Solar Cells at 300 K, Due to the Effects of Heavy Doping and Impurity Size. I

The effects of heavy doping and donor (acceptor) size on the hole (electron)-minority saturation current density JEo(JBo), injected respectively into the heavily (lightly) doped crystalline silicon (Si) emitter (base) region of n + p junction, which can be applied to determine the performance of solar cells, being strongly affected by the dark saturation current density: Jo≡JEo + JBo, were investigated. For that, we used an effective Gaussian donor-density profile to determine JEo, and an empirical method of two points to investigate the ideality factor n, short circuit current density Jsc, fill factor (FF), and photovoltaic conversion efficiency η, expressed as functions of the open circuit voltage Voc, giving rise to a satisfactory description of our obtained results, being compared also with other existing theoretical-and-experimental ones. So, in the completely transparent and heavily doped (P-Si) emitter region, CTHD(P-Si)ER, our obtained JEo-results were accurate within 1.78%. This accurate expression for JEo is thus imperative for continuing the performance improvement of solar cell systems. For example, in the physical conditions (PCs) of CTHD (P-Si) ER and of lightly doped (B-Si) base region, LD(B-Si)BR, we obtained the precisions of the order of 8.1% for Jsc, 7.1% for FF, and 5% for η, suggesting thus an accuracy of JBo (≤ 8.1%). Further, in the PCs of completely opaque and heavily doped (S-Si) emitter region, COHD(S-Si)ER, and of lightly doped (acceptor-Si) base region, LD(acceptor-Si)BR, our limiting η-results are equal to: 27.77%,..., 31.55%, according to the Egi-values equal to: 1.12eV ,..., 1.34eV, given in various (B,..., Tl)-Si base regions, respectively, being due to the acceptor-size effect. Furthermore, in the PCs of CTHD (donor-Si) ER and of LD(Tl-Si)BR, our maximal η-values are equal to: 24.28%,..., 31.51%, according to the Egi-values equal to: 1.11eV ,..., 1.70eV, given in various (Sb,..., S)-Si emitter regions, respectively, being due to the donorsize effect. It should be noted that these obtained highest η-values are found to be almost equal, as: 31.51%% 31.55%, coming from the fact that the two obtained limiting J -values are almost the same.


Introduction
The minority-carrier transport in the non-uniformly and heavily doped (NUHD), quasi-neutral, and uncompensated emitter region of impurity-silicon (Si) devices such as solar cells and bipolar transistors at temperature T 300 K , plays an important role in determining the behavior of many semiconductor devices . It should be noted that the minority-carrier saturation current density, J , injected into this emitter region strongly controls the common emitter current gain [4][5][6][7][8]. Thus, an accurate determination of this J or an understanding of minority-carrier physics inside heavily doped semiconductors is imperative for continuing the performance improvement of bipolar transistors, and that of solar cell systems, which is commonly characterized in terms of the parameters such as: the ideality factor n, short circuit current density J , fill factor FF, and photovoltaic conversion efficiency η, being Due to the Effects of Heavy Doping and Impurity Size. I expressed as functions of the open circuit voltage V [4]. Further, it should be noted that, in most fabricated silicon devices, the effective Gaussian donor-density profile ρ x , being proposed in next Equation (24), varies with carrier position x in the emitter region of width W [13,[18][19][20]22], and it decreases with increasing W, being found to be in good agreement with that used by Essa et al. [13]. As a result, many other physical quantities, given in this NUHD n(p)-type thin emitter region such as : the band gap narrowing (BGN), ΔE , Fermi energy E , apparent band gap narrowing (ABGN), ΔE , minority-hole (electron) mobility μ , minority-hole (electron) lifetime τ , and minority-hole (electron) diffusion length L , strongly depend on ρ x .
In the present paper, we determine an accurate expression for the minority-hole current density J , injected into the NUHD emitter region of n " − p junction silicon solar cells at 300 K, being also applied to determine the performance of such crystalline silicon solar cells.
In Section 2, we study the effects of impurity size [or compression (dilatation)], temperature and heavy doping, affecting the energy-band-structure parameters such as: the intrinsic band gap E % , intrinsic carrier concentration n % , band gap narrowing ΔE , Fermi energy E , apparent band gap narrowing ΔE , and effective intrinsic carrier concentration n % . In Section 3, an accurate expression for the optical band gap (OBG), E & , is investigated in next Equation (16), being accurate within 1.86%, as showed in Table 3. Some useful minority-carrier transport parameters such as: μ and L , being given in the heavily doped n-type emitter region, and μ , τ and the minority-electron saturation current density J ' , being given in the lightly doped p-type base region, are also presented in Section 4. Then, in Section 5, an accurate expression for the minorityhole saturation current density J , injected into the heavily doped emitter region of n " − p junction silicon solar cells at 300 K is established in Equation (39) or its approximate form given in Eq. (44), indicating an accuracy of the order of 1.78%, as seen in Table 4. Further, the total saturation current density: J = J + J ' , where J ' [1,7], determined in Equation (21), is the minority-electron saturation current density J ' , injected into the lightly doped base region of n " − p junction silicon solar cells, can be used to investigate the photovoltaic conversion effect, as presented in Section 6. Finally, some concluding remarks are given and discussed in Section 7.

Energy-Band-Structure Parameters in Donor (Acceptor)-Si Systems
Here, we study the effects of donor (acceptor) [d(a)]-size, temperature, and heavy doping on the energy-band-structure parameters of d(a)-Si systems, as follows.

Effect of d(a)-Size
In d(a)-Si-systems at T=0 K, since the d(a)-radius r *( ) , in tetrahedral covalent bonds is usually either larger or smaller than the Si atom-radius r , assuming that in the P(B)-Si system r +(') = r = 0.117 nm , with 1 nm = 10 ./ m , a local mechanical strain (or deformation potential energy) is induced, according to a compression (dilation) for r *( ) > r (r *( ) < r ) , respectively, due to the d(a)-size effect. Then, in the Appendix A of our recent paper [42], basing on an effective Bohr model, such a compression (dilatation) occurring in various d(a)-Si systems was investigated, suggesting that the effective dielectric constant, ε(r *( ) ), decreases with increasing r *( ) . This r *( ) -effect thus affects the changes in all the energyband-structure parameters, expressed in terms of ε(r *( ) ) , noting that in the P(B)-Si system ε(r +(') ), = 11.4. In particular, the changes in the unperturbed intrinsic band gap, E 5r +(') 6 = 1.17 eV, and effective d(a)-ionization energy in absolute values E * ( ) 5r +(') 6 = 33.58 meV, are obtained in an effective Bohr model, as [42]: E 5r *( ) 6 − E 5r +(') 6 = E * ( ) 5r *( ) 6 − E * ( ) 5r +(') 6 = E * ( ) 5r +(') 6 × :; Therefore, with increasing r *( ) , the effective dielectric constant ε(r *( ) ) decreases, implying that E 5r *( ) 6 increase. Those changes, which were investigated in our previous paper [42], are now reported in the following Table  1, in which the data of the critical d(a)-density N F( G) (r *( ) ) are also reported. This critical density marks the metal-toinsulator transition from the localized side (all the impurities are electrical neutral), N(N ) ≤ N F( G) 5r *( ) 6 , to the extended side, N(N ) ≥ N F( G) (r *( ) ), assuming that all the impurities are ionized even at 0 K. However, at T = 300 K, for example, all the impurities are thus ionized and the physical conditions, defined by: N(N ) > J F( G) (r *( ) ) and N(N ) < J F( G) (r *( ) ), can thus be used to define the n(p)type heavily and lightly doped Si, respectively. Table 1. The values of K L(M) , N(K L(M) ), and O PQ (K L(M) ), and critical impurity density J RS(RT) (K L(M) ), obtained in our previous paper [42], are reported here.

Temperature Effect
Being inspired from excellent works by Pässler [33,34], who used semi-empirical descriptions of T-dependences of band gap of the Si by taking into account the cumulative effect of electron-phonon interaction and thermal lattice expansion mechanisms or all the contributions of individual lattice oscillations [33][34][35], we proposed in our recent paper [43] a simple accurate expression for the intrinsic band gap in the silicon (Si), due to the T-dependent carrier-lattice interaction-effect, E % 5T, r *( ) 6, by where the values of E (r *( ) ) due to the d(a)-size effect are given in Table 1 and those of E % 5T = 300 K, r *( ) 6 tabulated in Table 2. Further, as noted in this Reference 43, in the (P, S)-Si systems, for 0 K ≤ T ≤ 3500 K, the absolute maximal relative errors of this E % -result were found to be equal respectively to: 0.22% and 0.15%, calculated using the very accurate complicated results given by Pässler [34]. Then, in the n-type HD silicon at temperature T, the effective mass of the majority electron can be defined by [31,32]: which gives: m = m (T = 0 K) = 0.3216 × m , m being the electron rest mass, and the effective mass of the minority hole yields [31,32]: which gives m k (T = 0 K) = m k = 0.3664 × m . Here, g k = 2 is the effective average number of equivalent valence-band edges. Now, the intrinsic carrier concentration n % is defined by n % C (T, r *( ) , g ) ≡ N (T, r * , g ) × N k (T, g k ) × exp ; . fh 5Z,= @(A) 6 x ? Z B (5) where, N (k) is the conduction (valence)-band density of states, given by [31,32]: where ℏ = h/2π is the Dirac's constant, k ' is the Boltzmann constant, and g is the effective average number of equivalent conduction-band edges. Moreover, for r * ≡ r + and at 300 K, some typical n % -results obtained for different g -values, calculated using Equation (5), are given as follows.
(i) If g = 6 , one then gets: n % = 10.7 × 10 / cm .W , being just a result investigated from a measurement of energy-band-structure parameters and intrinsic conductivity by Green [31].
(ii) If g = 5, one then obtains: n % = 9.77 × 10 / cm .W , according to a result given from a capacitance measurement of a pin diode biased under high injection, by Misiakos and Tsamakis [37]. basing on their updated fit of experimental data for the minority-carrier mobility and open-circuit voltage decay, which were given by Sproul and Green [36]. Further, from Equations (5, 2), in donor-Si systems and for T=300 K, the numerical results of n % and E % , calculated for g = 6, 5, and 4.9113, as functions of K L(M) , are tabulated in Table 2. From those results, one remarks that, for T=300 K, n % decreases with increasing r * since E % 5T, r * 6 increases, being due to the d(a)-size effect.

Heavy Doping Effect
First of all, in the donor-Si system, we define the effective intrinsic carrier concentration n % , by n % C w N 9 p w n % C 9 exp Y ' fA where n % C is determined in Equation (5). Here, we can also define the "effective doping density" by [8]: N ' "". w N/exp Y ' fA x ? Z _ so that N ' "". 9 p w n % C . Here, p is the density of minority holes at the thermal equilibrium and the ABGN is defined by: where N ƒ is defined in Equation (6) [18], and Yan and Cuevas (YC) [19], proposed their empirical expressions for the ABGN, being obtained in the P-Si system at 300 K, by: Then, in such the P-Si system at 300K, being inspired by the term: k ' T 9 ln † • • - ‡ given in Equation (9), and also by the result: ΔE ¡ƒ N given in Equation (14), we can now propose a modified (Mod.) YC-model for the ABGN so that its numerical results are found to be closed to those calculated by using Equation (9), as: having a same empirical form as that given in Equation (14).  Now, for g 6 , in d-Si systems at 300 K, our numerical ABGN ( ΔE )-results are calculated, using Equation (9). First, ours, obtained for the P-Si system, are plotted as a function of N in Figure 1 (a), in which, for a comparison, the other ones, calculated using Equations (10)(11)(12)(13)(14)(15), are also included. Secondly, in this P-Si system, the relative deviations between ours and the others are also plotted as functions of N in Figure 1 (b). Finally, in Figure 2 (c & , c C , ours are plotted in donor-Si systems as functions of N. Here, one observes that: (i) our numerical ABGN-results obtained using Equations (9,15) are found to be closed together as seen in Figure  1 (a), and their absolute maximal relative deviation yields: 3.03%, which occurs at N 1.2 9 10 C\ cm .W , as observed in Figure 1 (b), and (ii) in Figure 2 (c & , c C , for a given donor-Si system, due to the heavy doping effect, ours increase with increasing N, and for a given N, ours increase ( §) with increasing r * , due to the donor-size effect.
Then, in the following, it is possible to define the optical band gap (OBG), expressed in terms of the ABGN (or BGN), suggesting a conjunction between the electrical-and-optical phenomena.

Conjunction Between Electrical-and-Optical Phenomena
First of all, we define the optical band gap (OBG) by [25]: E & N, T, r * , g w E % T, r * # ΔE N, T, r * , g ( E N, T, r * , g where the intrinsic band gap E % is determined in Equation (2), the BGN ΔE is investigated in Equation (A9) of the Appendix B, and the Fermi energy E is given in Equation (A3) of the Appendix A, suggesting that the optical phenomenon is represented by E & . Furthermore, it is possible to establish a conjunction between the electrical and optical phenomena, obtained from Equations (9, 16), as: E & N, T, r * , g w E % T, r * # ΔE N, T, r * , g ( k ' T 9 ln ; N N ƒ T, r * , g B which can be rewritten, for example, replacing ΔE by ΔE ¢ *.¡ƒ N determined in Equation (15), as: Now, in the P-Si system, our numerical OBG-results, calculated using Equations (16,17)   Here, our best choice is g = 6 , meaning that at T ≥ 300 K, due to the high thermal agitation energy k ' T, all the six equivalent conduction-band edges are effective.

Minority-Carrier Transport Parameters
Here, in the heavily doped n-type emitter region and the lightly doped p-type base region of n " − p junction silicon solar cells, the minority-hole (electron) transport parameters are studied as follows.

Heavily Doped n-type Emitter-region Parameters
In order to determine the minority-hole saturation-current density J , injected into the heavily doped n-type emitterregion, we need to know an expression for the minority-hole mobility μ , being related to the minority-hole diffusion coefficient D , by the well-known Einstein relation: D = x ? Z × μ , where e is the positive hole charge. Here, in donor-Si systems at 300 K and for any g , since the minority-hole mobility depends on N [10], and also on g and ε(r * ) [11], we can propose: noting that as T = 300 K, g = 6, and r * ≡ r + , Equation (18) is reduced to that given by del Alamo et al. [10]. Moreover, Equation (18) indicates that, for a given N and with increasing r * , μ decreases, since ε(r * ) decreases as seen in Table 1, being due to the d-size effect, in good accordance with that observed by Logan et al. [9]. Further, from Equations (5,8,9,15,18), we can define the following minority-hole transport parameter F as [22,25]: where N ' "". is the "effective doping density" [8] and the ABGN is determined in Equation (9) for our ΔE -result or in Equation (15) for our approximate ΔE (¢ *.¡ƒ) -one.

Lightly Doped p-type Base-Region Parameters
Here, the minority-electron saturation current density injected into the lightly doped p-type base region, with an acceptor density equal to N , is given by [1,7]: where n % C (T, r *( ) , g = 6) is determined in Equation (5) and Here, in the acceptor-Si system, μ is the minorityelectron mobility, being determined by [3,11,16]: being reduced to the result obtained by Slotbottom and de Graaff [3,16], as T=300 K and r = r ' , and τ (N ) is the minority-electron lifetime, computed by [16,25]: Furthermore, Equation (22) indicates that, for a given N and with increasing r , μ decreases, since ε(r ) decreases, as seen in Table 1, in good accordance with that observed by Logan et al. [9].
Then, in P(B)-Si systems at 300 K and for g = 6 , Klaassen et al. confirmed, in Figures 1 and 2 of their paper [16], that the expressions (18,22) for minority-hole (electron) mobility μ ( ) are simple and accurate.
In the following, we will determine the minority-hole saturation-current density J , injected into the heavily doped n-type emitter-region of the n " − p junction solar cells.

Minority-Hole Saturation Current Density
Let us first propose in the non-uniformly and heavily doped (NUHD) emitter region of donor-Si devices our expression for the effective Gaussian donor-density profile or the donor (majority-electron) density, defined in the emitterregion width W, by: Ä (cm .W ) , 1 μm = 10 .[ cm , decreases with increasing W, in good agreement with the doping profile measurement on silicon devices, studied by Essa et al. [13]. Moreover, Equation (24) indicates that: (i) at the surface emitter: x=0, ρ(0) = N , defining the surface donor density, and (ii) at the emitter-base junction: x=W, ρ(W) = N (W) , which decreases with increasing W, as noted above. Here, we also remark that N (Éƒ') = 7 × 10 &… cm .W was proposed by Van Cong and Debiais (VCD) [22], and N (Ÿ ) = 2 × 10 &] cm .W , by Zouari and Arab (ZA) [17], for their Gaussian impurity density profile. Moreover, all the parameters given in Equation (24) were chosen such that the errors of our obtained J -values are minimized, as seen in next Table 4, and our numerical calculation indicates that, from Equation (24), we can determine the highest value of W, being equal here to 85 μm. Now, from Equations (8,9) or Equation (19), taken for 0 ≤ x ≤ W, and using Equation (24) Then, under low-level injection, in the absence of external generation, and for the steady-state case, we can define the minority-hole density by: and a normalized excess minority-hole density [or a relative deviation between p(x) and p (x)] by [22,25]: G g (·) (27) which must verify the two following boundary conditions proposed by Shockley as [2]: Here, n(V) is an ideality factor, S ( È ) is the hole surface recombination velocity at the emitter contact, V is the applied voltage, V Z ≡ (k ' T/e) is the thermal voltage, and the minority-hole current density J (x), being found to be similar to the Fick's law for diffusion equation, is given by [8,22]: where F(x) is determined in Equation (19), in which N is replaced by ρ(x), proposed in Equation (24).
Here, various results can be investigated as follows.
being found to be independent of S and C, since Ò ³,µ ¶ ¶.
Ò ³ is independent of S and C as observed in Equations (20,36), and noting that the ABGN-expression is determined by Equation (9) or by Equation (15). Now, in the P-Si system, for T = 300 K, r * ≡ r + and g = 6, 5, 4.9113, our two numerical J -results are calculated, using Equations (44,9) and (44,15), and given in Table 4, in which the CTER -condition, P ≪ 1 (or Ó à,µ ¶ ¶. Ó ³ (•) ≪ 1) , is fulfilled, and we also compare them with modeling and measuring J -results investigated by del Alamo et al. (ASS) [10,12]. One notes that their modeling J -result [10], based only on two independent parameters: N ' "" /D and L , can be obtained, for L , "". = W, from our above result (44). This could explain a great difference between their modeling results [10,12], being accurate within 36%, and ours, accurate within 1.78%, for g = 6, as those observed in the following Table 4.  [10,12], and also their relative deviations. The underlined |¬-|-values are the maximal ones. Table 4 indicates that: (i) the maximal relative deviations (RDs) in absolute values between our results (44,9) and the J -data [10,12] are found to be: 1.78% for g =6, 4.56% for g =5, and 5.07% for g =4.9113, and (ii) the maximal RDs in absolute values between our results (44,15) and the J -data [10,12] are given by: 2.42% for g =6, 5.49% for g =5, and 5.75% for g =4.9113. It suggests that our numerical results (44,9) for g =6 are the best ones, since they are accurate within 1.78%. Further, one notes that our ΔEexpression given in Equation (9) was obtained, taking into account all the physical effects such as: those of donor size, heavy doping and Fermi-Dirac statistics, while in Equation (15) our ΔE (¢ *.¡ƒ) -expression is only an empirical one. So, in the following, we will choose: g =6, T=300 K, and our ABGN-expression (9), for all the numerical calculations.    Some concluding remarks are obtained and discussed below.
(i) Figures 3(a & , a C ) Figures 3(b) and 4(b) show that, for a given N, J (or Ó à,µ ¶ ¶. Finally, it should be noted that in next Section 6 we must know the numerical results of dark saturation current density, defined by: J x W, N, r * , S, N , r w J x W, N, r * , S ( J ' N , r where J ' and J are determined respectively in Equations (21,39). Then, those are tabulated in the following Table 5, in which all the physical conditions are also presented. Table 5. Our numerical results of ù Q ù úQ ( ù Q , calculated using Equation (47), where ù Q •Š• ù Q are determined respectively in Equations (21,39), and those are obtained in the three following cases. It should be noted that these values of J will strongly affect the variations of various photovoltaic conversion parameters of n " − p junction silicon solar cells, such as: the ideality factor n, short circuit current density J , fill factor FF, and photovoltaic conversion efficiency η, being expressed as functions of the open circuit voltage, V [4], as investigated in the following. Our empirical treatment method used is that of two points. The first point is characterized by [27]: In the following, we will develop our empirical treatment method of two points, used to determine J and FF, basing on accurate results given in Equations (48) and (49).

Photovoltaic Conversion Effect
The well-known net current density J at T=300 K, expressed as a function of the applied voltage V, flowing through the n " − p junction of silicon solar cells, is defined by: For example, from the above remark given in Eq. (53) and from the first case reported in Table V, we can conclude that, with decreasing S and increasing W, both n and J decrease from the CTER to the COER. Therefore, from Equation (51), J thus increases from the CTER to the COER, since J is expressed in terms of þ k≡ gz × v . Then, the values of the fill factor FF for V = V &(C) can be found to be given by:  [27,23].
Moreover, in the case where both series resistance and shunt resistance have a negligible effect upon cell performance, z `(a),¹{µµ = 1, as proposed by Green [4]. Now, by applying a same above treatment method of two points, one has: , respectively [27,23].
Then, the photovoltaic conversion efficiency η can be defined by: where J and FF are determined respectively in Equations (51, 56), being assumed to be obtained at 1 sun illumination or at AM1.5G spectrum (P %F. = 0.100 Ã È a ) [27,28].
In summary, all above parameters such as: n, J , FF and η, defined in above, strongly depend on J , determined in Equation (47), which is thus a central result of the present paper.
Now, for given physical conditions such as: W, N, r * , S, N and r , and by taking into account all remarks given in Table 5 and also in above Equation (53), our numerical results of n, J , FF and η, expressed as functions V , are respectively computed by using Equations (52, 51, 56, 57), and reported in following Table 6 and Figures 7, 8  and 9.
In Figures 5 (a), (b) which are given also in these figures, and in Table 5 for the first case. Here, for a given V , and with decreasing S and increasing W, we observe that: (i) in the Figure 5 (a), the function n determined in Equation (52) (or the function J given in Table 5) decreases from the CTER to the COER (ii) in Figures 5 (b), 5 (c) and 5(d), the functions J , FF and η therefore increase from the CTER to the COER, and (iii) in Figure 5 according to the CTER, and they are also given in Table 5 for the third case. Here, the values of E % r * at 300 K are given in Table 2. Then, the numerical results of n, J , FF and η are calculated, using Equations (52, 51, 56, 57). Further, for a given V and with increasing r -values, it should be concluded that, due to the donor-size effect, (i) in the Figure 7 (a), the function n determined in Equation (52) (or the function J given in Table 5) decreases ( ), and (ii) in Figures 7 (b), (c), (d), the functions J , FF and η therefore increase ( §), and in particular, in Figure 7 (d), in the conditions of completely transparent and heavily doped (donor-Si) emitter-and-lightly doped (Tl-Si) base regions, the maximal η-values are equal to: Due to the Effects of Heavy Doping and Impurity Size. I 24.28 %,…, 31.51 %, at V =748 mV,…,703 mV, according to the E % -values equal to: 1.11 eV,…, 1.70 eV, obtained in various (Sb,…, S)-Si emitter regions, respectively, being due to the donor-size effect, which can be compared with those given in Figure 6 (d).

Concluding Remarks
We have developed the effects of heavy doping and impurity size on various parameters at 300 K, characteristic of energy-band structure, as given in Sections 2 and 3, and of the performance of crystalline silicon solar cells, being strongly affected by the dark saturation current density: J w J ( J ' , as given in Sections 4, 5 and 6. Then, some concluding remarks are obtained and discussed as follows. 1 Using the OPG (E & )-data given by Wagner and del Alamo [44], our E & -results, due to the heavy doping effect, and calculated using Equation (16), are found to be accurate within 1.86%, as observed in Table 3. 2 In the CTER-conditions, as those given in Table 4, and using the J -data, given by del Alamo et al. [10,12], by using Equation (44), our J -results, obtained in the heavily doped and completely transparent (P-Si) emitter region, are found to be accurate within 1.78%, while the modeled J -results, obtained by those authors, are accurate within 36% [10,12]. 3 For given physical conditions and using an empirical treatment method of two points, as developed and discussed in Section 6, both our two results (n and J ) have the same variations, which strongly affect other (V , J , FF, η)-results, as discussed in Eq. (53). Thus, J , determined in Equation (47), is a central result of our present paper. 4 In the CTER-conditions, as those given in Equation (58), and using various (J , FF, η)-data [23,24,[27][28][29], we get the precisions of the order of 8.1% for J , 7.1% for FF and 5% for η, suggesting thus a probable accuracy of J ' H 8.1% , since our J -results are accurate within 1.78%. 5 In the physical conditions of completely opaque and heavily doped (S-Si) emitter-and-lightly doped (acceptor-Si) base regions, as given in Eq. (60), and in the physical conditions of completely transparent and heavily doped (donor-Si) emitter-and-lightly doped (Tl-Si) base regions, as given in Eq. (61), our obtained maximal η-values, due to the impurity-size effect, are found to be equal respectively to: 27.77%, …, 31.55%, as seen in Figure 6 (d), and 24.28%, …, 31.51%, as observed in Figure 7 (d), suggesting that our obtained highest η -values are found to be almost equal, as: 31.51% ≃ 31.55% , since the two corresponding limiting J -values are almost the same, as given in Table 5, for second and third cases. In summary, being due to the impurity-size effects, our limiting value of η & =31.55%, as that given in Figure 6