2-1 Background Concepts
2-1.1 Carrier Relaxation Regimes
After an ultrashort pulse is sent into a semiconductor, it will be excited to higher energy level and then undergoes several stages of relaxation before it returns back to thermodynamic equilibrium [24]. There are four temporally overlapping regimes of the carrier relaxation:
(a) Coherent Regime
When a semiconductor is excited by an ultrashort pulse, the excitation in the semiconductor will have well-defined phase relationship within itself and with the electromagnetic field which created the excitations. The excitation can be either real or virtual. The scattering processes in semiconductors that destroy the coherence are exceedingly fast that requires pico- and femtosecond techniques to study. Coherent regime exhibits a lot of fascinating phenomena with basic quantum mechanics in semiconductors.
(b) Non-thermal Regime
In this regime, the excitation is real. After dephasing free electron-hole pairs and excitons in the coherent regime, the distribution of excitation (i.e. free electron-hole pairs or excitons) is likely to be non-thermal. It means the distribution function can not be represented by a temperature. This regime includes various processes such as carrier-carrier scattering or exciton-exciton scattering. Through the scatterings
above, it takes the non-thermal distribution to a hot, thermalized distribution.
(c) Hot-Carrier Regime
Energy redistribution within the carrier or exciton system is due to carrier-carrier or exciton-exciton scattering. The procedure leads to a thermalized distribution function of carriers or excitons which can be expressed by a temperature. The temperature is often higher than the lattice temperature, and might be different for various sub-systems. The times of thermalization depend on many factors, for example, carrier density. The electrons and holes thermalize among themselves in hundreds of femtoseconds, while they achieve an ordinary temperature in a few picoseconds. Those thermalized electron-hole pairs take hundreds of picoseconds to reach lattice temperatures, and this process is under interaction with several kinds of phonons in the semiconductor.
Investigation of this regime concentrates on the rate of cooling of carriers to the lattice temperature and the various scattering processes.
(d) Isothermal Regime
All the carriers, phonons, and excitons relax to equilibrium with one another at the end of hot-carrier regime. Yet there are still some electrons and holes exceeding the thermodynamic equilibrium. In isothermal regime, these excess electron-hole pairs or excitons recombine one another radiatively or non-radiatively then return the semiconductor to the thermodynamic equilibrium.
The schematic diagram of these four regimes is illustrated in Figure 2-1. The figure shows some typical processes occurring in each relaxation regimes. The time
the excitation, the density of excitation, and the lattice temperature…etc.
We should emphasize that many of the physical relaxation in the different regimes are occurring simultaneously and the four relaxation regimes is used for convenience to describe the dynamics of carrier relaxation. For instance, the processes that destroy coherence perhaps also contribute to thermalization of carrier distribution functions, and emission of phonons may take place while the electrons and holes are thermalizing to a hot distribution. The non-thermal carrier distribution function created by an ultrashort pulse is affected by the dephasing of coherent polarization during the pulse. Figure 2-1 is the relaxation regimes under different time scales.
Coherent Regime (≦200fs)
◆ Momentum scattering
◆ Carrier-carrier scattering
◆ Intervally scattering (Γ→L, X )
◆ Hole-optical-phonon scattering Non-thermal Regime (≦2ps)
◆ Electron-hole scattering
◆ Electron-optical-phonon scattering
◆ Intervally scattering (L, X →Γ)
◆ Carrier capture in quantum wells
◆ Intersubband scattering (ΔE >h ωLO) Hot-excitation Regime (~1-100ps)
◆ Hot-carrier-phonon interactions
◆ Decay of optical phonons
◆ Carrier-acousitc-phonon scattering
◆ Intersubband scattering (ΔE <h ωLO) Isothermal Regime (≧100ps)
◆ Carrier recombination
Fig. 2-1 Four temporally overlapping relaxation regimes in photoexcited semiconductors.
2-1.2 Scattering Processes in Semiconductors
There are various kinds of scattering mechanisms. The interactions of electrons, holes, excitons, and phonons will result in different relaxation times in a semiconductor. We will briefly discuss several scattering mechanisms in the following subsections [24].
(a) Carrier-Phonon Interactions
Interaction among carriers and phonons plays an important role in the exchange of energy and momentum between the lattice and carriers, and then determines the relaxation procedure of photoexcited semiconductors. Optical phonon and intervally scatterings proceed in a picosecond time scale, and they are the most frequent events in carrier-phonon interaction. On the contrary, acoustic phonon scattering occurs in a nanosecond time scale and is much less often.
(b) Carrier-Carrier Scattering
Carrier-carrier scattering is accountable for the thermalization of photoexcited non-thermal carriers. The process is judged by Coulomb interaction. It mainly determines the exchange of energy between carriers, and it contains electron-electron, hole-hole, and electron-hole scattering.
(c) Hot Phonons
When a laser beam is incident into a material, the density of photoexcitation will be moderate to high, and a lot of phonons may be generated as the carriers relax by the emission of phonons. The optical phonons near the zone center have a low
unlikely to leave the photoexcited volume. In this case, the decay time is found to be several picoseconds that a large non-equilibrium phonon population (hot phonons) is created at moderate to high photoexcitation density.
Other scattering processes specific to quantum wells are also significant, such as inter-subband scattering, capture of carriers from the barriers into wells, and real space transfer of carriers from quantum wells into the barriers. These are not the primary processes in our experiments, so we will not put emphasis on them in this article
2-2 Carrier-Induced Change in Refractive Index
2-2.1 Band Filling
According to Pauli exclusive principle, electrons and holes as Fermions can only occupy each quantum state once. Therefore with a spin-up and a spin-down carrier, each k state in a semiconductor band can be occupied twice. The carriers in quasi-equilibrium occupy the energetically lowest state first due to the principle of energy minimization. Finally, the states in the bottom of the conduction band will be filled with electrons and in the top of the valence band will be filled with holes.
When pumping energy is above the bandgap, carriers will be excited toward higher states and the absorption coefficient α will decrease. Figure 2-2 is a schematic of the bandfilling process [25].
Fig. 2-2 Bandfilling nonlinearity in a narrow-gap semiconductor.
(a) The absorption spectra at low temperatures for low (curve 1) and high (curve 2) carrier densities. (b) The change in the absorption coefficient, ∆α. (c) The
change in the refractive index, ∆n(ω).
The Fermi functions for the electrons (fe) and holes (fh) can only vary between 0 and 1. If we denote energy in the valence band by Ev and in the conduction band by Ec, then the absorption coefficient of an injected semiconductor is
α(N,P,E) =αo(E) [fv(Ev) - fc(Ec)],
where αo is the density of states near the bandgap in a pure direct-gap semiconductor with
no injection which is given by the square-root law: o( ) C g, g
E E E E E
α = E − ≥ , fv is the probability of an electron occupying a valence band state, and fc is the probability of a hole occupying a conduction band state. N and P are the concentrations of free electrons and holes, respectively. Band filling results in a change of the absorption coefficient∆α( , , )N P E =αo( )[ ( )E f Ev v − f Ec( )c −1 . Near the conduction and valence ] band edges, we know that fc > 0 and fv < 1 with carrier injection. This equation leads to a negative change of absorption coefficient, which means the band filling effect decreases α at a fixed incident photon energy.
The real and imaginary parts of the refractive index n-iκare related to each other
by Kramers-Kronig (KK) integrals: ( ) ( ') ' stands for the principle value of the integral. We define the change of refractive index as Δn (N,P,E) = n(N,P,E) - no(E), where no is the refractive index in the material with no injection. We can get the change of refractive index from the
formula above: ( , , ) ( , , ') '
2 P . Carrier injection results in
∆α, and ∆α leads to ∆n. After calculation, the refractive index increases at energies above Eg and vise versa.
2-2.2 Bandgap Renormalization
Fermions with parallel spin can not sit in the same unit cell. Since this situation would occur for a random distribution, but does not for fermions, we can conclude that the exchange energy increases the average distance between electrons with
parallel spin and consequently reduces their total repulsive Coulomb energy. The reduction of a repulsive energy term means a lowering of the total energy of the electron system.
The correlation energy describes that the electron-hole pair system can lower its energy, if the distribution of electrons and holes relative to each other is not random.
The change of Coulomb force leads to a change of atom’s potential, and then a change of band gap.
Wolff first derived the expression for the band gap renormalization:
( )( eh)/ and in the semiconductor, respectively, and neh is the concentration of free electrons or holes. After carrier excitation, ∆neh results in ∆Eg.
Bandgap reduction causes a red shift of the continuum absorption in semiconductors: ( eh, ) C g g( eh) C g
n E E E E n E E
E E
α
∆ = − − ∆ − − . It predicts that
∆α is always positive. The nearer to the bandgap, the more rapid decreasing of ∆α.
The change of refractive index is also calculated by applying the KK integrals. Due to bandgap renormalization, the largest change of refractive index is near the bandgap and it leads to a negative change of n at energies above the bandgap.
2-2.3 Free-Carrier Absorption
We have mentioned the changes of absorption coefficient and refractive index due to band filling and bandgap renormalization effects. Furthermore, in Drude model, a free carrier in the interband can absorb a photon and move to a higher energy state.
The free carrier absorption is also called plasma effect. The change of refractive
index is )( ) (8 2222
h e
o m
P m
N n c
n=− e +
∆ π ε
λ , whereλis the photon wavelength, me and mh
are the effective mass of electrons and holes, respectively. Free-carrier absorption causes a negative change of refractive index n.
We have introduced the concepts of carrier dynamics and the background of refractive index change. To find out the ultrafast behavior of InGaAs1-xNx SQW, the experimental setup of time-resolved photoreflectance will be demonstrated in the following chapter.