5 Ultrafast Nonlinear Plasmonics
179
It is based on the assumption that both the electron and lattice are thermalized at
different temperatures T e and T L , assumed to be homogeneous over the metal sample
(note that the assumption that the lattice temperature is always maintained by phonon
anharmonic interactions can certainly be questioned). This model can be generalized
to include non-homogeneous metal heating using spatial dependent temperatures and
adding a diffusion term in the metal. These extensions will not be considered here,
where the same assumption of spatial homogeneity and slow metal-environment
energy transfer will be assumed as for Eq. 5.11. The electron gas cooling dynamics
can then be simply modeled using the rate equation system:
C e (T e )
γ T e
γt = −G(T e − T L )
C L
γ T L
γt = G(T e − T L )
(5.20)
where G is the electron-phonon coupling constant. As C e , G can be assumed constant
for sufficiently small electron temperature rise ωT e ≤ 2,000 − 3,000 K, i.e., as
long as only conduction band electronic states are involved. It increases for larger
ωT e with increasing the density of available states for electron scattering due to
intervention of other electronic states [104]. As before, only moderate excitation will
be considered including only the conduction band electrons. The above equations are
then a consequence of the Boltzmann equation Eq. 5.11, and are obtained computing
the electron energy loss rate to the lattice in the thermalized regime (with T e , T L >
Θ D ):
C e
γ T e
γt
=
γωu e
γt
=
m
3/2
e
√
2E
π 2 3
E
γ f (E)
γt
e− ph
d E = −G(T e − T L ), (5.21)
with
G = ϕ
2 k B m 2
e q 2
D
16τπ 2 3 ,
(5.22)
where q D is the Debye wave vector [26, 66]. The above rate equation system can be
solved analytically [109]:
T eq ln
T e − T eq
T exc − T eq
+ ˜
T ln
T e − ˜
T
T exc − ˜
T
= −G
˜
T + T eq
2C L
t.
(5.23)
T exc is the temperature at time t th after which T e can be defined. For the sake of
simplicity t th = 0 has been used, i.e., instantaneous internal electron thermalization
is assumed and T exc = T 0 + ωT me
e . The final temperature, T eq , of the thermalized
electron-lattice system is then given by:
T eq =
T
2
s + T
2
exc + 2T s T 0
1/2 − T s ≈ T 0 +
T 2
exc − T 2
0
2T s
,
(5.24)
179
It is based on the assumption that both the electron and lattice are thermalized at
different temperatures T e and T L , assumed to be homogeneous over the metal sample
(note that the assumption that the lattice temperature is always maintained by phonon
anharmonic interactions can certainly be questioned). This model can be generalized
to include non-homogeneous metal heating using spatial dependent temperatures and
adding a diffusion term in the metal. These extensions will not be considered here,
where the same assumption of spatial homogeneity and slow metal-environment
energy transfer will be assumed as for Eq. 5.11. The electron gas cooling dynamics
can then be simply modeled using the rate equation system:
C e (T e )
γ T e
γt = −G(T e − T L )
C L
γ T L
γt = G(T e − T L )
(5.20)
where G is the electron-phonon coupling constant. As C e , G can be assumed constant
for sufficiently small electron temperature rise ωT e ≤ 2,000 − 3,000 K, i.e., as
long as only conduction band electronic states are involved. It increases for larger
ωT e with increasing the density of available states for electron scattering due to
intervention of other electronic states [104]. As before, only moderate excitation will
be considered including only the conduction band electrons. The above equations are
then a consequence of the Boltzmann equation Eq. 5.11, and are obtained computing
the electron energy loss rate to the lattice in the thermalized regime (with T e , T L >
Θ D ):
C e
γ T e
γt
=
γωu e
γt
=
m
3/2
e
√
2E
π 2 3
E
γ f (E)
γt
e− ph
d E = −G(T e − T L ), (5.21)
with
G = ϕ
2 k B m 2
e q 2
D
16τπ 2 3 ,
(5.22)
where q D is the Debye wave vector [26, 66]. The above rate equation system can be
solved analytically [109]:
T eq ln
T e − T eq
T exc − T eq
+ ˜
T ln
T e − ˜
T
T exc − ˜
T
= −G
˜
T + T eq
2C L
t.
(5.23)
T exc is the temperature at time t th after which T e can be defined. For the sake of
simplicity t th = 0 has been used, i.e., instantaneous internal electron thermalization
is assumed and T exc = T 0 + ωT me
e . The final temperature, T eq , of the thermalized
electron-lattice system is then given by:
T eq =
T
2
s + T
2
exc + 2T s T 0
1/2 − T s ≈ T 0 +
T 2
exc − T 2
0
2T s
,
(5.24)