1. Theory

Definition: The EKV model, from Enz, Krummenacher and Vittoz at EPFL, is a charge-based compact MOSFET model that expresses the drain current as the difference of a forward and a reverse current, each a function of the pinch-off voltage relative to the source or drain (EKV 2.6 report; Sallese et al.).
Bulk reference and symmetry: All voltages are referred to the substrate, and the forward and reverse terms have the same form, so the model is symmetric in source and drain by construction (EKV 2.6 report).
One expression for all inversion levels: The interpolation function $F (v) = [ln (1 + e^{v /2})]^{2}$ joins the exponential weak-inversion and square-law strong-inversion behaviour continuously, so moderate inversion is described without separate regions (EKV 2.6 report).
Inversion coefficient: Normalizing the current to the specific current $I_{S} = 2 n β V_{t}^{2}$ gives an inversion coefficient that sets the operating point, from weak inversion below 0.1 through moderate inversion near 1 to strong inversion above 10 (Sallese et al.).
Compactness: The intrinsic model needs few parameters, fewer than 20 for DC and RF in Sallese et al., while the full 2.6 model with temperature, noise, matching and impact ionization has about 40 (Sallese et al.; EKV 2.6 report).

2. Working Principle

Pinch-off voltage: The channel potential at which the inversion charge vanishes, $V_{P}$ , depends only on the gate voltage through the threshold, PHI and GAMMA, and it is linear in gate voltage with slope $1/ n$ to a good approximation (EKV 2.6 report; Sallese et al.).
Forward and reverse currents: The inversion charge at the source and drain ends follows from how far each terminal is below the pinch-off voltage, and the drain current is proportional to the difference of the normalized charges’ functions $F$ at the two ends (EKV 2.6 report; background knowledge for the charge picture).
Saturation: When the drain rises well above $V_{P}$ the reverse current vanishes, so saturation follows naturally rather than from a separate equation (EKV 2.6 report).
Short-channel effects: Velocity saturation enters through $V_{C} = U C R I T \cdot L$ , channel-length modulation through $Δ L$ , and charge sharing and reverse short-channel effects through LETA, WETA, Q0 and LK (EKV 2.6 report).

3. Applications

Low-power analog design: Sizing transistors by inversion coefficient and $g_{m} / I_{D}$ , especially in weak and moderate inversion where many low-power circuits operate (Sallese et al.).
RF modelling: EKV with charge-based NQS and noise models for RF CMOS, verified to about 10 $GHz$ on 0.35 $μ m$ devices (Sallese et al.).
Matching and temperature: Built-in mismatch parameters AVTO, AKP and AGAMMA and temperature parameters for analog design (EKV 2.6 report).
Current status: EKV3 and the charge-based design methodology remain in use for analog design and teaching, while foundry kits mostly ship BSIM4, PSP or BSIM-CMG (background knowledge; BSIM3 and BSIM4 page).

Pinch-Off Voltage and Normalized Currents

Name	Formula	Related Formulae	Variables/Constants
Pinch-off voltage and slope factor	$V_{P} = V_{G}^{'} - P H I - G A M M A [V_{G}^{'} + (\frac{G A M M A}{2})^{2} - \frac{G A M M A}{2}], V_{G}^{'} = V_{G} - V T O + P H I + G A M M A P H I$	$n = 1 + \frac{G A M M A}{2 V _{P} + P H I + 4 V _{t}}, V_{P} \approx \frac{V _{G} - V T O}{n}$	$V_{P} : p in c h - o f f v o l t a g e, V_{G}^{'} : e f f ec t i v e g a t e v o l t a g e$ $V_{G} : g a t e v o l t a g e r e f er r e d t o t h e s u b s t r a t e$ $V T O : l o n g - c hann e l t h r es h o l d$ $P H I : b u l k F er mi p o t e n t ia l t er m$ $G A M M A : b o d y - e f f ec t f a c t or$ $n : s l o p e f a c t or, V_{t} = k_{B} T / q : t h er ma l v o l t a g e$ $k_{B} : B o l t z mann co n s t an t$ $T : t e m p er a t u r e$ $q : e l e m e n t a r y c ha r g e; E K V 2.6 r e p or t : E q s . 33, 38 an d 39, V_{P} = - P H I f or V_{G}^{'} o f 0 or b e l o w, a l l v o l t a g es r e f er r e d t o t h e l oc a l s u b s t r a t e, c ha r g e s ha r in g e n t er in g V_{P} t h r o ug h a m o d i f i e d γ^{'}, an d a r e v er se s h or t - c hann e l s hi f t Δ V_{R S C E} (t h e V_{G}^{'} e x p r ess i o n i s g a r b l e d in t h e e x t r a c t an d g i v e n h er e in t h e s t an d a r d f or m, ba c k g r o u n d k n o w l e d g e); S a l l ese e t a l . : t h e s im pl i f i e d tw o - p a r am e t er r e l a t i o n V_{P} = (V_{G} - V_{T O}) / n_{V}, w i t h n_{V} f r o m ab o u t 1.6 in w e ak t o 1.2 in s t r o n g in v er s i o n$ $(co m p u t e d : t h e d e f a u l t G A M M A = 1.0 V^{1/2} an d P H I = 0.7 V g i v e n = 1.56 a t V_{P} = 0)$
Interpolation function and normalized currents	$F (v) = [ln (1 + e^{v /2})]^{2}, i_{f} = F (\frac{V _{P} - V _{S}}{V _{t}}), i_{r} = F (\frac{V _{P} - V _{D}}{V _{t}})$	$F (v) \to (\frac{v}{2})^{2} (v ≫ 0), F (v) \to e^{v} (v ≪ 0), y = 0.25 + i - 0.5, v = 2 y + ln y (inverse)$	$F : in t er p o l a t i o n f u n c t i o n b e tw ee n w e ak an d s t r o n g in v er s i o n$ $i_{f}, i_{r} : f or w a r d an d r e v er se n or ma l i z e d c u r r e n t s$ $V_{S}, V_{D} : so u r ce an d d r ain v o l t a g es r e f er r e d t o t h e s u b s t r a t e$ $v : n or ma l i z e d v o l t a g e$ $y : a ux i l ia r y v a r iab l e o f t h e in v er se; E K V 2.6 r e p or t : E q s . 40 t o 44, w i t h t h e s t r o n g - an d w e ak - in v er s i o n a sy m pt o t es an d t h e in v er se u se d in t h e co d e; t h e m o d e l i s sy mm e t r i c in so u r ce an d d r ain, i t s s in g l e e x p r ess i o n i s co n t in u o u s in a l l r e g i o n s, an d p - c hann e l d e v i ces a r e t r e a t e d a s p se u d o - n - c hann e l w i t h in v er t e d s i g n s o f V F B, V T O, T C V an d t h e v o l t a g es, an d a m o bi l i t y f a c t or η o f 1/2 f or N M O S an d 1/3 f or P M O S$ $(co m p u t e d : F (0) = (l n 2)^{2} = 0.48, t h e mi dd l e o f m o d er a t e in v er s i o n)$
Specific current and drain current	$I_{S} = 2 n β V_{t}^{2}, β = K P \frac{W _{e f f}}{L _{e q}}$	$I_{D S} = I_{S} (i_{f} - i_{r})$	$I_{S} : s p ec i f i c c u r r e n t$ $β : t r an sco n d u c t an ce f a c t or$ $K P : t r an sco n d u c t an ce p a r am e t er$ $W_{e f f}, L_{e q} : e f f ec t i v e w i d t h an d e q u i v a l e n t l e n g t h$ $I_{D S} : d r ain c u r r e n t; E K V 2.6 r e p or t : E q s . 65 an d 66, v a l i d in w e ak, m o d er a t e an d s t r o n g in v er s i o n an d in co n d u c t i o n an d s a t u r a t i o n, w i t h t h e r e v er se c u r r e n t cor r ec t e d f or m o bi l i t y e f f ec t s; S a l l ese e t a l . : I_{S} = 2 n μ C_{o x}^{'} U_{T}^{2} (W / L) an d t h e n or ma l i z e d c u r r e n t i = i_{f} - i_{r} a s an e f f i c i e n t c ha r a c t er i z a t i o n o f t h e o p er a t in g p o in t$ $(co m p u t e d : K P = 50 μ A/ V^{2}, W / L = 10 an d n = 1.3 (a ss u m e d) g i v e I_{S} = 0.87 μ A a t 300 K)$

Inversion Coefficient

Name	Formula	Related Formulae	Variables/Constants
Inversion coefficient and transconductance efficiency	$\frac{g _{m s} V _{t}}{I _{D}} = \frac{1}{0.5 + 0.25 + i _{f}}, g_{m} = \frac{g _{m s}}{n} (saturation)$	$I C = \frac{I _{D}}{I _{S}} \approx i_{f}, I C < 0.1 : weak, I C \approx 1 : moderate, I C > 10 : strong inversion$	$g_{m s} : so u r ce t r an sco n d u c t an ce$ $g_{m} : g a t e t r an sco n d u c t an ce$ $I_{D} : d r ain c u r r e n t$ $I C : in v er s i o n coe f f i c i e n t; E K V 2.6 r e p or t : G_{m} V_{t} / I_{D} = 1/ (0.5 + 0.25 + i_{f}) a s t h e ba s i s o f t h e s t a t i c, q u a s i - s t a t i c, N QS an d n o i se m o d e l s (r e a d a s t h e so u r ce t r an sco n d u c t an ce, w i t h g_{m} = g_{m s} / n in s a t u r a t i o n, m y r e a d in g); S a l l ese e t a l . : t h e n or ma l i z e d t r an sco n d u c t an ce - t o - c u r r e n t r a t i o, u se f u l f or ana l o g d es i g n an d in d e p e n d e n t o f t e m p er a t u r e an d M O S p a r am e t er s, w i t h w e ak in v er s i o n b e l o w 0.1, m o d er a t e in v er s i o n ce n t r e d a t 1 an d s t r o n g in v er s i o n ab o v e 10$ $(co m p u t e d : w i t h n = 1.3, g_{m} / I_{D} i s 29.5, 27.3, 18.4, 8.0 an d 2.8 V^{- 1} a t I C = 0.01, 0.1, 1, 10 an d 100, a g ain s t t h e w e ak - in v er s i o n l imi t 1/ (n V_{t}) = 29.8 V^{- 1})$

Short-Channel Effects and Parameters

Name	Formula	Related Formulae	Variables/Constants
Velocity saturation and saturation voltage	$V_{C} = U C R I T \cdot L_{e f f}, V_{D S S} = V_{C} [\frac{1}{4} + \frac{V _{t}}{V _{C}} i_{f} - \frac{1}{2}]$	$V_{D S S} \to V_{t} i_{f} (V_{C} \to \infty, half of V_{P} - V_{S} in the symmetric convention)$	$V_{C} : cr i t i c a l v o l t a g e f or v e l oc i t y s a t u r a t i o n$ $U C R I T : l o n g i t u d ina l cr i t i c a l f i e l d$ $L_{e f f} : e f f ec t i v e l e n g t h$ $V_{D S S} : s a t u r a t i o n v o l t a g e in t h e sy mm e t r i c co n v e n t i o n V_{d s} = (V_{D} - V_{S}) /2; E K V 2.6 r e p or t : V_{C} = U C R I T \cdot N S \cdot L_{e f f} w i t h N S t h e n u mb er o f ser i es d e v i ces, t h e V_{D S S} e x p r ess i o n (t h e s q u a r e r oo t o n i_{f} i s l os t in t h e e x t r a c t an d r es t or e d h er e so t ha t t h e l o n g - c hann e l l imi t i s cor r ec t, m y r e a d in g), an d t h e d e f a u l t U C R I T = 2.0 \times 1 0^{6} V/m$ $(co m p u t e d : a t i_{f} = 100, V_{C} i s 0.36 V f or 0.18 μ m an d 2 V f or 1 μ m, g i v in g V_{D S S} = 0.17 an d 0.23 V a g ain s t t h e l o n g - c hann e l 0.26 V, 33 an d 10 p er ce n t l o w er)$
Channel-length modulation and equivalent length	$Δ L = L A M B D A \cdot L_{C} ln (1 + \frac{V _{d s} - V _{i p}}{L _{C} \cdot U C R I T}), L_{C} = \frac{ε _{s i}}{C O X} X J$	$L^{'} = N S \cdot L_{e f f} - Δ L + \frac{V _{d s} + V _{i p}}{U C R I T}, L_{e q} = \frac{1}{2} [L^{'} + L^{' 2} + L_{min}^{2}], L_{min} = \frac{N S \cdot L _{e f f}}{10}$	$Δ L : c hann e l s h or t e nin g$ $L A M B D A : c hann e l - l e n g t h m o d u l a t i o n coe f f i c i e n t$ $L_{C} : c ha r a c t er i s t i c l e n g t h$ $V_{d s} : ha l f t h e d r ain - so u r ce v o l t a g e$ $V_{i p} : an in t er m e d ia t e p in c h - o f f - r e l a t e d v o l t a g e$ $ε_{s i} : s i l i co n p er mi tt i v i t y$ $C O X : o x i d e c a p a c i t an ce p er a r e a$ $X J : j u n c t i o n d e pt h, L^{'} : l e n g t h a f t er s h or t e nin g an d v e l oc i t y - s a t u r a t i o n cor r ec t i o n$ $L_{e q} : s m oo t h e d e q u i v a l e n t l e n g t h$ $L_{min} : l o w er l imi t; E K V 2.6 r e p or t : t h ese f or m s, w i t h d e f a u l t s L A M B D A = 0.5, X J = 0.1 μ m an d C O X = 0.7 \times 1 0^{- 3} F/ m^{2} (t h e e x t r a c t p r in t s L_{C} a s (ε_{s i} / C O X) X J, w hi c h i s n o t a l e n g t h; t h e s q u a r e r oo t i s r es t or e d, m y r e a d in g), pl u s s h or t - c hann e l c ha r g e s ha r in g t h r o ug h L E T A (0.1) an d W E T A (0.25) an d r e v er se s h or t - c hann e l e f f ec t t h r o ug h Q 0 an d L K (0.29 μ m)$ $(co m p u t e d : t h e d e f a u l t s g i v e L_{C} = 0.12 μ m)$
Mobility and default parameters	$β_{0} = K P \cdot N P \frac{W _{e f f}}{L _{e q}}, W_{e f f} = W + D W, L_{e f f} = L + D L$	$K P = μ_{0} C O X, C O X = \frac{ε _{o x}}{T O X}$	$β_{0} : t r an sco n d u c t an ce f a c t or b e f or e v er t i c a l - f i e l d m o bi l i t y r e d u c t i o n$ $N P : n u mb er o f p a r a l l e l d e v i ces$ $D W, D L : w i d t h an d l e n g t h cor r ec t i o n s$ $μ_{0} : l o w - f i e l d m o bi l i t y$ $ε_{o x} : o x i d e p er mi tt i v i t y$ $T O X : o x i d e t hi c k n ess (K P r e l a t i o n, ba c k g r o u n d k n o w l e d g e); E K V 2.6 r e p or t : v er t i c a l - f i e l d m o bi l i t y r e d u c t i o n t h r o ug h E 0 (d e f a u l t 1.0 \times 1 0^{12} V/m) or t h e s im pl er T H E T A (d e f a u l t 0), d e f a u l t s V T O = 0.5 V, G A M M A = 1.0 V^{1/2}, P H I = 0.7 V, K P = 50 μ A/ V^{2}, im p a c t i o ni z a t i o n I B A, I B B = 3.0 \times 1 0^{8} V/m an d I B N = 1.0, t e m p er a t u r e T C V = 1.0 \times 1 0^{- 3} V / K, B E X = - 1.5 an d U C E X = 0.8, ma t c hin g A V T O, A K P an d A G A M M A, f l i c k er n o i se K F an d A F, ini t ia l i z a t i o n f r o m T O X, N S U B, V F B, U O an d V M A X w h e n e l ec t r i c a l p a r am e t er s a r e n o t g i v e n, an d ab o u t 40 p a r am e t er s in a l l$ $(co m p u t e d : C O X = 0.7 \times 1 0^{- 3} F/ m^{2} cor r es p o n d s t o 49 nm o f S i O 2, an d K P = 50 μ A/ V^{2} t o μ_{0} = 714 c m^{2} / (V s))$

Important Questions

Ques: Does the EKV model need fewer than 20 parameters or about 40?

Both counts appear in EPFL publications, for different scopes. Sallese et al. state that the charge-based model uses fewer than 20 parameters, and that small-signal low-frequency behaviour needs only six independent real quantities, two transconductances and four transcapacitances. The EKV 2.6 report lists about 40 parameters, grouped as process and geometry, basic intrinsic (VTO, GAMMA, PHI, KP, E0, UCRIT), optional process, channel-length modulation and charge sharing, reverse short-channel effect, impact ionization, temperature, matching, flicker noise and setup. The intrinsic DC core is therefore well under 20, and the full model with temperature, noise, matching and impact ionization reaches about 40 (my reading). Either count is small compared with the hundreds of parameters in BSIM4, which is part of EKV’s appeal for analog design (background knowledge). Reference: J.-M. Sallese, W. Grabinski, A. S. Porret, M. Bucher, C. Lallement, F. Krummenacher, C. Enz and P. Fazan, ‘Advancements in DC and RF MOSFET Modeling with the EPFL-EKV Charge Based Model’, MIXDES 2001, Zakopane (fetch-tool summary); M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary).

Ques: How does the inversion coefficient help in sizing transistors?

Because transconductance efficiency depends only on the inversion coefficient. From the EKV relation $g_{m s} V_{t} / I_{D} = 1/ (0.5 + 0.25 + i_{f})$ with an assumed slope factor of 1.3, $g_{m} / I_{D}$ is 29.5 $V^{- 1}$ at an inversion coefficient of 0.01, 18.4 at 1, 8.0 at 10 and 2.8 at 100, against the weak-inversion limit of 29.8 $V^{- 1}$ (computed). Sallese et al. note that this normalized ratio is independent of temperature and MOS parameters, and they place weak inversion below 0.1, moderate inversion around 1 and strong inversion above 10. A designer chooses the coefficient for the required efficiency, speed and matching, then sets the width from the specific current: with $K P = 50$ $μ A/ V^{2}$ , a slope factor of 1.3 and $W / L = 10$ , $I_{S}$ is 0.87 $μ A$ (computed). Reference: M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary); J.-M. Sallese, W. Grabinski, A. S. Porret, M. Bucher, C. Lallement, F. Krummenacher, C. Enz and P. Fazan, ‘Advancements in DC and RF MOSFET Modeling with the EPFL-EKV Charge Based Model’, MIXDES 2001, Zakopane (fetch-tool summary).

Ques: Why does EKV refer all voltages to the substrate?

Because the MOSFET is physically symmetric, and a model referred to the source treats the two diffusions differently. The EKV 2.6 report refers the gate, source and drain voltages to the local substrate and writes the current as $I_{S} (i_{f} - i_{r})$ , with $i_{f} = F ((V_{P} - V_{S}) / V_{t})$ and $i_{r} = F ((V_{P} - V_{D}) / V_{t})$ . Exchanging source and drain simply swaps the forward and reverse terms and reverses the sign of the current, so the model is symmetric and continuous at zero drain voltage, which matters for switches, mixers and transmission gates (background knowledge for the circuit examples). A p-channel device is handled as a pseudo-n-channel device with inverted signs, the only other difference being the mobility factor of 1/2 for NMOS and 1/3 for PMOS. The same symmetry goal drives PSP’s symmetric linearization. Reference: M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary); Surface-Potential Models (PSP and HiSIM) page of this vault.

Ques: How does the slope factor change with bias?

It falls as the channel inverts, because the depletion capacitance shrinks relative to the oxide capacitance. In the EKV 2.6 report, $n = 1 + G A M M A / (2 V_{P} + P H I + 4 V_{t})$ , so the default GAMMA of 1.0 $V^{1/2}$ and PHI of 0.7 $V$ give 1.56 at zero pinch-off voltage (computed), and higher $V_{P}$ lowers it. Sallese et al. report a slope factor that varies from about 1.6 in weak inversion to about 1.2 in strong inversion, consistent with this dependence. The simplified relation $V_{P} = (V_{G} - V_{T O}) / n_{V}$ , with only two parameters, captures the gate dependence for hand analysis. Because $n$ also sets the subthreshold swing, $n V_{t} ln 10$ , a slope factor of 1.56 gives about 93 $mV$ per decade at 300 $K$ (computed). Reference: M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary); J.-M. Sallese, W. Grabinski, A. S. Porret, M. Bucher, C. Lallement, F. Krummenacher, C. Enz and P. Fazan, ‘Advancements in DC and RF MOSFET Modeling with the EPFL-EKV Charge Based Model’, MIXDES 2001, Zakopane (fetch-tool summary).

Ques: How much does velocity saturation lower the saturation voltage in EKV?

EKV uses a critical voltage $V_{C} = U C R I T \cdot L_{e f f}$ , with a default critical field of $2.0 \times 1 0^{6}$ $V/m$ , and a saturation voltage $V_{D S S} = V_{C} [1/4 + (V_{t} / V_{C}) i_{f} - 1/2]$ in its symmetric convention, where $V_{d s}$ is half the drain-source voltage. In the extract the square root on $i_{f}$ is missing, but without it the long-channel limit would be $V_{t} i_{f}$ , 2.6 $V$ at $i_{f} = 100$ , far above the pinch-off value, so the square root is restored (computed; my reading). At $i_{f} = 100$ , $V_{C}$ is 0.36 $V$ for 0.18 $μ m$ and 2 $V$ for 1 $μ m$ , giving 0.17 and 0.23 $V$ against the long-channel 0.26 $V$ , 33 and 10 per cent lower (computed). Short devices therefore saturate earlier. Reference: M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary).

Ques: How far has the EKV model been verified at RF?

Sallese et al. report RF measurements on devices from 300 $μ m$ down to 0.5 $μ m$ length in a 0.35 $μ m$ CMOS process: low-frequency checks on a 300 $μ m$ device from 75 $kHz$ to 30 $MHz$ , and high-frequency checks on a 10 $μ m$ NMOS and a 0.5 $μ m$ PMOS up to about 10 $GHz$ , with good agreement from moderate to strong inversion. They note that small-signal low-frequency behaviour needs only six independent real parameters, two transconductances and four transcapacitances, and they model polysilicon depletion by a factor $Γ_{p o l y}$ , comparing 2 with 50 for no depletion. The EKV 2.6 report builds the quasi-static, NQS and noise models on the same transconductance-to-current ratio. These checks are on older, thick-oxide technology, so modern RF use needs newer versions such as EKV3 (background knowledge). Reference: J.-M. Sallese, W. Grabinski, A. S. Porret, M. Bucher, C. Lallement, F. Krummenacher, C. Enz and P. Fazan, ‘Advancements in DC and RF MOSFET Modeling with the EPFL-EKV Charge Based Model’, MIXDES 2001, Zakopane (fetch-tool summary); M. Bucher, C. Lallement, C. Enz, F. Theodoloz and F. Krummenacher, ‘The EPFL-EKV MOSFET Model Equations for Simulation’, Technical Report, Model Version 2.6, EPFL (June 1997, revised September 1997 and July 1998) (equations partly garbled in the extract) (fetch-tool summary).

Quartz 5

Explorer

Charge-Based Models (EKV)