University of Paris-Sud
Thermal dissipation and self-heating at
nanometer-scale in silicon
T.T.Trang NGHIEM, Jérôme SAINT-MARTIN, Philippe DOLLFUS
1
2
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.
[Pop et al., 2001. IEDM Technical Digest. International



The theoretical study of these phenomena at the nanoscale is not
possible with macroscopic models.
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.
3
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
BTE for electron


Approche semi-classic -> Fct de distribution f(r,k,t)
Evolution de f(r,k,t) -> BTE
1
 
f
f
 v r f  F k f 
t
t
v  k E k
with E(k) is the dispersion relation
F
k
t
applied force


 G r , k ,t

Generation /recombinaison
net rate
coll
Variation of f due to the collisions
Time relaxation time: f
t

f  f equilibrium
coll

Direct resolution vs. Stochastic resolution
+ For electron: informatics resource cost (3 directions in reciprocal
space, even in 2D real space) -> stochastic resolution (suitable for
device simulation (even in 3D)).
+ For phonon: no force -> 2D real space -> could be resolved by direct
resolution.
ELECTRON TRANSPORT BY MC SIMULATION FOR
PHONON GENERATION
BTE -> N grand eq. Newtons
Analytic description
of E(k)
Particle MC Method
dk F

dt
dr 1
Particle trajectories
 k E  v ( k )
(propagation + collisions) dt
Scattering rates:  i  k 
during Dt
Electric force F = q E
Density calculation
Self consistent loop
each Dt
Poisson equation
Random selection:
• Time of free light tv
•Type of scattering
• Effect of scattering
  (r ).V (r, t )    (r, t )
Modelisation:
• Dispersion relation
• Scattering rate
ELECTRON TRANSPORT BY MC SIMULATION FOR
PHONON GENERATION
Electron dispersion
Vallée Δ : 6 non-parabolic ellipsoidal bands
 kt2 kl2 
Ek 1   Ek     
2  mt ml 
2
ml = 0.9163m0
mt = 0.1905m0
 = 0.5 eV-1
• Intra-valley scattering → acoustic phonon of small wavevector (Normal process).
• Inter-valley scattering → f- and g-type phonon (Umklapp process).
• Phonon type and energy are given by the selection rules : g-TA 11.4 meV, g-LA 18.8 meV,
g-LO 63.2 meV, f-TA 21.9 meV, f-LA 46.3 meV, f-TO 59.1 meV.
MONACO: All electron-phonon inter-valley interaction is treated inelastically.
Phonon w(q) dans MONACO
In MONACO
Ω
(rad/s)
Quadratic dispersion
Quadratic and isotropic dispersion relation :
w  w0  vs q  cq 2
Optical type
Acoustic type
0
E. Pop et al.,
JAP 2004
q (m-1)
2
a
Group velocity:
Vg 
d
dq
VLO,TO are verry small
- > The contribution of optical modes in heat transport
could be neglect.
Electron- phonon interaction
• The interaction which electron suffers are treated as perturbations ???
• Density of probability in unit of time
 scat
1 i

DP2  N q    g  E  q 
2 2

i =1 correspond to an emission,
i = -1 correspond to an absorption;
g(E) is density of state of the final state.
• Deformation potential : DP = D0 + D1*kP
When the selection rules forbid interaction
with the term of order 0, they allow it by
order 1.
g-TA
g-LA
g-LO
f-TA
f-LA
f-TO
ħω
11.4 meV
D1
3.0 eV
ħω
18.8 meV
D1
3.0 eV
ħω
63.2 meV
D0
3.4x108 eV/cm
ħω
21.9 meV
D1
3.0 eV
ħω
46.3 meV
D0
3.4x108 eV/cm
ħω
59.1 meV
D0
3.4x108 eV/cm
9
PHONON GENERATION
DEFORMATION POTENTIALS (108 eV/CM)
 scat
1

DP2  N q 
2

1
 g  E  q 
2
Ref.
f-TA
f-LA
f-TO
g-TA
g-LA
g-LO
Nash 1974
0.35
1.70
6.85
0.6
0.7
5.33
Canali 1975
0.15
3.4
4
0.5
0.8
3
Joergensen 1978
__
4.3
2
0.65
__
7.5
Brunetti 1981
0.3
2
2
0.5
0.8
11
Jacoboni1983,
Fischetti 1991
0.3
2
2
0.5
0.8
11
(18.96)
(47.39)
(59.03)
(12.06)
(18.53)
(62.04)
Yamada 1994
2.5
__
8
__
4
8
Pop.
0.5
3.5
1.5
0.3
1.5
6
2004 (used)
(19)
(51)
(57)
(10)
(19)
(62)
Obukhov 2009
~0
2.51
4.44
~0
~0
4.73
Wang
D1=0.18
(18.7)
D0=1.12
(44.9)
D0=4.24
(57.7)
D1=0.61
(11.9)
D1=1.22
(19.5)
D0=4.18
(62.1)
D1=3 .47
D0=3.4
D0=3.4
D1=3.47
D1=3.47
D0=3.4
(21.9)
(46.3)
(46.3)
(11.4)
(18.8)
(63.2)
2011
Our
simulation
Silicon bars under constant field
• Si bars doped to 1017 cm-3.
15
nm
1017 cm-3
F = cst
• Phonon generation net in Si bars under different field.
• Compare the dissipation by e-ph interaction with the Joule effect.
PHONON 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.
11
PHONON GENERATION
12
IN ULTRA THIN DG-MOSFET
S
1020
D
cm-3
1016
cm-3
1020 cm-3
16nm
Vg = 0.5V, Vds = 0.7V
Spatial distribution of emitted phonons
10nm
10nm
• Phonon absorption in the source and channel.
• Phonon emission mainly in the drain.
5 nm
PHONON GENERATION
13
IN ULTRA THIN DG-MOSFET
S
D
5 nm
16nm
10nm
MC simulation shows that the heat
is dissipated far into the drain.
Heat generation rate (W/cm3)
10nm
The classical (drift-diffusion) result:
a peak of dissipation is at the drainend of channel and there is no
generation in the drain.
X (nm)
14
HEAT TRANSPORT
THERMAL TRANSPORT MODEL
Boltzmann transport equation (BTE) per mode in the relaxation time approximation (RTA)
v (q).r N (r , q)  
Fourier equation
N (r ,q)  NT
scatt
 (q)
(r ,q)
 Ge ph (q)
 .D r Tscatt (r )  Pth ( r )  0
BTE using the 1st order spherical harmonic expansion
N ( r , q )  N s ( r , q)   p  pVg  q  r N s (r , q)

   q  
N Tscatt ( q)   exp 
  1
k
T
 B scatt  

1
 v q . q 2








D r 1 N (r , q)  N
(r , q)  G
(r , q)   q 
s
T
e

ph


3
scatt



• Fourier Eq. → Tscatt used in RT calculation for BTE → phonon distribution N  r , q 
•Total number of phonon
 N  r , q → Effective temperature Teff(r)
q
15
RELAXATION TIME MODEL
Holland – Asen Palmer
Umklapp processes
Normal processes
G
ω1,k1
Conservation d’énergie
q3  q1  q2
Holland
  q3     q1     q2 
PRB 1963

B

T
(
L
A
,
N
o
r
m
a
l

U
m
k
l
a
p
p
)

1
2
3
N
UL


B

T
(
T
A
,
N
o
r
m
a
l
)

1
4
NT
N
0
(
T
A
,U
m
k
la
p
p
p
o
u
r

1/2)


1 



 2
U
B

/
s
i
n
h
T
A
,U
m
k
la
p
p
p
o
u
r

1/2)

(
T
U

T
k
B


Asen-Palmer
PRB 1997
 L1  BL 2T 3 ( LA, Normal  Umklapp)
 T1  BT Te /T  2 (TA, Normal  Umklapp)
1
ω2,k2
ω3,k3
q3  G  q1  q2
Vecteur du
réseau
réciproque
HEAT TRANSPORT
16
OPTICAL AND ACOUSTIC PHONON SUB-SYSTEM EVOLUTION
Phonon dispersion
Group velocity of optical phonon  0.
Simplied BTE using the 1st order spherical harmonic
expansion for optical phonon
N (r , q)  N
(r , q)  G
(r , q)   q 
s
Tscatt
e  ph
fTA = f1
Decay of L/TO to L/TA
Probability
TA+LA
fLA = f2 + f3
BTE using the 1st order spherical harmonic expansion
for acousptic phonon
LA+
LA
TA+LA
Phonon energy (eV)
 v q . q 2








D r 1 N (r , q) 

 s
3


N
(r , q)  G
(r , q)  q   f LTO L /TAGe LTO   q 
Tscatt
e L /TA
17
HEAT TRANSPORT
Lx = Ly = 3x10-6 m
Nx = 100, Ny = 20
Tg = 310
K
Td = 300 K
Linear solution (well known) for a bar
with ΔT = 10K.
T = 300 K
P = 6.7e17 W/m3
F=50kV for Si 1e17 m-3)
T = 300 K
Obtained temperature with a power of
6.7x1014 W/cm3 in the middle of bar.
18
HEAT TRANSPORT
IN THIN DG - MOSFET
T = 300 K
T = 300 K
Adiabatic condition
S
C
D
Adiabatic condition
36 nm
Holland
36 nm
36 nm
Temperature of different phonon modes in
DG at Vg=0.5V, Vds=0.7 V
Asen-Palmer
• Acoustic phonons have the main role in heat transport.
• Bottleneck effect of optical phonons.
HEAT TRANSPORT
19
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.
20
HEAT TRANSPORT
NON-EQUILIBRIUM TRANSPORT
Vg=0.5V, Vds=0.7 V
At the drain
S
C
D
Far-from-equilibrium
state of 4 phonon modes
at the drain, especially
LO & TO.
21
HEAT TRANSPORT
NON-EQUILIBRIUM TRANSPORT
Vg=0.5V, Vds=0.7 V
25 nm from the source
S
C
D
Go far from the source
(or the drain), the 4
phonon modes tend to
the equilibrium state
(Bose-Einstein
distribution).
A FAIRE …
• BTE sans approximation :
v (q).r N (r , q)  
N (r ,q) NT
scatt
(r ,q)
 (q)
• Coupler électron-phonon dans Monaco
e-MC
Net phonon
generation
Entrée pour BTE
• Teff de phonons
• Distribution de phonon
Injecter T et la distribution de
phonon dans chaque maille.
• Scattering e-ph pour e-MC.
• Teff d’électrons
 Ge ph  q 