Tunnelling of plane waves through a square barrier

0.18 MB·English

by  J. Julve

Tunnelling of plane waves through a square barrier
J. Julve
8
0 IMAFF, Consejo Superior de Investigaciones Cient´ıficas, Serrano 113 bis, Madrid
0 28006,Spain
2
E-mail: [email protected]
n
a
J
F. J. de Urr´ıes
1
1 Departamento de F´ısica, Universidad de Alcala´ de Henares, Alcala´ de Henares

(Madrid), Spain
]
h
E-mail: [email protected]
p
-
t
n Abstract. The time evolution of plane waves in the presence of a 1-dimensional
a
u square quantum barrier is considered. Comparison is made between the cases of an
q infinite and a cut-off (shutter) initial plane wave. The difference is relevant when the
[
results are applied to the analysis of the tunnelling regime. This work is focused on

1 the analytical calculation of the time-evolved solution and highlights the contribution
v of the resonant (Gamow) states.
4
2
PACS numbers: 11.10.Ef, 11.10.Lm, 04.60
8
1
.
1
0
8
0
:
v
i
X
r
a
Plane wave tunnelling in a square barrier 2
1. Introduction
Thestudy ofthetraversalofpotentialbarriersbywave packets inthe1-dimensionalcase
isrelevantfortheelectrontransportthroughbarrierjunctions,thephysicsofwaveguides
and light transmission through Photonic Band Gaps, etc. On more theoretical grounds,
theresonances, transientexcitationsandthetimeofarrivalofwavepackets, withensuing
paradoxes of non-locality and super-luminal tunnelling, are further interesting aspects.
Relevant resonances are mostly expected in systems of two or more barriers
separated by a gap. Actually they generally occur in any simple barrier, and the plain
square barrier is most tractable, fully representative for many theoretical purposes and
devoid of bound states or other unessential features.
On the other hand, practical calculations face important limitations and the effort
of pushing the analytical calculation as far as possible is always rewarding. Significant
progress has been accomplished [1] for a Gaussian packet impinging on a square barrier,
where the Gaussian structure was exploited, and [2] for the shutter problem, where the
contribution of the resonant (Gamow) states in a double square barrier was worked out.
In this work we consider the time evolution, in the presence of a square barrier
of height V in the interval [0,L], of an initial plane wave with support on the whole
1-D space, aiming to work out the contribution of the resonant states. Our calculation
extends the results of the shutter problem, where the initial plane wave occupied the
space axis at one side of the barrier and was specially suited to study details of the
propagation of the wave front.
The reason for doing this is twofold. First, the simpler problem ofthe shutter initial
condition ψ (x) = θ( x)eikx has been assumed in the literature to be meaningful also
s
−
for the analysis of the transmission in the tunnelling regime when k k √2mV .
V
≪ ≡
Unfortunately such a cut-off initial wave function is highly non-monochromatic as its
Fourier transform shows:
1 1 1 1 π
ψ˜ (p) = ∞ dxe ipxψ (x) = PP + δ(k p)
s − s
√2π i √2π k p 2 −
Z−∞ − r
The Principal Part distribution around k is exceedingly wide so that the amplitude
of the momentum p = k skimming the barrier is (k k ) 1, a sizeable value (in
V V −
∝ −
modulus, actually it is > k 1) even for very small incident k , with a slow decrease for
V−
p > k . Secondly, by considering infinite plane waves the way is paved for further uses,
V
since superpositions of them build up any desired initial wave packet. In particular, the
compactresult obtainedfortheGaussianwave packet [1]couldbere-derivedpinpointing
the contribution of each single Fourier component.
In this work we follow the approach used by Peierls and Garc´ıa-Calder´on [3],
namely Laplace-transforming the time evolution equation into a second order linear
differential equation (SOLDE) in one variable, expressing its solutions in terms of the
Green function (GF) with resonant boundary conditions (RBCs) and then undoing the
Laplace transform.
Plane wave tunnelling in a square barrier 3
The first step is done in Section 2, where a brief discussion is made of the use of
the Green’s method for the solution of SOLDEs when the boundary conditions (BCs)
required for the GF and for the solution are different. The GF with RBCs features a
simple structure of isolated (resonance) poles, in terms of which the (p-dependent)
Laplace-transformed wave function is worked out in Section 3. Then the explicit
calculation of the inverse Laplace transform leading to the sought after t-dependent
solution is carried out in Section 4.
As a check of this cumbersome analytical calculation, the t = 0 (Section 5) and the
t = (Section 6) limits of the time-dependent solution ψ(x,t) are calculated: at t = 0
∞
one must recover the initial plane wave and at t = a suitable stationary solution
∞
must be reached. Then the conclusions are drawn in Section 7. Some notations and a
number of calculations and technical details are deferred to the Appendices.
2. The evolution equations
We consider the 1-D time-dependent Schr¨odinger equation
∂
(ih¯ H)ψ(x,t) = 0 (1)
∂t −
where the Hamiltonian H = p2/2m +V(x) corresponds to the square barrier potential
V(x) = θ(x)θ(L x)V and the solution satisfies the initial condition ψ(x,0) ψ (x) =
o
− ≡
eikx , k > 0.
The Laplace transform ψ¯(s) = dt e stψ(t) on the time variable, brings the
o∞ −
parabolic partial derivative differential equation (1) to the simpler SOLDE
R
∂2 2m
L ψ¯(x,p) [ +p2 V(x)] ψ¯(x,p) = iα eikx (2)
x ≡ ∂x2 − h¯2
where α 2m/h¯ and p2 = iαs (Appendix A). Besides the usual scattering in and out
≡
¯
solutions, the time-independent Schr¨odinger homogeneous equation L ψ = 0 has the
x
resonant solutions satisfying the homogeneous outgoing RBCs
¯ ¯ ¯ ¯
∂ ψ = ipψ(0) , ∂ ψ = ipψ(L) , (3)
x x=0 x x=L
| − |
which exist only for a denumerable set of isolated values p of p lying in the lower half
n
complex plane (Appendix A).
A solution to the inhomogeneous equation (2) for p2 R can be explicitly written
+
∈
down for each region of V(x) :
iα
ψ¯ (x,p) = Be ipx + eikx (x < 0)
I −
p2 k2
−
iα
ψ¯ (x,p) = M eip′x +N e ip′x + eikx (0 x L) (4)
II −
p2 k2 ≤ ≤
′
−
iα
ψ¯ (x,p) = Aeipx + eikx (x > L)
III
p2 k2
−
where the amplitudes A, B, M and N are functions of p completely determined by the
matching conditions at x = 0 and x = L and have a common denominator of the form
Plane wave tunnelling in a square barrier 4
D(p) 2eip′L 2e ip′L (notation in Appendix A). Out of all the terms of the general
−
≡ ⊖ −⊕
homogeneous solution, the choice of eipx (for x > L) or e ipx (for x < 0) is dictated by
−
the behaviour of the solution for x , since a small positive imaginary part in p
→ ±∞
is supposed when performing the inverse Laplace transform leading to ψ(x,t) , t > 0
(Appendix D).
However, this expression of the solution is not suited to perform the inverse Laplace
transform back to ψ(x,t) because of the nontrivial analytical form of the amplitudes
¯
above. Instead the GF approach lets us to express ψ(x,p) as a sum of isolated pole
terms plus other simple terms easier to deal with. To this end, (2) together with the
Green equation for G(x,x,p) can be written in the form
′
L ψ¯(x,p) iαeikx = 0 (5)
x
−
L G(x,x,p) δ(x x) = 0 , (6)
x ′ ′
− −
where G(x,x,p) is required to obey RBCs, namely
′
∂ G(x,x,p) = ipG(0,x,p) , ∂ G(x,x,p) = ipG(L,x,p) (7)
x ′ x=0 ′ x ′ x=L ′
| − |
It is expected that G(x,x,p) = L 1 will have poles at p = p , where the homogeneous
′ −x n
¯
equation L ψ = 0 with RBCs has non-trivial solutions.
x
The equations (5) and (6) may conveniently be given the short-hand notation
Ψ = 0 and Γ = 0 respectively. Then the Green method uses the integral equation
Ldx [ψ¯Γ GΨ] = 0 to obtain ψ¯(x,p) in terms of G(x,x,p). Only if ψ¯ and G obey the
0 − ′
same homogeneous BCs, the surface terms in this integral cancel out and one obtains
R
the familiar result ψ¯ (x,p) = Ldx iαG(x,x,p)eikx′.
II 0 ′ ′
¯ ¯ ¯
However the BCs for ψ(x,p) (involving ψ and ∂ ψ both at x = 0 and at x = L)
R x
stemming from(4)aredifferent from(3)andnon-homogeneous. Inthatcaseoneobtains
a modified expression for the region II, namely
L
ψ¯ (x,p) = dx iαG(x,x,p)eikx′
II ′ ′
Z0
α α
G(L,x,p)eikL G(0,x,p) (8)
− p+k − p k
−
¯
For the external regions I and III, the matching of ψ at x = 0 and x = L determine
the coefficients B and A respectively, so that
L
ψ¯ (x,p) = iαe ipx dx iαG(x,0,p)eikx′
I − ′ ′
Z0
α α
G(L,0,p)eikLe ipx G(0,0,p)e ipx (9)
− −
− p+k − p k
−
iα iα
e ipx + eikx
−
− p2 k2 p2 k2
− −
and
L
ψ¯ (x,p) = iαeip(x L) dx iαG(x,L,p)eikx′
III − ′ ′
Z0
α α
G(L,L,p)eip(x L)eikL G(0,L,p)eip(x L) (10)
− −
− p+k − p k
−
iα iα
eip(x L)eikL + eikx
−
− p2 k2 p2 k2
− −
Plane wave tunnelling in a square barrier 5
For comparison, only the last term in the r.h.s. of (8), the last three ones in (9) and the
third one in (10) appear in the shutter problem.
The crucial advantage of the method is that almost all the Green functions involved
in the equations above can be expanded as a sum of terms which are simple poles in p .
n
3. Analytical structure of the p-dependent solution
General theorems [4] and the explicit analytical derivation (Appendix B) of G(x,x,p)
′
show that G(x,x,p) 1/ p 0 as p in the complex plane for almost any
′
| | ∼ | | → | | → ∞
x [0,L] and x [0,L] . Then the Mittag-Leffler theorem tells that
′
∈ ∈
C (x,x)
n ′
G(x,x,p) = (11)
′
p p
n n
X −
The exception is for G(0,0,p) and G(L,L,p), the modulus of which grows as p for
| |
p in the lower half complex plane p and requires some special care (Appendix
| | → ∞
C).
The residues in the r.h.s of (11) can be easily computed [3] and one finds
C (x,x) = u (x)u (x)/N , where the functions u (x) belong to the denumerable
n ′ n n ′ n n
set of the resonant solutions satisfying L u (x) = 0 with RBCs
x n
∂ u (x) = ip u (0) , ∂ u (x) = ip u (L)
x n x=0 n n x n x=L n n
| − |
and N are suitable normalization factors (Appendix A).
n
The inverse Laplace transform
ψ(x,t) = 1 c+i∞ds estψ¯(x,p(s)) = 1 +∞dp p e−i2pm2tψ¯(x,p) , (12)
2πi Zc−i∞ 2πm Z−∞
written as an integral over the real momentum variable p (Appendix D), leads one to
¯
consider the pole expansion of p ψ(x,p) in the integrand, which can be worked out for
each region I, II and III (Appendix E):
p u (0) L
p ψ¯ (x,p) = iαe ipx n dx u (x)eikx′
I − ′ n ′
p p N
Xn − n n Z0
k p 1 u (L)u (0)
+α G(L,0, k)e ipxeikL α n n n e ipxeikL
− −
p+k − − p p k +p N
n n n n
X −
p 1 p2
+α G(0,0,0)e ipx α G(0,0,k)e ipx (13)
− −
k − kp k
−
p2 1 1 u2(0) iα 1 1
+α n e ipx + ( + )(eikx e ipx)
− −
p p p k p N 2 p k p+k −
n n n n n
X − − −
p u (x) L
p ψ¯ (x,p) = iα n dx u (x)eikx′
II ′ n ′
p p N
Xn − n n Z0
k p 1 u (L)u (x)
+α G(L,x, k)eikL αeikL n n n (14)
p+k − − p p k +p N
n n n n
X −
k p 1 u (0)u (x)
n n n
α G(0,x,k)+α
− p k p p k p N
n n n n
− X − −
Plane wave tunnelling in a square barrier 6
p u (L) L
p ψ¯ (x,p) = iαeip(x L) n dx u (x)eikx′
III − ′ n ′
p p N
Xn − n n Z0
p p2 α
α G(L,L,0)eip(x L)eikL + G(L,L, k)eip(x L)eikL
− −
− k p+k k −
p2 1 1 u2(L)
α n eip(x L)eikL (15)
−
− p p p k +p N
n n n n n
X −
p 1 u (0)u (L) k
+α n n n eip(x L) +α G(0,L,k)eip(x L)
− −
p p k p N k p
n n n n
X − − −
iα 1 1
+ ( + )(eikx eip(x L)eikL)
−
2 p+k p k −
−
In the third row of (14) and in the fourth row of (15) one recognizes the terms of the
shutter problem given in [2]. The shutter terms in the region I, not calculated in [2],
are given in the third and fourth rows of (13).
4. The time-dependent solution
For each of the terms in the expressions above, the integrals stemming from (12) can be
brought to the form of an integral representation of the error function erfc(z) so that
their inverse Laplace transform can be carried out thoroughly (Appendix F). We obtain:
u (0) L
ψ (x,t) = n ( dx u (x)eikx′) [AI]
I − N ′ n ′ n
n n Z0
X
p u (L)u (0)
+ikG(L,0, k)eikL [BI ] ieikL n n n [BI] (16)
− −k − n k +pn Nn n
X
1 1 1 1 u2(0)
+iG(0,0,0) [SI] iG(0,0,k) [SI]+i n [SI]
k 0 − k k p k p N n
n n n n
X −
1 1 1 1
eikx [SI] eikx [SI]+ [SI]+ [SI]
− 2 1 − 2 2 2 3 2 4
where the factors in square brackets embody the time (τ = t/2m) dependence of the
solution and are the result of the integrations over the momentum p, namely:
AIn = −eix4τ2(pney−2x,n erfc(y−x,n)− √eiππ4τ)
BIk = eix4τ2ey−2x,−k erfc(y x, k)
− − −
BnI = eix4τ2ey−2x,n erfc(y x,n)
−
SI = eiπ4 x eix4τ2
0 √πτ 2τ
SkI = −eix4τ2(k2ey−2x,k erfc(y−x,k)− √eiππ4τ(k − 2xτ)) (17)
SnI = −eix4τ2(p2ney−2x,n erfc(y−x,n)− √eiππ4τ(pn − 2xτ))
Plane wave tunnelling in a square barrier 7
SI = e iτk2 erfc(i√iτk)
1 − −
SI = e iτk2 erfc( i√iτk)
2 − − −
S3I = −eix4τ2ey−2x,k erfc(y−x,k)
S4I = −eix4τ2ey−2x,−k erfc(y−x,−k)
where we have introduced the variables y e iπ(4τ) 1(x 2τq) for q = p ,k, k.
x,q − 4 −2 n
≡ − −
Likewise
u (x) L
ψ (x,t) = n ( dx u (x)eikx′) [AII]
II − N ′ n ′ n
n n Z0
X
p u (L)u (x)
+ikG(L,x, k)eikL [BII ] ieikL n n n [BII] (18)
− −k − n k +pn Nn n
X
p u (0)u (x)
ikG(0,x,k) [SII]+i n n n [SII]
− k k p N n
n n n
X −
where
AInI = −pne−iτp2n
BII = e iτk2 erfc( i√iτk)
k − − −
−
BnII = −e−iτp2n erfc(i√iτpn) (19)
SII = e iτk2 erfc(i√iτk)
k − −
SnII = −e−iτp2n erfc(i√iτpn)
Finally
u (L) L
ψ (x,t) = n ( dx u (x)eikx′) [AIII]
III − N ′ n ′ n
n n Z0
X
eikL eikL
iG(L,L,0) [BIII]+iG(L,L, k) [BIII]
− k 0 − k k
−
1 1 u2(L)
ieikL n [BIII] (20)
− p k +p N n
n n n n
X
p u (0)u (L)
ikG(0,L,k) [SIII]+i n n n [SIII]
− k p N n
n n n
X −
1 1 1 1
eikx [CIII] eikx [CIII]+ eikL [CIII]+ eikL [CIII]
− 2 1 − 2 2 2 3 2 4
where
AInII = −ei(x−4τL)2(pneyx2−L,n erfc(yx−L,n)+ √eiππ4τ)
BIII = eiπ4 x−Lei(x−4τL)2
0 − √πτ 2τ
B−IIkI = −ei(x−4τL)2(k2eyx2−L,−k erfc(yx−L,−k)+ √eiππ4τ(−k + x2−τL))
BnIII = −ei(x−4τL)2(p2neyx2−L,n erfc(yx−L,n)+ √eiππ4τ(pn + x2−τL))
Plane wave tunnelling in a square barrier 8
SkIII = −ei(x−4τL)2eyx2−L,k erfc(yx−L,k)
SnIII = −ei(x−4τL)2eyx2−L,n erfc(yx−L,n) (21)
CIII = e iτk2 erfc( i√iτk)
1 − − −
CIII = e iτk2 erfc(i√iτk)
2 − −
C3III = −ei(x−4τL)2eyx2−L,−k erfc(yx−L,−k)
C4III = −ei(x−4τL)2eyx2−L,k erfc(yx−L,k)
In the equations (16), (18) and (20) the terms [S] are the ones arising in the shutter
problem. The particular values of the Green function involved are calculated in
Appendix B.
5. The short time limit
The t 0 limit is interesting both as a check of the calculation above and for the
→
study of the scattered wave at short times. Here we aim only to recover the initial wave
function ψ(x,0) = eikx at t = 0, which must happen in each of the regions I, II and
III.
Careful use of the limiting values of erfc(z) and/or w(z) = e z2erfc( iz) for z 0
−
− →
and for z in different directions of the complex z plane must be made. Notice
→ ∞
that y tends to in different directions for different q. Also the properties of the
x,q
∞
set of resonant functions u (x) as a basis of the space of solutions are crucial in region
n
II (Appendix G). Asymptotic expressions for erfc(z) and w(z) can be used to obtain
approximations to the form of ψ(x,t) in each region for small values of t, a task that
will be faced elsewhere.
In region I, the factor AI tends to 2iδ(x) and also SI , SI and SI tend to a
n − 0 k n
distribution concentrated in x = 0. The factors BI , BI , SI and SI vanish exactly in
k n 3 4
this limit and both SI and SI tend to 1 . There−fore, as required for x < 0 ,
1 2 −
lim ψ (x,t) = eikx
t 0 I
→
In region II, the factors AII , BII , BII , SII and SII tend to the value 1 so that
n −k n k n −
the [A] term in (18) gives
L u (x)u (x)
dx p n n ′ eikx′ = eikx (22)
′ n
N
Z0 n n
X
The [B] terms yield
p u (L)u (x)
ikG(L,x, k)eikL ieikL n n n
− − k +p N
n n n
X
1 u (L)u (x) p u (L)u (x)
= ikeikL n n ieikL n n n
k p N − k +p N
n n n n n n
X − − X
u (L)u (x)
= ieikL n n = 0 (23)
− N
n n
X
Plane wave tunnelling in a square barrier 9
and likewise for the [S] terms:
p u (0)u (x)
n n n
ikG(0,x,k) i
− k p N
n n n
X −
1 u (0)u (x) p u (0)u (x)
n n n n n
= ik i
k p N − k p N
n n n n n n
X − X −
u (0)u (x)
n n
= i = 0 (24)
N
n n
X
In region III, the factor AIII tends to 2iδ(x L) and also BIII, BIII and BIII
n − − 0 k n
−
approach distributions concentrated in x = L . Thus they vanish for x > L. The factors
SIII, SIII, CIII and CIII vanish exactly in this limit, whereas both CIII and CIII tend
n 3 4 1 2
to 1. Therefore the final result is that, for x > L , also
−
lim ψ (x,t) = eikx
t 0 III
→
We have quoted singularities at the points x = 0 and x = L respectively in the
regions I and III, but these points must be excluded from the domain of respectively
ψ (x,t) and ψ (x,t), as argued in Appendix F.
I III
6. The large time limit
For τ we see that also y as before, but in yet different directions of the
x,q
→ ∞ → ∞
complex plane for the different q.
In region I, the factors AI , BI , BI , SI , SI , SI and SI vanish, whereas
n k n 0 n 2 4
−
SI 2k2e iEkte ikx
k → − − −
SI 2e iEkt (25)
1 → − −
SI 2e iEkte ikx
3 → − − −
Using equation (B.4), some of the e ikx terms cancel out in (16) and one is left with
−
lim ψ (x,t) = e iEkt(eikx +R(k)e ikx) (26)
t I − −
→∞
= e iEktφin(x)
− r
that is, the asymptotic scattering in solution, where E = k2/2m (see Appendix H).
k
In region II, the factors AII, BII , BII and SII vanish, whereas SII 2e iτk2.
n k n n k → − −
−
Therefore, using (B.2), for 0 x L we have
≤ ≤
lim ψ (x,t) = 2ik G(0,x,k)e iτk2
t II −
→∞
= e iEktφin(x) (27)
− r
Thus the infinite plane wave evolves into the same final state of the shutter initial
condition also in region II.
For the region III, the only non-vanishing factors are
SIII 2 eik(x L)e iτk2
k → − − −
CIII 2 e iτk2 (28)
2 → − −
CIII 2 eik(x L)e iτk2
4 → − − −
Plane wave tunnelling in a square barrier 10
Then the terms corresponding to CIII and CIII in (20) cancel each other and, again
2 4
using (B.2), one is left with the same asymptotic solution of the shutter
lim ψ (x,t) = 2ik G(0,L,k)eik(x L)e iτk2
t III − −
→∞
= T(k)eikxe iEkt (29)
−
= e iEktφin(x)
− r
The finitetimebehaviour isthus madeupoftransient modeswhich quickly dampen
out, leaving (for k > 0) the scattering asymptotic solution with outgoing BC at x = L.
7. Conclusions
A solution for the time evolution of an infinite plane wave in the presence of a simple
square barrier has been worked out for each of the regions of the potential and Section
4 is the main result of this work. Among other terms, this solution contains a sum of
explicit analyticalcontributionscorresponding toeachofthe(infinitely many) resonance
poles. As in previous works in related problems [1] [2], only the location of these poles
needs to be obtained by numerical methods.
In the solution obtained the shutter terms have been pinpointed. Similarly, the
contributions to the time-evolved wave function coming from the segments of the initial
wave function lying inside and at the right of the barrier can be identified.
Asthemainapplicationofthisknowledgeweenvisagethepossibilityofstudyingthe
enhancement or the suppression of the transmission of the single Fourier components
of any realistic wave packet. This should provide new detailed insight on interesting
phenomena like the super-luminal tunnelling [5], the breakdown of energy conservation
[6] by transient interference in wave packet collisions with barriers [7] or the rising of
forerunners.
The explicit solution obtained for finite time is useful for deriving approximations
valid for short times, hence for the study of transient structures and forerunners. An
immediate result of the work is that the large time limit yields the same stationary
solutionoftheshutter. Thisagainshowsthattheresonancescontributeonlytotransient
structures of the scattered wave.
Acknowledgments
Work supported by MEC projects BFM2002-00834 and FIS2005-05304. The authors
are indebted to J. Le´on for suggestions and helpful discussions. J. Julve acknowledges
the hospitality of the Dipartimento di Fisica dell’Universit`a di Bologna, Italy, where
part of this work was done.
Appendix A. Resonant solutions
We adopt units such that h¯ = 1 . The homogeneous equation L u(x) = 0 with the
x
RBCs(3)hassolutionsonlyforadenumerablesetofvaluesp lyinginthelowercomplex
n