1. Theory

Definition: Electro-thermal TCAD solves a lattice heat equation together with the drift-diffusion system, so that device temperature follows the heat generated by current and recombination and feeds back into mobility, recombination and band structure (Kantner; Wagner; El Boukili).
Heat sources: Joule heating, recombination heating and the reversible Thomson and Peltier terms, the last arising from thermoelectric coupling between current and temperature gradients (Kantner; Wagner).
Thermal transport: Heat flows by lattice conduction with a temperature-dependent conductivity, about 1.3 to 1.5 $W/ (cm K)$ for silicon at room temperature and falling roughly as $T^{- 4/3}$ (Ioffe; Electronics Cooling).
Thermodynamic consistency: Using the Kelvin formula for the Seebeck coefficient makes the heat source consistent with the transport equations and removes spurious transient terms (Kantner).

2. Working Principle

Coupling loop: Current heats the lattice, temperature lowers mobility and raises leakage, which changes current, until a self-consistent solution is reached (Kantner; Temperature Scaling page).
Thermoelectric currents: Temperature gradients drive currents through the Seebeck term $P \nabla T$ , and currents carry heat through the Peltier coefficient (Kantner).
Boundaries set temperature: Heat sinks fix temperature and other surfaces are usually adiabatic, so the thermal path to the sink determines the rise (El Boukili; Kantner).
Time scales: Heat diffuses over micrometres in microseconds, far slower than electrical transients, so thermal and electrical time scales separate (Ioffe; computed).

3. Applications

Power devices and ESD: Self-heating, hot spots and thermal runaway (background knowledge; Electronics Cooling).
3-D integrated circuits: Temperature rise in stacked dies with longer heat paths (El Boukili).
Optoelectronics: Self-heating in lasers and LEDs with degenerate statistics (Kantner).
Thermoelectric devices: Peltier coolers and thermoelectric generators (Wagner).

Heat Equation and Sources

Name	Formula	Related Formulae	Variables/Constants
Lattice heat equation	$c_{V} \frac{\partial T}{\partial t} - \nabla \cdot (κ \nabla T) = H$	$\nabla \cdot (κ (T_{L}) \nabla T_{L}) + H (ϕ, ϕ_{n}, ϕ_{p}, T_{L}) = 0 (steady state)$	$c_{V} : v o l u m e t r i c h e a t c a p a c i t y$ $T, T_{L} : l a tt i ce t e m p er a t u r e$ $κ : t h er ma l co n d u c t i v i t y$ $H : h e a t g e n er a t i o n r a t e$ $ϕ, ϕ_{n}, ϕ_{p} : e l ec t r os t a t i c an d q u a s i - F er mi p o t e n t ia l s; K an t n er : t hi s h e a t t r an s p or t e q u a t i o n co u pl e d t o t h e d r i f t - d i f f u s i o n sy s t e m, w i t h e d g e co n d u c t i v i t i es a s ha r m o ni c a v er a g es o f n o d a l v a l u es; E l B o u k i l i : t h e s t e a d y f or m w i t h H f r o m J o u l e, r eco mbina t i o n an d T h o m so n an d P e l t i er h e a t (o pt i c a l ab sor pt i o n o mi tt e d); W a g n er : a t o t a l h e a t - f l o w e q u a t i o n f r o m e l ec t r o n, h o l e an d l a tt i ce co n t r ib u t i o n s w i t h a F o u r i er - l a w l a tt i ce f l ux, c a r r i er t h er ma l co n d u c t i v i t i es n e g l i g ib l e a g ain s t t h e l a tt i ce in n o n - d e g e n er a t e ma t er ia l$ $(n o co m p u t e d v a l u e)$
Heat generation terms	$H = \frac{∣ j _{n} ∣ ^{2}}{σ _{n}} + \frac{∣ j _{p} ∣ ^{2}}{σ _{p}} - T j_{n} \cdot \nabla P_{n} - T j_{p} \cdot \nabla P_{p} + q (φ_{p} - φ_{n} + Π_{p} - Π_{n}) R$	$j_{n} = - σ_{n} (\nabla φ_{n} + P_{n} \nabla T), P_{n} = - \frac{1}{q} \frac{\partial s}{\partial n}, Π = T P$	$j_{n}, j_{p} : c u r r e n t d e n s i t i es$ $σ_{n}, σ_{p} : co n d u c t i v i t i es$ $P_{n}, P_{p} : S ee b ec k coe f f i c i e n t s$ $s : e n t r o p y d e n s i t y$ $Π : P e l t i er coe f f i c i e n t s (Π = T P, ba c k g r o u n d k n o w l e d g e)$ $R : n e t r eco mbina t i o n r a t e; K an t n er : J o u l e, T h o m so n - P e l t i er an d r eco mbina t i o n h e a t in g w i t h t h e K e l v in f or m u l a f or t h e S ee b ec k coe f f i c i e n t, w hi c h r e m o v es t r an s i e n t co n t r ib u t i o n s, c u r r e n t s in t h er m o d y nami c f or m, an d a g e n er a l i z e d E in s t e in r e l a t i o n D_{n} = (k_{B} T M_{n} / q) g (n / N_{c} (T)) f or F er mi - D i r a c s t a t i s t i cs; W a g n er : J o u l e h e a t, r eco mbina t i o n h e a t t r an s f er r e d t o t h e l a tt i ce an d T h o m so n h e a t a s t h e s t e a d y - s t a t e co n t r ib u t i o n s$ $(co m p u t e d : 1 \times 1 0^{6} A/c m^{2} in ma t er ia l o f 100 S/cm g e n er a t es 1 \times 1 0^{10} W/c m^{3} o f J o u l e h e a t; e a c h r eco mbinin g p ai r r e l e a ses r o ug h l y t h e ban d g a p, ab o u t 1.12 eV in s i l i co n (ba c k g r o u n d k n o w l e d g e))$

Thermal Properties and Boundaries

Name	Formula	Related Formulae	Variables/Constants
Thermal conductivity of silicon	$κ (T) = 1.5 (\frac{T}{300})^{- 4/3} W/ (cm K)$	$κ (T) = \frac{1}{a + b T + c T ^{2}}, κ_{d o p e d} \approx 0.8 κ_{p u r e}$	$κ : l a tt i ce t h er ma l co n d u c t i v i t y, a, b$ $c : ma t er ia l co n s t an t s; E l ec t r o ni cs C oo l in g : han d b oo k v a l u es v a r y in g w i d e l y f r o m 68.8 t o 157 W/ (m K), tw o d a t a se t s a g r ee in g w i t hin 5 p er ce n t, t h e f i t 1.5 (T /300)^{- 4/3}, ab o u t 80 p er ce n t o f t h e p u r e v a l u e f or hi g h l y d o p e d s i l i co n, an d t e m p er a t u r e - d e p e n d e n t v a l u es n ee d e d f or e l ec t r o - t h er ma l d e v i ce m o d e l l in g; E l B o u k i l i : t h e in v er se - q u a d r a t i c f or m (co n s t an t s n o t l i s t e d); I o f f e : 1.3 W c m^{- 1} p er d e g r ee, p r a c t i c a l l y in d e p e n d e n t o f d o p in g ab o v e 100 K (see t h e Q / A)$ $(co m p u t e d : t h e f i t g i v es 1.50, 1.22, 1.02 an d 0.87 W/ (cm K) a t 300, 350, 400 an d 450 K)$
Heat capacity and thermal diffusivity	$D_{t h} = \frac{κ}{ρ c _{p}}$	$τ_{t h} \approx \frac{L ^{2}}{D _{t h}}$	$D_{t h} : t h er ma l d i f f u s i v i t y$ $ρ : d e n s i t y (2.329 g / c m^{3}, ba c k g r o u n d k n o w l e d g e)$ $c_{p} : s p ec i f i c h e a t$ $τ_{t h} : d i f f u s i o n t im e o v er l e n g t h L; I o f f e : s p ec i f i c h e a t 0.7 J / (g K), t h er ma l d i f f u s i v i t y 0.8 c m^{2} /s, m e l t in g p o in t 1412^{\circ} C (1685 K) an d l in e a r e x p an s i o n 2.6 \times 1 0^{- 6} p er d e g r ee$ $(co m p u t e d : κ = 1.3 W/ (cm K) w i t h ρ = 2.329 an d c_{p} = 0.7 g i v es 0.80 c m^{2} /s, ma t c hin g I o f f e^{'} s d i f f u s i v i t y, w hi l e 1.5 W/ (cm K) w o u l d g i v e 0.92; τ_{t h} = 12.5 ns, 1.25 μ s an d 125 μ s f or 1, 10 an d 100 μ m)$
Thermal boundary conditions	$T = T_{s ink} (heat sink, Dirichlet), κ \frac{\partial T}{\partial n} = 0 (adiabatic, Neumann)$	$- κ \frac{\partial T}{\partial n} = \frac{T - T _{e x t}}{R _{t h} A} (thermal resistance), R_{s p} \approx \frac{1}{4 κa} (disk source)$	$T_{s ink} : h e a t - s ink t e m p er a t u r e$ $T_{e x t} : e x t er na l t e m p er a t u r e$ $R_{t h} A : a r e a - s p ec i f i c t h er ma l r es i s t an ce$ $R_{s p} : s p r e a d in g r es i s t an ce o f a d i s k o f r a d i u s a o n a ha l f - s p a ce (t h er ma l - r es i s t an ce an d s p r e a d in g f or m s, ba c k g r o u n d k n o w l e d g e); K an t n er : i d e a l h e a t s ink s a t 300 K a t b o t h co n t a c t s; E l B o u k i l i : D i r i c h l e t h e a t s ink s a t t o p an d b o tt o m, h o m o g e n eo u s N e u mann e l se w h er e, an d v a r ian t s w i t h o n e s ink an d v a r y in g co n t a c t - w i r e co n d u c t an ce$ $(co m p u t e d : a 1 μ m r a d i u s h o t s p o t o n s i l i co n w i t h 150 W/ (m K) ha s R_{s p} = 1670 K/W, so 1 mW r ai ses i t b y 1.7 K)$

Important Questions

Ques: What does electro-thermal TCAD add to the device equations?

A heat equation for the lattice temperature, coupled both ways to the electrical equations. Kantner writes $c_{V} \partial T / \partial t - \nabla \cdot κ \nabla T = H$ , with the heat source built from the electrical solution, and El Boukili uses its steady form with a temperature-dependent conductivity. Wagner describes the total heat flow as the sum of electron, hole and lattice contributions, with the carrier contributions negligible against the lattice in non-degenerate semiconductors, so the lattice equation suffices. Temperature then enters every model: mobility falls, intrinsic density and leakage rise, and the band gap narrows (Temperature Scaling page). Kantner’s 1-D GaAs diode shows the effect on accuracy: with self-heating, both discretization schemes underestimated the total current by about 8 per cent on a coarse grid. Reference: M. Kantner, ‘Generalized Scharfetter-Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient’, arXiv:1911.00377 (2019) (fetch-tool summary); A. El Boukili, ‘Physical Modeling and Simulation of Thermal Heating in Vertical Integrated Circuits’, arXiv:1305.3249 (year not stated in the text) (fetch-tool summary); M. Wagner, ‘Simulation of Thermoelectric Devices’, doctoral dissertation, TU Wien, Section 3.5.10 ‘Electrothermal Transport Model’ (year not stated; equations truncated in the extract) (fetch-tool summary); Temperature Scaling page of this vault.

Ques: Which heat sources appear, and which dominate?

Joule heating usually dominates in conductive regions, recombination heating at junctions under forward bias. Kantner’s heat source is $∣ j_{n} ∣^{2} / σ_{n} + ∣ j_{p} ∣^{2} / σ_{p}$ for Joule heat, $- T j \cdot \nabla P$ for the Thomson-Peltier terms, and $q (φ_{p} - φ_{n} + Π_{p} - Π_{n}) R$ for recombination heat; Wagner names the same three. A current density of $1 \times 1 0^{6}$ $A/c m^{2}$ in 100 $S/cm$ material generates $1 \times 1 0^{10}$ $W/c m^{3}$ of Joule heat (computed), and each recombining electron-hole pair releases roughly the band-gap energy (background knowledge). The Peltier terms are reversible: they heat or cool depending on current direction, which thermoelectric coolers exploit, but in most power and logic devices they are small compared with Joule heat (background knowledge). Reference: M. Kantner, ‘Generalized Scharfetter-Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient’, arXiv:1911.00377 (2019) (fetch-tool summary); M. Wagner, ‘Simulation of Thermoelectric Devices’, doctoral dissertation, TU Wien, Section 3.5.10 ‘Electrothermal Transport Model’ (year not stated; equations truncated in the extract) (fetch-tool summary).

Ques: What is the thermal conductivity of silicon at 300 K?

About 1.3 to 1.5 $W/ (cm K)$ , depending on the source. Ioffe gives 1.3 W cm^-1 per degree, while the Electronics Cooling fit, drawn between two data sets that agree within 5 per cent, gives 1.5 $W/ (cm K)$ at 300 $K$ ; the same article notes that handbook values range from 68.8 to 157 $W/ (m K)$ , often without stating the temperature. A consistency check favours 1.5 only weakly: Ioffe’s own specific heat of 0.7 J/(g K) and diffusivity of 0.8 $c m^{2} /s$ with a density of 2.329 grams per cubic centimetre imply 1.30 $W/ (cm K)$ (computed), so Ioffe’s values are internally consistent. Measurements on pure silicon near room temperature are usually quoted at 1.48 to 1.56 $W/ (cm K)$ (background knowledge), so 1.5 is more likely for pure material and 1.3 suits doped or slightly warmer silicon (my reading). Reference: Ioffe Physico-Technical Institute, ‘Si - Silicon: Thermal properties’, NSM semiconductor database (fetch-tool summary); Electronics Cooling editors, ‘The Thermal Conductivity of Silicon’, Electronics Cooling (1 May 1998) (fetch-tool summary).

Ques: How fast does silicon thermal conductivity fall with temperature?

Roughly as $T^{- 4/3}$ . The Electronics Cooling fit $1.5 (T /300)^{- 4/3}$ $W/ (cm K)$ gives 1.22 at 350 $K$ , 1.02 at 400 $K$ and 0.87 $W/ (cm K)$ at 450 $K$ (computed), a 42 per cent fall over 150 $K$ . El Boukili uses the inverse-quadratic form $1/ (a + b T + c T^{2})$ common in TCAD. The fall comes from stronger phonon-phonon scattering (background knowledge). It creates positive feedback: a hot spot conducts heat away less well, so it heats further, which is why the article stresses temperature-dependent conductivity for electro-thermal device modelling and transient thermal impedance. Heavy doping lowers conductivity further, to about 80 per cent of the pure value, while Ioffe notes that above 100 $K$ doping has little effect. For a single steady-state number, the article advises evaluating the fit at the estimated average silicon temperature. Reference: Electronics Cooling editors, ‘The Thermal Conductivity of Silicon’, Electronics Cooling (1 May 1998) (fetch-tool summary); A. El Boukili, ‘Physical Modeling and Simulation of Thermal Heating in Vertical Integrated Circuits’, arXiv:1305.3249 (year not stated in the text) (fetch-tool summary); Ioffe Physico-Technical Institute, ‘Si - Silicon: Thermal properties’, NSM semiconductor database (fetch-tool summary).

Ques: How long does heat take to spread in silicon?

Nanoseconds over a micrometre, microseconds over ten. With Ioffe’s thermal diffusivity of 0.8 $c m^{2} /s$ , the diffusion time $L^{2} / D_{t h}$ is 12.5 $ns$ for 1 $μ m$ , 1.25 $μ s$ for 10 $μ m$ and 125 $μ s$ for 100 $μ m$ (computed). The device region therefore heats within nanoseconds of a current pulse, while the die and package respond over microseconds to milliseconds, which is why ESD and switching events are often nearly adiabatic and why transient thermal impedance grows with pulse length (background knowledge). These times are far longer than electrical ones, the dielectric relaxation time of about 0.1 ps on the Drift-Diffusion Device Equations page, so electro-thermal transients need implicit time integration over widely separated scales (background knowledge). Reference: Ioffe Physico-Technical Institute, ‘Si - Silicon: Thermal properties’, NSM semiconductor database (fetch-tool summary).

Ques: How much does self-heating matter in stacked 3-D integrated circuits?

Significantly, because upper dies lengthen the heat path. El Boukili simulates a two-die CMOS stack with 10 $μ m$ active layers and 4 $V$ on the gate and drain of the bottom transistor, with heat sinks at top and bottom and adiabatic sides, and finds the bottom layer, farthest from the sink, 17 $K$ hotter than a single-layer reference. He compares this with Puttaswamy and Loh’s processor study, where a planar IC peaked at 312 $K$ and a two-die 3-D IC at 328 $K$ , a 16 $K$ rise, while noting that the structures differ. Thicker layers raised the temperature further, and removing one heat sink or lowering wire conductance worsened it. A local hot spot also has its own spreading resistance: 1670 $K/W$ for a 1 $μ m$ radius on silicon, 1.7 $K$ per milliwatt (computed). Reference: A. El Boukili, ‘Physical Modeling and Simulation of Thermal Heating in Vertical Integrated Circuits’, arXiv:1305.3249 (year not stated in the text) (fetch-tool summary).

Ques: Why do thermoelectric terms need special discretization?

Because the Seebeck term changes the effective driving force inside the Scharfetter-Gummel flux. Kantner writes the currents as $- σ (\nabla φ + P \nabla T)$ with the Kelvin-formula Seebeck coefficient from the entropy density, and absorbs the thermal driving force into a generalized Einstein relation for Fermi-Dirac statistics. He compares two generalized SG schemes on a 1-D GaAs diode with $2 \times 1 0^{18}$ $c m^{- 3}$ doping: on a 13-node grid at 2 $V$ , the errors are about 13 and 15 per cent isothermally, and with self-heating both underestimate current by about 8 per cent; on fine grids the modified thermal-voltage scheme is an order of magnitude more accurate, equivalent to a four times coarser mesh. A naive discretization can violate equilibrium in a temperature gradient, the thermal analogue of the problem on the Discretization and Numerical Solution page (my reading). Reference: M. Kantner, ‘Generalized Scharfetter-Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient’, arXiv:1911.00377 (2019) (fetch-tool summary); Discretization and Numerical Solution page of this vault.

References

M. Kantner, “Generalized Scharfetter-Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient,” arXiv:1911.00377, 2019. Available: https://arxiv.org/abs/1911.00377
A. El Boukili, “Physical Modeling and Simulation of Thermal Heating in Vertical Integrated Circuits,” arXiv:1305.3249. Available: https://arxiv.org/abs/1305.3249
M. Wagner, “Simulation of Thermoelectric Devices,” doctoral dissertation, Institute for Microelectronics, TU Wien, Section 3.5.10 “Electrothermal Transport Model.” Available: https://iue.tuwien.ac.at/phd/mwagner/node45.html
Ioffe Physico-Technical Institute, “Si - Silicon: Thermal properties,” NSM Archive, Physical Properties of Semiconductors. Available: https://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html
Electronics Cooling editors, “The Thermal Conductivity of Silicon,” Electronics Cooling, May 1998. Available: https://electronics-cooling.com/1998/05/the-thermal-conductivity-of-silicon

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