1

Thermal dissipation and self-heating at nanometer-scale in silicon

T.T.Trang NGHIEM, Jérôme SAINT-MARTIN, Philippe DOLLFUS

University of Paris-Sud

22

CONTEXT

Miniaturization of transistors at the nanoscale.

The local self-heating, due to the emission of phonons by the hot carriers can lead to reductions in performance.

The theoretical study of these phenomena at the nanoscale is notpossible with macroscopic models.

[Pop et al., 2001. IEDM Technical Digest. International

To evaluate the phonon generation by electron-phonon interactions in semiconductor → MC method is efficient.Model of non-equilibrium phonon transport and phonon/phonon interactions.

33

OUTLINE

Phonon generation in DG-MOSFET•

Monte Carlo (MC) simulation of BTE for electrons

•

Phonon generation computed by including electron-phonon scattering

Heat transport in DG-MOSFET•

Thermal transport model : Direct solution of the BTE for phonons

•

Temperature distribution and non-equilibrium transport

Conclusion

44ELECTRON TRANSPORT BY MC SIMULATION FOR

PHONON GENERATION

( )2 22

12

t lk k

t l

k kE Em m

α⎛ ⎞

+ = +⎜ ⎟⎝ ⎠

h

• Intra-valley scattering → acoustic phonon of small wavevector (Normal process).

• Inter-valley scattering → f-

and g-type phonons (Umklapp processes).

6 non-parabolic ellipsoidal bands:

Electron dispersion

Electron-Phonon scattering

Particle MC solution of BTE

Phonon dispersion

20 sw w v q cq= + +

Quadratic and isotropic

dispersion relation :

Fr

1ft

2ft

( )0 0k tr ( )11 0 fk t t+

r

( )1 22 0 f fk t t t+ +r

•

Stochastic solution of the BTE• 1 particle ⇔ ( ) ( ),r t k t

rr

( ) ( )( ) ( )( ), , i ii

f r k t r r t k k tδ δ= − −∑rr

E. Pop et al., JAP 2004

55PHONON GENERATION IN SILICON BARS

Net phonon number generated by electrons in silicon bars doped to 1017

cm-3

Information on the energy spectra of phonon according to phonon types

• At the steady-state, Joule effect is given either by

+ the sum of the four dissipation modes

+ J.E.

• The highest contributions come from LA and TO.

66PHONON GENERATION IN ULTRA THIN DG-MOSFET

S D

• Phonon absorption in the source and channel.

• Phonon emission mainly in the drain.

Spatial distribution of emitted phonons Vg

= 0.5V, Vds

= 0.7V

5 nm

10nm 10nm16nm

1020

cm-3 1020

cm-31016

cm-3

77PHONON GENERATION IN ULTRA THIN DG-MOSFET

The classical (drift-diffusion) result: a peak of dissipation is at the drain-

end of channel and there is no generation far into the drain.

S D5 nm

10nm 10nm16nm

X (nm)

Hea

t gen

erat

ion

rate

(W/c

m3 )

MC simulation shows that the heat is dissipated far into the drain.

88HEAT TRANSPORT

THERMAL TRANSPORT MODEL

( ) ( )( ) ( )2.

1 ( , ) ( , ) ( , )3 r

scatt

v q qN r q N r q G r q qs T e ph

ττ

⎛ ⎞⎜ ⎟− Δ + = + ×

−⎜ ⎟⎜ ⎟⎝ ⎠

rr r r

( )1

( ) exp 1scattT

B scatt

qN q

k Tω

−⎛ ⎞⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

h

BTE BTE using the 1using the 1stst

order spherical harmonic expansionorder spherical harmonic expansion

Boltzmann transport equation (BTE) per mode in the relaxation tiBoltzmann transport equation (BTE) per mode in the relaxation time approximation (me approximation (RTARTA))

( , ) ( , )( ) . ( , ) ( )

( )T s c a t t

r e p h

N r q N r qv q N r q G q

qτ −

−∇ = − +r

r r rrr r r r

• Fourier Eq. → Tscatt

used in RT calculation for BTE → phonon distribution

•Total number of phonon → Effective temperature Teff

(r)

Fourier equation . ( ) ( ) 0r rr s c a t t thT Pκ Δ + =r rr

( )( , ) ( , ) ( , )s p p g r sN r q N r q V q N r qτ −= − ∇rrr r r r r

( ),N r qr

( ),q

N r q∑ r

9

RELAXATION TIME MODEL Holland –

Asen Palmer

1 2 3 ( , )NU LB T LA Normal Umklappτ ω− = +1 4 ( , )N TNB T TA Normalτ ω− =

1/21

21/2

0 ( , )

/ sinh ( , )UTU

B

TA Umklapp pour

B TA Umklapp pourk T

ω ω

τ ωω ω ω−

<⎧⎪= ⎛ ⎞⎨ >⎜ ⎟⎪

⎝ ⎠⎩

h

Holland Asen-Palmer

1 2 3 ( , )L LB T LA Normal Umklappτ ω− = +

1 /1 2 ( , )TT TB Te TA Normal Umklappθτ ω−− = +

PRB 1963 PRB 1997

10HEAT TRANSPORT OPTICAL AND ACOUSTIC PHONON SUB-SYSTEM

EVOLUTION

Decay of L/TO intto L/TA Decay of L/TO intto L/TA

( )( , ) ( , ) ( , )scatt

N r q N r q G r q qs T e ph τ= + ×−

r r r

Simplied BTE using the 1Simplied BTE using the 1stst

order spherical harmonic order spherical harmonic expansion for optical phononexpansion for optical phonon

Phonon dispersion

Group velocity of optical phonon ≈ 0.

( ) ( )( )

( ) ( )/

2.1 ( , )

3

( , ) ( , )/

r

LTO L TA e LTOscatt

v q qN r qs

N r q G r q q f G qT e L TA

τ

τ τ−> −

⎛ ⎞⎜ ⎟− Δ + =⎜ ⎟⎜ ⎟⎝ ⎠

+ × + ×−

rr

r r

Phonon energy (eV)

Pro

babi

lity

TA+LA

LA+

LA

TA+LA

BTE using the 1BTE using the 1stst

order spherical harmonic expansion order spherical harmonic expansion for acoustic phononfor acoustic phonon

fTA

= f1

fLA

= f2

+ f3

1111HEAT TRANSPORT

IN THIN DG -

MOSFET

• Acoustic phonons have the main role in heat transport.

• Bottleneck

effect of optical phonons.

Temperature of different phonon modes in DG at

Vg

=0.5V, Vds

=0.7 V

S DC

T = 300 KT = 300 K

36 nm 36 nm 36 nm

Adiabatic condition

Adiabatic condition

Holland Asen-Palmer

12HEAT TRANSPORT IN THIN DG -

MOSFET

Holland Asen-Palmer

Effective temperatures as a function of Vds

at Vg

= 0.5V

The higher Vds is, the higher the temperature is.

1313HEAT TRANSPORT

NON-EQUILIBRIUM TRANSPORT

S DC

Vg

=0.5V, Vds

=0.7 V

Far-from-equilibrium state of 4 phonon modes at the drain, especially LO & TO.

At the drain

1414HEAT TRANSPORT

NON-EQUILIBRIUM TRANSPORT

S DC

Vg

=0.5V, Vds

=0.7 V

25 nm from the source

Go far from the source (or the drain), the 4 phonon modes tend to the equilibrium state (Bose-Einstein distribution).

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CONCLUSION

Monte Carlo simulation to understand the heat generation by electron in silicon.

Heat transport model to determine the non-equilibrium phonon distribution in silicon.

Future: + Couple electron-phonon transport in MC simulation+ Exploitation for the study of thermoelectric effects in nanostructure

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Thank you for your attention !