As a re

Equation (2.9) takes no explicit account of any anisotropy in the ion scattering. sult, it usually overestimates the mobility values, particularly in the high dopant density range where impurity scattering is dominant.

=

=

The difficulty in using eq (2.9) for computing the ionized impurity scattering mobility lies in the choice of proper electron effective mass in eqs (2.7) and (2.9) when energy band structure is nonspherical. For silicon, the anisotropic scattering effect comes from electron mass anisotropy (e.g., mt 0.192 m and me = 0.98 m [11]) due to the ellipsoidal conduction band structure. Long [14] has made an extensive study of the validity of BrooksHerring formula for n-type silicon for dopant density less than 1016 cm-3 and temperatures below 100 K. He concluded that if the electron effective mass in eq (2.9) was used as an adjustable parameter, then good agreement between theory and experiment can be obtained. the present study [12], we have also made a detailed comparison among the theoretical models developed by Brooks-Herring [4,7], Conwell-Weisskopf [13], and Samoilovich et al. [9], and have found that the mobility values calculated from these models for n-type silicon are generally too large for 1013 <

-3 ND 1019 cm and 100 T < 500 K.+ To correct this discrep

ancy, we have made use of the results of Long [14], and Norton et al. [2], and obtained a mobility expression, similar to that of eq (2.9), for ionized impurity scattering:

Here G(b) is identical to eq (2.6) with the exception that b is replaced by b

O

In

(2.10)

* and m =

me.

0.98 m is used in eq (2.7) to obtain b The numerical coefficient in eq (2.10) was obtained directly from reference [14] for n-type silicon when the anisotropic scattering effect is included. The reader is referred to the original paper [14] for a complete description.

When electron-electron scattering effect is included in eq (2.10) for N > 2 × 1016 cm ̄3, the formula appears to yield the best theoretical fit to the measured mobility in the range of dopant density and temperature studied here.

Because of the analogy between a neutral donor atom and a hydrogen atom, the mobility for scattering by the neutral donors can be obtained by proper modification of the results for scattering of slow electrons by hydrogen atoms. Erginsoy first predicted a temperatureindependent mobility given by [15]:

This equation fails to predict the temperature dependence of neutral impurity scattering mobility observed for n-type silicon at low temperatures [2]. Sclar [16] sought to improve eq (2.11) by including the possibility of bound states in the electron-hydrogenic impurity scattering problem. He obtained the following expression for the neutral impurity scatter

Figure 2.

Neutral impurity scattering mobility as a function of donor density computed

from eq (2.11) (due to Enginsoy) - eq (2.12) (due to Sclar).

Equation (2.12) predicts that Hy varies as T1/2 for kT above the binding energy, EN The experimental data given by Norton et al. [2] for n-type silicon show such a dependence up to 50 K. Our mobility and resistivity calculations using eqs (2.11) and (2.12) show that the latter produces a better fit with the experimental data than the former. For comparison, the neutral impurity scattering mobilities computed from eqs (2.11) and (2.12) are shown in figure 2 for T = 300 K. Note that the mobility values predicted by eq (2.11) are somewhat smaller than those of eq (2.12).

2.4. Effect of Electron-Electron Scattering

The mobility formula given in eq (2.10) neglects the effect of electron-electron (e-e) scattering on the ionized impurity scattering mobility. Although e-e scattering does not affect the current density directly since it cannot alter the total momentum, it tends to randomize the way in which this total momentum is distributed among electrons with different energy. When the scattering mechanism is such as to lead to a nonuniform distribution, e-e scattering gives rise to a net transfer of momentum from electrons which dissipate momentum less efficiently to those which dissipate more efficiently, resulting in an overall greater rate of momentum transfer and lower mobility.

On the basis of the above argument, it is obvious that the size of the effect of e-e scattering on the mobility is a function of the energy dependence of the relaxation time. Thus, for neutral impurity scattering where the relaxation time is independent of energy, the mobility is not affected by e-e scattering. Ionized impurity scattering would be expected to be much more affected than lattice scattering since in the former case T is proportional to E3/2 while in the latter it is proportional to E-1/2.

Luong and Shaw [10] have analyzed the effect of e-e scattering on the ionized impurity mobility using a single-particle-like approximation from the time-independent Hartree-Fock theory. They have shown that with the correction for e-e scattering, the Brooks-Herring formula is reduced by a factor which can be expressed in closed form as [10]

where N is the ionized impurity density, n is the electron density, and μ is given by eq I (2.10). For uncompensated n-type silicon, the density of ionized donor impurities, N., is equal to the density of conduction electrons, n. Thus, eq (2.14) reduces to

μ = (1 - e-1)

=

HI

0.632 HI

(2.15)

The factor in eq (2.15) is in good agreement with a previous prediction [17] based on the Boltzmann theory. For n-type silicon, the effect of e-e scattering on the ionized impurity scattering becomes important for No > 2 × 1016 as will be discussed later.

-3

cm

The effect of e-e scattering on the lattice mobility has also been discussed in several classical papers [4,18,19]. It can be shown [18,19] that e-e scattering reduces the lattice scattering mobility by a maximum of 12 percent. This factor will be used in the present calculations.

3. CALCULATIONS OF ELECTRON MOBILITY AND RESISTIVITY IN N-TYPE SILICON

We now discuss the mobility calculations for n-type silicon in the range of dopant density from 1013 to 1019 cm-3 and temperature from 100 to 500 K.

The combined mobility due to both lattice and ionized impurity scattering contributions can ɔe calculated according to the mixed-scattering formula [18]:

HLI = μL [1 + x2 (Cix cosx + sinX (Six - )]

(3.1)

Note that eqs (3.1) and

Cix and Six are the cosine and sine integrals of X, respectively. (3.2) are applicable for donor densities less than 2 × 1016 cm where the effect of e-e scattering is negligible. For dopant densities greater than 2 x 1016 cm 3, the effect of e-e scattering is incorporated empirically in eq (3.2) as follows:

(i) High dopant density range (2 × 1017 < N, ND

< 1019 cm-3).

In this dopant density range, experimental evidence indicates that the full effect of e-e scattering should be taken into account in the mobility calculations, and eq (3.2) is replaced by:

(ii) Intermediate dopant density range (2 × 1016 ≤ N ≤ 2 × 1017 cm3).

In this dopant density range, the effect of e-e scattering on both lattice and ionized impurity scattering mobilities is increased gradually with increasing dopant density. To obtain the best fit to the measured mobility at 300 K in this transition range, the mobility reduction factors R(N) and S (N), for both H and H were derived empirically, assuming a linear dependence on the donor density. The results are given by:

D'

Note that the mobility reduction factors, R(N) and S (N), in eqs (3.7) and (3.9) decrease linearly from 1 to 0.88 and 0.632, respectively, as the dopant density, No, increases from 2 × 1016 to 2 × 1017 cm-3.

HLI

for 1013

Equations (3.1) through (3.10) allow the calculations of u N< 1019 cm-3, and 100 <T< 500 K. When neutral impurity scattering is included, the total electron mobility may be computed from the expression

=

(3.11)

where HLI is given by eq (3.1) and N is the mobility due to neutral impurity scattering as given in eq (2.12). A sum of the reciprocal mobilities for this case is expected to yield a considerably better approximation than it does for the reciprocal sum of HL and because neutral impurity scattering, being energy independent, does not affect any "contributions from HL and HI'

3.2.

Ionization of the Donor Impurity

In this section, formulations are given for computing the ionized donor density (or electron density) as a function of the total donor density in uncompensated n-type silicon. In order to compute the electron mobility and resistivity, it is necessary to know the amounts of ionized and neutral impurity atoms so that their scattering contributions can be calculated. The ionized donor densities for n-type silicon were computed by solving the charge balance equation for the Fermi energy by an iteration procedure. Since the minority carrier density (i.e., hole density in n-type silicon) is negligibly small for this case, the charge balance equation is simply

where

n ~ N+

ND

EE

1 + 2 exp((Ep - Ec + E)/kT]

and the electron density, n, is given by [20]

n = Nc/[exp (Ec- Ep)/kT+ 0.27], for Ep < 1.3 kT

E,

(3.12)

(3.13)

(3.14)

3.22 x 1019 cm -3 for n-type silicon at 300 K). The temperature dependence of the density of states effective mass was taken into account in accordance with the results of Barber [11].

Experimental evidence exists which shows that the donor ionization energy, ED, is not constant, but decreases with increasing dopant density. Hall coefficient measurements by Pearson and Bardeen [21] and more recently by Penin et al. [22] in heavily-doped silicon from 4 to 300 K show no evidence of an ionization energy at impurity densities greater than 3 × 1018 cm-3. For n-type silicon doped with phosphorus impurities, the dependence of donor ionization energy on the dopant density is [18]

where ε(o) = 0.045 eV is the ionization energy of the phosphorus donor; with a = 3.1 × 10−8, eq (3.15) gives a zero ionization energy for ND 3 x 1018 cm-3.

3.3.

=

Resistivity Analysis

Measurements of resistivity in silicon have been reported by previous investigators [23-35]. Irvin [25] first showed complete resistivity versus dopant density curves for both n- and ptype silicon at 300 K, using mostly previously published data. Recently Mousty et al. [26] reported the resistivity versus phosphorus density in n-type silicon. The electrical properties of heavily-doped silicon were also reported by Chapman et al. [28]. Most of these efforts focused on resistivity measurements near room temperature. A theoretical analysis of the resistivity as a function of dopant density for n-type silicon has been reported by Rode [3] and Norton et al. [2]; their calculations of resistivity were also confined to room temperature. In this work, we have extended the theoretical calculations of resistivity as a