1. Device Physics

General principle: In subthreshold operation only diffusion current is present. Ordinarily the forward and reverse diffusion currents are equal leaving a net current of 0. However, if we place a potential difference across the Source/Drain we can increase the diffusion in one direction producing a net current in one direction. This is seen in the exponential difference term below. We can also increase the gross diffusion by increasing the gate bias.

2. Basic Equation

$I_d = I_S \frac{W}L \exp{ \left [ \frac{\kappa V_{GB}}{U_T} \right ] } \left ( \exp{ \left [ \frac{-V_{SB}}{U_T} \right ] } - \exp{ \left [ \frac{-V_{DB}}{U_T} \right ] } \right )$
$I_d = I_S \frac{W}L \exp{ \left [ \frac{\kappa V_{GB}}{U_T} \right ] } \exp{ \left [ \frac{-V_{SB}}{U_T} \right ] } \left ( 1 - \exp{ \left [ \frac{-V_{DS}}{U_T} \right ] } \right )$
(See Regions of operation for saturation equation)

2.1 Diode connected transistors

pMOS

V_GB = V_DB
$I_d = I_S \frac{W}L \exp{ \left [ \frac{- \kappa V_{GB}}{U_T} \right ] } \left ( \exp{ \left [ \frac{V_{SB}}{U_T} \right ] } - \exp{ \left [ \frac{V_{GB}}{U_T} \right ] } \right )$
- Assuming κ ~ 1
$\ln(I_d) = \ln{ \left( I_S \frac{W}L \right) } + \left [ \frac{V_{SB} - \kappa V_{GB}}{U_T} \right ]$
$V_{SB} - \kappa V_{GB} = U_T \left( \ln(I_d) -\ln{ \left( I_S \frac{W}L \right) } \right)$
$\kappa V_{GB} = V_{SB} - U_T \left( \ln(I_d) -\ln{ \left( I_S \frac{W}L \right) } \right)$

3. Unifying models

Equations that work for both regions.

EKV Model:
$I_{ds} = I_F + I_R = I_S \frac{W}L \left[ \log^2{ \left( 1 + \exp{ \left{ \frac{\kappa(V_G - V_{TO}) - V_S }{2 U_T} \right} } \right) } - \log^2{ \left( 1 + \exp{ \left{ \frac{\kappa(V_G - V_{TO}) - V_D }{2 U_T} \right} } \right) } \right]$

ACM Model:
Don't have this yet.

SubThreshold.com A central location for SubThreshold design information.

1. Device Physics

2. Basic Equation

2.1 Diode connected transistors

pMOS

3. Unifying models

SubThreshold.com
A central location for SubThreshold design information.