104
Strain-Engineered MOSFETs
is still isotropic. The following expressions have been proposed for the electron and hole mobilities [33]:
1
1
/
1 2( /
1
0
1/2
,
1/2
m m
m m
F
F E
E
kT
F
F E
E
kT
ii
n
n
nl
nt
nl
nt
n
C
C i
n
C
C
µ = µ
+
−
+
− −
− −
−
(4.29)
1
1
/
1
/
1
0
0
0
1.5
1.5
1/2
,
1/2
,
m m
m m
F
E
E
F
kT
F
E
E
F
kT
P
P
Pl
P
Pl
Ph
Pl
Ph
V
Vl
P
V
Vh
P
(
)
(
)
µ = µ
+
µ
µ
−
+
+
−
+
−
−
(4.30)
where
0
n
µ and
0
P
µ are electron and hole mobility models without the strain.
m nl and m nt are the electron longitudinal and transfer masses in the subvalley.
,
E C i and E C are computed by Equations (4.29) and (4.30), respectively. The index i corresponds to a direction (for example, 11
n
µ is the electron
mobility in the direction of the X axis of the crystal system, and therefore
ΔE C,1 should correspond to the twofold subvalley along the X axis).
0
Pl
µ is
the mobility of light holes without the strain. m Pl and
,
m P h are the hole light
and heavy masses.
,
E V l and
,
E V h are computed also by Equation (4.18),
with the specification of light-hole and heavy-hole subvalley numbers in the
parameter file. F n and F p are quasi-Fermi levels of electrons and holes.
4.9.1 Strain-Induced Mobility Model under Electron–Phonon Interaction
In developing the strain-induced mobility model, we implemented strain
effects by considering scattering of mobile charges in process-induced
strained (PSS) n- and p-MOSFETs due to electron/hole-phonon interactions
in the strained Si channel. To obtain the strained-induced mobility, first we
calculated strain-induced interaction potential scattering by acoustic phonon. Then we used Fermi’s golden rule to obtain the electron/hole-phonon
scattering rates. Toward this, one needs to obtain the matrix element for electron–phonon scattering. The matrix element describes the coupling between
initial and final electronic states due to interactions with scattering charge
centres. Finally, we integrate the matrix elements over all final states to
obtain the scattering rate between electron/hole-phonons.
4.9.2 Strain-Induced Interaction Potential
Scattering by Acoustic Phonon
In a deformed Si substrate the coordinates of its lattice point are displaced.
If the radius vector of a lattice point in undeformed condition is r, and in
deformed condition is r′, then the displacement vector is given by [34]
( )
u r r r
= ′ −
(4.31)
Strain-Engineered MOSFETs
is still isotropic. The following expressions have been proposed for the electron and hole mobilities [33]:
1
1
/
1 2( /
1
0
1/2
,
1/2
m m
m m
F
F E
E
kT
F
F E
E
kT
ii
n
n
nl
nt
nl
nt
n
C
C i
n
C
C
µ = µ
+
−
+
− −
− −
−
(4.29)
1
1
/
1
/
1
0
0
0
1.5
1.5
1/2
,
1/2
,
m m
m m
F
E
E
F
kT
F
E
E
F
kT
P
P
Pl
P
Pl
Ph
Pl
Ph
V
Vl
P
V
Vh
P
(
)
(
)
µ = µ
+
µ
µ
−
+
+
−
+
−
−
(4.30)
where
0
n
µ and
0
P
µ are electron and hole mobility models without the strain.
m nl and m nt are the electron longitudinal and transfer masses in the subvalley.
,
E C i and E C are computed by Equations (4.29) and (4.30), respectively. The index i corresponds to a direction (for example, 11
n
µ is the electron
mobility in the direction of the X axis of the crystal system, and therefore
ΔE C,1 should correspond to the twofold subvalley along the X axis).
0
Pl
µ is
the mobility of light holes without the strain. m Pl and
,
m P h are the hole light
and heavy masses.
,
E V l and
,
E V h are computed also by Equation (4.18),
with the specification of light-hole and heavy-hole subvalley numbers in the
parameter file. F n and F p are quasi-Fermi levels of electrons and holes.
4.9.1 Strain-Induced Mobility Model under Electron–Phonon Interaction
In developing the strain-induced mobility model, we implemented strain
effects by considering scattering of mobile charges in process-induced
strained (PSS) n- and p-MOSFETs due to electron/hole-phonon interactions
in the strained Si channel. To obtain the strained-induced mobility, first we
calculated strain-induced interaction potential scattering by acoustic phonon. Then we used Fermi’s golden rule to obtain the electron/hole-phonon
scattering rates. Toward this, one needs to obtain the matrix element for electron–phonon scattering. The matrix element describes the coupling between
initial and final electronic states due to interactions with scattering charge
centres. Finally, we integrate the matrix elements over all final states to
obtain the scattering rate between electron/hole-phonons.
4.9.2 Strain-Induced Interaction Potential
Scattering by Acoustic Phonon
In a deformed Si substrate the coordinates of its lattice point are displaced.
If the radius vector of a lattice point in undeformed condition is r, and in
deformed condition is r′, then the displacement vector is given by [34]
( )
u r r r
= ′ −
(4.31)