%  FILE: MOS_Qs.m
%  Variation of space-charge density in the semiconductor Q_s
%     as a function of the surface potential V_s 
%     assume p-type semiconductor

% 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    =   4.0e15         ; % Units: 1/cm^3  <--- INPUT DOPING DENSITY

p_0    = N_a ; 
n_0    = n_i*n_i/p_0 ;
 
Phi_F  =  kToq * log( N_a/n_i)    ;
L_D    = sqrt( kToq*eps_si/q/N_a ) ;  % Debye length


V_s = linspace(-0.5,1,100)              ;          % <--- INPUT Depletion Bias Voltage


e_s = exp(-V_s/kToq) + V_s/kToq - 1 + (n_0/p_0) * (exp(V_s/kToq) - V_s/kToq + 1 )  ;
E_s = sqrt(2)*kToq/L_D * sqrt( e_s ) ;

Q_s = eps_si * E_s  ;


plot( V_s  , log10(Q_s) ) ;  hold on
plot( [Phi_F Phi_F] , [-9 -8 ] ) ;  hold on
plot( [2*Phi_F 2*Phi_F] , [-9 -7 ] ) ;  hold on
axis( [-0.4 1 -9 -4]);


% Depletion Approximation

Vdepl = linspace(0,1,100);
Wdepl = sqrt( 2*eps_si*Vdepl/q/N_a) ;
Qdepl = Wdepl * N_a * q ;

plot( Vdepl , log10(Qdepl) , ':' ); hold on