%  FILE: Vox_Vsi.m
%  Voltage drop across oxide and semiconductor
%     as a funcion of reverse bias
%  Input doping density
%


% Constants
eps_0 =   8.85e-14        ; % Units: F/cm
kToq  =   0.0259          ; % Units: V
q     =   1.6e-19         ; % Units: C
cm    =   1.0e4           ; % Units: micron  

% Parameters for Silicon
eps_si =  11.8 * eps_0    ; % Units: F/cm
n_i    =   1.5e10         ; % Units: 1/cm^3
N_a    =   1.0e16         ; % Units: 1/cm^3  <--- INPUT DOPING DENSITY

 
% Parameters for Oxide
eps_ox =   3.9 * eps_0     ; % Units: F/cm
d_ox   =  10.0             ; % Units: nm     <--- INPUT OXIDE THICKNESS
 
 

Phi_F  =  kToq * log( N_a/n_i)   ;
C_ox   =  eps_ox / (d_ox * 1e-7) ;
W_max  =  2 * sqrt( (eps_si*Phi_F) / (q*N_a) ) ;
Q_d    =  q * N_a * W_max  ;
V_T    =  Q_d / C_ox + 2 * Phi_F  ;


% Define Range of Depletion Bias Voltages
V  = linspace(0,1,21);

%  Find voltage drops across oxide and semiconductor
%
%     V = V_ox + V_si  --->  A * V_ox^2 + V_ox - V = 0
%
%                            A = (C_ox/eps_si)*(C_ox/2/q/N_a)

A = (C_ox/eps_si)*(C_ox/2/q/N_a) ;  a =1/(2*A) ;

V_ox = -a + sqrt(a*a + V/A) ;
V_si = V - V_ox             ;

VV= V_si - 2*Phi_F ;

W = sqrt( 2*eps_si*V/q/N_a);

C_si = eps_si ./ W ;

C = (C_ox .* C_si ) ./ (C_ox + C_si ) ;


% Display results
plot(V,V_ox,'k--',V,V_si,'k-',V,(V_ox+V_si),'k:')
hold on
plot( [V_T V_T], [0 1])
plot( [0 1], [Phi_F Phi_F],'r:')
plot( [0 1], [2*Phi_F 2*Phi_F],'r:')