Arican Journal o Mathematics and Comuter Science Research Vol. (9),. 189-195, October, 9 Available online at htt://www.academicjournals.org/ajmcsr ISSN 6-971 9 Academic Journals Full Length Research Paer Comarison o electron-lasmon scattering eect on low-ield electron mobility in and SiC H. Arabshahi and M. Rezaee Ron-Abadi Physics Deartment, Ferdowsi University o Mashhad, Mashhad, Iran. Acceted 8 Setember, 9 Iterative techniue is used to solve Boltzmann transort euation or calculating temerature and doing deendencies o electron mobility in and SiC materials. The two-mode nature o the olar otic honons is considered jointly with deormation otential acoustic, iezoelectric, ionized imurity and electron-lasmon scattering. Band non-arabolicity, admixture o unctions, arbitrary degeneracy o the electron distribution, and the screening eects o ree carriers on the scattering robabilities are incororated. It is shown that electron-lasmon scattering aects substantially the low-ield electron mobility in bul and SiC. It is ound that the electron mobility decreases monotonically as the temerature increases rom - 6 K. The low temerature value o electron mobility increases signiicantly with increasing doing concentration. The iterative results are in air agreement with other recent calculations obtained using the relaxation-time aroximation and exerimental methods. Key words: Electron-lasmon, relaxation-time, Boltzmann euation, non-arabolicity, degeneracy. INTRODUCTION Zinc oxide and sillicon carbide are o otential interest as a suitable materials or high temerature, high ower electronic devices either as the active material or as a suitable substrate or eitaxial growth o grou III-nitride comounds (Hamdani, 1997). Prior to advances in vaorhase growth yielding high uality bul o and SiC cystals [Loo, 5]; ractical alications or these materials were limited to those reuiring only oriented loycrystallities or owder [Kim, 1999]. With those large, direct band ga and wurtzite crystal structure, and SiC are similar to GaN. Furthermore, due to its relatively close match in lattice constants, it may be used as a substrate or GaN and AlN eitaxy (Tsuazai et al., 5). As a conseuence, there is renewed interest in the roerties o and SiC relevant or device alications (Maino et al., 1). The low-ield electron mobility is one o the most imortant arameters that determine the erormance o a ield-eect transistor. The urose o the resent aer is to calculate electron mobility or various temeratures and ionized-imurity concentrations. The ormulation itsel alies only to the central Γ valley conduction band. We have also consider band non-arabolicity, admixture o - tye valence-band Corresonding author. E-mail: arabshahi@um.ac.ir. wave unctions, degeneracy o the electron distribution to any arbitrary degree, and the screening eects o ree carriers on the scattering robabilities [Bellotti et al., 1999]. All the relevant scattering mechanisms, including the two-mode nature o the olar otic honon and electron-lasmon scattering are taen into account. The Boltzmann euation is solved iteratively or our urose, jointly incororating the eects o all the scattering mechanisms [Loo et al., 1998]. Our calculated results are comared with the available exerimental data on both temerature and the ree electron concentration deendence o mobility [Chen et al., 1998]. This aer is organized as ollows. Details o the iterative model, the electron scattering mechanism which have been used and the electron mobility calculations are resented in section and the results o iterative calculations carried out on structure are interreted in section. MODEL DETAILS To calculate mobility, we have to solve the Boltzmann euation to get the modiied robability distribution unction under the action o a steady electric ield. Here we have adoted the iterative techniue or solving the Boltzmann transort euation. Under the action o a
19 Ar. J. Math. Comut. Sci. Res. steady ield, the Boltzmann euation or the distribution unction can be written as v. r ef. coll Where ( / t) coll reresents the change o distribution unction due to the electron scattering. In the steady-state and under alication o a uniorm electric ield the Boltzmann euation can be written as ef. coll Consider electrons in an isotroic, non-arabolic conduction band whose euilibrium Fermi distribution unction is () in the absence o electric ield. Note the euilibrium distribution () is isotroic in sace but is erturbed when an electric ield is alied. I the electric ield is small, we can treat the change rom the euilibrium distribution unction as a erturbation which is irst order in the electric ield. The distribution in the resence o a suiciently small ield can be written uite generally as ( ) ( ) 1( )cosθ Where θ is the angle between and F and 1 () is an isotroic unction o, which is roortional to the magnitude o the electric ield. () satisies the Boltzmann euation and it ollows that: ef () () (1) { cosϕ1 [ si (1 ) si ] d 1 [ si (1 ) si ] d } i In general there will be both elastic and astic scattering rocesses. For examle imurity scattering is elastic and acoustic and iezoelectric scattering are elastic to a good aroximation at room temerature. However, olar and non-olar otical honon scattering are astic. Labeling the elastic and astic scattering rates with subscrits el and resectively and recognizing that, or any rocess i, s eli (, ) s eli (, ) euation 4 can be written as ( ) 1 ef 1 cos ϕ[ s (1 cos ϕ ) s d [ s el (5) (1 (1 ) s ) s (4) )] d ] d Note the irst term in the denominator is simly the momentum relaxation rate or elastic scattering. Euation 5 may be solved iteratively by the relation 1n ef 1 ( )[ n 1]cos ϕ[ s (1 ) s )] d ( ) (1 cos ϕ) s d [ s (1 ) s ] d el Where 1n () is the erturbation to the distribution unction ater the n-th iteration. It is interesting to note that i the initial distribution is chosen to be the euilibrium distribution, or which 1 () is eual to zero, we get the relaxation time aroximation 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 reuired accuracy, it is ossible to calculate uantities such as the drit mobility µ, which is given in terms o sherical coordinates by ( /1 αf ) 1d µ m F d Here, we have calculated low ield drit mobility in and SiC structures using the iterative techniue. In the ollowing sections electron-honon electron-imurity, and electron-lasmon scattering mechanisms will be discussed. Deormation otential scattering The acoustic modes modulate the inter atomic sacing. Conseuently, the osition o the conduction and valence band edges and the energy band ga will vary with osition because o the sensitivity o the band structure to the lattice sacing. The energy change o a band edge due to this mechanism is deined by a deormation otential and the resultant scattering o carriers is called deormation otential scattering. The energy range involved in the case o scattering by acoustic honons is rom zero to v, where v is the velocity o sound, since momentum conservation restricts the change o honon wave vector to between zero and, where is the electron wave vector. Tyically, the average value o is o the order o 1 7 cm -1 and the velocity o sound in the medium is o the order o 1 5 cms -1. Hence, v 1 mev, which is small comared to the thermal energy at room temerature. Thereore, the deormation otential scattering by acoustic modes can be considered as an elastic rocess excet at very low temerature. The deormation otential scattering rate with either honon emission or absortion or an electron o energy E in a non-arabolic band is given by Fermis golden rule as [Moglestue, 199; Jacoboni, 1989] 1/ Dac ( mt ml ) KBT E(1 αe) Rde( ) [(1 αe) 1/( αe) ] 4 πρv (1 α E) (8) (6) (7)
Arabshahi and Rezaee Ron-Abadi 191 Where D ac is the acoustic deormation otential, ρ is the material density and α is the non-arabolicity coeicient. The ormula clearly shows that the acoustic scattering increases with temerature. Piezoelectric scattering The second tye o electron scattering by acoustic modes occurs when the dislacements o the atoms create an electric ield through the iezoelectric eect. This can occur in the comound semiconductors such as the III-V and II-VI materials including and SiC which in act have a relatively large iezoelectric constant. The iezoelectric scattering rate or an electron o energy E in an isotroic, arabolic band has been discussed by Ridley [Ridley, 1997] who included the modiication o the Coulomb otential due to ree carrier screening. The screened Coulomb otential is written as e ex( r) V ( r) 4πε ε r s Where ε s is the relative dielectric constant o the material and is the inverse screening length, which under nondegenerate conditions is given by ne ε ε s K BT (1) Where n is the electron density. The exression or the scattering rate o an electron in a non-arabolic band structure retaining only the imortant terms can be written as [Moglestue, 199; Jacoboni, 1989]: R z ( ) m 4 K π K 8m γ ( E ) ln(1 ) 1 e av B ε ε s T γ 1 / 1 / 8 (9) ( E )(1 αe ) αe m γ ( E ) 1 αe (11) Where K av is the dimensionless so called average electromechanical couling constant. Polar otical honon scattering The diolar electric ield arising rom the oosite dislacement o the negatively and ositively charged atoms rovides a couling between the electrons and the lattice which results in electron scattering. This tye o scattering is called olar otical honon scattering and at room temerature is generally the most imortant scattering mechanism or electrons in III-V semiconductors, and this is also the case in and SiC desite the act that the otical honon energy is articularly high at 9 mev which suresses the honon oulation and also electrons must reach that energy beore honon emission is ossible. The scattering rate due to this rocess or an electron o energy E in an isotroic, non-arabolic band is (Moglestue, 199; Jacoboni, 1989) m e ωo 1 1 1 αe Ro( ) ( ) F ( E, E) 1/ o 8πε ε εs γ ( E) (1) { N, N 1} Where N o is the honon occuation number and the uer and lower cases reer to absortion and emission, resectively. For small electric ields, the honon oulation will be very close to euilibrium so that the average number o honons is given by the Bose- Einstein distribution. Non-olar otical honon scattering Non-olar otical honon scattering is similar to deormation otential scattering, in that the deormation o the lattice roduces a erturbing otential but in this case the deormation is carried by otical vibrations. The non-olar otical honon scattering rate in non-arabolic bands is given by [Moglestue, 199; Jacoboni, 1989] R no o [ N, N 1] 1/ Dod ( mt ml ) ( ) (1 αe ) γ 1/ ( E) o o π ρω o (1) Where D od is the otical deormation otential and E E±ωo is the inal state energy honon absortion (uer case) and emission (lower case). Imurity scattering This scattering rocess arises as a result o the resence o imurities in a semiconductor. The substitution o an imurity atom on a lattice site will erturb the eriodic crystal otential and result in scattering o an electron. Since the mass o the imurity greatly exceeds that o an electron and the imurity is bonded to neighboring atoms, this scattering is very close to being elastic. Ionized imurity scattering is dominant at low temeratures because, as the thermal velocity o the electrons decreases, the eect o long-range Coulombic interactions on their motion is increased. The electron scattering by ionized imurity centres has been discussed by Broos-
19 Ar. J. Math. Comut. Sci. Res. Herring (Chattoadhyay, 1981) who included the modiycation o the Coulomb otential due to ree carrier screening. The scattering rate or an isotroic, nonarabolic band structure is given by [Moglestue, 199; Jacoboni, 1989] 8π ne Rim ( ) 4 i s ( m t m ) 1/ l 1/ 1 α E γ ( E) 1 4 γ ( E)( m m ) t 1/6 l (14) / Where n i is the imurity concentration, is the screening length and s is the dielectric constant o the material. Electron-lasmon scattering The electron-lasmon interaction Hamiltonian can be written in random hase aroximation as (Di, 1991; Mansour, 1991) H int M( ac c a c c ) (15) a a Here, c and, c are the creation and annihilation oerators or lasmons and electrons, resectively. The matrix element e M.( ) / 8Ω εm ω ( ) (16) ω () Where is the disersion relation or lasmons, and are the lasmon and electron momenta, resectively, e and m are the charge and eective mass o an electron, ε the bacground dielectric constant, and Ω the real-sace volume. The irst term in arentheses in euation 15 describes the lasmon absortion rocess which obeys the energy conservation law as ε ε ω ( ) (17) Where ε is the energy o electron with momentum. In a similar manner, the lasmon emission rocess, in accordance with the second term in arentheses in euation 15, is governed by the energy conservation law which can be written as ε ε ω ( ) (18) Note that euation 17 describes the emission o lasmon with momentum -. To imart a more conventional orm to the energy conservation law, realce the variable o sum- mation in terms governing the lasmon emission in euation 1 by -. Then we can rewrite euation 15 as H int ( M a c M a c c ) c (19) The notation o euation 15 leads to the ollowing orm o the energy conservation law or the emission rocesses ε ε ω ( ) () From the Fermi Golden rule, we can calculate the electron-lasmon scattering rates or emission W e and absortion W a π Ωd W ) H i δ[ ε ε ω ( )] 8π int ± (1) e, a ( Where and are electron momenta in an initial state i and a inal state, resectively. Here and urther the uer signs in ormulae corresond to the lasmon emission, whereas the lower ones do to the lasmon absortion. By using euation 15 and the energy conservation reuirements in the orms o euations 18 and which are consistent with this notation o H int, euation 1 becomes W e, a π ( ) Ωd M [ ± ( ) ] δ ε ε ω 8π {( N 1) ;( N ) } em ab () Where N is the Bose-Einstein distribution unction or lasmons. The integration bounds with resect to are deined rom the ollowing conditions ω ( ) / m ω ( ) / m / m / m () Where is the electron momentum at the Fermi surace. RESULTS The electron-lasmon scattering is included only in the low eective mass Γ valley. So, we have just taen into account the temerature and electron concentration deendence o the electron mobility in the Γ valley, which arises due to the dierent scattering mechanisms. The electron mobility as a unction o temerature in bul and SiC materials or various tyes o scattering mechanisms such as ionized imurity, acoustic honon via
1 4 SiC Arabshahi and Rezaee Ron-Abadi 19 Electron Mobility (cm V -1 s -1 ) 1 1 1 1 1 without electron-lasmon scattering eect 6 9 Temerature (K) Figure 1. Calculated electron mobility in bul and SiC as a unction o temerature assuming a donor concentration o 1 m - without the electron-lasmon scattering. 1 4 LiUs exerimental reort (SiC) Electron Mobility (cm V -1 s -1 ) 1 SiC with electron-lasmon scattering eect 1 6 9 Temerature (K) Figure. Calculated electron mobility in bul and SiC as a unction o temerature assuming a donor concentration o 1 m - with the electron-lasmon scattering. deormation otential and olar otical honon but without electron-lasmon scattering eect is shown in Figure 1. As it is seen, the decrease in electron mobility at low temerature is caused in art by ionized imurity scattering versa the ste reduction at higher temerature is due to large otical honon scattering. Figure shows again the variation o electron mobility
194 Ar. J. Math. Comut. Sci. Res. 15 with electron-lasmon scattering eect Electron Mobility (cm V -1 s -1 ) 1 5 SiC LiUs exerimental reort (SiC) 1 1 1 1x1 1 4 Electron Concentration (m - ) Figure. Calculated low-ield electron drit mobility in bul and SiC as a unction o dierent donor concentration at room temerature including the electron-lasmon scattering in comarison with exerimental reort. 1 without electron-lasmon scattering eect Electron Mobility (cm V -1 s -1 ) 8 6 4 SiC LiUs exerimental reort (SiC) 1 1x1 1 4 1x1 5 Electron Concentration (m - ) Figure 4. Calculated low-ield electron drit mobility in bul and SiC as a unction o dierent donor concentration at room temerature without the electron-lasmon scattering in comarison with exerimental reort. as a unction o temerature adding electron-lamon scattering eect. As it can be seen, inclusion o the electron-lasmon scattering leads to the eecive heating o the hot-electron system. The calculated results or SiC are in comarison to the Liu et al. () (exerimental reort which shows good agreements). Figures and 4 show the calculated variation o the electron mobility as a unction o the donor concentration in bul and SiC crystal structure at room temerature with and without inclusion o electron-lasmon scattering eect, resectively. It can be seen that in both cases there is a air agreement with the Liu et al. ()
Arabshahi and Rezaee Ron-Abadi 195 (exerimental reort). Also the igures show that the mobility does not vary monotonically between donor concentrations o 1 1 and 1 4 m - due to the deendence o electron-lasmon scattering on donor concentration, but shows a maximum near 1 1 m - or both structures. Conclusion The comuted low-ield electron mobility in bul and SiC materials show that SiC has suerior electron transort roerties. It is shown that including the electronlasmon scattering eect increase the electron mobility o two structures by 1%, and Ohmic mobility dros by the same ercent. This is caused by combined eects o eective heating o electron gas by electron-lasmon scattering and redominantly orward eaed momentum relaxation or all electron momentum. Tsuazai A, Ohtomo A, Onuma T, Ohtani M, Kawasai M (5). Nat. Mater. 4: 4 Maino T, Segawa Y, Ohtomo A (1). Al. Phys. Lett. 78: 17. Bellotti E, Doshi BK, Brennan KF (1999). J. Al. Phys. 85 : 916. Loo DC, Reynolds DC, Sizelove JR, Harsch WC (1998). Solid State Commun. 15: 99. Chen Y, Bagnall DM, Koh HJ, Par K T, Zhu Z Q, Yao T (1998).J. Al. Phys. 84: 91. Moglestue C (199). Monte Carlo Simulation o Semiconductor Devices, Chaman and Hall. Jacoboni C, Lugli P (1989). The Monte Carlo Method or semiconductor and Device Simulation, Sringer-Verlag. Ridley BK (1997). Electrons and honons in semiconductor multilayers, Cambridge University Press. Chattoadhyay D, Queisser HJ (1981). Review o Modern Physics, 5, art1 Di K, Brennan K (1991).J. Al. Phys. 69: 97. Mansour N, Di K, Brennan K (1991). J. Al. Phys. 7 6854. Liu Y, Gorla CR, Liang S, Emanetoglu N, Wrabac M ().J. Electron Mater. 9: 69. ACKNOWLEDGEMENTS I would lie to than M. G. Paezi or her useul comments. REFERENCES Hamdani F (1997). High ield electron transort roerties in, Al. Phys. Lett. 7: 467. Loo DC (5). Monte Carlo study o electron transort in SiC, Semicond. Sci. Technol. S55. Kim SH, Lee JS, Choi HS, Lee YH (1999) Crystal growth o Zno and SiC, IEEE Electron Device Lett. : 11.