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master_equation.m
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master_equation.m
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function current = master_equation(Vd, Vg, n)
%master_equation computes the master equation
% current = master_equation(Vd, Vg, n) returns current as the sum of
% charge density for n electrons, drain voltage Vd and gate voltage Vg.
for i = -n : 1 : n
%ma(N+i+1,N+i+2) = - (gamma_L->dot + gamma_R->dot + gamma_dot->L + gamma_dot->R)
ma(n+i+1,n+i+2) = -(f_tunnel(1,i,Vd,Vg) + f_tunnel(2,i,Vd,Vg) + f_tunnel(3,i,Vd,Vg) + f_tunnel(4,i,Vd,Vg));
%ma(N+i+1,N+i+2) = gamma_L->dot + gamma_R->dot
ma(n+i+1,n+i+2-1) = f_tunnel(1,i-1,Vd,Vg) + f_tunnel(4,i-1,Vd,Vg);
%ma(N+i+1,N+i+2+1) = gamma_dot->L è gamma_dot->R
ma(n+i+1,n+i+2+1) = f_tunnel(2,i+1,Vd,Vg) + f_tunnel(3,i+1,Vd,Vg);
end
mma = [1,zeros(1,2*n+2);...
ma;zeros(1,2*n+2),1];
[p,o] = eig(mma); % produces a diagonal matrix o of eigenvalues and a full
% matrix p whose columns are the corresponding eigenvectors
% so that mma*p = p*o
[s,pos] = min(abs(diag(o))); % s is the minimum eigenvalue contained in
% the diagonal matrix o, pos is the index
% of the eigenvalue
p_n(:,1) = abs(p(:,pos))/sum(abs(p(:,pos))); % probability that n charges are into the dot
for i = 1:length(p_n)
% compute the current density as the multiplication between the probability p_n
% and the difference between tunneling rate dot->R and R->dot
charge_density(i) = p_n(i)*(f_tunnel(1,i-n-2,Vd,Vg) - f_tunnel(2,i-n-2,Vd,Vg));
end
% the sum of currentDensity is the current
current = sum(charge_density);
end