Simulating
the Energy Spectrum of
Quantum Dots
J. Planelles
PROBLEM 1. Calculate the electron energy spectrum
of a 1D GaAs/AlGaAs QD as a function of the size.
Hint: consider GaAs effective mass all over the structure.
L
0.25 eV
AlGaAs
AlGaAs
GaAs
m*GaAs = 0.05 m0
The single-band effective mass equation:
2 d 2
[
 V ( x)] f ( x)  E f ( x)
2
2m * dx
Let us use atomic units (ħ=m0=e=1)
1
d2
[
 V ( x)] f ( x)  E f ( x)
2
2(m * / m0 ) dx
Lb
V(x)
BC:
f(0)=0
L
Lb
0.25 eV
m*GaAs = 0.05 m0
f(Lt)=0
Numerical integration of the differential equation: finite differences
1 d2
[
 V ( x)] f ( x)  E f ( x)
2
2m * dx
Discretization grid
f1
f2
...
fn
x1
x2
...
xn
How do we approximate the derivatives at each point?
f ' ( xi )  f i ' 
fi
f i 1  f i 1
2h
fi-1
f i ' 1  f i ' 1
f ( xi )  f i 

2h
''
fi+1
''
f i 1  2 f i  f i 1
 ... 
h2
i-1
Step of the grid
i
h
i+1
h
FINITE DIFFERENCES METHOD
1 d2
[
 V ( x)] f ( x)  E f ( x)  BCs : f (0)  0
2m * dx 2
f ( Lt )  0


1. Define discretization grid
2. Discretize the equation:

0

h
Lt
1
f i ''  Vi f i  E f i
2m *
1
[ f i 1  2 f i  f i 1 ]  Vi f i  E f i
2m * h 2
3. Group coefficients of fwd/center/bwd points
i=n
i=1 i=2
L
n  t 1
h
1 
1 

 1


f 
 Vi  f i   
f  E fi

2  i 1
2
2  i 1
 2m * h 
 m*h

 2m * h 
i 1 f i 1  ai f i  bi 1 f i 1  E f i
i 1 f i 1  ai f i  bi 1 f i 1  E f i
Trivial eqs: f1 = 0, fn = 0.
Extreme eqs:
i=n
i=1 i=2
i2
....
 1 f1
0
....
 a2 f 2
....
 b3 f 3  E f 2
....
....
i  n  1   n  2 f n  2  an 1 f n 1  bn f n  E f n 1
0
Matriz (n-2) x (n-2) - sparse
We now have a standard diagonalization problem (dim n-2):
 a2 b3
 a b
3
4
 2

  

 an  2


 n2
  f2 
 f2 
  f 
 f 
  3 
 3 
     E  
 



bn 1   f n  2 
 f n2 
 f n 1 
an 1   f n 1 
i 1 f i 1  ai f i  bi 1 f i 1  E f i
 a2 b3
 a b
3
4
 2

  

 an  2


 n2
2
a2
0
3
a3
b3
...
...
b4
...
...
...
...
...
...
 n-2
a n-2
bn-2
0
a(n-1)
bn-1





bn 1 
an 1 
The result should look like this:
finite wall
n=4
n=3
n=2
n=1
infinite wall
Exercices:
a) Compare the converged energies with those of the particle-in-the-box with
infinite walls for the n=1,2,3 states.
 2 2 2
En 
2mL2
n
b) Plot the 3 lowest eigenstates for L=15 nm, Lb=10 nm. What is different from the
infinite wall eigenstates?
HOME WORK:
Calculate the electron energy spectrum of two coupled QDs as a function of their
separation S. Plot the two lowest states for S=1 nm and S=10 nm.
L=10 nm
L=10 nm
AlGaAs
AlGaAs
S
GaAs
GaAs
x
The result should look like this:
bonding
antibonding
bonding
antibonding