1. Theory

Definition: Model quality tests check that a compact model behaves correctly beyond fitting measured curves: symmetric between source and drain, smooth to high derivatives, charge conserving, physically consistent in capacitance and noise, and fast and robust in simulation (McAndrew; Scholten et al.; Chauhan et al.).
Why fits are not enough: Analog and RF circuits depend on derivatives of current and charge, so a model can match I-V data within 1 per cent and still predict wrong distortion, noise or charge (Scholten et al.; Chauhan et al.).
Benchmark suite: The Gummel symmetry test, AC symmetry test, harmonic slope-ratio test, capacitance reciprocity and sum rules, thermal-noise limits and charge-conservation circuits form the standard set used by the Compact Model Coalition and model developers (McAndrew; Scholten et al.; LCI; background knowledge for the CMC).
Criteria: Good models are analytical, symmetric, physically justified, scalable and easy to extract (Lu et al.).

2. Working Principle

Symmetry at zero drain bias: Driving the drain and source with $\pm V_{X}$ exposes any asymmetry or kink at $V_{D S} = 0$ , where many models switch between forward and reverse forms (Chauhan et al.; LCI; background knowledge for the switching).
Derivatives as a microscope: A small discontinuity in a high derivative is invisible in I-V curves but produces wrong harmonic slopes (Scholten et al.; computed).
Physical limits: Weak-inversion transconductance efficiency, Nyquist noise at $V_{D S} = 0$ and capacitance sum rules provide model-independent references (background knowledge; Scholten et al.).
Circuit tests: Simple circuits such as a binary current divider and a sample-and-hold expose matching and charge errors directly (LCI).

3. Applications

Model selection: Standardization decisions and comparisons between BSIM, PSP and EKV-type models (Scholten et al.; Chauhan et al.).
Model release checks: Regression tests of new model versions and model cards (McAndrew; background knowledge).
RF and mixed-signal design: Confidence in mixer distortion, switched-capacitor charge and noise predictions (McAndrew; Scholten et al.; LCI).
New devices: Applying the same tests to graphene and other emerging-device models (Lu et al.).

Symmetry Tests

Name	Formula	Related Formulae	Variables/Constants
Gummel symmetry test	$V_{D} = V_{X}, V_{S} = - V_{X} : I_{X} (- V_{X}) = - I_{X} (V_{X})$	$\frac{d ^{n} I _{X}}{d V _{X}^{n}} continuous at V_{X} = 0, n = 1, 2, \dots; odd- n derivatives even, even- n derivatives odd in V_{X}$	$V_{X} : sy mm e t r i c s w ee p v o l t a g e a ppl i e d a s + V_{X} t o t h e d r ain an d - V_{X} t o t h e so u r ce$ $I_{X} : d r ain c u r r e n t$ $n : d er i v a t i v e or d er; C ha u han e t a l . : B S I M 6 c h ec k e d f or I_{X} an d d er i v a t i v es t o f i f t h or d er in t h e t e x t an d se v e n t h in t h e f i g u r e, a l l co n t in u o u s a t V_{D S} = 0, n or ma l i z e d t o t h e i r ma x ima, in w e ak an d s t r o n g in v er s i o n; L C I (A C M) : t h e t es t s h o w s sy mm e t r y o f f or w a r d an d r e v er se o p er a t i o n an d co n t in u i t y o f c u r r e n t, c ha r g es an d d er i v a t i v es a r o u n d t h e or i g in; Lu e t a l . : a b e n c hma r k e v er y co m p a c t m o d e l m u s t p a ss, in c l u d in g g r a p h e n e F E T s w i t h a s m oo t hin g p a r am e t er a t V_{X} = 0; p a r i t y s t a t e m e n t, ba c k g r o u n d k n o w l e d g e$ $(n o co m p u t e d v a l u e)$
Modified DC and AC symmetry tests	$C_{GS} (V_{D S} = 0) = C_{G D} (V_{D S} = 0)$	$\frac{d ^{n} C}{d V _{X}^{n}} continuous at V_{D S} = 0$	$C_{GS}, C_{G D} : g a t e - so u r ce an d g a t e - d r ain c a p a c i t an ces; M c A n d r e w : t h e G u mm e l sy mm e t r y t es t a ppl i es o n l y t o D C c u r r e n t w i t h n e g l i g ib l e g a t e an d s u b s t r a t e c u r r e n t, so a m o d i f i e d D C t es t i s p r o p ose d f or t h ose c u r r e n t s an d a n e w A C t es t v er i f i es sy mm e t r y o f g a t e - t o - so u r ce an d g a t e - t o - d r ain c a p a c i t an ce, e x t e n s ib l e t o o t h er c a p a c i t an ce coe f f i c i e n t s, w i t h sy mm e t r y a f f ec t in g d i s t or t i o n in R F C M O S mi x er s; C ha u han e t a l . : c a p a c i t an ces an d d er i v a t i v es co n t in u o u s a t V_{D S} = 0 in B S I M 6 b y M c A n d r e w^{'} s A C t es t$ $(n o co m p u t e d v a l u e)$

Distortion and Derivative Tests

Name	Formula	Related Formulae	Variables/Constants
Harmonic slope-ratio test	$H_{k} \propto A^{k} \Rightarrow \frac{d ( H _{k} in dB )}{d ( A in dB )} = k$	$I = V + c V ∣ V ∣ \Rightarrow H_{3} \propto A^{2} (slope 2)$	$H_{k} : am pl i t u d e o f t h e k t h ha r m o ni c o f t h e d r ain c u r r e n t$ $A : s ma l l s i g na l am pl i t u d e a t V_{D S} = 0$ $c : coe f f i c i e n t o f a n o n - s m oo t h t er m (i l l u s t r a t i o n, ba c k g r o u n d k n o w l e d g e); S c h o l t e n e t a l . : a p a ss i v e - mi x er t es t s w ee p in g V_{R F} f r o m - 16 t o - 6 dB, P S P s h o w in g s l o p es o f 1, 2 an d 3 f or t h e f u n d am e n t a l, seco n d an d t hi r d ha r m o ni cs, an d B S I M 4 a t hi r d - ha r m o ni c s l o p e o f 2 (s h o u l d b e 3)$ $(co m p u t e d : f or I = V + 0.5 V ∣ V ∣ t h e t hi r d ha r m o ni c i s 8.5 \times 1 0^{- 6}, 3.4 \times 1 0^{- 5} an d 1.36 \times 1 0^{- 4} a t A = 0.01, 0.02 an d 0.04, a s l o p e o f e x a c tl y 2; f or I = V + 0.5 V^{3} i t i s 1.25 \times 1 0^{- 7}, 1.0 \times 1 0^{- 6} an d 8.0 \times 1 0^{- 6}, a s l o p e o f 3)$
Transconductance-efficiency and output-conductance checks	$\frac{g _{m}}{I _{D}} \leq \frac{1}{n U _{T}} (weak-inversion limit)$	$g_{D S} = \frac{\partial I _{D}}{\partial V _{D S}} smooth and positive, with smooth higher derivatives$	$g_{m} / I_{D} : t r an sco n d u c t an ce e f f i c i e n cy$ $n : s l o p e f a c t or$ $U_{T} : t h er ma l v o l t a g e$ $g_{D S} : o u tp u t co n d u c t an ce (l imi t, ba c k g r o u n d k n o w l e d g e; in v er s i o n - coe f f i c i e n t f or m s o n t h e E K V p a g e); S c h o l t e n e t a l . : l o n g - c hann e l g_{D S} f r o m 0 t o 3 V o n a 1 \times 1 0^{- 9} t o 1 \times 1 0^{- 3} S sc a l e, B S I M 4 b e ha v i o u r a sy s t e ma t i c l o n g - s t an d in g p r o b l e m w or se nin g f or hi g h er d er i v a t i v es, an d b e tt er P S P s h or t - c hann e l g_{D S}$ $(co m p u t e d : n = 1.3 a t 300 K l imi t s g_{m} / I_{D} t o 29.8 V^{- 1})$

Capacitance, Noise and General Criteria

Name	Formula	Related Formulae	Variables/Constants
Capacitance symmetry, reciprocity and sum rules	$C_{ij} = - \frac{\partial Q _{i}}{\partial V _{j}} (i \neq = j), \sum_{i} C_{ij} = \sum_{j} C_{ij} = 0 (conservation, signed matrix)$	$C_{i D} = C_{i S}, C_{D j} = C_{S j}, C_{ij} = C_{j i} (V_{D S} = 0)$	$C_{ij} : t r an sc a p a c i t an ce b e tw ee n t er mina l s i an d j$ $Q_{i} : t er mina l c ha r g e$ $V_{j} : t er mina l v o l t a g e (d e f ini t i o n s an d s u m r u l e, ba c k g r o u n d k n o w l e d g e; d e t ai l o n t h e C ha r g e an d C a p a c i t an ce M o d e l l in g p a g e); S c h o l t e n e t a l . : a t V_{D S} = 0 t h e m o d e l m u s t s h o w sy mm e t r y an d r ec i p r oc i t y; B S I M 4 f ai l s b o t h, w i t h w r o n g s i g n s o f C_{S D} an d C_{D S} an d C_{D D} an d C_{S S} e x cee d in g C_{GG}, a l so a t V_{D S} = 1 V, o n pl o t s o f - 0.2 t o 0.6 pF$ $(n o co m p u t e d v a l u e)$
Thermal-noise and charge-conservation checks	$γ (V_{D S} = 0) = 1, γ \to \frac{2}{3} (long-channel saturation)$	$Δ Q_{C, h o l d} = 0 (sample-and-hold test), \frac{I _{k + 1}}{I _{k}} = \frac{1}{2} (M2M divider)$	$γ : c hann e l t h er ma l - n o i se coe f f i c i e n t (S_{i d} = 4 k_{B} T γ g_{d 0})$ $Δ Q_{C, h o l d} : s p u r i o u s c ha r g e o n a h e l d c a p a c i t or$ $I_{k} : b r an c h c u r r e n t s o f a bina r y M O S F E T d i v i d er; S c h o l t e n e t a l . : γ a g ain s t V_{D S} w i t h r e f er e n ce l e v e l s 2/3 an d 1/2, B S I M 4 v i o l a t in g N y q u i s t^{'} s l a w a t V_{D S} = 0 an d P S P cor r ec t; L C I (A C M) : a s am pl e - an d - h o l d c i r c u i t v er i f y in g c ha r g e co n ser v a t i o n, an d a M O S F E T - o n l y bina r y d i v i d er (M 2 M) w i t h er r or s b e l o w 0.12 p er ce n t; n o i se f or m s o n t h e M O S F E T N o i se M o d e l s p a g e$ $(n o co m p u t e d v a l u e)$
General quality criteria	$ε_{r m s} \approx 1 per cent, t_{e v a l} \approx 10 μ s per DC point$	$continuity, symmetry, physical basis, few extractable parameters, scalability$	$ε_{r m s} : R M S f i tt in g er r or$ $t_{e v a l} : s im u l a t i o n t im e p er bia s p o in t; C ha u han e t a l . : t a r g e t s o f ab o u t 1 p er ce n t R M S er r or a f t er f i tt in g an d ab o u t 10 μ s p er D C p o in t, B S I M - C M G c i t e d a t ab o u t 5000 l in es o f V er i l o g - A an d o v er 50 p a r am e t er s; Lu e t a l . : nin e cr i t er ia, am o n g t h e m co n s i s t e n t b e ha v i o u r, sy mm e t r y, ana l y t i c a l f or m w i t h o u t d i f f er e n t ia l s or in t e g r a l s, s im pl i c i t y, e a s i l y c ha r a c t er i z e d p a r am e t er s, u p g r a d abi l i t y, p h y s i c a l j u s t i f i c a t i o n, s imi l a r i t y t o o t h er F E T m o d e l s an d t u nabi l i t y t o u n cer t ain d a t a; S c h o l t e n e t a l . : r u n t im e r a t i os a g ain s t B S I M 4 a cr oss s im u l a t or s an d c i r c u i t s i z es$ $(n o co m p u t e d v a l u e)$

Important Questions

Ques: What does the Gummel symmetry test check?

Whether the model treats source and drain identically and smoothly. The drain is driven at $+ V_{X}$ and the source at $- V_{X}$ , and the current $I_{X}$ and its derivatives are plotted through $V_{X} = 0$ . For a symmetric device $I_{X}$ is an odd function, so its odd-order derivatives are even and its even-order derivatives odd, and all must be continuous (background knowledge for the parity). Chauhan et al. show BSIM6 passing for derivatives up to the fifth order in the text and seventh in the figure, in weak and strong inversion, normalized to their maxima; the ACM group reports smooth derivatives in ELDO; and Lu et al. call the test a benchmark every compact model must pass, including graphene FETs. Models that switch formulas at $V_{D S} = 0$ show kinks in the second or third derivative. Reference: Y. S. Chauhan, M. Chalkiadaki, S. Venugopalan, M. A. Karim, N. Paydavosi, S. Jandhyala, J. P. Duarte, C. Enz, A. Niknejad and C. Hu, ‘Global Geometrical Scaling in BSIM6’, MOS-AK San Francisco (12 December 2012) (slides; equations garbled in the extract) (fetch-tool summary); Integrated Circuits Laboratory, Federal University of Santa Catarina, ‘ACM model: simulation results’ web page (fetch-tool summary); N. Lu, L. Wang, L. Li and M. Liu, ‘A review for compact model of graphene field-effect transistors’, arXiv:1703.09759 (fetch-tool summary).

Ques: Is the classic Gummel symmetry test sufficient?

No, it has two gaps. McAndrew points out that it tests only DC current and is valid only when gate and substrate currents are negligible, which is not true for thin-oxide devices with gate tunnelling or with impact ionization. He proposes a modified DC test that tolerates those currents and a new AC symmetry test of gate-to-source and gate-to-drain capacitance, extensible to the other capacitance coefficients. Chauhan et al. apply that AC test to BSIM6 and find the capacitances and their derivatives continuous at $V_{D S} = 0$ . The ACM group’s statement of the expected symmetry, written in the extract as $I_{D} (V_{X}) = - I_{D} (V_{X})$ , is evidently a transcription of $I_{D} (- V_{X}) = - I_{D} (V_{X})$ , since the literal form would force zero current (my reading). Reference: C. C. McAndrew, ‘DC and AC Symmetry Tests for MOSFET Models’, Technical Proceedings of the 2006 NSTI Nanotechnology Conference, Vol. 3, 745-748 (2006) (abstract only) (fetch-tool summary); Y. S. Chauhan, M. Chalkiadaki, S. Venugopalan, M. A. Karim, N. Paydavosi, S. Jandhyala, J. P. Duarte, C. Enz, A. Niknejad and C. Hu, ‘Global Geometrical Scaling in BSIM6’, MOS-AK San Francisco (12 December 2012) (slides; equations garbled in the extract) (fetch-tool summary); Integrated Circuits Laboratory, Federal University of Santa Catarina, ‘ACM model: simulation results’ web page (fetch-tool summary).

Ques: What does the harmonic slope-ratio test reveal?

Hidden non-smoothness at zero drain bias. In a passive mixer the transistor is biased at $V_{D S} = 0$ and driven by a small signal; for a smooth model the kth harmonic grows as the kth power of the amplitude, so on a log-log plot the fundamental, second and third harmonics have slopes 1, 2 and 3. Scholten et al. show PSP with slopes 1, 2 and 3, but BSIM4 with a third-harmonic slope of 2. A term like $V ∣ V ∣$ explains this: for $I = V + 0.5 V ∣ V ∣$ the third harmonic rises from $8.5 \times 1 0^{- 6}$ to $1.36 \times 1 0^{- 4}$ as A goes from 0.01 to 0.04, a slope of exactly 2, while for $I = V + 0.5 V^{3}$ it rises from $1.25 \times 1 0^{- 7}$ to $8.0 \times 1 0^{- 6}$ , a slope of 3 (computed). At small signals the non-smooth model overpredicts distortion by orders of magnitude. Reference: A. Scholten, G.-J. Smit, D. B. M. Klaassen, R. van Langevelde, G. Gildenblat, W. Wu and X. Li, ‘The PSP compact MOSFET model: An update’, MOS-AK Eindhoven (4 April 2008) (slides) (fetch-tool summary).

Ques: What capacitance properties should a MOSFET model satisfy?

Symmetry, reciprocity at zero drain bias and charge-conserving sum rules. Scholten et al. require at $V_{D S} = 0$ that $C_{i D} = C_{i S}$ , $C_{D j} = C_{S j}$ and $C_{ij} = C_{j i}$ , and show BSIM4 failing both symmetry and reciprocity, with wrong signs of $C_{S D}$ and $C_{D S}$ and with $C_{D D}$ and $C_{S S}$ larger than $C_{GG}$ , at 0 and 1 $V$ . In a charge-based model the transcapacitances form a matrix whose rows and columns sum to zero, because terminal charges sum to zero and depend only on voltage differences (background knowledge). Away from $V_{D S} = 0$ the matrix is non-reciprocal in general, $C_{G D} \neq = C_{D G}$ , which is physical, so reciprocity is only demanded at zero drain bias. The partitioning rules are on the Charge and Capacitance Modelling page. Reference: A. Scholten, G.-J. Smit, D. B. M. Klaassen, R. van Langevelde, G. Gildenblat, W. Wu and X. Li, ‘The PSP compact MOSFET model: An update’, MOS-AK Eindhoven (4 April 2008) (slides) (fetch-tool summary); Charge and Capacitance Modelling page of this vault.

Ques: How is a noise model checked for physical consistency?

At zero drain bias, where thermodynamics fixes the answer. With no drain voltage the channel is a resistor in equilibrium, so its noise must be the Nyquist value $4 k_{B} T / R$ , which corresponds to $γ = 1$ in $S_{i d} = 4 k_{B} T γ g_{d 0}$ ; in long-channel saturation $γ$ falls to 2/3 (background knowledge; MOSFET Noise Models page). Scholten et al. plot $γ$ against drain voltage with reference levels of 2/3 and 1/2 and state that BSIM4 violates Nyquist’s law while PSP behaves correctly. A wrong value at $V_{D S} = 0$ shows up directly in switch and mixer noise, which operate there. Higher $γ$ in short channels is real, but the zero-bias limit is not negotiable. The same test applies to induced gate noise and its correlation in RF models. Reference: A. Scholten, G.-J. Smit, D. B. M. Klaassen, R. van Langevelde, G. Gildenblat, W. Wu and X. Li, ‘The PSP compact MOSFET model: An update’, MOS-AK Eindhoven (4 April 2008) (slides) (fetch-tool summary); MOSFET Noise Models page of this vault.

Ques: How is charge conservation tested in a circuit?

With circuits that integrate any charge error. The ACM group uses a sample-and-hold circuit: if the model’s charges are not conserved, the held capacitor gains or loses charge every cycle, and here the model charged the capacitor correctly on each pulse. They also use an MOSFET-only binary current divider (M2M), in which a reference current is halved stage by stage through identical long-channel transistors sharing a substrate, and report errors below 0.12 per cent. Charge-based models with Ward-Dutton partitioning conserve charge by construction, whereas capacitance-based models such as the original Meyer model do not (background knowledge; Charge and Capacitance Modelling page). The M2M test also checks the model’s consistency across inversion levels. Reference: Integrated Circuits Laboratory, Federal University of Santa Catarina, ‘ACM model: simulation results’ web page (fetch-tool summary); Charge and Capacitance Modelling page of this vault.

Ques: What makes a good compact model beyond accuracy?

Lu et al. list nine criteria: consistent behaviour, symmetry reflecting the device, an analytical form without differentials or integrals, simplicity, parameters that are easy to characterize or even guess, upgradability, physical justification, similarity to other FET models and tunability to uncertain data. Chauhan et al. add practical targets for BSIM6: about 1 per cent RMS error after fitting and about 10 $μ s$ per DC point, contrasted with BSIM-CMG at about 5000 lines of Verilog-A and over 50 parameters. Scholten et al. compare runtimes with BSIM4 across simulators and circuit sizes. Speed matters because large circuits evaluate the model billions of times, and robustness matters because discontinuities cause convergence failures (background knowledge). Reference: N. Lu, L. Wang, L. Li and M. Liu, ‘A review for compact model of graphene field-effect transistors’, arXiv:1703.09759 (fetch-tool summary); Y. S. Chauhan, M. Chalkiadaki, S. Venugopalan, M. A. Karim, N. Paydavosi, S. Jandhyala, J. P. Duarte, C. Enz, A. Niknejad and C. Hu, ‘Global Geometrical Scaling in BSIM6’, MOS-AK San Francisco (12 December 2012) (slides; equations garbled in the extract) (fetch-tool summary); A. Scholten, G.-J. Smit, D. B. M. Klaassen, R. van Langevelde, G. Gildenblat, W. Wu and X. Li, ‘The PSP compact MOSFET model: An update’, MOS-AK Eindhoven (4 April 2008) (slides) (fetch-tool summary).

References

Y. S. Chauhan, M. Chalkiadaki, S. Venugopalan, M. A. Karim, N. Paydavosi, S. Jandhyala, J. P. Duarte, C. Enz, A. Niknejad and C. Hu, “Global Geometrical Scaling in BSIM6,” MOS-AK Workshop, San Francisco, Dec. 2012. Available: https://www.mos-ak.org/sanfrancisco_2012/presentations/T07_Chauhan_MOS-AK_121212.pdf
Integrated Circuits Laboratory, Federal University of Santa Catarina, “ACM model: simulation results.” Available: https://lci.ufsc.br/acm/simulation.html
N. Lu, L. Wang, L. Li and M. Liu, “A review for compact model of graphene field-effect transistors,” arXiv:1703.09759. Available: https://arxiv.org/abs/1703.09759
C. C. McAndrew, “DC and AC Symmetry Tests for MOSFET Models,” in Technical Proceedings of the 2006 NSTI Nanotechnology Conference and Trade Show, vol. 3, pp. 745-748, 2006. Available: https://briefs.techconnect.org/papers/dc-and-ac-symmetry-tests-for-mosfet-models/
A. Scholten, G.-J. Smit, D. B. M. Klaassen, R. van Langevelde, G. Gildenblat, W. Wu and X. Li, “The PSP compact MOSFET model: An update,” MOS-AK Workshop, Eindhoven, Apr. 2008. Available: https://www.mos-ak.org/eindhoven/papers/02_PSP_MOS-AK_Eindhoven.pdf

Quartz 5

Explorer

Model Quality and Benchmark Tests