146 lines
3.7 KiB
Matlab
146 lines
3.7 KiB
Matlab
clear all;
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% Physical constants in MKS units
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hbar = 1.054e-34;
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q = 1.602e-19;
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% Energy parameters in eV; included are the single-electron charging
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% energy U0; the kBT product; the equilibrium Fermi level mu; and the
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% energy levels cal_E1 and cal_E2, where "cal_E" is short-form for "calligraphic E"
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% Please especially note that ALL ENERGY VARIABLES IN THIS CODE
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% ARE IN eV (NOT joules); the equations from class must be adjusted
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% accordingly, multiplying or dividing appropriate terms by a factor of q
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% YOU MUST ENTER THE APPROPRIATE VALUES OF cal_E1 and cal_E2
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U0 = 0.025;
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kBT = 0.025;
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mu = 0;
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cal_E1 = 0.2879 / 2;
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cal_E2 = 0.2879 * 3 / 2;
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% Capacitance parameters
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alpha_G = 0.5;
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alpha_D = 0.5;
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alpha_S = 1 - alpha_G - alpha_D;
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% Energy grid in eV, from -1 eV to 1 eV
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NE = 501;
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E = linspace(-1,1,NE);
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dE = E(2) - E(1);
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% Coupling coefficients, which are now grids over E, with
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% gamma_1 equal to 0.005 eV for E > 0, and 0 for E <= 0, and
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% with gamma_2 equal to 0.005 eV for all E; the 1e-6
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% term is included to avoid divide-by-zero errors
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gamma_1 = 0.005*(E + abs(E)) ./ (E + E + 1e-6);
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gamma_2 = 0.005*ones(1,NE);
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gamma = gamma_1 + gamma_2;
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% Reference number of electrons in the channel, assumed to be zero in
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% this code
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N0 = 0;
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% Voltage values to consider for the final plots
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NV = 121;
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VV = linspace(-0.4,0.8,NV);
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dV = VV(2) - VV(1);
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% Loop over voltage values and compute number of electrons and current
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% for each voltage value in a self-consistent manner
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for count = 1:NV
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% Set terminal voltages
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VG = 0;
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VD = VV(count);
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VS = 0;
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% Values of mu1 and mu2; notice that the usual factor of q multiplying
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% the voltages is omitted, because in this code, energy is in eV
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mu1 = mu - VS;
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mu2 = mu - VD;
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% Compute source and drain Fermi functions
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f1 = 1./(1+exp((E - mu1)./kBT));
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f2 = 1./(1+exp((E - mu2)./kBT));
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% Value of Laplace potential in eV
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UL = - (alpha_G*VG) - (alpha_D*VD) - (alpha_S*VS);
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% Initial value of Poisson part in eV
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UP = 0;
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% Iterate until self-consistent potential is achieved by monitoring
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% the Poisson part (the Laplace part does not change)
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dUP = 1;
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while dUP > 1e-6
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% Lorentzian SHIFTED density of states for levels 1 and 2,
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% each normalized so that its integral is unity
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D1 = (0.01/(2*pi))./((E - (UL + UP) - cal_E1).^2+(0.01/2)^2);
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D1 = D1./(dE*sum(D1));
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D2 = (0.01/(2*pi))./((E - (UL + UP) - cal_E2).^2+(0.01/2)^2);
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D2 = D2./(dE*sum(D2));
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% Total density of states
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D = D1 + D2;
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% Compute number of channel electrons
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N(count) = dE*sum( ((gamma_1./gamma).*f1 + (gamma_2./gamma).*f2).*D );
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% Newly calculated Poisson part of self-consistent potential
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UPnew = U0*( N(count) - N0 );
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% Change in Poisson part between iterations
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dUP = abs(UP - UPnew);
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% New guess for next iteration, found by adding a fraction of the
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% difference between iterations to the old guess
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UP = UP + 0.1*(UPnew - UP);
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end
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% Compute the current in A after the self-consistent potential
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% has been achieved; notice the extra factor of q preceding the
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% equation, which is needed since the gammas are in eV
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I(count) = q*(q/hbar)...
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*dE*sum((f1-f2).*D.*gamma_1.*gamma_2./gamma);
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run q4b_subplots;
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end
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% Plotting commands, including lines to modify the linewidth
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% and Fontsize, just to make the plots look nicer; you don't
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% need to worry about how these work
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figure(1); h = plot(VV,N,'k'); grid on;
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set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('DRAIN VOLTAGE [V]'); ylabel('NUMBER OF ELECTRONS');
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figure(2); h = plot(VV,I,'k'); grid on;
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set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('DRAIN VOLTAGE [V]'); ylabel('CURRENT [A]');
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