S. D. RISTI~: et al.: On the Carrier Concentrations and Diffusivities in Si 499

phys. stat. sol. (a) 152, 499 (1995)

Subject classification: 71.25 and 72.20; S5.11

Faculty of Electronic Engineering, University of Nii‘)

On the Effective Intrinsic Carrier Concentrations and Diffusivities in Heavily Doped Silicon

BY S. D. RISTIC, Z. D. PRIJIC, S. V. ~ I V A N O V I C , S. 2. MIJALKOVIC, and A. P. TRAJKOVIC

(Receiued July 24, I995)

It is demonstrated that the concept of the apparent electrical band-gap narrowing allows the transport equations in heavily doped silicon to be written in “nondegenerate” form. In these equations heavy doping effects are incorporated through the effective carrier diffusivities. A discussion is presented of the relationships between these diffusivities and the effective intrinsic carrier concentrations. It is found that for the case of current flow the effective carrier diffusivities are not significantly altered from their thermal equilibrium values. Quantitative results for the apparent electrical band edge shifts, band-gap narrowing, corresponding effective intrinsic carrier concentrations, and effective carrier diffusivities are provided for phosphorus-doped silicon up to loz1 ~ m - ~ .

On a montre que le concept du retrecissement electrique apparent de la bande interdite permettait que les equations de transport du silicium fortement dope soient ecrites en forme equivalente comme au cas du silicium nondegenert. Dans ces equations-ci, les effets de fort dopage sont introduits par I’intermediaire des coefficients effectifs de la diffusion de porteurs de charges. On a discute le rapport entre ces coefficients de diffusion et les concentrations intrinseques effectives de porteurs de charges. On a montre pour le courant electrique que les coefficients effectifs de la diffusion de porteurs de charges ne changent pas considerablement par rapport a leurs valeurs en equilibre thermique. On donne des rksultats quantitatifs pour les deplacements electriques apparents des limites de bandes, pour les retrecissements de la bande interdite auxquels correspondent les concentrations intrinseques effectives de porteurs et les coefficients effectifs de la diffusion pour le silicium dope dc phosphore jusqu’a 10’’ ~ m - ~ .

1. Introduction

The introduction of the effective intrinsic carrier concentration n,, [l] into the transport equations has been recognized as a very convenient concept for modeling of heavy doping effects in silicon devices [2 to 41. The definition of n,, is physically founded on the band-gap narrowing (BGN) phenomenon [3,5,6] which has been treated by a large number of papers in the literature (an extensive reference list can be found in, e.g., [7]). Although several detailed theoretical BGN models [S, 91 and representative sets of experimental data have been proposed [7, 10, 111, the proper incorporation of BGN into the transport equations is still a matter of debate and criticism (for a recent comprehensive review on this topic see, e.g., [12]).

A practical problem which arises here is that an in-depth treatment of BGN yields results which cannot be quantitatively deduced easily. On the other hand, empirical BGN models, although quite convenient for practical use [lo, 111, in most cases cannot give full insight

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500 S. D. RISTIC et al.

into BGN and carrier transport phenomena, primary due to different experimental techniques and simplified assumptions involved in their derivation. Moreover, a dilemma which appears concerning BGN is where to join such a complex effect into transport equations [13] such that they retain their form as simple as possible, thus preserving convenience for the efficient solution. It is the aim of this paper to suggest such an approach, which is essentially based on the appropriate treatment of carrier diffusivities which appear in the transport equations. As will be shown in Section 2 this yields the “nondegenerate” transport equations with diffusivities related to the effective intrinsic carrier concentrations. In order to illustrate the accessibility of these parameters the results of their quantitative determination are given in Section 3. Finally, concluding remarks are summarized in Section 4.

2. Theory

Starting from the fundamental work of Marshak and van Vliet [14, 151, Lundstrom et al. [16] have developed transport equations in the following form:

d(A AG) dn dx d X

J , = qp,nF - ,u,n--- + k T p n - - ,

d((1 - A)AG) dP J , = qP,PF + p,,p _ _ _ ~ - - k T p p -, dx dX

where A is the asymmetry coefficient, F the electric field, AG the apparent electrical BGN [12, 171, while all other symbols have their usual meaning. The apparent electrical BGN is defined with respect to band edges E,, and E,, of nondegenerate silicon,

AG = (Ec0 - EZPP) + (E;”” - Eva) 1 AE;” + (3)

and it determines the effective intrinsic concentration, i.e. the p n product in thermal equilibrium [5, 181,

(3 pono = n t = n: exp (4)

with n, being the intrinsic concentration in nondegenerate silicon. It is to be noted that the asymmetry factor is related to the apparent conduction band shift by

On the other hand, Marshak et al. [ 191 have introduced the effective intrinsic concentrations for each type of carriers as

nien = ni exp (g) = ni exp (s) ,

On the Carrier Concentrations and Diffusivities in Heavily Doped Silicon 501

By using (5) to (7) the transport equations (1) and (2) can be recast into the “nondegenerate” form,

dn dx

J , = w , n F + qD,, - >

dP J , = WpPF - 4 D , e - - 7 dx

where the effective diffusivities D,, and D,, are defined as

(9)

with D, and D , being the carrier diffusivities which obey the Einstein relation for nondegenerate silicon (D,/p, = D,/p, = kT/q). As can be seen from (10) and (1 l), the BGN effect is incorporated into the effective diffusivities through the effective intrinsic carrier concentrations, thus allowing comfortable manipulation with transport equations (8) and (9).

Basically, for the calculation of the effective diffusivities one needs to know how to calculate the apparent band edge shifts. Namely, preservation of the apparent electrical BGN concept allows the equilibrium carrier densities to be expressed in Shockley-like form,

no = N c e x p ( - E:” - E = N.exp( - EcO kT - EF )exP(=-). AE:,, (12)

where N , and N, denote the effective densities of states for the conduction and valence band (as defined for nondegenerate silicon), respectively, and E , denotes the Fermi level.

Introducing E” = E,, - E and assuming an n-type heavily doped semiconductor, from (12) and (13) the following expressions for the apparent conduction and valence band shifts can be derived:

+ m 1 g,(E”) exp (- g ) dE” - x

AE;,, = kT In c~ . ~, 1 e,,(E”) exp (- g) dE“ 5

In (15) g, and Q , , denote densities of states for the valence band of degenerate and nondegenerate silicon, respectively. The term (Eco - EF) which appears on the right-hand side of (14) can be calculated from the expression for the equilibrium electron concentration,

33*

502 S. D. R I S T I ~ et al.

under the assumption that it nearly equals the impurity concentration N, at room temperature,

where E’ = E - E,, and Q , is the density of states for the conduction band of degenerate silicon. Therefore, dependencies of the apparent band edge shifts on the impurity concentra- tion can be calculated by self-consistent solution of (14) to (16). Once these dependencies are known, the effective intrinsic carrier concentrations for each type of carriers can be determined on the basis of (6) and (7). It is obvious that calculation of the effective diffusivities can be performed in this way for thermal equilibrium. Their disturbances in the nonequilib- rium (i.e. current flow) case are discussed in the next section, on the basis of the obtained calculations.

3. Calculation Results

Results presented in this section apply to phosphorus-doped silicon in the range lo1’ 5 N, 5 10’’ ~ r n - ~ at room temperature. Densities of states which refer to the

No (cm-3) - Fig. 1. Apparent band edge shifts vs. impurity concentration

On the Carrier Concentrations and Diffusivities in Heavily Doped Silicon 503

Fig. 2. Density of states in the valence band vs. impurity concentration and energy

nondegenerate case are assumed to be parabolical, while those for the degenerate case are approximated using Kane's expressions [20] combined with the functions proposed by Slotboom [21], Morgan [22], and Stern [23] (all these models as well as suggestions for the calculation procedure are summarized by Selberherr in [4]). The dependence of the static dielectric constant on the impurity concentration is also included [24].

Calculated dependencies of the apparent band edge shifts on the impurity concentration are shown in Fig. 1. From this figure one observes two facts: (i) the apparent conduction band edge shift always is larger than the apparent valence band edge shift, primarily contributing to the BGN value at higher impurity concentration, (ii) the apparent valence band edge shift exhibits a maximum around an impurity concentration of ND z 4 x 1019 C ~ I - ~ , which is attributed to the behaviour of the density of states function in the valence band. Namely, as can be seen from Fig. 2, this function (calculated with respect to Ev,) spreads inside the band gap as the impurity concentration increases, reaches its maximum around the above-mentioned value of N , and starts to shrink towards E,, for higher impurity concentrations. The symmetry coefficient A calculated from (5) is plotted in Fig. 3 and, up to this authors' knowledge, this is the first quantitative evaluation of this parameter in the literature.

The obtained values for the apparent band edge shifts are used in (6) and (7) to calculate the effective intrinsic concentrations nien and niepr shown in Fig. 4, along with nie calculated from (4). On the ni, versus ND curve (which is actually the exponentially transformed AG versus N , dependence) we observe a small hump in the range 10'9cm-3 < N ,

504 S. D. RISTI~. et al.

l.OO ~

0.95

t 0’90 0.85 1 0.80

T

0.75

0.70

0.65

0.60

1017 10’8 1019 1020 1021

N~ (cm”) - Fig. 3. Asymmetry coefficient vs. impurity concentration

< lo2’ ~ m - ~ . This hump can also be observed on experimental electrical BGN data of del Alamo and Swanson ([lo], Fig. 13) and of other authors (for a collection see, e.g., [25], Fig. 4, [26], Fig. 2, [ l l] , Fig. 5 ) in approximately the same doping range.

Considering the nonequilibrium case, the effective electron diffusivity may be calculated from (10) by replacing n by no, since in n-type heavily doped silicon the electron concentration is not disturbed significantly from its equilibrium value. However, even for very low applied external electrical fields the hole concentration p is at least several orders of magnitude above its equilibrium value po. Therefore, one has to consider how the effective hole diffusivity is altered with respect to its equilibrium value. As shown in the Appendix, (11) can be rewritten as

where M < 1. Thus, the effective hole diffusivity can be approximated by its equilibrium value by replacing p in (11) by po. For such a case, using (4) and taking into account that nicnniep = n;e, the effective hole diffusivity can be expressed as

On the Carrier Concentrations and Diffusivitics in Heavily Doped Silicon 505

10

1 1017 10'8 1019 1020 1 O 2 l

N" (cm-3) - Fig. 4. Effective intrinsic concentrations vs. impurity concentration

The effective carrier diffusivities calculated from (10) and (18) by using the results shown in Fig. 4 are shown in Fig. 5, along with that obtained when Maxwell-Boltzmann statistics applies for both carriers (nondegenerate case). For the latter calculations mobility models from [27] have been used. Ratios of the effective to the nondegenerate diffusivities from Fig. 5 are plotted in Fig. 6. This figure clearly illustrates the changes in diffusivities caused by heavy doping effects, with a particular hint on the fact that the effective hole diffusivity is almost doubled in comparison to the nondegenerate value around the impurity concentration of 10' cm-3 (which is the concentration frequently encountered in modern semiconductor devices).

It is noteworthy that the connection between the effective electron and hole diffusivities can be also estimated. Namely, from (10) and (18), after some manipulation, the following is obtained:

1 - 2 ; i ~ ; ~ an,

Using the expression for the effective impurity concentration [ 11,

506 S. D. R I S T I ~ et al.

No (cm-3) - Fig. 5. Effective (D,,, Dpe) and nondegenerate (D“, Dp) diffusivities vs. impurity concentration

equation (19) can be rewritten as

a ln ND N D C f f 8ND

Therefore, the effective hole diffusivity can be expressed in terms of the effective electron diffusivity and effective impurity concentration as

4. Conclusion

An evaluation of the effective carrier diffusivities which preserves the “nondegenerate” form of the transport equations in heavily doped silicon has been proposed. This is achieved by using the effective intrinsic concentrations for each type of carriers, which are obtained on the basis of the apparent electrical band-gap narrowing (BGN) definition. The procedure for calculating these quantities has been described and its applicability is demonstrated by

On the Carrier Concentrations and Diffusivities in Heavily Doped Silicon 507

Fig. 6. Ratios of the effective to nondegenerate diffusivities from Fig. 5 vs. impurity concentration

quantitative results for phosphorus-doped silicon up to lo2' ~ m - ~ . The presented results cover calculated values of the apparent band edge shifts, apparent BGN, effective intrinsic carrier concentrations, and effective diffusivities. It is believed that the provided quantitative insight into the above-mentioned parameters can be also useful in view of device simulation oriented problems.

Appendix

The normalized diffusivity for holes as minority carriers from (11) is given by

d n i e p ~

Introducing [28]

P Po

( = -,

508 S. D. RISTI~: et al.

from (Al) it follows that

where M is given by

dx

Selvakuniar [28] has proved that the hole current (as the minority carrier current) can be expressed as

J = - q D dl p P 0 ~- > dx

so that (A4) and (AS) yield

where the superscript zero denotes the case when there is no current flow through the semiconductor, so that the diffusion component of the current J," (diff) is counterbalanced by the drift component of the hole current J:(dtift). Hence, J,"(diff) > J,, in addition to the fact that p % po. Since D:e should stay nearly within the same order of magnitude as D,, from (A6) it follows that M < 1, i.e. from (A3) it may be concluded that

In other words, as is the case for electrons, when there is a current flow, the effective hole diffusivity is negligibly changed with respect to the thermal equilibrium case (Jp = 0).

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290 (1973).

On the Carrier Concentrations and Diffusivities in Heavily Doped Silicon 509

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