% BJT Equilibrium Energy Band Diagram Generator

% This program plots out the BJT equilibrium energy band diagram



% Original version authored by Aaron Luft as part of a course project for Prof. Gerry Neudeck

% Major revisions by R. F. Pierret



DOPING=[1e18 -1e16 1e15];   % E, B, and C type and doping concentrations (- = n-type)

WB=1.0e-4;   %Total base width in cm; 1.0e-4cm=1micrometer

close



%Constants

T=300;        % Temperature in Kelvin

k=8.617e-5;   % Boltzmann constant eV/K

e0=8.85e-14;  % permittivity of free space (f/cm)

q=1.602e-19;  % charge on an electron (coul)

KS=11.8;      % Dielectric constant of Si at 300K

ni=1.0e10;    % intrinsic conc. of Silicon at 300K

EG=1.12;      % Silicon band gap (eV)

%end constants



%General Computations and Manipulations

NE = DOPING (1);                % Emitter doping and type

NB = DOPING (2);                % Base doping and type

NC = DOPING (3);                % Collector doping and type



sE = sign (NE);

sB = sign (NB);

sC = sign (NC);



NE = abs(NE);           % Emitter doping

NB = abs(NB);           % Base doping

NC = abs(NC);           % Collector doping



Ei_emitter  = [ (sE * k * T * log ( NE / ni ) ) ...

	       (-sB * k * T * log ( NB / ni ) ) ];

Ei_collector = [ (sB * k * T * log ( NB / ni ) ) ...

	       (-sC * k * T * log ( NC / ni ) ) ];



Vbi  = [ (sum (Ei_emitter)) (sum (Ei_collector)) ];

svbi = sign (Vbi);

Vbi  = abs  (Vbi);



			% Depletion width on emitter side of EB junction

xE  = sqrt(2*KS*e0/q*NB*Vbi(1)/(NE*(NB+NE)));

			% Depletion width on base side of EB junction

xBeb = sqrt(2*KS*e0/q*NE*Vbi(1)/(NB*(NE+NB)));



			% Depletion width on base side of CB junction

xBcb  = sqrt(2*KS*e0/q*NC*Vbi(2)/(NB*(NC+NB)));

			% Depletion width on collector side of EB junction

xC = sqrt(2*KS*e0/q*NB*Vbi(2)/(NC*(NB+NC)));

W = WB-xBeb-xBcb;



if W < 0

error('For the given DOPING and WB the base is totally depleted.')

end

  

if ( xC > xE )          % Adjust the x-axis for optimum looking plot

HIGH_X = 1.5;           

LOW_X = xC/xE;

else

HIGH_X = xE/xC;

LOW_X  = 1.5;

end



VMAX = 3;                        % Maximium Plot Voltage

plot ( [-LOW_X*xE HIGH_X*xC+WB ] , [ 0 VMAX ] , 'i');

hold on;



% EB JUNCTION

xlft = -LOW_X*xE;               % Leftmost x position

xrght = xBeb + W/2;             % Rightmost x position



x = linspace(xlft, xrght, 200);

sVx = -svbi(1) * sE * sB;



Vx1=sVx * (Vbi(1)-q*NB.*(xBeb-x).^2/(2*KS*e0).*(x<=xBeb)).*(x>=0);

Vx2=sVx * 0.5*q*NE.*(xE+x).^2/(KS*e0).*( x>=-xE & x<0 );

Vx=Vx1+Vx2;     % V as a function of x



EF=Vx(1)+VMAX/2-sE*k*T*log(NE/ni);  % Fermi level



Ec = -Vx+EG/2+VMAX/2;

Ev = -Vx-EG/2+VMAX/2;

Ei = -Vx+VMAX/2;



LEc = Ec (1);

LEv = Ev (1);

LEi = Ei (1);



% Plot V vs x

plot ( x, Ec );                               % Ec

plot ( x, Ev );                               % Ev

plot ( x, Ei, 'w:');                        % Ei

plot ( [xlft 0], [ EF EF ], 'w' );      % EF on left

plot ( [ 0 0 ], [ 0.15 VMAX-0.15 ], 'w--' );    % Junction center





% CB JUNCTION

xlft = -xBcb-W/2;               % Leftmost x position

xrght = HIGH_X*xC;      % Rightmost x position



x = linspace(xlft, xrght, 200);

sVx = -svbi(2) * sC * sB;



Vx1=sVx * (Vbi(2)-q*NC.*(xC-x).^2/(2*KS*e0).*(x<=xC)).*(x>=0);

Vx2=sVx * 0.5*q*NB.*(xBcb+x).^2/(KS*e0).*( x>=-xBcb & x<0 );

Vx=Vx1+Vx2;     % V as a function of x



OFFSET = (Ec(200))-(-Vx(1)+EG/2+VMAX/2);

Ec = (-Vx+EG/2+VMAX/2) + OFFSET;

Ev = (-Vx-EG/2+VMAX/2) + OFFSET;

Ei = (-Vx+VMAX/2) + OFFSET;



x = x + WB;



% Plot V vs x

plot ( x, Ec );                         % Ec

plot ( x, Ev );                         % Ev

plot ( x, Ei, 'w:');                            % Ei

plot ([0 xrght+WB], [EF EF], 'w');  %EF on right

plot ( [ WB WB ], [ 0.15 VMAX-0.15 ], 'w--' );  % Junction center



if ( sC == -1 )

    RIGHT = 'N';

else

    RIGHT = 'P';

end



if ( sB == -1 )

    MIDL = 'N';

else

    MIDL = 'P';

end



if ( sE == -1 )

    LEFT = 'N';

else

    LEFT = 'P';

end



A = -LOW_X*xE/2;

B = WB + ((HIGH_X*xC+WB)-WB)/2;



text ( A , 2.5, LEFT );

text ( WB/2, 2.5, MIDL );

text ( B , 2.5, RIGHT);





text ( x(200), Ec (200), 'Ec' );

text ( x(200), Ei (200), 'Ei' );

text ( x(200), Ev (200), 'Ev' );

text ( x(200), EF , 'EF' );



REG = [ LEFT(1) MIDL(1) RIGHT(1) ];

TITLE = [ ('Energy band diagram for the ') (REG) (' device') ];

title (TITLE);