Articol ATEE 2013
Articol ATEE 2013
Articol ATEE 2013
Wavelet Solution
of the Time Independent Schrodinger Equation
for a Rectangular Potential Barrier
2
AbstractElectronics industry is a major user of MetalInsulator-Metal (MIM) structures. When the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the
MIM structures, we are dealing with the Fowler-Nordheim field
emission. In MIM structures we have to calculate the currents
densities that go through. Unfortunately, the calculation methods
currently used lead to values that get predictions of emission
current density too low. The current density is calculated based
on the transmission coefficient through the barrier, which is determined by solving the Schrodinger equation. One of the oldest
and most efficient methods of solving the Schrodinger equation
is the WKB method. This paper proposes the wavelet method
for solving the 1D Schrodinger equation, that is: determining
the wave function using the harmonic multiresolution analysis.
KeywordsSchrodinger equation, dyadic wavelets, potential
barriers
I. I NTRODUCTION
As the size of the structures used in electronics industry goes
towards nanaoscopic scale, the importance of quantum effects
must be seriously considered. Metal-Insulator-Metal (MIM)
structures begin to assert themselves in microelectronics industry against those based on silicon because they allow a
substantial reduction in production costs and achieve higher
speeds compared to the latter [1].
In the case of MIM structures with a dielectric thickness of
a few nanometers, conduction can occur in two ways.
At high temperatures and electric field strengths smaller
than 108 V/m, the electrons, due to increased thermal agitation,
gain enough energy to climb the potential barrier that occurs at
metal-insulator interface. This conduction mechanism is called
Schottky injection [2] [3].
At normal temperatures and electric field intensity that
exceeds 108 V/m, the potential barrier, corresponding to the
metal-insulator contact becomes very thin and there will
appear a quantum type phenomenon called tunneling, when
electrons, although they do not have enough energy to climb
the potential barrier, they manage to pass through the barrier.
When the tunneling phenomenon occurs in metal-insulator
contact barrier, as in the MIM structures, this is a Fowler Nordheim injection [2] [3].
In the case of devices that are based on the phenomenon of
Fowler - Nordheim injection [1] [4] [5], the emission density
ii) for x = a
C (a) = R (a),
0
C (a) = R (a) .
(9)
SCHR ODINGER
EQUATION FOR THE
RECTANGULAR POTENTIAL BARRIER
Let us consider the one-dimensional time independent
Schrodinger equation
~2 d2
+ U (x)(x) = E(x),
2m dx2
where the potential energy U (x) is by the form
(
U0 , 0 x a
U (x) =
, U0 > 0 ,
0
, otherwise
(1)
q
ix 2mE
~2
+k2 exp
ix
2mE
~2
R = |r|2 , T = |t|2
(11)
with
(2)
r = Br + Bl 1,
t = Br expia(kk0 ) +Bl expia(k+k0 ) .
(12)
(3)
if x < 0 ,
(4)
if x > a .
(5)
and
(6)
Fig. 2. The wave function after tunneling the rectangular barrier when a = 1.
where k 2 = 2m(U~02E) .
In this case, the solution has the form
C (x) = Br expikx +Bl expikx .
(7)
L (0) = C (0).
(8)
HW
b HW (0) = 0.
a wavelet) and its energy EHW =
(x)
= 1,
2
The Calderons admissibility constant is
HW
2
Z
b ()
(23)
0 < CHW =
d = ln(2) < .
||
R
More, according to multiresolution analysis the followings are
satisfied
Z
Z
HW
HW
(x)dx = 1 ,
(x)dx = 0 .
(24)
R
D [(x)] = 2 2 (2x) ,
T [(x)] = (x 1) .
(13)
(14)
HW
HW
HW
, : R C ,
def
(x) =
exp2ix 1
2ix
(15)
respectively,
def
HW
HW
(16)
then, we can easily get Fourier transform for (15) and (16),
i.e.,
b HW () = ( + 2)
(18)
b HW () = () .
2 4
1,
() =
0,
elsewhere.
HW
(19)
cHW () = ( + 4)
(26)
d
HW () = ( + 6) .
(27)
,
=
= ,
=
=1
E
D
D HW
E
D HW
E
D HW
E
HW
, HW = HW ,
HW =
, HW = 0
=
,
HW
HW
HW
HW
HW ,
HW
E
D HW
,
HW = 0
HW
,
HW , HW
HW , HW
E
D HW
HW = 0 .
,
HW
(28)
According to (21), the proof is trivial.
SW
The Shannon scaling function (x) is a particular case of
(15), [21] [17]
h HW x i
SW
(x) = Re
= sinc(x)
(29)
2
and also, according to the multiresolution analysis, the ShanSW
non wavelet function (x) can be expressed as
SW
SW
SW
1
(x) = 2 (2x 1)
x
.
(30)
2
Their Fourier transforms are
(20)
HW
(25)
(22)
b SW () = ( + 3)
(31)
respectively,
h
i
b SW () = exp i
2
+ 3 ( + 3) .
2
(32)
= 0.
S CHR ODINGER
EQUATION FOR THE RECTANGULAR
POTENTIAL BARRIER
A. Problem formulation
Let us consider the time-independent Schrodinger equation
00 (x) = K(x) (x) ,
(34)
HW
HW
HW
(44)
where
2m
K(x) = 2 (U (x) E) .
~
(35)
HW
d2
dx2
+
HW
HW
d
dx
+
HW
+ 2HW HW
y 00 + (y 0 )2 = K(x) .
(36)
(37)
(38)
where
hZ
h h (x) +
n k n k (x) ,
n, kZ
+ 2
HW HW
W(x2 )
A expW(x1 )
C1 = B exp
x2 x1
(40)
d2
dx2
d2
dx2
HW !2
!2
d
dx
+ HW
HW
HW
+ HW
+ HW
d2
dx2
!2
HW
2
+ HW
d
dx
+ HW
d
dx
HW
+ 2HW
HW
3
3
56HW 5
2HW 10
HW HW + 14HW HW =
3
3
b
K()d
2
2
b
K()d
.
4
(46)
SW
SW
SW
(47)
d2
dx2
2
+ SW
SW
+ SW
d
dx
d2
dx2
SW
d
dx
+ SW
!2
SW
+ 2SW SW
!2
+
(48)
SW
d d
dx
dx
= K(x) .
SW
HW
d d
+
dx
dx
SW
W(x1 )
x1 B expW(x2 )
C2 = x2 A exp
.
x2 x1
HW
HW
HW
HW
HW dHW
d d
d d
d
+ 2HW HW
+ 2
HW HW
+
dx
dx
dx
dx
dx
dx
HW
HW
HW
HW
d
d
d d
+ 2HW HW
= K(x) .
dx
dx
dx
dx
SW
(41)
!2
(45)
HW
HW
HW ,
HW and
By inner product with functions , ,
according to (21), (28), (42), (43), results the coefficients HW ,
HW and their complex conjugates
HW , HW (for more technical
details of computation see [11] [16] [15] [18]). Finally, they
are determined by solving the following nonlinear system of
equations
Z 2
3
b
Z0 0
2
b
K()d
8
HW
HW + 6HW
HW + 54HW HW + 22HW HW = 3
HW
HW
HW
W(x) =
HW
(x) + (x) +
(x) .
W (x) = HW (x) +
HW
HW
HW
SW
Z
2
2 + 3 2
7 + 15 2
1
b
SW SW +
SW =
K()d
SW +
2
3
7 2
3
1
2SW =
3 SW
4
2
Z
b
K()
exp
2
i
2
d +
b
K()
exp
i
2
d
(49)
b
In the relaions above was denoted by K()
the Fourier
transform of (35).
C. Results
This paper considers three cases for the rectangular potential
barrier, when the ends of the barrier are in x1 = 0, x2 = a = 1,
a second one, when x1 = 0 and x2 = a = 3 and when x1 = 0
and x2 = a = 10. The wave function was calculated, with
the analytical method and with the wavelet methods, for these
values of x. The absolute error was calculated for each value
of in x1 andx2 as xnum xAN , with xnum representing
the numerical solution and xAN being the exact solution. The
results are listed in the tables below.
From Figures 3, 4, 5, 6, one can observe that inside as
well as outside the potential barrier, the harmonic wavelet
solution and the Shannon wavelet solution represent a good
approximation of the analytical solution.
The transmission coefficient
2)
Tr = (x2 ) (x
(50)
SW
x1 = 0
1.642 0.742i
1.642 0.742i
1.642 0.742i
1.642 0.742i
1.642 0.742i
1.642 0.742i
1.642 0.742i
x2 = a = 1
0.144 + 0.125i
0.144 + 0.125i
0.144 + 0.125i
0.144 + 0.125i
0.144 + 0.125i
0.144 + 0.125i
0.144 + 0.125i
Fig. 3. Real part of analytical solution by dash line and real part of harmonic
solution by solid line for a = 3.
TABLE II
A BSOLUTE ERROR
Absolute error for x1 = 0 and x2 = 1
Method
HW
SW
7 1010 + 1 1010 i
1 109 3 1010 i
0
1 1010 1 1010 i
TABLE III
VALUES OF FOR x1 = 0 AND x2 = a = 3
x
AN
HW
SW
x1 = 0
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
x2 = a = 3
0.002 + 0.002i
0.002 + 0.002i
0.002 + 0.002i
0.002 + 0.002i
0.002 + 0.002i
0.002 + 0.002i
0.002 + 0.002i
Fig. 4. Real part of analytical solution by dash line and real part of harmonic
solution by solid line for a = 10.
TABLE IV
A BSOLUTE ERROR
Method
HW
SW
1 1011 + 7 1012 i
5 1011 i
TABLE V
VALUES OF FOR x1 = 0 AND x2 = a = 10
x
AN
HW
SW
x1 = 0
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
1.666 0.745i
x2 = a = 10
1.36 109 + 1.22 109 i
1.37 109 + 1.23 109 i
1.37 109 + 1.23 109 i
1.4 109 + 1.23 109 i
1.4 109 + 1.225 109 i
1.368 109 + 1.229 109 i
1.369 109 + 1.225 109 i
Fig. 5. Real part of analytical solution by dash line and real part of Shannon
solution by solid line for a = 3.
TABLE VI
A BSOLUTE ERROR
Method
HW
SW
1 1011 + 3 1012 i
9 1013 + 2 1012 i
TABLE VII
T RANSMISSION COEFFICIENT FOR x2 = 1, x2 = 3 AND x2 = 10
Method
AN
HW
SW
x2 = 1
0.036
0.036
0.036
x2 = 3
1 105
1 105
1 105
x2 = 10
3 1018
3 1018
3 1018
Further results, regarding wavelet based wave function approximation for other potential barrier types, will be reported
elsewhere.
ACKNOWLEDGMENT
This work has been supported by the Sectorial Operational
Programme Human Resources Development 2007-2013 of the
Romanian Ministry of Labour, Family and Social Protection
through the Financial Agreement POSDRU/107/1.5/S/76813.
R EFERENCES
Fig. 6. Real part of analytical solution by dash line and real part of Shannon
solution by solid line for a = 10.
V. CONCLUSION
This paper considers a numerical wavelet based method to
solve the time independent Schrodinger equation, considering
a rectangular potential barrier. The numerical solution is
compared to the analytical one for four values of x describing
the barrier. These values are important for determining the
transmission coefficient and the density current.
While the analytical method offers solutions only for the
rectangular potential barrier, the numerical wavelet based
method allows solutions for more complex potential barrier
types, where no analytical method is available. From the
results and considering the small obtained error (see Tables
II,IV,VI) it is obvious that the dyadic wavelet based method
offers a very good approximation of the solution compared to
the analytical method. Also, considering the representations
from Figures 3, 4, 5, 6 one can conclude that the wavelet
solution offers a precise approximation for the wavefunction,
inside and outside the potential barrier compared to the analytical one.