Brazilian Journal of Physics, vol. 29, no. 4, December, 1999
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High Temperature Behavior of Subpicosecond Electron Transport Transient in 3C- and 6H-SiC E. F. Bezerra, E. W. S. Caetano, V. N. Freire,
Departamento de Fsica, Universidade Federal do Ceara, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceara, Brazil
E. F. da Silva Jr.,
Departamento de Fsica, Universidade Federal de Pernambuco, Cidade Universitaria, 50670-901 Recife, Pernambuco, Brazil
and J. A. P. da Costa
Departamento de Fsica Teorica e Experimental, Universidade Federal do Rio Grande do Norte, Caixa Postal 1641, 59072-970 Natal, Rio Grande do Norte, Brazil
Received February 8, 1999 A study of the subpicosecond high- eld electron transport transient in 3C- and 6H-SiC polytypes at high lattice temperatures is performed. Electric eld intensities up to 1000 kV/cm and lattice temperatures of 300, 673, and 1073 K are considered. It is shown that the transient regime behavior depends on the electric eld intensity as well as on the lattice temperature, but it is always shorter or of the order of 0.3 ps. For the lowest lattice temperature, an electron velocity overshoot can always occur when the electric eld intensity is higher than 300 kV/cm, but it decreases and can even disappear when the lattice temperature is raised.
I Introduction In the past decade, we have witnessed the growing technological importance of silicon carbide (SiC) as a new material with a number of physical and electrical properties which surpasses, in certain conditions, those of crystalline silicon. The use of SiC in new generations of advanced devices is stimulated by its high stability, strong chemical bonding, large thermal conductivity and breakdown elds, high saturation velocity and low leakage current. SiC may be grown in either cubic or hexagonal crystalline forms, presenting a large variety of polytypes (3C-, 2H-, 4H- and 6H-SiC are some of the most common forms). Among the wide range of applications of SiC polytypes are suitable for high-power, high-temperature, high-frequency, radiation-hard, and light-emitting devices. Ab initio studies of SiC electronic properties showed that their polytypes present a wide range of band gaps (1.27-3.36 eV) and di erent values for the carrier e ective mass as a result of signi cant di erences in their band structures [1]. 3C-SiC (Eg =2.39 eV) and 6H-SiC (Eg = 2.86 eV) have been
more extensively studied due to potential applications associated to their higher saturation velocities, particularly in the domain of high-temperature and highspeed/high- eld nanostructures and devices circuitry. On the other hand, non-stationary physical conditions are often imposed on the electron transport mechanisms, which lead to electron drift velocity and energy time transients before steady-state transport conditions are attained. Cubic SiC high- eld transport in the steady-state was studied more than two decades ago by Ferry [2] through the balance equation method. In recent years, a number of new and interesting SiC high- eld electron transport studies have been addressed by several authors [3-8]. Steady-state transport behavior at room temperature was studied in 3C-SiC using Monte Carlo simulation within a single equivalent isotropic valley picture [3], and considering both parabolic and nonparabolic band schemes. It was shown that the electron drift velocity presents a gentle peak at electric elds around 400 kV/cm in the nonparabolic band case. Similarly, Monte Carlo calculations [4, 5] of electron
786 transport in 4H-SiC indicate a mobility reduction as the temperature is increased in the 100-700K range. Other studies performed using the Monte Carlo method [6, 7] further investigated the steady-state transport properties in 6H-, 4H-, and 3C-SiC, mainly at room temperature. The hydrodynamic balance equation method was recently used to study the high temperature hot electron steady-state transport behavior in 3C- and 6H-SiC [8]. The calculations were performed within the single equivalent isotropic valley approach for lattice temperatures of 300, 600, and 1073K, and have also considered band nonparabolicity. The behavior of the electron drift velocity and energy with the external electric eld indicate a shift at high elds of a smooth peak in the drift velocity-electric eld relation as the temperature is raised. In all of these works, the main scattering mechanisms considered in determining the steadystate high- eld electron transport in the 4H-, 6H-, and 3C-SiC polytypes were the acoustic deformation potential, polar-optical phonon, intervalley phonon, ionizedimpurity and impact ionization. To date, most of the investigations on the SiC high eld transport were restricted to steady-state phenomena, principally the determination of the saturated electron drift velocity. However, it is also of scienti c and technological interest to have information on the ultrafast transport transient properties of silicon carbide polytypes. This need arises because SiC thermal stability at high temperatures is a remarkable characteristic for advanced technological applications, while the subpicosecond time response to high electric elds make SiC presently to be one of the most important semiconductors for high-frequency applications in high power and high temperature operating devices. In this work, it is presented a study of high lattice temperature e ects on the subpicosecond transient behavior of the electron drift velocity and energy in 3C- and 6H-SiC as they attain their steady-state. Boltzmann-like transport equations in the momentum and energy relaxation time approximation are used to calculate the time evolution of the mean electron drift velocity and energy within a single equivalent isotropic valley picture for both polytypes. Acoustic, polar optical and intervalley phonons, as well as neutral, ionized impurities and impact ionization are the main electron scattering mechanisms considered in the transport calculations. The transient regime is shown to be shorter than 0.3ps for electric elds up to 1000kV/cm. For the lowest lattice temperature, an overshoot in the electron
E.F. Bezerra et al. drift velocity always occurs when the applied electric eld intensity is higher than 300kV/cm.
II Model and transport equations The model used for the 3C- and 6H-SiC polytypes during the calculations performed in this study consider a simple isotropic valley picture for their band structures [3]. The electron e ective mass in the equivalent conduction band mc is taken as 0.346m0 for 3C-SiC and 0.3m0 for 6H-SiC, where m0 is the free electron mass. This procedure has been successfully adopted in a recent Monte Carlo simulation of the 3C-SiC high eld steady-state transport at 300K, and in the high temperature hot electron steady-state transport calculations for 6H- and 3C-SiC which have being performed using the hydrodynamic balance equation method [3, 8]. In addition, the model accounts for band nonparabolicity through the hyperbolic approximation, in which the following relation is valid: [3, 8, 9] h 2k2 = "(1 + ") ; (1) 2mc where is the nonparabolic band coecient, which is 0.323eV,1 for 3C-SiC, and = (1 , mc =m0)2 for 6H-SiC [8]. The time evolution of the mean electron drift velocity v(t) and energy "(t) is obtained by solving numerically their two coupled Boltzmann-like transport equations in the momentum and energy relaxation time approximation, as given below: dv(t) = qEF , v(t) ; dt mc p (")
(2)
d"(t) = qv(t)E , "(t) , "L ; (3) F dt " (") where EF is the electric eld intensity; "L = 3kB TL =2 is the average electron thermal energy at the lattice temperature TL ; p (") and " (") is the momentum and energy relaxation time, respectively; kB is the Boltzmann constant, and q is the electric charge of the electron. The calculations are performed for electric elds of 200, 600, and 1000kV/cm. They are assumed to be applied in the < 111 > direction for the sake of disregarding electron redistribution among equivalent valleys, and to minimize the electron population in higherlying subsidiary minima. The relaxation times p (")
Brazilian Journal of Physics, vol. 29, no. 4, December, 1999 and " (") are obtained through the steady-state relations v EF and " EF determined for 3C-SiC and 6H-SiC at the lattice temperatures TL =300K, 673K, and 1073K, as described elsewhere [8]. Therefore, the electron scattering mechanisms considered here are acoustic-deformation-potential, polar-optical phonon, intervalley phonon, and ionized impurity scattering. The use of the steady-state relations v EF and " EF to calculate the relaxation times p (") and "(") is a successful scheme pioneered by Shur [10], which was thereafter used by other researchers [11, 12], either to study transient behavior of carriers in GaAs or as a way to save computational time during the modeling of a submicrometer gate eld-e ect transistor including e ects of nonsteady-state dynamics. Recently, this technique allowed the study of high-magnetic- eld effects on the terahertz mobility of hot electrons in n-type InSb [13]. While Monte Carlo transport simulations of the transient regime is very time consuming, the approach used here to solve Eqs. (2) and (3) requires very short computation time since at most a standard Runge-Kutta scheme of fourth order is necessary for their numerical solution.
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Figure 1. Subpicosecond behavior of the mean electron drift velocity in 3C-SiC for lattice temperatures of 300 K (top), 673 K (middle), and 1073 K (bottom). The applied electric eld intensities are 200 kV/cm (dotted line), 600 kV/cm (dashed line), and 1000 kV/cm (solid line).
III Results and Discussions Fig. 1 and 2 (Fig. 3 and 4) present, respectively, the time evolution of the mean drift velocity and energy of electrons in 3C-SiC (6H-SiC) at lattice temperatures of 300 K (top), 673 K (middle), and 1073K (bottom). The curves were obtained for electric eld intensities of 200kV/cm (dotted lines), 600kV/cm (dashed lines), and 1000kV/cm (solid lines). The transient behavior of the mean drift velocity and energy is always shorter than 0.3ps for both SiC polytypes, being a little bigger in 3C-SiC ( 0:3ps) than in 6H-SiC ( 0:2ps). With the lattice temperature increasing from 300K to 1073K, one can observe a strong reduction of the maximum values for both the electron mean drift velocity and energy, which is a consequence of the scattering mechanisms strength. For high enough electric elds, the mean electron drift velocity in both polytypes presents an overshoot, as shown in Figs. 1 and 3. At 300K, the electron velocity overshoot appears when the electric eld intensity is higher than 300kV/cm. However, the overshoot e ect is substantially reduced when the lattice temperature increases, and can even disappear when the lattice
Figure 2. Subpicosecond behavior of the mean electron energy in 3C-SiC for lattice temperatures of 300 K (top), 673 K (middle), and 1073 K (bottom). The applied electric eld intensities are 200 kV/cm (dotted line), 600 kV/cm (dashed line), and 1000 kV/cm (solid line).
788 temperature is high enough (see dotted lines in both the gures). For a given electric eld and instant of time, v(t) is always higher in 3C-SiC than in 6H-SiC. This is particularly true in the velocity overshoot case, which is more pronounced in the former than in the latter. It has been suggested that an overshoot e ect in the electron drift velocity may occur for low elds only when the mean momentum relaxation rate of the electron is larger than its mean energy relaxation rate [14, 15]. The overshoot e ect can also be due to the intervalley transfer mechanism and the related change in the electron e ective mass associated to the intervalley scattering process [14, 15].
Figure 3. Subpicosecond behavior of the mean electron drift velocity in 6H-SiC for lattice temperatures of 300 K (top), 673 K (middle), and 1073 K (bottom). The applied electric eld intensities are 200 kV/cm (dotted line), 600 kV/cm (dashed line), and 1000 kV/cm (solid line).
Fig. 2 and 4 depicts, respectively, the time evolution of the mean electron energy in 3C-SiC and 6H-SiC towards the steady state. When the electric eld intensity is low, the energy gained by the electrons from the heating action of the electric eld (see the rst term in the right side of Eq. (3)) is mainly transferred to the phonons through the energy dissipation term (second term in the right side of Eq. (3)). Since at the beginning the momentum dissipation is small, the growth rate of the mean electron drift velocity is higher than the growth rate of the mean electron energy, and consequently v(t) increases faster than "(t). For high eld
E.F. Bezerra et al. intensities, the dissipative term is not strong enough to drain the excess energy transmitted from the electric eld to the electrons. As a consequence, this excess is retained as electron thermal energy, and the growth rate of the electron mean energy increases strongly. At high lattice temperatures, the variation of the growth rate of the mean electron energy is smaller because the electron energy dissipation is stronger. Finally, it is worth to highlight the long term behavior of both the mean electron drift velocity and energy presented in Figs. 1-4 is in very good agreement with steady-state calculations performed either through Monte Carlo simulations or using the hydrodynamic balance equation method [8, 6].
Figure 4. Subpicosecond behavior of the mean electron energy in 6H-SiC for lattice temperatures of 300 K (top), 673 K (middle), and 1073 K (bottom). The applied electric eld intensities are 200 kV/cm (dotted line), 600 kV/cm (dashed line), and 1000 kV/cm (solid line).
IV Summary The high- eld transport behavior in the transient regime of electrons in 3C-SiC and 6H-SiC at lattice temperatures of 300K, 673K, and 1073 K was studied for electric elds intensities up to 1000 kv/cm. The calculations were performed within a simple equivalent isotropic valley picture, and nonparabolic band e ects were considered through the hyperbolic model. The
Brazilian Journal of Physics, vol. 29, no. 4, December, 1999 transient regime was shown to occur in less than 0.3ps for both SiC polytypes. It was demonstrated also the possibility of existence of an overshoot in the electron drift velocity, which depends strongly on the applied electric eld intensity, the lattice temperature and the SiC polytype. The overshoot e ect presents a remarkable reduction as the lattice temperature is raised. For the electron mean energy, it was shown that its growth rate is substantially reduced with the increase of the lattice temperature.
Acknowledgments
E. F. Bezerra and E. W. S. Caetano are sponsored by graduate fellowships from the Brazilian National Research Council (CNPq) at the Physics Department of the Universidade Federal do Ceara. V. N. F. and E. F. S. Jr. would like to acknowledge the nancial support received during the development of this work from the Science Funding Agencies of the Ceara and Pernambuco states in Brazil (FUNCAP and FACEPE, respectively), the Brazilian National Research Council (CNPq), and the Ministry of Planning (FINEP) under contract PADCT/SDRI # 77.97.1120.00 3235/97.
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