1. Theory

Definition: Temperature scaling makes compact-model parameters functions of device temperature, so one model card covers the operating range, typically -40 to 125 $^{\circ} C$ or wider (BSIM-BULK 106.2.0; background knowledge for the range).
Physical drivers: The band gap narrows and the intrinsic density rises steeply with temperature, phonon scattering lowers mobility and saturation velocity, and resistances change with their own coefficients (Ioffe; LIGO notes; BSIM-BULK 106.2.0).
Model forms: Compact models use simple laws referenced to a nominal temperature TNOM: linear or power-law shifts of threshold, power laws for mobility, and Arrhenius-type laws for saturation currents (BSIM-BULK 106.2.0; Diode Compact Models page).
Self-heating: The same equations are evaluated at the device temperature including self-heating, which is why temperature and thermal models are coupled (ASM-HEMT page; background knowledge).

2. Working Principle

Threshold voltage: A higher temperature moves the Fermi level toward midgap and lowers the threshold, captured by KT1 and its body-bias term KT2 (BSIM-BULK 106.2.0; background knowledge for the mechanism).
Mobility: More lattice vibration means more scattering, so $U 0 (T / T_{N O M})^{U T E}$ with negative UTE (BSIM-BULK 106.2.0).
Competing effects: Falling threshold raises current and falling mobility lowers it, so a bias point exists where they cancel (background knowledge; computed).
Leakage: Junction and subthreshold leakage scale with $n_{i}$ or $n_{i}^{2}$ and grow by orders of magnitude over the range (Ioffe; computed).

3. Applications

Corner and temperature sign-off: Simulating circuits at -40, 25 and 125 $^{\circ} C$ with one card (background knowledge; BSIM-BULK 106.2.0).
Bias design: Choosing the ZTC overdrive for temperature-stable references and amplifiers (computed; background knowledge).
Leakage and standby power: Hot-chip leakage estimates (Ioffe; computed).
Extraction: Temperature parameters fitted after the room-temperature model, as in the extraction sequence on the Parameter Extraction Methods page (background knowledge).

Band Gap and Intrinsic Density

Name	Formula	Related Formulae	Variables/Constants
Band gap of silicon (Varshni form)	$E_{g} (T) = E_{g 0} - \frac{α T ^{2}}{T + β}$	$E_{g} (T) = 1.16 - \frac{7.02 \times 1 0 ^{- 4} T ^{2}}{T + 1108} (SPICE)$	$E_{g} : ban d g a p (e V)$ $E_{g 0} : v a l u e a t 0 K$ $α, β : V a r s hni co n s t an t s$ $T : t e m p er a t u r e (K); I o f f e : E_{g} = 1.17 - 4.73 \times 1 0^{- 4} T^{2} / (T + 636) an d 1.12 eV a t 300 K; L I GO n o t es : E_{g 0} = 1.166 eV, a = 4.73 \times 1 0^{- 4} eV/K an d b = 636 K; S P I C E f or m o n t h e D i o d e C o m p a c t M o d e l s p a g e (see t h e Q / A)$ $(co m p u t e d : a t 300 K t h e t h r ee f or m s g i v e 1.1245, 1.1205 an d 1.1151 eV; t h e I o f f e f or m f a l l s b y 0.25 m e V p er k e l v in a t 300 K an d r e a c h es 1.0969 eV a t 400 K)$
Effective densities of states and intrinsic density	$n_{i} = N_{c} N_{v} exp (- \frac{E _{g}}{2 k _{B} T})$	$N_{c} = 6.2 \times 1 0^{15} T^{3/2} c m^{- 3}, N_{v} = 3.5 \times 1 0^{15} T^{3/2} c m^{- 3}$	$n_{i} : in t r in s i c c a r r i er d e n s i t y$ $N_{c}, N_{v} : e f f ec t i v e d e n s i t i es o f s t a t es in t h e co n d u c t i o n an d v a l e n ce ban d s$ $k_{B} : B o l t z mann co n s t an t; I o f f e : t h ese f or m s w i t h M = 6 e q u i v a l e n t v a l l ey s an d n_{i} = 1 \times 1 0^{10} c m^{- 3} a t 300 K; L I GO n o t es : t h e s am e n_{i} f or m w i t h m_{e}^{} = 1.08 m_{0} an d m_{p}^{} = 0.81 m_{0}, g i v in g 8.86 \times 1 0^{9} c m^{- 3} a t 300 K an d n o t in g t ha t t h e T^{3/2} p r e f a c t or ma tt er s l i ttl e b ec a u se i t i s o u t s i d e t h e e x p o n e n t ia l$ $(co m p u t e d : I o f f e^{'} s o w n N_{c}, N_{v} an d E_{g} g i v e 8.68 \times 1 0^{9} c m^{- 3} a t 300 K, 1.93 \times 1 0^{10} a t 310 K an d 4.6 \times 1 0^{12} a t 400 K; n_{i} d o u b l es e v er y 8.6 K n e a r r oo m t e m p er a t u r e)$

MOSFET Temperature Parameters

Name	Formula	Related Formulae	Variables/Constants
Threshold and flat-band voltage	$V_{t h} (T) = V_{t h} (T_{N O M}) + (K T 1 + K T 2 V_{b se f f}) [(\frac{T}{T _{N O M}})^{K T 1 E X P} - 1]$	$V_{f b} (T) = V_{f b} (T_{N O M}) - K T 1 (\frac{T}{T _{N O M}} - 1), N F A C T O R (T) = N F A C T O R + T F A C T O R (\frac{T}{T _{N O M}} - 1)$	$V_{t h} : t h r es h o l d v o l t a g e$ $T_{N O M} : n o mina l t e m p er a t u r e, K T 1 : t h r es h o l d t e m p er a t u r e coe f f i c i e n t, K T 2 : i t s b o d y - bia s d e p e n d e n ce$ $V_{b se f f} : e f f ec t i v e b o d y bia s (V_{b r e f f} a s e x t r a c t e d, r e a d a s V_{b se f f}), K T 1 E X P : e x p o n e n t$ $V_{f b} : f l a t - ban d v o l t a g e$ $N F A C T O R : s u b t h r es h o l d f a c t or$ $T F A C T O R : i t s coe f f i c i e n t; B S I M - B U L K 106.2.0 : t h ese e q u a t i o n s, E T A 0 (T) = E T A 0 + T E T A 0 (T / T_{N O M} - 1), an d l e n g t h sc a l in g o f U T E, U A 1, U D 1 an d A T t h r o ug h U T E L, U A 1 L, U D 1 L an d A T L$ $(co m p u t e d : an a ss u m e d K T 1 = - 0.11 V w i t h K T 1 E X P = 1 l o w er s V_{t h} b y 36.7 mV f r o m 300 t o 400 K, - 0.37 mV/K)$
Mobility	$U 0 (T) = U 0 (T_{N O M}) (\frac{T}{T _{N O M}})^{U T E}$	$U A (T) = U A [1 + U A 1 (T - T_{N O M})], U C (T) = U C [1 + U C 1 (T - T_{N O M})], U D (T) = U D (\frac{T}{T _{N O M}})^{U D 1}$	$U 0 : l o w - f i e l d m o bi l i t y$ $U T E : m o bi l i t y t e m p er a t u r e e x p o n e n t$ $U A, U C, U D : m o bi l i t y - d e g r a d a t i o n coe f f i c i e n t s, U A 1, U C 1, U D 1 : t h e i r t e m p er a t u r e coe f f i c i e n t s; B S I M - B U L K 106.2.0 : t h ese e q u a t i o n s an d U C S (T) = U C S (T / T_{N O M})^{U C S T E}; A S M - H E M T u ses t h e s am e p o w er l a w f or U 0 (A S M - H E M T p a g e)$ $(co m p u t e d : U T E = - 1.5 (a ss u m e d, p h o n o n - l imi t e d) l o w er s U 0 b y 35 p er ce n t f r o m 300 t o 400 K)$
Saturation velocity and series resistance	$V S A T (T) = V S A T (T_{N O M}) (\frac{T}{T _{N O M}})^{- A T}$	$R D S W (T) = R D S W (T_{N O M}) (\frac{T}{T _{N O M}})^{P R T}$	$V S A T : s a t u r a t i o n v e l oc i t y$ $A T : i t s t e m p er a t u r e e x p o n e n t in B S I M - B U L K$ $R D S W : so u r ce - d r ain r es i s t an ce p er w i d t h$ $P R T : i t s e x p o n e n t (a l so a ppl i e d t o R D S W M I N, R D W, R S W an d t h e i r minima f or R D S M O D = 1); B S I M - B U L K 106.2.0 : t h ese p o w er l a w s (B S I M 4 u ses l in e a r f or m s V S A T - A T (T / T_{N O M} - 1) an d R D S W + P R T (T / T_{N O M} - 1), ba c k g r o u n d k n o w l e d g e; see t h e Q / A)$ $(co m p u t e d : P R T = 1 (a ss u m e d) r ai ses R D S W b y 33 p er ce n t f r o m 300 t o 400 K)$

Bias Points and Junctions

Name	Formula	Related Formulae	Variables/Constants
Zero-temperature-coefficient bias point	$\frac{\partial I _{D}}{\partial T} = 0 \Rightarrow (V_{GS} - V_{t h})_{Z T C} = \frac{2 T _{N O M}}{∣ U T E ∣} \frac{d V _{t h}}{d T} (square law)$	$I_{D} \propto μ (T) (V_{GS} - V_{t h} (T))^{2}$	$(V_{GS} - V_{t h})_{Z T C} : o v er d r i v e a t w hi c h t h e d r ain c u r r e n t i s t e m p er a t u r e in d e p e n d e n t, d er i v e d f r o m t h e s q u a r e l a w w i t h a p o w er - l a w m o bi l i t y an d a l in e a r t h r es h o l d s hi f t (ba c k g r o u n d k n o w l e d g e); b e l o w i t t h e f a l l in g t h r es h o l d d o mina t es an d c u r r e n t r i ses w i t h t e m p er a t u r e, ab o v e i t t h e f a l l in g m o bi l i t y d o mina t es an d c u r r e n t f a l l s$ $(co m p u t e d : U T E = - 1.5 w i t h 1 mV/K g i v es a Z T C o v er d r i v e o f 0.40 V; w i t h 0.37 mV/K i t i s 0.15 V)$
Junction and bipolar saturation currents	$I S (T) = I S (\frac{T}{T _{0}})^{X T I / N} exp [(\frac{T}{T _{0}} - 1) \frac{q E _{g}}{N k _{B} T}]$	$I S \propto n_{i}^{2} \propto T^{3} exp (- \frac{E _{g}}{k _{B} T})$	$I S : s a t u r a t i o n c u r r e n t$ $X T I : t e m p er a t u r e e x p o n e n t$ $N : e mi ss i o n coe f f i c i e n t$ $E_{g} : a c t i v a t i o n e n er g y$ $T_{0} : m e a s u r e m e n t t e m p er a t u r e (p r o p or t i o na l i t y t o n_{i}^{2}, ba c k g r o u n d k n o w l e d g e); f or m s an d e x am pl es o n t h e D i o d e C o m p a c t M o d e l s an d E b er s - M o l l an d G u mm e l - P oo n M o d e l s p a g es$ $(co m p u t e d : n_{i}^{2} r i ses b y a f a c t or 4.9 f or a 10 K r i se f r o m 300 K, c l ose t o t h e f a c t or 4.4 o f t h e S P I C E d i o d e l a w w i t h E G = 1.11 eV o n t h e d i o d e p a g e)$

Important Questions

Ques: Which band-gap formula for silicon should be used?

They differ only slightly. Ioffe gives $E_{g} = 1.17 - 4.73 \times 1 0^{- 4} T^{2} / (T + 636)$ $eV$ and quotes 1.12 $eV$ at 300 $K$ ; the LIGO notes use the same constants with $E_{g 0} = 1.166$ $eV$ ; SPICE uses $1.16 - 7.02 \times 1 0^{- 4} T^{2} / (T + 1108)$ . At 300 $K$ these give 1.1245, 1.1205 and 1.1151 $eV$ (computed), a spread of 9 meV. The Ioffe and LIGO forms are the commonly cited Varshni fit for silicon (background knowledge), and their 300 $K$ value matches the quoted 1.12 $eV$ best, so they are more likely accurate as physics; the SPICE form is kept for compatibility, because model cards are extracted with it. A 9 meV change in $E_{g}$ alters $n_{i}$ by about 20 per cent at 300 $K$ (computed). Reference: Ioffe Physico-Technical Institute, ‘Si - Silicon: Band structure and carrier concentration’, NSM semiconductor database (fetch-tool summary); ‘Carrier density of intrinsic semiconductors’, LIGO document T1500619 (calculation notes; author and date not stated) (fetch-tool summary); Diode Compact Models page of this vault.

Ques: Is the intrinsic carrier density of silicon at 300 K $1 \times 1 0^{10}$ or below $9 \times 1 0^{9}$ ?

The best modern value is just below $1 \times 1 0^{10}$ . Ioffe quotes $1 \times 1 0^{10}$ $c m^{- 3}$ , but its own formulas, $N_{c} = 6.2 \times 1 0^{15} T^{3/2}$ , $N_{v} = 3.5 \times 1 0^{15} T^{3/2}$ and its band gap, give $8.68 \times 1 0^{9}$ $c m^{- 3}$ (computed). The LIGO notes, with effective masses of 1.08 and 0.81 free-electron masses, obtain $8.86 \times 1 0^{9}$ $c m^{- 3}$ . The widely used older value of $1.45 \times 1 0^{10}$ $c m^{- 3}$ came from early measurements and is now considered too high, and the accepted modern value, from Altermatt and others, is about $9.65 \times 1 0^{9}$ $c m^{- 3}$ (background knowledge). The calculated values therefore bracket the modern value from below, and $1 \times 1 0^{10}$ is a rounded figure. Because currents scale as $n_{i}^{2}$ , using $1.45 \times 1 0^{10}$ instead of $9.65 \times 1 0^{9}$ overstates diffusion leakage by 2.3 times (computed). Reference: Ioffe Physico-Technical Institute, ‘Si - Silicon: Band structure and carrier concentration’, NSM semiconductor database (fetch-tool summary); ‘Carrier density of intrinsic semiconductors’, LIGO document T1500619 (calculation notes; author and date not stated) (fetch-tool summary).

Ques: How fast does leakage rise with temperature?

Very fast, because $n_{i}$ is exponential in $E_{g} / (2 k_{B} T)$ . With Ioffe’s band gap and densities of states, $n_{i}$ rises from $8.68 \times 1 0^{9}$ $c m^{- 3}$ at 300 $K$ to $1.93 \times 1 0^{10}$ at 310 $K$ and $4.6 \times 1 0^{12}$ at 400 $K$ (computed), doubling every 8.6 $K$ . Diffusion leakage scales as $n_{i}^{2}$ , a factor 4.9 for 10 $K$ (computed), while generation leakage in depletion regions scales as $n_{i}$ (background knowledge). The SPICE diode law with XTI = 3 and EG = 1.11 $eV$ gives a factor 4.4 over the same 10 $K$ (Diode Compact Models page). From 25 to 125 $^{\circ} C$ , diffusion leakage therefore rises by several orders of magnitude, which is why standby power is specified hot and why junction temperature limits power devices. Reference: Ioffe Physico-Technical Institute, ‘Si - Silicon: Band structure and carrier concentration’, NSM semiconductor database (fetch-tool summary); Diode Compact Models page of this vault.

Ques: How does BSIM model the threshold-voltage shift with temperature?

With a linear coefficient relative to TNOM. BSIM-BULK writes $V_{t h} (T) = V_{t h} (T_{N O M}) + (K T 1 + K T 2 V_{b se f f}) [(T / T_{N O M})^{K T 1 E X P} - 1]$ , with the flat-band voltage shifted by $- K T 1 (T / T_{N O M} - 1)$ and temperature terms for the subthreshold factor and DIBL. With an assumed KT1 of -0.11 $V$ and KT1EXP = 1, the threshold falls by 36.7 $mV$ from 300 to 400 $K$ , -0.37 $mV/K$ (computed), a typical magnitude. The shift comes mainly from the Fermi potential, which moves toward midgap as $n_{i}$ rises (background knowledge). KT2 lets the coefficient depend on body bias, and KT1EXP allows curvature over wide ranges. The extract writes the body-bias variable as $V_{b r e f f}$ , read here as $V_{b se f f}$ . In extraction, KT1 is fitted from linear-region curves of a large device at several temperatures, after the room-temperature parameters are fixed (background knowledge). Reference: H. Agarwal, C. Gupta, H.-L. Chang, S. Khandelwal, J. P. Duarte, Y. S. Chauhan, S. Salahuddin and C. Hu, ‘BSIM-BULK106.2.0 MOSFET Compact Model Technical Manual’, University of California, Berkeley (2017), Section 5.1 (band-gap and junction subsections not in the extract) (fetch-tool summary).

Ques: Why does MOSFET drain current rise with temperature at low gate drive but fall at high gate drive?

Because two effects compete. Mobility falls, as $(T / T_{N O M})^{U T E}$ with negative UTE in BSIM-BULK, lowering the current at all biases, while the threshold voltage falls, raising the overdrive. In the square law $I_{D} \propto μ (T) (V_{GS} - V_{t h} (T))^{2}$ the two cancel at an overdrive of $2 T_{N O M} ∣ d V_{t h} / d T ∣ / ∣ U T E ∣$ (background knowledge, computed derivation). For UTE = -1.5 this is 0.40 $V$ with 1 $mV/K$ and 0.15 $V$ with 0.37 $mV/K$ (computed). Below this zero-temperature-coefficient point the current rises with temperature, which can cause thermal runaway in parallel devices; above it the current falls, so power MOSFETs and digital gates slow down when hot. Analog designers bias near the ZTC point when temperature stability matters. Reference: H. Agarwal, C. Gupta, H.-L. Chang, S. Khandelwal, J. P. Duarte, Y. S. Chauhan, S. Salahuddin and C. Hu, ‘BSIM-BULK106.2.0 MOSFET Compact Model Technical Manual’, University of California, Berkeley (2017), Section 5.1 (band-gap and junction subsections not in the extract) (fetch-tool summary).

Ques: Is saturation-velocity scaling a power law or linear?

It depends on the model generation. BSIM-BULK 106.2.0 uses $V S A T (T) = V S A T (T_{N O M}) (T / T_{N O M})^{- A T}$ , a power law in which AT is an exponent, and power laws for the series resistance with PRT and for mobility with UTE. BSIM4 uses linear forms, $V S A T - A T (T / T_{N O M} - 1)$ with AT in metres per second and $R D S W + P R T (T / T_{N O M} - 1)$ (background knowledge). The two are not a disagreement about physics but different parameterizations, so an AT value cannot be copied between them: in BSIM4 a typical AT of $3.3 \times 1 0^{4}$ $m/s$ is a velocity, while a BSIM-BULK AT is dimensionless (background knowledge for the typical value). Over 300 to 400 $K$ , a power law with AT = 0.5 and a linear law matching it at both ends differ by under 1 per cent in between (computed), so both describe the data; the power law stays positive at any temperature. Reference: H. Agarwal, C. Gupta, H.-L. Chang, S. Khandelwal, J. P. Duarte, Y. S. Chauhan, S. Salahuddin and C. Hu, ‘BSIM-BULK106.2.0 MOSFET Compact Model Technical Manual’, University of California, Berkeley (2017), Section 5.1 (band-gap and junction subsections not in the extract) (fetch-tool summary).

References

Ioffe Physico-Technical Institute, “Si - Silicon: Band structure and carrier concentration,” NSM Archive, Physical Properties of Semiconductors. Available: https://www.ioffe.ru/SVA/NSM/Semicond/Si/bandstr.html
“Carrier density of intrinsic semiconductors,” LIGO Document T1500619 (author and date not stated). Available: https://dcc-llo.ligo.org/public/0123/T1500619/001/CarrierDensityIntrinsicSi.pdf
H. Agarwal, C. Gupta, H.-L. Chang, S. Khandelwal, J. P. Duarte, Y. S. Chauhan, S. Salahuddin and C. Hu, “BSIM-BULK106.2.0 MOSFET Compact Model Technical Manual,” University of California, Berkeley, 2017. Available: https://simetrix.co.uk/supplementary/Docs/BSIM-BULK/BSIM-BULK_106.2.0_technical_manual.pdf

Quartz 5

Explorer

Temperature Scaling

1. Theory

2. Working Principle

3. Applications

Band Gap and Intrinsic Density

MOSFET Temperature Parameters

Bias Points and Junctions

Important Questions

Ques: Which band-gap formula for silicon should be used?

Ques: Is the intrinsic carrier density of silicon at 300 K $1 \times 1 0^{10}$ or below $9 \times 1 0^{9}$ ?

Ques: How fast does leakage rise with temperature?

Ques: How does BSIM model the threshold-voltage shift with temperature?

Ques: Why does MOSFET drain current rise with temperature at low gate drive but fall at high gate drive?

Ques: Is saturation-velocity scaling a power law or linear?

References

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Temperature Scaling

1. Theory

2. Working Principle

3. Applications

Band Gap and Intrinsic Density

MOSFET Temperature Parameters

Bias Points and Junctions

Important Questions

Ques: Which band-gap formula for silicon should be used?

Ques: Is the intrinsic carrier density of silicon at 300 K 1×1010 or below 9×109?

Ques: How fast does leakage rise with temperature?

Ques: How does BSIM model the threshold-voltage shift with temperature?

Ques: Why does MOSFET drain current rise with temperature at low gate drive but fall at high gate drive?

Ques: Is saturation-velocity scaling a power law or linear?

References

Graph View

Table of Contents

Backlinks

Ques: Is the intrinsic carrier density of silicon at 300 K $1 \times 1 0^{10}$ or below $9 \times 1 0^{9}$ ?