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Matlab Simulation for Semiconductors Lab Report

Purpose of the Experiment


The purpose of this experiment was to help us familiarize, execute and verify various semiconductor parameters (Minority carrier concentration, Fermi Level) using MATLAB simulation software. We were supposed to write codes for the different problems and then execute them with MATLAB so as to obtain certain output results.

Equipment Used
In this experiment we used:

ï‚· MATLAB Simulation Software

Introduction


Semiconductors normally have an array of parameters that we should be familiar with so as to have a comprehensive understanding about semiconductors. Parameters such as Fermi Level, Minority Carrier Concentration, Conductivity, Drift, Diffusion, and Carrier Lifetime are extremely important when learning semiconductor Physics.

In this report, we were required to perform the simulation of some parameters using MATLAB Simulation software and identify the nature of the output so as to determine if it conforms with the theoretical knowledge we are conversant with.

Theory


In general, semiconductors fall into two categories: Intrinsic semiconductors and extrinsic semiconductors. An Intrinsic Semiconductor is one that is made of the semiconductor material in its extremely pure form. Common examples of such semiconductors are: Pure germanium and silicon which have forbidden energy gaps of 0.72eV and 1.1eV respectively. The energy gap is so small that even at ordinary room temperature, there are many electrons which possess sufficient energy to jump across the small energy gap between the valence and the conduction bands.

Those intrinsic semiconductors to which some suitable impurity or doping agent or dopant has been added in extremely small amounts (about 1 part in 108 ) are called extrinsic or impurity semiconductors. Depending on the doping material used, extrinsic semiconductors can be sub- divided into two classes:

 

i.     N-type semiconductors 


ii.   P-type semiconductors 


Fermi Level in an Intrinsic Semiconductor

It can be proved that an in an intrinsic semiconductor, Fermi energy level ??lies in the middle of the energy gap i.e. midway between the conduction and valence bands.

Minority Carriers

In a piece of pure germanium or silicon, no free charge carriers are available at0°?. However, as its temperature is raised to room temperature, some of the covalent bonds are broken by heat energy and as a result, electron-hole pairs are produced. These are called thermally-generated charge carriers. Ordinarily, their number is quite small.

Experimental Procedure and Data


For this experiment, we were required to use MATLAB Simulation software to solve the following problems.

1.        Write a Matlab program to confirm that the ??versus T curve for Geand Si graphed in Figure 2.20 can be generated by employing the empirical fit relationships below. Check over the temperature range of 275? ≤ ? ≤ 375?.

 

 

 

2.   Generate a MATLAB program to compute and plot the Fermi functions, f(E), and 1-f(E) versus ∆? = ? − ??for values ∆? that is over the range −0.5?? ≤ ∆? ≤ 0.5?? for varying temperature settings where Temperature=150,250,350,450 and 550K. Make sure that each f(E) versus ∆? curve at each temperature is superimposed on the same plot. Discuss what you see in your plots as the Fermi function, f(E), and 1-f(E) varies with temperature and energy. 


3.   Consider GaAs at T=300K with??= 0. (a) Plot the position of the Fermi energy level with respect to the intrinsic Fermi energy level as a function of the acceptor impurity concentration over the range of 1014 ≤ ??≤ 1017?��−3. (b) Plot the position of the Fermi energy level with respect to the valence-band energy over the same acceptor impurity concentration given in part (a). 


4.   The temperature of a sample of Ge is T=300K and the acceptor doping concentration is ??= 0. Plot the minority carrier concentration (on a log-log plot) versus ??over the range1015 ≤ ??≤ 1018??−3. 


Results and Discussion


Based on the problems given, we wrote the MATLAB code and generated output graphs for each section as shown below.

MATLAB Code and Graph for Question 1

%% Part A : Intrinsic carrier concentration Vs Temperature

clc;


clearall;


closeall;


%% Temp Range


T=275 :1:375;
k= 1.38*10^-23;


e=1.609*10^-19;


ni_si = (9.15*10^19).*((T./300).^2) .* exp(-0.5928*e./(k.*T));

ni_Ge = (1.76*10^16).*((T).^(3/2)).*exp(-0.392*e./(k.*T));

plot(T,ni_si, 'b');


xlabel('Temperature T');


ylabel('Intrinsic concentration')


legend('Silicon')


figure;


plot(T,ni_Ge, 'r');


xlabel('Temperature T');


ylabel('Intrinsic concentration')


legend('Germenium')

 

MATLAB Code and Graph for Question 2

 

 

 

%% Part B

%Fermi Function Calculation,f(\DeltaE,T)

 %Constant


k=8.617e-5;


%Computation proper

forii=1:4; T=100*ii;

kT=k*T;

dE(ii,1)=.5*kT;

forjj=1:101

f(ii,jj)=1/(1+exp(dE(ii,jj)/kT));

dE(ii,jj+1)=dE(ii,jj)+0.1*kT;

endend

dE=dE(:,1:jj); %This step strips the extra dE value


%Plotting result


close


plot(dE',f'); grid; %Note the transpose (') to form data columns

xlabel('E-EF(eV)');

ylabel('f(E)');

text(0.05,0.2,T(400));

text(-.03,.1,T(100));

 

 

MATLAB Code and Graph for Question 3

%% part 3:

Nv=7*10^18;


Na=10^14:10^14:10^17;


Ef_Ev = ( 0.0259 ).*log(Na* 7*10^18);

figure;


plot(Na,Ef_Ev);


xlabel('Na');


ylabel('Ef-Ev')

 

MATLAB Code and Graph for Question 4

%% Part 4:

Nc = 4.7*10^17;


Nv= 7.0*10^18;

Nd=10^15:10^15:10^18;


ni = 2.26 *10^6;


Ec_Ef = 0.0259 *log(Nc./Nd);


Na= Nc* exp((Ec_Ef)./0.0259);


Po = ni^2./Na;
figure;
loglog(Nd,Po);


gridon;


xlabel('Nd');


ylabel(' Minority carrier concentration')

 

Discussion

The Effect of Temperature on Intrinsic Carrier Concentration

When we speak of an intrinsic semiconductor several factors come to mind:

1.   It is extremely pure, containing an insignificant amount of impurities. 


2.   The properties of the material depend only on the element(s) the semiconductor is made 
of. 


3.   For every electron created, a hole is created also, no = po = ni. 


For an electron-hole pair to be created in an intrinsic semiconductor, a bond must be broken in

the lattice, and this requires energy. An electron in the valence band must gain enough energy to

jump to the conduction band and leave a hole behind. nirepresents the intrinsic carrier

concentration, or we can see it as the number of bonds broken in an intrinsic semiconductor.

As the temperature is increased, the number of broken bonds (carriers) increases because there is more thermal energy available so more and more electrons gain enough energy to break
free. Each electron that makes it to the conduction band leaves behind a hole in the valence band and there is an increase in both the electron and hole concentration. As the temperature is decreased, electrons do not receive enough energy to break a bond and remain in the valence

band. If electrons are in the conduction band they will quickly lose energy and fall back to the valence band, annihilating a hole. Therefore, lowering the temperature causes a decrease in the intrinsic carrier concentration, while raising the temperature causes an increase in intrinsic carrier concentration.

Whereas the density of states tells one how many states exist at a given energy E, the Fermi function f(E) specifies how many of the existing states at the energy E will be filled with an electron or equivalently.

F(E) specifies, under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron.

Mathematically speaking, the Fermi function is simply a probability distribution function. In mathematical symbols,

 

 

 

Where ??=Fermi energy or Fermi level

K=Boltzmann constant (k=8.167x10^-5eV/K)

T=temperature in Kelvin

 

 

The Fermi function, f(E), specifies the probability of electrons occupying states at a given energy E. The probability that a state is empty (not filled) at a given energy E is equal to 1-f(E). Before concluding the discussion here, it is perhaps worthwhile to reemphasize thus the Fermi function applies only under equilibrium conditions. The Fermi function, however, is universal in the sense that it applies with equal validity so all materials—insulators, semiconductors, and metals. Although introduced in relationship to semiconductors, the Fermi function is not dependent in any way on the special nature of semiconductors but is simply a statistical function associated with electrons in general.

Signatures of all members of the group