MOSFET Depletion Region Electric Field and Charge Density Forms
[pmath] Phi = {q_total / varepsilon_s} [/pmath]
[pmath] E circ A = {rho A x/ varepsilon_s} [/pmath]
[pmath] E = {rho}/{varepsilon_s} {x} [/pmath]
[pmath] dE/dx = {rho}/{varepsilon_s} [/pmath]
[pmath] {{d^2{phi}}/dx^2} = -{rho}/{varepsilon_s} [/pmath] which is Poisson's equation.
Substitute the depletion region charge density
[pmath] {rho}= q(p-n+N_D-N_A)[/pmath] at the surface
[pmath]0 = q(p-n+N_D-N_A)[/pmath] far from the surface
Subtracting:
[pmath] {rho}= q(p-p_0)-q(n-n_0)[/pmath]
| Depletion Derivation Line | Inversion Derivation Line |
| [pmath] {rho}= q(p-p_0)=q{p_0}[e^{{-phi}/{phi_t}}-1] [/pmath] | [pmath] {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] [/pmath] [pmath] {{{n_i}^2}/{p_0}^2}=e^{-2{phi_F}/{phi_t}} [/pmath] |
| [pmath] {{d^2{phi}}/dx^2} = {{-qp_0}/{varepsilon_s}} {[e^{{-phi}/{phi_t}}-1]} [/pmath] | [pmath] {{d^2{phi}}/dx^2} = {{qp_0}/{varepsilon_s}}{e^{{-2{phi_F}}/{phi_t}}}{[e^{{phi}/{phi_t}}-1]} [/pmath] |
| [pmath] 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}}} [/pmath] | [pmath] 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}}} [/pmath] |
| [pmath] ({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}} {[{phi_t}e^{{-phi}/{phi_t}}+{phi}]} [/pmath] | [pmath] ({d{phi}}/{dx})^2 = {{2qp_0}/{varepsilon_s}}{e^{-2{phi_F}/{phi_t}}} {[{phi_t}e^{{phi}/{phi_t}}+{phi}]} [/pmath] right hand side evaluated 0 to [pmath] {phi_s} [/pmath] |
| [pmath] ({d{phi}}/{dx})^2 = {{2qp_0{phi_t}}/{varepsilon_s}} {[e^{{-phi_s}/{phi_t}}+{phi_s}/{phi_t}-1]} [/pmath] | [pmath] ({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]} [/pmath] |
| [pmath] ({d{phi}}/{dx})^2 = {{2qN_A}/{varepsilon_s}}{[{phi_t}e^{{-phi_s}/{phi_t}}+{phi_s}-{phi_t}]} [/pmath] | [pmath] ({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}]} [/pmath] |
Identity Table
| [pmath] Phi = {qN_a / varepsilon_s} [/pmath] | [pmath] E_surface = sqrt { {2 phi_s Phi_s} } [/pmath] |
| [pmath] x_d = sqrt { {2 phi_s } / {Phi_s} } [/pmath] |
[pmath] E_surface = Phi_s x_d [/pmath] |
| [pmath] {C_D/C_ox}={varepsilon_S}/{t_D} {/} {varepsilon_ox}/{t_ox} [/pmath] |
[pmath] gamma = {varepsilon_s}/{1} {/} {varepsilon_ox}/{t_ox} {sqrt{2 Phi_s }} [/pmath] |
0 Comments