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

GaAs

AlGaAs AlGaAs0.25 eV

m*GaAs = 0.05 m0

)()()](*2

[2

22

xfExfxVdx

d

m

The single-band effective mass equation:

L

0.25 eV

m*GaAs = 0.05 m0

V(x)

)()()]()/*(2

1[

2

2

0

xfExfxVdx

d

mm

Let us use atomic units (ħ=m0=e=1)

LbLb

f(0)=0BC: f(Lt)=0

)()()](*2

1[

2

2

xfExfxVdx

d

m

h

fffxf iiii 2

)( 11''

h

fffxf iiii 2

)('1

'1''''

Numerical integration of the differential equation: finite differences

Discretization grid

x1 x2xn

How do we approximate the derivatives at each point?

i-1 i+1i

h h

fi

fi-1

fi+1

211 2

...h

fff iii

Step of the grid

f1 f2 ... fn

...

1. Define discretization grid

2. Discretize the equation:

FINITE DIFFERENCES METHOD

)()()](*2

1[

2

2

xfExfxVdx

d

m

0)(

0)0(:

tLf

fBCs

iiiiii fEfVfffhm

]2[*2

1112

i=1 i=ni=2

h0 Lt

iiiii fEfbfafb 11

1h

Ln t

iiii fEfVfm

''

*2

1

iiiii fEfhm

fVhm

fhm

12212 *2

1

*

1

*2

1

3. Group coefficients of fwd/center/bwd points

1

2

3

2

1

2

3

2

1

2

3

2

n

n

n

n

n

n

f

f

f

f

E

f

f

f

f

ab

bab

bab

ba

We now have a standard diagonalization problem (dim n-2):

Trivial eqs: f1 = 0, fn = 0.

Matriz (n-2) x (n-2) - sparse

Extreme eqs:

11121 nnnnn fEfbfafbni0

232212 fEfbfafbi 0

iiiii fEfbfafb 11

i=1 i=ni=2

The result should look like this:

n=1

n=2

n=3

n=4

finite wall

infinite wall

PROBLEM 1 – Additional questions

a) Compare the converged energies with those of the particle-in-the-box with infinite walls for the n=1,2,3 states.

b) Use the routine plotwf.m to visualize the 3 lowest eigenstates for L=15 nm, Lb=10 nm. What is different from the infinite wall eigenstates?

22

22

2n

mLEn

PROBLEM 2. 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

GaAs

AlGaAs

L=10 nm

GaAs

AlGaAsS

x

bonding antibonding bonding antibonding

The result should look like this:

PROBLEM 3. Calculate the electron energy spectrum of N=20 coupled QDs as a function of their separation S.

Plot the charge density of the n=1,2 and n=21,22 states for S=1 nm and L=5 nm.

The result should look like this (numerical instabilities aside):

n=1

n=2

n=21

n=22

PROBLEM 4. Write a code to calculate the energies of an electron in a 2D cylindrical quantum ring with inner radius Rin and outer radius Rout, subject to an axial magnetic field B.

Calculate the energies as a function of B=0-20 T for a structure with (Rin,Rout)=(0,30) nm –i.e. a quantum disk- and for (3,30) nm –a quantum ring-. Lb=10 nm. Discuss the role of the linear and quadratic magnetic terms in each case.

ρ

B

V(ρ,Φ)GaAs

AlGaAs

Hint 1: after integrating Φ, the Hamiltonian reads (atomic units):

)()()(8

1

*2

1 22

2

2

2

2

fEfVB

BMM

m zz

with Mz =0, ±1, ±2... the angular momentum z-projection. 1 atomic unit of magnetic field = 235054 Tesla.

GaAs

AlGaAs

Hint 2: describe the radial potential as

where ρ=0 is the center of the ring

ρ

Hint 3: use the following BC

f(Lt)=0 (i.e. fn=0)

If Mz=0, then f’(0)=0 (i.e. f1=f2)

If Mz≠0, then f(0)=0 (i.e. f1=0)

LbRout

Rin

The results should look like this:

Disk(Rin=0)

Ring(Rin=3 nm)