fundamental of nanoscience (computer project i) by ravi ... · ravi sharma dulal . computer...

Chalmers University

Fundamental

of

NanoScience

(Computer project I)

By

Ravi Sharma Dulal

Computer Project, Part 1

Problem C1.1 (Transmission through a rectangular potential)

Use the propagation matrix approach to calculate the transmission coefficient T of

a rectangular potential step with width L=1 nm and a height Vo=0.3 eV. Plot T as a

function of energy. Do you see any oscillations in the propability? Why? Change

the barrier width and height. What happens? Compare your results with theory. Do

they agree?

Consider a potential of height Vo= 0.3 eV.

Wavefunction parameterization:

xikxik

I BeAe 11

xikxik

II DeCe 22

xikxik

III GeFe 11

fig 1: Potential barrier with height Vo

Where,

221

pmEk ,

)(22

VoEmk

me is mass of electron and E is energy of electron.

We can calculate transmission coefficient by using propagation matrix which

relates the waves on the right hand side to those left of the barrier. Here, we have

matrix for up front (Pu), propagation in barrier (Pp) and down front (Pd).

,

Final propagation matrix can be calculated by multiplying all above matrices

together.

P = Pu*Pp*Pd

Then, Transmission coefficient is defined as,

Theoretically, transmission coefficient can be calculate directly,

Fig 2: Numerical matrix calculation for transmission coefficient (L=1 nm)

Yes, I can see the oscillations in the probability. For E>Vo, term becomes

oscillating which leads to the fluctuations in T(E) and isolated energies for which

transmission occurs with complete certainty i.e. T(E)=1. Such transmission

resonances arise from the wave interference and constitute further evidence for

the wave nature of the matter. We see transmission resonance when K2L=nΠ.

Within the barrier, wave function exponentially damped over the distance 1/k2.

Quantum mechanical phenomenon called tunneling occurs easily if barrier width is

very thin, i.e. smaller than decay length 1/K2.

Fig 3a: T(E) for barrier width L=3nm(Theoretical calculation)

Fig 3b: T(E) for barrier width L=3nm(Numerical calculation)

If we keep the barrier height constant and change the barrier width, we see

significance change in oscillations. When barrier width is narrow, even electron

with lower energy can pass through the barrier by quantum tunneling. Wider the

barrier less observable the quantum tunneling is. We see sharp front when E=V for

wider barrier. However, wavy behavior increases with increase in barrier width.

Fig 3c: T(E) for barrier width L=5nm(Theoretical calculation)

Fig 3d: T(E) for barrier width L=5nm(Numerical calculation)

If we keep the barrier width constant and vary the barrier height, we can see that

passing limit shifting towards higher energies on increasing barrier height. This one

is more close to classical expectation. Wavy behavior is observed more

prominently at higher barrier height and higher energies (E>V).

Fig 4: T(E) for barrier width L=1nm and different height(Numerical calculation)

Problem C1.2(Bound states)

Modify the program from problem 1 to treat bound states in a rectangular

potential well. Extract the bound state energies from the transmission coefficient

and compared with theory (e.g. the bound states of a particle in a box). When is

this comparison relevant?

Consider a potential well with two levels V1 and V2. Wave function

parameterization:

xik

L Ae 1

xCSinkxBCoskC 22

xik

R De 1

fig 5: Potential well with height V1

Exponentially growing terms are not allowed states so that they have been

removed. When E<V1, this is the case of bound particle. If we plot the transmission

spectrum manually, we will see the discrete energy values. Here, I use same

propagation matrix method and process is same as previous problem for potential

barrier. Just by interchanging k1 and k2, we can get discrete energy states.

Fig 6a: T(E) for finite potential well with L=1nm (Numerical calculation)

Fig 6b: T(E) for finite potential well with L=5nm(Numerical calculation)

Theoretically we can also use another equation,

This above equation can be solved graphically by taking two functions for RHS and

LHS and plotting this with respect to energy, we can get solutions. All these

solutions are valid because above equations are derived by discarding

exponentially growing terms.

Fig 6c: T(E) for finite potential well with L=1nm(Theoretical calculation)

Fig 6d: T(E) for finite potential well with L=5nm(Theoretical calculation)

Comparison between both methods:

V1=0.3eV, L=1nm (1 solution)

Matrix calculation Theoretical calculation 0.1198 0.1198

V1=0.3eV, L=1nm (5 solutions)

Matrix calculation Theoretical calculation

0.114 0.011

0.046 0.0457 0.102 0.1018

0.179 0.178 0.267 0.267

Comparison of the bound states in a rectangular potential well to the bound states

of a particle in a box is relevant only when E<V1. And LL+2δ where δ=1/k1 is decal

length for rectangular potential well. Outside the wall, wave function decays

exponentially.

Appendix

1.Matlab program for fig 2

%******************************************************************* % % % % % This Matlab script will calculate the transmission probability % for a wave impending on a step potential of height V = 0.3 eV, % using the propagation matrix method. % % % % V1 V2 % % |----------------------| % | | % | | % | | % | | % ----------------------| |---------------------- % %*******************************************************************

%% Define constants

hbar = 1.0545716e-34; % Planck's constant [Js] mass = 9.109382e-31; % Electron mass [kg] eCharge = 1.6021764e-19; % Electron charge [C] L=1e-9;

% Potential heights in the two regions [eV]

V1 = 0.0; V2 = 0.3;

% Number of energy values to consider n = 1000;

% Vector of n energies [eV]

energyVector = linspace(0,3.0,n);

% Vector of n calculated transmission probabilities

transmissionVector = zeros(1,n); % initially put to zero

% Loop over energy values

for index = 1:n

energy = energyVector(index);

% Energy convertions from eV to J

energyJoules = energy*eCharge; V1Joules = V1*eCharge; V2Joules = V2*eCharge;

% Calculate wave numbers in the different regions

k1 = sqrt(2*mass*(energyJoules - V1Joules))/hbar; k2 = sqrt(2*mass*(energyJoules - V2Joules))/hbar;

% Construct the propagation matrix Pu= 1/(2*sqrt(k1*k2))*[(k1+k2) (k1-k2); (k1-k2) (k1+k2)]; Pp=[exp(-(1i*k2*L)) 0; 0 exp(+(1i*k2*L))]; Pd= 1/(2*sqrt(k1*k2))*[(k1+k2) (k2-k1); (k2-k1) (k1+k2)]; pMatrix = Pu*Pp*Pd;

% Calculate the transmission probability and store it in % transmissionVector

transmission = 1/abs(pMatrix(1,1))^2; transmissionVector(index) = transmission; end

% Plot transmission probability as a function of energy plot(energyVector, transmissionVector); % Add labels to axes xlabel('Energy [eV]'); ylabel('Transmission probability'); % Add figure title title('Transmission probability for a step potential of height 0.3 eV and

Length 1 nm'); % Scale axes

2. Matlab Program for fig 3b and fig 3d

%% Define constants



V1 = 0.0; V2 = 0.3;







for L=3e-9:2e-9:5e-9 for index = 1:n








transmission = 1/abs(pMatrix(1,1))^2; transmissionVector(index) = transmission; end figure(); hold all; % Plot transmission probability as a function of energy plot(energyVector, transmissionVector); % Add labels to axes xlabel('Energy [eV]'); ylabel('Transmission probability'); % Add figure title title('Transmission probability for a step potential of height 0.3 eV'); % Scale axes end

3. Matlab program for fig 3a and 3c

%% Define constants



V1 = 0.0; V2 = 0.3;



energyVector = [linspace(0,0.29,n/2) linspace(0.301,3.0,n/2)];










if energy > V2 transmission = 1/(1 + ((((k1^2-k2^2)/(2*k1*k2))^2)*(sin(k2*L))^2)); elseif energy <V2 transmission = 1/(1 + ((((k1^2+k2^2)/(2*k1*k2))^2)*(sin(k2*L))^2)); end transmissionVector(index) = transmission; end

figure(); % Open a figure canvas

% Plot transmission probability as a function of energy plot(energyVector, transmissionVector); % Add labels to axes xlabel('Energy [eV]'); ylabel('Transmission probability'); % Add figure title title('Transmission probability for a step potential of height 0.3

eV(Theoritical)'); % Scale axes end

4. Matlab program for fig 4

%% Define constants



V1 = 0.0; V2 = 0.3;



energyVector = linspace(0.0,5.0,n);




for V2 = 0.1:0.3:3.0 for index = 1:n






% Construct the propagation matrix Pu= 1/(2*sqrt(k1*k2))*[(k1+k2) (k1-k2); (k1-k2) (k1+k2)]; Pp=[exp(-(i*k2*L)) 0; 0 exp(+(i*k2*L))]; Pd= 1/(2*sqrt(k1*k2))*[(k1+k2) (k2-k1); (k2-k1) (k1+k2)]; pMatrix = Pu*Pp*Pd;



hold all; % Open a figure canvas

% Plot transmission probability as a function of energy plot(energyVector, transmissionVector); % Add labels to axes xlabel('Energy [eV]'); ylabel('Transmission probability'); % Add figure title title('Transmission probability for a step potential of length 1nm and

different height'); % Scale axes end

5. Matlab program for fig 6a and 6b

%******************************************************************* % % Fundamentals of Nanoscience [FKA131] % Example file, computer project % % This Matlab script will calculate the transmission probability % for a wave impending on a step potential of height V = 0.3 eV, % using the propagation matrix method. % % % % V1 V2 V1 % % -----------------------| |--------------- % | | % | | % | | % | | % |---------------------| % %*******************************************************************

%% Define constants



V1 = 0.3; V2 = 0.0;
















% Open a figure canvas figure();

% Plot transmission probability as a function of energy plot(energyVector, transmissionVector); % Add labels to axes xlabel('Energy [eV]'); ylabel('Transmission probability'); % Add figure title title('Transmission probability for a finite potential well of height 0.3 eV

and L= nm'); % Scale axes end

6. Matlab program for fig 6c and 6d

%% Define constants



V1 = 0.3; V2 = 0;





yb = zeros(1,n); % initially put to zero ya = zeros(1,n);


for index = 1:n





k1 = sqrt(2*mass*(-energyJoules + V1Joules))/hbar; k2 = sqrt(2*mass*(energyJoules - V2Joules))/hbar;

ya(index)=k1/k2;

yb(index) = -(((k1/k2)*(cos(k2*L))-

sin(k2*L))/(((k1/k2)*sin(k2*L))+cos(k2*L))); end

% Open a figure canvas figure(); % Plot transmission probability as a function of energy plot( energyVector, yb,'r',energyVector, ya);

% Add labels to axes xlabel('Energy [eV]'); ylabel('y'); % Add figure title title('Theoritical calculation for discrete bound energy state');

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