MatLab homework - root solving

Simulation on the Nanoscale ENG-M03: Assignment 3

General information

In the following you will be asked to complete a set of tasks using MATLAB. To help you

accomplish these tasks many questions will include MATLAB code (blue text), this can be

directly copied and pasted into MATLAB. Each question will carry a mark which is indicated

at the questions end. For each question document your answers, be that a numerical or

graphical output in an associated word document, which will be submitted as part of your

continual assessment at the of the semester.

1. Infinite quantum well

In this exercise we will develop a suite of programs to calculate the energy eigen-values

and eigen-vectors within both a infinite and finite quantum well system. To achieve this

we will be utilising the root finding and bracketing algorithms developed in Lab 2.

The simplest illustration of the basic postulates of quantum mechanics is the case of a

particle confined to a one-dimensional infinite quantum well of width 2a say. The

impenetrable walls/sides of the well/box can be modelled by the potential:

( ) | |

Or (1)

( ) | |

Using the above potential Schrödinger’s equation may be expressed in the following form for

the interval| | as

( )

( )

(2)

Which has solutions of the form:

( ) ( ) ( )

(3)

Where A and B are integration constants and

√

(4)

As the sides of the box are infinite we require ( ) for | | Recalling from lectures

that the wavefunction, ( ), must be everywhere continuous except where the potential

has an infinite discontinuity, hence, at we have

( ) ( ) ( ) ( )

} (5)

Infinite quantum well indicating the first three eigenvalues (dotted lines)

and corresponding eigen-functions (full line) for a well width of 50Å.

There are two non-trivial sets of solutions of equations 5. Either ( ) or

( ) . In either case

(6)

with or respectively. This is the crucial step because it is where the

energy levels appear. By imposing the boundary conditions (5) on the solution (3) we

immediately restrict the parameter to the discrete set of values (6). Substituting (6) into

(4) yields

(7)

The energy E, which is a continuous parameter in classical mechanics, is here ‘quantised’ to

certain discrete energy levels.

More useful information is provided by the wavefunction (3), if we determine the

integration constants A and B by the normalisation procedure discussed in lectures, i.e. for n

even we require:

∫| ( )|

∫ (

)

(8)

From which we find

and similarly

, yielding

( ) (

)

√

( ) (

)

√

} (9)

(a) Create a function that evaluates equation 7 for user-defined and

(b) Create a function that evaluates equation 9 for user-defined and

(c) For n = 1:4 and well widths = 1 and 10nm, calculate the eigenvalues and plot the

corresponding eigenvectors.

[4 marks]

Finite Potential Well

The quantised energy levels for motion of carriers in the -direction within the well are

obtained by solving Schrödinger’s equation for electrons in both the well and the barrier

regions of the structure.

For the well region

( )

( )

(10)

And for the barrier regions

( )

( )

( )

(11)

V is the height of the potential well and ( ) is the dependent part of the envelope

function, ( ) ( ) ( ) of the Schrodinger wavefunction of the

electron.

The electron wavefunctions are:

in the well in the barrier

odd solutions    zkAzF we sin    zkBzF be  exp even solutions    zkCzF we cos    zkBzF be  exp

where 2*

2 Emk ww

 and   2*2 EVmk bb  with *

w m and

*

b m the effective

masses of the carriers at the band edge within the quantum well and the barrier

respectively.

Graph showing the three lowest eigenenergy solutions to a 1eV deep, 50Å wide

quantum well. The eigenenergies are shown as the dotted lines, whereas full

lines denote their corresponding wavefunctions.

At the junction of the well and barrier the carrier wave-function and its derivative must be

continuous:

(12)

Assuming the well interfaces are located at and where is then the width of the

well

Solutions:

Then if we are at the interface and use the boundary conditions in (12) we may find

odd solutions of the form:

√ ( )

√

(13)

And even solutions of the form:

√ ( )

√

(14)

(a) Write two functions to evaluate equations (13) and (14) and plot these functions

between the energies 0 and 0.3eV to suppress the y-axis to -4:4. Use

and , where

kg.

(b) Write a script that calls either the bisection or Newton algorithms in conjunction

with the bracketing routine of Lab 2 to locate the first two odd (equation (13)) and

even solutions (equation (14)) within a finite quantum well with the following

parameters:

i. Depth, eV and width, .

ii. Depth, eV and width, .

iii. Depth, eV and width, .

iv. Depth, eV and width, .

[16 marks]