Comparison of Low Field Electron Transport Properties in Compounds of groups III-V Semiconductors by Solving Boltzmann Equation Using Iteration Model

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1 International Journal of Engineering Invention ISSN: , Volume 1, Iue (September 1) PP: Comparion of Low Field Electron Tranport Propertie in Compound of group III-V Semiconductor by Solving Boltzmann Equation Uing Iteration Model H. Arabhahi 1, A. Pahaei and M. H.Tayarani 1 Phyic Department, Payame Noor Univerity, P. O. Box , Tehran, Iran Phyic Department, Higher Education Khayyam, Mahhad, Iran Abtract Temperature and doping dependencie of electron mobility in InP, InA,GaP and GaA tructure have been calculated uing an iterative technique. The following cattering mechanim, i.e, impurity, polar optical phonon, acoutic phonon and piezoelectric are included in the calculation. The electron mobility decreae monotonically a the temperature increae from 1 K to 5 K for each material which i depended to their band tructure characteritic. The low temperature value of electron mobility increae ignificantly with increaing doping concentration. The iterative reult are in fair agreement with other recent calculation obtained uing the relaxation-time approximation and experimental method. Keyword Iterative technique; Ionized impurity cattering; Born approximation; electron mobility I. INTODUCTION The problem of electron tranport propertie in emiconductor have been extenively invetigated both theoretically and experimentally for many year. Many numerical method available in the literature (Monte Carlo method, Iterative method, variation method, elaxation time approximation, or Mattieen' rule) have lead to approximate olution to the Boltzmann tranport equation [1-]. In thi paper iterative method i ued to calculate the electron mobility of InA, GaA, InP and Gap. Becaue of high mobility InP ha become an attractive material for electronic device of uperior performance among the IIIphophate emiconductor.therefore tudy of the electron tranport in group IIIphophate i neceary. GaP poee an indirect band gap of.6 ev at room temperature wherea InP have a direct band gap about 1.8 ev, repectively. InP and InA offer the propect of mobility comparable to GaA and are increaingly being developed for the contruction of optical witche and optoelectronic device. The low-field electron mobility i one of the mot important parameter that determine the performance of a fieldeffect tranitor. The purpoe of the preent paper i to calculate electron mobility for variou temperature and ionizedimpurity concentration. The formulation itelf applie only to the central valley conduction band. We have alo conider band nonparabolicity and the creening effect of free carrier on the cattering probabilitie. All the relevant cattering mechanim, including polar optic phonon, deformation potential, piezoelectric, acoutic phonon, and ionized impurity cattering. The Boltzmann equation i olved iteratively for our purpoe, jointly incorporating the effect of all the cattering mechanim [-4].Thi paper i organized a follow. Detail of the iteration model, the electron cattering mechanim which have been ued and the electron mobility calculation are preented in ection II and the reult of iterative calculation carried out on Inp,InA,GaP,GaA tructure are interpreted in ection III. II. MODEL DETAIL To calculate mobility, we have to olve the Boltzmann equation to get the modified probability ditribution function under the action of a teady electric field. Here we have adopted the iterative technique for olving the Boltzmann tranport equation. Under the action of a teady field, the Boltzmann equation for the ditribution function can be written a f v. r t Where ( f / t) ef f. coll f f ( ) t coll (1) repreent the change of ditribution function due to the electron cattering. In the teady-tate and under application of a uniform electric field the Boltzmann equation can be written a ef. f f ( ) t coll () Conider electron in an iotropic, non-parabolic conduction band whoe equilibrium Fermi ditribution function i f () in the abence of electric field. Note the equilibrium ditribution f () i iotropic in pace but i perturbed when an electric field i applied. If the electric field i mall, we can treat the change from the equilibrium ditribution function a a 56

2 perturbation which i firt order in the electric field. The ditribution in the preence of a ufficiently mall field can be written quite generally a f ( ) f ( ) f1( )co () Where θ i the angle between and F and f 1 () i an iotropic function of, which i proportional to the magnitude of the electric field. f() atifie the Boltzmann equation and it follow that: ef f co f S (1 f ) S f d f S (1 f ) S f d t i 1 i i 1 i i (4) In general there will be both elatic and atic cattering procee. For example impurity cattering i elatic and acoutic and piezoelectric cattering are elatic to a good approximation at room temperature. However, polar and nonpolar optical phonon cattering are atic. Labeling the elatic and atic cattering rate with ubcript el and repectively and recognizing that, for any proce i, eli (, ) = eli (, ) equation 4 can be written a ef f f 1 co [ S(1 f) S f] d f 1( ) (5) (1 co ) S d [ S (1 f ) S f ] d el Note the firt term in the denominator i imply the momentum relaxation rate for elatic cattering. Equation 5 may be olved iteratively by the relation ef f f 1 co [ n 1][ S(1 f) S f] d f 1n( ) (6) (1 co ) S d [ S (1 f ) S f ] d el Where f 1n () i the perturbation to the ditribution function after the n-th iteration. It i intereting to note that if the initial ditribution i choen to be the equilibrium ditribution, for which f 1 () i equal to zero, we get the relaxation time approximation reult after the firt iteration. We have found that convergence can normally be achieved after only a few iteration for mall electric field. Once f 1 () ha been evaluated to the required accuracy, it i poible to calculate quantitie uch a the drift mobility which i given in term of pherical coordinate by * m F (7) Here, we have calculated low field drift mobility in III-V tructure uing the iterative technique. In the following ection electron-phonon and electron-impurity cattering mechanim will be dicued. Deformation potential cattering The acoutic mode modulate the inter atomic pacing. Conequently, the poition of the conduction and valence band edge and the energy band gap will vary with poition becaue of the enitivity of the band tructure to the lattice pacing. The energy change of a band edge due to thi mechanim i defined by a deformation potential and the reultant cattering of carrier i called deformation potential cattering. The energy range involved in the cae of cattering by acoutic phonon i from zero to v, where v i the velocity of ound, ince momentum conervation retrict the change of phonon wave vector to between zero and, where i the electron wave vector. Typically, the average value of i of the order of 1 7 cm -1 and the velocity of ound in the medium i of the order of 1 5 cm -1. Hence, v ~ 1 mev, which i mall compared to the thermal energy at room temperature. Therefore, the deformation potential cattering by acoutic mode can be conidered a an elatic proce except at very low temperature. The deformation potential cattering rate with either phonon emiion or aborption for an electron of energy E in a non-parabolic band i given by Fermi' golden rule a [,5] de (1 E) D ( ac /1 F) f d ( m v f * * 1/ t ml ) 4 1/ ( E) d K T E(1 E) B E(1 E) (8) 1 Where D ac i the acoutic deformation potential, i the material denity and i the non-parabolicity coefficient. The formula clearly how that the acoutic cattering increae with temperature Piezoelectric cattering The econd type of electron cattering by acoutic mode occur when the diplacement of the atom create an electric field through the piezoelectric effect. The piezoelectric cattering rate for an electron of energy E in an iotropic, parabolic band ha been dicued by idley [4]. 57

3 The expreion for the cattering rate of an electron in a non-parabolic band tructure retaining only the important term can be written a [,5]: 1 (9) 1 ( ) Where i the relative dielectric contant of the material and K av i the dimenionle o called average electromechanical coupling contant. Polar optical phonon cattering The dipolar electric field ariing from the oppoite diplacement of the negatively and poitively charged atom provide a coupling between the electron and the lattice which reult in electron cattering. Thi type of cattering i called polar optical phonon cattering and at room temperature i generally the mot important cattering mechanim for electron in III-V.The cattering rate due to thi proce for an electron of energy E in an iotropic, non-parabolic band i [-5] e m * 1 1 PO 1 E (1) 8 PO E F, PO E E N op, Nop 1 Where E = E'±_w po i the final tate energy phonon aborption (upper cae) and emiion (lower cae) and Nop i the phonon occupation number and the upper and lower cae refer to aborption and emiion, repectively. For mall electric field, the phonon population will be very cloe to equilibrium o that the average number of phonon i given by the Boe- Eintein ditribution. Impurity cattering Thi cattering proce arie a a reult of the preence of impuritie in a emiconductor. The ubtitution of an impurity atom on a lattice ite will perturb the periodic crytal potential and reult in cattering of an electron. Since the ma of the impurity greatly exceed that of an electron and the impurity i bonded to neighboring atom, thi cattering i very cloe to being elatic. Ionized impurity cattering i dominant at low temperature becaue, a the thermal velocity of the electron decreae, the effect of long-range Coulombic interaction on their motion i increaed. The electron cattering by ionized impurity center ha been dicued by Broo Herring [6] who included the modification of the Coulomb potential due to free carrier creening. The creened Coulomb potential i written a e exp( qr) V ( r) (11) 4 r Where i the relative dielectric contant of the material and q i the invere creening length, which under no degenerate condition i given by ne q (1) KBT PZ im Where n i the electron denity. The cattering rate for an iotropic, nonparabolic band tructure i given by [,5] 4 Nie (1 E) b Ln(1 b) (1) * / m ( ( E)) 1 b av e K BTK m ( ) * 1 1 * 8m ( E) b (14) q Where N i i the impurity concentration. III. ESULTS We have performed a erie of low-field electron mobility calculation for GaP, InP, GaA and InA material. Low-field motilitie have been derived uing iteration methode.the electron mobility i a function of temperature and electron concentration. Figure 1 how the comparion the electron mobility depend on the Temperature at the different electron concentration in bul Gap,GaA,InP and InA material. figure 1 how that electron mobility at the definite temperature for the InA emiconductor i gained about 8117cmv-1-1 and for GaA, InP, Gap about 488,47,76 cmv-1-1.alo electron mobility decreae quicly by temperature increaing from to 5 for all the different electron concentration becaue temperature increaing caue increae of phonon energy too. So it caue a trong interaction 58

4 Electron drift mobility (cm / V-) Electron drift mobility (cm / V-) between electron and thee phonon that it reult i increae of electron cattering rate and finally decreae of electron mobility. Our calculation reult how that the electron mobility InA i more than other material thi increaing i becaue of mall effective ma. InA Donor electron denity=1 16 ( cm - ) 15 1 GaA 5 InP GaP Temperature (K) Fig 1. Change the electron mobility Function in term of temperature in bul InA, GaA, InP and GaP at the electron concentration 116 (cm-). Figure how comparion the electron mobility of InA, GaA, InP and GaP at the different Temperature.Our calculation reult how that the electron mobility InA i more than GaA,InP and GaP 1 Temperature= K 8 InA 6 4 GaA InP GaP 5.E+17 1.E+18 Donor electron denity ( cm - ) Fig. Change the electron mobility Function in term of room temperature in bul InA, GaA, InP and GaP at the different electron concentration. Figure how comparion the electron mobility depend on the electron concentration at the different Temperature in bul GaP and InP material. Semiconductor mobility decreae by electron concentration increaing becaue electron increaing caue increae of ionized impurity center in crytal that it caue time more electron under the influence of the Coulomb potential of impurity center located that it reult i increae of electron cattering rate and finally decreae of electron mobility 59

5 Electron drift mobility (cm / V-) Electron drift mobility (cm / V-) 4 GaP:T= K GaP:T= K InP:T= K InP:T= K 1 5.E+17 Temperature (K) 1.E+18 Fig. Change the electron mobility Function in term of electron concentration in bul InP and GaP at the different Temperature 15 InA:Ionized impurity deniy =1 16 [cm - ] InA:Ionized impurity deniy =1 17 [cm - ] GaA:Ionized impurity deniy =1 16 [cm - ] GaA:Ionized impurity deniy =1 17 [cm - ] Temperature (K) Fig 4. Change the electron mobility Function in term of temperature in bul GaA and InA at the different electron concentration. Figure 4 how comparion the electron mobility depend on temperature at the different electron concentration in bul InA and GaA material..our calculation reult how that the electron mobility InA i more than GaAS IV. CONCLUSION 1. InA emiconductor having high mobility of GaA,GaP and InP becaue the effective ma i mall compared with all of them. The ionized impurity cattering in all the emiconductor InA,GaA,InP and GaP at all temperature i an important factor in reducing the mobility. 6

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