Iterations of an indirect equation using MATLAB

I am stuck on plotting a graph of surface potential(shy_s) Vs Gate voltage(vgb). I just have to solve this equation below and find the root for every iterations

vgb=vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t))

where

shy_f=0.347; %shy_f=shy_t*ln(Na/ni)

shy_t=0.0259;   %Thermal voltage = KT/e ; where k = 1.3806*10^-23 @ 300 K       

es=11.7*8.85*10^-12; 

Na=10^10; %[unit]=[m^-3)

cox=6.93*10^-12; %[unit]=[F/m^2] and t_ox=550 A

q=1.6*10^-19;

vfb=0;

gama=(sqrt(2*q*es*Na)/cox);

Here I have to find the value of shy_s(surface potential) for different values of vgb(gate voltage).

So I tried different methods to solve it, such as

a=zeros(1,100);

b=zeros(1,100);

for vgb=0:0.1:10

shy_s=0;

% Say 

p=shy_s;

% And

j=vgb-vfb-((sqrt(2*q*es*10^10))/cox).*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t));

D=p-j;

if D>0

  for  shy_s=0:0.1:30;开发者_Go百科

    D=p-j;

    if D<0

        a=shy_s;

        break

    end

  end

elseif D<0

  for shy_s=0:0.1:30

    D=p-j;

    if D>0

        a=shy_s;

        break

    end

  end

end

b(1,vgb)=a;

end

plot(vgb,b)

At this manner the following error shows up:

??? Subscript indices must either be real positive integers or logicals.

Error in ==> shy_s_vs_vgb_latest2 at 78 b(1,vgb)=a;

Again I tried to use rather a simpler technique-

vgb=fzero(@(shy_s)vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-(2.*shy_f))/shy_t)),2)

but it says-

Exiting fzero: aborting search for an interval containing a sign change because complex function value encountered during search. (Function value at -0.56 is -0.56+62.1585i.) Check function or try again with a different starting value.

vgb =

NaN

Another relation can be used for the same purpose

(vgb-vfb-shy_s)/gama)^2 = shy_s+shy_t.*(exp((shy_s-2*shy_f)/shy_y))+shy_t.*(exp(-shy_s/shy_t)-1) 

Although not likely the best solution, a quick and dirty trick is to the following:

opt = optimset('TolFun',1e-8);
vgb=@(shy_s) vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t));
b = fminsearch(@(shy_s) abs(vgb(shy_s)-VAL),10,opt);

with VAL being the number you wish to find the inverse for.

Here how to use the function fzero for a simple iteration with out doing much -

for i=1:length(vgb)

c=@(shy_s)((vgb(i)-vfb-shy_s)/gama)-sqrt(abs(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t)));

shy_s=fzero(c,[-3 10])

a(i)=shy_s

end

• 'a' gives the correct iterated value!