Transcript No Slide Title

A three-dimensional wave function for an
ideal, finite potential quantum well.
Dr. A. Zahedi
Francis K. Rault
Prof. R.E. Morrison
Abstract:
Outside The Well:
For Values: x<  L / 2
The quantum well has been studied for many years and the understandings have proven to be
very beneficial for society. The undertaking of this paper was to extend the knowledge and
understanding of the well in three dimensions.
y<  L / 2
z<  L / 2
T Lower Region (x, y, z)=
Re-
This paper successfully attempts to find a solution, for an ideal finite three-dimensional well.
The solution is based on an existing one-dimensional solution.
2m

(
(Vo ( x )  E X ) |x| + (Vo ( y )  EY )
|y| +
(Vo ( z )  EZ )
|z| )
Define: R = A -x A -y A –z
For Values: x> L / 2
Using a proposed one-dimensional solution and extending it to three dimensions, a satisfactory
model was developed. The model is purely mathematical and satisfies the fundamental criteria.
That is, it is a solution of the Schrodinger wave equation and the wave function and its first
derivatives are continuous at the boundary regions. The solution also takes into account
degenerate energy levels.
y> L / 2
z> L / 2
T Upper Region (x, y, z)=
Se-
2m

(
(Vo ( x)  E X ) |x| + (Vo ( y )  EY ) |y| + (Vo ( z )  EZ )
|z| )
Define: S = A +x A +y A +z
The constants can be determined using appropriate boundary conditions.
Theoretical Approach:
Results:
For a three-dimensional solution one must solve Schrodinger’s Wave equation:
Normalized Probability Density Function.
2m( E  Vo )
 2  2  2
+
+ 2 +
=0
2
2
2
x
z
y
For Values: |x|  L / 2 , |y|  L / 2 , |z|  L / 2 .
Begin with Aronstien and Stroud 1-D definition:
Wave functions:
Region 1: Lower Well Region.
Region 2: Inner Well Region.
Region 3: Upper Well Region.
x  Re gion1  A e
x
x Re gion2  T x e
 2 mVo  x  Ex 
x

i 2 mE x
x

x Re gion3  Ax e
 T x e
Cubic 10 nm quantum well.
Particle in energy level 1.
Likely location of finding
the particle within the well: Centre of
well.
i 2 mE x
x

 2mVo  x  Ex 
x

Transcendental Equation Relationship:
Shows relationship between Well length (L), Energy (E) and Well Potential (Vo).
Takes into account degenerate energy levels.
n is principal quantum number.
Normalized Upper Region Probability Density Function.
For Values: x> L / 2
   n
  sin 1   
P
2m E L

2

2
y> L / 2
Normalized Lower Region Probability Density Function.
z> L / 2
For Values: x<  L / 2
y<  L / 2
z<  L / 2
2m Vo L

2
P
Complete wave-functions for the respective regions:
Making the assumptions:
1- The three dimensions are independent to one another.
2- The complete wave function T (x, y, z) for the individual regions is the super position of
the three wave function in the three respective dimension. T (x, y, z)= n (x). n (y). n (z)
3- The total energy E is equal to the summation of the x,y,z energy levels. E = Ex +Ey + Ez .
The total potential of the well is the summation of the individual potentials in the respective dimensions.
Vo(x,y,z)= Vo(x) + Vo(y) + Vo(z)
N.B: Energy is a scalar quantity.
4- m is the effective mass of the particle within the quantum well.
Cubic well. Length 90 nm
Particle in energy level 2.
For: |x|  L / 2 , |y|  L / 2 , |z|  L / 2 .
i 2m
T Inner Region (x, y, z) = D e

 i 2m
Le

E
(
E
(
Both Upper and Lower region probability density functions show a slight chance of
finding the particle outside the well; due to tunneling effect.
E
x +
x
y +
y
E
x +
x
E
z)
z
E
y +
y
z)
+
Energy levels for a GaAs/AlGaAs valence quantum well (x=0.25) Lengths Lx=Ly=Lz= 10nm.
Energy Level(Ex)
Heavy Hole (meV)
Light Hole (meV)
Theory
Exp.
Theory
Exp.
1
6.17
7.04
23.86
21.48
2
24.40
28.07
87.83
81.48
3
54.00
61.11
4
93.10
104.44
-
+
z
Where:
i 2m
Fe

(
E
 i 2m

Ke
i 2m

Ge
Ie
x
E
(
E
y
E
x +
x
E
y -
z)
z
E
y y
+
z)
z
+
D=( T
x
+
T
y
+
T
z
+
H=( T
x
+
T
y
-
T
z
-
), I= ( T
K=( T
x
-
T
y
-
T
z
+
), L= ( T
), F=( T
x
+
x
-
T
T
x
+
y
+
y
+
T
T
T
y
-
z
- ),
z
+
T
G=( T
), J=( T
x
-
x
+
T
y
-
T
T +y T
z
-
z
+
),
),
z
- ).
Likelihood of finding particle is at point (3,3,3).
E
(
 i 2m

Je
i 2m

He
x +
Energy Level Predictions:
E
y +
y
E
z)
z
+
The total energy of the particle is:
(
E
E
(
 i 2m

x x
(
E
x x
x
-
x
x x
E
E
E
y +
y
y y
y y
E
E
E
z)
z
z)
z
z)
z
+
+
E = Ex + Ey + Ez
2 2
2
2
2
2 x  2 2 y 
2 z  2 2 2   x 2  y  z 2 


E=
+
+
=
mL x
mLz
m Ly
m  Lx Ly
Lz 
Conclusion:
The proposed solution for the finite ideal three-dimensional well has satisfied the requirements under the
appropriate boundary conditions and quantum postulates. The proposed equation does solve the
Schrodinger wave equation. The equations produce valid results in accordance with current standing
theory, in regards to both wave function and energy levels. It gives rise to some interesting implications and
possible applications for the field of photovoltaics and other quantum electronic areas.
Electrical and Computer Systems Engineering
Postgraduate Student Research Forum 2001