Effects of Donor Size and Heavy Doping on Optical, Electrical and Thermoelectric Properties of Various Degenerate Donor-Silicon Systems at Low Temperatures

: In various degenerate donor-silicon systems, taking into account the effects of donor size and heavy doping and using an effective autocorrelation function for the potential fluctuations expressed in terms of the Heisenberg uncertainty relation and also an expression for the Gaussian average of (cid:1) (cid:2) (cid:3)(cid:4) (cid:5)(cid:6) , a ≥ 1 (cid:1) (cid:2) being the kinetic energy of the electron, calculated by the Kane integration method (KIM), we investigated the density of states, the optical absorption coefficient and the electrical conductivity, noting that this average expression calculated by the KIM was found to be equivalent to that obtained by the Feynman path-integral method. Then, those results were expressed in terms of (cid:1) (cid:3)(cid:4)(cid:7)(cid:8)/(cid:10)(cid:11) for total electron energy (cid:1) (cid:12) 0 , vanished at the conduction-band edge: (cid:1) (cid:14) 0 , and for (cid:1) (cid:15) 0 exhibited their exponential tails, going to zero as (cid:1) → (cid:17)0 and (cid:17) ∞ , and presenting the maxima, in good accordance with an asymptotic form for exponential conduction-band tail obtained by Halperin and Lax, using the minimum counting methods. Further, in degenerate d-Si systems at low temperatures, using an expression for the average of (cid:1) (cid:23) , p ≥ 3/2, calculated using the Fermi-Dirac distribution function, we determined the mobility, electrical conductivity, resistivity, Hall factor, Hall coefficient, Hall mobility, thermal conductivity, diffusion coefficient, absolute thermoelectric power, Thomson coefficient, Peltier coefficient, Seebeck thermoelectric potential, and finally dimensionless figure of merit, which were also compared with experimental and theoretical results, suggesting a satisfactory description given for our obtained results.


Introduction
Donor (acceptor)-silicon d (a)-Si system at a given temperature T, doped with a given d (a)-density N, assuming that all the impurities are ionized, is the base material of modern semiconductor devices [1][2][3][4][5][6].Then, due to the Fermi-Dirac statistics, there are three cases may be classified as: non-degenerate (T >> T and N<< N ), moderately degenerate (T > T and N>>N ) and degenerate (T << T and N>>N )-cases, T and N being respectively the degenerate temperature defined in Eq. (15) and critical impurity density.
In Section 2, we studied the effects of donor size [or compression (dilatation)], temperature, and heavy doping on the energy-band-structure parameters.At T = 0 K, with increasing values of donor radius r , since the effective dielectric constant ε r decreases, due to the donor-size effect, the effective donor ionization r , unperturbed intrinsic band gap !r , and critical donor density N ( ) increase, as seen in Table 1.Then, for a given " , the effective intrinsic band gap !# (" , T), due to the T-effect, decreases with increasing T, as given in Eq. (3).Finally, due to the heavy doping effect (HDE), for a given r , the effective electron mass m & ' (N, r ) increases with increasing N, as given in Eq. (8), and for given " ( and T, the reduced band gap !(N, T, r ) decreases with increasing N, as given in Eq. (10).
In Section 3, the effective autocorrelation function for potential fluctuations, W , was determined and in Eq. (B.6) the appendix B, being a central result of the present paper, as noted in Eq. (20).It was suggested that W ( → ±∞) → η , and η being respectively the total electron energy and the energy parameter characteristic of the conduction-band tail states, and W ( → ±0) → 0 .Therefore, the density of states, the optical absorption coefficient and the electrical conductivity, being proportional to our result (20), vanished at the conduction-band edge = 0, as given in Eqs.(23,26).Those results were also compared with other theoretical results, obtained at − = 0, in the small time approximation [21,29,30] and in the full ground-state case and deep-tail approximation [21], which were found to be constant, being not correct, as discussed also in Eq. (26).Then, for ≤ 0, their exponential tails were obtained in Figures 1, 4 and 7, in which they increased with increasing r for a given value of − , due to the donor-size effect, and further they went to zero as → −0 and − ∞ and presented the maxima, being found to be in good accordance with an asymptotic form for exponential conduction-band tail, obtained by Halperin and Lax [19], using the minimum counting methods.
In Section V, we determined the critical donor density, as given in Table 1, suggesting that its numerical results are in good agreement with the corresponding data given in Ref. 12, and it increases with increasing r , due to the donor-size effect [12].Then, for ≤ 0 , the exponential band-tail behaviors were investigated and reported in Table 4, and also in Figures 1 and 2a (b).
In Section 6, various optical functions were determined in band-to-band transitions ( ≥ 0) as found in Figs.3a, 3b, 3c, being compared with other theoretical and experimental works [33-35, 38, 44-48], and also the exponential optical absorption-coefficient tail behaviors were investigated when ≤ 0, as seen in Table 7, and Figures 4 and 5a (b).
In Section 8, for ≥ 0, using also the function G , we studied various thermoelectric functions, and reported their numerical results in Table 15 and Figures 9a, 9b, 9c, 9d, 9e, and 9f, noting that for ≤ 0 we could also study the exponential thermoelectric function-tail behaviors by a same treatment, as those obtained in Sections 5-7.
Finally, some concluding remarks were given and discussed in Section 9.

Energy-Band-Structure Parameters
Here, we study the effects of donor size, temperature, and heavy doping on the energy-band-structure parameters.

Donor-Size Effect
In donor-Si systems at T=0 K, since the d-radius, r , in tetrahedral covalent bonds is usually either larger (or smaller) than the Si atom-radius, r , assuming that in the P-Si system r 5 = r = 0.117 nm, with nm = 10 9 m, a local mechanical strain (or deformation potential-or-strained energy) is induced, according to a compression (dilation) for r > r (r < r ), respectively, or to the donor size (r )-effect.In the Appendix A of our recent paper [12], basing on an effective Bohr model, such a compression (dilation) occurring in various donor (d)-Si systems was investigated, suggesting that the effective dielectric constant, < (" ), decreases with increasing " .This donor size (" )-effect affects the changes in all the energy-band-structure parameters or the electronic properties of various donor-Si systems, expressed in terms of < (" ), as those investigated in our recent paper [12], noting that < (r 5 ) = 11.4 .In particular, the changes in the unperturbed intrinsic band gap, !(r 5 ) = 1170 meV , effective donor ionization energy, (r 5 ) = 33.58meV , and critical donor (P)-density, N (5) = 3.5 × 10 B cm D , of the P-Si system at 0 K, are obtained in an effective Bohr model, as [12] !(r ) − !(r 5 ) = (r ) − (r 5 ) = (r 5 ) × EF G H (I J ) G H (I K ) L − 1M, and in a simple generalized Mott model, by [12] N (I K ) = N (5) × F G H (I J ) G H (I K ) L D .
Therefore, with increasing " , the effective dielectric constant < (" ) decreases, implying thus that !(r ) , (r ) and N (I K ) increase.Those changes, given in our previous paper [12], are now reported in the following Table 1, in which the numerical results of critical donor density, due to the exponential band tail (EBT)-effect, N ( ),N 'OP , being obtained in the next Section V, are also included for a comparison.Here, N is normally equal to 1, but it will be chosen as: N = N = 1.0028637416, so that the obtained Various Degenerate Donor-Silicon Systems at Low Temperatures results of N (I K ),N R 'OP would be more accurate.

Table 1.
The following values of " ( , < S , T SU , V WU (" ( ), and X Y(() -data, given in our previous paper [12], are now reported in this TABLE, in which we also include the numerical results of X Y((),N Z[\ , where N = 1 ]" N U = 1.0028637416, calculated using Eqs.(41,42), and their absolute relative errors defined by: `. Here, gh ≡ 10 9 h.Moreover, it should be noted that in donor-Si systems such the !# (r , 300 K)-increase with increasing r observed in Table I, !# being the effective intrinsic band gap given in next Eq.( 3), well agrees with a result obtained recently by Ding et al. [42].In fact, in their study of the optical properties of isolated silicon nano-crystals (nc-Si) with the size of 2r s# = 4.2 nm (≫ 2r 5 = 2 × 0.117 nm) embedded in a SiO matrix, they showed that !# (r s# , 300 K) = 1.79 eV ≫ !# (r 5 , 300 K) = 1.12 eV given in the bulk crystalline Si at room temperature, being also due to the size effect (r s# ≫ r 5 ).Now, the effective Bohr radius can be defined by

Donor
In Eq. ( 1), m * is the effective electron mass given in the Si, being equal to: (i) the effective mass m = 0.3216 × m , m being the free electron mass, defined for the calculation of m & ' (N), as determined in next Eq.( 8), due to the heavy doping effect (HDE), (ii) the reduced effective mass: for the optical absorptioncoefficient calculation, where m = 0.3664 × m is the effective hole mass in the silicon [12], (iii) m & ' (N), given in next Eq.( 8), for the determination of the density of states, as given in Section 5, and finally (iv) the conductivity effective mass: m .= 0.26 × m for the electrical conductivity calculation [6], as used in Section 7.
Then, in the degenerate case ( N > N ( ) ), denoting the Fermi wave number by: k .(N) ≡ (3~ N/g ) /D , where g = 3 is the effective average number of equivalent conduction-band edges [11,12], the effective Wigner-Seitz radius r € characteristic of the interactions is defined by being proportional to N /D .Here, • = (4/9~) /D , and k .means the averaged distance between ionized donors.

Temperature Effect
Here, in d-Si systems, being inspired from recent works by Pässler [8,9], we can propose an accurate expression for the effective intrinsic band gap as a function of r and T, as For example, in the (P, S)-Si systems, for 0 ≤ T (K) ≤ 3500, the absolute maximal relative errors of !# are equal to: 0.22%, 0.15%, respectively, calculated using the accurate complicated results given by Pässler [9].

Heavy Doping Effect (HDE)
HDE on h S Now, using Eq.(2) for r € (N, r , m * = m ), the ratio of the inverse effective screening length k € to Fermi wave number k .at 0 K is defined by [12] It is noted that, in the very high electron-density limit [or in the Thomas-Fermi (TF)-approximation], R € is reduced to being proportional to N /k .It should be noted that the effective screening length k € P. is very larger than the averaged distance between ionized donors k .(i.e., this is the TF-condition given in the very degenerate case, N ≫ N ( ) ), and in the very low electron-density limit [or in the Here, when the relative spin polarization ¡ is equal to zero (paramagnetic state), ¢' means the majority-electron correlation energy (CE), determined by as [11,12] ¢' (N, r ) = ƒOE.£¤¥¥šOE.OE¦OE£| where a 5 = 2•1 − Ln(2)'/~ and b 5 = −0.093288 .Then, due to such the HDE, the effective electron mass can be approximated by [11,12] HDE on W± In the degenerate case, the optical band gap is defined by where . is the Fermi energy determined at any N and T in Eq. (C.1) of the Appendix C, with an accuracy equal to: 2.11 × 10 l , and ! is the reduced band gap defined as !(N, T, " ) ≡ !# (T, " ) − BGN(N, " ), (10) where the intrinsic band gap !# (T, " ) is determined in Eq.
(3) and the band gap narrowing (BGN) is determined below.
In our recent paper [12], an accurate formula for the BGN was investigated, being expressed in the following spinpolarized chemical potential-energy contributions, as: the exchange energy of an effective electron gas, the majority-electron correlation energy of an effective electron gas, (iii) the minority-hole correlation energy, (iv) the majority electron-ionized donor interaction screened Coulomb potential energy, and finally (v) the minority hole-ionized donor interaction screened Coulomb potential energy.
It should be noted that the two last contributions (iv) and (v) are found to be the most important ones.Therefore, an approximate form for the BGN can be proposed by which is a very simplified form compared with our previous complicated expression for BGN [12].
Here, the values of effective dielectric constant ε (r ) are given in Table 1 and the electron effective mass m & ' (N, r ), due to the heavy doping effect, is determined in Eq. (8).Further, the empirical parameter C = 8.5 × 10 D (eV) has been chosen so that the absolute maximal relative error |MRE| of our result (9), calculated using the optical band-gap ( ! )-data for P-Si systems at 20 K obtained by Wagner, [7] are found to be minimized.
In a degenerate P-Si system, with use of the next Eq.( 43), obtained for the definition of effective density of free electrons given in the conduction band, N * ≃ N − N (5) , where the value of N ( 5) is given in Table I, our present results of !(N * , T = 20 K, r 5 ), computed using Eqs.(9,11), and their absolute relative errors |REs|, calculated using the !-data at 20 K [12], are obtained and reported in Table 2, in which our previous accurate !(N * )-results and their |REs| are also included [12], for a comparison.
Table 2. Numerical results of optical band gap at T=20 K, WS ( X * ) , expressed in eV, being investigated in our recent paper [12], and determined in Eq. ( 9), and finally their absolute relative errors |_V|, calculated using the WS -data [7] This table indicates that the maximal value of |REs| , obtained from our present !(N * )-result, is found to be equal to 1.7%, which can be compared with that equal to 1.2% obtained from our previous !(N * )-result [12].HDE conditions Finally, in degenerate d-Si systems, the energy parameter Å , being characteristic of the exponential conduction-band tail, is determined in Eq. (B.4) of the Appendix B as where k € / is determined in Eq. ( 4).Moreover, in highly degenerate case (N ≫ N ( ) ) or in the Thomas-Fermi approximation, k € / ≃ k € P. / , determined in Eq. ( 5), Å is found to be proportional to N È/ .Then, from Eq. ( 12) and next Eq.( 15), we can obtain another heavy doping condition as being proportional to N /l in this highly degenerate case.In summary, in the highly degenerate case (N ≫ N ( ) ) and from Eqs. (2,4,13), one has where . is the Fermi energy at 0 K, defined by In Eq. ( 15), m * is the electron effective mass, defined in Eq. (1), and in this highly degenerate case one has a low Tcondition as: T ≪ T ≡ .(N)/k O , T and k O being the degeneracy temperature and the Boltzmann's constant, respectively.

Effective Autocorrelation Function and its Applications
In the degenerate d-Si systems, the total screened Coulomb impurity potential energy due to the attractive interaction between an electron charge, −Ë, at position r Ì and an ionized donor charge: +Ë at position R Í ÎÎÎÌ randomly distributed throughout the Si crystal, is defined by where Ò is the total number of ionized donors, V is a constant potential energy, and v Ñ (r) is a screened Coulomb potential energy for each d-Si system, defined as Further, using a Fourier transform, the v Ñ -representation in wave vector k ÎÌ -espace is given by where Ω is the total Si-crystal volume and k € is the inverse screening length determined in Eq. ( 4).Moreover, in Eqs.(16,17), V is defined as a constant so that 〈V(r)〉 = 0 , reflecting a charge neutrality, where the notation 〈… 〉 denotes the configuration average [25,58].In fact, from Eq. ( 17), one has indicating that from Eq. ( 16) one obtains: 〈V(r)〉 = 0. Therefore, the effective autocorrelation function for potential fluctuations can thus be defined by [25,58] W (×r Ì − r â ÎÎÌ ×) ≡ 〈V(r)V(r â )〉 ≡ 〈V(r)〉 × 〈V(r â )〉 + 〈〈V(r)V(r â )〉〉 = 〈〈V(r)V(r â )〉〉, where 〈〈V(r)V(r â )〉〉 denotes the effective second-order cumulant, and r(t) ÎÎÎÎÎÎÎÌ and r â (t â ) ÎÎÎÎÎÎÎÎÎÎÌ are the electron positions at the times t and t â , noting that the cumulant is just the average potential energy, which may be absorbed by a redefinition of the zero energy.Then, the expression for W is determined in Eq. (B.6) of the Appendix B, as Here, R € (N) is given in Eq. ( 4), η is determined in Eq. ( 12), the constant ℋ will be chosen in next Section V as: ℋ = 5.4370, such that the determination of the density of electrons localized in the conduction-band tail would be accurate, and finally ν ≡ ‚HR , where is the total electron energy and . is the Fermi energy at 0 K, determined in Eq. (15).Now, we calculate the ensemble average of the function: ×y * is the kinetic energy of the electron and V(r) is determined in Eq. ( 16), using the two following integration methods, which strongly depend on the effective autocorrelation function for potential fluctuations W (ν , N), determined in Eq. (19).

Kane Integration Method
In heavily doped d-Si systems, the effective Gaussian distribution probability is defined by So, in the Kane integration method (KIM), the Gaussian average of ( − V) ≡ is defined by [14] Then, by variable changes: s = ( − V)/ √ • H and x = − /çW , and using an identity [15]: where D (x) is the parabolic cylinder function, Γ(T + ) is the Gamma function, one thus finds This result (20) will used to study the optical, electrical, and thermoelectric properties of various degenerate d-Si systems, depending on W defined in Eq. ( 19) and the variable x, expressed also in terms of W , as where .and Å are determined in Eqs.(15,12), respectively.Therefore, the effective autocorrelation function for potential fluctuations W , defined in Eq. (19), is thus a central result of the present paper.

Feynman Path-Integral Method
In the Feynman path-integral method (FPIM), the ensemble average of ( − V) is defined by • is found to be proportional to the averaged Feynman propagator given the dense donors [16].
Thus, as → +0, from Eq. ( 20), one gets: being in good agreement with our result obtained in Eq. (A3) of the Appendix A.
It should be noted that, as ≤ 0, the ratios ( 26) and ( 27) can be taken in an approximate form as so that: F (ò , r , a) → H (ò , r , a) for 0 ≤ ò ≤ 16 , and F (ò , r , a) → K (ò , r , a) for ò ≥ 16.For that, in next sections V and VI, the constants c and c may be respectively chosen as: c = 10 ln and c = 80 when a = 1 , being used to the study of reduced density of exponential conduction-band-tail states, and c = 20 Èn and c = 300 when a = 5/2 , for the study of reduced optical absorption coefficient and exponential tails of electrical conductivity.
It should be noted that the important results (20) obtained for any -values, (24) for ≥ 0, and (28-31) for ≤ 0, can be used to determine the density of states and the optical, electrical and thermoelectric functions in Sections V-VIII, respectively.

Low Temperature Effect, Due to the Fermi-Dirac Distribution Function
The Fermi-Dirac distribution function (FDDF) is given by where .(N, T) is the Fermi energy determined in Eq. (A10) of the Appendix C. So, the average of , calculated using the FDDF, can be defined as Further, one remarks that, at 0 K, − = ( − .), ( − .) being the Dirac delta ( )-function and . is the Fermi energy at T=0 K defined in Eq. (15).Therefore, G ( . ) = 1.
Then, at low T, by a variable change • ≡ ( − .)/(k O T), Eq. ( 32) yields where C ≡ p(p − 1) … (p − β + 1)/β! and the integral I is given by , vanishing for old values of ú.Then, for even values of ú = 2n, with n=1, 2, …, one obtains Now, using an identity [15]:  32), we get in the degenerate case the following ratio: It should be noted that our previous expression for G (!) [58] can now be corrected, replacing β by 2n and the Bernoulli numbers B / by |B |.Further, Jaffe [49] proposed the following result: where . Now, using an identity: Eq. ( 35) is found to be identical to our above result (34), which is a more practical result.Then, some usual results of G (y) are given in Table 3. [15], and some expressions for $ % (&), obtained from Eq. ( 34) at low T and for

Table 3. The values of absolute Bernoulli numbers |# S |
These functions G (y) obtained in Table 3 will be applied to determine the electrical and thermoelectric properties of the degenerate d-Si systems, being given in Sections 7 and 8, respectively.

Determination of Critical Donor Density
In degenerate d-Si systems at T=0 K, due to the heavy doping effect (HDE), using Eq. ( 20) for a=1, 〈 〉 Šëì , the density of states ,( ) is given by: where m & ' N is the electron effective mass, due to the HDE, determined in Eq. ( 8), and the variable x is defined in Eq. ( 21), as Here, . is determined in Eq. ( 15) for m * m & ' N , m & ' N being the electron effective mass due to the HDE and determined in Eq. ( 8), and the value of Heisenberg empirical parameter ae was defined in the Appendix B and proposed here as: ae =5.4370, so that the following determination of the critical density of electrons localized in the exponential conduction-band tail would be accurate.Further, from Eq. ( 24), one also has Going back to the functions: H , K and F , given respectively in Eqs.(26)(27)(28), in which the factor is now replaced by: where the reduced density of exponential-tail states: F ò , r , a = 1 ≡ F ò , r , for a simplicity of presentation, is determined in Eq. (28).Then, in d-Si systems at 0 K and for N = 5 1 10 n cm D , our results of the functions F ò , r obtained for each r -value, are plotted as functions of ò in Figure 1. Figure 1 shows that: (i) our results of F ( ) increase with increasing r for a given ò , due to the donor-size effect, and (ii) present the maxima at ò = ò (ì) and go to zero as ò → 0 and ∞, being found to be in good agreement with theoretical results obtained by Lifshitz [18], Friedberg and Luttinger [20], our results given in Eq. (A.3) of the Appendix A, and in particular with an asymptotic form for exponential conduction-band tail, obtained for 0 ≲ ò ≲ ∞, by Halperin and Lax [19], using the minimum counting methods.
Finally, our numerical results of energy parameter, (N; r , a = 1), obtained in the small interval: 1.2 ≤ ò ≤ 1.25, using Eq.(31), are plotted as functions of N in Figures 2a and 2b, indicating that, for a given N, increases with increasing r -values, due to the donor-size effect.
Then, by a variable change: ò ≡ ‚HR , Eq. ( 39) yields where Here, and N is normally equal to 1, but it can be an empirical parameter, being chosen as: N = N = 1.0028637416 such that the obtained values of N 'OP would be accurate.
Hence, in the degenerate d-Si system, replacing N, given in the parabolic conduction band of an effective electron gas, by the effective density of free electrons defined here by: N * = N − N ,N 'OP ≥ 0. So, in this system, the Fermi energy given in Eq. ( 15) is now rewritten as where the Fermi wave number k .and m & ' are respectively determined in Eqs.(2,8).One notes here that .(N * ) vanishes at N * = 0, or at the critical donor density defined by: N = N ( ),N 'OP ≡ N ,N 'OP (N = N ( ),N 'OP , r ) , at which the metal-insulator transition thus occurs.Then, the numerical results of N ( ),N 'OP , for N = 1 and N = 1.0028637416 , and their absolute relative errors |RE| , calculated using the N ( ) -data given in Table I  given in the parabolic conduction band of the degenerate d-Si systems can be approximated by [22] N Here, this notion of effective density of free electrons N * defined by Eq. ( 43) should be equivalent to that of (N − N ) given in the n-type compensated Si, in which N is the total density of donors (or majority electrons) and N is the total density of acceptors (or minority holes), assuming that all the impurities are ionized [22].Finally, in degenerate d-Si systems, in which N > N ( ) and T ≤ 77 K or T ≪ T , T being the degeneracy temperature defined in Eq. ( 15), this result (43) will be used in all the following Sections.

Optical Properties
The problem of exponential optical absorption-coefficient tails has by now a rather long history.We will limit our study here to the degenerate d-Si systems, although the band structure of random alloys and amorphous materials is a problem with many common features [41].
Optical properties of any medium can be described by the complex refraction index ℕ and the complex dielectric function <, defined by: ℕ ≡ g − ôN and < ≡ < − ô< , where ô = −1 and < ≡ ℕ , and by the optical absorption coefficient O, which is related to the imaginary part of <: < , the refraction index n, the extinction coefficient N, and the conductivity σ Q , due to the electro-optical effect, as [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] α(E) One remarks that the real part of < is defined by and the normal-incidence reflectance R(E), by which are the optical dispersion relations since in general the values of those optical functions are expressed as functions of the multi-photon energy [46], E ≡ ℏω, 2ℏω , 3 ℏω , 4ℏω,….In the present work, we only focus our attention to the case of photon energy E ≡ ℏω.Here, -q, ℏ, |\(V)|, ], < U , and J(V) respectively represent the electron charge, Dirac's constant, matrix elements of the velocity operator between valence-and-conduction bands in n-type semiconductors, photon frequency, permittivity of free space, velocity of light, and joint density of states (JDOS).One remarks here that: (i) if some optical functions are known such as: (J, n, |v| ), (n, κ), or (ε , ε ), then, all other ones are determined, and (ii) in n-type semiconductors, all the optical functions will be expressed in terms of the total energy of the electron, defined by: where the band gap given in the degenerate d-Si systems, !, can be equal to: !# , ! and !, defined in Eqs.(3, 9, 10), respectively.Now, we determine the accurate expressions for the optical functions obtained in band-to-band transitions at E ≥ !, and for the exponential optical absorption-coefficient tails and also their behaviors at E ≤ !, due to the effects of temperature, donor size and heavy doping, being also compared with corresponding experimental and theoretical results.

Optical Functions Obtained in Band-to-Band
Transitions at ^≥ _` or ≥ À First of all, one remarks from Eq. ( 37) that if replacing the density-of-states effective mass, g /D × m & ' , by the reduced effective mass m I = 0.171 × m , defined in Eq. ( 1), the density of state ,( ≥ 0) becomes the joint density of states, given in Eq. ( 44), as Then, we use a transformation, replacing ( ) / by: ‚HR ïƒ × ( ) ( / ) §( )ÓaÓ .So, J(E) yields for a ≥ 1 as and from Eq. ( 44), one gets Further, for any E or , using Eq. ( 20 where x is defined in Eq. ( 21), as , ò ≡ ‚HR .Here, the Fermi energy .
Furthermore, in the d-Si systems, from Eqs. (44,48) one can determine the extinction coefficient N , obtained for a=5/2, as We now propose an improved FB-M (IFB-M).
, where the values of empirical parameters: A #(.O) , B #(.O) and C #(.O) , are given in the FB-M for the Si [40], and simply replacing the band-gap energy != 1.06 eV [40] by !, which can be equal to: !# , ! and !, which are determined respectively in Eqs.(3,10,11), we can now propose, as that done by O'Leary et al. for very large values of E, [39] so that κ ë.O ì (E → ∞ goes to 0 as E D , in good accordance with both experimental [36] and theoretical [41] I, giving a correct asymptotic behavior of n ë.O ì (E).
Here, n is the factor to be determined so that the function n ë.O ì (E) for E ≥ 6 eV is continuous at E = 6 eV, depending on T, " , and N.
For example, in intrinsic d-Si systems at 298 K, in which ! = !#(T = 298 K, " ) = 1.125 eV is determined in Eq. ( 3), the values of n (r ) are evaluated and tabulated in Table 5.
Table 5.In intrinsic donor-Si systems at 298 K, the numerical results of Factor g U (" ( ), being due to the donor-size effect and expressed as functions of donorradius " ( , are determined so that the function g de[ 1 (V) given in Eq. ( 52) for V ≥ 6 'f is continuous at V = 6 'f.As noted in Eqs.(44)(45)(46), if from Eqs. (51,52) the values of κ ë.O ì (E) and n ë.O ì (E) are evaluated, all other optical functions can thus be determined.So, at 298 K and 1.5 ≤ E(eV) ≤ 6 , in the intrinsic P-Si systems, in which ! = !#(T = 298 K, " ) = 1.125 eV is evaluated using Eq.(3), our results of all the optical functions and the corresponding ones obtained from the FB-M, and the absolute errors of those, calculated using the optical-function data obtained by Aspnes and Studna [33], are tabulated in the Table 6.Table 6.In intrinsic P-Si systems at 298 K and for 1.5 ≤ V('f) ≤ 6, our numerical results of all the optical functions (OF) are calculated, using Eqs.(44-46, 51, 52) obtained in our IFB-M, and using the OF-data obtained by Aspnes and Studna [33], their absolute maximal relative errors (|g_Vh|) determined at the photon energy E (eV) are also evaluated and tabulated in this Table, in which the corresponding |g_Vh| obtained in FB-M are also included.The underlined |g_V|-value is the maximal one for each optical function.

MRE
Table 6.In intrinsic P-Si systems at 298 K and for 1.5 ≤ V('f) ≤ 6 , our numerical results of all the optical functions (OF) are calculated, using Eqs.(44-46, 51, 52) obtained in our IFB-M, and using the OF-data obtained by Aspnes and Studna [33], their absolute maximal relative errors (|MREs|) determined at the photon energy E (eV) are also evaluated and tabulated in this Table, in which the corresponding |MREs| obtained in FB-M are also included.
Table 6 indicates that our results given in our IFB-M are found to be more accurate than those obtained in the FB-M.Further, our numerical calculation indicates that, for a given E, since N ë.O ì (E) given in Eq. ( 51) is expressed in terms of (E − ! ) , if ! increases (decreases), then other functions such as: (E − ! ) , κ ë.O ì (E) , ε (ë.O ì) (E) and ε (ë.O ì) (E) decrease (increase), respectively.This useful remark will be used in our IFB-M to explain all the following results.

Figure 3b. In intrinsic donor-Si systems, our results of <
de[ 1 V , in absolute values, decrease with increasing " ( . In degenerate P-Si systems at T=4.2 K, in which ! ≡ !N , being the optical band gap determined in Eq. ( 9), increases with increasing N, due to the heavy-doping effect.
So, for a given E, the absolute values of •E E !N ' and < ë.O ì E decrease with increasing N , in good accordance with experiments by Aspnes et al. [34], and Vina and Cardona [35], as seen in the following Figure 3c.

Figure 3c. In degenerate P-Si systems, our results of <
de[ 1 V , in absolute values, decrease with increasing X.
Finally, in degenerate P-Si systems, in which ! ≡ !N , being the reduced band gap determined in Eq. ( 10), decreases with increasing N, due to the heavy-doping effect.Consequently, for a given E, the absolute values of •E !N ' and < ë.O ì E increase with increasing N. Now, identifying our above results (50, 51) and using Eq. ( 52), we can propose an useful expression for |v E | as , for E ! or for 0.
(53)  Figure 4 shows that: (i) our results of F ( ¥ ) increase with increasing r for a given ò , due to the donor-size effect, and (ii) present the maxima at ò = ò (ì) and go to zero as ò → 0 and ∞, being found to be in good agreement with theoretical results obtained by Lifshitz [18], Friedberg and Luttinger [20], our results given in Eq. (A.3) of the Appendix A, and in particular with an asymptotic form for exponential conduction-band tail, obtained for 0 ≲ ò ≲ ∞, by Halperin and Lax [19], using the minimum counting methods.
Finally, our numerical results of energy parameter, N; r , a = 5/2) , obtained in the small interval: 1.2 ò 1.25, using Eq. ( 31), are plotted as functions of N in Figures 5a and 5b, indicating that, for a given N, increases with increasing r -values, due to the donor-size effect.

Electrical Properties
Here, m * ≡ m .= 0.26 1 m .Then, the electrical functions, obtained in the two cases: 0 and 0, will be considered as follows.

Electrical Functions Obtained as À
In the effective electron gas at 0 K [66], denoting the relaxation time by t, the mobility is defined by the conductivity σ (or resistivity ρ ≡ 1/σ ), given in the Drude model, by the Hall conductivity x & , by and finally, from Eqs. (55,56), the Hall coefficient at 0 K, by This result ( 57) is not correct for the degenerate donor (d)-Si systems at low temperatures, where N may be replaced by the total effective density of free electrons given in the conduction band, N * ≃ N − N ( ) , as that given in Eq. ( 43), in which the values of critical donor density N ( ) are given in Table I.In those degenerate d-Si systems, the relaxation time can be defined by where ℏk/(m .× m) is the electron velocity, C is an empirical parameter, ~(C × k) is the scattering cross section, and finally the factors x # are included to represent the high donor-density conditions when k = k ., as those given in Eq. ( 14), such that y( ‚H ) < 1.
We now report and discuss the results of t, being obtained by Van Cong and Mesnard method (VCMM) [58] and also by Yussouff and Zittarz [59], as follows.
By a Green function (G)-method, assuming that the Gaussian ensemble average of GG as: 〈GG〉 ≡ 〈G〉 × 〈G〉 + ΔG ≃ 〈G〉 × 〈G〉, Van Cong and Mesnard obtained [58] which can be replaced into Eqs.(54,55), respectively, to obtain which is proportional to .D/ , where the Fermi energy .(N * ) is determined in Eq. ( 42).Furthermore, by qualitative arguments, based on the diagram method, then for the lowest order in inverse screening length k € , Yussouff and Zittarz [59] obtained where α |} is the dimensionless function, being not well determined.However, this qualitative argument method is useful to investigate the accurate t -result, as that given below.
Our numerical calculation indicates that the (μ, σ)-results, obtained from Eqs. (60,61) do not well agree with the corresponding experimental ones [50,54,60].Thus, there is a need of performing those results.
In this performed VCM-method (PVCMM), proposing the total correction: , being proportional to .
/ , where T O is the effective Bohr radius determined in Eq. ( 1), and also using our result (34) for G ( . ) ≡ with p=3/2, then the results (60) and ( 61) are now performed as where μ ê¢ìì and σ ê¢ìì are respectively determined in Eqs.(60,61) and the function Gš (y) is given in Table III, with Further, the Hall coefficient is defined by where the Hall factor is found to be given by Furthermore, the Hall mobility is given by We now propose our present method (PM) to determine all the electrical functions as follows.
First of all, one remarks that in Section 6 all the optical functions, given in Eq. ( 44) and obtained in d-Si systems, are found to be proportional to or to ., as = . .Then, in the PM, we propose both principal parts of μ and σ, being Various Degenerate Donor-Silicon Systems at Low Temperatures found to be proportional to . .Further, using now the total correction given by:

‚HR
, which is proportional to ./ , and also using our result (34) for G ( . ) , given for p=2 and p=3/2, we propose the expression for electron mobility, obtained for p=2 and p=3/2, as where (0.85 is the empirical parameter chosen to minimize the absolute deviations between the numerical results of μ 5ì and the corresponding μ-data, and the functions G y and Gš y are given in Table III.Then, the expression for electrical conductivity is given by where σ ( . ) = , being proportional to . .Further, the Hall coefficient is defined by where the Hall factor is given by . ( 70 6a and 6b.
Figures 6a and 6b indicate that those results of r & are positive, decrease with increasing N, increase with increasing r for a given N, and tend towards 1 at very high values of N, in good agreement with the result obtained in an effective electron gas [66].

Table 8.
In the As-Si system at T=10 K and for X = 2.7 1 10 9 h D , the numerical results of Hall mobility … " X * and Hall coefficient |_ " X * |, obtained in the PM and PVCMM, and their absolute relative errors, |_Vh|, calculated using the corresponding data obtained by Morin and Maita [50], … " ("E" = 155 ( Y/ †× ‡Y. and ×_ " ("E" × = 0.33 h D /ˆ , are calculated and tabulated.| are equal to 17% and 3% obtained for the PM, and 0.6% and 3% for PVCMM, respectively, confirming thus the use of N * for the effective density of free electrons given in the conduction band when N : N ( ) , given in Eq. ( 43).
In the P (As)-Si systems at T=4.2 K, N (5) = 3.52 × 10 B cm D and N ( €) = 8.58 × 10 B cm D , as given in Table 1, the numerical results of resistivity ρ(N * ) = 1/ σ(N * ), σ(N * ) being determined in Eq. ( 63) for the PVCMM and in Eq. ( 68) for the PM, are tabulated in Table 9, in which their absolute relative errors |REs|, calculated using the data obtained by Chapman et al. [54], are also included, suggesting that the maximal |REs| of ρ(N * ) are equal to 10% (11%), obtained respectively for the PM (PVCMM).Table 9.In the P (As)-Si systems at T=4.2 K, the numerical results of resistivity ‹(X * ), obtained for the PM and PVCMM and expressed in •10 l ]ℎh × h', are tabulated in this Table IX, in which their absolute relative errors |_Vh|, calculated using the data obtained by Chapman et al. [54], are also included, suggesting that the maximal |_Vh| of ‹(X * ) are equal to 10% (11%), obtained respectively for the PM (PVCMM).
In the P-Si system at T=77 K and for N (5) = 3.52 × 10 B cm D , the numerical results of conductivity x(N * ) , obtained respectively from Eqs. (63,68) for the PVCMM and PM, are tabulated in this Table 10, in which their absolute relative errors |REs|, calculated using the x-data obtained by Finetti and Mazzone [60], are also included.This indicates that the maximal |RE| of σ(N * ) are equal to 12% and 14% for PM and PVCMM, respectively.Table 10.In the P-Si system at T=77 K, the numerical results of conductivity x(X * ), obtained respectively for the PVCMM and PM, are tabulated in this Table X, in which their absolute relative errors |_Vh|, calculated using the x-data obtained by Finetti and Mazzone [60], are also included, indicating that its maximal |_Vh| are equal to 12% and 14% for PM and PVCMM, respectively.¾ (¿À The underlined |_V|-value is the maximal one.
As noted above, in the following, we will only present the numerical results of various electrical and thermoelectric functions obtained in the PM, since those obtained in the PVCMM can also be investigated by a same treatment.

Donor
) and obtained respectively from Eqs. (67,71), are tabulated here.This indicates that … " = … at X = 10 h D .Table 11 indicates that (i) at a given " , the resistivity decreases with increasing N, and (ii) at a given N, it increases with increasing " .That means: ρ(r sª ) < ρ(r 5 ) < ρ(r € ) < ⋯ < ρ(r sÕ ) < ρ(r s ) , in good agreement with the observations by Logan et al. [53].Table 12 suggests that (i) for a given r , the mobility and the Hall mobility decrease with increasing N, (ii) for given N, they decrease with increasing " , since … (or … & ) is proportional to σ ≡ 1/ρ, where ρ increases with increasing r , as observed in above Table XI, (iii) for given N and r , μ & : μ , and finally μ & = μ for N = 10 cm D , since the Hall factor r & is equal to 1, as that given in the effective electron gas [66].Now, in degenerate (d)-Si systems at 77 K, from the generalized Einstein relation [62][63][64][65][66][67], it is interesting to present in following Table 13 our numerical results of diffusion coefficients: D N * , T, r , D N * , T, r , and D N * , T, r , determined respectively in Eqs.(A15, A16, A17) of the Appendix C, being related to the mobility μ N * , T, r given in Eq. (67).Table 13 indicates that: (i) for a given r , D and D increase with increasing N, (ii) for a given N, since D, D and μ, being expressed in terms of x ≡ 1/ρ, where ρ increases with increasing r , as observed in above Table 11, our results of D and D thus decrease with increasing r , due to the donor-size effect, and finally (iii) for N = 10 cm D , all the results of D, D and D are found to be almost the same, suggesting that the asymptotic behaviors of D and D are correct.

Behaviors of Electrical Functions Obtained for À
First of all, it should be noted from Eqs. (26,68) for any the conductivity can be rewritten in a general form as where σ ( . ) = is proportional to ., and 〈E 〉 Šëì is determined in Eq. ( 26) for a = 5/2 and a= 2, respectively.
Here, as 0, using the functions: H , K and F , given respectively in Eqs.(26-28) for a=5/2 and 2, the conductivity, given in Eq. ( 72), is now rewritten by Figure 7.Our results of electrical conductivity x(ò S , " ( ) increase with increasing " ( for a given ò S , due to the donor-size effect, and present the maxima at ò S = ò S(1) and go to zero as ò S → 0 and ∞.
So, our numerical results of exponential tails of the electrical conductivity x(ò , r at 0 K and for N = 5 1 10 n cm D , calculated using Eq. ( 73), are plotted in Figure 7, as functions of ò .
Figure 7 shows that: (i) our results of x ò , r increase with increasing r for a given ò , due to the donor-size effect, and (ii) present the maxima at ò = ò (ì) and go to zero as ò → 0 and ∞, being found to be in good agreement with theoretical results obtained by Lifshitz [18], Friedberg and Luttinger [20], our results given in Eq. (A.3) of the Appendix A, and in particular with an asymptotic form for exponential conduction-band tail, obtained for 0 ≲ ò ≲ ∞, by Halperin and Lax [19], using the minimum counting methods.
Finally, our numerical results of energy parameter, (N; r ), obtained in the small interval: 1.37 ≤ ò ≤ 1.42, using Eq. ( 31), are plotted as functions of N in Figures 8a and  8b, indicating that, for a given N, increases with

Thermoelectric Properties
When the electron-electron and electron-phonon interactions are neglected, the Kubo formulae for the thermal transport coefficients [51], derived by very general arguments of Luttinger [55], are found to be reduced to the Greenwood ones [52].Then, the phenomenological equations are written as [58]  , are determined in Eq. ( 34) and given in Table III, calculated using the Fermi-Dirac distribution function (FDDF), and using also the expression for electrical conductivity as a function of , derived from Eq. ( 72) for ≥ 0, as the Onsager relations are found to be given as follows.
First, one has [58,61] L ( ) ≡ 〈x( , " )〉 . .= σ ( . ) ~〈 which is just the result obtained in Eq. ( 68).Then, one gets (i) First, our numerical calculation indicates that, in the degenerate (P)-Si system, for N = 10 cm D and at T=3 K and 300 K, noting that at 300 K the degenerate temperature T is equal to 7895 K ≫ 300 K, our results of K P are equal to 8 1 10 l and 0.125 W/(cm.K), in good agreement with the experimental results obtained by Slack [68]: 5 1 10 l and between 0.1 and 0.2 W/(cm.K), respectively.
(ii) Second, at N 10 cm D and T=3 K, the values of relative deviations between our results of K P (N * , T, r /•T 1 σ N * , T, r ', calculated using Eqs.( 68) and (80), and the constant: , being obtained from the Wiedemann-Frank law for metals [58,61], are tabulated in Table 15, indicating that our result (80) well verifies this law, with a precision of the order of 6.52 1 10 m .The underlined |_V|-value is the maximal one for each donor-Si system.
(iii) Finally, our numerical calculation shows that, in degenerate (d)-Si systems, for N = 10 cm D and in the temperature range from T=3 to 300 K, the maximal value of absolute deviations between K P N * , T, r given in Eq. ( 80) and its approximate form K P N * , T, r ≃ C Š " 1 T is found to be equal to 9.9 1 10 l , in good agreement with our previous result [58,61].Then, those are plotted in Figure 9a as functions of T, suggesting that at a given T the thermal conductivity K P decreases with increasing r , due to the donor-size effect.

Figure 9a.
Our results of ¤ \ X * , q, " ( ≃ ˆ f 1 q are plotted as functions of T, suggesting that at a given T the thermal conductivity ¤ \ decreases with increasing " ( , due to the donor-size effect.
In degenerate (d)-Si systems, for N = 10 cm D and in the temperature range from T=3 to 300 K, our numerical calculation indicates that: (i) the maximal value of absolute relative deviations between Q determined in Eq. ( 81) and its approximate form: C ¦ 1 T is found to be equal to 6.16 1 10 D , and (ii) the maximal value of absolute relative deviations between T s determined in Eq. ( 82) and its approximate form: C s 1 T is equal to 0.019.So, our numerical results of Q ≃ C ¦ 1 T and T s ≃ C s 1 T are plotted in Figures 9b and 9c, as functions of T, respectively, suggesting that at a given T, Q and T s both decrease with increasing r , due to the donor-size effect.

Figure 9b.
Our results of ¨≃ ˆ© 1 q are plotted as functions of T, suggesting that, at a given T, Q decreases with increasing " ( , due to the donor-size effect.

Figure 9c.
Our results of q ª ≃ −ˆª 1 q are plotted as functions of T, suggesting that at a given T, q ª decreases with increasing " ( , due to the donor-size effect.where R € ≡ ŽH ‚H is determined in Eq. ( 4), being in good accordance with our results (23,26).Furthermore, for a=1 and B = B Ÿ.Ÿ = π, the first-and-second terms of the last member of Eq. (A3) are found to be identical to the L-and-FL results, respectively.

Figure 1 .
Figure 1.Our results of 0 S increase with increasing " ( for a given ò S , due to the donor-size effect, and present the maxima at ò S = ò S(1) and go to zero as ò S → 0 and ∞.

Figure 3a .
Figure 3a.In the intrinsic P-Si system, our results of < ( )(de[ 1) (V) , obtained in absolute values, increase with increasing q.

Figure 4 .
Figure 4. Our results of 0 S( ¥ ) increase with increasing " ( for a given ò S , due to the donor-size effect, and present the maxima at ò S = ò S(1) and go to zero as ò S → 0 and ∞.

Figures 5 .
Figures 5. Our results of energy parameter, SU X; " ( , T = 5/2) are plotted as functions of N, indicating that, for a given N, SU increases with increasing " ( -values, due to the donor-size effect. ) Furthermore, the Hall mobility yields μ &(5ì N * , T, r = μ 5ì 1 r & 5ì .(71) Our numerical calculation indicates that in degenerate d-Si systems the r & -behaviors obtained in PVCMM and PM, using Eqs.(65, 70), are almost the same.So, in the PM, our numerical results of Hall factors r & obtained in various d-Si systems at 77 K, using Eq.(70), are plotted as functions of N in Figures

Figures 8 .
Figures 8. Our results of energy parameter, SU X; " ( , are plotted as functions of N, indicating that, for a given N, SU increases with increasing " ( -values, due to the donor-size effect. )where J Ì is the electric current density, J ÎÎÎÌ is the energy current, • Î ÎÌ is the electric field, and L (#) is the transport coefficient determined in an isotropic system.Now, using the average of 〈 〉 . .≡ G (y) 1 ., where the expressions for G (y), y =

Finally, in
the following Figures 9d, 9e and 9f, our numerical results of Peltier coefficient P P , Seebeck thermoelectric potential S O , and dimensionless figure of merit ZT, calculated using Eqs.(83-85), are plotted as functions of T, respectively.

Figure 9d .
Figure 9d.Our results of Peltier coefficient « \ are plotted as functions of T, suggesting that at a given T, « \ increases with increasing " ( , due to the donor-size effect.

Figure 9e .
Figure 9e.Our results of Seebeck thermoelectric potential ¬ [ are plotted as functions of T, suggesting that at a given T, ¬ [ decreases with increasing " ( , due to the donor-size effect.

Figure 9f .
Figure 9f.Our results of dimensionless figure of merit ZT are plotted as functions of T.

Table 13 .
In degenerate (d)-Si systems at 77 K, our numerical results of diffusion coefficients

Table 15 .
For X = 10 h D and T=3 K, the values of the relative deviations (RD) between our results of f \1z obtained in various degenerate donor-Si systems, and the constant: = 2.443 1 10 B ¢. £. ¤ , obtained from the Wiedemann-Frank law for metals, indicating a perfect agreement between those results.