International Journal o the Physical Sciences Vol. 5(11), pp. 1752-1756, 18 September, 21 Available online at http://www.academicjournals.org/ijps ISSN 1992-195 21 Academic Journals Full Length Research Paper Numerical calculation o the electron mobility in and semiconductors using the iterative method M. Rezaee Ron-Abadi Physics Department, Ferdowsi University o Mashhad, Mashhad, Iran. E-mail: ronabad@um.ac.ir. Accepted 21 August, 21. The electron mobility o and semiconductor compounds were calculated by using the iteration method. We considered polar optical phonon scattering, deormation potential acoustic, piezoelectric and ionized impurity scattering mechanisms. Band nonparabolicity, admixture o p unctions, arbitrary degeneracy o the electron distribution, and the screening eects o ree carriers on the scattering probabilities are incorporated. We investigated temperature and doping dependencies o mobility or the given compounds. The electron mobility o the two materials was ound to be similar, though the characteristics were on the whole superior. It is also ound that the electron mobility decreases monotonically as the temperature increases rom 1 to 6 K or two crystal structures. The low temperature value o electron mobility increases signiicantly with increasing doping concentration. Key words: Electron mobility, piezoelectric, ionized impurity, nonparabolicity. INTRODUCTION The transport properties o and have been a subject o extensive investigation in recent years (Chen et al., 1998). There has been a great interest in the study o charge carrier transport o and due to the wide band gap, which has become one o the most applied insulators in thin-ilm electro luminescence devices (Loo et al., 1998). Most previous studies have assumed the electron distribution unction o cool temperature with high-energy tail ends at less than 4 ev. Maino et al. (21) have used a single parabolic band in the irst conduction band. This is at very high ield unrealistic. The ull band details o the irst two conduction bands as well as ull-order treatment o the electron phonon interaction have been included by Brennan (1998). Electron transport at low electric ield was calculated or the irst time by solving the Boltzmann transport equation, exactly under generalized Fermi-Dirac statistics or the most scattering events. Electron mobility and resistivity were calculated as a unction o electron concentration at both 3 and 77 K (Di and Brennan, 1991). Mansour et al. (2) used a model that included only the irst conduction in a nonparabolic three valleys model. Chattopadhyay and Queisser (1981) have developed a model including a nonparabolic three valleys o the irst conduction band and a single valley in the second conduction band. Using the pseudopotential method, the density o states in the last case was included. Most o the pervious theoretical wor on transport properties o and, in high electric ield, and at room temperature has been mainly based on Monte Carlo method. The temperature and doping dependencies o electron mobility in and are o particular relevance to the large eort at present to producing AC and DC electro luminscent devices. AC electro luminscent devices were made by encapsulating large band gap semiconductors such as : Mn or : Mn by two insulates layers, typically Y 2 O 3, on either side o the semiconductor layers. An AC bias applied across the device alternately accelerates the electrons rom one semiconductor insulator interace to the other. An electro luminscent phenomenon occurs when the ree carriers are accelerated to suiciently high energy, such that impact excitation o the Mn center becomes possible. In and semiconductor compounds the dominant scattering mechanisms is primarily the polar optic phonon scattering mechanism. The others are ionized impurity scattering, acoustic phonon deormation potential scattering and acoustic phonon piezoelectric scattering mechanisms (Moglestue, 1993). As a result o being elastic process o acoustic phonon deormation potential scattering, acoustic piezoelectric scattering and
Ron-Abadi 1753 ionized impurity scattering, their mobilities can be calculated with the relaxation time approach. But i the electron energy can be compared with the phonon energy, this is important and such a scattering is astic scattering. The compensated changing in the energy is taen place ater astic scattering. Since in astic scattering process the phonon energy is the bigger than the electron energy. Thus, relaxation time approach is not valid or astic scattering process (Jacoboni and Lugli, 1989). Because o this the calculation o polar optic phonon scattering is used the other numerical method instead o relaxation time approach. In this wor it was used an iterative method or the calculation. MODEL DETAILS The present calculations o electron mobility are based on solving Boltzmann transport equation with iterative method (Liu et al., 2). Rodes iterative technique provides a compact method o solution o the Boltzmann equation in the low ield regime (Ridley, 1997). To derive Rodes method, we start with the Boltzmann transport equation or the case o steady-state conditions and no spatial gradients. + v. r ef +. Where ( / t) coll represents the change o distribution unction due to the electron scattering. In the steady-state and under application o a uniorm electric ield the Boltzmann equation can be written as ef. t coll Consider electrons in an isotropic, non-parabolic conduction band whose equilibrium Fermi distribution unction is () in the absence o electric ield. Note the equilibrium distribution () is isotropic in space but is perturbed when an electric ield is applied. I the electric ield is small, we can treat the change rom the equilibrium distribution unction as a perturbation which is irst order in the electric ield. The distribution in the presence o a suiciently small ield can be written quite generally as ) ( ) + ( )cosθ (3) ( 1 Where θ is the angle between and F and 1() is an isotropic unction o, which is proportional to the magnitude o the electric ield. () satisies the Boltzmann Equation 2 and it ollows that ef coll { cosϕ 1 [ si (1 ) + si ] d 1 [ si (1 ) + si ] d } i In general there will be both elastic and astic scattering processes. For example impurity scattering is elastic and acoustic and piezoelectric scattering are elastic to a good approximation at room temperature. However, polar and non-polar optical phonon scattering are astic. Labeling the elastic and astic scattering rates with subscripts el and respectively and recognizing that, or any process i, s eli(, ) s eli(, ) equation 4 can be written as (1) (2) (4) ( ) 1 ef + 1cosϕ [ s(1 ) + s)] d (5) (1 cosϕ ) s d + [ s (1 ) + s ] d el Note the irst term in the denominator is simply the momentum relaxation rate or elastic scattering. Equation 5 may be solved iteratively by the relation ef + 1( )[ n 1]cos ϕ[ s(1 ) + s )] d 1 ( ) (6) n (1 cosϕ ) s d + [ s (1 ) + s ] d el Where 1n() is the perturbation to the distribution unction ater the n-th iteration. It is interesting to note that i the initial distribution is chosen to be the equilibrium distribution, or which 1() is equal to zero, we get the relaxation time approximation result ater the irst iteration. We have ound that convergence can normally be achieved ater only a ew iterations or small electric ields. Once 1() has been evaluated to the required accuracy, it is possible to calculate quantities such as the drit mobility µ, which is given in terms o spherical coordinates by 3 ( /1 + 2αF ) 1d µ * 3m F 2 3 d (7) Here, we have calculated low ield drit mobility in and crystal structures using the iterative technique. The eects o piezoelectric, acoustic deormation, polar optical phonons and ionized impurity scattering have been included in the model. It is also assumed that the electrons remain in the -valley o the Brillouin zone. Further we supposed that the materials have the isotropic nonparabolic band structure. We too into account electron screening and mixing o s and p wave unctions. LOW-FIELD TRANSPORT RESULTS IN AND In the wor, electron mobility or and compounds are calculated using iterative method. We investigated temperature and doping dependencies o electron mobility in range o 15 to 5 K. The electron concentration o and compounds was taen as 1 22, 1 23 and 1 24 m -3. Further in this wor we calculated the mobility or various concentrations. Important parameters used throughout the calculations are listed in Table 1. Figure 1 shows the calculated electron drit mobilities versus temperature and donor concentration or and compounds. The electron drit mobilities at room temperature that we ind are 125 and 15 cm 2 V -1 s -1 or and structures, respectively, or an electric ield equal to 1 4 Vm -1 and with a donor concentration o 1 23 m -3. The material parameters used to calculate the electron drit mobilities are tabulated in Table 1. The results plotted in igure 1a indicate that the electron drit mobility o is lower than that or the structure. This is largely due to the higher valley 3
1754 Int. J. Phys. Sci. Table 1. Important parameters used in our calculations or and materials which are taen rom reerences (Chattopadhyay and Queisser, 1981). Density ρ (gm -3 ) 475 542 Longitudinal sound velocity v s. (ms -1 ) 5668 458 Low-requency dielectric constant ε s 9.6 9.2 High-requency dielectric constant ε 5.7 15 Acoustic deormation potential (ev) 4.9 4.5 Polar optical phonon energy (ev).426.314 -valley eective mass (m * ).28.17 -valley non-parabolicity (ev -1 ).69.67 6 14 4 2 (a) Electron Mobility (cm -2 /V.s) 12 1 8 (b) 2 3 4 5 6 1E22 1E23 1E24 Donor Concentration (m -3 ) Figure 1. (a) Electron drit mobility o and structures versus temperature. Donor concentration is approximately 1 23 m -3, (b) Electron drit mobility o and structures versus donor concentration at room temperature. eective mass in the structure. Figure 1b shows the calculated variation o the electron mobility as a unction o the donor concentration or both and crystal structures at room temperature. The mobility does not vary monotonically between donor concentrations o 1 22 m -3 and 1 24 m -3 due to the dependence o electron scattering on donor concentration, but shows a maximum at 1 22 m -3 or both compounds. In order to understand the scattering mechanisms which limit the mobility o and under various conditions, we have perormed calculations o the electron drit mobility when particular scattering processes are ignored. The solid circle curve in Figure 2 shows the calculated mobility or material including all scattering mechanisms whereas the open circle and star curves show the calculated mobility without ionized impurity and polar optical scattering, respectively. It can be seen that below 25 K the ionized impurity scattering is dominant while at the higher temperatures electron scattering is predominantly by optical modes. Thus the mared reduction in mobility at low temperatures in Figure 2 can be ascribed to impurity scattering and that at high temperatures to polar optical phonon scattering. The temperature variation o the electron drit mobility in and compounds or dierent donor concentrations is shown in Figure 3. It is evident rom this igure that the curves approach each other at very high temperatures, where the mobility is limited by longitudinal optical phonon scattering, whereas the mobility varies inversely with donor concentration at low temperatures as we would expect rom the oregoing discussion. Conclusions In conclusion, we have studied the electron transport characteristic associated with and compounds. Temperature dependent and ree electron concentration dependent o the electron mobility in both structures have
Ron-Abadi 1755 1 1 All scattering mechanisi Without polar optical scattering W ithout impurity scattering 1 1 2 3 4 5 Figure 2. Comparison o electron drit mobility material with donor concentration o 1 23 m -3 and when individual scattering processes are ignored. 6 6 4 2 1 22 m -3 1 23 m -3 1 24 m -3 4 2 1 22 m -3 1 23 m -3 1 24 m -3 1 2 3 4 5 1 2 3 4 5 Figure 3. Calculated low-ield electron drit mobility o and compounds as unctions o temperature or dierent donor concentrations. been compared. It has been ound that the low-ield electron mobility is signiicantly higher or the structure than structure due to the lower electron eective mass in this crystal structure. Several scattering mechanisms have been included in the calculation. Ionized impurities have been treated beyond the born approximation using a phase shit analysis. Screening o ionized impurities has been treated more realistically using a multi-ion screening ormalism, which is more relevant in the case o highly compensated III-V semiconductors. ACKNOWLEDGMENTS This wor is supported by the Ferdowsi University o Mashhad through a contract with Vive president or Research and Technology.
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