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Thermal dissipation and self-heating at nanometer-scale in silicon
T.T.Trang NGHIEM, Jérôme SAINT-MARTIN, Philippe DOLLFUS
University of Paris-Sud
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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.
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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.
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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
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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 !