how to calculate value of Vs if Qs+Qsc=0

Qs=-e*(1+Ga*exp(-(Ecs-Eas)/K*T)*exp(Ecb-Ef)/K*T*exp(-eVs/K*T))^-1
Qsc=sqrt(2)*(Nb+Pb)*e*Ld(cosh(Ub+eVs/K*T)/cosh(ub)-(eVs/K*T)*tanh(Ub)-1)^1/2

if Qs+Qsc=0

calculate Vs

Answer 1

[John D'Errico]

First, I must assume that eVs is what you mean by Vs, because there is NO variable Vs in those equations.

So next, assuming all of those other variables are unknown symbolic coefficients, then no analytical solution will exist. I doubt I even need to check that.

Nothing stops you from putting it into the symbolic toolbox and checking. Just add the two right hand sides. You then have ONE equation in the variable Vs. See if solve gives a solution.

Only if all of the other variables have KNOWN numerical values will it be possible to search for a solution. Again, just add the two right hand sides, and use fzero. Choose a decent starting value, as there may be multiple solutions. I don't expect infinitely many solutions here, but there may be two or three at most, depending on the other variables. I can't plot it to make a better guess though.

[Anil Kumar]

Firstly Thank You John for your time. Actually all other variables used in this equation are known excep Vs; even value of e is also known. Now I am wondering whether I have to calculate manualy the value of Vs by keeping it on right hand side(for this i have to do some maths calculation i.e. solving this equation for Vs on right hand side and all other variables on LHS. My query is can it be done by coding so that I just apply Qs+Qsc=0 and in this condition I can get value of Vs(that i want). Hope I clarified it!! Thank you in advance! will wait for reply!

[Anil Kumar]

p=10^-4
T=300
Nd=10^18
q0=0.1*10^-19
nunegative=10^13
nunuetral=10^13
M=32*1.67*10^-27
k1=K*T
k1=0.0259*10^-19
Beta=Beta0*[1+1/Ga*nunegative/nunuetral*exp(-(Ecs-Ef)/k*T)]^-1*(1+1/Ga*exp(Ef - Eas/k*T))
Theta=Beta*p/1+Beta*p
Ga=2
y=Ecb-Ef
y=0.21*1.6*10^-19
x=Ecs-Eas
x=0.8*1.6*10^-19
Nc=2.5*10^19
Nv=1.04*10^19
Eg=3.6*1.6*10^-19
Ni=sqrt(Nc*Nv*exp(-Eg/K*T))
Nd=10^18
vs=e*Vs/K*T
Nb=Nd/2+sqrt(Nd^2/2+Ni^2)
Pb=Ni^2/Nb
ub=log(Ni^2/Nb)
Qs=-e*(1+Ga*exp(-(x)/k1)*exp(Ecb-Ef)/k1*exp(-eVs/k1))^-1;
Qsc=sqrt(2)*(Nb+Pb)*e*Ld(cosh(Ub+eVs/k1)/cosh(ub)-(eVs/k1)*tanh(Ub)-1)^1/2;

z=0
function c=f(z)
z=Qs+Qsc
fun=@f
Vs=0.1;
w=fzero(fun,Vs)

[Anil Kumar]

This is what I did after your response. But will this code give me values of Vs when Qs+Qsc=0?

[John D'Errico]

So e is actually a variable.

Please learn to format your code when you type it in. I had to modify your comment to make it readable.

https://www.mathworks.com/matlabcentral/answers/13205-tutorial-how-to-format-your-question-with-markup#answer_18099

But, NO. That will do nothing, as you wrote it.

Qs and Qsc may seem like functions. But they are not anything of the sort.

As well, you you define f as a function, thin in f, you create a function handle fun that references itself, and then seem to call fzero. I'd suggest you read through some of the examples on hot to use fzero, as well as do some reading about functions.

As well, you use the variable Vs in multiple places in there, but have never defined it. That implies you are thinking about Vs as a symbolic unknown.

That means you should probably start out with this:

syms Vs

Next, I tried to solve it for you, but you have never defined the variables K or k.

Finally, since you appear to be working in symbolic form, then you will want to use solve, thus something like this:

solve(Qs - Qsc == 0,Vs)

If you want to use fzero, then you need to put the above code in a function as a function of Vs. Fzero passes in trial values for Vs.

[Anil Kumar]

    %-------Quantitative evolution of chemisorption Processes on
    %Metal oxide Semiconductor--------------

    % ----------Discreption of the Programme-----
    % This Matlab program calculate the relation between the total surface
    % coverage area and the surface potential of the metal oxide
    % semiconductors.The surface potential depends upon the gas pressure and
    % temperature during the chemnisorption process.This program uses the
    % Wolkienstien's theory of adsorption which is more adequate than the
    % conventional Langmuir model.

    % Variable used-----
    % P- Pressure of the gas
    % T-Temperature
    % q0= Energy released during the weak adsorption
    % nunegative= frequency of oscillation of the adions
    % nunuetral= frequency of oscillation of the nuetral adsorbates
    % M=  Molecular mass of the oxygen atom
    % Eg= Energy band gap of the semiconductor
    % Ga= degeneracy factor for the chemisorption induces states
    % Ni= Intrinsic carrier concentration
    % K= Boltazmann constant
    % Vs= Surface potential
    % Nd= Dopant concentration
    % Nb= Bulk electron concentration
    % Pb= Bulk hole concentration
    % Ld= Debye length
    % Qs= Surface charge density
    % Qsc= Space charge density

    p= 1
    T=300;
    Nd=10^24;
    q0=0.1*10^-19;
    M=32*1.67*10^-27;
    Ga=2;
    Nstar=10^19
    es=5.4*8.85*10^-14;
    e=1.6*10^-19;
    x=0.21*1.6*10^-19;  %(Ecb-Ef)=x
    y=0.8*1.6*10^-19;    %(Ecs-Eas)=y
    Nc=2.5*10^25;
    Nv=1.04*10^25;
    Eg=2.42*1.6*10^-19;
    Ni=sqrt(Nc*Nv*exp(-Eg/k1));
    Nb=Nd/2+sqrt(Nd^2/2+Ni^2);
    Pb=Ni^2/Nb;
    ub=k1*log(Ni^2/Nb)
    Ld=sqrt(es*k1/(e^2*Nd))
    nstar=10^15;
    nunegative=10^13;
    nunuetral=10^13;
    M=32*1.67*10^-27;
    k1=4.14*10^-21;
    y1=x-y;
    k=1;
    S0=10^-54;
    z1 =y+e*0.5;
    alpha =k*S0/(2*pi*M*k1)^1/2;
    M=32*1.67*10^-27;
    q0=0.1*1.6*10^-19;

      Beta0 =alpha*(nunegative)^-1*exp(q0/k1)
      Beta =Beta0/[1+(1/Ga)*(nunegative/nunuetral)*exp(-x/k1)*exp(e*Vs/k1)]*(1+1/Ga*exp(y/k1)*exp(-(x)/k1)*exp(e*Vs/k1))
      Theta=(Beta.*p/1+Beta.*p)
      plot(p,Theta)
  Qs=-e/[1+Ga*exp(-(y)/k1)*(exp(x/k1)*exp(-e*Vs/k1))]*Theta*Nstar
  Qsc=(sqrt(2)*(Nb+Pb)*e* 2.1989e-07*sqrt(cosh( -2.6594e-19+(e*Vs/k1))/cosh( -2.6594e-19)-(e*Vs/k1)*tanh( -2.6594e-19)-1))

[Anil Kumar]

Dear John Sir, I have to solve the non linear equation Qs+Qsc=0 as a function of Vs from the below code that I wrote. I want a value of Vs for which the above condition is fullfilled. I tried the method you told by following method but could not get the relevant value. I am learning MATLAB hence found it difficult to solve these kind of non-linear equations. Please enlighten me in this regard. Thank you in anticipation! will be a kind help for me as I am stuck here!! Love and respect from India

syms Vs
eqn= -e/[1+Ga*exp(-(y)/k1)*(exp(x/k1)*exp(-e*Vs/k1))]*Theta*Nstar+(sqrt(2)*(Nb+Pb)*e* 2.1989e-07*sqrt(cosh( -2.6594e-19+(e*Vs/k1))/cosh( -2.6594e-19)-(e*Vs/k1)*tanh( -2.6594e-19)-1))==0;
Vs=solve(-e/[1+Ga*exp(-(y)/k1)*(exp(x/k1)*exp(-e*Vs/k1))]*Theta*Nstar+(sqrt(2)*(Nb+Pb)*e* 2.1989e-07*sqrt(cosh( -2.6594e-19+(e*Vs/k1))/cosh( -2.6594e-19)-(e*Vs/k1)*tanh( -2.6594e-19)-1))==0,Vs)