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1. Introduction
The wide-band gap semiconductor materials such as silicon carbide, gallium nitride, aluminum nitride, diamond, and others, are often referred to as third-generation semiconductor materials. Aluminum nitride (AlN) with a large direct band gap of 6.2 eV is a typical representative of these materials. Due to its ultra-wide direct band gap, high carrier mobility, large breakdown field, high thermal conductivity, high volume resistivity, high chemical and thermal stabilities, high dielectric strength, and low dielectric loss, make AlN an excellent material for novel electronic and opto-electronic device applications: i) laser diodes (LDs), deep-ultraviolet (DUV) solid-state light-emitting diodes (LEDs); ii) high frequency, high-power, and high-temperature electronic applications; iii) for packaging and substrates for high-density or high-power assemblies of microelectronic components [1,2,3].
The study of photoinjected (or photoexcited) carriers in AlN-based semiconductors can provide information regarding both their phonon and carrier dynamics properties, which are important for a better device design. Basic research regarding the ultrafast transport transient of hot carriers is relevant for technological applications in sub-micron devices.
In this paper, an analysis was performed to investigate the ultrafast transport properties of photoinjected carriers in wurtzite AlN subjected to electric fields up to 80 kV/cm. The optical and transport properties of semiconductors have been studied mainly by using Nonequilibrium Green’s Functions Techniques [24], Monte Carlo simulation [25,26], balance equation approach [27,28], Boltzmann transport equations [26,29,30], etc. In this paper was used the “Nonequilibrium Statistical Operator Method” (NESOM) [31-35]. The NESOM is a powerful formalism that seems to offer an elegant and concise way for an analytical treatment in the theory of irreversible processes, adequate to deal with a large class of experimental situations, and physically clear picture of irreversible processes. The NESOM is also practical and efficient in the study of the optical and carrier dynamics in semiconductors [36-42]. More specifically, a Non-Equilibrium Quantum Kinetic Theory [43] derived from NESOM was used in this paper.
2. The evolution equations
We consider a pure WZ AlN sample, with a concentration n of electron-hole pairs created by an intense laser light pulse. The photoinjected carriers were far from equilibrium with the lattice but thermalized between themselves by Coulomb interaction after a fraction of picosecond. A constant electric field Ɛ was applied to the sample to accelerate the carriers as well as relax the energy and momentum to the phonon field simultaneously. The sample was in contact with a thermal reservoir at temperature T 0.
The following variables were used to describe the nonequilibrium thermodynamic state of the system: i) The reciprocal of the carrier nonequilibrium temperature, β*c(t)=(1/kBT*c)ii) The reciprocal of the longitudinal optical phonon nonequilibrium temperature, β*LO(t)=1/(kBT*LO); iii) The reciprocal of the acoustic phonon nonequilibrium temperature, β*AC(t)=1/(kBT*AC); iv) The variables, -β*c(t)μ*e(t) and -β*c(t)μ*h(t), where μ*e(t) and μ*h(t) are the electron and hole chemical potentials, respectively; v) The variables: -β*c(t)ve and -β*c(t)vh, where v e and v h are the electron and hole drift velocities, respectively. Here, k B is the Boltzmann constant. Note that the term “nonequilibrium temperature”, symbolized by T*, is also referred to as “quasi-temperature”.
The nonlinear coupled set of integro-differential generalized transport equations that govern the time-evolution of these nonequilibrium thermodynamic variables is based on nonlinear quantum kinetic theory, which is based on a nonequilibrium statistical ensemble method [43], and the Markovian approximation is used [44]. The equations of evolution for the carriers’ linear momentum, carriers’ energy and LO, TO, AC phonons energy are as follows [43]:
In Eqs. (1) and (2) the first term on the right side is the force produced by the electric field (J(1)Pe=-NeE, J(1)Ph=NeE), where N = nV , with n being the concentration of electron-hole pairs, and V the volume of the sample. In Eqs. (1) and (2) the second term on the right side is the rate of momentum transferred to the phonons [43].
In Eq. (3) the first term on the right side accounts for the rate of energy pumped to the carrier system owing to the the electric field,
and the second term on the right side represents the rate of excess energy transferred to the phonons [43].
The populations of the LO- and AC- phonons [43] are expressed as follows:
where ω (LO) and ω (AC) are the frequencies of the (LO) or (AC) phonons, respectively. We take the “Einstein model” dispersionless frequency relation for LO phonons, that is: hbarωq,LO=ℏω0 (i.e. all LO phonons have the same angular frequency ω 0), and the “Debye model” dispersionless frequency relation for AC phonons, that is: ℏωq,AC=ℏqvs, where v s is the sound velocity. Note that the AC and LO phonons were considered to be internally thermalized, disregarding possible differentiated distribution of populations in reciprocal space as a result of what can be termed as Frohlich-¨ Cherenkov effect [45-48]. The inhomogeneous distribution in reciprocal space is restricted to a very small region of the Brillouin zone [45-48] and such effect can be neglected in the present study.
In Eqs. (4) and (5) the first term on the right side are the gain of energy pumped on the phonons (LO and AC, respectively) by the “hot” carriers, and the second term on the right side in Eqs. (4) and (5) are the transfer, via anharmonic effects, of the energy of LO-phonons to the AC-phonons [43]
where τ op,an is the relaxation time obtained from the band width from Raman scattering experiments [49] and V cell is the volume of the unit cell.
The second term on the right side in Eq. (5) is the same of Eq. (4), but with the opposite sign. The last term in Eq. (5) is the contribution of thermal diffusion to the reservoir (thermal bath) at temperature T 0.
Finally, it noteworthy that, to close the system of equations, we must express the linear momentum of the carriers, and the energies of the phonons and carriers, as follows [43]:
3. Results
The calculations were performed in WZ AlN for photoinjected carriers with a concentration n = 1018 cm−3 and an excess energy of 0.6 eV per pair gained from photon absorption. After Coulomb thermalization, the initial carrier temperature was approximately T c (0) = 2321 K. The thermal bath temperature considered was T 0 = 300 K, and the carrier scattering mechanisms considered in the numerical calculations included piezoelectric, deformation acoustic and polar optical (or Fröhlich’s interaction). Table I summarizes the material parameters of the WZ AlN used in numerical calculations in this paper. In Table I, m 0 is the free electron mass.
TABLE I Parameters of wurtzite AlN.
| Parameter | Value |
| Electron effective mass, m*e | 0.48m0 [50] |
| Hole effective mass, m*h | 3.53m0 [40] |
| Band gap, Eg | 6.2 eV [51] |
| Intervalley energy separation | 0.7 eV [51] |
| Lattice parameter, a | 3.11 Å [52] |
| Lattice parameter, c | 4.98 Å [52] |
| Static dielectric constant, ε0 | 8.5 [53] |
| High frequency dielectric constant, ε∞ | 4.77 [53] |
| Optical phonon energy, ℏω0 | 99.2 meV [53] |
| Mass density, ρ | 3.23 g/cm3 [54] |
| Longitudinal sound velocity, v sl | 10127 m/s [55] |
| Transversal sound velocity, v st | 6333 m/s [55] |
| Acoustic deformation potential, E 1 | 9.5 eV [55] |
| Piezoelectric constant, h pz | 0.92 C/m2 [50] |
Figure 1 shows, for four values of the electric field intensities, the time evolution of the nonequilibrium temperature of the carriers, describing a extremely rapid process of relaxation of energy to the lattice. It can be noticed that for E≲80 kV/cm. the carriers cool down to a steady state in less than 0.4 ps.
FIGURE 1 Time evolution of carriers nonequilibrium temperature in photoinjected WZ AlN for four electric field intensities.
Figure 2 shows, for an electric field strength of 80 kV/cm, the evolution of the rate of change of the carriers nonequilibrium temperature with respect to time, dT*c/dt. The rate of change (in arbitrary units) is negative for up to 0.05 ps. The arrow in Fig. 2 indicates this point. For values greater than 0.05 ps the rate of change becomes positive. The behavior of the curves for the other electric field values are similar.
FIGURE 2 Time evolution of rate of change of carriers nonequilibrium temperature in photoinjected WZ AlN, in arbitrary units, for an electric field of 80 kV/cm.
Note that Eq. (3) comprises two terms: J(1)Ec and J(2)Ec. The first term, J(1)Ec, is a source term. It is the energy received of the carriers due to applied electric field. The second term, J(2)Ec, is the dissipation of energy: the carriers lose his initial excess of energy (gained in photon absorption processes) for the phonons. In larger electric fields J(1)Ec is also larger, and the total dissipation of energy (J(1)Ec+J(2)Ec) is slower that for smaller values of electric fields.
Figure 3 shows the time evolution of the electron (Fig. 3a) and hole (Fig. 3b) drift velocities for four electric field intensities. After a transient regime in the order of picoseconds, a steady state was achieved. Similar to Fig. 1, as larger the electric field more time the system takes to reach the steady state. During the transient regime, it can be noticed the behavior of the maxima existence for holes, with one of the maxima corresponding to a velocity drift overshoot for fields larger than approximately 40 kV/cm. The structured ultrafast hole drift velocity in photoinjected WZ AlN is explained in this paper uniquely through the crossover of the evolution curves for the transport and momentum relaxation times, whose definition is based on the nonequilibrium variables used to describe the system [56]. The values obtained for the hole drift velocities are much smaller than those of electrons. This fact can be ascribed to the holes having a higher effective mass than the electrons: m*h/m*e≃7.3 (see Table I).
FIGURE 3 Time evolution of drift velocity: a) electrons and b) holes, in photoinjected WZ AlN for four electric field intensities.
Using the results of Fig. 3, we can derive the mobility, M e (h), of the electrons and holes, as given by M e (h) = |v e(h) |/|Ɛ|. For a electric field of 20 kV/cm the electron and hole mobility are approximately Me≃213 cm2/V·s and Mh≃21.3 cm2/V·s, respectively. These values are among the range of theoretical and experimental data obtained for the WZ AlN [15,50,53,57].
Figure 4a) displays the time evolution of the nonequilibrium temperature of the LO-phonons for four different values of the electric field intensities. We can verify that the LO-phonon quasitemperature does not increase appreciably above equilibrium temperature T 0: approximately 3% for an electric field of 80 kV/cm, the higher field we have considered. Figure 4b) shows the time evolution of the nonequilibrium temperature of the AC-phonons for four different values of the electric field intensities. It is possible to verify that the temperature increase of the AC-phonons are irrelevant.
4. Conclusions
In this study, we obtain equations that govern the time evolution of the nonequilibrium state of highly photoinjected double plasma in the semiconductor WZ AlN subjected to electric field intensities to 80 kV/cm. For this study, we used a nonlinear quantum transport theory derived from the Nonequilibrium Statistical Operator Method [31-35]. We presented a study regarding ultrafast transport transient characteristics, focusing on the dependency of the nonequilibrium temperatures (carriers and phonons) as well as the electron and hole drift velocities on the electric field strength. Maxima of the hole drift velocity in its transient regime should be observable depending on the evolution of the nonequilibrium macroscopic state of the photoinjected carriers in WZ AlN. The transient regime occurred on a picosecond scale. The longitudinal optical phonons are slightly heated up in scattering events involving Frohlich interaction with the carriers,¨ and the temperature increase of the AC-phonons was irrelevant.
