MOSFET Depletion Region Electric Field and Charge Density Forms

$Phi = {q_total / varepsilon_s}$

$E circ A = {rho A x/ varepsilon_s}$

$E = {rho}/{varepsilon_s} {x}$

$dE/dx = {rho}/{varepsilon_s}$

${{d^2{phi}}/dx^2} = -{rho}/{varepsilon_s}$ which is Poisson's equation.

Substitute the depletion region charge density

${rho}= q(p-n+N_D-N_A)$ at the surface

0 = q(p-n+N_D-N_A) far from the surface

Subtracting:

${rho}= q(p-p_0)-q(n-n_0)$

Depletion Derivation Line	Inversion Derivation Line
${rho}= q(p-p_0)=q{p_0}[e^{{-phi}/{phi_t}}-1]$	${rho}= -q(n-n_0)=-q{n_0}[e^{{phi}/{phi_t}}-1]=-q{{{n_i}^2}/{p_0}^2}{p_0}[e^{{phi}/{phi_t}}-1]$ ${{{n_i}^2}/{p_0}^2}=e^{-2{phi_F}/{phi_t}}$
${{d^2{phi}}/dx^2} = {{-qp_0}/{varepsilon_s}} {[e^{{-phi}/{phi_t}}-1]}$	${{d^2{phi}}/dx^2} = {{qp_0}/{varepsilon_s}}{e^{{-2{phi_F}}/{phi_t}}}{[e^{{phi}/{phi_t}}-1]}$
$int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}} int{0}{phi_s}{{[e^{{-phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}$	$int{0}{phi_s}{2{{d^2{phi}}/dx^2}{d{phi}/{dx}}} = {{-2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} int{0}{phi_s}{{[e^{{phi}/{phi_t}}-1]} {{d{phi}}/{dx}}}$
$({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}} {[{phi_t}e^{{-phi}/{phi_t}}+{phi}]}$	$({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} {[{phi_t}e^{{phi}/{phi_t}}+{phi}]}$ right hand side evaluated 0 to ${phi_s}$
$({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}} {[e^{{-phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}$	$({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[e^{{phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]}$
$({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{[{phi_t}e^{{-phi_s}/{phi_t}}+{phi_s}-{phi_t}]}$	$({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}}{[{phi_t}e^{{phi_s}/{phi_t}}+{phi_s}-{phi_t}]}$

Identity Table

$Phi = {qN_a / varepsilon_s}$	$E_surface = sqrt { {2 phi_s Phi_s} }$
$x_d = sqrt { {2 phi_s } / {Phi_s} }$
${C_D/C_ox}={varepsilon_S}/{t_D} {/} {varepsilon_ox}/{t_ox}$	$gamma = {varepsilon_s}/{1} {/} {varepsilon_ox}/{t_ox} {sqrt{2 Phi_s }}$