## A Matlab Codes

The MATLAB codes for the toy models can also be obtained at www.nanohub.purdue.edu.

14.7.A.1 Discrete One-Level Model

E0 = - 5 . 5; E f = -5;gam1 = 0 .2 ; gam2 = 0.2;U = 1 ; %Constants (all MKS, except energy which is in eV) hbar = 1.06e-34;q = 1.6e-19;IE = (2*q*q)/hbar;kT = .025; % Bias (calculate 101 voltage points in [-4 4] range) nV = 101;VV = linspace(-4,4,nV);dV = VV(2)-VV(1); N0 = 2/ (1 + exp((E0-Ef)/kT) ) ; for iV = 1:nV% Voltage loop UU = 0;dU = 1;

V = VV(iV);mu1 = Ef-(V/2);mu2 = Ef+(V/2); while dU>1e-6%SCF E = E0+UU;

f1 = 1/(1+exp((E-mu1)/kT));f2 = 1/(1+exp((E-mu2)/kT)); NN = 2* ( (gam1*f1) + (gam2*f2))/(gam1 + gam2) ; % Charge Uold = UU;UU = Uold+(.05*((U*(NN-N0))-Uold)); dU = abs (UU-Uold) ; [V UU dU] ;

end curr = IE*gam1*gam2 *(f2-f1)/(gam1+gam2) ; II (iV) = curr ;N (iV) = NN; [V NN] ;

G = diff(II)/dV;GG = [G(1) G] ; % Conductance h = plot(VV,II*10^6,'k');% Plot I-V

14.7.A.2 Discrete Two-Level Model

Ef = -5;E0 = [-5.5 -1.5];gam1 = [.2 ,2];gam2 = [.2 .2];U = 1*[1 1;1 1];

% Constants (all MKS, except energy which is in eV) hbar = 1.06e-34;q = 1.6e-19;IE = (2*q*q)/hbar;kT = .025; n0 = 2 . / (1 + exp( (E0-Ef) ./kT) ) ;

nV = 101; VV = linspace (-6 , 6 , nV) ; dV = VV (2 )-VV (1) ;Usc = 0; for iV = 1 : nV dU = 1;

V = VV (iV) ; mu1 = Ef+(V/2);mu2 = Ef-(V/2); while dU>1e-6

f1 = 1./(1 + exp( (E-mu1) ./kT) );f2 = 1./(1 + exp( (E-mu2)./kT) ) ;

Uold = Us c;Us c = Uold+(.1*(( (n-n0)*U')-Uold) ) ;

G = diff (II)/dV;GG = [G(1) G] ; h = plot(VV,II) ; % Plot I-V

14.7.A.3 Broadened One-Level Model

% Toy model, restricted solution with broadening % Inputs (all in eV)

E0 = - 5 . 5; E f = -5;gam1 = 0 . 2; gam2 = 0.2;U = 1.0; % Constants (all MKS, except energy which is in eV) hbar = 1.06e-34;q = 1.6e-19;IE = (2*q*q)/hbar;kT = .025; % Bias (calculate 101 voltage points in [-4 4] range) nV = 101;VV = linspace(-4,4,nV);dV = VV(2)-VV(1); N0 = 2/(1 + exp((E0-Ef)/kT) ) ; for iV = 1:nV% Voltage loop UU = 0;dU = 1;

nE = 400;% Numerical integration over 200 points id = diag(eye(nE)) ' ;

EE = linspace(-10,0,nE) ;dE = EE(2)-EE(1); f1 = 1./(1+exp((EE-id*mu1)/kT)); f 2 = 1./(1 + exp( (EE-id*mu2)/kT)) ; while dU>1e-4% SCF E = E0+UU;

NN = 2*sum(g.*conj(g) .*(gam1* f1+gam2* f2))/(2*pi)*dE; Uold = UU;UU = Uold+(.2*((U*(NN-N0))-Uold)); dU = abs (UU-Uold) ; [V UU dU] ;

end curr = IE*gam1*gam2 *sum((f2-f1) .*g.*conj(g))/(2 *pi)*dE; II (iV) = real (curr) ;N(iV) = NN; [V NN curr E mu1 mu2];

G = diff(II)/dV;GG = [G(1) G] ; % Conductance h = plot(VV,II,'.'); Plot I-V

14.7.A.4 Unrestricted Discrete One-Level Model

% Toy model unrestricted solution % Inputs (all in eV)

E0 = - 5 . 5 ; E f = -5;gam1 = 0 .2 ; gam2 = 0.2;U = 1 ; % Constants (all MKS, except energy which is in eV) hbar = 1.06e-34;q = 1.6e-19;IE = (q*q)/hbar;kT = .025; % Bias (calculate 101 voltage points in [-4 4] range) nV = 101;VV = linspace(-4,4,nV);dV = VV(2)-VV(1); N0 = 1/(1+exp((E0-Ef)/kT)); for iV = 1:nV% Voltage loop

V = VV(iV);mu1 = Ef-(V/2);mu2 = Ef+(V/2); while (dU1+dU2)>1e-6% SCF

f11 = 1/(1 + exp( (E1-mu1)/kT) ) ; f21 = 1/( 1 + exp( (E1-mu2)/kT) ) ; f 12 = 1/(1 + exp( (E2-mu1)/kT) ) ;f22 = 1/(1 + exp( (E2-mu2)/kT) ) ; NN1 = ((gam1*f12)+(gam2*f22))/(gam1+gam2); NN2 = ((gam1*f11)+(gam2*f21))/(gam1+gam2); Uold1 = U1;Uold2 = U2 ;

U1 = Uold1+(.05*( (2 *U*(NN1-N0))-Uold1)) ; U2 = Uold2+(.05*( (2 *U*(NN2-N0))-Uold2)) ; dU1 = abs (U1-Uold1) ;dU2 = abs (U2-Uold2 ) ;

end curr1 = IE*gam1*gam2*(f21-f11)/(gam1+gam2); curr2 = IE*gam1*gam2*(f22-f12)/(gam1+gam2); (iV) = curr1 ; I2 (iV) = curr2 ; N1(iV) = NN1;N2(iV) = NN2 ; [V NN1 NN2 ] ;

G = diff(I1 + I2)/dV;GG = [G(1) G] ; % Conductance h = plot(VV,I1 + I2, '-x) ; Plot I-V

14.7.A.5 Unrestricted Broadened One-Level Model

% Toy model, unrestricted solution with broadening % Inputs (all in eV)

E0 = - 5 . 5 ; E f = -5;gam1 = 0 . 2 ; gam2 = 0.2;U = 1; % Constants (all MKS, except energy which is in eV) hbar = 1.06e-34;q = 1.6e-19;IE = (q*q)/hbar;kT = .025; % Bias (calculate 101 voltage points in [-4 4] range) nV = 101;VV = linspace(-4,4,nV);dV = VV(2)-VV(1); N0 = 1/(1+exp((E0-Ef)/kT));

nE = 200;% Numerical integration over 200 points id = diag(eye(nE))';

EE = linspace(-9,-1,nE);dE = EE(2)-EE(1); for iV = 1:nV% Voltage loop

V = VV(iV);mu1 = Ef-(V/2);mu2 = Ef+(V/2); f1 = 1./(1+exp((EE-id*mu1)/kT));

f 2 = 1./(1 + exp( (EE-id*mu2)/kT)) ; while (dU1+dU2)>1e-3% SCF

NN1 = sum(g1.*conj(g1) .*(gam1*f1 + gam2 *f2))/(2 *pi)*dE; NN2 = sum(g2.*conj(g2) .*(gam1*f1 + gam2 *f2))/(2 *pi)*dE; Uold1 = U1;Uold2 = U2 ;

U1 = Uold1+(.2*((2*U*(NN2-N0))-Uold1)); U2 = Uold2+(.2*((2*U*(NN1-N0))-Uold2));

dU1 = abs (U1-2*U* (NN2-N0) ) ; dU2 = abs (U2-2*U* (NN1-N0) ) ;

end curr = IE*gam1*gam2 *sum((f2-f1) .*(g1.*conj(g1) + .. .

G = diff(II)/dV;GG = [G(1) G] ; % Conductance h = plot(VV,II,' - x);% Plot I-V

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