用mtalab确定PN结势垒区内电场分布和碰撞电离率随反偏电压的变化关系。 （1）基本目标：突变结分析 （2）标准目标：突变结＋线性缓变结分析

确定PN结势垒区内电场分布和碰撞电离率随反偏电压的变化关系可以使用MATLAB进行数值模拟求解。以下是突变结分析和线性缓变结分析的MATLAB代码示例：

突变结分析：

% Parameters
eps_0 = 8.854e-12; % vacuum permittivity
eps_r = 11.68; % relative permittivity of silicon
q = 1.602e-19; % electron charge
N_d = 1e16; % donor concentration
N_a = 1e16; % acceptor concentration
V_bias = linspace(-10, 10, 1000); % bias voltage range
W = 1e-6; % junction width

% Calculate depletion layer width
d_n = sqrt(2*eps_0*eps_r*q*N_a)/q;
d_p = sqrt(2*eps_0*eps_r*q*N_d)/q;
W_d = d_n + d_p;

% Calculate electric field
E = zeros(1, length(V_bias));
for i = 1:length(V_bias)
    if V_bias(i) <= 0
        E(i) = -q*N_a*d_n/eps_0/eps_r;
    else
        E(i) = q*N_d*d_p/eps_0/eps_r;
    end
end

% Plot electric field
figure;
plot(V_bias, E);
xlabel('Reverse bias voltage (V)');
ylabel('Electric field (V/m)');
title('Electric field vs. reverse bias voltage for PN junction with abrupt junction');

% Calculate collision ionization rate
E_th = 3e4; % threshold electric field for collision ionization
alpha = zeros(1, length(V_bias));
for i = 1:length(V_bias)
    if E(i) > E_th
        alpha(i) = 1.5e16*(E(i)/E_th - 1)*exp(-7.5e-10*(E(i)/E_th - 1));
    else
        alpha(i) = 0;
    end
end

% Plot collision ionization rate
figure;
semilogy(V_bias, alpha);
xlabel('Reverse bias voltage (V)');
ylabel('Collision ionization rate (1/m)');
title('Collision ionization rate vs. reverse bias voltage for PN junction with abrupt junction');

线性缓变结分析：

% Parameters
eps_0 = 8.854e-12; % vacuum permittivity
eps_r = 11.68; % relative permittivity of silicon
q = 1.602e-19; % electron charge
N_d = 1e16; % donor concentration
N_a = 1e16; % acceptor concentration
V_bias = linspace(-10, 10, 1000); % bias voltage range
W = 1e-6; % junction width

% Calculate depletion layer width
d_n = sqrt(2*eps_0*eps_r*q*N_a)/q;
d_p = sqrt(2*eps_0*eps_r*q*N_d)/q;
W_d = d_n + d_p;

% Calculate electric field
E = zeros(1, length(V_bias));
for i = 1:length(V_bias)
    x = linspace(-W/2, W/2, 1000);
    N = N_a*ones(size(x));
    N(x > -d_n & x < d_p) = N_d;
    psi = -q/eps_0/eps_r*1/2*(N_a*x(x < -d_n).^2 + (N_d - N_a)*d_n^2 + N_d*x(x > d_p).^2);
    E(i) = -diff(psi)./diff(x);
end

% Plot electric field
figure;
plot(V_bias, E);
xlabel('Reverse bias voltage (V)');
ylabel('Electric field (V/m)');
title('Electric field vs. reverse bias voltage for PN junction with linearly graded junction');

% Calculate collision ionization rate
E_th = 3e4; % threshold electric field for collision ionization
alpha = zeros(1, length(V_bias));
for i = 1:length(V_bias)
    if E(i) > E_th
        alpha(i) = 1.5e16*(E(i)/E_th - 1)*exp(-7.5e-10*(E(i)/E_th - 1));
    else
        alpha(i) = 0;
    end
end

% Plot collision ionization rate
figure;
semilogy(V_bias, alpha);
xlabel('Reverse bias voltage (V)');
ylabel('Collision ionization rate (1/m)');
title('Collision ionization rate vs. reverse bias voltage for PN junction with linearly graded junction');