New Critical Density in Metal-Insulator Transition, obtained in n(p)- Type Degenerate InP 1-x As x (Sb x ), GaAs 1-x Te x (Sb x ,P x ), CdS 1-x Te x (Se x )- Crystalline Alloys. (II)

: By basing on the same physical model and treatment method, as used in our recent works [1, 2]

In the following, we will determine those functions: N CDn(CDp) (r d(a) , x) and N CDn(CDp) EBT � r d(a) , x�.

CRITICAL DENSITY IN THE MOTT MIT
Such the critical impurity density N CDn(CDp) (r d(a) , x), expressed as a function of r d(a) and x, is determined as follows.

Effect of x-concentration
Here, the values of the intrinsic energy-band-structure parameters [1,2], such as: the effective average number of equivalent conduction (valence)-band edges g c(v) (x), the unperturbed relative effective electron (hole) mass in conduction (valence) bands m c(v) (x)/m o , m o being the electron rest mass, the reduced effective mass m r (x)/m o , the unperturbed relative dielectric static constant ε o (x), the effective donor (acceptor)-ionization energy E do(ao) (x), and the isothermal bulk modulus B do(ao) (x) ≡ E do(ao) (x) (4π/3)×�r do(ao) � 3 , at r d(a) = r do(ao) , are given respectively in Table 1 in Appendix 1.

Effects of impurity size, with a given x
Here, one shows that the effects of the size of donor (acceptor) d(a)-radius, r d(a) , and the xconcentration strongly affects the changes in all the energy-band-structure parameters, which can be represented by the effective relative static dielectric constant ε(r d(a) , x) [1,2,13], in the following.
At r d(a) = r do(ao) , the needed boundary conditions are found to be, for the impurity-atom volume V= (4π/3) × �r d(a) � 3 , V do(ao) = (4π/3) × �r do(ao) � 3 , for the pressure p, as: p o = 0, and for the deformation potential energy (or the strain energy) σ, as: σ o = 0. Further, the two important equations, used to determine the σ-variation: ∆σ≡ σ−σ o = σ, are defined by: and p=− dσ dV . giving: . Then, by an integration, one gets: , , for r d(a) ≥ r do(ao) , and for r d(a) ≤ r do(ao) , Therefore, from above Equations ( 3) and ( 4), one obtains the expressions for relative dielectric constant ε(r d(a) , x) and energy band gap E gn(gp) �r d(a) , x�, as: where −q is the electron charge.
Then, the critical donor (acceptor)-density in the Mott MIT, N CDn(NDp) (r d(a) , x), is determined, using an empirical Mott parameter, M n(p) = 0.25, for each the conduction (valence) band, as: noting that M n(p) could be chosen in general case so that the obtained numerical N CDn(NDp) (r d(a) , x)results, being found to be in good agreement with the corresponding experimental ones.
In the following, these obtained numerical results can also be justified by calculating the numerical results of the density of electrons (holes

Physical model
, N being the total impurity density, assuming that all the impurities are ionized even at 0 K, the effective reduced Wigner-Seitz radius r sn(sp) , characteristic of interactions, is defined by: So, the ratio of the inverse effective screening length k sn(sp) to Fermi wave number k Fn(kp) is defined by: These ratios, R snTF(spTF) and R snWS(spWS) , are determined in the following.
being proportional to N −1/6 .Secondly, N < N CDn(NDp) (r d(a) ), according to the Wigner-Seitz (WS)-approximation, the ratio R snWS(spWS) is reduced to [3][4][5][6][7]: .So, n(p)-type degenerate X(x)-crystalline alloys, the physical conditions are found to be given by : Here, ±E Fno(Fpo) is the Fermi energy at 0 K, and η n(p) is defined in next Eq.( 15), Then, the total screened Coulomb impurity potential energy due to the attractive interaction between an electron (hole) charge, −q(+q), at position r ⃗, and an ionized donor (ionized acceptor) charge: +q(−q) at position R ȷ ���⃗ , randomly distributed throughout X(x)-crystalline alloys, is defined by: where ℕ is the total number of ionized donors (acceptors), V o is a constant potential energy, and the screened Coulomb potential energy v j (r) is defined as: where k sn(sp) is the inverse screening length determined in Eq. (11).
Further, using a Fourier transform, the v j -representation in wave vector k �⃗ -espace is given by where Ω is the total X(x)-crystalline alloy volume.
European Journal of Applied Science, Engineering and Technology www.ejaset.com191 Then, the effective auto-correlation function for potential fluctuations, W n(p) �ν n(p) , N, r d(a) � ≡ 〈V(r)V(r ′ )〉, was determined, [8,9] as : Here, E is the total electron energy, and the empirical Heisenberg parameter ℋ n(p) = 0.47137 was chosen above such that the determination of the density of electrons localized in the conduction(valence)-band tails will be accurate, noting that as E → ±∞ , �ν n(p) � → ∞, and therefore, In the following, we will calculate the ensemble average of the function: being the kinetic energy of the electron (hole), and V(r) determined in Eq. ( 16), by using the two following integration methods, which strongly depend on W n(p) �ν n(p) , N, r d(a) , x�.

Kane integration method (KIM)
Here, the effective Gaussian distribution probability is defined by: So, in the Kane integration method, the Gaussian average of 2 is defined by Then, by variable changes: �, and using an identity: ) is the Gamma function, one thus has: Feynman path-integral method (FPIM) Here, the ensemble average of 2 is defined by noting that as a=1, (it) 2 � is found to be proportional to the averaged Feynman propagator given the dense donors (acceptors).Then, by variable changes: �, for n(p)-type respectively, and then using an identity: 2 〉 KIM being determined in Eq. ( 16).
In the following, with the use of asymptotic forms for KIM can be obtained in the two following cases.

CONCLUSION
In those Tables 2-8, some concluding remarks are given and discussed in the following.(3)-In those Tables 2-8, one notes that the maximal value of |RD| is found to be given by: 2.92 × 10 (4) Finally, once N CDn(CDp) is determined, the effective density of free electrons (holes), N * , given in the parabolic conduction (valence) band of the n(p)-type degenerate X(x)-crystalline alloy, can thus be defined, as the compensated ones, by: EBT , needing to determine the optical, electrical, and thermoelectric properties in such n(p)-type degenerate X(x)-crystalline alloys, as those studied in n(p)-type degenerate crystals [3][4][5][6][7][8][9].

N
), respectively, noting that the relative deviations in absolute values are defined by: |RD| ≡ �1 −
−7 , meaning that N CDn EBT ≅ N CDn .In other words, such the critical d(a)-density N CDn(NDp) (r d(a)) , x), is just the density of electrons (holes), being localized in the EBT, N CDn(CDp) EBT ( r d(a) , x), respectively.