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  1. 1. Mohammad Khooshebast / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.633-636 The Scattering Mechanisms and Electronic Properties of SiC Semiconductor M. Khooshebast and H. Arabshahi Department of Physics, Payame Noor University, Tehran, IranAbstractWe have studied the band structure properties of atoms by C atoms in wurtzite crystal structure. ThereSiC in wurtzite crystal structures. In our are two important parameters which should becalculations we have adopted a pseudopotential optimized correctly for saving time and also to haveapproach based on the Density Functional Theory an acceptable precision in calculations. These(DFT). We have calculated the band structure and parameters are cutoff-energy and kgrid-cutoff wheredensity of state (DOS). The result shows that the their optimized values were chosen. SiC canelectronic band structure and density of state data crystallize in zincblende, wurtzite and rombohedralfor SiC in wurtzite crystal structures are structures, which have slightly different materialcomparable with their experimental calculations. properties and substantially different band structuresIn the second part we have calculated the effective [3]. However, the wurtzite phase of SiC is the moremass of electron in three valley model. And finally stable bulk form of the material and the commonin the third part of this survey we have studied form for epitaxial layers. The wurtzite crystalpiezoelectric, deformation potential and optical structure of SiC is shown in figure 1. It can bephonon scattering mechanisms. regarded as an interpenetration of two hexagonally close packed (HCP) sublattices with a relativeKey words: Pseudopotential; Density Functional displacement along the c-axis by uc where u is the soTheory; Effective Mass; Electron Scattering. called internal parameter and c is the lattice constant for the c-axis. The distance between in-plane atomicBand Structure Calculation neighbors is denoted by a. For an ideal wurtzite First principles or ab-initio approaches structure c/a=(8/3)1/2 =1.633 and u=3/8=0.375 and theprovide a method for modeling systems based solely atoms have the regular tetrahedral coordination ofon their atomic coordinates and the Z numbers of the nearest neighbors that occurs for the zincblendedifferent atoms. These techniques rely on the fact that structure [4]. The difference between the twothere should be one unique charge density or structures is in the next nearest neighbordistribution which describes the ground state of a arrangement. The first Brillouin zone of wurtzite SiCsystem [1]. This reduces the problem of solving for which is a hexagonal prism of volume is shown inthe electronic structure of a system from a 3N figure 1 as well together with the locations of thedimensional problem to one that only depends on the high-symmetry points , K, M and U [5,6].charge density. A number of approaches have beendeveloped to properly reduce a system to theminimum energy electronic configuration and abinitincludes several of these. First principles codes suchas abinit have proved useful to topics ranging fromthe composition of the planetary core to the electricalproperties of single molecule. Abinit can calculate a)the forces on atoms in a structure and use thisinformation to relax the system. This has providedcritical understanding in how surfaces reconstruct,how absorbates interact with surface sites, and howmagnetic impurities affect neighbor lattice sites.Typically calculations can consider 30-40 atomscomfortably. Larger systems can be done, but forfairly large systems (greater than 150 atoms), parallelcalculations are essential [2-6]. In addition tominimizing the total energy of a structure, abinit canalso calculate band structures, density of states, b)magnetic properties, and phonon dispersion curves. figure1. a) The crystal structure of wurtzite SiC andWe made a 6×6×6 super cell of the unit cell which b) the first brillouin zone of a wurtzite.contain 12 atoms and then substituted some of the Si 633 | P a g e
  2. 2. Mohammad Khooshebast / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.633-636The calculated band structure of 2H-SiC in wurtzite 1. Effective Masscrystal structure has been shown in figure 2. The In this section we shall present a generalobtained band gap for 2H-siC is about 2.2 eV. For model for the band structure which enables us toSiC the top of the valance band is at the K point, and interpret the macroscopic properties of theit has an indirect band gap. The real band gap in this semiconductors of interest in devices.crystal will be somewhat larger on the order of about The model consists of one conduction band,1 eV. with three sets of minima in the region around the minima of the conduction band, the function 25 E (k ) can be approximated by a quadratic function 20 of k : 2 1 15 E (k )   ( ) ij k i k j (1) Energy(eV) 10 i, j 2 m 1 1 2E ) ij  2 ( 5 Where ( ) (2) 0 m  k i k j -5 Is the inverse effective mass tensor, and k is A G K L measured from the centers of the valleys. In the model considered here E (k ) is assumed infigure2. The band structure of 2H- following from [9]:SiC 2  k l2 k t2  E (k )     (3) 2  ml mt  Equation (3) represents a band with ellipsoidal equi-energetic surfaces, with tensor effective mass. The ellipsoids have rotational symmetry around the crystallographic directions which contain the centers of the valleys. Here, k l and kt are he longitudinal and transverse components of k with respect to these directions; 1 1 and are the longitudinal and transverse ml mt fi components, respectively, of the inverse effectivegure 3. The calculated total density of states in 2H- mass tensor. For values of k from the minima ofSiC. conduction band, the energy deviates from the simple quadratic expression, and nonparabolicity occurs. ForTable 1 summarizes the values of lowest indirect the conduction band, a simple analytical way offorbidden gaps (Eg) for 2H-SiC from our model, as introducing nonparabolicity is to consider an energy-well as theoretical results from LDA, GW and wave vector relation of type [10]experimental results. The discrepancies of the energy E (1  E )   (k ) (4)gap between the GW and experiment are around 0.01 Where  (k ) is given by the right-had sideeV. Our result is in very good agreement with theexperimental data [7] and pseudopotential ab initio of Eqs. (3).  is a nonparabolicity parameter [11-13].results including the GW approximation [8] as listed By using equations (3) and (4) the effectivein Table 1. mass in conduction bad for 2H-SiC has been calculated in  ,  and U valleys and the result isTable 1: Values of indirect gap (in eV) of SiC. shown in table (1) valley   U Table1: Calculated effective mass for 2H-SiC in ,Effective 0.08 m 0 0.16 m 0 0.8 m 0  and U valley.massMethod This LDA GW Experiment 3. Scattering Mechanisms Work As usual in semi-classical transport, theBand 2.23 2.10 3.31 3.30 dynamics of electron interactions is assumed to beGap 634 | P a g e
  3. 3. Mohammad Khooshebast / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.633-636independent of the applied field, and the collisionsare assumed to occur instantaneously.All scattering calculations presented here will becarried out with a first order perturbative approach,consequently only two-body interactions will beanalyzed. The electronic transitions of interest forcharge transport in semiconductors can be classifiedas intra valley when the initial and final states lie inthe same valley, or inter valley when they lie indifferent valleys. The most important sources of scatteringthat determine these transitions in the bulk ofhomogeneous crystals are phonons and impurities.The interaction of phonons with charge carriers isdue to the deformation of the otherwise perfectcrystal produced by phonons through the deformationpotential mechanism or through the electrostaticforces produced by the polarization waves that Fiaccompany the phonons. It is useful to present a gure4. Energy dependence of the piezoelectricphysical discussion of the main features of each scattering rate in gamma valley of SiC at differentscattering process with particular attention to their temperatures.influence on carrier transport.a) Acoustic Scattering With Piezoelectric andInteraction Deformation Potential The maximum energy transfer for anelectronic interaction with acoustic phonons is, ingeneral, much smaller than the electron energy, andthus very often acoustic scattering is treated as anelastic process. The electrostatic nature of thismechanism leads to a scattering efficiency thatdecreases at increasing carrier energy; its importanceis restricted to low temperature and low fieldsituations, especially in high-purity materials. Indealing with hot-electron problems, where itsinfluence is negligible, the piezoelectric scatteringshould always be considered together with the Fiscattering due to deformation-potential interaction. gure5. Energy dependence of the deformation-As is shown in figures 4 and 5 you can see the potential scattering rate in gamma valley ofpiezoelectric and deformation-potential scattering SiC at different temperatures.rates as a function of electron energy for acoustic b) Optical-Phonon Scattering with Deformation-phonons for SiC semiconductor in different Potential Interaction.temperatures. Usually the equivalent temperature of the As is shown in figure 4, by increasing the optical phonons s assumed to be constant since thetemperature the scattering rate increases and for high dispersion relation of such kind of phonons is quiteenergy electrons, the scattering rate decreases. flat for the q values involved in electronic intra valleyAs is shown in figure 5, by increasing the transitions. As we shall see, this kin of scattering istemperature the scattering rate increases, and by isotropic.increasing the electron energy, the scattering rateincreases from 0 to 7  10 13 c) Optical-Phonon Scattering With Polar in room temperature. Interaction The electrostatic nature of interaction is such that forward scattering is favored, so that this mechanism is strongly anisotropic. The treatment of this scattering is simplified by the constancy of the phonon energy in the transition. At high electron energies the total scattering rate for polar optical scattering decreases with increasing energy. 635 | P a g e
  4. 4. Mohammad Khooshebast / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.633-636Optical phonon scattering rate in phonon emissionand absorption as a function of energy in  valley for References2H-SiC are shown in figures 6 and 7 for different [1] X. Gonze, J.-M. Beuken, R. Caracas, F.temperatures. Detraux, M. Fuchs, G.-M. Rignanese, L.As you can see in figure 6 electron can absorb Sindic, M. Verstraete, G. Zerah, F. Jollet, M.phonon in a high rate specially in low energies. By Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-increasing the temperature, the scattering rate Y. Raty, D. C. Allan, Comput. Mater. Sci.increases too. By increasing the electron energy from 25, 478 (2002). [2] A. M. Rappe, K. M. Rabe, E. Kaxiras, J. D.0 to 0.4 eV the scattering rate varies from 5.2  10 12 Joannopoulos, Phys. Rev. B 41, 1227 (1990) 12to 3 10 . [3] Numerical Data and FunctionalAs you can see in figure 7 the probability of emission Relationships in Science and Technology,is obviously zero when E   op , since the carrier edited by O.Madelung, Landolt-Bornstein, New Series, Group 3, vol. 17, Pt. adoes not have enough energy to emit the phonon. It is (Springer-Verlag, New York), (1982)also known that by increasing the temperature, the [4] A. R. Verma and P.Krishna, Polymorphismscattering rate increases. and Polytypism in Crystals (Wiley, New York, 1966), Chap. 5. [5] Properties of Advanced semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe, M. E. Levinshtein, S. L. Rumyantsev, and M. S. Shur, Eds. New York: Wiley, 2001, pp. 31-47. [6] M. Ruff, H. Mitlehner, and R. Helbig, “SiC devices: hysics and numerical simulation”, IEEE Trans. Electron Dev., vol. 41, pp. 1040-1053, 1994 [7] K. H. Hellwege (ed.), Landolt-Börnstein, Numerical and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982). [8] B. Wenzien, P. Käckell, F. Bechstedt, and G. Cappellini, Phys. Rev. B 52, 10897 (1995).Figure6. Energy dependence of the optical phonons [9] Smith, R. A.: Semiconductors. Cambridge:scattering rate in gamma valley of SiC at different Cambridge University Press 1964.temperatures for phonon absorption. [10] Conwell, E. M., Vassel, M.O.: Phys. Rev. 166,797(1968). [11] Fawcett, W., Boardman, D. A., Swain, S.: J. Phys. 55, 645 (1983). [12] Paige, E. G. S.: The Electrical Conductivity of Germanium (Gibson, A. F., Burgess, R. E., eds.). London: Heywood. 1964.Figure7. Energy dependence of the optical phononscattering rate in the gamma valley of 2 SiC at different temperatures for phonon emission. 636 | P a g e