MOSFET Threshold Voltage Formula Derivation

At threshold voltage Vt the surface potential is

${phi} = {2 phi_F}$

Given this condition threshold voltage is:

$V_T0 = V_FB + {2 phi_F} + {{varepsilon_s}{E_s}}/C_ox$ ….the 3rd term is V=Q/C

Reorganizing terms:

$V_T0 = V_FB + {2 phi_F} +{{varepsilon_s}/{1}{/}{varepsilon_ox}/{t_ox}{E_s}}$

Substituting the identity for Es:

$V_T0= V_FB + {2 phi_F} +{{varepsilon_s}/{1}{/}{varepsilon_ox}/{t_ox}{sqrt {{2 phi_s Phi_s}}}}$

$V_T0= V_FB + {2 phi_F} +{{varepsilon_s}/{1}{/}{varepsilon_ox}/{t_ox}{sqrt {{4{phi_F}{Phi_s}}}}}$

$V_T0= V_FB + {2 phi_F} +{{varepsilon_s}/{1}{/}{varepsilon_ox}/{t_ox}{sqrt{2 Phi_s}sqrt{2 phi_F}}}$

Substituting for the body effect factor identity gamma:

$V_T0= V_FB + {2 phi_F} + {gamma}{sqrt{2 phi_F}}$

Equation 2.1.59 can be found in: Tsividis:Operation and Modeling of the MOS Transistor :Page 110

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

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