Bandgap changes in excited intrinsic (heavily doped) Si and Ge semiconductors

ARTICLE IN PRESS Physica B 405 (2010) 1139–1149

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Bandgap changes in excited intrinsic (heavily doped) Si and Ge semiconductors H. Van Cong  Universite´ de Perpignan Via Domitia, LAMPS (EA 4217), De´partement de Physique, 52, Avenue Paul Alduy, 66 860 Perpignan, France

a r t i c l e in fo

abstract

Article history: Received 14 May 2009 Received in revised form 26 October 2009 Accepted 3 November 2009

Our results for the bandgap changes in highly excited intrinsic (heavily doped-HD) Si (Ge) for any majority-carrier density N and temperature T have been investigated and expressed in terms of (i) the bandgap narrowing (BGN 4 0) due to many-body carrier–carrier interactions and screening effect on carrier-impurity (or electron–hole) potential energies, (ii) the bandgap widening (BGW o 0) due to the effects of Fermi Dirac statistics, and (iii) the apparent BGN defined by ABGN  BGN + BGWo BGN. Since those ABGN and BGN can be extracted from respective electical-and-optical measurements, this relation thus suggests a conjunction between electrical-and-optical bandgaps (EBG-and-OBG). Then, our results for BGN, ABGN, OBG, and EBG have been computed and also compared with other theoretical and experimental ones, giving rise to a satisfactory description of both electrical-and-optical data in those materials. Furthermore, in the p-type HD base of Si1  xGex hetero bipolar transistors for xr 0.3, using a same assumption taken by Eberhardt and Kasper (EK) [Mater. Sci. Eng. B 89 (2002) 93–96], we have obtained the results for ABGN and EBG, which are found to be in good accordance with the respective EK-ones. & 2009 Elsevier B.V. All rights reserved.

Keywords: Si Ge SiGe Intrinsic bandgap Fermi energy Bandgap narrowing Optical bandgap Electrical bandgap

1. Introduction The bandgap changes in excited intrinsic (EI)-and-heavily doped (HD) Si and Ge semiconductors at temperatures T, are given in the following relation, in which the apparent bandgap narrowing (ABGN) [1,2], DEg;A 4 0, may be expressed in terms of (i) the intrinsic carrier concentration, ni, and the effective intrinsic carrier concentration, ni,e, and (ii) the bandgap narrowing (BGN) due to many-body interactions, DEg 4 0, and the bandgap widening (BGW) due to the effects of Fermi–Dirac statistics, DEg;FD o0, as

DEg;A ¼ 2kB T  Lnðni;e =ni Þ ¼ DEg þ DEg;FD ;

ð1Þ

where kB is Boltzmann’s constant, and the BGW is defined by

DEg;FD  kB T  LnðN=NcðvÞ ÞEF :

ð2Þ

It should be noted that relation (1) gives rise to a conjunction between BGN and ABGN or optical-and-electrical phenomena occurring in those materials [2]. In Eq. (2), EF is the penetration of the Fermi energy level into the majority band, N is the density of electron–hole (e–h) pairs  Tel.: + 33 4 68 66 22 36; fax: + 33 4 68 66 22 34.

E-mail address: [email protected] 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.11.016

defined in EI Si(Ge) or the total impurity density in HD Si(Ge) assuming that all the impurities are ionized even at 0 K, and Nc(v) means the conduction (valence)-band density of states (DOS) defined by [3] 2

NcðvÞ  ½gs gcðvÞ ðmdcðvÞ kB T=2p‘ Þ3=2 ¼ 2:540933 1019 ½mdcðvÞ T=3003=2 ðcm3 Þ;

ð3Þ

where ‘  h=2p, h being Planck’s constant, mdcðvÞ is the DOS effective mass found in each conduction (valence) band to free electron mass mo, gs =2 is the spin degeneracy factor defined with spin up and spin down, gcðvÞ;Si ¼ 6ð2Þ [4] and gcðvÞ;Ge ¼ 4ð1Þ[4–6] are the average numbers of equivalent conduction (valence)-band edges 2=3 in the n(p)-type Si and Ge, respectively, and finally mdcðvÞ  gcðvÞ mdcðvÞ means the DOS effective mass in such materials. Thus, in this Eq. (3), Nc(v) is expressed as functions of mdcðvÞ ; gcðvÞ and T. Moreover, in Eq. (1), the intrinsic carrier concentration is defined by ni 

pffiffiffiffiffiffiffiffiffiffiffi Nc Nv expðEg;I =2kB TÞ;

ð4Þ

where EgI(T) is the intrinsic bandgap. It should be noted in Eqs. (1) and (2) that in the EI Si (Ge) one has DEg ¼ DEgn þ DEgp and EF = EFn +EFn and in the n(p)-type HD

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Si(Ge), DEg and EF are simply replaced by DEgnðpÞ and EFnðpÞ , respectively. In those semiconductors, the photoluminescence (PL) spectra consist of a number of different lines arising from electrons and holes recombining from states such as free excitons, excitons bound to impurities, and electron–hole condensate [7–10]. Here, we are primarily concerned with free-exciton and condensate emission. Since Si(Ge) are indirect bandgap semiconductors, radiative recombination may be accompanied by phonon emission; the PL spectra thus consist in general of phonon replicas involving the momentum-conserving transverse-acoustic(optical) TA(O)phonon with respective energy Ep;TAðOÞ or longitudinalacoustic(optical) LA(O) with respective energy Ep;LAðOÞ . The effective bandgaps of given phonon-mode replicas are defined as follows. First of all, the low-energy threshold or reduced bandgap, Eg0  Eg Ep , is defined by Eg0  EgI DEg ;

ð5Þ

which is the difference between the bottom (top) of perturbed conduction (valence)-band edges. Then, the high-energy cutoff, equivalent chemical potential or optical bandgap (OBG) energy, 0 ¼ Eg;O Ep ¼ Eg0 þ EF , is determined by Eg;O 0 Eg;O

¼ Egl DEg ¼ EF ¼ EgI þ m;

ð6Þ

where m  DEg þ EF is the chemical potential defined in the EI Si (Ge) and in the n(p)-type HD Si (Ge) m is simply replaced by mnðpÞ  DEgnðpÞ þ EFnðpÞ . 0  Egx Ep , is Further, the free exciton low-energy threshold, Egx defined by 0 Egx  EgI Ex Ep ;

ð7Þ

where Ex is the exciton Rydberg. Here, in the EI Si (Ge) at very low temperatures one, respectively, has Ex ¼ 12:87ð2:65Þ meV, Egl ðT ¼ 0Þ ¼ 1170ð743:7Þ meV and data of free exciton low-enery 0 ¼ 1096:9ð712:9Þ meV [10]. Therefore, the values of threshold Egx phonon energy are found to be EpðTOÞ;Si ¼ 60:23 ðmeVÞ and EpðLAÞ;Ge ¼ 26:548 ðmeVÞ. Then, from Eqs. (1), (2), and (6), if denoting the electrical bandgap (EBG) in the n(p)-type HD Si (Ge) by E0gnðpÞ;E  EgnðpÞ;E Ep ¼ EgI DEgnðpÞ;A , where DEgnðpÞ;A is defined in Eq. (1), the above conjunction (1) can now be rewritten by the one between the OBG and EBG as mdv ðTÞ ¼

carrier concentration ni, the Fermi energy EF, and the BGN: DEg , from which other ones can also be determined and computed. The aim of the present paper is to investigate the accurate results for these parameters, and also compare those with existing data and other theoretical (or empirical) results, giving rise to a satisfactory description of both optical-and-electrical data observed in EI-and-HD Si (Ge) at any N and T.

2. Intrinsic carrier concentration As seen in Eqs. (3)–(7), the intrinsic carrier concentration EgI ðTÞ depends not only on the intrinsic bandgap EgI ðTÞ but also on the 2=3 DOS effective masses mdcðvÞ  gcðvÞ mdcðvÞ defined in Eq. (3). In order to evaluate ni we thus need to know those values of EgI ðTÞ and mdcðvÞ , which are chosen as follows. First of all, we use here accurate approximate forms for intrinsic bandgap EgI ðTÞ in meV at any T, which were recently ¨ investigated by Passler [11], taking into account not only the change in the crystal volume with T resulting from the lattice expansion but also the electron–phonon interaction at constant volume, for the Si as EgI;Si ðTÞ ¼ 117072½ð1 þ ½2T=4462:2011 Þ1=2:20111 and for the Ge by EgI;Ge ðTÞ ¼ 743:752½ð1 þ ½2T=2532:2726 Þ1=2:2726 1:

Then, is expressed here in terms of EgI ðTÞ, and longitudinal and transverse effective masses to free electron mass mo, associated with the ellipsoidal constant energy surfaces in the Si as [3,9] "   #1=3 EgI ðT ¼ 0Þ 2 2=3  mdc ðTÞ ¼ 6  0:9163  0:1905  ; ð12Þ EgI ðTÞ where EgI ðTÞ is determined by Eq. (10), and similarly in the Ge by "   #1=3 EgI ðT ¼ 0Þ 2 mdc ðTÞ ¼ 42=3  1:580  0:082  ; ð13Þ EgI ðTÞ where EgI ðTÞ is determined in Eq. (11). Further, we will use the approximate form for mdv in the Si obtained by Lang et al. [12], using the exact calculation including the full nonspherical-and-nonparabolic nature of the valence band structure, as

1 þ 4:683382  103  T þ 2:286895  104  T 2 þ 7:469271  107  T 3 þ 1:727481  109  T 4

ð8Þ

Note that in Eq. (8), if replacing DEg;A by its empirical form, extracted from electrical measurements in n-and-p types Si doped with N r 1020 cm3 , by Klaassen et al. [1], assuming that the ABGN obtained in the N-type Si is found to be equal to the one measured in the p-type Si,

DEKSG g;AðSiÞ ðNÞ ¼ 6:92

   pffiffiffiffiffiffiffiffi   N N 2 þ Ln þ0:5 ;  Ln 1:3  1017 cm3 1:3  1017 cm3 0

ð9Þ

the OBG can then be determined and denoted by EðKSGÞ g;O . In summary, as seen in Eqs. (1) and (2), there are three fundamental parameters to be determine such as the intrinsic

ð11Þ

mdc

0:443587 þ 3:609528  103  T þ 1:173515  106  T 3 þ 3:025581  107  T 4

0 0 ¼ Eg;E þ kB T  LnðN=NcðvÞ Þ: Eg;O

ð10Þ

!2=3 ;

ð14Þ

and mdv in the Ge [3,9] can be expressed in terms of ligh hole (heavy hole) effective masses of the valence bands, spin–orbit split-off energy and the first Kohn–Luttinger parameter [13], as mdv ðTÞ ¼ ð0:343=2 þ 0:0423=2 þ13:43=2  e3296 meV=2kB T Þ2:3 :

ð15Þ

In the Si, the values of EgI ðTÞ, mdc ðTÞ, and mdv ðTÞ, calculated for any T using Eqs. (10), (12), and (14), respectively, are tabulated in Table 1, in which we also include the respective data given by Green [3] to calculate the relative errors (RE) defined by RE  1ðResult=DataÞ and expressed in %. Table 1 indicates that our chosen results are accurate to within 0.11% for EgI ðTÞ, 0.48% formdc ðTÞ and 2.65% for mdv ðTÞ, giving us some confidence in our above analytic forms for those structure– band parameters at any T. Then, from those accurate structure–band parameters of the Si given in Table 1, our values of ni(T) at any T are computed using

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Table 1 In the Si, the values of EgI ðTÞ, mdc ðTÞ, and mdv ðTÞ, calculated for any T using Eqs. (10), (12), (14), respectively, are tabulated in Table 1, in which we also include the respective data given by Green [3] to calculate the relative errors (RE) expressed in %. T (K)

mdc [3]

mdc (RE)

mdv [3]

mdv (RE)

Eg;I ðmeVÞ [3]

4.2 50 100

1.06 1.06 1.06

1.0618 (  0.17) 1.0625 (  0.24) 1:0651 ð0:48Þ

0.59 0.69 0.83

0.5874 (0.44) 0.6898 (0.03) 0.8314 (  0.17)

1170.0 1169.0 1164.9

150 200 250 300 350

1.07 1.08 1.08 1.09 1.10

1.0694 1.0752 1.0821 1.0900 1.0987

0.95 1.03 1.10 1.16 1.19

0.9457 1.0328 1.0999 1.1525 1.1943

1157.9 1148.3 1136.7 1124.2 1110.4

1124.8 (  0.05) 11116 ð0:11Þ

400 450

1.11 1.12

1.1079 (0.19) 1.1177 (0.21)

1.23 1.29

1.2281 (0.15) 1:2558 ð2:65Þ

1096.8 1083.2

1097.7 ( 0.08) 1083.4 ( 0.02)

500

1.13

1.1279 (0.19)

1.29

1.2786 (0.88)

1069.5

1068.7 (0.08)

(0.06) (0.45) (  0.20) (0.00) (0.12)

Table 2 Our values of ni (cm  3) at any T are computed using Eq. (4) and reported in Table 2, in which we also include those given in Ref. [2] and the respective data [3] used to calculate the RE (%). T (K)

ni-data [3]  686

ni (RE)

ni (RE) [2]  686

4.2 50

3.14  10 1.64  10  41

3.134  10

100

1.95  10  11

1:98  1011 ð1:5Þ

1

1:68  10

41

(0.45) (  0.27) (0.01) (  0.21) (  0.36)

EgI (RE)

Gamma function, and in the non-degenerate case (Z b0) to another one: Fj ðZÞ  Zj þ 1 =ðj þ 1Þ. The reduced carrier density for parabolic conduction (valence) bands in n(p)-type doped semiconductors at temperature T can be defined by N=NcðvÞ  unðpÞ ¼ F1=2 ðZnðpÞ Þ=Gð32Þ;

(0.2)

ð2:5Þ

1

150 200 250 300 350

3.16  10 5.03  104 7.59  107 1.07  1010 3.92  1011

3.19  10 ( 1.2) 5.04  104 (  0.2) 7 7.51  10 (1.1) 1.06  1010 (0.7) 3:85  1011 ð1:7Þ

3:73  1011 ð4:8Þ

400 450 500

6.00  1012 5.11  1013 2.89  1014

5.92  1012 (1.2) 5.11  1013 (0.0) 2.92  1014 ( 1.2)

5.82  1012 (3) 5.13  1013 (0.4) 3.03  1014 (4.7)

5.13  104 (2) 7.38  107 (2.8) 1.03  1010 (3.9)

Eq. (4) and reported in Table 2, in which we also include the ones for 200rT(K)r500 given in Ref. [2] and the respective data [3] used to calculate the RE (%). Table 2 indicates that for two T-ranges: 4.2 rT(K) r500 and 200 rT(K)r500 our results for ni are accurate to within 2.5% and 1.7%, respectively, which also gives us some confidence in our above analytic form (4) for ni at any T. Further, our present results at any T are much more accurate compared with our respective recent ones for 200 rT(K) r500 [2], which are accurate to within 4.8% and calculated using Varshni–Thurmond’s model for intrinsic bandgap [14,15] and taking into account the effects of nonparabolicity in the bands.

3. Fermi energy First of all, the Fermi–Dirac integrals Fj(Z) for any order j 4 1 and reduced Fermi energy Z  EF =kB T is defined by [16] Z 1 xj dx ; ð16Þ Fj ðZÞ  ZÞ 1 þ expðx 0 which has been approximated to an accurate form for any 0.5 rj r12 and Z, to within 3.7%, by Aymerich-Humet et al. [17] as !1 ðj þ 1Þ2j þ 1 expðZÞ þ ; ð17Þ Fj ðZÞ ffi Gðj þ 1Þ ½b þ Z þðjZbjc þac Þ1=c j þ 1 2 1=2 1 , b=1.8 +0.61j, and c ¼ where paffiffiffi ¼ ½1 þ 15 4 ðj þ 1Þ þ 40ðj þ 1Þ  j 2 þð2 2Þ2 . Moreover, result (17) reduces in the degenerate case (Z 50) to a correct asymptotic result: Fj ðZÞ  Gðj þ 1Þ  eZ , Gðj þ1Þ being the

ð18Þ

where Nc(v) is determined in Eq. (3). It should be noted that by a reversion method one can determine ZnðpÞ ðunðpÞ Þ as a function of un(p). So, based on two correct asymptotic forms for ZnðpÞ ðunðpÞ Þ, a simple approximate expression for ZnðpÞ ðunðpÞ Þ at any un(p) has been obtained by Van Cong and Debiais [16], with an accuracy of the order of 0.0211%, as

ZnðpÞ 

EFnðpÞ ðN; T; mdcðvÞ Þ kB T

¼

GðunðpÞ Þ þ 0:0005372u4:82842262  FðunðpÞ Þ nðpÞ 1þ 0:0005372u4:82842262 nðpÞ

;

ð19Þ reducing in the highly degenerate case ðZnðpÞ b 1; unðpÞ b 1Þ to 2=3

4=3

8=3

FðunðpÞ Þ ¼ AunðpÞ ð1 þ BunðpÞ þ CunðpÞ Þ2=3 ; ð20Þ pffiffiffiffi 2=3 where A ¼ ½3 p=4 , B ¼ ð1=8Þðp=AÞ2 , and C= (62.3739855  1920)(p/A)4, and then in the non-degenerate case ðZnðpÞ 5 0; unðpÞ 5 1Þ to GðunðpÞ Þ ¼ LnðunðpÞ Þ þ23=2 unðpÞ expðD  unðpÞ Þ; D ¼ 23=2 ½p1ffiffiffiffi 27

ð21Þ

3 16  ¼ 0:01400097.

where We remark that as T-0 K our asymptotic result (20) reduces to the T-independent one as  2=3 16:78459 N EFnðpÞ ðN; T ¼ 0; mdcðvÞ Þ   ðmeVÞ ð22Þ  mdcðvÞ 1019 cm3 depending on the values of mdcðvÞ for a given N. Basing on this result (22), our present values of Fermi energies (meV) in the EI Si(Ge) for a given N= n = p and T =0 K, (EFn, EFp and EF =EFn +EFp), are computed using again Eqs. (12, 14) for mdcðvÞ in the Si, and Eqs. (13, 15) for mdcðvÞ in the Ge, and then tabulated in Table 3, in which we also report the respective exact results (exact) calculated using exact values of mdcðvÞ given in the Si [3], and the Fermi energy data obtained by Hammond et al. (HMM) for EI Si [7] and those by Thomas et al. (TPRH) for EI Ge [8]. Table 3 shows that our present results for Fermi energies are found to be accurate to within 0.9% and 0.04% for the EI Si and Ge, respectively. In summary, such high accuracies of our results for intrinsic carrier concentration ni and Fermi energy EFn(p) give us some confidence in the use of above fundamental material parameters such as: intrinsic bandgap EgI(T), average numbers of equivalent

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Table 3 Based on Eq. (22), our present values of Fermi energies (meV) in the EI Si(Ge) are computed using Eqs. (12) and (14) for mdcðvÞ in the Si and Eqs. (13) and (15) for mdcðvÞ in the Ge, respectively, and compared with respective exact results (exact) calculated using exact values of mdcðvÞ given in the Si [3], and the data obtained by Hammond et al. (HMM) in the Si [7] and those by Thomas et al. (TPRH) in the Ge [8]. mdc For the Si excited with Exact 1.0600 Present 1.0618 HMM 1.0800

EFn

mdv

N = 3.3  1018 cm  3 7.60737 0.5900 7.59446 0.5874 7.46649 0.5500 17

EFp

EF ðREÞ

13.66747 13.86397 14.66147

21.27484 21.45843 (  0.9%) 22.12796 (  4.3%)

3

For the Ge excited with N = 2.4  10 cm Present 0.5547 2.51744 0.3567 TPRH 0.5586 2.50000 0.3581

3.91543 3.90000

6.43287 6.43000 (0.04%)

conduction (valence) bands gc(v) and effective DOS masses in each conduction (valence) bands mdc(v).

mx =0.123(0.046), exciton Bohr radius: ax = 49(177)  10  8 cm, exciton Rydberg: Ex = 12.87(2.65) meV, and effective numbers of equivalent conduction (valence) bands: geff = 2(1), due to the effective screening process as pointed out by Rosenbaum et al. [24], respectively. Moreover, in HD and HEI Si (Ge) at 0 K, the exciton (or impurity) density N is related to the Wigner–Seitz radius rs, expressed in units of ax, and also to the Fermi wave number kF, by   3 1=3 1=3 rs  =ax ¼ ðakF gcðvÞ ax Þ1 ; ð23Þ 4pN where a = (4/9p)1/3. The effective values of exciton Rydberg Ex and exciton mass mx, being denoted, respectively, by Ex ðrs Þ ¼ Ex  mx ðrs Þ=mx and mx ðrs Þ, both decrease with increasing rs. That is given in the following Landau’s formula for mx , used by Yasuhara and Ousaka (YO) [25], mx ðrs Þ E ðrs Þ  x mx E 2 x

4. Bandgap narrowing Recently, Grivickas et al. [18] demonstrated a novel spatially and time-resolved spectroscopy approach for examining the fundamental absorption edge in Si at low temperatures and presenting an experimental evidence for the existence of excitonic states above the excitonic Mott transitions in both Ntype HD Si and highly excited intrinsic (HEI) Si (e.g., the system of two components). Such an existence of excitonic states is also supported by Pankove and Aigrain [19] and Pankove [20] in their cathodoluminescence studies of n(p)-type HD GaAs, in which the donor-to-acceptor transition probability is found to be dominantly high, by electron-beam-excitation technique allowing the generation of a high density of e–h pairs about 1.5  1018 cm  3, and also by Bergersen et al. [21] in their photoluminescence studies of the e–h droplets and impurity band states in HD Si (P) for a doping range 1.2  1017 4  1019 cm  3, in which their obtained results have been explained by the recombination of charge carriers inside an e–h drop and of a free hole with an electron loosely bound to an impurity site. Then, with the use of assumptions of parabolic bands and a photoexcited carrier density much smaller than completely ionized donor (acceptor) density NdðaÞ ffi N, taken by Wagner and del Alamo [22] in determining the BGN: DEgnðpÞ from the n(p)-type HD photoluminescence and photoluminescence excitation, the result of Grivickas et al. [18] for the chemical potential, mn  DEgn þEFn , obtained in N-type HD Si is found to be approximately equal to the respective one given in the conduction band of the HEI Si. Such an excellent idea can now be applied to the following model of three components: N-type HD Si with n= Nd =N, p-type HD Si with p = Na =N, and an HEI Si for a density of electron–hole pairs (n= p= N), so that the sum of two chemical potentials given from n-and-p HD types, mn þ mp  ðDEgn þ EFn Þ þðDEgp þ EFp Þ, may be approximately equal to the chemical potential given in the HEI Si, m  DEg þEF . Therefore, EF =Efn + EFp and DEg ¼ DEgn þ DEgp , suggesting that many-body carrier–carrier interactions and screening effects on carrier-impurity (or electron–hole) potential energies given in both n(p)-type HD-and-HEI Si may be treated by a same way, because above the Mott transition the excitons become ionized and the impurities are also ionized even at 0 K, being due to the same screening effect of Coulomb attraction [23]. That can also be applied to the Ge case. Furthermore, in order to investigate the BGN in doped Si (Ge), we need to know the values of other fundamental band–structure parameters given, respectively, in the Si (Ge) such as [9] exciton mass to free electron mass mo:

8

2

3

1=3 < 1=3 ars geff 2ars geff R2s 1 5 ¼ 41 1 þ 41 þ 2 p : 2 pRsðTFÞ

2 ln4

3931 = 5 5 ; 1=3 2 2 1 þ½ars geff Rs =pRsðTFÞ  ; 1=3

½ars geff R2s =pR2sðTFÞ 

ð24Þ

where Rs ðrs Þ  ks =kF is the ratio of the inverse screening length ks to kF. Here, based on the works given in Refs. [25–27], an accurate form for Rs has been developed and given in Eq. (A.6) of Appendix A. Then, in the Si, for example, our values of Rs and mx =mx are computed using Eqs. (A.6) and (24), respectively, and tabulated in Table 4, in which we also include the respective results for Rs(TF) given in the well-known Thomas–Fermi (TF) approximation, and those for Rs(YO) and mxðYOÞ =mx obtained by Yasuhara and Ousaka (YO) [25], using a systematic consideration of vertex corrections in accordance with the Pauli principle up to higher orders in computing the Landau interaction function or the self-energy. From Table 4, some cluding remarks can be discussed as follows: (i) First, our results of ratio Rs at any rs (or N) are found to be decreased with increasing N and lower than the unit, agreeing thus to the screening condition as discussed in Appendix A, while the respective YO ones, Rs(YO), which are more accurate in comparison with those given in the TF model, Rs(TF), are found to be only valid for N44  1018(cm  3).That gives us some confidence in the use of our accurate result (A.6) for Rs. (ii) Our results of mx ðrs Þ=mx ¼ Ex ðrs Þ=Ex are increased with increasing N, and reduced to 0 as N-0 and to the unit as N-N (or rs-0), where all the values of Rs, Rs(YO), and Rs(TF) tend to zero, and ðmx =mx Þ ffiðmxðYOÞ =mx Þ-1. In the following, taking into account such our accurate screening model (A.6), various important contributions to the BGN in the Table 4 In the Si, our values of Rs and mx =mx are computed using Eqs. (A.6) and (24), respectively, and compared with respective results for RsðTFÞ and those for RsðYOÞ and mxðYOÞ =mx obtained by Yasuhara and Ousaka (YO) [25]. rs N (1020 cm  3)

587.06 10  10

0.79677 0.04

0.17166 4

0.07397 50

0.03433 500

RsðTFÞ RsðYOÞ [25] mxðYOÞ =mx [25]

31.33

1.15 1.06 0.95

0.54 0.53 0.97

0.35 0.35 0.98

0.24 0.24 0.99

0.91 0.02

0.79 0.92

0.53 0.97

0.36 0.98

0.25 0.99

Rs mx =mx

ARTICLE IN PRESS H. Van Cong / Physica B 405 (2010) 1139–1149

n(p)-type Si (Ge) due to the doping effect (DE) [2,4,5,22,28–34] are reported as

DEgnðpÞ ðNÞ ¼ 

CðmajminÞ XC SE SE X X X X 1=3 ðrs gcðvÞ Þ ðrs Þ ðk ¼ kF Þ ðk ¼ 0Þ cðvÞ

cðvÞ

cðvÞ

Ge–P(In) systems [35], EDðAÞ;Si ¼ 45ð45Þ and EDðAÞ;Ge ¼ 12ð11:2Þ, respectively, as EDðAÞ ¼ bnðpÞ 

cðvÞ

ð25Þ being determined and computed, using the following Eqs. (26)–(31). In the second member of Eq. (25), these four contributions to the BGN valid at any rs (or N) are explained and also discussed in two particular cases: high carrier-density limit (HCDL), rs-0 or N-N, and high carrier-density range (HCDR): 4  1018 rN (cm  3)r4  1020, being used in various optical measurements in the n(p)-type HD Si (Ge), as follows. (i) The exchange-correlation (XC) conduction (valence)-band shift is given for the one majority band at k =kF by   XC X 2anðpÞ rs4 d½Ec ðrs Þ=rs3  ; ð26Þ  ðrs Þ ¼ Ex ðrs Þ  þ aprs 3 drs cðvÞ where the correlation energy of an electron gas Ec(rs) [27] is determined in Eq. (A.3) of Appendix A, and an(p),Si = 0.685 and an(p),Ge =0.85(0.98) are the dimensionless reduction factors due to the band structure [9]. In the HCDL, result (26) is proportional to N1/3 since Ex ðrs -0Þ ¼ Ex , as that given in Ref. [4], while in the , where no = N/1018 HCDR this yields: 6:896ð7:292Þn0:32740ð0:32515Þ o 3 (cm ), accurate to within 0.2(0.1)% for the n(p)-type Si, and 5:1783ð9:2265Þno0:32714ð0:33044Þ to within 0.15(0.07)% for the n(p)type Ge, respectively. (ii) The majority–minority correlation energy shift was calculated at k= 0 by Thuselt [28], using the random-phase approximation as " # Cðmaj:;min:Þ X Gð3=4Þ  21=2  ð43p2 Þ1=4 1:0038 ; ð27Þ ðrs Þ ¼ Ex ðrs Þ  ¼  3=4 Gð5=4Þ  rs3=4 rs cðvÞ which is found to be slightly higher than the respective one obtained by Mahan [29] using the plasmon-pole approximation, 3=4 Ex ðrs Þ  0:95=rs . Note that in the HCDL the numerical result (27) is proportional to N1/4 as that given in Ref. [4], while in the accurate to within HCDR this yields 6:274ð8:255Þn0:2626ð0:2626Þ o 0.51(0.57)% for the n(p)-type Si, and 4:0243ð5:6913Þn0:25293ð0:25293Þ o to within 0.11(0.10)% for the n(p)-type Ge, respectively. (iii) Finally, the conduction (valence)-band shift at any k due to the screening effect (SE) was calculated from the impurity– electron (hole) interactions, and commonly obtained by Parmenter, using the second-order perturbation approximation [30], and by Wolff using a simplest diagrammatic analysis method by means of second-order process [31] as " # SE X R3sðTFÞ p  ð28Þ ðk ¼ 0Þ ¼ Ex ðrs Þ  bnðpÞ   3  3=2 Rs 2rs geff cðvÞ and 2 SE X   ðk ¼ kF Þ ¼ Ex ðrs Þ  bnðpÞ  4 cðvÞ

p 3=2

2rs

geff



R3sðTFÞ R3s

3 1 5  ; 1 þ R42

ð29Þ

s

where Rs(TF) and Rs are determined in Eqs. (A.1) and (A.6) of Appendix A, noting that Eq. (A.5) gives the limiting value of Rs as rs-N as Rs ðrs -1Þ ¼ RsðWSÞ ðrs -1Þ ¼ 0:9337

ð30Þ

and bn(p) are the empirical parameters to be determined as follows. Taking into account the limiting result (30), our result (28) as rs-N can also be used to determine the data of donor (acceptor) ionization energies ED(A) expressed in meV for Si–P(B) and

1143

¼ bnðpÞ 

Ex  mdcðvÞ p R3sðTFÞ  Limit 3=2 rs -1 2r mx R3s s geff " #3=2 Ex  mdcðvÞ 4a  geff  mx p1=3 22=3 ð0:9337Þ2

ð31Þ

giving rise to the values of above dimensionless reduction factors as bnðpÞ;Si ¼ 0:6412ð0:5628Þ and bnðpÞ;Ge ¼ 0:91ð0:5225Þ. Furthermore, in the HCDL, according to the Thomas–Fermi (TF) approximation, our result (28) at k= 0 is reduced to 3=2 bnðpÞ  p=2rs geff  Ex , being proportional to N1/2, agreeing with that obtained by Berggren and Sernelius [33] using the Green’s function by means of second-order perturbation theory, and our 1=3 1=2 result (29) at k= kF is reduced to bnðpÞ  ageff =2rs  Ex , being proportional to N1/6, agreeing with that obtained by Van Cong et al. [32] using the second-order cumulant approximation. But, in , the HCDR, our result (28), respectively yields 18:6ð16:4Þn0:245ð0:245Þ o being accurate to within 11.0(10.7)% for the n(p)-type Si, and to within 9(10)% for the n(p)-type Ge, and 8:97ð5:17Þn0:392ð0:391Þ o , our result (29), respectively, yields 3:08ð2:714Þn0:1048ð0:1040Þ o accurate to within 3.6(3.6)% for the n(p)-type Si, and correct to within 2.9(3.0)% for the n(p)0:770ð0:442Þn0:141ð0:141Þ o type Ge. In summary, by using Eqs. (26)–(31), our result (25) valid at any N for the BGN in the n(p)-type Si (Ge), due to the DE, is thus determined, showing that the TF screening model is inaccurate at low values of N. Furthermore, in the HCDR, our result (25) can also be approximated to a simple approximate result (SApR) for the n(p)-type Si as

DEgnðpÞ;Si ðNÞ ¼ 35:34ð34:85Þn0:2535ð0:2615Þ ; o

ð32Þ

accurate to within 3.7%(3.1%), and that for the n(p)-type Ge as

DEgnðpÞ;Ge ðNÞ ¼ 20:995ð20:7Þn0:2998ð0:3055Þ ; o

ð33Þ

accurate to within 3.6%(2.2%), suggesting that, in each material: Si (or Ge), the results of BGN calculated in the n-and-p types are found to be almost the same, because of almost equal values of band–structure parameters used. It should be noted that Eqs. (25, 32, 33) for the BGN are our central results found in the present work, and in particular Eqs. (32, 33) in the HCDR are our remarkable accurate ones because of their very simple forms. Finally, replacing N by N*  N*(N,T), determined using Eq. (B.7) of Appendix B, into Eqs. (19, 23–33), we can now determine the Fermi energy and the BGN at any N and T. Therefore, all the optical-and-electrical properties for the EI (HD) Si (Ge) defined in Eqs. (1)–(11) are also determined and computed at any N and T, using our central result (25) for the BGN. So, in this HCDR and from a conjunction between ABGN and BGN given in Eq. (1), one has

DEgnðpÞ;A ðN Þ ¼ DEgnðpÞ ðN  Þ þ kB T  LnðN =NcðvÞ ÞEFnðpÞ ðN Þ;

ð34Þ

where the BGN: DEgnðpÞ ðN  Þ is determined in Eqs. (32) and (33) for the n(p)-type Si and Ge, respectively, and the Fermi energy: EFnðpÞ ðN  Þis computed from Eq. (19). Then, in particular, taking into account an approximate form for the ABGN given in the n(p)-type Si, fitted to various electrical data by Klaassen et al. [1], as that  given in Eq. (9) for Nr1020 cm  3: DEKSG g;AðSiÞ ðN Þ, our result (34) obtained in the HCDR can be approximated to a very simple form

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H. Van Cong / Physica B 405 (2010) 1139–1149

denoted by a generalized KSG’s (GenKSG’s) model as  KSG  DEGenKSG gnðpÞ;AðSiÞ ðN Þ ¼ DEg;AðSiÞ ðN ÞAnðpÞ 



BnðpÞ

N 1:3  10

17

cm3

;

ð35Þ

where An(p) = 10.483(6.78)  10  6 meV and Bn(p) = 1.95(1.97), being correct to within 10.5%(5.3%), respectively, and the second term of  the second member of Eq. (35) means the decrease in DEKSG g;AðSiÞ ðN Þ due to the effects of the Fermi–Dirac statistics, as seen in Figs. 1 and 2 for the n(p)-type Si, respectively. Moreover, those relative deviations (RDs) between our results (34) and (35) can also be seen in Fig. 3. Further, from our result (34), from which we replace DEgnðpÞ;A  by DEGenKSG gnðpÞ;AðSiÞ ðN Þ given in Eq. (35), we can get an approximate form for the BGN as  GenKSG    DEGenKSG gnðpÞ;Si ðN Þ ¼ DEgnðpÞ;AðSiÞ ðN Þ þ EFnðpÞ ðN ÞkB T  LnðN =NcðvÞ Þ:

ð36Þ

The values of RDs between our results (32) and (36) are given in Fig. 4, suggesting a maximal value of RD is found to be 5.6%, giving us some confidence in our above analytic form (36) for the BGN. Furthermore, Eq. (6) for the equivalent chemical potential or for the OBG can be rewritten in the EI Si (Ge)-case as 0 Eg;O ðN Þ  Eg;O ðN  ÞEp ¼ Eg0 ðN Þ þ ½EFn ðN Þ þ EFp ðN Þ;

Fig. 3. Relative deviations of apparent bandgap narrowing between our results (34, 35) in n(p)-type HD Si vs. N.

ð37Þ

Fig. 4. Relative deviations of bandgap narrowing between our results (32, 36) in n(p)-type HD Si vs. N.

Fig. 1. Apparent bandgap narrowing in the N-type HD Si vs. N.

where Eg0 ðN Þ  EgI ðTÞDEgn ðN ÞDEgp ðN Þ is the low-energy threshold defined in Eq. (5), the intrinsic bandgap EgI(T) is determined in Eq. (10) for Si and in Eq. (11) for Ge, and the BGN: DEgnðpÞ ðN Þ and Fermi energy: EFnðpÞ ðN  Þ are determined, respectively, by Eqs. (25) and (19) for any N and T, and in the n(p)type HD Si (Ge) by E0gnðpÞ;O ðN; TÞ  Eg;O ðN; TÞEp ¼ EgI ðTÞDEgnðpÞ ðN Þ þ EFnðpÞ ðN Þ;

ð38Þ

where the BGNs: DEgnðpÞ;SiðGeÞ ðN Þ are computed, respectively, by Eqs. (25, 32, 33) for Si(Ge). Finally, the above conjunction (8) between the OBG and EBG occurring in the n(p)-type Si is now rewritten by 0

  ðN; TÞ ¼ EgI ðTÞDEGenKSG EðGenKSGÞ gnðpÞ;AðSiÞ ðN Þ þ kB T  LnðN =NcðvÞ Þ; gnðpÞ;OðSiÞ

ð39Þ

 where the ABGN: DEGenKSG gnðpÞ;AðSiÞ ðN Þ is determined in Eq. (35). The values of RDs between our results (38) and (39) are given in Fig. 5, suggesting a maximal value of RD is found to be 0.7%, giving us some confidence in this simple analytic form (39). Furthermore, from Eq. (8), the EBG defined in the base of Si1  xGex hetero bipolar transistor (HBT), heavily doped (HD) with boron impurities, is given by

Fig. 2. Apparent bandgap narrowing in the p-type HD Si vs. N.

E0gp;EðSiGeÞ ðN; T; xÞ  EgIðSiGeÞ ðT; xÞDEgp;AðSiGeÞ ðN; T; xÞ:

ð40Þ

ARTICLE IN PRESS H. Van Cong / Physica B 405 (2010) 1139–1149

1145

Table 5 Our results of RE for the EBG, calculated using Eqs. (43) and (44) and the respective data [36], and denoted by REs (43) and (44), respectively, compared with the respective RE (42)-results, calculated using Eq.(42). x

N (109 cm  3) Data (meV) RE (42)

0.154 0.186 0.222 0.254 0.284

4.26 5.78 5.17 5.49 5.35

937 908 888 866 848

2.98  10  3  6.44  10  4 1.55  10  3 6.74  10  4 1.04  10  3

RE (43)

RE (44)

1.56  10  3  2.20  10  3 6.26  10  4 8.77  10  6 8.14  10  4

8.26  10  4  2.71  10  3  3.10  10  5  5.95  10  4 1.65  10  4

Table 6 Our results of ABGN, calculated using Eqs. (43) and (44) and compared with both experimental and theoretical EK-results [36].

Fig. 5. Relative deviations of optical bandgap between our results (38, 39) in n(p)type HD Si vs. N.

Then, Eberhardt and Kasper (EK) [36] recently assumed the ABGN: DEgp;AðSiGeÞ ðN; T; xÞ for Ge content xr0.3 to be the same like in Si

x

N (109 cm  3)

Data (meV)

Eq. (42)

Eq. (43)

Eq. (44)

0.154 0.186 0.222 0.254 0.284

4.26 5.78 5.17 5.49 5.35

188.78 217.79 237.78 259.79 277.79

190.31 215.93 237.90 259.10 277.40

190.56 216.10 238.66 260.11 278.79

189.56 215.33 237.76 259.27 277.93

[1] as DEKSG gp;AðSiÞ ðNÞ; see also Eq. (9). Therefore, their best fit to their own data was found to be given by T 2 EEK gI;SiGe ðT; xÞ ¼ EgI;Si ðTÞ750x þ238x ;

ð41Þ

where they used the expression for ETgI;Si ðTÞ, being obtained by Thurmond [15]. As a result, from Eqs. (40, 41), the ABGN, 0

ðEKÞ DEEK gp;AðSiGeÞ , and the EBG, Egp;EðSiGeÞ , given in the p-type HD base of

the Si1  xGex HBT at T=295 K is given by 0

KSG 2 ETgI;Si ðTÞEðEKÞ ðN; T; xÞ  DEEK gp;AðSiGeÞ ðN; xÞ ¼ DEgp;AðSiÞ ðNÞ þ 750x238x gp;EðSiGeÞ

ð42Þ being T-independent. Now, following such a treatment used by EK, we can obtain from Eqs. (34), (35) and (40), the two respective results given in such a p-type HD base of Si1  xGex HBT for x r0.3 as EgI;Si ðTÞE0gp;EðSiGeÞ ðN; T; xÞ  DEgp;AðSiGeÞ ðN; T; xÞ ¼ DEgp;AðSiÞ ðN  Þ þ759x234x2 ;

ð43Þ 

where EgI,Si(T) and DEgp;AðSiÞ ðN Þ are, respectively, determined by Eqs. (10) and (34), and 0

GenKSG GenKSG  EgI;Si ðTÞEðGenKSGÞ gp;EðSiGeÞ ðN; T; xÞ  DEgp;AðSiGEÞ ðN; T; xÞ ¼ DEgp;AðSiÞ ðN Þ

þ 759x234x2 ;

ð44Þ

where the ABGN, DEGenKSG gp;AðSiÞ ðN; TÞ, is determined by Eq. (35), according thus to our expression for intrinsic bandgap as EgI;SiGe ðT; xÞ ¼ EgI;Si ðTÞ759x þ 234x2

ð45Þ

being in good accordance with the above result (41). Our results of RE at 295 K for the EBG and ours of ABGN obtained in the p-type HD base of Si1  xGex HBT , calculated using Eqs. (43)–(44) and the respective EK-data [36], are reported in Tables 5 and 6, in which we also report the respective RE values and the ABGN ones, calculated using Eq. (42). Table 5 indicates that for the EBG our results obtained from Eqs. (43) and (44) are accurate to within 0.220% and 0.271%, while the EK-results obtained from Eq. (42) are correct to within 0.298%. Further, from Table 6, for the ABGN, our results obtained from Eqs. (43) and (44) are in perfect accordance with both experimental and theoretical EK-results [36], noting that their theoretical ones [36] can also be calculated using Eq. (42).

Fig. 6. Relative deviations of electrical bandgap between results (42, 43) and (42, 44) in the p-type HD base of Si1  xGex HBT at T= 295 K and for x= 0.3 vs. N.

Moreover, for another comparison, the relative EBG-deviations between Eqs. (42, 43) and (42, 44) in the p-type HD base of Si1  xGex HBT at T=295 K and x =0.3, calculated and expressed as functions of N are reported in Fig. 6, giving rise to a maximal relative deviation in absolute value equal to 0.24%. That gives us some confidence in our simple analytic forms (43)–(44) for xr0.3. In a forthcoming work, we hope to study the ABGN and EBG in the n(p)-type HD base of Si1  xGex HBT for 0 rxr1. Now, in order to compare our numerical results with existing experimental ones [5,7,8,22], we will focus our attention to the Nrange: Nr4  1020 cm  3. In those optical experiments, some remarks are given as follows: (i) In the Si(Ge) excited, respectively, with N= 33.3(2.4)  1017 cm  3, the phonon assistance is thus needed [5,7], and the values of phonon energy are Ep(TO);Si = 60.23(meV) and Ep(LA);Ge =26.548 ( meV), as discussed in Eq. (7). In those cases, one has from Eq. (6) 0 0 þ Ep , where EgO is measured. or Eq. (37): Eg;O ¼ Eg;O (ii) However, in HD indirect-gap Si (Ge), it is possible to conserve momentum by a scattering process such as electron– electron scattering [5] or by impurity scattering [19], and the

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H. Van Cong / Physica B 405 (2010) 1139–1149

photon assistance is not needed in these cases [20], explaining, from the photoluminescence spectra of N-type HD Si (Ge) observed by Wagner [22] and Haas [5], the high-energy cutoff 0 0 ) of the no-phonon replica. As a result, Egn;O ¼ Eg;O (or OBG Egn;O because of Ep = 0 for the N-type HD Si (Ge). But, in p-type HD Si, 0 þ EpðTAÞ from the data of highone can determine: Egp;O ¼ Egp;O 0 [22] and the TA-phonon energy: Ep(TA) = 15.8 energy cutoff Egp;O (meV) [20].

5. Numerical results and concluding remarks The two materials such as EI Si(Ge) and n(p)-type HD Si (Ge) are considered here. 5.1. EI Si (Ge)-materials Here, at 0 K at which N* = N, our values of DEg ¼ DEgn þ DEgp , Eg0 , 0 , calculated using Eqs. (25) and (37), are tabulated in and Eg;O Table 7, in which we also report the respective data [7,8]. 0 are Table 7 indicates that our values of DEg , Eg0 , and Eg;O accurate to within 2.1%. Table 7 0 Our present values of DEg , Eg0 , and Eg;O , expressed in meV, calculated using Eqs. (25) and (37), and accompanied with RE (%) computed using respective data given in the EI Si by Hammond et al. (HMM) [7], and in the EI Ge by Thomas et al. (TPRH) [8].

DEg (RE)

Eg0 (RE)

0 Eg;O (RE)

For the Si excited with N = 3.33  1018 cm  3 HMM 103.5 1066.5 Present 101.2 (2%) 1068.8 (  0.2%)

1088.7 1090.8 (  0.2%)

For the Ge excited with N = 2.4  1017 cm  3 TPRH 37.43 704.67 Present 36.63 (2.1%) 705.46 (  0.1%)

711.1 711.9 (  0.1%)

Then, from our result (37) given for equivalent chemical 0 ðN; TÞ, we can define the chemical potential or for OBG, Eg;O 0 ðN; TÞEgI ðTÞ, which can be computed potential as mðN; TÞ  Eg;O at any N and T. Thus, we are in position to study accurately the critical densities and temperatures, nc and Tc, of the e–h liquid/e–h plasma phase transition by using the following 0 ðN; TÞ or m(N, T) by conditions expressed simply in terms of Eg;O  @mðN; TÞ  @2 mðN; TÞ  ð46Þ N ¼ nc ;T ¼ Tc ¼ N ¼ nc ;T ¼ Tc ¼ 0 @N  @N 2 indicating that the function m(N, T) has an inflection point at N=nc and T= Tc. Our results computed using conditions (46) are reported in Table 7, suggesting that these critical values for EI Si are found to be Tc = 18.5 K and nc = 1.8905  1018 cm  3, and those for EI Ge yield Tc =26 K and nc =7.649  1016 cm  3. Now, as noted above, from our result (38), from which the BGN is determined by Eq. (25), given in the n(p)-type doped Si(Ge), we can also define the chemical potential as mnðpÞ ðN; TÞ  E0gnðpÞ;O ðN; TÞEgI ðTÞ, which can also be computed at any N and T. Then, following a same treatment but replacing m(N,T) in Eq. (46) by mn(p)(N,T), we thus obtain in the metal– insulator transition (MIT) of the n(p)-type doped Si an inflection point at Tc =18.5 K and nc;nðpÞ ffi 2:08ð1:679Þ  1018 cm3 , and in the MIT of the n(p)-type doped Ge at Tc = 26 K and nc;nðpÞ ffi 7:52ð8:95Þ  1016 cm3 , being approximately equal to the respective ones obtained in EI doped Si (Ge). Furthermore, because of the Mott transition [37], occurring is found to be roughly equal to when the screening length k1 s (ax,Si(Ge)/1.19), where the values of exciton Bohr radius ax,Si(Ge) are already given above, the carrier density–temperature (N–T) plane is divided into an area where bound states are possible as k1 s 4ðax;SiðGeÞ =1:19Þ and another area without bound states as k1 s oax;SiðGeÞ (i.e., in the high-density e–h liquid). Our numerical calculation indicates that such a Mott transition condition gives for EI Si (Ge): Tc(K)=18.5(26) and nc;Mott ffi 203:2ð6:39Þ 1016 cm3 ,

Table 8 Critical densities nc and temperatures Tc of the e–h liquid/e–h plasma phase transition for the EI Si (Ge) with N =n =p. Tc = 18.5 K and nc =1.8905  1018 cm  3 N (1018 cm  3) m (meV) @m ð1019 meV cm3 Þ @N 2 @ m ð1041 meV cm6 Þ @N 2 Tc = 26 K and nc = 7.649  1016 cm  3 N (1016 cm  3) m (meV) @m ð1018 meV cm3 Þ @N 2 @ m ð1035 meV cm6 Þ @N 2

1.8907  79.04706  8.31479

1.8906  79.04698  8.31479

1.8905  79.04689  8.31479

1.8904  79.04681  8.31479

6.22678

1.20242

 3.82321

 8.85022

7.651  33.01965  4.71599

7.650  33.01961  4.71625

7.649  33.01956  4.71650

7.648  33.01951  4.71674

5.07206

.18539

 2.04273

 2.05508

Table 9 In the n-type HD Ge at 80 K, using the BGN-and-OBG data [5], the REs of our results (33) and (38) for BGN and OBG, denoted, respectively, by RE (33) and RE (38), are compared with those obtained from an empirical form for BGN by Jain and Roulston (JR) [4], denoted by RE (JR). N  nð1018 cm3 Þ

2.4

4.5

9.6

19.5

21.7

43

N ð1018 cm3 Þ BGN data (meV) RE (33) (%) RE (JR) (%) OBG data (meV) RE (38) (%)

2.31

4.38

9.47

19.36

21.56

42.84

27 0.1  5.2 715 0.03

31  5.5  13.9 719 0.3

40  2.9  15.3 723 0.2

47  8.6  26.5 734 0.6

54 2.4  14.5 731  0.09

70 7.5  13.5 747  0.3

ARTICLE IN PRESS H. Van Cong / Physica B 405 (2010) 1139–1149

respectively, which can be compared with those obtained above, and in particular, with a recent value of the Mott transition carrier density for the Si taken by Grivickas et al. [18]: nc,Mott =2  1018 cm  3 (Table 8).

5.2. n(p)-Type HD Si(Ge)-materials In the N-type HD Ge at 80 K, using the BGN-and-OBG data expressed as functions of N [5], the REs of our simple form (33) for BGN and those of our result (38) for OBG, are computed and tabulated in Table 9, in which we also report those calculated from an empirical form for BGN obtained by Jain and Roulston (JR) [4]. Table 9 indicates that our results (33) and (38) are correct to within 8.6% and 0.6%, respectively, being more accurate compared with the respective ones for BGN obtained with a maximal RE in absolute value (26.5%) by JR [4]. In the n(p)-type HD Si at T=20 K and 300 K, using the BGN-andOBG data expressed as functions of N [22], the REs of our simple approximate results (32, 36) for BGN, and those of our results (38, 39) for OBG, are evaluated and tabulated in Tables 10 and 11, in which we also report those calculated from a theoretical BGN result obtained by van Teeffelen et al. (TPEJ) [34].

1147

Tables 10 and 11 suggest that our results (32, 36) for BGN and (38, 39) for OBG in the n(p)-type HD Si, are accurate: (i) at T= 20 K, to within 11.3%(10.4%) for BGN and 0.88%(12.4%) for OBG in the n(p)-type HD Si, being found to be more accurate compared with the respective ones for the BGN at 0 K, obtained with maximal REs in absolute values:18.8%(17.0%) by TPEJ [34], respectively, and (ii) at T=300 K, to within 13.6%(17.2%) for the BGN and 2.50%(1.84%) for OBG in N-type HD Si, and 11.7%(12.4%) for the BGN and 2.08%(2.37%) for OBG in p-type HD Si. Moreover, in those tables, it should be noted that such optical data obtained at 300 K by Wagner and del Alamo [22] are very approximated since for the 300 K spectra the direct extraction of OBG is not possible due to the large thermal spread of the carriers which smears out the high-energy cutoff. Thus, they have calculated the OBG at 300 K from the low-temperature data, using the approximate values of band–structure parameters in the intrinsic Si, as given in the following: (i) For the difference in intrinsic bandgap at 20 and 300 K, a high value of 60 meV has been taken [22], while our respective

Table 10 In the n-type HD Si at T= 20 and 300 K, using the BGN-and-OBG data expressed as functions of N [22], the REs of our results (32) and (36) and (38) and (39) for BGN and OBG, denoted, respectively, by REs (32) and (36) and REs (38) and (39), are computed and compared with those calculated from a theoretical BGN result at 0 K obtained by van Teeffelen et al. (TPEJ) [34], denoted by RE (TPEJ). n ð1018 cm3 Þ

4

8.5

15

50

80

150

At 20 K N  ð1018 cm3 Þ BGN data (meV) RE (32) (%) RE (TPEJ) (%) OBG data (meV) RE (38) (%)

3.89 54.3 8.1 18:8 1138 0.87

8.39

14.88

49.84

79.81

149.74

56.7  6.7 2.2 1133 0.86

74.6 6.1 11.6 1129 0.76

96 0.8 0.5 1131 0:88

99.0  8.3  11.7 1132 0.52

141.9 11:3 5 1133  0.68

7.06

12.47

45.42

75.32

144.62

55.2  5.1  4.9 1028  0.48  0.48

77 12.9 12.4 1033  0.71  0.67

92  1.1  2.8 1050  0.76  0.61

93 13:6 17:2 1056  1.08  0.77

116.5  7.0  12.9 1059 2:5 1:84

At 300 K N  ð1018 cm3 Þ BGN data (meV) RE (32) (%) RE (36) (%) OBG data (meV) RE (38) (%) RE (39) (%)

3.14 53.6 11.9 14.9 1020  0.14  0.30

Table 11 In the p-type HD Si at T= 20 and 300 K, using the BGN-and-OBG data expressed as functions of N [22], the REs of our results (32) and (36) and (38) and (39) for BGN and OBG, denoted, respectively, by REs (32) and (36) and RE (38) and (39), are computed and compared with those calculated from a theoretical BGN result obtained by van Teeffelen et al. (TPEJ) [34], denoted by RE (TPEJ). p ð1018 cm3 Þ

6.5

11

15

26

60

170

400

At 20 K N  ð1018 cm3 Þ BGN data (meV) RE (32) (%) RE (TPEJ) (%) ROBG data (meV) RE (38) (%)

6.39 56  1.1 3.0 1142 0.7

10.89 64  1.7 1.0 1140 0.5

14.88 65.7  7.5  5.6 1139 0.3

25.87 88.6 7.9 8.0 1142 0.2

59.83 92 10:4  13.3 1142  1.5

169.71 145.3 8.2 2.5 1162  4.8

399.47 155.7  7.2 17:0 1178 12:4

At 300 K N  ð1018 cm3 Þ BGN data (meV) RE (32) (%) RE (36) (%) ROBG-data (meV) RE (38) (%) RE (39) (%)

5.27 60.8 1.1 12:4 1036 0.90 0.84

9.09

12.47

23.25

55.24

164.37

393.68

64.8 4.2 4.6 1044 0.99 0.97

76.4 11:7 12.1 1048 1.00 0.98

82.5 3.8 4.6 1051 0.59 0.53

99.8 0.3 0.9 1062 0.56 0.51

133.2 0.6  1.8 1086 0.48 0.78

156.6  6.2  4.1 1102 2:08 2:37

ARTICLE IN PRESS 1148

H. Van Cong / Physica B 405 (2010) 1139–1149

accurate value, calculated using Eq. (10), only yields 45 meV. (ii) They also used [22] mdc ð20 KÞ ¼ 1:063 and mdc ð300 KÞ ¼ 1:18, while our respective accurate values, as seen in Table 1, mdc ð20 KÞ ¼ 1:062 and mdc ð300 KÞ ¼ 1:09. As a result, such large relative deviations between our results for the BGN and OBG at room temperature and the respective data [22], reported in above Tables 10 and 11, can thus be accepted. In summary, we have developed a theory for the bandgap changes in EI (HD) Si(Ge) for any majority-carrier density N and temperature T, due to the many-body carrier–carrier interactions and screened carrier-impurity (or on electron–hole) potential energies. Our treatment method is based on the following: (i) the accurate formula (19) for the Fermi energy at any N and T, (ii) the accurate values of band–structure parameters such as 2=3 DOS effective masses mdcðvÞ  gcðvÞ mdcðvÞ calculated using Eqs. (12)–(15), effective exciton mass mx calculated using Landau’s formula (24), and intrinsic bandgap EgI(T) calculated using Eqs. (10, 11), (iii) our accurate screening model at any N developed in Eq. (A.6) of Appendix A, instead of the Thomas–Fermi model currently used in the literature, and (iv) the effective majority-carrier density N*(N,T) due to the effects of Fermi–Dirac statistics, being developed in Eq. (B.7) of Appendix B.

Lien, Igne s, and Ho Van Hoa, for their continuous encouragements during the realization of this work.

Appendix A. An accurate screening model 1=3

The Fermi wave number is defined by kF  1=ðars geff ax Þ. The two asymptotic forms for ratio of the inverse screening length ks to kF, Rs  ks/kF, can be obtained as follows: (i) In the Thomas–Fermi (TF) approximation, valid only in the high carrier-density limit (HCDL, rs-0), one has a well-known result: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4=3 ðA:1Þ RsðTFÞ  ksðTFÞ =kF ¼ 4ars geff =p; pffiffiffiffiffiffiffiffiffiffi 1=3 where ksðTFÞ ¼ 2geff =ðax pars Þ. (ii) In the Wigner–Seitz (WS) approximation, valid only in the low carrier-density case (LCDC, rs b1) Ichimaru [26] obtained the expression for the ratio of WS-wave number ks(WS) to kF for the d½r 2 E ðr Þ one band, Iðrs Þ  23p a s drcs s , where Ec(rs) is the correlation energy of an electron gas given in the paramagnetic state, one thus gets for geff bands: 1=3

RsðWSÞ  ksðWSÞ =kF ¼ Iðrs geff Þ:

Here, we use an accurate simple form for Ec(rs), obtained by Van Cong [27], as Ec ðrs Þ ¼ EcðLDLÞ ðrs Þ þ

Therefore, our expressions for BGN, BGW and ABGN, and a conjunction between BGN and ABGN (or OBG and EBG) have been established, computed and compared with existing data and other theories, giving rise to a satisfactory description of both electricaland-optical data in those materials. Moreover, it should be noted that (a) our results (25, 32, 33) for the BGN are found to be the central ones in this work, (b) in particular, in the n(p)-type HD Si at room temperature, our result (34) for the ABGN obtained in the N-range: 4  1018 rN (cm  3)r4  1020 has been approximated to a very simple form (35) for the ABGN, being expressed in terms of KSG’s result valid only at Nr1020 cm  3 [1], and taking into account the effects of Fermi–Dirac statistics, as seen in Figs. 1 and 2, and a conjunction between BGN and ABGN (or OBG and EBG) has been reported in Figs. 4 and 5, and (c) using a same assumption taken by Eberhardt and Kasper (EK) [36]: the ABGN given in the p-type HD base of Si1  xGex HBT at 295 K and for x r0.3 is the same like in the p-type HD Si given by KSG’s model [1], and from our results (34, 35) for the ABGN given in the HD Si, we have obtained the results (43, 44) for the ABGN and EBG in the p-type HD base of Si1  xGex HBT, which can be compared with the respective EK-result [36], being also reported in Eq. (42), giving rise to a perfect agreement between the numerical results of (42–44) and respective EK-data [36], as seen in Tables 5 and 6 and Fig. 6.

Acknowledgements The author is grateful to the reviewer of the original manuscript for his (her) helpful remarks, which have greatly improved the physics and presentation of the present work. He also thanks l’Inge´nieur Ge´ne´ral de l’Armement Yann Pivet, Profs. Sylvain Brunet, Olivier Henri-Rousseau, Jean-Louis De´jardin, and Yu.P. Kalmykov, and Drs. Kim-Cuong, Axel, Cam-Tu, Cam-Van, Anh-

ðA:2Þ

EcðHDLÞ EcðLDLÞ ½1 þ 0:0359045435rs1:6814966787 1:005

;

ðA:3Þ

reducing in the HCDL to a correct result given in the random phase approximation, Ec(HDL)(rs)=(2/p2)[1ln(2)] ln(rs)0.093288, and in the LCDC to another correct result: Ec ðrs Þ ¼

0:87553 : 0:0326 þ rs

ðA:4Þ

In particular, in the low carrier-density LCDL (rs-N), taking into account Eq. (A.4), result (A.2) is reduced to a constant: RsðWSÞ ðrs -1Þ ¼ 0:87553a þð3=2pÞ ¼ 0:9337:

ðA:5Þ

Basing on those two correct asymptotic results, we now propose an accurate form for Rs  ks/kF as Rs 

1=3 ks br g ¼ RsðWSÞ þ ½RsðTFÞ RsðWSÞ e s eff o 1; kF

ðA:6Þ

where the empirical parameter b may be chosen as b 40.5, so that this ratio is reduced to two above correct asymptotic results as given in Eqs. (A.1) and (A.2) in the HCDL and LCDC, respectively. In the present work b is chosen to be bSi = 1 and bGe =10, giving rise to a satisfactory description of both electrical and optical data in HD Si(Ge). Moreover, our result (A.6) means that for any rs the screening length k1 s is larger than the averaged distance between (or the average number of particles involved in the carriers k1 F is found to be higher than unit), that defines sphere of radius k1 s the screening condition.

Appendix B. Effective majority-carrier density N*(N,T) Here, we study the change in the Wigner–Seitz radius rs or the majority-carrier density N defined in Eq. (23), which can be due to Fermi–Dirac statistics effects. In an electron gas, for given N and T, the kinetic energy Ko(N,T) is defined by Ko ðN; TÞ  kB T

F3=2 Z ; F1=2 Z

ðB:1Þ

where the Fermi–Dirac integrals Fj(Z) are computed using Eq. (17) 2=3 and the reduced Fermi energy ZnðpÞ ðN; T; gcðvÞ mdcðvÞ Þ determined in

ARTICLE IN PRESS H. Van Cong / Physica B 405 (2010) 1139–1149 2=3

Eq. (19) is now replaced by ZnðpÞ ðN; T; gcðvÞ mx Þ  ZnðpÞ for a same value of exciton mass mx. By taking into account the asymptotic forms for Fj(Z) as those given in Eq. (17), Ko ðN; TÞ thus reduces in the degenerate limit (DL) to a correct result expressed in exciton Rydberg (Ex) units as Ko ðN; T-0Þ=Ex ¼ 3=5a2 r22 , and in the nondegenerate limit (NDL) to another correct result: Ko ðN-0; TÞ ¼ 3kB T=2. Furthermore, the kinetic chemical potential mK ðN; TÞ can therefore be defined by r 4 dðKo =rs3 Þ mK ðN; TÞ  s ; 3 drs

ðB:2Þ

which reduces in the NDL to the same result as given above

mK ðN-0; TÞ ¼

3kB T ¼ Ko ðN-0; TÞ 2

ðB:3Þ

and in the DL to the Fermi energy defined by mKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN; T-0Þ ¼ EF ðN;ffi T-0Þ  a2 rs2 Ex according to rs  p Ex =a2 mK ðN; T-0Þ. Therefore, the Wigner–Seitz radius for any N and T due to Fermi–Dirac statistics effects can also be defined by a similar form given in conduction (valence) bands, respectively, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ex  ; ðB:4Þ ðN; TÞ  rscðvÞ a2 mKcðvÞ ðN; TÞ where mKc(v)(N,T) is determined in Eq. (B.2), noting result (B.4)  reduces in the DL to rscðvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðN; T-0Þ  rs , and in the NDL to  ðN; 0; TÞ ¼ 2Ex =3a2 kB T Moreover, from Eqs. (23) and (B.4), rscðvÞ  ðN; TÞ3 we can also define the if denoting rcðvÞ ðN; TÞ ¼ 3=4p½ax rscðvÞ averaged majority-carrier density for any N and T by

r ðN; TÞ ¼ ðrc þ rv Þ=2; reducing in the DL to N, and in the NDL to  3=2 3 3a2 kB T ro ðTÞ  r ðN-0; TÞ ¼ : 4p 2Ex a2x

ðB:5Þ

ðB:6Þ

As a result, we can now define for any given N and T the effective majority-carrier density, which can be measured by the Hall effect as N ðN; TÞ  r ðN; TÞro ðTÞ þ ni ðTÞ

ðB:7Þ

reducing in the DL to N, according to the ‘‘non-compensated’’ case, and in the NDL to the intrinsic carrier concentration ni(T) determined by Eq. (4), according to the completely ‘‘compensated’’ case. Therefore, N ðN; TÞ o N for N4ni(T).

1149

So, if replacing N by N*(N,T) determined in Eq. (B.7) into Eqs. (19, 23–36), we then obtain the numerical results for Fermi energy and BGN at any N and T.

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