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The Flow of PMOS’s Mobility (Part2)

The Flow of PMOS’s Mobility (Part2). Student ：光心君 Date ： 2010/04/15. 推導 Poisson equation 為矩陣表示. Proof: 經整理變矩陣程式. V 3. V 1. 推廣. V 2. 推廣. A. V. Rho. V0=ARho  initial guess(V0=A -1 R). Poisson equation 的 Boundary Condition. Code:

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The Flow of PMOS’s Mobility (Part2)

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  1. The Flow of PMOS’s Mobility(Part2) Student：光心君 Date：2010/04/15

  2. 推導Poissonequation為矩陣表示 • Proof: • 經整理變矩陣程式 V3 V1 推廣 V2 推廣 A V Rho • V0=A\Rho initial guess(V0=A-1R)

  3. Poissonequation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

  4. 推導Poissonequation為矩陣表示 • 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; J=2 J=N-1

  5. 推導Poissonequation為矩陣表示 %****************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

  6. 推導Schrödingerequation為矩陣表示 From subprogram: shhole01R.m %**************************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) dd=1/2/(dx^2); % (a.u.)^-2 • Xscalexstart~xend中有N個元素的行向量

  7. 推導Schrödingerequation為矩陣表示 • 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 • Interp1做一維的內插法 • HrHartreeenergy ,the atomic unit of energy.

  8. 推導Schrödingerequation為矩陣表示 • Time independent equation: • 其中H=,且 pf 推廣 V=∞ Ψ0=0 V=∞ ΨN+1=0 xstart xend

  9. 推導Schrödingerequation為矩陣表示 • 矩陣形式 %******** 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 推廣

  10. 推導Schrödingerequation為矩陣表示 • 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; • 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)

  11. 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)

  12. 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)); else Nep(j)=+Nep0*exp(-beta*V0(j)); Nen=+ni^2./Nep(只考慮classicalcase) xstart=0.0; xend=fregion;

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