INTRODUCTION
Irradiation of metallic and semiconducting thin films surfaces with ultra short laser pulses leads to a disturbance of the freeelectron gas out of thermal equilibrium. The nonequilibrium dynamics of the electron gas has been an area of intense research both theoretically and experimentally for the past couple of decades (Fujimoto et al., 1984; Jensen et al., 2003). The advent of nanosecond, picosecond and femtosecond laser pulses has greatly enhanced the study of the fundamental processes of electronelectron interaction as well as electronphonon interaction in these materials (Esarey et al., 1993; James, 1983; Feldstein et al., 1997; Papadogiannis et al., 2001; Rethfeld et al., 2002).
Nonequilibrium between electrons and phonon temperatures (the lattice) in the picosecond and femtosecond time regimes has been very important for metals because it has been widely studied (both theoretically and experimentally) and applied (Fujimoto et al., 1984; Shoenlein, 1996; Shwoerer et al., 2001; Usesaka et al., 2000; Eiguren et al., 2001; Papadogiannis et al., 2001; Rethfeld et al., 2002; Hafz et al., 2003). In a metal however, the free electrons gas first absorbed the energy from the incident laser while the lattice remains cold and transfer the energy to the electrons below through ElectronElectron Collision Time (EECT) (Rethfeld et al., 2002).
After a while, energy is distributed among the free electrons by electronelectron
collision leading to the thermalization of the electron gas. A distribution
function is assumed to be thermalized depending on the features one wishes to
observe. The temperature of the free electron will first rise when there is
sufficient nonequilibrium between the electron temperature (T_{e})
and the lattice temperature (T_{l}) due to the large mass difference
between the electrons and phonons; typically, a few tens of picosecond will
remain at low temperature (Rethfeld et al., 2002). The process involved
with the phenomena mentioned above continued for ultra short laser pulses to
a temperature higher by a few thousands of Kevin for a short period of time
where local temperature equilibrium is reached. The temperature in this region
becomes gradually equal to ambient temperature through relative long procedures
in which heat diffusion also plays a key role (Lin and Cheng, 1981; Wu et
al., 2001) due energy transfer from electrons to phonons through ElectronPhonon
Coupling Time (EPCT). The decoupling is only possible when the free electrons
receive energy from the laser at a rate faster than they are able to transfer
to the lattice through EPCT coupling. In metal the EPCT is quite strong and
that is why it is very difficult to decouple the two temperatures (electron
temperature, T_{e} and lattice temperature T_{l}) when
a metal is irradiated by high power laser of nano to pico second pulse width.
Conceptually, electron phonon coupling helps the free electron gas assembly
to thermally equilibrate (or thermalize) to a single temperature, T_{e}
at a given instant of time t, over a time range of Δt; (approximately equal
to the collision time) around t, so that the electron energy distribution over
a time t + Δt around t can be described on the average by Fermi function
with a single temperature as T, with Δt ≤ τ. T is a function of t
in these situations and τ is the relaxation time.
The development of photon devices requires the material with both fast optical response speed and large temperature stability (Jensen et al., 2003) provided the damage threshold value of the material is not reached. The semiconducting thin film of IndiumAntimonide (InSb) becomes an obvious choice since in this paper we have been able to theoretically delay the pulse width and increase the concentration of the charge carrier with a view of obtaining high average nonequilibrium temperature and enhanced free carrier concentration in the conduction band of the material. For a freeelectron laser of high brightness, many time repetitions and low emittance, Fujimoto et al. (1984) and others (Jensen et al., 2003) have used tungsten material or metal related materials to obtain nonequilibrium of a few thousand Kelvin. However, the concentration in metal is fixed and can only be irradiated with a femtosecond laser pulses, which hardly yield the above condition for Freeelectron laser applications. This idea motives us to theoretically investigate the nonequilibrium system of an intrinsic IndiumAntimonide thin film semiconducting material by considering the Anisimov et al. (1974) coupled differential equations and applications of appropriate boundary conditions and tailoring the characteristic of semiconductors to our advantage with a view of determining nonequilibrium transient hot electron time profile and the concentration time profile of this material. The focus in this study is therefore geared towards carrying out a theoretical investigation of the possibility of creating very large temperature difference between electrons and lattice (so as to create an appreciable carrier concentration) of semiconducting thin films of different thickness of material for different pump CO_{2 }laser of different shapes such as Gaussian pulse for time scales of one nanosecond.
THEORETICAL MODEL
When a laser pulse is incident normally on a thin film material, there is an exchange of energy between the electrons and the lattice through electronphonon interaction governed by a set of coupled nonlinear differential equations given as follows De and Musongong (2007).
We employ C_{e} and C_{l} as functions of t(i) and z(j), C_{e}
and C_{l} are the electronic specific heat capacity per unit volume
and lattice specific heat respectively. K_{t} is the thermal conductivity
of the material and g is the electronphonon coupling. For a thin semiconducting
film of IndiumAntimonide with pump laser incident along the zdirection, the
transient electron heating is at z = 0.
Since the lattice receives energy from electrons through electron phonon coupling
of both z = 0 and z>0
is valid.
A (z, t) is the source term which is actually the zgradient of the incident
laser pulse probe intensity and R being the reflectivity at the surface of the
material thin film (IndiumAntimonide) at time t. It is natural that there will
be a temperature differential due to heating of the material. This has been
taken into consideration as the material mounted on a substrate with experimental
set up as in De and Musongong (2007). The laser power is assumed to be uniform
over the incident surface. Even within the skin depth (δ) of the metal,
electrons at different layers receive energy from the laser pulse at different
rates because of the
term of the intensity within the metal. The free electrons
at a given layer after receiving energy from the laser within Eq.
2. Therefore:
The boundary conditions that exist at the surface are
for Gaussian pulse
Both T_{e} and T_{l} are functions of and z.
where, NΔz = d
The electrons at the surface (z = 0) absorb energy from the incident laser
pulse through electron phonon coupling. The energy is conducted to electrons
in the layers below z = 0 through electronic thermal conductivity. Part of the
energy is transferred to the lattice through the term g(T_{e}T_{l})
where as the radiation can take place through motion. The electronphonon coupling
factor g is given as:
where:
m 
= 
The effective mass of the electrons or holes. 
n_{e} 
= 
The electronic concentration. 
V_{s} 
= 
The velocity of sound in the metal. 
τ_{ep} 
= 
The relaxation time. 
The validity of the two temperatures model (i.e., T_{e} and T_{l})
is obviously limited to times longer than the electronelectron collision time
τ_{ee} for every material. For this to happen, the metal with suitable
femtosecond electronphonon relaxation time τ_{ep} becomes an obvious
choice suitable since there is strong electronphonon coupling. We decided to
investigate theoretically some of the parameters such as the charge carrier
concentration n_{e} that appeared in Eq. 7 as a constant,
the transient electron hot temperature (that should be maintained even when
the laser source is switched off) and C_{e}, the electronic specific
heat capacity per unit volume of IndiumAntimonide with a view to extending
the nonequilibrium to the semi conducting material and in the nanosecond (ns)
time regime as an obvious choice. In this study, we considered the electronphonon
coupling g as dynamic rather than a constant as assumed by Fujimoto et al.
(1984). Preliminary calculations showed that for semiconducting thin film,
if the concentration of the charge carrier is varied, implies enhancement in
the electronphonon coupling and since τ_{ep }is given by the expression:
for semiconductor, while a and b are constants and determined experimentally, photoemission will be enhanced.
ANALYSIS
Thermal enhancement technique, involves laser pumping of electrons from valence
band to conduction band and emitting those excited electrons by a delayed laser
pulse in addition to those usual photoelectrons from valence band. Here, in
order to obtain thermally enhanced photoemission using ultra short laser pulses;
time resolved measurement of the electron and lattice temperatures has been
studied by James (1983) for some of the semiconductors through impact ionization
led exponential growth using CO_{2} laser. The impact ionization gives
an idea of obtaining a nonequilibrium transient hot electron temperature and
the lattice temperature using semiconductors. We have decided to employ Eq.
17 in our simulations. The idea here is that, if significant
temperature difference can be achieved it will lead to enhanced photoemission
when using a suitable delayed pulse probe laser. The analytical solution of
(1) and (2) along with the boundary conditions is nearly impossible since (1)
and (2) are coupled nonlinear differential equations. For a given thickness
d, relaxation time τ and gradient of the laser intensity A (z,t) we apply
numerical finite difference method. We took N = 1000 and further to several
thousands, being the divisions of both the time t(i), τ = 1 nanosecond
(ns)and the thickness d(j). We took i = 0, 1, 2, 3,…N and j = 0, 1, 2,
3,…N where (i,j) are functions of time and thickness, respectively. We
calculated the values of T_{e} (i,j) and T_{l}(i,j) and obtained
the average temperature +T_{e}(i), and +T_{l}(i),, respectively
for intrinsic InSb thin film Fig. 39. Here
we have assumed
for simplification and better approximation. We took different
thicknesses of material with different peak pulses of averagely low intensities.
RESULTS AND DISCUSSION
To apply our method of calculation for enhanced concentration of intrinsic semiconducting thin film with a variation in the electronic specific heat capacity per unit volume of Indium Antimonide, we apply the derivation specific heat C_{e} at a temperature T_{e} while adopting the equipartition approach and the quantum approach for the absorption of pulse intensity from the heating CO_{2} laser. The electronphonon interactions and absorptions in terms of the Fermi Function f(k) is given in Rethfeld et al. (2002).
The computed specific heat capacity of the sample is shown in
Fig. 1 and 2. Both methods have however shown significant
increase in the specific heat capacity when the temperature is increased.

Fig. 1: 
Temperature dependence of specific heat capacity applying
classical approach for InSb 

Fig. 2: 
Temperature dependence of specific heat capacity applying
quantum approach for InSb. 
This
idea gives a way of controlling C_{e} in semiconductors, rather than
the constant value assumed by Fujimoto et al. (1984) and Jensen et
al. (2003) for metals.
There is a direct relationship of C_{e} with temperature as it increases progressively and at low temperature the curve seems very sluggish is both cases. This is expected since InSb has a very low value of energy gap ε_{g}.
Here E_{g} is the energy gap, M_{e}*, M_{h}* are the effective mass of the electron and hole, respectively. M_{e}*, M_{h}*, ε_{g} are given in De and Musongong (2007).
Figure 39 show calculations of nonequilibrium
electronlattice temperatures time profile for various values of InSb thin film
thicknesses, pulsed laser power and doping concentration. Figure
3 is for doped InSb and Fig. 49 is
for intrinsic InSb.

Fig. 3: 
280 A^{0} thick of InSb thin film irradiated with
10 MW cm^{–2} laser pulse with a doped concentration of N_{d}
= 10^{21} m^{–3} for Gaussian pulse with τ = 1 n sec 

Fig. 4: 
1.2 μm thick of InSb thin film irradiated with 10^{–2}
MW cm^{–2} laser pulse with intrinsic concentration for Gaussian
pulse 

Fig. 5: 
0.9 μm thick of InSb thin film irradiated with 10^{–2}
MW cm^{–2} laser pulse with intrinsic concentration of for Gaussian
pulse 

Fig. 6: 
1.2 μm thick of intrinsic InSb thin film irradiated with
0.1 MW cm^{–2} laser pulse with a intrinsic concentration for a
Gaussian pulse 
For intrinsic InSb the calculations are made using the quantum expression
of C_{e} Fig. 3 shows a 280 A^{0} thick of
InSb thin film irradiated with 10^{8} W m^{–2} laser pulses.
Using Eq. 18 and our method of calculating
C_{e }and the electronphonon coupling, it is seen from Fig.
39, that the results obtained here predicts significant
nonequilibrium between electron and lattice temperatures in nanosecond time
regime. The results showed that <T_{e}(t)>_{max} depends
on the thickness of the thin film, the incident laser pulse intensity and the
duration of the pulse width.

Fig. 7: 
1.1 μm thick of InSb thin film irradiated with 0.1 MW
cm^{–2} laser pulse with intrinsic concentration of for Gaussian
pulse 

Fig. 8: 
0.9 μm thick of InSb thin film irradiated with 0.1 MW
cm^{–2} laser pulse with intrinsic concentration of for Gaussian
pulse 
It is worth mentioning here that the result of
the quantum derivation of C_{e }fits well in to the calculations of
intrinsic InSb with a significant nonequilibrium electron lattice temperature
as obtained in Fig. 49. Here, the full wave at half maximum (FWHM) is about 1 nanosecond with a maximum
electron temperature ranging from 760 K. to about 5000 K depending on incident
laser power intensity and thin film thickness. It could be observed that the
transient hot electrons remain significantly hot (in comparison to the lattice
temperature) essentially for time scale of the order of 0.5 nanosecond after
the cessation of the incident nanosecond laser pulse. This feature could be
different for other semiconductors such as GaAs, Ge etc. Such differences are
expected since the transient electron temperature time profile depends on the
electronic properties of the semiconducting materials that include the effective
masses, band gap etc. Figure 9 shows the simulation of +T_{e}(t),
for InSb thin films of thickness 0.9, 1.1 and 1.2 μm for a Gaussian pulse
peakpower of 0.1 MW cm^{–2} of the incident laser intensity.

Fig. 9: 
d_{1} = 1.2 μm, d_{2} = 1.1 μm,
d_{3} = 0.9 μm thick of InSb thin film irradiated with 0.1
MW cm^{–2} laser pulse with intrinsic concentration for Gaussian
pulse 

Fig. 10: 
Transient hot concentrationtime N_{e}(t) profile
for intrinsic InSb using T_{e}t in Fig. 9 

Fig. 11: 
Exponential growth rate with transient electron temperature
of ntype InSb. λ = 9.6 μm 

Fig. 12: 
Exponential growth rate with transient electron temperature
of ntype InSb. λ = 10.6 μm 

Fig. 13: 
Exponential growth rate with transient electron temperature
of ptype InSb. Wavelength λ = 9.6 μm 
The curves depict a transient evolution of non equilibrium electronlattice
temperature of the order, above 3000 K for both thickness of the thin film.
The curves show +T_{e}(z,t),_{max} dependent on thickness of
film as well as time when the peak power and duration are both held constant.
We applied the +T_{e}(z,t),_{max} relations above to calculate
the transient hot electron concentration after the cessation of the 1 nanosecond
(ns)incident laser as shown in Fig. 10.
Using the study in James (1983) we proceed to calculate transient hot Concentrationtime
profile for doped IndiumAntimonide material by first calculating the exponential
growth rate as function of transient electron/hole temperatures and subsequent
as a function of time. The results of the calculations are shown in the Fig.
1114.
Figure 11 shows the exponential growth rate verses transient
hot electrons temperature irradiated with high intensity CO_{2} laser
of 9.6 μm wavelength of InSb.

Fig. 14: 
Exponential growth rate with transient electron temperature
of ptype InSb. Wavelength λ = 10.6 μm 

Fig. 15: 
Exponential growth rate as function of time hot electrons
in ntype InSb. Wavelength λ = 10.6 μm 
In Fig. 12, ntype InSb is irradiated with 10.6 μm
wavelength CO_{2 }laser of high intensity pulse power. It is shown that
the gradient of the plot is very stiff due to very short range of temperature.
This could be attributed to the chemical properties of the material.
Figure 13 shows the exponential growth rate verses transient
hot holes temperature irradiated with high intensity CO_{2} laser of
9.6 μm wavelength of InSb. The values of the exponential growth rate of
hot holes in the semiconductor material are quite high but the transient holes
temperature range is short. In Fig. 14, the exponential growth
rate verses transient hot holes temperature irradiated with high intensity CO_{2}
laser of 9.6 μm wavelength of the semiconductor material are quite high
but the transient temperature range is very short.
In Fig. 15, ntype InSb is irradiated with 10.6 μm
wavelength CO_{2 }laser of high intensity pulse power. Here, the exponential
growth rate is a function of time. It is shown that the gradient of the plot
is very stiff inline with the short range of temperature as shown in Fig.
1114.

Fig. 16: 
Transient electron concentrationtime N_{e}(t) profile
of ntype InSb irradiated with CO_{2} laser intensity of wavelength
λ = 10.6 μm 
Following James (1983) one has calculated the
exponential growth rate α(s^{1}) as a function of T_{e} for various thicknesses of the semiconductors irradiated with laser pulse of
different wavelengths, for ntypes and ptypes (Fig. 1114)
and then based on calculations presented here one has obtained T_{e}(t)
profile which then gave us the α(s^{1}) as a function of time
for the CO_{2} laser pulse irradiation with a given intensity and duration
of a semiconducting thin film of given thickness.
The density of transient hot electrontime profile from impact ionization led
exponential growth has been theoretically calculated and presented in Fig.
16. The result shows a gradual increase in the concentration with time as
the laser source is heating.
In Fig. 16, ntype InSb is irradiated with 10.6 μm
wavelength CO_{2 }laser of high intensity pulse power. It is shown that
the concentration gradually built up within the time range of zero to about
0.5 nanosecond before a sudden growth sets in. The plot N_{e}(t) verses
time is shown to terminate at maximum. (i.e., N_{e}(t) = 10^{20 }m^{–3}).
This is not so as the concentration will decay back to the initial concentration
as time increases.
Finally our theoretical investigation has shown that semiconducting film offers an ideal medium of observing nonequilibrium between transient electron temperature and the lattice temperatures in the nanosecond laser pulse time regime. For practical application of these findings, what is important is that one needs to have thin film of proper thickness matching the peak power available laser pulse intensity such that T_{e} stays sufficiently hot (say above 3000 K) over a period three to four times the width of the pump laser pulse, so that a suitably delayed probe laser will be able to photoemit the hot electrons out of the surface (held at high negative potential with the respect to accelerating anode).
CONCLUSIONS
It is shown that nonequilibrium temperature between T_{e} and T_{l}
can be achieved with InSb semiconducting thin films both for intrinsic and
extrinsic material, which could lead to increase carrier concentration from
the concentration verses time profile without affecting the material properties.
It is also shown that using our theoretical method formulated for calculating
the transient hot carrier concentration in intrinsic InSb semiconducting material
and extending that in James (1983) both results agreed perfectly well (Fig.
10 and 16) and is strongly noted here that, among the
semiconductors tested with this method only InSb shows significant increase
in concentration both using intrinsic and extrinsic material. The experimental
set up for this research is found in De and Musongong (2007). It is worth mentioning
without reservation that this could lead to enhanced photoemission for Freeelectron
laser applications and other nano physics technology. The calculation of the
true enhancement is another subject of discussion as the process is a twostep
multiphoton generation in Milonni and Eberly (1988). When we apply the RichardsonLaueDushman
expression for the thermionic part, we note the effect is very small. However,
this method has increased the substrates list to be used for charge carrier
concentration for free electron laser applications for high brightness and low
emittance. From what has been presented above on the N_{e}(t) versus
time profile for different thicknesses of different semiconducting materials
and the exponential growth rate as function of hot electron temperature and
time, one can make an approximate estimate of the enhanced electronconcentration.
This is done as follows: The FWHM (full wave at half maximum) of the T_{e}(t)
versus time profile is usually around 1 to 4 nanosecond for film thicknesses
of 100 A^{0} to 1.2 micron though, it increases with film thicknesses
(for higher film thicknesses, as seen before higher laser power is required
to achieve enhanced electron temperature). The exponential growth rate varies
on the average from 10^{5} to 10^{10} as electron temperature
increases. Also, we computed the final concentration N_{e}(t). We noted
that the value of the N_{e}(t)/N_{o} (which is the enhancement
factor), using t of the order of FWHM is found to vary from 10^{5} to
10^{9}. In other words, one can enhance the free electron concentrations
in semiconductor conduction band for a time scale of 1 to 4 nanosecond, using
our technique of electron pumping from valence band to conduction band with
intense CO_{2} laser pulse. This is expected to significantly enhance
the photoemission by a suitably delayed probe laser with hv >Φ work
function. The delay should be such that the probe laser is switched on after
the cessation of the pump laser but within the FWHM period (it is to be noted
that as per our calculations the electrons remain significantly hot for time
scale of FWHM when the pump laser pulse ceases). It is worth to mention that
the thin film semiconducting materials (because of the peculiar properties
of the semiconductors) offer a more reliable lasing action rather than the traditional
tungsten metal that has been used Fujimoto et al. (1984) and co workers.
This promises to be a prodigious amount of work to be done in the area of solid
state physics. The discoveries here are merely a humble beginning of an unlimited
wealth of scientific knowledge in the field of laser technology.