1. Theory

Definition: Energy-transport and hydrodynamic models extend drift-diffusion by adding an energy balance for each carrier type, so that carrier temperature can exceed the lattice temperature and transport depends on carrier energy rather than only on the local field (Grasser et al.; Aste and Vahldieck).
Moment hierarchy: Taking moments of the Boltzmann transport equation gives drift-diffusion (two moments), energy transport (three or four), the full hydrodynamic model with convective terms, and six-moment models (Grasser et al.).
Why needed: In channels shorter than about 200 $nm$ , carriers do not reach steady state with the field, so velocity overshoot and non-local heating appear that drift-diffusion can only mimic by raising the saturation velocity more than twice (Grasser et al.).
Closure problem: Each moment equation introduces the next higher moment, which must be closed by an assumption such as Fourier’s law for heat flux; these closures limit accuracy (Grasser et al.).

2. Working Principle

Energy gain and loss: Carriers gain energy $- e v \cdot E$ from the field and lose it to the lattice at a rate set by the energy relaxation time (Aste and Vahldieck).
Lag behind the field: Because energy relaxes over about 0.3 $ps$ , carriers entering a high-field region stay cool and fast for a few tens of nanometres, giving velocity overshoot (Grasser et al.; computed).
Hot-carrier mobility: Mobility and diffusivity depend on carrier temperature, so they follow energy rather than field (Grasser et al.).
Convective terms: The full hydrodynamic model keeps inertia and can become supersonic with shock waves; energy-transport drops these terms (Aste and Vahldieck; Grasser et al.).

3. Applications

Short-channel MOSFETs and HBTs: Velocity overshoot and hot-carrier distributions (Grasser et al.).
Ballistic diodes: Benchmark n+ n n+ structures for comparing transport models (Aste and Vahldieck).
Hot-carrier reliability and impact ionization: Energy-based generation models need carrier temperature (Grasser et al.; background knowledge for the reliability link).
High-frequency transients: Full hydrodynamic models for picosecond dynamics in GaAs and silicon devices (Aste and Vahldieck).

Moment Equations

Name	Formula	Related Formulae	Variables/Constants
Energy-balance equation	$\frac{\partial ω}{\partial t} + \nabla \cdot (v ω) = - e n v \cdot E - \nabla \cdot (n k_{B} T v) + \nabla \cdot (κ \nabla T) - \frac{ω - \frac{3}{2} n k _{B} T _{L}}{τ _{w}}$	$ω = \frac{3}{2} n k_{B} T + \frac{1}{2} n m^{*} v^{2}$	$ω : e l ec t r o n e n er g y d e n s i t y$ $v : m e an v e l oc i t y$ $e : e l e m e n t a r y c ha r g e$ $n : d e n s i t y$ $E : f i e l d$ $T : e l ec t r o n t e m p er a t u r e$ $T_{L} : l a tt i ce t e m p er a t u r e$ $κ : e l ec t r o n t h er ma l co n d u c t i v i t y$ $τ_{w} : e n er g y r e l a x a t i o n t im e$ $m^{*} : e f f ec t i v e ma ss; A s t e an d V ah l d i ec k : t hi s e n er g y ba l an ce (w i t h k f or k_{B}) an d t h e e n er g y - d e n s i t y c l os u r e f or p a r ab o l i c ban d s; G r a sser e t a l . : a l oc a l e n er g y ba l an ce g i v in g a c a r r i er t e m p er a t u r e se t b y t h e e n er g y r e l a x a t i o n t im e, w i t h t h e c a r r i er t e m p er a t u r e r e a c hin g tw i ce t h e l a tt i ce t e m p er a t u r e a t f i e l d s o f or d er 10 kV/cm an d f i e l d s ab o v e 1 M V / c m in a d v an ce d d e v i ces$ $(co m p u t e d : a l oc a l ba l an ce T = T_{L} + (2 q τ_{w} μ E^{2}) / (3 k_{B}) (ba c k g r o u n d f or m) w i t h τ_{w} = 0.3 ps an d μ = 1400 c m^{2} / (V s) r e a c h es 2 T_{L} a t 9.6 kV/cm, ma t c hin g t h e r e v i e w^{'} s or d er o f ma g ni t u d e)$
Full hydrodynamic momentum balance	$\frac{\partial j}{\partial t} + (\nabla \cdot j) v + (j \cdot \nabla) v = - \frac{e}{m ^{}} n E - \frac{e}{m ^{}} \nabla (\frac{n k _{B} T}{e}) - \frac{j}{τ _{p}}$	$\frac{\partial n}{\partial t} + \nabla \cdot j = 0, \nabla \cdot (ϵ \nablaΦ) = e (n - N_{D})$	$j = n v : p a r t i c l e f l ux d e n s i t y$ $τ_{p} : m o m e n t u m r e l a x a t i o n t im e$ $Φ : p o t e n t ia l; A s t e an d V ah l d i ec k : t h ese e q u a t i o n s; G r a sser e t a l . : t h e f u l l h y d r o d y nami c m o d e l r ese mb l es t h e E u l er e q u a t i o n s w i t h h e a t co n d u c t i o n an d i s h y p er b o l i c in s u p er so ni c r e g i o n s, w hi l e n e g l ec t in g t h e co n v ec t i v e t er m s g i v es t h e p a r ab o l i c e n er g y - t r an s p or t f or m v a l i d f or s u b so ni c f l o w u se d in m os t s im u l a t or s; t h e d r i f t - d i f f u s i o n l imi t o n t h e D r i f t - D i f f u s i o n D e v i ce E q u a t i o n s p a g e$ $(n o co m p u t e d v a l u e)$

Relaxation Times, Closure and Mobility

Name	Formula	Related Formulae	Variables/Constants
Relaxation times (Baccarani-Wordemann)	$τ_{p} = \frac{m ^{} μ _{0}}{e} \frac{T _{0}}{T}, τ_{w} = (\frac{1}{2} + \frac{3 k _{B}}{2 m ^{} v _{s}^{2}} \frac{T ^{2}}{T + T _{0}}) τ_{p}$	$τ_{w} \approx 0.3 to 0.42 ps (silicon, room temperature)$	$μ_{0} : l o w - f i e l d m o bi l i t y$ $T_{0} : l a tt i ce t e m p er a t u r e$ $v_{s} : s a t u r a t i o n v e l oc i t y; A s t e an d V ah l d i ec k : t h ese f or m s, r e l a x a t i o n t im es d e p e n d in g o n d o p in g, l a tt i ce t e m p er a t u r e an d e l ec t r o n t e m p er a t u r e, w i t h m^{*} = 0.26 m_{e}, v_{s} = 1.03 \times 1 0^{5} m/s an d a l o w - f i e l d m o bi l i t y f r o m 80 t o 1430 c m^{2} / (V s) w i t h N_{r e f} = 1.12 \times 1 0^{17} c m^{- 3}; an d in a G a A s c hann e l an im p u l se r e l a x a t i o n t im e n e a r 0.3 ps an d e n er g y r e l a x a t i o n f r o m 0.3 t o 1 ps; G r a sser e t a l . : r oo m - t e m p er a t u r e s i l i co n τ_{w} o f a f e w t e n t h s o f a p i coseco n d, A g os t in e l l i e t a l . u p t o ab o u t 0.68 ps (co n s i d er e d t oo hi g h), H a s na t e t a l . ab o u t 0.42 ps, 0.3 ps in t h e n o n l oc a l i t y ana l y s i s, an d a 1 p er ce n t c han g e in τ_{w} c han g in g d r ain c u r r e n t b y o n l y 0.01 t o 0.17 p er ce n t$ $(co m p u t e d w i t h t h e A s t e p a r am e t er s : τ_{p} = 0.211 ps an d τ_{w} = 0.367 ps a t T = T_{0} = 300 K, an d 0.063 an d 0.434 ps a t T = 1000 K)$
Heat flux closure	$κ = (\frac{5}{2} + r) n \frac{k _{B}^{2} μ ( T )}{e} T$	$Q = - κ \nabla T (Fourier law)$	$κ : e l ec t r o n t h er ma l co n d u c t i v i t y (W i e d e mann - F r an z f or m)$ $r : sc a tt er in g e x p o n e n t (- 2.1 f or s i l i co n a t 300 K)$ $Q : h e a t f l ux; A s t e an d V ah l d i ec k : t hi s κ; G r a sser e t a l . : B l o t e k j a e r^{'} s ma cr osco p i c r e l a x a t i o n - t im e m o d e l s c l ose d w i t h F o u r i e r^{'} s an d t h e W i e d e mann - F r an z l a w s, c a l l e d a maj or in co n s i s t e n cy, h e a t - f l ux f a c t or s a d j u s t ab l e in d e p e n d e n tl y in t h r ee - m o m e n t m o d e l s, a h e a t - f l ux r e d u c t i o n f a c t or o f 0.2 u se d in o n e M O S s t u d y t o l imi t c a r r i er s p r e a d in g, an d t h e hi g h es t - m o m e n t c l os u r e a s t h e main c a u se o f s p u r i o u s v e l oc i t y o v er s h oo t$ $(co m p u t e d : r = - 2.1 r e d u ces t h e p r e f a c t or 5/2 + r t o 0.4)$
Carrier-temperature-dependent mobility	$μ (T) = μ_{0} \frac{T _{L}}{T} (Hansch-type, homogeneous)$	$λ_{w} \approx v_{s} τ_{w} (energy relaxation length)$	$μ (T) : m o bi l i t y a s a f u n c t i o n o f e l ec t r o n t e m p er a t u r e$ $λ_{w} : d i s t an ce o v er w hi c h c a r r i er s r e l a x t h e i r e n er g y (f or m s, ba c k g r o u n d k n o w l e d g e); G r a sser e t a l . : t h e B a cc a r ani m o d e l p ai r e d w i t h an e n er g y r e l a x a t i o n m o d e l, an d t h e H an sc h m o d e l, w hi c h c a pt u r es t h e h y s t er es i s p r o p er l y b u t g i v es l a r g er er r or s ab o v e ab o u t 100 kV/cm; f i e l d - d e p e n d e n t m o bi l i t y f or d r i f t - d i f f u s i o n o n t h e T C A D M o bi l i t y M o d e l s p a g e$ $(co m p u t e d : v_{s} τ_{w} = 30 nm f or 1 \times 1 0^{5} m/s an d 0.3 ps, an d 43 nm f or 0.42 ps)$

Important Questions

Ques: Why does drift-diffusion fail in short-channel devices?

Because carriers cannot follow the field instantly. Drift-diffusion assumes the carrier gas is in equilibrium with the lattice, so mobility depends only on the local field. Grasser et al. note that transport becomes non-stationary below about 200 $nm$ and that drift-diffusion can only be fitted to short devices by raising the saturation velocity above twice its bulk value. Carriers relax their energy over a length of about $v_{s} τ_{w}$ , 30 $nm$ for $1 \times 1 0^{5}$ $m/s$ and 0.3 $ps$ (computed), so in a channel only a few such lengths long the field changes faster than the energy can follow. Carriers then move faster than the saturated velocity where the field rises, velocity overshoot, and remain hot after the field drops. Energy-transport and hydrodynamic models capture this by solving for the carrier energy. Reference: T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary).

Ques: What does the energy-balance equation add?

A carrier temperature. Aste and Vahldieck write the electron energy balance with field heating $- e n v \cdot E$ , pressure work, heat conduction $\nabla \cdot (κ \nabla T)$ and relaxation $(ω - \frac{3}{2} n k_{B} T_{L}) / τ_{w}$ , with the energy density $ω = \frac{3}{2} n k_{B} T + \frac{1}{2} n m^{*} v^{2}$ . In a uniform field the balance reduces to $T = T_{L} + 2 q τ_{w} μ E^{2} / (3 k_{B})$ (background knowledge), which reaches twice the lattice temperature at 9.6 $kV/cm$ for 0.3 $ps$ and 1400 $c m^{2} / (V s)$ (computed), consistent with the critical field of order 10 $kV/cm$ that Grasser et al. quote. Advanced devices reach fields above 1 MV/cm, where electrons are far hotter than the lattice, so mobility, diffusivity and impact ionization must be expressed through T rather than E. Reference: A. Aste and R. Vahldieck, ‘Time-Domain Simulation of the Full Hydrodynamic Model’, International Journal for Numerical Modelling 16, 161-174 (2003), arXiv:physics/0312021 (fetch-tool summary); T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary).

Ques: What energy relaxation time should be used for silicon?

A few tenths of a picosecond, though published fits disagree. Grasser et al. quote room-temperature silicon values of a few tenths of a picosecond, cite Agostinelli et al.’s fit to Fischetti’s data with a maximum near 0.68 $ps$ , which they consider too high, and Hasnat et al.’s near 0.42 $ps$ , and use 0.3 $ps$ in their own analysis. The Baccarani-Wordemann form used by Aste and Vahldieck gives 0.367 $ps$ at 300 $K$ and 0.434 $ps$ for 1000 $K$ electrons (computed). In a GaAs channel they find 0.3 to 1 $ps$ . The value matters less for current than for internal quantities: a 1 per cent change shifts drain current by only 0.01 to 0.17 per cent (Grasser et al.), so 0.3 to 0.4 $ps$ is the more likely range for silicon (my reading). Reference: T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary); A. Aste and R. Vahldieck, ‘Time-Domain Simulation of the Full Hydrodynamic Model’, International Journal for Numerical Modelling 16, 161-174 (2003), arXiv:physics/0312021 (fetch-tool summary).

Ques: Do the energy-balance and full hydrodynamic models agree?

For DC current, yes; for internal dynamics, no. Aste and Vahldieck simulated a 0.3 $μ m$ silicon n+ n n+ diode with $1 \times 1 0^{18}$ $c m^{- 3}$ contacts and a $2 \times 1 0^{15}$ $c m^{- 3}$ channel at 1 $V$ . The hydrodynamic electron flow became supersonic between 0.109 and 0.146 $μ m$ , forming a shock wave, with a peak velocity of 2.61 $v_{s}$ against 1.87 $v_{s}$ in the energy-balance model, 28 per cent lower (computed), and only about 30 per cent above saturation in drift-diffusion. Electron temperatures and current-voltage curves of the two energy models agreed closely. In a GaAs diode, the energy-balance current responded instantly to a 1 $ps$ pulse while the hydrodynamic current lagged. They conclude that energy balance gives reasonable static characteristics but wrong internal dynamics. Reference: A. Aste and R. Vahldieck, ‘Time-Domain Simulation of the Full Hydrodynamic Model’, International Journal for Numerical Modelling 16, 161-174 (2003), arXiv:physics/0312021 (fetch-tool summary).

Ques: What is spurious velocity overshoot?

An artefact of energy-transport models. Grasser et al. describe velocity peaks in Blotekjaer-based energy-transport simulations, for example at the drain end of a MOSFET channel or in a ballistic diode, where Monte Carlo simulations show none. They argue that it is not caused only by the mobility model but mainly by the closure of the highest moment, the heat flux, since Blotekjaer’s Fourier and Wiedemann-Franz closure is inconsistent with the underlying distribution. A six-moment model reduces the spurious peak, and a closure derived from Monte Carlo removes it. Spurious overshoot also makes energy-transport models overestimate carrier spreading into the substrate and drain current; one MOS study used a heat-flux reduction factor of 0.2 to suppress it. Reference: T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary).

Ques: How is the electron heat flux modelled?

By Fourier’s law with a Wiedemann-Franz conductivity. Aste and Vahldieck use $κ = (\frac{5}{2} + r) n k_{B}^{2} μ (T) T / e$ with $r = - 2.1$ for silicon at 300 $K$ , so the prefactor is only 0.4 (computed), and the heat flux is $- κ \nabla T$ . Grasser et al. call this closure a major inconsistency of Blotekjaer’s macroscopic relaxation-time models, because it is not derived from the same distribution function as the lower moments, and they note that three-moment models allow the heat-flux factors to be adjusted independently, which makes results depend on tuning. Reducing the heat flux limits spurious carrier spreading but is a fit, not physics (my reading). Higher-moment models replace the closure by an extra equation, and about 12 moments were needed to match spherical-harmonics solutions in one study. Reference: A. Aste and R. Vahldieck, ‘Time-Domain Simulation of the Full Hydrodynamic Model’, International Journal for Numerical Modelling 16, 161-174 (2003), arXiv:physics/0312021 (fetch-tool summary); T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary).

Ques: When is Monte Carlo simulation needed instead of moment models?

When the energy distribution is far from a displaced Maxwellian. Grasser et al. summarize Monte Carlo comparisons: energy-transport models overestimate carrier spreading into MOS bulk and drain current, Banoo and Lundstrom found energy transport dramatically overestimating current and velocity in short channels, Tomizawa et al. found relaxation-time models overestimating non-stationary dynamics, and Nekovee et al. found poor ballistic-diode predictions, though Grasser et al. attribute that to the closure used rather than to the moment method itself. Near the channel start, drift energy can reach about 30 per cent of the total, and partition assumptions cause about 15 per cent differences in submicrometre MOSFETs. Monte Carlo is therefore the reference for hot-carrier injection, impact ionization tails and quasi-ballistic transport, with moment models calibrated against it (background knowledge for the calibration role). Reference: T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, ‘A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation’, Proceedings of the IEEE 91(2), 251-272 (2003) (equation bodies blank in the extract) (fetch-tool summary).

References

T. Grasser, T.-W. Tang, H. Kosina and S. Selberherr, “A Review of Hydrodynamic and Energy-Transport Models for Semiconductor Device Simulation,” Proceedings of the IEEE, vol. 91, no. 2, pp. 251-272, Feb. 2003. Available: https://www.iue.tuwien.ac.at/pdf/ib_2003/JB2003_Grasser_1.pdf
A. Aste and R. Vahldieck, “Time-Domain Simulation of the Full Hydrodynamic Model,” International Journal for Numerical Modelling, vol. 16, pp. 161-174, 2003, arXiv:physics/0312021. Available: https://arxiv.org/abs/physics/0312021

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Hydrodynamic and Energy-Balance Transport