Chapter 6: Electronic Materials
What You Will Learn:
Relevance to modern electronics
Brief introduction to quantum mechanics
Numerical Solutions of the Schrodinger equation
Applications to energy levels of quantum wells
1
6.3 Numerical Simulation of Schrodinger Equation
Most potential wells cannot be solved analytically
Need a numerical solution algorithm for solving schrödinger's Eq. for any V(z)

2

2
2 m z
2
  V  z   E 

2

2
2 m z
2
 V  z   E    0
Preliminaries:
z0
(1) num erical m esh:

2
(2 )
z
2

z
2 z
n z
……
N z
  z   z   2  z     z   z 
z
2
2
The Shooting Method
Discrete schrödinger Equation becomes:


2m 

2
 z   z   2  z     z   z  
z
   V  z   E 

2
z  0
Re-arranging:

 z   z   2
z 
2m
2
  z  V
2
z 
E  
 z     z  z 
OR
2
 2m

  z   z    2   z  V  z   E   2  


 z     z  z 
Careful! Represent this
as a real number in a
computer, not an integer
“Shooting
equation”
3
Shooting Method: Procedure
Start with 1st two
values of 
  z  z 
 z
Predict the next
increment in space
  z  z 
Continue till the
end of domain

  z  z 
 z
  z  z 

  z  z 
 z
  z  z 
This is an
“effective infinity”
z0
z  z
z  2z
z  3 z
z  4z
z 4 N  z
Boundary Conditions on the Wave Functions
Mathematical Solution can be obtained for any energy “E” in Schrödinger
equation
We only accept those solutions that satisfy boundary conditions:
 z  0 &

z
 0 as z   "  " ("  "  N  z of sim ulat ion b ox)
Need to run the interative procedure on previous page through many timeseach for a different energy-until the computed solution satisfy the boundary
conditions.
5
Initial Conditions Wave Functions at the Origin
Need to know two initial points of wave function to carry at the shooting
method:
V(z)
These can be deduced from the form of V (z)
V(z)=V(-z)
z
Most quantum well potentials are symmetric:

even party
 z  z
  z  0  0

  z   z   1


"o d d p arty"
  z   
z

 2m 

2
  z   z    2   z   V  0   E   2    0 




  z  0  1


Possible since scaling a wave function   z  by a constant   
still makes it a valid solution of Schrödinger's Equations

6
An Flow Chart for Computing the Shooting Method
Initialize variables, arrays & parameters
(i.e. set , m ,  z , size of
system (well), etc)
Main loop changing trial energies
Iteration of shooting method(from starting
conditions at z  0 & z   z  to  z  N  z
No
Are B.C.S
satisfied?
Yes
END
7
An Algorithm For the Shooting Method
!se t p a ra m e te rs a n d va ria b le
 1 .0 5 4 5 8 9  1 0
;:  3 4
m  9 .1 0 9 5 3 1 1  1 0
z  1  10
-1 0
In f - e ff

!m a x im u m m e sh p oin ts
z
 (0 :N )
 kg 
m 
E upper  1 0 0 0  m e V
N=
31
!store s w a ve fu n c tion in a r ra y
E in c  1 e V
!e n e rgy in c re m e n t  E
E
!tria l e n e rgy
!R e la x 
0 :
N
continued

8
An algorithm For the Shooting Method
For E = 0 to E upper in steps of  E
 0  0
!assum es odd ponity w ave states
 1   1
!assum es odd ponity w ave states
For i= 2 to N  z do,
com pute   i   shooting-fu nction   i  1  ,  i  2  
END
R eturn   N  For inspection of B .C .'S (e.g. vis)
if(boundary conditions of ψ  N  exit loop,otherw ise repeat)
END
!outside loop
print final w ave function and energy lev el
9
“Parabolic” Quantum Well
This is also known as the “simple harmonic oscillator”
Imagine an electron “stuck” to a collection of nearby by atoms by a
“Quantum-Spring”!
V  z   cz
Solution known analytically
2

En   n 

1

2
2c
m
As we have said from the beginning all good computer models must first be tested
against a known result first!
10
6.4 Numerical Calculation of Harmonic Quantum Well
Energy Levels
Take
 z
V z  
 100
2
 at z=1

 eV  

 v  z=1
Å
Å
1

Expect E n   n  
2

a w a y from c e n te r of w e ll 

  1e V  typ ic a l of solid 
2e
100  10
 10
m    9.109534  10 kg 
2
3
 39.039 m eV
Odd parity wave functions (states) turn out to correspond to
n=2,4,6,…levels
11
F90 Code for Computing Energy Levels of Quantum Wells
Code is found in directory Part_III-electronic_materials
12
Computing the Quantum Well Energy States
What to expect from code output:
All energies that lead to the wave function going through zero correspond
to acceptable energy states of the quantum well
n=2
  N z, E   0
n=4
n=6
n=8 n=10 n=12
bound
states
0
E(meV)
1000
13
Computed Results For a Parabolic Potential
1000000
800000
600000
Series1
400000
200000
  N z, E 
0
-200000
0
200
400
600
800
1000
-400000
-600000
-800000
-1000000
Energy Level
14
Accuracy of Calculations and Open Questions
  E and  z chosen Somewhat arbitrarily
choose small enough but not too small to slow down calculation
Effective
"  " : how far out is “far enough”
 Don’t make arrays too large
OK in 1-D
Can be impossible in 2-D
Two-for-one: in the process of “shooting”, can also print and wave functions!
15