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Problem 4.14

Approach:

STEP 1: Using MATLAB (or some other mathematics software), build the E-k

relationship from 0 to π/d. This can be done using the following MATLAB script

(copy-paste into MATLAB for better readability):

%Problem 4-14 MATLAB Solution

%STEP 1

%-------DECLAIR CONSTANTS----------

%Physical Constants

hbar = 1.055e-34; %hbar in units of J.s

m0 = 9.109e-31; %mass of an electron in kg

eV = 1.602e-19; %Electron Volt in J

%Problem Specific Constants

a = 3e-10; %a,b,d from Kronig-Penny

Model in m

b = 3e-10;

d = a + b;

U0 = 10*eV; %Height of periodic potenial

square wave in J.

%------Determine the positive half of E1-------------

E = 0*eV; %Initialize E (first

iteration of while loop will be small for F(E)

dE = 0.00001*eV; %Set dE to a small

increment

F_E = 100; %Set to a number larger than

1 to initialize while loop

taken.

%Determine E1(0), i.e. the first point.

while abs(F_E)>1

E = E+dE;

F_E=cos(a*sqrt(2*m0*E/hbar^2))*cosh(b*sqrt(2*m0*(U0-

E)/hbar^2))+(U0-2*E)/(2*sqrt(E*(U0-E)))...

*sin(a*sqrt(2*m0*E/hbar^2))*sinh(b*sqrt(2*m0*(U0-E)/hbar^2));

end

E1(1) = E; %The first point of E1.

k(1) = acos(F_E)/d;

%Determine the rest of the E-k structure

n = 2; %index for E and k.

while abs(F_E)<1

E = E+dE;

F_E=cos(a*sqrt(2*m0*E/hbar^2))*cosh(b*sqrt(2*m0*(U0-

E)/hbar^2))+(U0-2*E)/(2*sqrt(E*(U0-E)))...

*sin(a*sqrt(2*m0*E/hbar^2))*sinh(b*sqrt(2*m0*(U0-E)/hbar^2));

E1(n) = E;

k(n) = acos(F_E)/d;

n = n+1;

end

plot(k,E1); %Plot the results.

xlabel('k in units of /m'); %Caption plot.

ylabel('E in units of J');

hold on; %hold on the plot.

Giving the following plot (see Figure 4.17(b)):

Step 2: Now, we want to determine a good value for the effective mass near the start

of E1. To do this we will solve the given equation for m* for the first firth of the E-k

curve, and then take the average. This is done by adding the following MATLAB

script (copy-paste into MATLAB for better readability):

%STEP 2

k_size = size(k); %Determine the size of k

k_size = k_size(2); %to find m_eff for 1/5 of

this

for i = 2:1:round(k_size/5) %for loop to find fits for

m*, note that we

%can't use the first point

%E1(1)-k(1), as it would

cause k=0,

%E1(i)-E1(1)=0.

m(i-1)= hbar^2*k(i)^2/(2*(E1(i)-E1(1)));

end

m_eff = mean(m); %The effective mass is taken

as the average

for i=1:1:k_size %Determine the parabolic fit

values for E

E_fit(i)=E1(1)+hbar^2*k(i)^2/(2*m_eff);

end

plot(k,E_fit,'k--'); %Plot these values on same

plot in dotted black.

title(['E-k using (4.37) vs. parabolic fit with m* =

',num2str(m_eff/m0), 'm0']); %title plot

This gives the following plot:

Thus m* = 8.35m0 = 7.6e-30kg