/'5
and, if classically forbidden regions exist, the following purely imaginary quantities
q(z)dz,
wl(x2) = ½f/"
(5e)
2
q(z)dz,
w2(x2) = ½ f r2 -
w3(x4)
=
(5f)
rK1
½fr q(z)dz,
(5g)
4
q(z)dz.
w4(x4) = ½ ( F4--r~
(5h)
2
The contours of integration to be used in (5a-h) lie on a two-sheet Riemann surface and are shown in figs. 1-4. In the notations introduced above a subscript on w is used to indicate that the integration contour encloses the turning point (tl, t2, t 3 or t4) with the same subscript.
630
N, FR()MAN AND O. DAMMERT
V(xl-E
(a)
x, :
F~
(b)
~
'-._.~
I
I
I
t2
t3
tz,
i
I ......
I
t2
t3
t4
tl
Ct I
X2 m ~ -i ~
(c)
X3 C :
•
' • ' i --:-"
(d} '
(e)
(f)
tl
q2 :jq]~
~KI
t2
//," q2:~ Z- qi~ / /
t3
q~:.jlql~
t4
/ q~:e-IT qi ~- -'// q~:.lqi~
(g)
Fig. I. Both barriers overdense, a Qualitative b e h a v i o u r o f V(x)--E. b-f C o n t o u r s o f integration for obtaining w(x). D a s h e d lines indicate parts o f the c o n t o u r s lying on the second R i e m a n n sheet. g Phase o f q~(z) on the real axis o f the first R i e m a n n sheet.
"
gg9
~.
~'~
~.o~ ~'~ S~
0
2
~J
/
x
=-n
_7 /
--o
X
~"
x
m
632
N. I~ROMANAND (). DAMMERT V(xI-E
XI
~I
I
t
~
X3 ~
X
X5
{a)
t~ (b)
"--'"
t2
tl
t3
t4 ,c)
C" .: tI
F'2
t2 t3
t4 (d)
,, r.,
?~
G
t3
{e)
tl
FKI
" .
.
.
.
, t3
K2
t4
Fig. 3. The left-hand barrier overdense, the right-hand barrier underdense, a Qualitative behaviour o f V(x)--E. b-e C o n t o u r s o f integration for obtaining w(x). Dashed lines indicate parts of the contours lying on the second Riemann sheet, f P h a s e o f q½(z)on the real axis o f the first Riemann sheet.
TUNNELING AND TRANSMISSION
633
V(x)-E (a}
.
.
.
.
.
.
:
=X
/ '(2 (b) t3
t4
I
,,,I
t3
t4
t! %2 (c)
~
3
7K I ~ tl
t2
~ ill
........
r K , ~
: .....
x4
(d) t~
"T; a
tt
t2 (e)
tl
t2 I
~
~
I
1
!
3it
1
y
J
Fig. 4. T h e left-hand u n d e r d e n s e , the r i g h t - h a n d barrier overdense, a Qualitative b e h a v i o u r o f V(x) --E. b-e C o n t o u r s o f integration for o b t a i n i n g w(x). D a s h e d lines indicate p a r t s o f the c o n t o u r s lying o n the s e c o n d R i e m a n n sheet, f P h a s e o f q~(z) o n t h e real axis o f t h e first R i e m a n n sheet.
634
N . F R O M A N A N D (). DAMiVIERT
It is also convenient to introduce the following notations
K 1 =- ½i~ q(z)dz,
(6a)
,g F K 1
K2 =--½i(
q(z)dz,
(6b)
•] F K 2
q(z)dz,
L----Re½~
(positive)
(7)
•] F L
where the paths of integration are the closed loops depicted in figs. 1-4. The quantities K1, K2 and L are all real. In fact, Kt and Kz are positive in the case of tunneling (subbarrier penetration) but negative in the case of super-barrier transmission, whereas L is always positive. In the case of sub-barrier penetration for both barriers the integral in (7) is real, and the symbol Re on the r.h.s, of (7) can be left out. In the case of superbarrier transmission for both barriers the closed-loop integral occurring in (7) can be written
-½ f rrq(z)dz = L +½i(Kt + K2).
(8)
It can be shown [cf. ref. 14)] that certain terms in the expressions for Y2,(z) do not contribute to the values of the quantities K1, Ka and L, defined by eqs. (6a, b) and (7) and entering into the final formula for the transmission coefficient. This fact considerably economizes the numerical work required for calculating the transmission coefficient by means of our final formula. Since q(z) is an analytic function on the Riemann surface under consideration, the contours of integration in eqs. (5), (6) and (7), may of course be deformed in a way that is convenient for performing numerical calculations. The contours of integration in figs. 1-4 are chosen so as to give a close analogy between all the possible cases of sub-bariier and/or super-barrier transmission. The formulae (5), (6) and (7) are the same in all cases. Only the contours of integration are different. If we imagine the vertical cuts in the super-barrier case to be rotated in the clockwise direction until they coincide with the real axis, we obtain the same scheme of integration contours as in the sub-barrier case. When super-barrier transmission occurs for one or both of the barriers, one can also use other paths of integration, as has been done in the above-mentioned work 12) on the double-oscillator. One should then let the cut between two complex conjugate transition points proceed along the Stokes' line t connecting these points. We then t For a symmetric barrier this Stokes' line coincides with the straight line connecting the two complex conjugate transition points.
TUNNELING AND TRANSMISSION
635
realize that the formulae (5a, b, c, d) and (7) can also be written as follows
wl(xl) = fs[1_oq(x)dx,
(9a)
q(x)dx,
(9b)
= f~[~_oq(x)dx,
(9c)
f[~ q(x)dx,
(9d)
wz(xa) =
wa(x3)
w4(xs) =
f f 1+0
2+0
/'s2 - o
L = -- Js~+o
q(x)dx,
(10)
tltz
where s a and s2 are the points on the real axis where the Stokes' lines and t3 t4, respectively, cross the real axis. Cf. fig. 2 in ref. 12). We shall now turn our attention to the function defined by (2). It is convenient to express where x is real, in terms of the quantities defined by (5a-h). With a convenient choice t of the lowex limit of integration in the integral (2) defining we have
w(z)
w(x),
w(z)
W(Xl) = wl(xl),
(lla)
w(x3) = w~(x3)-izq = w3(x~)-iX~-L,
(1tb)
w(xs) = w , ( x s ) - i(~q - X 2 ) - ~
01c)
and, if classically forbidden regions exist, w(x2) = wl(x2) = w2(x2)- iIq,
(lld)
w(x4) = w 3 ( x 4 ) - iKx - L
(lle)
= w4(x4) - i(K1 - K 2 ) - L .
3. E x a c t f o r m u l a e for the t r a n s m i s s i o n and r e f l e x i o n coefficients 3.1. G E N E R A L
CASE
The transmission problem for a single potential barrier has been treated by Fr6man and FrSman 15) in the first-order JWKB approximation and, in greater detail, with the aid o f the higher-order phase-integral approximations 14). Some of the results obtained in these investigations will be utilized in the present treatment of the twobarrier problem. t For the first-order JWKB approximation and the case of sub-barrier penetration through the left-hand barrier this choice amounts to choosing the lower limit of integration in (2) to be the transition point tl.
636
N. FRIDMAN AND (). DAMMERT
We shall make similar assumptions concerning the behaviourof V(x) for large values of Ixl as in chapter 9 of ref. 15). Thus we assume that Q2(x) and hence also q2(x) is positive for sufficiently large values of [xl and that the integral/~(xl, xs) defined by eq. (10) in ref. ~2) converges as xl tends to - c o and xs tends to + ~ . From this it follows similarly as on p. 92 in ref. 1 s) that, if z does not coincide with a singularity or a zero of q2(z), the limits F(-oo, z), F(z, +co), and F ( - c o , + o o ) exist and are finite, which in turn implies that the limits a~( ± oo) exist and are finite, and that f l (x) andf2(x) for increasing values of lxl approach two linearly independent solutions of the Scba'6dinger equation. These solutions may be interpreted as waves travelling in opposite directions. Far to the left as well as far to the right of the doublebarrier f , (x) represents a wave moving from the left to the right. Assuming that there is only an outgoing wave far to the right of the double-barrier, we obtain the following expressions for the transmission coefficient T and the reflexion coefficient R r =
a , ( + o o ) 2exp a,(-- oo) {2(K, - K2)},
(12a)
R=
az(-°°)
(12b)
<(-oo)
2.
Since there is only a transmitted wave to the right of the double-barrier, we have a2(+ co) = 0, and from formula (3.16) in ref. ,5) it follows that
a,(z) = F**(z, + co)at(+ co), az(z) = Fz,(z, + co)at(+ oo).
(13a) (13b)
From the symmetry relations (5.7a, b) in ref. ,5), which can be generalized to the higher-order phase-integral approximations, and eqs. (1 la, c) in the present paper, we obtain the symmetry relations
F2z(x,, xs) = exp {2(K2-K,)}F**(xl, Xs),
(14a)
F,2(x,, Xs) = exp {2(K2- Ka)}F**(Xl, xs).
(14b)
With the aid of (14a, b) the relation det F(x,, xs) = 1 [of. eq. (3.19) in ref. 15)] can be written
[Fzl(x,, Xs)l a = [FI,(x,, xs)l z-exp {2(K,-Kz)}.
(15)
Using (13a, b) and (15), we obtain from (12a, b) the following exact expressions for T and R T = exp {2(K, - Kz)} = exp { - 2 ( K , + K2) } I F , , ( - o o , +oo)1 z I F , l ( - o o , +oo) exp {-2K1}I 2'
(16a)
R = Fza(-°°' +oo) z = [ r l l ( _ o o ' +oo)12_exp {2(K _K2)} = 1 - T . (16b) F , l ( - o o , +oo)
IFl1(-~,
+oo)i z
TUNNELING AND TRANSMISSION
637
To be able to utilize the results obtained for a single barrier in refs. ,4,15), we shall express the denominator in (16a, b) in terms of quantities referring to the two separate barriers. To prepare this and for later reference we shall first consider the matrices F(xl, x3) and F(X3, x s ) . We have chosen the phase of q~(z) and the constant lower limit of integration in the integral (2) defining w(z) such that the matrix F(xl, xa) in the present paper corresponds for the first-order JWKB approximation to the matrix F(xl, x2) treated in subsects. 6.4 and 6.5 ofref. 15) and for the phase-integral approximations of an arbitrary order to the matrix F(xt, x2) in ref. ,4). In accordance with eqs. (6.17a, b) and (6.32a, b) in ref. ~5) the symmetry relations for the matrix F(xl, x3) in the present paper are t
F*2(x,, x3) exp {2K1},
(17a)
F21(x~, x3) = --F*a(Xl , x3) exp {2K1},
(17b)
F l l ( x , , x3) = -
where K, is defined by (6a) and is positive for the case of sub-barrier penetration and negative for super-barrier transmission.. Using (17a, b), we can write the relation det F(xl, x3) = 1 as follows
IF~2(xt, x3)[ 2 =
If22(xx, x3)12-t-exp {-2/£I}.
(18)
From this formula it follows that
F~z(xl, x3)
= [[Fz2(X,, x3)12+exp {-2K1}] ~ exp {i(½zc+2al)},
(19)
where cr~ is a real quantity defined by the formula o-, = ½[arg Fa2(x,, xa)-½7z].
(20)
This definition of at corresponds precisely to the definition (36c) in ref. i2). The symmetry relations for the matrix F(x3, xs) are F22(x3, Xs) = --F*l(x3, xs) exp {2K2},
(21a)
-F~,(xa,
(21b)
F12(x3, xs) =
xs) exp { 2 K z - 4 K , } ,
where K2 is defined by (6b). These symmetry relations can be obtained either directly as described in the previous footnote, or from the symmetry relations (17a, b) by means of the same arguments as will be used in sect. 4 to obtain the estimates of F(xa, xs) from those of F(x,, X3). By means of (21a, b) the relation det F(x3, xs) = 1 can be written as follows IF21(xa, xs) exp {-2/£1}12 -- ]F,~(x3, xs)lZ+exp {-2K2}.
(22)
t The symmetry relations (17a, b) and {21a, b) can be obtained by applying the symmetry relations (5.7a, b) in ref. ,5), generalized to apply to the phase-integral approximations of any order, and using the expressions ( l l a , b, c) for w(x).
638
?4. ]FR{}MAN AND O. DAMMERT
From this formula it follows that
Fz,(X3, xs) = [[Fll(x3, xs)]2+exp {-2K2}] ~ exp {2K1 +i(½~z-2L+2a2)}, (23) where 0"2 is a real quantity defined by 0-2 = ½[arg Fa 1(x3, xs) - ½n + 2L].
(24)
Although 0.1 and a2 are analogous quantities for the two barriers, the analytical expression for 0"2 is different from that for a 1 due to the fact that in the expression for w(x) the transition point tl has a preferred role compared to the transition point t3 (cf. the footnote on p. 635). The values of 0.1 and 0.2 depend only very slightly on the positions of the points xl, x3, and xs, provided no one of these points lies too close to a transition point. If, in particular, the two barriers are of approximately equal shape, we have 0-1 ~ 0-2. For a symmetric double-barrier we can achieve that 0-1 is exactly equal to 0-2 by choosing xl, x3 and x5 as described in the next subsection. According to the multiplication rule (3.27) in ref. 15) we have the relation r ( x l , xs) = F(xl, x3)r(x3, xs).
(25)
Using (17a, b), (19) and (23), we obtain from (25) the formulae
Fll(xl, xs) = - e x p {2K~}{F*2(x~, x3)F~(x3, xs) + [[F2a(xt, xa)]2 +exp {-2K~}]~[lFll(x3, xs)[ 2 + exp { - 2/£2} l~exp { - 2i(L- 0-1- 0-2)}} = - exp {2K1 - i[arg Fzz(Xl, x 3 ) - arg F l l (x3, xs)]} × {IF22(x1, x3)Flx(x3, Xs)[ q-[[Fz2(x 1, x3)12 +exp {-2321}] ~ x []Fl~(x3, xs)[ 2 +exp {-2/£2}] ~ exp
{-2i~}},
(26a)
Fzl(xl, xs) = i exp {2K1-2i0-1}{[[F22(xl, x3)12 + exp {-2K1}]@F11(x3, xs) +Fz2(xl, x3)[lFx1(x3, x5)12 +exp {-2Kz}] ~ x exp { - 2i(L- 0-1- a2)}}. (26b) The quantity c~in the last member of (26a) is defined by the formula c~ = L-(0.1 +0-2)-½[arg
F2z(x~, x a ) - a r g Fll(xa, xs)].
(27)
From (26a) we obtain
IFll(x~, xs) exp {-2K~}I 2 = {[]F22(x,, x3)lZ+exp
{-2K~}]-~[lFll(x3, Xs)[2+exp {-2K2}] @
-iF2 (xl, + 4[F22(xl,
xa)Fll (xa, xs)] [[F2z(xl, x3)] 2 + exp { - 2K1} ]~-
x [IFl~(xa, xs)12+exp {-2/£2}] -~ cos 2 c~.
(28)
TUNNELING AND TRANSMISSION
639
Inserting this expression with x~ = - o o and x 5 = + oo into the last member of (16a), we obtain an exact formula for T, which will be used in the following for analysing the transmission properties of the double-barrier. 3.2. T H E C A S E W H E N Q 2 ( - z ) = Q2(z)
As has been shown in ref. 14) it follows from the relation Q2(-z) = Q2(z) that qZ(-z) = q2(z). The results obtained in appendix A of ref. 12) can therefore be used for obtaining further symmetry relations between elements of the F-matrices. Using eqs. (A.5a) and (A.10) in ref. lz) and the symmetry relations (17a, b) and (21a, b) in the present paper, and noting that with the phase of q~(z) chosen as in figs. 1-4 we have q(xl) = q ( - x l ) , we obtain
Fll(xl, 0) = g2*z(0, xs),
(29a)
Flz(xt, O) = Fzl(O, xs) exp { - 2 K 1+2iL},
(29b)
F21(xl, O) = F12(0, x5) exp {2Kx -2iL},
(29c)
F22(xl, 0) = Fj*I(0, xs),
(29d)
provided xl = - x s . We remark that from (A.12) in ref. 12) and (14b) in the present paper it follows that F~z(Xl, x5) exp {-iL} and Fzt(xl, Xs) exp {iL} are purely imaginary quantities for the symmetric case considered in this subsection, provided xa = - x s . Furthermore, if xl = - x 5 and x 3 = 0 we have exactly al = 0"2 according to the definitions (20) and (24) and the symmetry relation (29b). We shall now obtain an exact formula for the transmission coefficient when Q2(-z) = QZ(z). Putting/£1 = Kz = K, xl = - x 5 = - o % x3 = 0, and 0"1 = 0"2 = a in (28), and using the symmetry relation (29d), we obtain from (16a) T -~ = 1 + 4 exp {4K}IFz2 ( - o0, 0)1211F22(-oo, 0)[Z+exp { - 2 K } ] cos 2 e.
(30)
This formula shows that the transmission coefficient is exactly equal to unity if the energy E satisfies the equation cc = (n+l)zr, (31) where n is an integer. Using the definition (27) of c~and the symmetry relation (29d), we can write (31) as follows L-[2a+arg
F22(--oo, 0)] = (n+½)~.
(32)
This is thus an exact equation for the determination of the resonance energies, i.e. the energies corresponding to complete transparency. From the formulae (7), (3) and (4) it is seen that L, which is positive, is a monotonically increasing function of the energy E when the first-order phase-integral approximation is used. Since for n > 0 the quantities Y2,(z) are considered to be small compared to unity, we expect L to be a monotonically increasing function of E also for the higher-order phase-integral
640
N. t'ROMAN AND O. DAMMERT
approximations. According to the estimates which will be given in sect. 4 the quantity 2o-+arg Fz2(-oe, 0) is small compared to unity for particle energies that are not near to the peaks of the barriers. Its change with E can then be neglected compared to the change of L, and hence the left-hand member of (32) is a monotonically increasing function of E. Thus we realize that eq. (32) has precisely one root for every non-negative value of n and that an approximate value of this root is obtained from the equation L = (n+½)r~, (33) except when the root of this equation turns out to lie close to the energy corresponding to a barrier peak.
4. Estimates of F(xl, xs) and F(xs, xs) In the following we shall assume that the well between the two barriers is not too narrow. More precisely, we shall assume that there is a region between the two barriers where the phase-integral approximations under consideration can be used. We can then utilize the results obtained for a single barrier 1s, t 4). According to eq. (43a, b) in ref. 14) we can write the estimates for F22(xl, x3) and Fi2(xi, x3) in the following unified way, which covers the case of sub-barrier penetration as well as the case of super-barrier transmission, Fz2(xl, x3) = 1 +exp {½([Kil-Ki)}O(pl),
(34a)
Flz(xl, xa) = i exp {2io'1} x [1 +exp {-2Ki}l~-[1 +exp {-½(IKll-K1)}O(pi)+O(Plz)].
(34b)
Here #t --/~(xl, x3) is a//-integral pertaining to the left-hand barrier and defined as in (44) in ref. 14). The integration is to be performed along the appropriate one of the paths depicted in figs. 6.4, 6.5 and 6.6 in ref. i 5). In the following we assume that /~l exp {½(]Kl]-K~)} is small compared to unity. Hence, the term O(#~) in (34b) can be omitted since this term is then at the most of the order of magnitude /A exp {-½(IKI[-K~)}. The matrices f(xl, x3) and F(x3, xs) are similar to each other in the sense that they connect two points lying on opposite sides of a potential barrier, but their elements do not directly correspond to each other. Evidently the phase of q(z) in the regions of the real axis pertaining to one barrier differs by ~z from the phase of q(z) in the corresponding regions pertaining to the other barrier. Furthermore, in the formulae (11) for w(x) the transition point tl has a preferred position compared to the transition point t a. Both these circumstances can, however, easily be taken account of, and the estimates of the elements of F(x 3, xs) can be obtained from those of F(xl, x3). Noting that, according to p. 20 in ref. is), the effect of adding ~ to the phase of q(z)is that F12(x ~, x3) and F22(x~, x3) should be compared with F21(x3, xs), and Fll(x3, xs), respectively, and utilizing the fact that, according to p. 21 in ref. 15) the quantities F12(xl, xa) exp {2iw(x3)} and F2t(x3, xa) exp {--2iw(x3)} as well as
TUNNELING AND TRANSMISSION
641
the diagonal elements of both of the F-matrices are independent of the choice of the lower limit of integration in the integral (2) defining w(z), we find that * F,t(x3, xs) corresponds to F22(x~, xa) and that F21(x 3, xs)exp {--2/(1 +2iL} corresponds to F12(Xl, x3). From (34a, b) we therefore obtain the following estimates F~I (x3, xs) = 1 + exp {21-(I/{21-K2)}O(p2), Fz,(X3, xs) exp { - 2 <
(35a)
+2/L} = i exp {2io-1}[1 +exp {-2/(2}1 ~ x [1 +exp {-½([K2I-K2)}O(/~2)+O(P2)],
05b)
where g2 =/~(x3, xs) is the #-integral from x3 to x s along an appropriate path similarly as for the estimates (34a, b). In the following we assume that #a exp {½(IKg]-K2)} is small compared to unity. Hence, the term O(# 2) in (35b) can be omitted for the reason explained in the comments to (34a, b). In ref. 1,v) it has been shown that if [Kt] is sufficiently large the quantity 0.1 is at the most of the order of magnitude of the integral ¢q, the integration being performed in the sub-barrier case along a path proceeding as depicted in fig. 6.4 in ref. ,s) and in the super-barrier case along the real axis. In the first-order approximation the quantity o-1 is rather small compared to ½re also for small values of [KI[, i.e. for particle energies in the neighbourhood of the top of tile barrier (see fig. 5), but when the higher-order phase-integral approximations are used, the quantity o-1 is very large compared to unity when [K,] is sufficiently small [cf. pp. 50 and 54 in reL 1s) and p. 89 in ref. 12)]. Similar remarks apply to the quantity 0.> An approximate analytic expression for 0.1 as a function of K1 and for o-2 as a function of 722 pertaining to the first-order approximation will be given in sect. 7. 5. Approximate formulae for the transmission coefficient
Using (28), (34a) and (35a), we obtain from (16a) the formula T -
1
A + B cos 2
,
(36)
where A = ([exp {2K~} + 11~ [exp {2K2} + 1 ]•- exp {K I + K2}) 2 x [1 + e x p {½(IX~l-K~)}O(/~z)+exp {½(]K2[-K2)}O(/~2)],
(37a)
B = 4 exp {K1 +K2} [exp {2K1}+ 1]~[exp {2/<2}+1] ~ x [I +exp {½([Kl[-K1)}O(/q)+exp {½([K2[--Kz)}O(/~2)].
(37b)
Since we are assuming that /~l exp {½(IKll-K1)}+p2 exp {½([KgI--Kz)} << 1,
(38)
* The symmetry relations (29a, b, c, d) although valid only in the special case when Q2(--z) = Q2(z), illuminate the correspondence between the matrices F(xl, xa) and B(x3, xs).
642
N. ]FI/OMAN AND O. DAMMERT
we obtain from (36) and (37a, b) the approximate formula Y - ' ~ ([exp {2K~}+ 1]~[exp {2K2} + 1 ] ~ - e x p {K~ +K2}) 2 + 4 exp {K1 +K2} [exp {2K~} + l li[exp {2/(2} + 1]÷ cos 2 ~,
(39)
which, introducing the notation S = [1 + exp { - 2K 1}1~ [1 + exp { - 2K2 } l~,
(40)
exp {-- 2(K1 + K2)} r ~ ( S - 1) 2 + 4S cos ~-c~ "
(4t)
we may also write
The relative error of this formula is at the most of the order p~ exp {½(IKll-K~)} + gz exp {½([Kal-K2)}. Sometimes the relative error actually amounts to the order of magnitude of this upper bound, but often it is much smaller. Consequently we may sometimes expect good results when applying (39) even to cases where the condition (38) is not fulfilled.
O'(KI~) _10"I
~
110"05 ' 1
-.3
-2
2
a
/
K/rr
-1
0.~ Fig. 5. Graphical representation of the function a(K/a). The full-drawn curve is obtained from eq. (57). The dashed curve is obtained by inserting the approximation (58) in the iast member of (57). Specializing formula (39) to the first-order 3WKB approximation, replacing ~ by L - 0 - 1 - 0-a [cf. eq. (27)], and substituting for 0-1 and 0-2 the expressions which will be given in sect. 7, we arrive at the formula given by Connor 5), while, if we replace cz by L and consider positive values of K1 and K2, we obtain Ponomarev's formula 6).
TUNNELING
643
AND TRANSMISSION
However, the replacement of c~ by L is not a very good approximation in the neighbourhood of the top of a barrier, where the term 0-1 +0- 2 on the r.h.s, of (27) may be important (cf. fig. 5). Also, for large positive values of K1 and K2 the replacement of c~ by L - 0 - 1 -er2 or by L may be dangerous when one is close to a resonance, since the half-width A ~ of the resonance is then much smaller than 1~ - ( L - 0-1 - 0-2)[ or ]~ - L [ , unless one uses phase-integral approximations of rather high orders. In subsect. 3.2 we have shown that for a symmetric double barrier there are certain energies, the so called resonance energies, which correspond to complete transparency of the double barrier. We also gave an exact relation in dosed form, namely eq. (31) or eq. (32), for the location of these energies. The approximate formula (39), valid for any smooth double barrier, yields T ~ 1 if K1 = K2 when c~ = (n + ½)Tr. However, whereas the resonance value of T is exactly equal to unity in the case of a symmetric double barrier, it is only approximately equal to unity if the double barrier is not symmetric but, incidentally, KI = /(2 when e = (n + ½)n. In the non-symmetric case we cannot in a simple way obtain exact expressions for the location of the energies corresponding to the transmission maxima. When exp { - 2 K 1 } and exp { - 2 / ( 2 } are not both small compared to unity, the quantities A, B and ~ all change appreciably with energy when E changes by the half-width of a m a x i m u m peak. There is then no simple way of determining the position of the maxim u m analytically. However, for particle energies far below the peaks of both barriers the quantity A is very small compared to B, and we realize from (36) and (37a, b) that the transmission coefficient exhibits narrow maxima. In this case A, B and c~all change only very slightly when E changes by the half-width of the maximum, and the maxima therefore lie very close to the energies for which cos 2 ~ = 0, i.e. c~ = (n+½)Tr where n is an integer. Since A is very small compared to B in the case under consideration, the term B cos 2 ~ will be much larger than the term A in the denominator of (36), except when cos c~ is very close to zero. This occurs in a narrow energy interval, in which we may regard A and B as constant. The maxima of the transmission coefficient, which we refer to as resonances also in the non-symmetric case, will appear as high but very narrow peaks in a diagram displaying T as a function of E. Confining our discussion of the resonances for a non-symmetric double barrier to the case when exp { - 2 K 1 } and exp { - 2 K 2 } are both small compared to unity, we obtain from (39) the following approximate formula for the transmission coefficient r-1 ~
1+
cosh 2 ( K 1 - K2).
(42)
exp { - 2 K 1 } + ¼ exp {--2K2} The maxima occur when a = (n + ½)rc, where n is a non-negative integer. The transmission coefficient at resonance is equal to 1/cosh 2 ( K 1 - / ( 2 ) and thus depends only on IK1-K21. When I~-(n+½)rc[ << ½~r eq. (42) can be written
T -1
1+
¼exp{-2K1}+¼exp{-2K2
cosh 2 (KI - K2).
(43)
644
N. ][ZR()MAN AND O. DAMMERT
From this formula it is seen that the total half-width of the resonance is Ac~ ~ ½ exp {-2K1}+½ exp {-2K2}.
(44)
To obtain the half-width on the energy scale we use eqs. (27), (7), (3) and (4), getting AE~
AL Ae --~-=
Ac~
A~
rL
f~3 | / 2 r a [E-- V ( x ) ] d x OEJ,2 I/ h 2
o
~3E
~E
~E L.
has
f,'a21/ dx
[2 [e-v(x)] m
Inserting the expression (44) for A ~, we obtain hv hv AE ~ - - e x p { - 2 K 1 ) + exp ( - 2 K 2 } ,
(45)
where
_l = 2 ( t3
dx
1
(46)
V is the time for a whole classical oscillation within the region tzt 3, and v, therefore, is the classical frequency of oscillation for a particle with energy E in the region between the two barriers. Let us now replace c~ by L in formula (42). When assessing the accuracy of the formula thus obtained one must be aware of the fact that the approximation a ~ L shifts the resonance curve slightly fi'om the position of the resonance and that this shift, although small, may be much larger than the half-width of the resonance curve, unless one uses phase-integral approximations of rather high orders. This means that although (42) gives a very good description of T also in the neighbourhood of a resonance, the replacement of c~ by L in (42) introduces in the neighbourhood of the resonance peak an error which is expected to be more marked the sharper is the resonance peak. Since we are assuming the energy of the impinging particle to lie well below the peaks of both barriers (i.e. K1 and/(2 both to be positive and sufficiently large), eqs. (27), (34a) and (35a) in the present paper together with eq. (51) in ref. 1,~) yield = L + O(/~1) + O(p2),
(47)
where/~1 and/~2 are integrals along paths of integration such asin fig. 6.4 in ref. 15). If # l + # z << Icos LI
(48a)
exp { - 2 K 1 } + exp { - 2/£2} << lcos L],
(48b)
we also have
TUNNELING
AND TRANSMISSION
645
by virtue of eq. (46) in ref. 14), and from (41) we get T N~"exp { - 2 ( K 1 + K2)} 4 cos 2 L
(49)
[t should be noted that for the first-order JWKB approximation the expression (49) for T can also be derived with the aid of the connection formulae (8.19), (8.20) and (8.21) in ref. t 5), and for the higher-order phase-integral approximations the expression (49) can be derived by means of the corresponding connection formulae given in ref. 16). In the case under consideration, i.e. when both barriers are overdense and sufficiently thick and lcosL{ >>/zl +/z2, we may use these connection formulae for passing through everyone of the four classical turning points. Starting with an outgoing wave to the right of the double-barrier, we then obtain in a quite straightforward way the corresponding approximate solution to the left of the double-barrier. The condition IcosLI >> #1 +#2 has to be fulfilled in order that we shall be able to trace the solution from the well between the two barriers further towards the left across the turning points t 2 and t 1. Using the approximate solution obtained in this way, we obtain the approximate formula (49) for T. For certain symmetric double barriers Skorupski 17) solved the Schr6dinger equation numerically and obtained accurate values of the transmission coefficient for different values of the energy E. He also calculated numerical values of T from the approximate formula (41) with c~ replaced by L. Comparison between these results revealed an extremely good accuracy of formula (41) when phase-integral approximations of higher order were used. For values of the energy above the barrier peaks, where the transmission coefficient is close to unity and the conventional JWKB approximation does not give any reflection, formula (41) yielded very good values for the reflection coefficient when the higher-order approximations were used. Skorupski's calculations include cases of physical interest where the first-order JWKB approximation is not good, while the higher-order phase-integral approximations are very accurate. In his calculations Skorupski also investigated the accuracy of the approximate equation (33) for locating the positions of the resonances. It turned out that although the error in the position obtained is very small compared with the distance to neighbouring resonances, it is yet, in general, much larger than the extremely small halfwidth of the resonance unless one uses phase-integral approximations of rather high order (the ninth order in a case considered by Skorupski). 6. The wave function
According to (4.3a, c) in ref. 1~) the following estimates are valid for the matrix
F(xs, + oo)
Ftl(xs, + oo) = Fzl(Xs, + oo) =
1 + O(#3),
(50a)
exp {2(K1-Ka)} 0(#3),
(50b)
646
N. F R O M A N A N D O. DAMMERT
where/Z3 = g(xs, + co) denotes the/z-integral along the real axis from xs to + oo. Estimates for the matrix f(x3, + co) are obtained from (35a, b) by putting xs = + co. For the matrix F(xl, + co) we obtain from (26a, b), (27), (34a) and (35a), denoting/zt +/z2 by/z, the estimates
F~t(xl, + o~) = - e x p {2Kt}{1 + [1 +exp {-2K1}]3[1 + e x p {-2K2}] 3 x exp { - 2 i ( L - o - , - o ' 2 ) } + [exp {½(IKt[-K1)}+exp {½([K2I-K2)}IO(/Z)},
(51a)
Fa,(x,, +oo) = i exp {2K,-2io',}{[1 +exp {-2K~}]3+ [1 +exp {-2/£2}] 3 x exp { - 2 i ( L - o - , - o - / ) } + exp {½(IK,[-K,)+½([K2I-K2)}O(#)}.
(51b)
We shall assume that far to the right of the system of barriers there is only an outgoing wave, i.e. that a2(+ oo) = 0. From the relations (3.25a) and (3.26) in re£ ,s), i.e. 0(z) = f(z)a(z) and a(z) = F(z, zo)a(zo), and the formulae (la, b), (lla, b, c), (50a, b), (35a, b) with xs = + 0% and (51a, b) in the present paper one can obtain approximate expressions for the wave function in the classically allowed regions of the real axis, except in the vicinity of the transition points t,, t2, t3 and t 4. Choosing a t ( + 0~) = --exp { K z - K , +i(L+¼~)}, and noting that w4(x,)is positive, whereas we(x3) and wl(x,) are negative, we get (,'(xs) ~ lq-3(xs)] exp
O(x3)
{i[[w4(xs)l +¼=1},
(52a)
exp {K2 + to'2}[q-3 (x3)[ exp {--i[[wz(xa)l+¼Tc--(L--o'z)]}
+ exp
{Kz + io-z} [1 + exp {-2Ke}] 3 Iq - (x3)lexp {+i[]w2(xa)[+¼~z-(L-o'2)]}, (52b)
0(xt) ~ exp {K, + K2} {1 + [1 + exp { - 2K1 } ]3 [1 + exp { - 2K2} ]3 x exp {--2i(L--o'1--o'2)}}[q-3(xt)[exp
{-i[[w~(xt)[-¼~z-L]}
+exp {K, +K2}{[1 +exp { - 2 K , } ] 3 exp {i(L-2o-,)} + [1 +exp {-2/(2}] 3 • xexp {--l(L-2o-2)}}lq
--3
(x,)lexp
{+i[[w1(x,)l-¼7c]}.
(52c)
For particle energies far below the peak of the right-hand barrier one can utilize the connection formula (19) in ref. * 6) for tracing the wave function to the interior of the classically forbidden region between t3 and t 4. Thus, if exp {-2K2} << 1, we obtain from (52a) -3 O(x4) ~ [q (x4)[ exp {Iw4(x4)l}. (52d) Analogously, if the particle energy lies far below the top of the left-hand barrier and is not in the neighbourhood of a resonance, i.e. if exp { - 2 K t } << 1 and Icos L[ >> #1 +/zz [cf. (48a)], we obtain from (52b), after having made the appropriate approximations in this expression by neglecting exp { - 2 K , } and o'1 compared to
TUNNELING AND TRANSMISSION
647
unity [cf. (46) and (51) in ref. 14)] 0(x2) ~ {1 + [1 +exp {-2K2}1 ~ exp {--2i(L--o-2)}} x exp {K2 q-iL}lq-~(x2)l exp {Iw2(x2)I}.
(52e)
If, in particular, the particle energy lies well below the top of the right-hand barrier also, the quantities exp {-2Kz} and o-2 are small compared to unity [cf. (46) and (51) in ref. 14)], and eq. (52e) becomes O(x2) ~ 2 cos L exp { K z } l q - ~ ( x 2 ) l e x p {Iw2(x2)]}.
(52e')
7. Transmission coefficient in first-order approximation. Brief review
In some physical situations, e.g. in the current theory of fission, where only a rather rough description of the transmission properties of the double-barrier is needed, the first-order approximation is sufficient. For the benefit of those who would like to use the above results in the first-order approximation only, without studying the whole treatment given above, we shall now specialize the general formula for the transmission coefficient and give approximate expressions for 0-1 and o-z pertaining to this particular case. The transmission coefficient is given by (39) or (41). In the first-order approximation we have K t = +_
Q(z)dz
,
(53a)
Q(z)dz
,
(53b)
1
K z = +__ 3
where Q ( z ) is defined by (4). The plus sign refers to the sub-barrier case and the minus sign to the super-barrier case. We may replace the exact expression (27) for c~ by c~ ~ L-(~l~-~r2),
(54)
except when the energy is well below the lowest barrier maximum and is furthermore in the neighbourhood of a resonance. In that case the half-width A c~of the resonance is in general much smaller than the terms neglected in the approximate expression (54) for e. In other words, the use of (54) instead of the exact expression (27) for e means that the resonance peak will be displaced, the displacement being in general much larger than the half-width of the resonance. The quantity L is given by L=
l Re ftta Q ( z ) d z 2
.
(55)
An approximate expression for o-1 as a function of K1/~z can be obtained from the definition (20) by substituting for F12(xl, x3) the expression for F12(- m, + ce) per-
648
N. FR()MA.N AND 0. DAM M E R T
taining to a parabolic barrier 1s). The corresponding expression for a2 is quite analogous except that/£1 should be replaced by Kz. Thus,
al ~ a(Ka/rc),
(56a)
a z "~ ~r(K2/rc),
(56b)
with a(K/zc) given by a(K/n)=-argF(¼+i~n)+
2nKlnlK - 2 nI
2nK+½[arctge-K--¼n]
_12 I KlnI K-~z __Krc+ a r g F ( ½ - i K ) l .
(57)
The function a(K/rc) is depicted in fig. 5. Cf. also the work by Ford, Hill, Wakano and Wheeler 19) on quantum effects near a barrier maximum, where the approximate formula [their eq. (16a)] ( ~) K ½argF ½-i ~ ---ln 2~r
E(K) ~ 2~ + (1__] ] ~ \4y] _1
7 = 1.78107 ....
(58)
is given and checked numerically (their table 1). The last naember of (57) corresponds to the expression given by Connor 5) for the phase factor ½~o(e)in his notation. We remark that for a single parabolic barrier represented by the potential V(x) Vo--½mo)Zx z we have [refs. 5,19)] =
K _ Vo-E 7z
(59)
hco
We are much indebted to Professor P. O. Fr6man for many valuable suggestions concerning the manuscript. We would also like to thank Dr. A. Skorupski for his interest in carrying out numerical calculations in connection with this paper. His results greatly stimulated our further work on the problem. References 1) 2) 3) 4) 5) 6)
D. Bohm, Quantum theory (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961) L. V. Iogansen, JETP (Soy. Phys.) 18 (1964) 146 L. V. Iogansen, JETP (Soy. Phys.) 20 (1965) 180 M. S. Child, Molec. Phys. 12 (1967) 401 J. N. L. Connor, Molec. Phys. 15 (1968) 37 L. L Ponomarev, Lectures on quasiclassics ITF-67-53, Institute for Theoretical Physics, Acad. Sc. Ukr.SSR (Kiev 1968) 7) A. Zwaan, Intensit~iten im Ca-Funktenspektrum (Academisch Proefschrift, Joh. Ensched6 en Zonen, Haarlem, 1929) 8) V. M. Strutinsky, Nucl. Phys. A95 (1967) 420; V. M. Strutinsky and S. Bjornholm, Proc. Syrup. on nuclear structure, Dubna, i968
TUNNELING AND TRANSMISSION 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
649
S. G. Nilsson et al., Nucl. Phys. A l l 5 (1968) 545 A. V. Ignatyuk, N. S. Rabotnov and G. N. Smirenkin, Phys. Lett. 29B (1969) 209 N. Frtiman, Ark. Fys. 32 (1966) 541 N. Fr~Sman, Ark. Fys. 32 (1966) 79 N. FrOman and U. Myhrman, Ark. Fys., to be published N. Fr~Sman and P. O. Fr6man, Nucl. Phys. A147 (1970) 606 N. Fr~Sman and P. O. FfiSman, JWKB Approximation, contributions to the theory (NorthHolland, Amsterdam, 1965) N. Fr~man, to be published A. Skorupski, to be published M. Soop, Ark. Fys. 30 (1965) 217 K. W. Ford, D. L. Hill, M. Wakano and J. A. Wheeler, Ann. of Phys. 7 (1959) 239