ORIGINAL ARTICLE Electron Mobility in InP at Low Electric Field Application

Transcription

1 International Archive o Applied Science and Technology Volume [] March : 99-4 ISSN: Society o Education, India Webite: OIGINAL ATICLE Electron Mobility in InP at Low Electric Field Application M. Mahmoodi and A. Feyzi Phyic Department, Payame Nour Univerity o Fariman, Fariman, Ira ABSTACT Temperature and doping dependencie o electron mobility in InP emiconductor ha been calculated uing an iterative technique. The ollowing cattering mechanim, i.e, impurity, polar optical phonon, acoutic phonon and piezoelectric are included in the calculation. It i ound that the electron mobility decreae monotonically a the temperature increae rom K to 6 K. The low temperature value o electron mobility increae igniicantly with increaing doping concentration. The iterative reult are in air agreement with other recent calculation obtained uing the relaxation-time approximation and experimental method. Keyword-: Polar optical; ionized impurity cattering; electron mobility INTODUCTION InP i a direct band gap emiconductor, and thereore ha a high breadown ield and low thermal generation rate. Thee propertie combined with good thermal conductivity and tability mae InP an attractive material or high power, high temperature and radiation harh environment electronic device. Monte Carlo imulation predict a pea electron velocity o 5 m - and a aturation electron velocity o.7-5 m - [-5]. Thi mae poible high requency operation o InP device. For the above tated reaon, InP i o great interet or power FET and optoelectronic device tructure. InP baed ield-eect tranitor have been reported to exhibit continuou wave output up to 4.9 W/mm [6] and high requency operation [6] at T = 55 GHz and max= GHz. The low-ield electron mobility i one o the mot important parameter that determine the perormance o a ield-eect tranitor. The purpoe o the preent paper i to calculate electron mobility or variou temperature and ionized-impurity concentration. The ormulation itel applie only to the central valley conduction band. We have alo conider band non-parabolicity, admixture o p-type valence-band wave unction, degeneracy o the electron ditribution to any arbitrary degree, and the creening eect o ree carrier on the cattering probabilitie. All the relevant cattering mechanim, including the two-mode nature o the polar optic phonon cattering, are taen into account. The Boltzmann equation i olved iteratively or our purpoe, jointly incorporating the eect o all the cattering mechanim. Our calculated reult are compared with the available experimental data on both temperature and the ree electron concentration dependence o mobility. MODEL DETAILS To calculate mobility, we have to olve the Boltzmann equation to get the modiied probability ditribution unction under the action o a teady electric ield. Here we have adopted the iterative technique or olving the Boltzmann tranport equation. Under the action o a teady ield, the Boltzmann equation or the ditribution unction can be written a, v. t Where r ef. coll ( ) t coll () ( / t) repreent the change o ditribution unction due to the electron cattering. In the teady-tate and under application o a uniorm electric ield the Boltzmann equation can be written a, ef. ( ) t coll () IAAST Volume [] March ~ 99 ~ Society o Education, India

2 Conider electron in an iotropic, non-parabolic conduction band whoe equilibrium Fermi ditribution unction i () in the abence o electric ield. Note the equilibrium ditribution () i iotropic in pace but i perturbed when an electric ield i applied. I the electric ield i mall, we can treat the change rom the equilibrium ditribution unction a a perturbation which i irt order in the electric ield. The ditribution in the preence o a uiciently mall ield can be written quite generally a, ( ) ( ) ( ) co () Where θ i the angle between and F and () i an iotropic unction o, which i proportional to the magnitude o the electric ield. () atiie the Boltzmann equation and it ollow that, ef co S ( ) S d S ( ) S d t i i i i i (4) In general there will be both elatic and inelatic 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 non-polar optical phonon cattering are inelatic. Labeling the elatic and inelatic cattering rate with ubcript el and inel repectively and recognizing that, or any proce i, eli(, ) = eli(, ) equation 4 can be written a, ( ) ef co [ Sinel ( ) Sinel ] d (5) ( co ) S d [ S ( ) S ] d el inel inel Note the irt term in the denominator i imply the momentum relaxation rate or elatic cattering. Equation 5 may be olved iteratively by the relation, ef co [ n ][ S ( ) ( co ) S d [ S ( ) S Mahmoodi and Feyzi ] d inel inel (6) n S d el inel( ) inel ] where n () i the perturbation to the ditribution unction ater the n-th iteration. It i intereting to note that i the initial ditribution i choen to be the equilibrium ditribution, or which () i equal to zero, we get the relaxation time approximation reult ater the irt iteration. We have ound that convergence can normally be achieved ater only a ew iteration or mall electric ield. Once () ha been evaluated to the required accuracy, it i poible to calculate quantitie uch a the drit mobility, which i given in term o pherical coordinate by, * m F ( / F) d d (7) Here, we have calculated low ield drit mobility in GAN and InN tructure uing the iterative technique. In the ollowing ection electron-phonon and electron-impurity cattering mechanim will be dicued. Deormation potential cattering The acoutic mode modulate the inter atomic pacing. Conequently, the poition o the conduction and valence band edge and the energy band gap will vary with poition becaue o the enitivity o the band tructure to the lattice pacing. The energy change o a band edge due to thi mechanim i deined by a deormation potential and the reultant cattering o carrier i called deormation IAAST Volume [] March ~ ~ Society o Education, India

3 Mahmoodi and Feyzi potential cattering. The energy range involved in the cae o cattering by acoutic phonon i rom zero to v, where v i the velocity o ound, ince momentum conervation retrict the change o phonon wave vector to between zero and, where i the electron wave vector. Typically, the average value o i o the order o 7 cm - and the velocity o ound in the medium i o order 5 cm -. Hence, v ~ mev, which i mall compared to the thermal energy at room temperature. Thereore, the deormation potential cattering by acoutic mode can be conidered a an elatic proce except at very low temperature. The deormation potential cattering rate with either phonon emiion or aborption or an electron o energy E in a non-parabolic band i given by Fermi' golden rule a [-5], de D ac * * / t ml ) 4 ( m v ( E) /( E) K T B E( E) E( E ) (8) Where D ac i the acoutic deormation potential, i the material denity and i the nonparabolicity coeicient. The ormula clearly how that the acoutic cattering increae with temperature. Piezoelectric cattering The econd type o electron cattering by acoutic mode occur when the diplacement o the atom create an electric ield through the piezoelectric eect. The piezoelectric cattering rate or an electron o energy E in an iotropic, parabolic band ha been dicued by idley [4]. The expreion or the cattering rate o an electron in a non-parabolic band tructure retaining only the important term can be written a [-6]: ( ) PZ av e K BTK m ( ) * (9) Where i the relative dielectric contant o the material and K av i the dimenionle o called average electromechanical coupling contant. Polar optical phonon cattering The dipolar electric ield ariing rom the oppoite diplacement o the negatively and poitively charged atom provide a coupling between the electron and the lattice which reult in electron cattering. Thi type o cattering i called polar optical phonon cattering and at room temperature i generally the mot important cattering mechanim or electron in III-V emiconductor, and thi i alo the cae in GaN and InN depite the act that the optical phonon energy i particularly high at ~ 9 mev which uppree the phonon population and alo electron mut reach that energy beore phonon emiion i poible. The cattering rate due to thi proce or an electron o energy E in an iotropic, non-parabolic band i [-6], F PO PO e m * PO E 8 E, EN, N op op E () Where E = E'±hw po i the inal tate energy phonon aborption (upper cae) and emiion (lower cae) and Nop i the phonon occupation number and the upper and lower cae reer to aborption and emiion, repectively. For mall electric ield, the phonon population will be very cloe to equilibrium o that the average number o phonon i given by the Boe- Eintein ditribution. IAAST Volume [] March ~ ~ Society o Education, India

4 Impurity cattering Thi cattering proce arie a a reult o the preence o impuritie in a emiconductor. The ubtitution o an impurity atom on a lattice ite will perturb the periodic crytal potential and reult in cattering o an electron. Since the ma o the impurity greatly exceed that o 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 o the electron decreae, the eect o long-range Coulombic interaction on their motion i increaed. The electron cattering by ionized impurity centre ha been dicued by Broo Herring [5] who included the modiication o the Coulomb potential due to ree carrier creening. The creened Coulomb potential i written a, e exp( qr) V ( r) 4 r Mahmoodi and Feyzi () Where i the relative dielectric contant o the material and q i the invere creening length, which under non-degenerate condition i given by ne q () KBT Where n i the electron denity. The cattering rate or an iotropic, non-parabolic band tructure i given by [-6], im 4 Nie ( E) b Ln( b) (4) * / b m ( ( E)) b 8m ( E) * (5) q where N i i the impurity concentration. ESULTS We have jut taen into account the temperature and electron concentration dependence o the electron mobility in the valley, which arie due to the dierent cattering mechanim. The eect o temperature on the electron mobility i hown in igure. 6 Electron Mobility [cm /V] 5 4 Ionized impurity deniy = 6 cm - Ionized impurity deniy = 8 cm Temperature (K) Fig.. Change the electron mobility in term o temperature in bul InP at the dierent electron concentration Figure how that ample with le electron denity at all temperature ha more mobility and with increaing the temperature the mobility decreae in the deinite temperature. The ample with more electron denity ha le mobility becaue in thi ample the number o ionized impurity IAAST Volume [] March ~ ~ Society o Education, India

5 Mahmoodi and Feyzi center i more and electron or more time will be aected by coulomb potential. The ionized impurity catter rate i higher in thi ample. Figure how comparion the electron mobility o InP and AlP at dierent electron concentration. 6 Electron mobility [cm /V] 4 InP AlP Temperature (K) Fig.. Change the electron mobility in term o Temperature in bul InP and AlP at the electron concentration 6 cm -. Our calculation reult how that the electron mobility at the deinite temperature K or the InP emiconductor i gained about 5 cm v - - and or AlP about 4 cm v - -, alo that the electron mobility InP i more than AlP. Thi i due to a mall electron eect ma. Figure how the electron mobility depend on temperature in bul InP material. the electron concentration at the dierent 5 Low -ild mobility [cm /(V)] T T 4 T E 6 E 7 E8 Ioniz ed im purity deniy [cm - ] Fig.. Change the electron mobility in term o electron concentration in bul InP at the dierent temperature. Figure 4 how comparion the electron mobility o InP and AlP at the dierent temperature.our calculation reult how that the electron mobility InP i more than AlP. T = K In P Low-ild mobility [cm /(V)] T = 4 K In P T = K A lp T = 4 K A lp E 6 E 7 E 8 Io n ize d im p u rity d e n iy [c m - ] Fig. 4. Change the electron mobility in term o electron concentration in bul InP and AlP at the dierent temperature. IAAST Volume [] March ~ ~ Society o Education, India

6 Mahmoodi and Feyzi Figure 4 how that The electron mobility decreae by the electron concentration increaing becaue electron increaing caue increae o ionized impurity center in crytal that it caue time more electron under the inluence o the Coulomb potential o impurity center located that it reult i increae o electron cattering rate and inally decreae o electron mobility. CONCLUSION Uing an iterative method, it wa hown that the electron cattering in the valley ubtantially aect the electron mobility and tranport propertie in InP. It i hown that the electron mobility increae and Ohmic mobility drop by the ame percent. Thi i caued by combined eect o eective heating o electron ga by electron cattering and predominantly orward peaed momentum relaxation or all electron momenta. InP emiconductor having high mobility o AlN becaue the eective ma i mall compared with AlN. EFEENCES. Bhatta, udra Praad, "Electron Spectrocopic Study o Indium Nitride Layer" (8). Phyic & Atronomy Diertation. Paper.. Jacoboni C, Lugli P (989). The Monte Carlo Method or emiconductor and Device Simulation, Springer-Verlag.. Mogletue C (99). Monte Carlo Simulation o Semiconductor Device,Chapman and Hall 4. idley BK (997). Electron and phonon in emiconductor multilayer,cambridge Univerity Pre. 5. Chattopadhyay D, Queier HJ (98). eview o Modern Phyic, 5,part. 6. H. Arabhahi, (6).Comparion o SiC and ZnO Field Eect Tranitor or High Power Application, Modern PhyicLetterB,,pp.7- IAAST Volume [] March ~ 4 ~ Society o Education, India