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The Flow of PMOS’s Mobility�(Part2)�

Student：光心君

Date：2010/04/15

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推導Poisson equation為矩陣表示

  • Proof:

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  • 經整理變矩陣程式

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V1

V2

V3

推廣

推廣

A

V

Rho

  • V0=A\Rho 🡪initial guess(V0=A-1R)

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Poisson equation的Boundary Condition

  • Code:

A = zeros(N,N); % Matrix for 2nd differential operator

A(1,1)=1/dx0^2; %boundary condition Vsurface=Vs

A(N,N)=1/dx(N-1)^2;

  • Explain:

Initial Condition 🡪 V(1)=Vs , V(N)=0

1.V(1)=Vs 2.V(N)=0

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  • for j=2:N-1
  • avgdx=(dx(j-1)+dx(j))/2; 🡪 avgdx=Δx
  • A(j,j-1) = 1/dx(j-1)/avgdx; 🡪A(j,j-1) = 1/ Δx2
  • A(j,j) = -(1/avgdx)*(1/dx(j-1)+1/dx(j)); 🡪A(j,j)=-2/ Δx2
  • A(j,j+1) = 1/dx(j)/avgdx; 🡪 A(j,j+1) = 1/ Δx2
  • end;

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推導Poisson equation為矩陣表示

J=2

J=N-1

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%****************CALCULATED PARAMETERS****************

  • A(1,1)=1/dx0^2; %boundary condition Vsurface=Vs

A(N,N)=1/dx(N-1)^2;

%**************POISSON EQUATION SETUP*****************************

  • Rho(1)=Vs/dx0^2; %bondary condition on the surface

Rho(N)=0; %bondary condition V(N)=0

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推導Poisson equation為矩陣表示

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%**************************scale set up *****************************

  • xscale = linspace(xstart,xend,N).'; % New scale cm

dx0=(xend-xstart)/real(N);

dx= dx0/au; % Mesh separation in a.u.

au = 0.5262E-8; % atomic unit in cm(波耳氫原子半徑)

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dd=1/2/(dx^2); % (a.u.)^-2

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  • Xscale🡪 xstart~xend中有N個元素的行向量

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推導Schrödinger equation為矩陣表示

From subprogram: shhole01R.m

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  • 找由1至N所對應的potential值🡪以帶入薛丁格方程式

%**************potential set up*****************************

V=zeros(N,1);

% Potential in Hr (Hr = 27.212; % 1 Hartree in eV)

for j=2:(N-1)

V(j) = interp1(xscaleI,VI,xscale(j))/Hr;

end

V(1)=20; %boundary condition

V(N)=20; %boundary condition

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  • Interp1🡪做一維的內插法
  • Hr🡪 Hartree energy ,the atomic unit of energy.

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推導Schrödinger equation為矩陣表示

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  • Time independent equation:

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  • 其中H= ,且

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推導Schrödinger equation為矩陣表示

pf

推廣

xstart

xend

V=∞

Ψ0=0

V=∞

ΨN+1=0

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  • 矩陣形式

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推導Schrödinger equation為矩陣表示

推廣

%******** Schrodinger Equation ***************

H = zeros(N,N);% light hole

for j=2:(N-1)

H(j,j) = V(j)+2*dd/m1;🡪

end

H(N,N)=V(N)+2*dd/m1;

H(1,1)=V(1)+2*dd/m1;

for j=2:N

H(j-1,j) = -dd/m1;

H(j,j-1) = -dd/m1; 🡪

end

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  • 解波函數Ψ 與E
  • [Y,D]=eig(H); % Eigen vectors(Y) and Eigen values(D)

[lambda1,key1] =sort(diag(D));

%sort:以行為單位,將每一行的向量由小到大排列

Y1 = Y(:,key1); 🡪取key1行的一整列的元素

E1=lambda1*Hr+Ev1;

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  • Y=eigenvector : [Ψ]N ×1(Ψ以行向量的方式儲存在矩陣Y裡)
  • D=eigenvalue : (Ei存放在矩陣D的對角元素)

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推導Schrödinger equation為矩陣表示

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計算hole density

  • 藉由Ψ去計算n(x)

%**************** Calculating hole densities ******************

  • for j=1:N 🡪 hole density /cm2

p1(j)=Do1*k*T*log(1+exp((Ef-E1(j))/k/T)); %hole den in heavy

p2(j)=Do2*k*T*log(1+exp((Ef-E2(j))/k/T)); %hole den in light

end

Do1 = md1*m0/3.1415/(hb)^2/6.24146E11; %density of state (#/eV/cm2)

Do2 = md2*m0/3.1415/(hb)^2/6.24146E11; %density of state (#/eV/cm2)

  • for j=1:N jj=1:N🡪hole density at each valley /cm3

YY1(j,jj)=(Y1(j,jj))^2*p1(jj)/dx0; %hole den. (heavy band in #/cc)

YY2(j,jj)=(Y2(j,jj))^2*p2(jj)/dx0; %hole den. (light band in #/cc)

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計算濃度 for Quantum case

  • 求R: hole density /cm3

R(j) = R(j)+(YY1(j,jj)+YY2(j,jj));

  • if xscale(j)>=xstart & xscale(j)<=xend;

Nep(j) = interp1(xscaleO,R0,xscale(j));

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else

Nep(j)=+Nep0*exp(-beta*V0(j));

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Nen=+ni^2./Nep(只考慮classical case)

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xstart=0.0;

xend=fregion;

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