ACM2 2.0.0: model equations and characteristics

Advanced Compact MOSFET model 2 (ACM2), version 2.0.0: source models/acm2.va, module acm2_va. This page gives every equation of the released Verilog-A code, in the order in which the code evaluates them, with its parameters (Sections 1 to 16), and plots of the characteristics of an IHP SG13G2 NMOS device (Section 17). Parameter extraction from your own I-V data: Google Colab notebook.

1. Model overview

ACM2 is a charge-based model: the drain current and the terminal charges are functions of the normalized inversion charge densities at the source and drain ends of the channel, $q_S$ and $q_D$, obtained from the unified charge-control model (UCCM). Currents, charges and their derivatives are single-piece expressions, valid from weak to strong inversion, and the model is symmetric: source and drain are interchangeable. Five DC parameters set the drain current: $V_{T0}$ (vt0), the sheet specific current $I_{SH}$ (ish), the slope factor $n$ (n), the DIBL coefficient $\sigma$ (sigma) and the velocity-saturation coefficient $\zeta$ (zeta).

NMOS frame. One module serves both device types. The code multiplies the terminal voltages by type (+1 NMOS, −1 PMOS), evaluates the NMOS equations, and multiplies the branch currents by type again:

$$V_{GB}=\mathrm{type}\cdot V(g_i,b),\quad V_{DB}=\mathrm{type}\cdot V(d,b),\quad V_{SB}=\mathrm{type}\cdot V(s,b),\quad V_{DS}=\mathrm{type}\cdot V(d,s),$$

$$V_{GD}=V_{GB}-V_{DB},\qquad V_{GS}=V_{GB}-V_{SB}.$$

Equivalent circuit. Terminals $d$, $g$, $s$, $b$; $g_i$ is the internal gate node, present when rg > 0. The branch contributions of the code are:

The intrinsic charges satisfy $Q_G+Q_B+Q_D+Q_S=0$: the model conserves charge.

2. Parameters

Instance parameters (device line)

Model parameters (.model line). Last column: value in the IHP SG13G2 NMOS card of the plots (Section 17).

Alias: trise = dtemp.

The card is acm2n_L0p13 (cards/ihp_sg13g2/acm2_nmos_perL.lib, extracted at W = 10 µm, L = 0.13 µm). Parameters that a card does not set keep the defaults above (temperature, flicker-noise and mismatch parameters are not extracted).

3. Constants, geometry and thermal voltage

$$\varepsilon_{ox}=\mathrm{epsrox}\cdot\varepsilon_0,\qquad C_{ox}=\frac{\varepsilon_{ox}}{t_{ox}},\qquad W_e=m\,w,\qquad l_c=l-\mathrm{dlc}\;\;(\mathrm{dlc}<l),$$

$$T=T_{sim}+\mathrm{dtemp},\qquad \phi_t=\frac{kT}{q}.$$

$W_e$ is the total width of the $m$ parallel devices. The charge length $l_c$ replaces $l$ in the charges, capacitances, mobility and noise; the drawn length $l$ sets the specific current and the mismatch. $T_{sim}$ is the simulator temperature ($temperature). The drain current uses the smoothing constant $\epsilon=0.1$ (Section 8).

4. Specific current, temperature and mismatch

The specific current of one device scales with its drawn geometry through the sheet specific current $I_{SH}$ (ish): one card covers every width at its length.

$$I_S=I_{SH}\,\frac{w}{l}.$$

With $\Delta T=T-T_{ref}$ and the area $A=W_e\,l$ of the $m$ devices:

$$V_{T0}(T)=V_{T0}+\alpha_{VT0}\,\Delta T+\frac{A_{VT0}}{\sqrt{A}},\qquad I_S(T)=I_S\left(1+\frac{2-\alpha_{IS}}{T}\,\Delta T\right)+\frac{I_S\,A_K}{\sqrt{A}},$$

$$\sigma(T)=\sigma+\alpha_\sigma\,\Delta T,\qquad \zeta(T)=\max\left(\zeta+\alpha_\zeta\,\Delta T,\;0\right).$$

$\alpha_{VT0}$, $\alpha_{IS}$, $\alpha_\sigma$, $\alpha_\zeta$ are alphavt0, alphais, alphasigma, alphazeta. The instance parameters $A_{VT0}$ (avt0) and $A_K$ (ak) shift $V_{T0}$ and $I_S$ by amounts that scale with $1/\sqrt{\mathrm{area}}$ (Section 16). $\zeta(T)=0$ removes velocity saturation. From here on, $\sigma$ and $\zeta$ stand for $\sigma(T)$ and $\zeta(T)$.

5. Mobility and saturation velocity

The mobility follows from the specific current, $I_S=\mu\,n\,C_{ox}\,\phi_t^2\,w/(2\,l_c)$, written with the width $w$ of one device, so that it does not depend on $m$:

$$\mu=\frac{I_S(T)\;l_c}{C_{ox}\,n\,\phi_t^2\;w/2}.$$

The velocity-saturation coefficient is $\zeta=\mu\,\phi_t/(l_c\,v_{sat})$; the charges (Section 10) use the transit-time term

$$\frac{l_c}{v_{sat}}=\frac{l_c^2\,\zeta(T)}{\mu\,\phi_t},$$

which is zero when $\zeta(T)=0$.

6. Terminal voltages and pinch-off voltage

The type-scaled voltages of Section 1 enter the pinch-off voltage, where $\sigma$ adds the drain-induced barrier lowering (DIBL) through the drain and source potentials:

$$V_P=\frac{V_{GB}-V_{T0}(T)+\sigma\,(V_{DB}+V_{SB})}{n}.$$

With $V_S=V_B=0$, $V_P=(V_{GB}-V_{T0}+\sigma V_{DS})/n$: a higher drain voltage raises $V_P$ as a lower threshold voltage would.

7. Normalized charge densities

The normalized inversion charge density is $q=Q_I'/(-n\,C_{ox}\,\phi_t)$, where $Q_I'$ is the inversion charge per unit area. At the source end, the UCCM

$$\frac{V_P-V_{SB}}{\phi_t}=q_S-1+\ln q_S$$

is solved with the Lambert function $W$, the solution of $w\,e^{w}=z$:

$$q_S=W\!\left[\exp\left(\frac{V_P-V_{SB}}{\phi_t}+1\right)\right].$$

Algorithm 443 [5] evaluates $W(z)$ with a rational first estimate $w_0$ and one correction:

$$w_0=\begin{cases}\dfrac{z+\frac{4}{3}z^2}{1+\frac{7}{3}z+\frac{5}{6}z^2}, & z<0.7385,\\[2ex] \ln z-24\,\dfrac{\ln^2 z+2\ln z-3}{7\ln^2 z+58\ln z+127}, & z\ge 0.7385,\end{cases}$$

$$z_n=\ln z-w_0-\ln w_0,\qquad W(z)\simeq w_0\left[1+\frac{z_n}{1+w_0}\cdot\frac{2(1+w_0)\left(1+w_0+\frac{2}{3}z_n\right)-z_n}{2(1+w_0)\left(1+w_0+\frac{2}{3}z_n\right)-2z_n}\right].$$

Saturation charge. Velocity saturation limits the drain-end charge to $q_{sat}=q_S+1+\frac{1}{\zeta}-\sqrt{\left(1+\frac{1}{\zeta}\right)^2+\frac{2q_S}{\zeta}}$. The code evaluates the equivalent form without cancellation, which also holds for $\zeta=0$ ($q_{sat}=0$):

$$q_{sat}=q_S-\frac{2\,q_S}{1+\zeta+\sqrt{(1+\zeta)^2+2\,\zeta\,q_S}}.$$

Drain charge. The UCCM written between source and drain, with both charges measured from $q_{sat}$,

$$\frac{V_{DS}}{\phi_t}=(q_S-q_D)+\ln\frac{q_S-q_{sat}}{q_D-q_{sat}},$$

gives

$$q_D=q_{sat}+W\!\left[(q_S-q_{sat})\exp\left(q_S-q_{sat}-\frac{V_{DS}}{\phi_t}\right)\right].$$

At $V_{DS}=0$, $q_D=q_S$; in saturation ($V_{DS}\gg\phi_t$), $q_D\rightarrow q_{sat}$.

8. Drain current

$$I_D=m\,\frac{I_S(T)\,(q_S+q_D+2)\,(q_S-q_D)}{1+\sqrt{\zeta^2\,(q_S-q_D)^2+\epsilon^2}},\qquad \epsilon=0.1.$$

The denominator is the velocity-saturation term $1+\zeta\,|q_S-q_D|$ with $|x|$ replaced by $\sqrt{x^2+\epsilon^2}$, so that it stays differentiable at $V_{DS}=0$, where it equals $1+\epsilon$. The extraction notebook (colab/ACM2_Parameter_Extraction.ipynb) uses the same expression, so ish is extracted with this factor. The drain-to-source branch carries $\mathrm{type}\cdot I_D$; the drain terminal current also contains $g_{min}V(d,b)$ (Section 14).

9. Transconductances (operating-point outputs)

The simulator differentiates $I_D$ exactly. For the operating-point outputs gm_op, gms_op and gmd_op (of the $m$ devices), the code evaluates analytical expressions, with the normalized virtual-charge shift (also used by the charges, Section 10)

$$q_{idsat}=\frac{I_D}{W_e\,v_{sat}\,n\,C_{ox}\,\phi_t}=\frac{\zeta\,I_D}{2\,m\,I_S(T)},$$

$$\begin{aligned}g_{ms}=\frac{2\,m\,I_S(T)}{\phi_t\left[1+\zeta(q_S-q_D)\right]^2}\Bigg\{&\left[1+q_S+\frac{\zeta}{2}(q_S-q_D)^2\right]\left(1-\frac{\sigma}{n}\right)\frac{q_S-q_{idsat}}{1+q_S-q_{idsat}}\\&+\left[1+q_D-\frac{\zeta}{2}(q_S-q_D)^2\right]\frac{\sigma}{n}\,\frac{q_D-q_{idsat}}{1+q_D-q_{idsat}}\Bigg\},\end{aligned}$$

$$\begin{aligned}g_{md}=\frac{2\,m\,I_S(T)}{\phi_t\left[1+\zeta(q_S-q_D)\right]^2}\Bigg\{&\left[1+q_S+\frac{\zeta}{2}(q_S-q_D)^2\right]\frac{\sigma}{n}\,\frac{q_S-q_{idsat}}{1+q_S-q_{idsat}}\\&+\left[1+q_D-\frac{\zeta}{2}(q_S-q_D)^2\right]\left(1-\frac{\sigma}{n}\right)\frac{q_D-q_{idsat}}{1+q_D-q_{idsat}}\Bigg\},\end{aligned}$$

$$\begin{aligned}g_m=\frac{2\,m\,I_S(T)}{n\,\phi_t\left[1+\zeta(q_S-q_D)\right]^2}\Bigg\{&\left[1+q_S+\frac{\zeta}{2}(q_S-q_D)^2\right]\frac{q_S-q_{idsat}}{1+q_S-q_{idsat}}\\&-\left[1+q_D-\frac{\zeta}{2}(q_S-q_D)^2\right]\frac{q_D-q_{idsat}}{1+q_D-q_{idsat}}\Bigg\}.\end{aligned}$$

These outputs are analytical estimates; the small-signal plots of Section 17 are numerical derivatives of $I_D$.

10. Charges

With the linearity factor

$$\alpha=\frac{1+q_D-q_{idsat}}{1+q_S-q_{idsat}},$$

the total inversion, bulk and gate charges of the $m$ devices are

$$Q_I=-n\,\phi_t\,C_{ox}W_el_c\left[\frac{2}{3}\cdot\frac{1+\alpha+\alpha^2}{1+\alpha}\,(q_S+1-q_{idsat})-1\right]-\frac{l_c}{v_{sat}}\,I_D,$$

$$Q_B=-\frac{n-1}{n}\left[Q_I+C_{ox}W_el_c\left(V_{GB}-V_{T0}(T)\right)\right],\qquad Q_G=-Q_B-Q_I.$$

The second term of $Q_B$ is the bulk reference charge for a constant slope factor; $V_{T0}(T)$ fixes the charge origin at a given temperature. The charges are therefore referred to their values at $V_{GB}=V_{T0}(T)$ (hence $Q_G<0$ and $Q_B>0$ below threshold); only their time derivatives enter the simulation (Section 1). The inversion charge is split between drain and source with the Ward–Dutton partition [7]:

$$Q_D=-n\,\phi_t\,C_{ox}W_el_c\left[\frac{2}{15}\,(q_S+1-q_{idsat})\,\frac{2+4\alpha+6\alpha^2+3\alpha^3}{(1+\alpha)^2}-\frac{1}{2}\right]-\frac{1}{2}\cdot\frac{l_c}{v_{sat}}\,I_D,\qquad Q_S=Q_I-Q_D.$$

The charge densities at the channel ends, used by the flicker noise, are $Q_{IS}'=-n\,C_{ox}\,\phi_t\,q_S$ and $Q_{ID}'=-n\,C_{ox}\,\phi_t\,q_D$.

11. Capacitances

Exact capacitances. The simulator stamps the time derivatives of the charges of Sections 10 and 12, so the small-signal capacitances are the exact derivatives $C_{xy}=-\partial Q_x/\partial V_y$ ($x\neq y$). $C_{dg}\neq C_{gd}$: the capacitances of a charge-based model are nonreciprocal.

Operating-point estimates. For the outputs Cgs_op … Cm_op the code evaluates analytical expressions:

$$C_{gso}=\frac{2}{3}W_el_cC_{ox}\,\frac{(1+2\alpha)(q_S-q_{idsat})}{(1+\alpha)^2(1+q_S-q_{idsat})},\qquad C_{gdo}=\frac{2}{3}W_el_cC_{ox}\,\frac{(\alpha^2+2\alpha)(q_D-q_{idsat})}{(1+\alpha)^2(1+q_D-q_{idsat})},$$

$$C_{dso}=-\frac{4}{15}\,n\,W_el_cC_{ox}\,\frac{(1+3\alpha+\alpha^2)(q_S-q_{idsat})}{(1+\alpha)^3(1+q_S-q_{idsat})},\qquad C_{sdo}=-\frac{4}{15}\,n\,W_el_cC_{ox}\,\frac{(\alpha+3\alpha^2+\alpha^3)(q_D-q_{idsat})}{(1+\alpha)^3(1+q_D-q_{idsat})},$$

$$C_{gs}=C_{gso}\left(1-\frac{\sigma}{n}\right)-C_{gdo}\frac{\sigma}{n}+\frac{l_c}{v_{sat}}\,\frac{g_{ms}}{3n}\,\frac{(1-\alpha)^2}{(1+\alpha)^2},$$

$$C_{gd}=C_{gdo}\left(1-\frac{\sigma}{n}\right)-C_{gso}\frac{\sigma}{n}-\frac{l_c}{v_{sat}}\,\frac{g_{md}}{3n}\,\frac{(1-\alpha)^2}{(1+\alpha)^2},$$

$$C_{gb}=\frac{n-1}{n}\left(W_el_cC_{ox}-C_{gs}-C_{gd}\right),\qquad C_{bs}=(n-1)\,C_{gs},\qquad C_{bd}=(n-1)\,C_{gd},$$

$$C_{ds}=C_{dso}\left(1-\frac{\sigma}{n}\right)-\frac{l_c}{v_{sat}}\,\frac{g_{ms}}{30}\,\frac{(3\alpha+7)(1-\alpha)^2}{(1+\alpha)^3},$$

$$C_{sd}=C_{sdo}\left(1-\frac{\sigma}{n}\right)+\frac{l_c}{v_{sat}}\,\frac{g_{md}}{30}\,\frac{(3+7\alpha)(1-\alpha)^2}{(1+\alpha)^3},\qquad C_m=\frac{C_{sd}-C_{ds}}{n}.$$

These are analytical estimates; the capacitances of Section 17 are the exact derivatives of the charges, obtained from transient currents.

12. Overlap and junctions

Overlap. $C_{gs,ov}=C_{gd,ov}=C_{ox}\,W_e\,l_d$ ($l_d$ = ld), with the charges $q_{ovs}=C_{gs,ov}V_{GS}$ and $q_{ovd}=C_{gd,ov}V_{GD}$.

Junctions. Each drain and source has a bottom junction (area ad, as) and a side-wall junction (perimeter pd, ps, depth xj), per device; the $m$ devices add their junctions. With $V=V_{DB}$ (drain) or $V_{SB}$ (source), reverse biased for $V>0$:

$$C_j(V)=\begin{cases}C_0\left(1+\dfrac{V}{\phi_0}\right)^{-m_j}, & V>0,\\[1.5ex] C_0\left(1-m_j\,\dfrac{V}{\phi_0}\right), & V\le 0,\end{cases}$$

with $C_0=\mathrm{cj0}\cdot\mathrm{ad}$, $\phi_0=$ phi_zero, $m_j=$ xd_mj for the bottom junction and $C_0=\mathrm{cj0sw}\cdot\mathrm{xj}\cdot\mathrm{pd}$, $\phi_0=$ phi_zerosw, $m_j=$ xd_mjsw for the side wall (as, ps for the source). The charges are the integrals of $C_j$ from 0 to $V$, so that $dQ/dV=C_j(V)$ on both sides of $V=0$:

$$q_{dep}(V)=\begin{cases}\dfrac{C_0\,\phi_0}{1-m_j}\left[\left(1+\dfrac{V}{\phi_0}\right)^{1-m_j}-1\right], & V>0\quad\left(C_0\,\phi_0\ln\left(1+\dfrac{V}{\phi_0}\right)\ \text{if}\ |1-m_j|<10^{-6}\right),\\[2ex] C_0\left(V-\dfrac{m_j\,V^2}{2\,\phi_0}\right), & V\le 0,\end{cases}$$

$$q_{jd}=m\left[q_{dep}^{bottom}(V_{DB})+q_{dep}^{side}(V_{DB})\right],\qquad q_{js}=m\left[q_{dep}^{bottom}(V_{SB})+q_{dep}^{side}(V_{SB})\right].$$

Cjdb_op and Cjsb_op are $m\,(C_j^{bottom}+C_j^{side})$: exact derivatives of these charges.

Terminal charges. With overlap and junctions, the charges at the four terminals are $Q_g=Q_G+q_{ovd}+q_{ovs}$, $Q_d=Q_D-q_{ovd}+q_{jd}$, $Q_s=Q_S-q_{ovs}+q_{js}$ and $Q_b=Q_B-q_{jd}-q_{js}$ in the NMOS frame (times type for the physical charges).

13. Noise

Drain-current noise from drain to source, power spectral densities in A²/Hz:

$$S_{th}=-\frac{4kT\,\mu\,Q_I}{l_c^2},\qquad S_{fl}(f)=\frac{q^2\,N_{ot}\,\mu}{l_c^2\,n\,C_{ox}}\,\ln\left(\frac{nC_{ox}\phi_t-Q_{IS}'}{nC_{ox}\phi_t-Q_{ID}'}\right)\frac{I_D}{f}=\frac{q^2\,N_{ot}\,\mu}{l_c^2\,n\,C_{ox}}\,\ln\left(\frac{1+q_S}{1+q_D}\right)\frac{I_D}{f}.$$

$Q_I<0$ is the total inversion charge of Section 10, so $S_{th}\ge 0$. The logarithm and $I_D$ have the sign of $q_S-q_D$, so $S_{fl}\ge 0$ in both directions and zero at $V_{DS}=0$. $N_{ot}$ (n_ot) is the equivalent oxide-trap density of the flicker-noise model [6]; the IHP cards keep its default. The gate resistance adds the thermal noise $4kT\,m/r_g$ of the gate branch (Section 14). Here $k$ and $q$ are P_K and P_Q.

14. Gate resistance and $g_{min}$

rg is the gate resistance of one device; the $m$ parallel gates give $r_g/m$: $I(g,g_i)=V(g,g_i)\,m/r_g$, with the thermal noise $4kT\,m/r_g$. rg = 0 connects $g$ to $g_i$ directly (no internal node). The IHP cards set rg = 0: give the gate resistance of each device on its instance line.

The simulator conductance $g_{min}$ ($simparam("gmin"), ngspice option gmin) is connected from drain and source to bulk: $g_{min}V(d,b)$ and $g_{min}V(s,b)$, with the physical voltages.

15. Temperature

Section 4 gives the temperature laws. The IHP cards do not extract the coefficients alphavt0, alphais, alphasigma and alphazeta: they keep the defaults of Section 2.

16. Mismatch

avt0 (V·m) and ak (m) shift $V_{T0}$ and $I_S$ of one instance by $A_{VT0}/\sqrt{W_el}$ and $I_S\,A_K/\sqrt{W_el}$ (Section 4); for a mismatch analysis, assign them per instance (avt0 may be negative, ak ≥ 0).

17. Characteristics

The figures are computed with ngspice 47 from models/acm2.va compiled by OpenVAF-Reloaded, with the IHP SG13G2 NMOS card acm2n_L0p13 (cards/ihp_sg13g2/acm2_nmos_perL.lib): W = 100 µm, L = 0.13 µm, ad = as = 20 µm², pd = ps = 100 µm, 27 °C, and $V_S=V_B=0$ unless a terminal is swept; Fig. 2 also uses the PMOS card of the same length. The derivatives are numerical: seven-point centered differences with a 1-mV step.

DC

Small signal

Capacitances

RF

Noise

References

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[2] A. I. A. Cunha, M. C. Schneider, and C. Galup-Montoro, "An MOS transistor model for analog circuit design," IEEE J. Solid-State Circuits, vol. 33, no. 10, pp. 1510–1519, Oct. 1998.

[3] O. da Costa Gouveia-Filho, A. I. A. Cunha, M. C. Schneider, and C. Galup-Montoro, "Advanced compact model for short-channel MOS transistors," in Proc. IEEE Custom Integrated Circuits Conf., 2000.

[4] M. C. Schneider and C. Galup-Montoro, CMOS Analog Design Using All-Region MOSFET Modeling. Cambridge, U.K.: Cambridge Univ. Press, 2010, doi: 10.1017/CBO9780511803840.

[5] F. N. Fritsch, R. E. Shafer, and W. P. Crowley, "Algorithm 443: Solution of the transcendental equation $we^w=x$," Commun. ACM, vol. 16, no. 2, pp. 123–124, Feb. 1973, doi: 10.1145/361952.361970.

[6] A. Arnaud and C. Galup-Montoro, "A compact model for flicker noise in MOS transistors for analog circuit design," IEEE Trans. Electron Devices, vol. 50, no. 8, pp. 1815–1818, Aug. 2003, doi: 10.1109/TED.2003.815143.

[7] S.-Y. Oh, D. E. Ward, and R. W. Dutton, "Transient analysis of MOS transistors," IEEE Trans. Electron Devices, vol. 27, no. 8, pp. 1571–1578, Aug. 1980, doi: 10.1109/T-ED.1980.20072.