1. Theory

Definition: Surface-potential compact models compute the silicon surface potential at the source and drain ends of the channel from the Poisson equation and express charges, current and capacitances in terms of these two potentials, instead of a threshold voltage; PSP and HiSIM are the main examples (Gildenblat et al.; Miura-Mattausch et al.).
Origins: PSP merged Penn State’s SP model with Philips MOS Model 11 and was elected the industry-standard compact MOSFET model by the Compact Model Council in December 2005; HiSIM, the Hiroshima-university STARC IGFET Model, comes from Hiroshima University and STARC (EEPower; PSP 102.5 note; Miura-Mattausch lecture).
Single expression for all regions: Because the surface potential follows one equation from accumulation through depletion and weak inversion to strong inversion, there is no threshold boundary and no smoothing function, and the model is continuous in all derivatives by construction (EEPower; Gildenblat and Chen).
Symmetry: PSP’s symmetric linearization of the inversion charge around the midpoint potential makes the model symmetric in source and drain, so it passes the Gummel symmetry test, important for analog, RF and distortion simulation (Gildenblat and Chen).
Cost: Solving the implicit surface-potential equation accurately is the main computational cost, handled by an explicit approximation within 10 $nV$ in SP and PSP and by iteration in HiSIM (Gildenblat and Chen; Miura-Mattausch et al.).

2. Working Principle

Surface-potential equation: Gauss’s law at the gate and the charge-sheet solution of the Poisson equation relate gate voltage to surface potential, with the channel quasi-Fermi potential running from the source to the drain voltage (background knowledge; MOS Capacitor page).
Pinning in strong inversion: Above threshold the surface potential stays within a few thermal voltages of $2 ϕ_{F} + V$ , so the inversion charge, not the potential, absorbs further gate drive (background knowledge; computed in the table).
Current from two potentials: The drift and diffusion currents integrate exactly in the charge-sheet model, so the current depends only on the surface potentials at the two ends; PSP linearizes the inversion charge between them for a compact form (Gildenblat and Chen; Gildenblat et al.).
Short-channel physics: Mobility degradation, velocity saturation, DIBL and channel-length modulation enter through parameters such as MUE, THEMU, THESAT, CF and ALP in PSP, and through pocket-implant and channel-shortening terms in HiSIM (PSP 103.1 note; Miura-Mattausch et al.).

3. Applications

Analog and RF design kits: PSP in foundry design kits for analog, RF and mixed-signal CMOS, where symmetry, continuity of derivatives and accuracy above 50 $GHz$ matter (EEPower; background knowledge).
High-voltage and SOI devices: HiSIM_HV for LDMOS and HiSIM-SOI and SOTB variants (background knowledge).
Statistical and variability modelling: Surface-potential parameters tied to doping and oxide thickness for process-variation models (Miura-Mattausch lecture).
Current status: PSP remains a CMC standard for planar bulk CMOS alongside BSIM4, while multi-gate devices use BSIM-CMG (background knowledge; BSIM3 and BSIM4 page).

Surface Potential and Charges

Name	Formula	Related Formulae	Variables/Constants
Surface-potential equation	$(V_{GB} - V_{F B} - ψ_{s})^{2} = γ^{2} ϕ_{t} [e^{- ψ_{s} / ϕ_{t}} + \frac{ψ _{s}}{ϕ _{t}} - 1 + e^{- (2 ϕ_{F} + V) / ϕ_{t}} (e^{ψ_{s} / ϕ_{t}} - \frac{ψ _{s}}{ϕ _{t}} - 1)]$	$γ = \frac{2 q ε _{s i} N _{A}}{C _{o x}}, ϕ_{F} = ϕ_{t} ln \frac{N _{A}}{n _{i}}, V = V_{C B} (quasi-Fermi potential, V_{S B} to V_{D B})$	$V_{GB} : g a t e - b u l k v o l t a g e$ $V_{F B} : f l a t - ban d v o l t a g e$ $ψ_{s} : s u r f a ce p o t e n t ia l$ $γ : b o d y f a c t or, ϕ_{t} = k_{B} T / q : t h er ma l v o l t a g e$ $ϕ_{F} : F er mi p o t e n t ia l o f t h e s u b s t r a t e$ $V : c hann e l q u a s i - F er mi p o t e n t ia l r e l a t i v e t o b u l k, f r o m V_{S B} a t t h e so u r ce t o V_{D B} a t t h e d r ain$ $q : e l e m e n t a r y c ha r g e$ $ε_{s i} : s i l i co n p er mi tt i v i t y$ $N_{A} : d o p in g$ $C_{o x} : o x i d e c a p a c i t an ce p er a r e a$ $n_{i} : in t r in s i c d e n s i t y$ $k_{B} : B o l t z mann co n s t an t$ $T : t e m p er a t u r e (im pl i c i t c ha r g e - s h ee t f or m so l v e d a t b o t h c hann e l e n d s, ba c k g r o u n d k n o w l e d g e; M O S p h y s i cs o n t h e M O S C a p a c i t or p a g e); G i l d e nb l a t an d C h e n : S P u ses an ana l y t i c a l a pp r o x ima t i o n o f t h e so l u t i o n w i t h a ma x im u m ab so l u t e er r or b e l o w 10 nV, a cc u r a t e e n o ug h f or t r an sco n d u c t an ces, w i t h a p er t u r ba t i o n Δ ϕ_{s} f or p o l y s i l i co n d e pl e t i o n; P S P 102.5 d e f a u l t s : T O X O = 2 \times 1 0^{- 9} m, E P S R O X O = 3.9, N S U B O = 3 \times 1 0^{23} m^{3}, V F B O = - 1 V an d D P H I B O = 0$ $(co m p u t e d : t h e P S P d e f a u l t s g i v e C_{o x} = 0.0173 F/ m^{2}, γ = 0.18 V^{1/2} an d 2 ϕ_{F} = 0.89 V w i t h an a ss u m e d n_{i} o f 1 \times 1 0^{16} m^{3}; so l v in g t h e e q u a t i o n a t V = 0 g i v es ψ_{s} = 0.39, 0.84, 1.03, 1.07 an d 1.09 V f or V_{GB} - V_{F B} = 0.5, 1.0, 1.5, 2.0 an d 2.5 V, so ab o v e in v er s i o n ψ_{s} r i ses o n l y 60 mV p er v o l t o f g a t e d r i v e)$
Inversion charge from the surface potential	$Q_{in v} = - C_{o x} (V_{GB} - V_{F B} - ψ_{s}) - Q_{B} (ψ_{s}), Q_{B} = - γ C_{o x} ϕ_{t} (e^{- ψ_{s} / ϕ_{t}} + \frac{ψ _{s}}{ϕ _{t}} - 1)$	$Q_{in v} = - C_{o x} [V_{GB} - V_{F B} - ψ_{s} - γ ψ_{s} - ϕ_{t}] (depletion-charge approximation)$	$Q_{in v} : in v er s i o n c ha r g e p er a r e a$ $Q_{B} : d e pl e t i o n c ha r g e p er a r e a, o t h er sy mb o l s a s ab o v e (c ha r g e - s h ee t r e l a t i o n s, ba c k g r o u n d k n o w l e d g e); t h e s am e s u r f a ce p o t e n t ia l s g i v e t h e c ha r g es a t so u r ce an d d r ain f r o m w hi c h c u r r e n t an d c a p a c i t an ces f o l l o w; G i l d e nb l a t an d C h e n : G u mm e l sy mm e t r y g i v es C_{g s} = C_{g d} a t V_{d s} = 0 in t h e C - V r es u l t s$ $(n o co m p u t e d v a l u e)$

PSP Current and Parameters

Name	Formula	Related Formulae	Variables/Constants
Drain current with symmetric linearization (PSP and SP)	$I_{d s} = \frac{W}{L} μ C_{o x} (q_{im} + α ϕ_{t}) Δ ψ (normalized charge q_{i} = - Q_{in v} / C_{o x})$	$q_{i} (ψ) \approx q_{im} - α (ψ - ψ_{m}), ψ_{m} = \frac{ψ _{s 0} + ψ _{s L}}{2}, Δ ψ = ψ_{s L} - ψ_{s 0}, α = - \frac{d q _{i}}{d ψ _{s}}_{ψ_{m}}$	$I_{d s} : d r ain c u r r e n t w i t h o u t v e l oc i t y s a t u r a t i o n$ $q_{im} : n or ma l i z e d in v er s i o n c ha r g e a t t h e mi d p o in t p o t e n t ia l ψ_{m}$ $α : l in e a r i z a t i o n coe f f i c i e n t$ $ψ_{s 0}, ψ_{s L} : s u r f a ce p o t e n t ia l s a t so u r ce an d d r ain$ $Δ ψ : t h e i r d i f f er e n ce (n or ma l i z e d f or m s, ba c k g r o u n d k n o w l e d g e a f t er G i l d e nb l a t e t a l .); G i l d e nb l a t an d C h e n : l in e a r i z a t i o n o f t h e in v er s i o n c ha r g e a r o u n d t h e p o t e n t ia l mi d p o in t, d o n e sy mm e t r i c a l l y t o p r eser v e G u mm e l sy mm e t r y, w i t h β = μ (W / L) C_{o x}, t h e m o bi l i t y e v a l u a t e d a t t h e mi d p o in t an d a v e l oc i t y - s a t u r a t i o n t er m w i t h V_{c} = L v_{s} / μ; c ha r a c t er i s t i cs s a t i s f y t h e G u mm e l sy mm e t r y t es t; G i l d e nb l a t e t a l . (2006) : f u r t h er d e v e l o p m e n t an d v er i f i c a t i o n o f t h e sy mm e t r i c l in e a r i z a t i o n m e t h o d in P S P$ $(n o co m p u t e d v a l u e)$
Mobility, velocity saturation and output conductance parameters	$μ = \frac{U _{0}}{1 + ( M U E \cdot E _{e f f} ) ^{T H E M U} + C S ( \dots )} (structure)$	$T H E S A T : velocity saturation, C F : DIBL, A L P, A L P 1, V P : channel-length modulation$	$μ : e f f ec t i v e m o bi l i t y$ $U_{0} : z er o - f i e l d m o bi l i t y$ $M U E : m o bi l i t y r e d u c t i o n coe f f i c i e n t$ $E_{e f f} : e f f ec t i v e v er t i c a l f i e l d$ $T H E M U : m o bi l i t y r e d u c t i o n e x p o n e n t$ $C S : C o u l o mb sc a tt er in g p a r am e t er (s t r u c t u r e f r o m t h e p a r am e t er d e f ini t i o n s, m y r e a d in g; G i l d e nb l a t an d C h e n g i v e a g a r b l e d f or m w i t h a C o u l o mb t er m e v a l u a t e d a t t h e p o t e n t ia l mi d p o in t); P S P 102.5 g l o ba l d e f a u l t s : U O = 5 \times 1 0^{- 2} m^{2} / (V s), M U E O = 0.5 m / V, T H E M U O = 1.5, T H E S A T O = 0 V^{- 1}, C F B O = 0 V^{- 1}, A L P L = 5 \times 1 0^{- 4}, V P O = 0.05 V an d R S W 1 = 2500 Ω; P S P 103.1 l oc a l d e f a u l t s : M U E = 0.5 m / V, T H E M U = 1.5, C S = 0, T H E S A T = 1 V^{- 1}, C F = 0, A L P = 0.01, A L P 1 = 0 V, V P = 0.05 V an d T R = 21^{\circ} C (see t h e Q / A o n T H E S A T)$ $(n o co m p u t e d v a l u e)$

HiSIM and Extraction

Name	Formula	Related Formulae	Variables/Constants
HiSIM surface potentials and current	$I_{d s} = f (ϕ_{S 0}, ϕ_{S L}) (charge-sheet, gradual-channel)$	$ϕ_{S L} \to ϕ_{S 0} + V_{d s} beyond pinch-off, with ϕ (Δ L) for channel shortening$	$ϕ_{S 0}, ϕ_{S L} : s u r f a ce p o t e n t ia l s a t t h e so u r ce an d d r ain e n d s$ $Δ L : c hann e l s h or t e nin g in p in c h - o f f; M i u r a - M a tt a u sc h e t a l . (2005) : H i S I M o b t ain s t h e s u r f a ce p o t e n t ia l s b y so l v in g t h e P o i sso n e q u a t i o n i t er a t i v e l y t o g e t h er w i t h G a u s s^{'} s l a w, w hi c h d oes n o t mak e s im u l a t i o n s l o w er t han B S I M 3 v 3 b ec a u se a s in g l e e q u a t i o n i s v a l i d f or a l l bia s co n d i t i o n s; t h e c ha r g e - s h ee t an d g r a d u a l - c hann e l a pp r o x ima t i o n s g i v e t h e d e v i ce p er f or man ce ana l y t i c a l l y a s a f u n c t i o n o f ϕ_{S 0} an d ϕ_{S L}; c hann e l - l e n g t h m o d u l a t i o n i s m o d e l l e d b y Δ L, w i t h ϕ (Δ L) e x t r a c t e d f r o m m e a s u r e d g_{d s}; p oc k e t - im pl an t an d s h or t - c hann e l p a r am e t er s s u c h a s sc p, sc, p g d, r p oc k an d c l m; M i u r a - M a tt a u sc h (l ec t u r e) : H i S I M, t h e H i r os hima - u ni v er s i t y S T A R C I GF E T M o d e l, so l v es t h e P o i sso n e q u a t i o n t o o b t ain t h e s u r f a ce p o t e n t ia l s, a l l o w in g d e v i ce v a r ia t i o n s t o b e m o d e l l e d in a w a y s imi l a r t o 2 D d e v i ce s im u l a t i o n$ $(n o co m p u t e d v a l u e)$
Parameter counts and extraction	$N_{p a r} (local and global models)$	$ε_{r m s} = \frac{1}{N} \sum_{i} (\frac{I _{m e a s, i} - I _{m o d, i}}{I _{m e a s, i}})^{2}$	$N_{p a r} : n u mb er o f m o d e l p a r am e t er s$ $ε_{r m s} : r e l a t i v e r m s f i tt in g er r or o v er N d a t a p o in t s (er r or d e f ini t i o n, ba c k g r o u n d k n o w l e d g e); G i l d e nb l a t an d C h e n : a l oc a l S P m o d e l n ee d s u p t o 28 p a r am e t er s an d t h e g l o ba l v er s i o n 68, o f w hi c h 35 a r e n ee d e d f or a t y p i c a l ma t u r e p r ocess, w i t h v er i f i c a t i o n a t 218, 298 an d 423 K an d o n 10/0.13 μ m d e v i ces; M i u r a - M a tt a u sc h e t a l . : H i S I M 1.1 ha s 72 p a r am e t er s, a tw o - s t a g e g e n e t i c a l g or i t hm e x t r a c t e d 32 o f t h e m (6 t h e n 26) in ab o u t 23 h o u r s w i t h a p o p u l a t i o n o f 450 o v er 6000 g e n er a t i o n s an d 352 d a t a p o in t s p er d e v i ce, r e a c hin g r m s I - V er r or s o f 0.41 t o 0.93 p er ce n t o n t a r g e t d e v i ces (ma x im u m 2.59 p er ce n t) an d 0.49 t o 0.79 p er ce n t o n o t h er d e v i ces, w h er e a s in g l e - s t a g e r u n s t u c k in l oc a l minima$ $(co m p u t e d : 35 o f 68 g l o ba l S P p a r am e t er s i s 51 p er ce n t; 32 o f 72 H i S I M p a r am e t er s i s 44 p er ce n t)$

Important Questions

Ques: Is the default velocity-saturation parameter THESAT in PSP 0 or 1 per volt?

The two PSP technical notes give different values for different parameters. PSP 102.5 lists THESATO = 0 $V^{- 1}$ among the global-model defaults, while PSP 103.1 lists THESAT = 1 $V^{- 1}$ among the local-model parameters at the reference temperature. In PSP the local parameters describe one device, and the global parameters, with suffixes such as O, L and W, generate the local values from geometry (background knowledge), so THESATO is the geometry-independent part of a global expression whose length terms add the actual velocity saturation (my reading). The two defaults therefore do not conflict: 1 $V^{- 1}$ is the more meaningful default for a single short device, and 0 the neutral start for a global fit. Model cards must not mix the two parameter sets. The note’s reference temperature of 21 $^{\circ} C$ also differs from SPICE’s usual 27 $^{\circ} C$ . Reference: G. D. J. Smit, A. J. Scholten, D. B. M. Klaassen and R. van der Toorn, ‘PSP 102.5’, Technical Note NXP-TN-2013-0031 (December 2013) (CEA-Leti PSP support copy; model-equation chapter not in the excerpt) (fetch-tool summary); ‘PSP 103.1’, Technical Note NXP-R-TN-2008/00299 (June 2009) (CEA-Leti PSP support copy; model-equation chapter not in the excerpt) (fetch-tool summary).

Ques: Why does PSP use symmetric linearization of the inversion charge?

Because a compact current expression must be symmetric in source and drain to pass the Gummel symmetry test. The exact charge-sheet current needs the inversion charge as a function of surface potential, which is not linear. Gildenblat and Chen linearize it around the midpoint of the source and drain potentials, $q_{i} \approx q_{im} - α (ψ - ψ_{m})$ , which gives a current proportional to $(q_{im} + α ϕ_{t}) Δ ψ$ (background knowledge for the normalized form) and treats the two ends identically. Linearizing around the source instead, as threshold-based models effectively do, breaks the symmetry and produces artefacts in the higher derivatives of the current at zero drain bias. Their characteristics satisfy the Gummel test, and $C_{g s} = C_{g d}$ at zero drain voltage. Reference: G. Gildenblat and T.-L. Chen, ‘Overview of An Advanced Surface-Potential-Based Model (SP)’, invited paper, MSM 2002 (TechConnect volume; equations garbled in the extract) (fetch-tool summary); G. Gildenblat, X. Li, W. Wu, H. Wang, A. Jha, R. van Langevelde, G. D. J. Smit, A. J. Scholten and D. B. M. Klaassen, ‘PSP: An Advanced Surface-Potential-Based MOSFET Model for Circuit Simulation’, IEEE Transactions on Electron Devices 53(9), 1979-1993 (2006), doi:10.1109/TED.2005.881006 (abstract only).

Ques: How accurately must the surface potential be computed?

Very accurately, because the inversion charge in weak inversion depends exponentially on it, and derivatives of current set transconductance, capacitance and distortion. Gildenblat and Chen report that the analytical approximation used in SP has a maximum absolute error below 10 $nV$ , accurate enough for transconductances. HiSIM instead solves the Poisson equation iteratively together with Gauss’s law, and Miura-Mattausch et al. state that this does not make simulation slower than BSIM3v3, since a single equation covers all biases. The pinning of the potential in strong inversion shows why: with the PSP default doping and oxide, the surface potential rises only from 1.03 to 1.09 $V$ as the gate drive goes from 1.5 to 2.5 $V$ (computed), so errors of even a millivolt would distort the charge. Reference: G. Gildenblat and T.-L. Chen, ‘Overview of An Advanced Surface-Potential-Based Model (SP)’, invited paper, MSM 2002 (TechConnect volume; equations garbled in the extract) (fetch-tool summary); M. Miura-Mattausch, N. Sadachika, M. Murakawa, S. Mimura, T. Higuchi, K. Itoh, R. Inagaki and Y. Iguchi, ‘RF-MOSFET Model-Parameter Extraction with HiSIM’, NSTI-Nanotech 2005 (WCM) (fetch-tool summary).

Ques: What advantage do surface-potential models claim over threshold-voltage models?

Physical continuity across all regions. EEPower reports that PSP is based on the surface potential across the whole operating range, while the earlier standard models simplify the two limits, subthreshold and strong inversion, and join them mathematically. As the BSIM3 and BSIM4 page shows, those models use smoothing functions such as $V_{g s t e f f}$ and $V_{d se f f}$ for the transition. A surface-potential model needs no such joins, so moderate inversion, which analog circuits often use, is described by the same physics. EEPower also quotes accurate prediction up to frequencies well above 50 $GHz$ and verification on processes down to 65 $nm$ . Miura-Mattausch adds that surface potentials let device variations be modelled in a way similar to 2D device simulation. The cost is the surface-potential solution itself. Reference: J. Shepard, ‘Philips and Penn State submit new MOSFET model’, EEPower (29 March 2005) (fetch-tool summary); IEEE Malaysia EDS/MTT-S/SSC chapter, Distinguished Lecture by M. Miura-Mattausch, ‘Surface-Potential-Based Compact Modeling for Accurate Prediction of Circuits Performances’ (23 November 2010) (event page) (fetch-tool summary); BSIM3 and BSIM4 page of this vault.

Ques: How many parameters do surface-potential models need, and how are they extracted?

Tens. Gildenblat and Chen state that a local version of SP needs up to 28 parameters and the global version 68, of which 35 are needed for a typical mature process, about half (computed). Miura-Mattausch et al. report that HiSIM 1.1 has 72 parameters for circuit simulation and extracted 32 of them, about 44 per cent (computed), with a two-stage genetic algorithm: 6 basic parameters such as flat-band voltage and doping first, then 26 more, in about 23 hours, against several days for a human expert. The rms I-V errors were 0.41 to 0.93 per cent on target devices and 0.49 to 0.79 per cent on other devices, while a single-stage search stuck in local minima. The physical basis means that parameters such as doping and oxide thickness can often be taken from the process. Reference: G. Gildenblat and T.-L. Chen, ‘Overview of An Advanced Surface-Potential-Based Model (SP)’, invited paper, MSM 2002 (TechConnect volume; equations garbled in the extract) (fetch-tool summary); M. Miura-Mattausch, N. Sadachika, M. Murakawa, S. Mimura, T. Higuchi, K. Itoh, R. Inagaki and Y. Iguchi, ‘RF-MOSFET Model-Parameter Extraction with HiSIM’, NSTI-Nanotech 2005 (WCM) (fetch-tool summary).

Ques: How does HiSIM model channel-length modulation?

Through the surface potential beyond pinch-off. Miura-Mattausch et al. describe the channel in saturation as shortened by $Δ L$ , with the potential running from the source value $ϕ_{S 0}$ through the drain-side value $ϕ_{S L}$ and $ϕ (Δ L)$ to $ϕ_{S 0} + V_{d s}$ at the drain. The unknown $ϕ (Δ L)$ is extracted from the measured channel conductance $g_{d s}$ against $V_{d s}$ , so the output conductance follows from the potential distribution rather than from an empirical Early voltage. Short-channel and pocket-implant effects enter through parameters such as scp, sc, pgd, rpock and clm, fitted in the second extraction stage. This contrasts with PSP’s channel-length modulation parameters ALP, ALP1 and VP, whose defaults in PSP 103.1 are 0.01, 0 $V$ and 0.05 $V$ . Reference: M. Miura-Mattausch, N. Sadachika, M. Murakawa, S. Mimura, T. Higuchi, K. Itoh, R. Inagaki and Y. Iguchi, ‘RF-MOSFET Model-Parameter Extraction with HiSIM’, NSTI-Nanotech 2005 (WCM) (fetch-tool summary); ‘PSP 103.1’, Technical Note NXP-R-TN-2008/00299 (June 2009) (CEA-Leti PSP support copy; model-equation chapter not in the excerpt) (fetch-tool summary).

Quartz 5

Explorer

Surface-Potential Models (PSP and HiSIM)