C. Analytical MOS Model

The formulae to derive the drain current for the analytical model used in this work are listed below in a sequence suitable for computation purposes. (C.2)-(C.7) are well-known formulae which are also used in common SPICE models [37]. (C.10) and (C.12) are, with some minor modifications, taken from [17]. All constants and parameters are listed in Table C.1.

Table C.1: Constants and parameters of the analytical MOS model used in this work.

constants
${\epsilon}_{\mathrm{ox}}$	permittivity constant in the oxide
${\epsilon}_{\mathrm{si}}$	permittivity constant in silicon
$V_\mathrm{T}$	temperature voltage
$q$	electron charge
primary model parameters
$\mu _0$	zero field, zero substrate-doping electron mobility
$N_{\mathrm{sub}}$	substrate doping
$t_{\mathrm{ox}}$	gate oxide thickness
$n_{\mathrm{i}}$	intrinsic carrier concentration in the substrate
$V_{\mathrm{contact}}$	gate contact work-function difference
$v_{\mathrm{sat}}$	electron saturation velocity
secondary model parameters
$\mu$	effective electron mobility
$KP$	transconductance factor
$\phi$	twice the bulk Fermi potential
$\gamma$	body effect parameter
$V_{\mathrm{th,0}}$	threshold voltage at $V_\mathrm{ds}$ = 0 V
$E_{\mathrm{c}}$	critical field for velocity saturation
$L_{\mathrm{DIBL}}$	DIBL-characteristic length
auxiliary model parameters
$L$	device length
$W$	device width
$F_{\mathrm{DIBL}}$	DIBL factor (is set to 0 if DIBL is disabled)
$E_{\mathrm{DIBL}}$	DIBL exponent

The model can be accessed via primary or secondary model parameters. The primary parameters listed are physically based parameters which reflect the device geometry and the material properties. The secondary parameters are model internal parameters which can be set directly or will be evaluated from the primary parameters by default.

The electron mobility is calculated using the ionized impurity scattering model by Scharfetter and Gummel [46]

$$\mu = \frac{\mu _0}{\sqrt{1+\frac{\displaystyle \rule[-.8ex]... ...\rule{0mm}{1.9ex} N_{\mathrm{sub}}}{\displaystyle 350}}}} \; .$$ (C1)

The transconductance parameter is defined as

$$KP=\mu \cdot \frac{\epsilon _{\mathrm{ox}}}{t_{\mathrm{ox}}}~ \; ,$$ (C2)

and $\phi$, which has the value of twice the bulk Fermi potential, reads

$$\phi =2\cdot V_{\mathrm{T}}\cdot \ln \frac{N_{\mathrm{sub}}}{n_{\mathrm{i}}} \; .$$ (C3)

The body effect parameter is calculated by

$$\gamma =\frac{t_{\mathrm{ox}}}{\epsilon _{\mathrm{ox}}}\cdot... ...t{2\cdot q\cdot \epsilon _{\mathrm{si}}\cdot N_{\mathrm{sub}}}$$ (C4)

and the threshold voltage for zero drain-source voltage by

$$V_{\mathrm{th,0}}=V_{\mathrm{contact}}+\frac{\phi}{2}+\gamma \cdot \sqrt{\phi} \; .$$ (C5)

The bulk depletion width reads

$$w_{\mathrm{d}}=\sqrt{\frac{2\cdot \epsilon _{\mathrm{si}}\cdot \phi}{q\cdot N_{\mathrm{sub}}}}$$ (C6)

and the critical field for carrier velocity saturation is calculated by

$$E_{\mathrm{c}}=\frac{2\cdot v_{\mathrm{sat}}}{\mu} \; .$$ (C7)

To include the DIBL effect a DIBL-characteristic length is introduced

$$L_{\mathrm{DIBL}}=\sqrt{\frac{\epsilon _{\mathrm{si}}}{\epsilon _{\mathrm{ox}}}\cdot w_{\mathrm{d}}\cdot t_{\mathrm{ox}}}$$ (C8)

and the effective threshold voltage is calculated by a formula similar to that used in the BSIM3v3 model [13]

$$V_{\mathrm{th}}=V_{\mathrm{th,0}}-F_{\mathrm{DIBL}}\cdot V_{... ...mathrm{DIBL}} \cdot L}{L_{\mathrm{DIBL}}} \right) \right) \; .$$ (C9)

The slope factor is

$$n= \left (1-\frac{\gamma}{2\cdot \sqrt{V_{\mathrm{gs}} - V_{\mathrm{th}} + (\frac{\gamma}{2} + \sqrt{\phi})^2}} \right) ^{-1}$$ (C10)

and a velocity-saturation factor is introduced using the Heavyside function $\sigma$ [53]

$$F_{\mathrm{sat}}= \left( 1+\frac{V_{\mathrm{gs}}-V_{\mathrm{... ...dot \sigma \left( V_{\mathrm{gs}}-V_{\mathrm{th}} \right) \; .$$ (C11)

Finally, the drain current is calculated by

$$I_{\mathrm{d}}=2\cdot n\cdot \frac{W}{L}\cdot KP\cdot V_\mat... ...ot V_{\mathrm{t}}} \right) \right) \cdot F_{\mathrm{sat}} \; .$$ (C12)