A semiclassical approach to Coulomb scattering of conduction electrons on ionized impurities in nondegenerate semiconductors

In the proposed model of mobility, the time of electron–ion interaction equals the time taken by the conduction electron to pass a spherical region, corresponding to one impurity ion in crystal, and the minimum scattering angle is determined after Conwell–Weisskopf. We consider the acts of electron scattering on ions as independent and incompatible events. It is shown in the approximation of quasimomentum relaxation time, that for nondegenerate semiconductors, the mobility m i , limited by the elastic scattering by impurity ions with the concentration Ni , is proportional toT/Ni ; the Hall factor equals 1.4. The calculated dependences of the mobility of the majority charge carriers upon their concentration for different temperatures T agree well with known experimental data. It is shown, that the Brooks–Herring formula mBH}T /Ni gives overestimated values of mobility. Comparison of the calculations of mobility in degenerate semiconductors with experimental data also yieldsm i,mBH . © 2003 American Institute of Physics. @DOI: 10.1063/1.1573735 #


I. INTRODUCTION
The concentration of ionized impurities in crystal is usually evaluated from the comparison of experimental charge carrier mobility i , limited by the elastic scattering by ions, with the calculated BH by the Brooks-Herring ͑BH͒ formula ͑see, for example, Refs. 1-6͒. But, as noted in Ref. 7, the concentration of donors, found using BH from the Hall mobility of electrons in the n part of the InSb ingot with a p-n junction measured at Tϭ77 K, is several times greater than those found from the distribution curve of donors along the ingot. The conclusion that i Ͻ BH is confirmed by the comparison of experimental data for different semiconductors with the calculations using the BH formula ͑see Ref. 8͒. The fact that i Ͻ BH is also confirmed by the simulation of the dependence of the drift mobility on the doping impurity concentration using the Monte Carlo method. 9 In the BH model, the interaction time of the conduction electron with one impurity ion is nominally taken to be infinite. In the Conwell-Weisskopf ͑CW͒ model 2-6,10 the minimum scattering angle ͑maximum impact parameter͒ is introduced. It is known, that in numerical simulation of the dependence of the drift velocity of the electrons on the electric-field strength in n GaAs 11 the CW model gives better agreement with experiment than the BH model. When simulating the scattering by impurity ions in semiconductors by the Monte Carlo method, 12 the Ridley approximation 4 is used, which reconciles the BH and CW models by introducing the geometrical probability that the ion is the nearest scattering center for the conduction electron. Moreover, as pointed out in Ref. 6: ''. . . it is difficult to observe experimentally a i ϰT 3/2 behavior over a wide temperature range.'' The purpose of this work is to calculate the electron ͑hole͒ mobility in relaxation time approximation under the condition, that the act of elastic pure Coulomb scattering by an ion continues only during the time taken by the conduction electron to pass a spherical region of the crystal, corresponding to one impurity ion, and the minimum scattering angle is defined in accordance with CW.
Let us consider the crystalline semiconductor of volume V, which contains impurity ions with concentration N i . Thus, the number of ions in the crystal is N i V. We will describe the electron in the effective-mass approximation by the wave function in the plane wave form k ϭV Ϫ1/2 exp(ikr), where k is the quasiwave vector, r is the coordinate of the electron. In such a description, the electron plane wave interacts with all impurity ions of the crystal.
The transport relaxation time k of the quasiwave vector k of the electron with the kinetic energy Eϭ(បk) 2 /2m for elastic, independent, and incompatible scattering on N i V impurity ions of the same type in the crystal is given by: 2,3,13 where N i is the concentration of impurity ions, W k Ј k is the electron scattering probability per unit time into the angle ϭ(k,kЈ ) induced by the spherically symmetric Coulomb potential of an ion, ␦ k Ј k is the Kronecker delta, ␦((kЈ Ϫk)/k) is the dimensionless Dirac delta function introduced to take into account the conservation law of quasiwave vector absolute value (kЈϭk) in elastic scattering, បϭh/2 is Plank's constant, and m is the electron effective mass.
In the first order of perturbation theory, the ratio of the probability of electron transition between k and kЈ states a͒ with E and EЈ energies to the ''duration'' t c of an impurity ion influence U k Ј k upon the electron is equal: 14 when the probability of elastic scattering of electron by an ion t c W k Ј k Ӷ1.

II. MAIN IDEA OF THE PROPOSED MODEL
Let us consider a doped semiconductor with impurity ions of one type in the charge state Z 0. The first assumption is that the impurity ion with the charge Ze does not possess internal degrees of freedom and scatters the electron with quasimomentum បk elastically. The second assumption is that the conduction electron ͑regarded as a reasonably well-localized wave packet͒ interacts with one impurity ion only during the time t c Ϸ2R/vϭ2Rm/បk, where v is the electron velocity, Rϭ(4N i /3) Ϫ1/3 is the radius of the spherical region of the crystal associated with an ion. 15 After interaction with an ion during the time t c , the electron scatters on the next ion over the same period of time and so on, i.e., scattering occurs permanently. 16 Then, the minimum scattering angle i of the electron on the Coulomb potential Ze/4⑀r, where ⑀ϭ⑀ r ⑀ 0 is the static permittivity of a nondoped crystal, ⑀ 0 is the electric constant, and the duration t c ϭR(2m/E) 1/2 of the scattering act are bound up by a relation from classical mechanics: 17 where the maximum impact parameter bϭR is determined from the relation 4R 3 /3ϭ1/N i . Note, that introduction of the finite interaction time t c of the electron with an impurity ion and minimum scattering angle i makes the problem classical.
So, when determining the action sphere of an impurity ion upon the conduction electron, we follow the CW model, 2-6,10 but additionally take into account the finiteness of time t c of the Coulomb interaction of the electron with an ion. The lower bound on the angle i and upper on the time t c make consecutive scattering acts of the conduction electron by different impurity ions independent and incompatible events. 18 According to Ref. 19, the condition of validity of classical mechanics while obtaining formula ͑3͒ is the minuteness of the electron de Broglie wavelength in comparison to the size of the scatterer: 2/kӶ2R. It should be noted that at scattering, the form of the wave function does not change, only the direction of the electron quasiwave vector changes.
For the energy of Coulomb interaction of a conduction electron with an ion U(r)ϭϪZe 2 /4⑀r we have: 14 where បqϭប͉kЈϪk͉ϭ2បk sin(/2) is the absolute value of electron quasimomentum transfer at elastic scattering (͉kЈ͉ϭ͉k͉) on the angle ; k Ј * ϭV Ϫ1/2 exp(ϪikЈr).
According to Ref. 14, the sufficient condition for the validity of formula ͑4͒ is kRӷ1. The same condition, in Ref. 19, denotes the range of validity for formula ͑3͒.
For t c Ӷប/͉EЈϪE͉, when in the scattering act on an impurity ion, the electron changes only the direction of motion, 20 formula ͑2͒ is transformed according to Ref. 14 in the following way: 21 where U(k,) is found from Eq. ͑4͒. Substitution of Eq. ͑5͒ into Eq. ͑1͒ taking into account Eqs. ͑4͒ and ͑3͒ gives Note, that from Eqs. ͑2͒-͑5͒, the averaged by angles frequency i of elastic collisions of the electron having the kinetic energy E with ionized impurities can be written in the form: The impurity scattering limited mobility of a conduction electron i ϭe͗͘/m is determined by the average relaxation time 4 -6,22 where f ϭ͕1ϩexp͓(EϪE F )/k B T͔͖ Ϫ1 is the Fermi-Dirac function, E F is the Fermi energy, and k B T is the thermal energy. Let us consider the case, when in n-type crystal, the impurity ions of different types are presented, so that the electroneutrality equation takes the form: nϭ ͚ j ͚ z ZN jz , where n is the concentration of conduction electrons, and N jz is the concentration of ions of type j in the charge state Z( j). We are assuming that the minimum scattering angles i for the impurities of type j in the charge states ͉Z( j)͉Ͼ1 and ͉Z͉ϭ1 are the same, i.e., i (Z( j))ϭ i (͉Z͉ϭ1). Then, according to Eq. ͑3͒, for the maximum impact parameter b jz ϭR jz of electron scattering on the ion with the charge Z( j)e, we have b jz ϭR jz ϭ͉Z( j)͉R 1 , where b 1 ϭR 1 is the maximum impact parameter for the ion in the charge state ͉Z͉ϭ1. The concentration of all impurity ions in the crystal ͚ j ͚ z 0 N jz satisfies the condition: . The time of interaction of the electron with an impurity ion ͑charge Ze) is t c Ϸ2R jz /v ϭ2R jz m/បk, where R jz ϭ͉Z͉R 1 for all j and Z 0. In formulas ͑6͒ and ͑8͒, the presence of a different sort of ions may be considered by substituting: It should be noted, that if the anisotropy of quasimomentum relaxation time in a multivalley semiconductor at elastic scattering on the ionized impurities can be neglected, then the mobility of electrons, say, in n Si is i *ϭe ͗͘/m c ϭ(m/m c ) i , where mϭm d /6 2/3 ϭ0.36m 0 and m c ϭ0.28m 0 are the density of states and electric conductivity effective masses for one of the six equivalent valleys, respectively; m 0 is the mass of a free electron.

III. COMPARISON OF THE CALCULATIONS WITH EXPERIMENTAL DATA
The calculation of the electron mobility in nondegenerate gas of electrons will be carried out assuming, that Ϫ1 ϭ l Ϫ1 ϩ i Ϫ1 , where l is the phonon scattering limited electron mobility in the crystal, and i is the impurity ion scattering limited mobility.
Different experimental data 26 -30 on the dependence of the mobility of electrons on their concentration nϭ (1 ϪK)N at Tϭ77-80 K in n InSb crystals with average compensation ratio KϷ0.15 is shown in Fig. 1. Here, K is the ratio of concentration of totally ionized hydrogenlike acceptors (ZϭϪ1) to donors (Zϭϩ1); N i ϭ(1ϩK)N is the concentration of impurity ions and N is the concentration of donors. The mobilities of electrons ͑when l was estimated by Refs. 7 and 26 as 1.5ϫ10 6 cm 2 V Ϫ1 s Ϫ1 ) were calculated using formulas ͑9͒, ͑10͒, and ͑8͒ with the following parameters: ⑀ r ϭ17.8 and mϭ0.013m 0 . 31 The arrow in Fig. 1 indicates the concentration nϭN c ϵ2(2mk B T) 3/2 /(2ប) 3 of conduction electrons, above which their degeneration begins.
As seen in Fig. 1, formula ͑8͒ describes experimental data on mobility over a wide range of majority carrier concentrations. The condition k t RϾ1, where បk t ϭ(3mk B T) 1/2 is thermal quasimomentum of electrons, Rϭ(4(1 ϩK)N)/3) Ϫ1/3 , is satisfied. Note, that the BH ͑9͒ and CW ͑10͒ formulas give overestimated values of mobility. It should be noted, that for n InSb crystals in Fig. 1, the density-of-states effective mass of electrons does not depend on their concentration. 31 Note, that our calculations for other nondegenerate semiconductors with isotropic effective masses of conduction electrons also confirm the conclusions of papers ͑Refs. 7-9 and 12͒. Thus, the model of majority charge carrier mobility, limited by elastic scattering on randomly distributed in crystal ionized impurities, is presented. The finiteness of electron ͑hole͒ interaction time with an ion and impact parameter limitations are considered simultaneously. It is shown, that the mobility of charge carriers in a crystalline semiconductor, calculated using our model agrees more closely to experimental data over a wide range of concentrations of the doping impurities and temperatures, than BH ͑9͒ and CW ͑10͒ formulas.