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 Issue Eur. Phys. J. Appl. Phys. Volume 90, Number 1, April 2020 10102 12 Semiconductors and Devices https://doi.org/10.1051/epjap/2020190332 26 June 2020

## 1 Introduction

Transport properties of heavily doped semiconductors get significantly altered because of the presence of band tails in the forbidden energy gap. Most of the existing theories of various transport coefficients do not incorporate the effect of band tails and use parabolic density of states function in their calculations of transport coefficients. The main difficulty encountered in the calculations of transport coefficients which incorporate the density of states having band tails is due to the dependence of the modified density of states function on the doping concentration and as a consequence of this one cannot calculate Fermi energy for a given doping concentration.

Kane [1] has derived an expression of the modified density of states function in a heavily doped silicon with doping concentration >1024/m3 at room temperature. But this expression is so complicated that it is not useful for making any calculation. Slotboom [2] has, however, suggested an approximated form of the density of states that is useful for making transport calculations.

This paper which is the first of its kind to carried out calculations of electrical and thermal coefficients which incorporate the effect of band tails in heavily doped n-type silicon. We assumed ellipsoidal and multi-valleyed constant energy surfaces in p-space. If the minimum in the direction of (100) occurs for the value (p 1,0, 0, 0) of the momentum vector p, the energy E near the minimum will be given by(1)where is longitudinal effective mass and is transverse effective mass.

There are the corresponding equations for E at points near (±p 1,0, 0, 0), (0, ± p 2,0, 0), (0, 0, ± p 3,0). We make the following transformations to get spherical energy surfaces [3].(2)

By the help of these relations equation (1) becomes(3)

When the minimum value of energy in the conduction band occurs for a value of p corresponding to an internal point of the first Brillouin zone, the number of quantum states in the absence of band tails in p-space in the range from E to E + dE is(4)where ρ(E) is the density of states and Mv is the number of equivalent valleys in the conduction band. By setting the bottom of the conduction band EC  = 0 and define density of states effective mass as(5)

Then the expression of ρ(E) can simply be simplified as(6)that represents the parabolic total density of states in the conduction band.

### 1.1 Density of states for heavily doped n-type Silicon

A semiconductor is considered heavily doped when the impurity band associated with the doped impurity merges with either the conduction or valence band [4].

We can illustrate the formation of band tails based on the fact that in thermal equilibrium, the Fermi level is constant throughout the semiconductor, but the spatial variations in local donor concentration causes the spatial fluctuation in EC (x) and EV (x). The macroscopic (average) density of states in Figure 1 then shows the tail states below (above) the parabolic conduction band (valence) band. The study by [5] on degenerate donor-silicon system used an expression of the Gaussian average of Ek being the kinetic energy of the electron calculated by using Kane integration method. His results were expressed in terms of E (total energy) vanished at the conduction band edge and for E < 0 exhibited exponential band tails an asymptotic form of exponential conduction-band tail obtained by Halperin and Lux (cited in [5]).

Figure 1 shows emergence of band tails for heavily doped silicon and distortion of parabolic energy structure for heavily doped silicon which are effect of lattice disorder on the band structure caused by rigid shifts due to many-body effects (i) electron-electron interaction (ii) electron-donor interaction (iii) electron-hole interaction were briefly reviewed by [6]. Actually Figure 1 involves many-body effects as well as results of previous studies of band tails and impurity bands.

Kane [1] has shown that the variation in local electrostatic potential is sufficiently slow; the average density of states function for the conduction band could be expressed as follow(7)where(8)and(9)

The standard deviation of the Gaussian distribution for the impurity potential energy is(10)

For a screened coulomb potential of impurity atoms with ∈ d is the dielectric constant of the given semiconductor. The Thomas-Fermi screening length according to [3,7] is(11)

We obtain the following expression of the electron concentration n,(12)where,(13)

It is more convenient to introduce normalized electron concentration nn given by(14)

In this study the semi-classical and quantum treatments are applied in the calculations of scattering mechanisms under the assumptions of the electron concentrations from 2 × 1018–2 × 1020/cm3 and in the temperature range 77–300 K.

 Fig. 1The Figure shows band tails and an impurity (donor) band superimposed on the parabolic conduction and valence band due to energy band distortion of heavily doped silicon.

## 2 Application of Boltzmann equation and relaxation time

All quantities of interest to us may be expressed immediately in terms of the distribution function f(k,r,t). The Boltzmann equation is expressed as(15)

To proceed to first order we suppose(16)

Information can be derived by considering a homogeneous thermal equilibrium at temperature T in the absence of applied fields. In that case the left side of Boltzmann equation vanishes identically. To study the motivation behind the relaxation time approximation we go right back to the original Boltzmann equation and consider a time dependence but spatially homogeneous situation in the absence of applied fields. Thus Boltzmann equation reduces to(17)

Now let us suppose that(18)where f 0 is the thermal equilibrium distribution and f 1 is a small perturbation. Then we have(19)

We would expect f 0 to be stable so that the solution to Boltzmann equation, starting from any given initial perturbation f 1(k, 0) at time t = 0, should decay away to zero with a time constant which is characteristic of the collision process. In the relaxation time approximation we suppose that has the simplest form which will yield this behavior(20)

Based on Boltzmann equation it will be appropriate to present the derivation of collision time due to ionized impurity scattering according to the work of Cowell and Weisskopf [8]. The Boltzmann equation in the presence of a d.c. electric field along x-direction i.e. E=Exî may be written as(21)where me is the effective mass of an electron in that direction, b is the number of electron entering the volume element dV and a is the corresponding number of electrons leaving the volume element per second.

Conwell and Weisskopf [8] made the following assumptions (i) mass of the scattering center is infinite (ii) collision between electron and ionized impurity atom can be treated to a first approximation as independent of all other ions.

One can refer the derivations made by [8] for the quantity(22)

Substituting equations (20) into (19) and noting that , we obtain(23)

Using equation (16) we have(24)

The corresponding equation for θ = 0 is(25)

Thus equation (21) gives(26)

In most of the temperature range of interest one can treat the logarithmic term as a constant and thus the energy dependence of relaxation time due to ionized impurity scattering may be written as(27)where ϵ is the dimensionless electron kinetic energy and(28)

Shockley and Bardeen [9] derived the following expression for relaxation time due to acoustic scattering(29)where(30)where ρ is density of semiconductor; v is velocity of acoustic phonon; E is kinetic energy of electron; EI is deformation potential of constant of the semiconductor.

Electron–electron scattering is briefly discussed in [10] and at low temperature the electron–electron interaction time (τ e-e) is supposed to be significantly smaller than the electron–phonon interaction time (τ e-ph). According to [11] the probability of electron-electron scattering at a temperature of 1 K is about a factor ≅10−10 is smaller than that of electron-defect scattering. In our treatment of transport processes only the scattering from defects is considered due the following additional reasons.

In case when electrons get scattered by ionized impurities as well as the acoustic phonons, the relaxation time τ of electrons [3] is given by(31)

However, comparison of equations (27) and (29), in the temperature range of 77–300 K and electron concentrations n ≥ 2 × 1024/m3, show that is one order of magnitude smaller than and we, therefore, use τ = τ i, in our derivations of the expressions for electrical and thermal conductivities. According to [10] phonon system dominates at higher temperatures for solids.

### 2.1 Electrical current density and electrical conductivity

The general expression for electrical current density [12] is(32)

Consider the applied d.c. electric field is along x-direction, and the x-component of the electric current density is given by(33)

Using the fact that f = f 0 + vxfx and f 0 doesn't contribute to the current density, the expression of Jx reduces to(34)

Making transformation from rectangular to spherical coordinate in velocity space for single band of carriers, we have(35)

From equations (20) and (21), one can show the anisotropic part of the distribution function is(36)

Now the expression of the current density becomes(37)where change of variable is made from v to ε = E/kBT. Substituting the expression for τ(E), the relaxation time for ionized impurity scattering equations (27) into (37), we obtain the following expression for the electrical conductivity(38)

### 2.2 Thermal current and thermal conductivity

In the presence of an external d.c. field Ex and a temperature gradient dT/dx along the x-direction, the Boltzmann transport equation equation (17) can be written as(39)

Now can be expressed as(40)

We substitute equations (40) into (39) for to obtain(41)

Solving this for fx , we get(42)

We use this expression to calculate the thermal current density, it can be written [12] as(43)where we used the fact that f 0 doesn't contribute to the current density and vcosθ = vx and v 2 dvsinθdθdϕ = d 3 v.

We substitute equations (42) in (40), to obtain(44)

Since the thermal conductivity is defined [13] under the boundary condition that Jx  = 0, we obtain the following relationship between Ex and dT/dx.(45)

Now substituting equations (42) and (45) successively into equation (43), and integrating over θ and ϕ we obtain the expression for Cx following the same arrangements for the derivation of Jx and making change of variable from v to ε (46)

The expression for the electronic thermal conductivity Ke can be written as

(47)

## 3 Derivation of equations for numerical calculation of transport coefficients

In all these equations the integration is over the interior of the first Brillouin zone. Thus the primary task of electron transport theory lies the calculation of the distribution function within the first Brillouin zone with density of charge carries (electron concentration) for the parabolic density of states is given by(48)where,(49)is well known tabulated Fermi-integral. One can see p =1/2 corresponds to F1/2, p = 2 corresponds for F2 and in the same way families of Fermi-integral like F3 and F4 so on are defined.

Equation (48) corresponds to equation (12) for non-parabolic (modified density of states with band tails) density of states.

One can express equation (38) for electrical conductivity and equation (47) for thermal conductivity in terms of n in equation (48) for parabolic density of states(50) (51)

These expressions for σ and Ke in this form can be used to obtain Weidmann-Franz ratio (Ke/σT) for parabolic density of states(52)

We can obtain expressions for σ, Ke, and Ke/σT for the case of modified density of states using the corresponding expressions, i.e., equations (38), (47), and (52) obtained based on standard model with parabolic density of states (which doesn't incorporate the effect of band tails) by substituting equation (12) for modified density of states and by extending the integration limits from −∞ to . This yields the following expressions for σ, Ke, Ke/σT (53)

We can write them in dimensionless form as(54)

where δ is the standard deviation of the Gaussian distribution given by equation (10) and the integral functions ψ 0, , , are given by(55) (56) (57)

The integral functions ψ 0, , are given by equations (13), (55), (56) and (57) respectively have been computed numerical for various concentrations using sun ultra 5 work station computer with mathematica software 5.0 [14] installed in it and these values are used to calculate and for various values of doping concentration. The Appendix part contains Table A2 and A3 which show table of values of all integrals in the above expressions.

## 4 Numerical calculation of electrical and thermal conductivities and Weidmann-Franz ration

It is obvious from equation (54) that to calculate and i.e., the dimensionless electrical conductivity, thermal conductivity and Weidmann-Franz ratio, we need to know the numerical value of the dimensional Fermi energy η for a given doping concentration. But as it can be seen from equations (12) and (13) that it is not possible to calculate η analytically for a given concentration nn . Because of the fact that η occurs inside the definite integrals of equation (12) and also because nn depends on the standard deviation of Gaussian distribution δ. However it possible to calculate η for a given electron concentration using an iterative method.

For a given carrier concentration nn we choose an arbitrary value of η as first approximation and then calculate the RHS (right hand side) of equation (12) using given values of nn and first approximation value of η. Then for a second approximation of η, we either decrease or increase the value of first approximation of η and then for the second approximation value of η, we recalculate the RHS. In case the value of RHS for the second approximation of η is closer to the given value of nn , we go on changing η in the same direction till the difference between RHS and LHS (given value of nn ) of equation (12) becomes negligible, i.e., 1 in 106. In case the second approximation of η yields a value of RHS of equation (12), which is further away from the value of LHS, we change in the opposite direction and continue to vary it till the difference between RHS and LHS of equation (12) is negligibly small and two sides of the equation are almost equal. This value of η for which LHS and RHS of equation (12) are equal is the Fermi energy for the given electron concentration. Thus we can compute Fermi energy and also various definite integrals occurring in the expressions of and for various electron concentrations. The calculated values of and for various values of η have been tabulated. In our numerical calculations the following values of various parameters of n-type silicon ϵ d  = 11.8, T = 300 K, and (m 0 is free mass of electron).

## 5 Discussion of the result

In the last section we have described the procedure to numerically calculate Fermi energy η, the electrical conductivity σ, the electronic conductivity Ke and Weidmann-Franz ratio Ke/σT in heavily doped n-type silicon having band tails in the conduction band. We have calculated η, Ke , σ, and Ke/σT for various values of electron concentrations in the range of 2 × 1024/m3 to 2 × 1026/m3 at 300 K. We have considered the electron-ionized impurity scattering as the sole mechanism of electron scattering as our calculations show that even at 300 K, electron collision frequency due to acoustic phonon scattering is one order of magnitude smaller than that due to impurity scattering. Our theory assumes that all the impurities are ionized, i.e., we assume 100% ionization of impurities. This is justified in view of ionization energy (Ed= 0) for Nd ≥ 1024/m3. For sake of comparison we have also made the calculations of η, σ, Ke , Ke/σT for the same values of electron concentrations for the case when the effect of band tails is ignored. Note that in the interpretation of the graphs, we observe in the higher regions of the concentration metallic conductivity occurs and that is why linear fitting was used in that region.

One can see in Figure 2 that the broken-line shows η for the case which takes into account the band tails of the conduction band. For the sake of comparison the solid-line shows for the case when the effect of band tails is ignored. Note that doping introduces, in addition to a Fermi level shift, considerable changes of the conduction mechanism as well.

The variation of normalized electrical conductivity for parabolic density of states as indicated by solid-line in Figure 3 that can be fitted with a straight line of linear equation y = 0.388x + 4.4979. But the variation of normalized electrical conductivity for modified density of states having band tails can be fitted with a straight line of linear equation y = 0.0822x + 3.0041. The goodness of fit R 2 = 0.5281. Thus the difference in the variation between the two cases is given by the difference in the two slopes 38.0%–8.22% ≈ 30%. It means the electrical conductivity exceeds for the first case of parabolic density of states as much as 30% compared to the second case for modified density of states having band tails.

One can see the variation of dimensionless thermal conductivity for parabolic density of states as indicated by the solid-line in Figure 4 that can be fitted to a straight line with linear equation y = 1.2709x + 18.063. But the variation of dimensionless thermal conductivity for modified density of states can be fitted to a straight line with linear equation y = 0.2523x + 12.277. The goodness of fit R 2 = 0.9978. Thus the difference in the variation between the two cases is given by the difference in the two slopes 127.09%–25.23% ≈ 101.86%. It means the thermal conductivity exceeds for the first case of parabolic density of states as much as 101.86% compared to the second case for modified density of states having band tails.

One can see the variation of dimensionless Weidmann-Franz ratio for parabolic density of states as indicated by solid-line in Figure 5 that can be fitted to a straight line with linear equation y = −0.0235x + 3.9858. But the variation of dimensionless thermal conductivity for modified density of states can be fitted to a straight line with linear equation y = −0.0169x + 4.0613. The goodness of fit R 2 = 0.9847. Thus the difference in the variation between the two cases is given by the difference in the two slopes 2.35%–1.69% ≈ 0.66%. It means the Weidmann-Franz Ratio exceeds for the first case of parabolic density of states as much as 0.66% compared to the second case for modified density of states having band tails.

The result of this study particularly on the electronic thermal conductivity correlates with the results of other studies on phonic one as one can see from the study by [15] carried out a first-principles calculation on the lattice thermal conductivity of silicon considering both phonon–phonon and electron-phonon interactions, and predicted a large reduction (45%) at room temperature when the carrier concentration is in the range of 1019–1021 cm−3.

Important progress towards a more predictive treatment of K (single crystal thermal conductivity) in doped Si was made recently by Liao et al cited in [16] who performed an ab initio study of n- and p-doped Si and showed that, electron-phonon scattering at a carrier concentration of ρ ≅ 1021 cm−3 can result in a ≅ 45% reduction in K at room temperature. The calculations reproduce how K is lower in p-doped samples than in n-doped ones, in agreement with the experiments. However, they do not capture the magnitude of the reduction observed in B-doped p-type single crystal Si, which at a doping level of 5 × 1020 cm−3 amounts to more than 70%. The study of [15], based on combined treatment of phonon scattering by electrons and point defects explains the thermal conductivity reduction in highly-doped Si, make predictions for the highly-doped cases (1020 and 1021 cm−3) as there is no experimental thermal conductivity data available for such high doping levels. At room temperature, more than 60% reduction is found as compared to the bulk value at 1020 cm−3 doping level and 90% at 1021 cm−3 for B doping, and 40% and 80% for P doping, respectively.

 Fig. 2The variation of dimensionless normalized Fermi energy with normalized electron concentration: The solid line is for parabolic density of states and the broken-line is for modified density of states.
 Fig. 3The variation of dimensionless electrical conductivity with normalized electron concentration: the solid-line is for parabolic density of states and the broken-line is for modified density of states.
 Fig. 4The variation of dimensionless thermal conductivity with normalized electron concentration: the solid-line is for parabolic density of states and the broken-line is for modified density of states.
 Fig. 5The variation of Weidmann-Franz ratio with normalized electron concentration: The solid-line is for parabolic density of states and the broken-line is for modified density of states.

## 6 Conclusion

In this study the semi-classical and quantum treatments are applied in the calculations of scattering mechanisms. The electron concentration varies from 2 × 1018–2 × 1020/cm−3 and the temperature ranges from 77–300 K which is suitable for practical applications as one doesn't want strongly temperature dependent devices. It is assumed that impurity scattering mechanism to be dominant in this concentration and temperature range compare to acoustic phonon scattering. The comparison is made between the calculations of the transport coefficients and for parabolic density of states in one side and for modified density of states having band tails on the other side shows significant variation as much as 30%, 101.86%, and 6.6% respectively. Actually at higher temperature it is difficult to do experimental research and we can rarely get experimental data for higher temperature range where phonic conduction is dominant. DFT calculations involving Botzmann equation can be used to find significant corrections of calculations based on classical treatment of electrical and thermal conductivity.

## Appendix A

Table A1

Dimension of parameters appeared in numerical calculation.

Table A2

Calculated values for parabolic density of states.

Table A3

Calculated values for modified density of states having band tails.

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Cite this article as: Mulugeta Habte Gebru, Electrical and thermal conductivity of heavily doped n-type silicon, Eur. Phys. J. Appl. Phys. 90, 10102 (2020)

## Supplementary material

Supplementary material provided by the author. (Access here)

## All Tables

Table A1

Dimension of parameters appeared in numerical calculation.

Table A2

Calculated values for parabolic density of states.

Table A3

Calculated values for modified density of states having band tails.

## All Figures

 Fig. 1The Figure shows band tails and an impurity (donor) band superimposed on the parabolic conduction and valence band due to energy band distortion of heavily doped silicon. In the text
 Fig. 2The variation of dimensionless normalized Fermi energy with normalized electron concentration: The solid line is for parabolic density of states and the broken-line is for modified density of states. In the text
 Fig. 3The variation of dimensionless electrical conductivity with normalized electron concentration: the solid-line is for parabolic density of states and the broken-line is for modified density of states. In the text
 Fig. 4The variation of dimensionless thermal conductivity with normalized electron concentration: the solid-line is for parabolic density of states and the broken-line is for modified density of states. In the text
 Fig. 5The variation of Weidmann-Franz ratio with normalized electron concentration: The solid-line is for parabolic density of states and the broken-line is for modified density of states. In the text

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