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  1. 1. Hadi Arabshahi and Azar pashaei / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.2200-2204 Comparison of Low Field Electron Transport Characteristics in InP,GaP,Ga0.5In0.5P and In As0.8P0.2 by Solving Boltzmann Equation Using Iteration Model Hadi Arabshahi and Azar pashaei Physics Department, Payame Noor University, Tehran, Iran Physics Department, Higher Education Khayyam, Mashhad, IranAbstract Temperature and doping dependencies industry. In this communication we present iterativeof electron mobility in InP, GaP and calculations of electron drift mobility in low electricGa0.52In0.48P structures have been calculated field application. Because of high mobility InP hasusing an iterative technique. The following become an attractive material for electronic devicesscattering mechanisms, i.e, impurity, polar of superior performance among the III phosphatesoptical phonon, acoustic phonon and semiconductors (Besikci et al., 2000).Thereforepiezoelectric are included in the calculation. study of the electron transport in group IIIIonized impurity scattering has been treated phosphates is necessary. GaP possesses an indirectbeyond the Born approximation using the phase- band gap of 0.6 eV at room temperature whereasshift analysis. It is found that the electron InP have a direct band gap about 1.8 eV,mobility decreases monotonically as the respectively.temperature increases from 100K to 500K for The low-field electron mobility is one ofeach material which is depended to their band the most important parameters that determine thestructures characteristics. The low temperature performance of a field-effect transistor. The purposevalue of electron mobility increases significantly of the present paper is to calculate electron mobilitywith increasing doping concentration. The for various temperatures and ionized-impurityiterative results are in fair agreement with other concentrations. The formulation itself applies onlyrecent calculations obtained using the relaxation- to the central valley conduction band. We have alsotime approximation and experimental methods. considered band nonparabolicity and the screening effects of free carriers on the scatteringKEYWORDS: Iterative technique; Ionized probabilities. All the relevant scatteringimpurity scattering; Born approximation; electron mechanisms, including polar optic phonons,mobility. deformation potential, piezoelectric, acoustic phonons, and ionized impurity scattering. TheINTRODUCTION Boltzmann equation is solved iteratively for our The ternary semiconductor, Ga 0.5In 0.5P purpose, jointly incorporating the effects of all theoffers a large direct band gap of the III-V compound scattering mechanisms [3,4].This paper is organizedsemiconductor material which is lattice matched to a as follows. Details of the iteration model , thecommonly available substrate like GaP or InP. electron scattering mechanism which have beenOwing to its relatively large band gap, this material used and the electron mobility calculations arehas been cited for its potential use in visible light presented in section II and the results of iterativeemitters such as light emitting diodes and lasers [7- calculations carried out on Inp,Gap,Ga 0.5 In 0.5P and8]. Ga0.52In0.48P has further potential usage as a InAs0.8P0.2 structures are interpreted in section III.visible light detector as well as in high temperatureand/or high power electronics. The problems of II. Model detailelectron transport properties in semiconductors have To calculate mobility, we have to solve thebeen extensively investigated both theoretically and Boltzmann equation to get the modified probabilityexperimentally for many years. Many numerical distribution function under the action of a steadymethods available in the literature (Monte Carlo electric field. Here we have adopted the iterativemethod, Iterative method, variation method, technique for solving the Boltzmann transportRelaxation time approximation, or Mattiessens rule) equation. Under the action of a steady field, thehave lead to approximate solutions to the Boltzmann Boltzmann equation for the distribution function cantransport equation [1,2]. However, carrier transport be written asmodeling of Ga0.50In0.50P materials has onlyrecently begun to receive sustained attention, now f eF f  v. r f  . k f  ( ) coll (1)that the growth of compounds and alloys is able to t  tproduce valuable material for the electronics 2200 | P a g e
  2. 2. Hadi Arabshahi and Azar pashaei / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.2200-2204 Where (f / t ) coll represents the change of have found that convergence can normally be achieved after only a few iterations for small electric distribution function due to the electron scattering. fields. Once f 1 (k) has been evaluated to the In the steady-state and under application of a required accuracy, it is possible to calculate uniform electric field the Boltzmann equation can be quantities such as the drift mobility , which is written as given in terms of spherical coordinates by eF f . k f  ( ) coll  t   (k / 1  2 F ) f1 d 3 k 3 Consider electrons in an isotropic, non-  parabolic conduction band whose equilibrium Fermi   * 0  3m F (7) distribution function is f0(k) in the absence of k 2 f 0 d 3k electric field. Note the equilibrium distribution f0(k) 0 is isotropic in k space but is perturbed when an Here, we have calculated low field drift electric field is applied. If the electric field is small, mobility in III-V structures using the iterative we can treat the change from the equilibrium technique. In the following sections electron-phonon distribution function as a perturbation which is first and electron-impurity scattering mechanisms will order in the electric field. The distribution in the be discussed. presence of a sufficiently small field can be written Deformation potential scattering The acoustic quite generally as modes modulate the inter atomic spacing. Consequently, the position of the conduction and f (k )  f 0 (k )  f1 (k ) cos  (3) valence band edges and the energy band gap will vary with position because of the sensitivity of the band structure to the lattice spacing. The energy Where θ is the angle between k and F and f1(k) is change of a band edge due to this mechanism is an isotropic function of k, which is proportional to defined by a deformation potential and the resultant the magnitude of the electric field. f(k) satisfies the scattering of carriers is called deformation potential Boltzmann equation 2 and it follows that: scattering. The energy range involved in the case of eF  f0  t i    1 i 0     cos  f  S  (1  f )  S f d k  f  S (1  f  )  S f   d k  (4) i 0 3 1 i  0  i 0   3  scattering by acoustic phonons is from zero to 2vk  , where v is the velocity of sound, since momentum conservation restricts the change of phonon wave vector to between zero and 2k, where In general there will be both elastic and inelastic k is the electron wave vector. Typically, the average scattering processes. For example impurity value of k is of the order of 107 cm-1 and the velocity scattering is elastic and acoustic and piezoelectric of sound in the medium is of the order of 105 cms-1. scattering are elastic to a good approximation at Hence, 2vk  ~ 1 meV, which is small compared room temperature. However, polar and non-polar to the thermal energy at room temperature. optical phonon scattering are inelastic. Labeling the Therefore, the deformation potential scattering by elastic and inelastic scattering rates with subscripts acoustic modes can be considered as an elastic el and inel respectively and recognizing that, for process except at very low temperature. The any process i, seli(k’, k) = seli(k, k’) equation 4 can deformation potential scattering rate with either be written as phonon emission or absorption for an electron of  eF f 0    f1cos [ Sinel (1  f 0 )  Sinel f 0 ] d 3k   energy E in a non-parabolic band is given by Fermisf1 (k )   k (5) golden rule as [3,5]  (1  cos ) Sel d 3k     [ Sinel (1  f 0 )  Sinel f 0] d 3k   2 D 2 ac (mt ml )1/ 2 K BT E (1  E ) *2 * Note the first term in the denominator is Rde  simply the momentum relaxation rate for elastic  v 2  4 E (1  2E ) scattering. Equation 5 may be solved iteratively by the relation (1  E ) 2  1 / 3(E ) 2  (8)  eF f 0    f1cos [n  1][Sinel (1  f 0 )  Sinel f 0 ] d 3k   Where Dac is the acoustic deformation f1n (k )   k (6)   (1  cos )Sel d 3k     [Sinel (1  f0 )  Sinel f0] d 3k   potential,  is the material density and  is the non- parabolicity coefficient. The formula clearly shows Where f 1n (k) is the perturbation to the that the acoustic scattering increases with distribution function after the n-th iteration. It is temperature interesting to note that if the initial distribution is Piezoelectric scattering chosen to be the equilibrium distribution, for which The second type of electron scattering by acoustic f 1 (k) is equal to zero, we get the relaxation time modes occurs when the displacements of the atoms approximation result after the first iteration. We create an electric field through the piezoelectric 2201 | P a g e
  3. 3. Hadi Arabshahi and Azar pashaei / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.2200-2204effect. The piezoelectric scattering rate for an free carrier screening. The screened Coulombelectron of energy E in an isotropic, parabolic band potential is written ashas been discussed by Ridley [4] . e2 exp(q0r ) (11)The expression for the scattering rate of an electron V (r )  4  0 s rin a non-parabolic band structure retaining only theimportant terms can be written as [3,5]: Where  s is the relative dielectric constant of the material and q0 is the inverse screening length, 2 e 2 K B TK av m* which under no degenerate conditions is given byR PZ (k )   1 2 1  2  ne2 2 2   s 2 q0  2 (12)  0 s K BT 21     3 ( )  2 1 Where n is the electron density. The scattering rate  for an isotropic, nonparabolic band structure is (9) given by [3,5] Ni e4 (1  2E )  b  (13) Where  s is the relative dielectric constant Rim  3/ 2  Ln(1  b)  1 b 32 2m   s ( ( E ))   * 2of the material and Kav is the dimensionless socalled average electromechanical coupling constant .Polar optical phonon scattering 8 m*  ( E ) (14) b The dipolar electric field arising from the  2q0 2opposite displacement of the negatively andpositively charged atoms provides a coupling Where Ni is the impurity concentration.between the electrons and the lattice which results inelectron scattering. This type of scattering is called III. RESULTSpolar optical phonon scattering and at room We have performed a series of low-fieldtemperature is generally the most important electron mobility calculations for Inp,Gap,Ga 0.5Inscattering mechanism for electrons in III-V .The 0.5P and InAs0.8P0.2 materials. Low-field motilitiesscattering rate due to this process for an electron of have been derived using iteration method. Theenergy E in an isotropic, non-parabolic band is [3,5] electron mobility is a function of temperature and  e2 2m* PO  1 1  1  2E  electron concentration.  RPO k     8         E  (10)   FPO E , EN op , N op  1 Where E = E±_wpo is the final stateenergy phonon absorption (upper case) and emission(lower case) and Nop is the phonon occupationnumber and the upper and lower cases refer toabsorption and emission, respectively. For smallelectric fields, the phonon population will be veryclose to equilibrium so that the average number ofphonons is given by the Bose- Einstein distribution.Impurity scattering This scattering process arises as a result ofthe presence of impurities in a semiconductor. Thesubstitution of an impurity atom on a lattice site willperturb the periodic crystal potential and result inscattering of an electron. Since the mass of theimpurity greatly exceeds that of an electron and theimpurity is bonded to neighboring atoms, thisscattering is very close to being elastic. Ionizedimpurity scattering is dominant at low temperaturesbecause, as the thermal velocity of the electronsdecreases, the effect of long-range Coulombicinteractions on their motion is increased. Theelectron scattering by ionized impurity centers hasbeen discussed by Brooks Herring [6] who includedthe modification of the Coulomb potential due to 2202 | P a g e
  4. 4. Hadi Arabshahi and Azar pashaei / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.2200-2204 GaP InP Ga 0.5In 0.5 P InAs0.8P0.2 m* 0.3m 0.073 0.105 0.036 Nonparabolicity(ev) 0.2 0.7 0.45 2.26 Density ρ(gr cm--3) 4.138 4.810 4.470 5.496 Longitudinal sound 5.850 5.3 4.33 4.484 velocity vs(10 5cms--1) Low-frequency 11.1 12.4 11.75 14.16 dielectric constant ε0 High-frequency 9.11 9.55 9.34 11.71 dielectric constant ε∞ Polar optical phonon 0.051 0.06 0.045 0.024 energy (eV) Acoustic deformation 3.1 8.3 7.2 5.58 potential D(eV) Figures 1 shows the comparison theelectron mobility depends on the Temperature at and InAs0.8P0.2 at the electron concentration 1016the different electron concentration in bulk (cm-3).Gap,GaAs,InP and InAs materials. Figure 1 shows Figure 2 shows the calculated variation of thethat electrons mobility at the definite temperature electron drift mobility as a function of free electron300 k for the Inp semiconductors is gained about concentration for all crystal structures at room11606 cm2v-1s-1 and for Gap,Ga 0.5In 0.5P and temperature. The mobility does not varyInAs0.8P0.2 about 5020,8960 and 26131 cm2v-1s-1. monotonically between free electron concentrationsAlso electrons mobility decrease quickly by of 1016 cm-3 and 1018 cm-3.Our calculation resultstemperature increasing from 200 to 500 K for all show that the electron mobility InAs 0.8P 0.2 isthe different electron concentrations becausetemperature increasing causes increase of phonons InAs0.8P0.2 Temperature= 300 Kenergy too. So it causes a strong interaction 25000between electrons and these phonons that its result Electron drift mobility (cm / V-s)is increase of electrons scattering rate and finally 20000decrease of electrons mobility. Our calculation 2results show that the electron mobility InAs 0.8P0.2is more than 3other materials this increasing is 15000because of small effective mass. Ga0.5In0.5P 16 -3 Donor electron density=10 ( cm ) 10000 InP 50000 InAs0.8P0.2 Electron drift mobility (cm2 / V-s) 40000 GaP 5000 30000 5.00E+017 1.00E+018 -3 InP Donor electron density ( cm ) 20000 Ga0.5In0.5P 10000 Fig 2. Changes the electron mobility Function in GaP terms of room temperature in bulk Inp,Gap,Ga 0.5In 0 200 250 300 350 400 450 500 0.5P and InAs0.8P0.2 at the different electron Temperature (K) concentration.Fig 1. Changes the electron mobility Function interms of temperature in bulk Inp,Gap,Ga 0.5In 0.5P 2203 | P a g e
  5. 5. Hadi Arabshahi and Azar pashaei / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.2200-2204 Fig.5. Electron drift mobility versus temperature in 16 -3 Inp,Gap,Ga 0.5In 0.5P and InAs0.8P0.2 with Donor electron density=10 ( cm ) including elastic scattering. 400 InAs0.8P0.2 Electron drift mobility 10 (cm2 / V-s) CONCLUSION 300 In conclusion, we have quantitatively obtained temperature-dependent and electron 4 concentration dependent electron mobility in 200 Inp,Gap,Ga 0.5In 0.5P and InAs0.8P0.2 materials InP using an iterative technique. The theoretical values show good agreement with recently obtained 100 experimental data. InAs0.8P0.2 semiconductor Ga0.5In0.5P having high mobility of Inp,Gap and Ga 0.5In 0.5P GaP because the effective mass is small compared with 0 all of them . The ionized impurity scattering in all 200 250 300 350 400 450 500 the Gap,Ga 0.5In 0.5P and InAs0.8P0.2 at all Temperature (K) temperatures is an important factor in reducing the mobility. REFERENCES Fig.4. Electron drift mobility versus temperature in [1] Arabshahi H, Comparison of SiC and ZnO Inp,Gap,Ga 0.5In 0.5P and InAs0.8P0.2 with field effect transistors for high power including inelastic scattering applications ,Modern Phys. Lett. B, 23 (2009) 2533-2539. [2] J. Fogarty, W. Kong and R. Solanki, Solid 16 -3 State Electronics 38 (1995) 653-659. 6 Donor electron density=10 ( cm ) InAs0.8P0.2 [3] Jacoboni C, Lugli P (1989). The Monte Carlo Method for semiconductor andElectron drift mobility 10 (cm2 / V-s) 5 Device Simulation, Springer-Verlag. [4] Ridley BK (1997). Electrons and phonons 4 in semiconductor multilayers,Cambridge4 University Press. 3 InP [5] Moglestue C (1993). Monte Carlo Simulation of Semiconductor 2 Devices,Chapman and Hall Ga0.5In0.5P [6] Chattopadhyay D, Queisser HJ (1981). 1 Review of Modern Physics,53,part1 GaP [7] S Nakamura, M Senoh and T Mukai, 0 Appl. Phys. Lett. 62, 2390 (1993). 200 250 300 350 400 450 500 [8] D L Rode and D K Gaskill, Appl. Phys. Temperature (K) Lett. 66, 1972 (1995). 2204 | P a g e