1. Theory

Definition: MVSG (MIT Virtual Source GaNFET) is a physics-based, charge-based compact model for GaN HEMTs in RF and high-voltage applications, built on the virtual-source transport model developed for scaled silicon transistors (Waterloo; Radhakrishna, S.M.).
Virtual source: The drain current is the product of the charge and velocity at the top of the source barrier, with a saturation function for the linear-to-saturation transition, using about ten mostly physical parameters (Khakifirooz et al.).
GaN extension: Non-linear access regions, self-heating, drift-diffusion transport for long-channel high-voltage devices and self-consistent charges were added at MIT (Radhakrishna et al. 2012 and 2013).
Standard model: MVSG is a CMC standard model with field plates, gate leakage, breakdown, trapping, p-GaN gates and noise, now maintained at the University of Waterloo (Waterloo).

2. Working Principle

Charge at the virtual source: A single smooth expression gives the channel charge from weak to strong accumulation, shifted by DIBL (Khakifirooz et al.).
Velocity: In short channels the carriers at the virtual source move at a quasi-ballistic velocity $v_{x o}$ ; in long channels the saturation voltage grows as $v_{x o} L / μ$ and the model blends to drift-diffusion with pinch-off (Khakifirooz et al.; Radhakrishna et al. 2013).
Access regions as transistors: The ungated access regions are treated as transistors with an implicit gate whose overdrive reproduces their sheet charge, so their current saturates like a channel (Radhakrishna et al. 2013).
Charges: Ward-Dutton partitioning of the self-consistent channel charge gives terminal charges consistent with the current (Radhakrishna et al. 2013).

3. Applications

RF power and low-noise amplifiers: Load-pull, intermodulation and noise-figure design, as validated in Radhakrishna’s thesis (Radhakrishna, Ph.D.).
Power conversion: Switching slew rates of GaN DC-DC converters, including field plates, temperature and trapping (Radhakrishna, Ph.D.).
Technology assessment: Projection of the cut-off frequency with scaling and identification of bottlenecks (Radhakrishna, S.M.).
Process design kits: CMC-standard GaN models alongside ASM-HEMT (Waterloo; ASM-HEMT page).

Virtual-Source Core

Name	Formula	Related Formulae	Variables/Constants
Virtual-source drain current	$\frac{I _{D}}{W} = Q_{i x o} v_{x o} F_{s}$	$V_{GS}^{'} = V_{GS} - I_{D} R_{S}, V_{D S}^{'} = V_{D S} - I_{D} (R_{S} + R_{D})$	$I_{D} : d r ain c u r r e n t$ $W : w i d t h$ $Q_{i x o} : c hann e l c ha r g e p er a r e a a t t h e v i r t u a l so u r ce (t h e t o p o f t h e so u r ce ba r r i er)$ $v_{x o} : c a r r i er v e l oc i t y t h er e$ $F_{s} : s a t u r a t i o n f u n c t i o n, V_{GS}^{'}, V_{D S}^{'} : in t r in s i c v o l t a g es$ $R_{S}, R_{D} : so u r ce an d d r ain r es i s t an ces; K hak i f i r ooz e t a l . : t hi s c u r r e n t, t e n p a r am e t er s (s i x f r o m s t an d a r d m e a s u r e m e n t s : C_{in v}, s u b t h r es h o l d s w in g, D I B L, w e ak - in v er s i o n c u r r e n t, t o t a l r es i s t an ce a t V_{D S} = 0 an d e f f ec t i v e l e n g t h; t h r ee f i tt e d p h y s i c a l p a r am e t er s : l o w - f i e l d m o bi l i t y, p a r a s i t i c r es i s t an ce an d v_{x o}; an d t h e e m p i r i c a l β), an ini t ia l v_{x o} o f ab o u t 1 \times 1 0^{7} cm/s f or s i l i co n, an d a n o n - i t er a t i v e ser i es - r es i s t an ce a pp r o x ima t i o n; R a d hak r i s hna (S . M .) : v i r t u a l - so u r ce t r an s p or t f r o m sc a l e d s i l i co n t r an s i s t or s a ppl i e d t o s h or t - c hann e l G a N H E M T s w i t h a f e w p h y s i c a l, in d e p e n d e n tl y m e a s u r ab l e p a r am e t er s$ $(co m p u t e d : f or an a ss u m e d G a N ba r r i er w i t h C_{g} = 4.26 \times 1 0^{- 3} F/ m^{2}, 2 V o f o v er d r i v e (Q_{i x o} = 8.5 \times 1 0^{- 3} C/ m^{2}) an d v_{x o} = 1.5 \times 1 0^{7} cm/s, I_{D} / W = 1.28 A/mm in f u l l s a t u r a t i o n, b e f or e a ccess - r e g i o n l imi t s)$
Virtual-source charge and threshold	$Q_{i x o} = C_{in v} n ϕ_{t} ln [1 + exp (\frac{V _{GS}^{'} - ( V _{T} - α ϕ _{t} F _{f} )}{n ϕ _{t}})]$	$V_{T} = V_{T 0} - δ V_{D S}^{'}, F_{f} = \frac{1}{1 + e x p [ ( V _{GS}^{'} - ( V _{T} - α ϕ _{t} /2 ) ) / ( α ϕ _{t} ) ]}$	$C_{in v} : g a t e c a p a c i t an ce p er a r e a in in v er s i o n (a cc u m u l a t i o n o f t h e 2 D E G f or a H E M T)$ $n : s u b t h r es h o l d f a c t or$ $ϕ_{t} : t h er ma l v o l t a g e$ $V_{T} : t h r es h o l d v o l t a g e$ $V_{T 0} : i t s v a l u e a t z er o d r ain bia s$ $δ : D I B L coe f f i c i e n t$ $α : e m p i r i c a l s hi f t b e tw ee n w e ak an d s t r o n g in v er s i o n t h r es h o l d$ $F_{f} : F er mi f u n c t i o n b l e n d in g t h e tw o r e g im es; K hak i f i r ooz e t a l . : t h ese e q u a t i o n s, a cc u r a t e t o ab o u t 15 p er ce n t o v er se v e n or d er s o f ma g ni t u d e o f c ha r g e u p t o 2.5 \times 1 0^{13} c m^{2} an d t o b e l o w ab o u t 5 p er ce n t in s t r o n g in v er s i o n f or b o d y f a c t or s o f 0.1 t o 1 V^{1/2}$ $(co m p u t e d : n = 1.3 g i v es a s u b t h r es h o l d s w in g o f 77 mV/dec a t 300 K)$
Saturation function and saturation voltage	$F_{s} = \frac{V _{D S}^{'} / V _{D S A T}}{[ 1 + ( V _{D S}^{'} / V _{D S A T} ) ^{β} ] ^{1/ β}}$	$V_{D S A T} = V_{D S A T s} (1 - F_{f}) + ϕ_{t} F_{f}, V_{D S A T s} = \frac{v _{x o} L _{C}}{μ}$	$F_{s} : t r an s i t i o n f r o m l in e a r t o s a t u r a t i o n$ $V_{D S A T} : s a t u r a t i o n v o l t a g e$ $β : e m p i r i c a l e x p o n e n t$ $V_{D S A T s} : s t r o n g - in v er s i o n v a l u e$ $L_{C} : e f f ec t i v e c hann e l l e n g t h$ $μ : l o w - f i e l d m o bi l i t y; K hak i f i r ooz e t a l . : t h ese f or m s w i t h β o f ab o u t 1.8 (n F E T) an d 1.4 (pF E T) f or t h e e x pl i c i t so l u t i o n an d 2.5 an d 2.0 f or t h e a pp r o x ima t e ser i es - r es i s t an ce so l u t i o n, an d m o bi l i t i es o f 200 t o 300 c m^{2} / (V s) f or e l ec t r o n s an d 100 t o 250 c m^{2} / (V s) f or h o l es in s t r ain e d s i l i co n; R a d hak r i s hna e t a l . (2013) : t h e M V S G - H V m o d e l e x t e n d s t h e q u a s i - ba l l i s t i c s h or t - c hann e l m o d e l t o d r i f t - d i f f u s i o n t r an s p or t, han d l in g d r ain - e n d c ha r g e m o d u l a t i o n b y p in c h - o f f an d v e l oc i t y s a t u r a t i o n s m oo t h l y w i t h g a t e l e n g t h$ $(co m p u t e d : F_{s} a t V_{D S}^{'} = V_{D S A T} i s 0.61, 0.68, 0.71 an d 0.76 f or β = 1.4, 1.8, 2.0 an d 2.5; V_{D S A T s} i s 0.14 V f or s i l i co n w i t h 1 \times 1 0^{7} cm/s, 35 nm an d 250 c m^{2} / (V s), 0.10 V f or G a N w i t h 1.5 \times 1 0^{7} cm/s, 0.1 μ m an d 1500 c m^{2} / (V s), an d 3.0 V f or t h e s am e G a N v a l u es a t 3 μ m)$

Access Regions and Charges

Name	Formula	Related Formulae	Variables/Constants
Implicit-gated access regions	$I_{a cc} = W Q_{a cc} v_{a cc} F_{s, a cc}, Q_{a cc} = q n_{s, a cc} = C_{g} V_{im p}$	$R_{s h} = \frac{1}{q n _{s, a cc} μ _{a cc}}$	$I_{a cc} : a ccess - r e g i o n c u r r e n t$ $Q_{a cc} : a ccess c ha r g e p er a r e a$ $v_{a cc} : c a r r i er v e l oc i t y in t h e a ccess r e g i o n$ $F_{s, a cc} : i t s s a t u r a t i o n f u n c t i o n$ $n_{s, a cc} : a ccess 2 D E G d e n s i t y$ $V_{im p} : im pl i c i t g a t e o v er d r i v e$ $R_{s h} : a ccess s h ee t r es i s t an ce$ $μ_{a cc} : a ccess m o bi l i t y (f or m s, m y r e a d in g o f t h e d escr i pt i o n); R a d hak r i s hna e t a l . (2013) : a ccess r e g i o n s m o d e l l e d a s im pl i c i t - g a t e d t r an s i s t or s w i t h n o p h y s i c a l g a t e, t h e i r im pl i c i t g a t e o v er d r i v e l ink e d t o t h e s h ee t r es i s t an ce; R a d hak r i s hna e t a l . (2012) : a ccess r e g i o n s a s n o n - l in e a r r es i s t or s, v e l oc i t y q u a s i - s a t u r a t i o n a t ab o u t 5 kV/cm, a 105 nm d e v i ce w i t h 0.5 μ m a ccess r e g i o n s an d T L M l e n g t h s o f 5 t o 25 μ m, a ccess r e g i o n s r a t h er t han t h e c hann e l l imi t in g t h e ma x im u m c u r r e n t, an d g_{m} f a l l in g a f t er i t s p e ak$ $(co m p u t e d : n_{s, a cc} = 1 \times 1 0^{13} c m^{2} an d 1500 c m^{2} / (V s) (a ss u m e d) g i v e R_{s h} = 416 Ω/ sq an d, w i t h C_{g} = 4.26 \times 1 0^{- 3} F/ m^{2}, V_{im p} = 3.76 V; t h e a ccess s a t u r a t i o n c u r r e n t q n_{s} v_{s a t} i s 1.60 A/mm f or 1 \times 1 0^{7} cm/s; μ E a t 5 kV/cm i s 7.5 \times 1 0^{6} cm/s)$
Terminal charges	$Q_{D} = \int_{0}^{L} \frac{x}{L} Q_{i} (x) W d x, Q_{S} = \int_{0}^{L} (1 - \frac{x}{L}) Q_{i} (x) W d x$	$Q_{G} = - (Q_{S} + Q_{D}) (plus access and fringe charges)$	$Q_{D}, Q_{S}, Q_{G} : d r ain, so u r ce an d g a t e c ha r g es, Q_{i} (x) : c hann e l c ha r g e p er a r e a a l o n g t h e c hann e l$ $x : p os i t i o n$ $L : c hann e l l e n g t h (W a r d - D u tt o n in t e g r a l s, ba c k g r o u n d k n o w l e d g e); R a d hak r i s hna e t a l . (2013) : t er mina l c ha r g es so l v e d se l f - co n s i s t e n tl y f r o m c u r r e n t co n t in u i t y an d W a r d - D u tt o n p a r t i t i o nin g; W a t er l oo : se l f - co n s i s t e n t c u r r e n t an d c ha r g e w i t h bia s - d e p e n d e n t f r in g in g c a p a c i t an ces; W a r d - D u tt o n p a r t i t i o nin g o n t h e C ha r g e an d C a p a c i t an ce M o d e l l in g p a g e$ $(n o co m p u t e d v a l u e)$

Model Features

Name	Formula	Related Formulae	Variables/Constants
MVSG feature set	$I_{D} = f (channel, access, field plates, T_{j}, traps)$	$T_{j} = T_{amb} + R_{t h} P_{d i ss} (static)$	$T_{j} : c hann e l t e m p er a t u r e$ $R_{t h} : t h er ma l r es i s t an ce$ $P_{d i ss} : d i ss i p a t e d p o w er (s t a t i c f or m, ba c k g r o u n d k n o w l e d g e); W a t er l oo : se l f - co n s i s t e n t c hann e l c u r r e n t an d c ha r g e, so u r ce an d d r ain a ccess r e g i o n s, f i e l d - pl a t e c u r r e n t s an d c ha r g es o n b o t h so u r ce an d d r ain s i d es, g a t e l e ak a g e, c hann e l b r e ak d o w n, bia s - d e p e n d e n t f r in g in g c a p a c i t an ces, l a y o u t - d e p e n d e n t g a t e r es i s t an ce, se l f - h e a t in g an d t h er ma l co u pl in g, g a t e - l a g an d d r ain - l a g t r a pp in g, g_{m} d i s p er s i o n, p - G a N g a t e s t a c k s an d n o i se; a C M C s t an d a r d m o d e l, w i t h s t an d a r d m o d e l s f r ee t o t h e p u b l i c 18 m o n t h s a f t er r e l e a se, main t ain e d b y L . W e i a t W a t er l oo; R a d hak r i s hna e t a l . (2012) : se l f - h e a t in g in t h e c hann e l; R a d hak r i s hna (P h . D .) : v a l i d a t i o n a g ain s t D C an d p u l se d c u r r e n t s, c ha r g es, S - p a r am e t er s, s w i t c hin g, l o a d - p u l l, p o w er s w ee p s, in t er m o d u l a t i o n an d n o i se f i g u r e, an d b e n c hma r k s o n a D C - D C co n v er t er, p o w er am pl i f i er s, L N A s an d V C O s$ $(n o co m p u t e d v a l u e)$

Important Questions

Ques: How does the virtual-source model compute the drain current?

As charge times velocity at the virtual source. Khakifirooz et al. write $I_{D} / W = Q_{i x o} v_{x o} F_{s}$ , where $Q_{i x o}$ is a smooth charge expression valid from weak to strong inversion, $v_{x o}$ the carrier velocity at the top of the source barrier and $F_{s}$ a saturation function. Six of its ten parameters come from standard measurements, three are fitted physical quantities (mobility, parasitic resistance and $v_{x o}$ ) and one, $β$ , is empirical. Radhakrishna adapted it to short-channel GaN HEMTs because it needs only a few physical, independently measurable parameters. For an assumed 25 $nm$ AlGaN barrier, 2 $V$ of overdrive and $v_{x o} = 1.5 \times 1 0^{7}$ $cm/s$ , the saturated current is 1.28 $A/mm$ (computed), before the access regions reduce it. Reference: A. Khakifirooz, O. M. Nayfeh and D. A. Antoniadis, ‘A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters’, IEEE Transactions on Electron Devices 56(8), 1674-1680 (2009) (Table I not in the extract) (fetch-tool summary); U. Radhakrishna, ‘A compact transport and charge model for GaN-based high electron mobility transistors for RF applications’, S.M. thesis, MIT (2013), advisor D. A. Antoniadis (abstract only) (fetch-tool summary).

Ques: How can a virtual-source model describe long-channel high-voltage GaN HEMTs?

By blending quasi-ballistic and drift-diffusion transport. In the original model the strong-inversion saturation voltage is $V_{D S A T s} = v_{x o} L_{C} / μ$ (Khakifirooz et al.), which is 0.10 $V$ for an assumed 0.1 $μ m$ GaN channel with $1.5 \times 1 0^{7}$ $cm/s$ and 1500 $c m^{2} / (V s)$ but 3.0 $V$ at 3 $μ m$ (computed). Long channels therefore do not reach velocity saturation at moderate drain bias; they saturate by pinch-off at the drain end instead. Radhakrishna et al. (2013) extended MVSG to this regime in the MVSG-HV model, handling drain-end charge modulation by pinch-off and velocity saturation smoothly as a function of gate length, so the model scales from no velocity saturation to full velocity saturation. They validated it on a device with a 3 $μ m$ gate and 9.5 $μ m$ gate-drain spacing, typical of high-voltage switches. Reference: A. Khakifirooz, O. M. Nayfeh and D. A. Antoniadis, ‘A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters’, IEEE Transactions on Electron Devices 56(8), 1674-1680 (2009) (Table I not in the extract) (fetch-tool summary); U. Radhakrishna, T. Imada, T. Palacios and D. A. Antoniadis, ‘MIT Virtual Source GaNFET - High Voltage (MVSG-HV) Model: A Physics-based Compact Model for HV-GaN HEMTs’, MIT MTL Annual Research Report 2013 (fetch-tool summary).

Ques: Were MVSG access regions modelled as resistors or as transistors?

Both, at different stages. The 2012 MIT report describes the access regions as non-linear resistors, with velocity quasi-saturation at about 5 $kV/cm$ , and finds that they, not the intrinsic channel, limit the maximum output current of a 105 $nm$ device with 0.5 $μ m$ access regions. The 2013 MVSG-HV report models them as implicit-gated transistors with no physical gate, whose implicit overdrive is linked to the sheet resistance. The two are consistent: a gateless transistor with a fixed charge is a non-linear resistor that saturates (my reading). For an assumed access density of $1 \times 1 0^{13}$ $c m^{2}$ and 1500 $c m^{2} / (V s)$ , the sheet resistance is 416 $Ω/ sq$ and the equivalent overdrive under a 25 $nm$ barrier 3.76 $V$ (computed). The transistor form reuses the channel equations, so access saturation and its temperature dependence come out of the same model. Reference: U. Radhakrishna, L. Wei and D. A. Antoniadis, ‘A Virtual-source-based Transport Model for GaN based HEMTs including Non-linear Access Region Behavior and Self-heating’, MIT MTL Annual Research Report 2012 (fetch-tool summary); U. Radhakrishna, T. Imada, T. Palacios and D. A. Antoniadis, ‘MIT Virtual Source GaNFET - High Voltage (MVSG-HV) Model: A Physics-based Compact Model for HV-GaN HEMTs’, MIT MTL Annual Research Report 2013 (fetch-tool summary).

Ques: Why does the transconductance of a GaN HEMT fall after its peak?

Mainly because the access regions saturate. Radhakrishna et al. (2012) report that in a 105 $nm$ RF HEMT the access regions, not the intrinsic channel, limit the maximum output current, and that $g_{m}$ decreases with gate voltage after reaching a peak. With an access density of $1 \times 1 0^{13}$ $c m^{2}$ and a saturation velocity of $1 \times 1 0^{7}$ $cm/s$ , the access region can carry at most about 1.60 $A/mm$ (computed), comparable to the 1.28 $A/mm$ the channel could deliver at 2 $V$ of overdrive (computed). As the drain current approaches this limit, the access resistance rises steeply, the internal gate overdrive grows less than the applied one, and $g_{m}$ falls. Self-heating, which lowers mobility at high current, adds to the drop (background knowledge). A constant series resistance cannot reproduce this, which is why both MVSG and ASM-HEMT use current-dependent access models. Reference: U. Radhakrishna, L. Wei and D. A. Antoniadis, ‘A Virtual-source-based Transport Model for GaN based HEMTs including Non-linear Access Region Behavior and Self-heating’, MIT MTL Annual Research Report 2012 (fetch-tool summary); ASM-HEMT Model page of this vault.

Ques: Which value of beta should the virtual-source model use?

It depends on how series resistance is handled. Khakifirooz et al. give $β$ of about 1.8 for nFETs and 1.4 for pFETs when the internal voltages are solved explicitly, and 2.5 and 2.0 when their non-iterative series-resistance approximation is used. The difference is not a disagreement but a refit: the approximation folds part of the resistance into the saturation function, so a sharper transition is needed (my reading). At $V_{D S}^{'} = V_{D S A T}$ , $F_{s}$ equals 0.68 for $β = 1.8$ and 0.76 for 2.5 (computed), so using the wrong value changes the knee current by about 11 per cent (computed). For GaN models, $β$ should be refitted to the output characteristics around the knee rather than taken from silicon values. Reference: A. Khakifirooz, O. M. Nayfeh and D. A. Antoniadis, ‘A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters’, IEEE Transactions on Electron Devices 56(8), 1674-1680 (2009) (Table I not in the extract) (fetch-tool summary).

Ques: What effects does the CMC version of MVSG include?

The Waterloo group, which maintains the model, lists self-consistent channel current and charge, source and drain access regions, field-plate currents and charges in flexible source- and drain-side configurations, gate leakage, channel breakdown, bias-dependent fringing capacitances, layout-dependent gate resistance, self-heating and thermal coupling, gate-lag and drain-lag trapping, $g_{m}$ dispersion, p-GaN gate stacks and noise. MVSG is a Compact Model Coalition standard; members get the latest versions, and standard models become free to the public 18 months after release. Radhakrishna’s 2016 thesis validated the model against DC and pulsed currents, charges, S-parameters, switching, load-pull, power sweeps, intermodulation and noise figure, and benchmarked it on a DC-DC converter, power amplifiers, LNAs and VCOs. The p-GaN gate support makes it applicable to normally-off power devices (background knowledge). Reference: Waterloo Emerging Integrated Systems Group, ‘MVSG model’, University of Waterloo web page (updated 6 October 2023) (fetch-tool summary); U. Radhakrishna, ‘Modeling gallium-nitride based high electron mobility transistors: linking device physics to high voltage and high frequency circuit design’, Ph.D. thesis, MIT (2016), advisor D. A. Antoniadis (abstract only) (fetch-tool summary).

Ques: How accurate is the virtual-source charge approximation?

Accurate enough for circuit use across all regions. Khakifirooz et al. report that their single expression for $Q_{i x o}$ stays within about 15 per cent of the exact charge over seven orders of magnitude, up to $2.5 \times 1 0^{13}$ $c m^{2}$ , and within about 5 per cent in strong inversion for body factors of 0.1 to 1 V^{1/2}. The largest errors occur in moderate inversion, where the blending function $F_{f}$ switches between the weak- and strong-inversion thresholds through the empirical $α$ . In subthreshold the charge falls as $exp [V_{GS}^{'} / (n ϕ_{t})]$ , so a factor $n = 1.3$ gives 77 $mV/dec$ at 300 $K$ (computed). For GaN HEMTs the same form describes the 2DEG under the barrier, whose densities near $1 \times 1 0^{13}$ $c m^{2}$ (background knowledge) are within the validated range. Reference: A. Khakifirooz, O. M. Nayfeh and D. A. Antoniadis, ‘A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters’, IEEE Transactions on Electron Devices 56(8), 1674-1680 (2009) (Table I not in the extract) (fetch-tool summary).

References

A. Khakifirooz, O. M. Nayfeh and D. A. Antoniadis, “A Simple Semiempirical Short-Channel MOSFET Current-Voltage Model Continuous Across All Regions of Operation and Employing Only Physical Parameters,” IEEE Transactions on Electron Devices, vol. 56, no. 8, pp. 1674-1680, 2009. Available: https://dspace.mit.edu/handle/1721.1/52366
U. Radhakrishna, “A compact transport and charge model for GaN-based high electron mobility transistors for RF applications,” S.M. thesis, Massachusetts Institute of Technology, 2013. Available: https://dspace.mit.edu/handle/1721.1/82394
U. Radhakrishna, T. Imada, T. Palacios and D. A. Antoniadis, “MIT Virtual Source GaNFET - High Voltage (MVSG-HV) Model: A Physics-based Compact Model for HV-GaN HEMTs,” MIT MTL Annual Research Report, 2013. Available: https://mtlsites.mit.edu/annual_reports/2013/people/ujwal-radhakrishna
U. Radhakrishna, L. Wei and D. A. Antoniadis, “A Virtual-source-based Transport Model for GaN based HEMTs including Non-linear Access Region Behavior and Self-heating,” MIT MTL Annual Research Report, 2012. Available: https://mtlsites.mit.edu/annual_reports/2012/tag/lan-wei
Waterloo Emerging Integrated Systems Group, “MVSG model,” University of Waterloo, updated Oct. 2023. Available: https://uwaterloo.ca/waterloo-emerging-integrated-systems-group/mvsg-model
U. Radhakrishna, “Modeling gallium-nitride based high electron mobility transistors: linking device physics to high voltage and high frequency circuit design,” Ph.D. thesis, Massachusetts Institute of Technology, 2016. Available: https://dspace.mit.edu/handle/1721.1/105951

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MVSG Model