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Density functional theory study of stable configurations of substitutional and interstitial C and Sn atoms in Si and Ge crystals
Journal of Crystal Growth 463 (2017) 110–115
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crys
Density functional theory study of stable configurations of substitutional and interstitial C and Sn atoms in Si and Ge crystals Hiroki Koyama ⇑, Koji Sueoka Department of Communication Engineering, Okayama Prefectural University, 111 Kuboki, Soja, Okayama 719–1197, Japan
a r t i c l e
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Article history: Received 24 December 2016 Received in revised form 28 January 2017 Accepted 30 January 2017 Available online 2 February 2017 Communicated by T.F. Kuech Keywords: Semiconductor materials Solar cells Density functional theory Formation energy Atomic configuration
a b s t r a c t Group IV semiconductor compounds, e.g., Si and Ge containing substitutional C (Cs) and/or Sn (Sns) atoms (mono-doping and co-doping) with contents of several % are attracting attention for application to solar cells because they are good for the environment and have an affinity with Si materials. In this study, we evaluate the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial C (Ci) atoms by means of density functional theory calculations. The Hakoniwa method proposed by Kamiyama et al. (2016) is applied to a 64-atom supercell to obtain the thermal equilibrium ratio of Ci to the total C atoms. The results of the analysis are fourfold. First, the isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Second, a Ci atom is formed in a [0 0 1] oriented Ci-Cs pair in Ge crystals with the ratio of 7.7% to total C atoms at 450 °C when the concentration of uniformly distributed C atoms is about 3%. Third, the difference of the formation energy of Ci and Cs in Si decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%. Fourth, the co-doping of C and Sn suppresses the formation of Ci atoms in Si and Ge crystals. The results obtained here are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for solar cell application. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Multi-crystalline Si crystals are relatively inexpensive, so currently they are the main crystals used in solar cells. However, to achieve higher engineering efficiency in solar cells, the sensitivity zones of wavelengths in sunlight need to be expanded. One alternative candidate is group III–V semiconductors with multijunction solar cells composed of films with different band gaps [1]. However, some III–V elements negatively affect the environment and lack affinity with Si materials. Group IV semiconductor compounds, e.g., Si and Ge containing C and/or Sn atoms (monodoping and co-doping) with contents of several %, are attracting attention for their potential to control band structures and band gaps [2,3]. One of the technological problems with this is how to control the atomic configuration, i.e., the substitutional and interstitial sites of doped C and/or Sn atoms in the host Si and Ge crystals [4–13]. The substitutional C and Sn atoms help control the band gaps, but interstitial C and Sn atoms form complexes with other impurities and/or intrinsic point defects, which results in the formation of carrier traps in the band gap. First principles anal⇑ Corresponding author. E-mail address: [email protected] (H. Koyama). http://dx.doi.org/10.1016/j.jcrysgro.2017.01.054 0022-0248/Ó 2017 Elsevier B.V. All rights reserved.
ysis based on density functional theory (DFT) have been performed to obtain the basic data on atomic configurations, such as bond strain and stable sites of substitutional C and Sn atoms in bulk Si or Ge crystals [5–8,10,13]. In this study, we examined the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial Ci atoms by means of DFT calculations. The Hakoniwa method proposed by Kamiyama et al. [14] was applied to a 64-atom supercell to obtain the ratio of Ci to total C atoms. A clear difference between Si and Ge was found. We considered not only the mono-doping but also the co-doping of C and Sn. The phase diagrams of Si-C or Si-Sn allow very little solubility (fractions of percent maximum) of the dopant and rather phase separate into Si and SiC or Si and Sn. One possibility of enhancing the solubilities of C and Sn is to take advantage of non-equilibrium and surface effects in thin films [6,15]. Therefore, most of the structures considered in the present work are probably metastable. However, the results we obtained are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for solar cell application. Previous experimental and theoretical works [16–18] have considered the effects of co-doping Sn and C in Si crystals in terms of the interactions of the dopants with vacancies, and suppression of
H. Koyama, K. Sueoka / Journal of Crystal Growth 463 (2017) 110–115
A-center defect complexes. Such defect complexes should be the scope in a more comprehensive study in the future. 2. Calculation details 2.1. DFT calculation for Ef of C and Sn atoms in Si (Ge) crystals DFT calculations were performed within the generalized gradient approximation (GGA) for electron exchange and correlation using the CASTEP code [19]. The wave functions were expanded with plane waves, and the ultra-soft pseudo-potential method [20] was used to reduce the number of these waves. The cut-off energy was 340 eV. The expression proposed by Perdew et al. [21] was used for the exchange-correlation energy in the GGA. A density mixing method [22] and the Broyden-Fletcher-GoldfarbShanno (BFGS) geometry optimization method [23] were used to optimize the electronic structure and atomic configurations, respectively. A three-dimensional periodic boundary condition was used with cubic 64-atom supercells for Si and Ge crystals. We carried out k-point sampling at 2 2 2 and 4 4 4 special points of the Monkhorst–Pack grid [24] for the Si and Ge 64-atom cells, respectively. For mono-doping of C or Sn atoms in Si, formation energies (Ef) of substitutional m Cs atoms or n Sns atoms in the Si 64-atom cell are calculated with Eqs. (1a) and (1b) as
Ef ðCs;m Þ ¼ Etot ðSi64-m Cs;m Þ ½Etot ðSi64 Þ-m=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ;
111
one Cs and one Ci in the 64-atom cell including several C and Sn atoms. A ‘‘gap problem” in DFT calculations [25,26] occurs when an incomplete description of the electron self-interaction by the exchange-correlation potential causes the calculated band gap based on GGA to be less than half that of the experimental value [27,28]. A number of methods have been proposed to overcome this problem. GW approximation [29], which is based on manybody perturbation theory, can calculate accurate defect levels [30], but the computational resources required for even a 64atom supercell are too high. More practical methods, such as the scissors operator scheme, the marker method [31], the LDA + U method [32], the use of adapted pseudopotentials [33], and the application of ad hoc extrapolation schemes [34], have been proposed. More recently, hybrid DFT has become popular as an accurate method of calculation [35,36]. Compared with general DFT calculations, these methods and functionals improve the band gap with higher levels of accuracy [37,38]. However, this does not guarantee high levels of accuracy for the calculated defect levels, such as the formation energy of impurities. Chroneos et al. [39,40] found that conventional DFT calculations (e.g., GGA-PBE with ultrasoft pseudopotentials), used in conjunction with aggressive trials designed to improve the theoretical approach, yielded reliable results for some neutral impurities and complexes in Ge crystal. In this work, we use GGA-PBE with ultrasoft pseudopotentials so as to generate useful systematic data on the stable sites of C and Sn atoms in Si and Ge crystals.
ð1aÞ 2.2. Ratio of each configuration of two C atoms
Ef ðSns;n Þ ¼ Etot ðSi64-n Sns;n Þ ½Etot ðSi64 Þ n=64Etot ðSi64 Þ þ n=64Etot ðSn64 Þ;
ð1bÞ
where Etot(Si64-mCs,m) and Etot(Si64-nSns,n) are the total energies of a Si 64-atom supercell including m Cs atoms and n Sns atoms, respectively. Etot(Si64), Etot(C64), and Etot(Sn64) are the total energies of a Si 64-atom supercell, a C 64-atom supercell, and a Sn 64-atom supercell, respectively. We also considered the case of one interstitial Ci included in the total m C atoms. Formation energies (Ef) of one Ci and substitutional (m1) Cs atoms in the Si 64-atom cell are calculated with Eq. (2) as
Ef ðCi;1 Cs;m1 Þ ¼ Etot ðSi64-mþ1 Ci;1 Cs;m1 Þ ½fEtot ðSi64 Þ ðm 1Þ=64Etot ðSi64 Þg þ m=64Etot ðC64 Þ;
ð2Þ
where Etot(Si64-m+1Ci,1Cs,m1) is the total energy of a Si 64-atom supercell including one Ci atom and (m1) Cs atoms. For co-doping of C and Sn atoms in Si, we considered Cs and Sns atoms and/or one Ci atom. The Ef of substitutional m Cs atoms and n Sns atoms in a Si 64-atom cell are calculated with Eq. (3) as
Ef ðCs;m Sns;n Þ ¼ Etot ðSi64-mn Cs;m Sns;n Þ ½Etot ðSi64 Þ ð3Þ
where Etot(Si64-mnCs,mSns,n) is the total energy of a Si 64-atom supercell including m Cs atoms and n Sns atoms. The Ef of one interstitial Ci atom and (m1) Cs atoms and n Sns atoms in Si is calculated with Eq. (4) as
Ef ðCi;1 Cs;m1 Sns;n Þ ¼ Etot ðSi64-mþ1n Ci;1 Cs;m1 Sns;n Þ ½Etot ðSi64 Þ ðm þ n 1Þ=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ þ n=64Etot ðSn64 Þ;
Pi ¼
Z¼
ðm þ nÞ=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ þ n=64Etot ðSn64 Þ;
As discussed later in Section 3.1, the Ef of the isolated Ci is rather high compared to that of the isolated Cs. Thus, for two C atoms in the 64-atom supercell, we neglected the formation of two Ci atoms and calculated the ratio of each atomic configuration of two C atoms ((Cs and Cs) and (Cs and Ci)) in Si and Ge 64-atom supercells by using the Hakoniwa method [14]. This is the case for the uniform distribution of C atoms at 3.12% in Si and Ge crystals. There are 35 configurations of two C atoms (9 of (Cs and Cs) and 26 of (Cs and Ci)) in the 64-atom cell. Unfortunately, no reliable data are available for the formation entropy of each atomic configuration of (Cs and Cs), (Cs and Ci). Therefore, it is assumed as first order approximation that the formation entropy is constant in the analysis that follows. The ratio Pi of i-th atomic configuration (i = 1–9 for Ci-th at 1st–9th from the other Cs, and i = 10–35 for s st th Ci-th at 1 –26 from Cs) can be calculated with Eq. (5): i
" # Eif 1 i w exp z kB T
X i
"
Eif exp kB T
ð5Þ
# ð6Þ
where wi and Eif are the coordination number and formation energy of the i-th configuration, respectively, Z is the partition function, kB is the Boltzmann’s constant, and T is the absolute temperature. In the present work, we used T = 723 K (450 °C), which is the typical temperature of Si film growth, and T = 300 K (27 °C). 3. Results and discussion
ð4Þ
where Etot(Si64-m+1nCi,1Cs,m1Sns,n) is the total energy of a Si 64-atom supercell including one Ci and (m1) Cs atoms and n Sns atoms. Similar calculations were performed for a Ge 64-atom supercell. By using the calculated Ef, we obtained the formation energy of
3.1. Ef of isolated atom in Si and Ge crystals For interstitial Ci and Sni atoms, we considered [0 0 1] dumbbell (D)-site, [1 1 0] D-site, tetrahedral (T)-site, and hexagonal (H)-site. The calculated results showed that Ci (Sni) is the most stable at
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Table 1 Calculated Ef of isolated Cs, Ci ([0 0 1] D-site), Sns, and Sni (T-site) atom in Si and Ge crystals. Matrix
Ef (eV) Cs
Ci
Sns
Sni
Si Ge
1.45 2.27
3.56 4.17
0.72 0.28
5.40 3.88
Table 2 and Ci-th around Cs atom in Si and Ge 64-atom cell. Coordination number of each Ci-th s i In the case of Ci-th , the atomic configuration between Ci-th and Cs differs in Si and Ge, i i as the structure of Ci at [0 0 1] D-site is affected by Cs atom. A [0 0 1] oriented Ci-Cs pair is the Ci2nd in both Si and Ge. i-th
Cs
Ci in Si
Ci in Ge
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
4 12 12 3 12 12 4 3 1
24 6 48 24 24 24 24 48 12 48 48 48 24 24 24 24 24 48 24 48 24 48 24 2 12 12
24 6 24 48 24 24 12 24 48 48 48 24 48 24 24 24 24 24 48 48 24 24 48 2 12 12
Fig. 1. Calculation model of 64-atom cubic cell including classified atomic configurations of Ci-th according to distance from other Cs atom. s
[0 0 1] D-site (T-site). Table 1 summarizes the calculated Ef of isolated Cs, Ci ([0 0 1] D-site), Sns, and Sni (T-site) atoms in Si and Ge crystals. The isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Due to the very large Ef difference between Sns and Sni, we consider only Sns atoms hereafter. 3.2. Eif and Pi of two interacting atoms in Si and Ge 64-atom supercell i-th i-th 3.2.1. Eif of Cs-Ci-th s , Cs-Sns , and Sns-Sns Fig. 1 shows the calculation model, namely, the 64-atom cubic cell including the classified atomic configurations of Ci-th according s to the distance from the other Cs atom. In the 64-atom model, there are 1st, 2nd, . . ., and 9th Cs sites from the other Cs atom. Table 2 summarizes the coordination number of each Ci-th in the s 64-atom Si (Ge) supercell. For Cs-Sni-th and Sns-Sni-th s s , the classifications are the same as Cs-Ci-th s . To avoid any adverse effect from the cell constraint, some of the pairs are moved to the central region of the 64-atom model. i-th Fig. 2(a), (b), and (c) shows the Eif of Cs-Ci-th s , Cs-Sns , i-th and Sns-Sns at each atomic configuration in the Si (Ge) 64-atom cell, respectively. The Ef of Cs-Ci-th and Sns-Sni-th is large in the s s zigzag-bond (1st, 2nd, 5th, and 8th) including both Cs (Sns) atoms, while the Ef of Cs-Sni-th is the lowest at the nearest neighbor (1st) s site. The stress superposition of Cs-Cs (Sns-Sns) in the zigzagbond and the stress compensation of Cs-Sns at the 1st site are what changes their formation energies [13]. Comparing Si and Ge, the Ef of Cs-Cs (Sns-Sns) in Si is smaller (larger) than that in Ge. The Ef of Cs-Sns in Si is slightly smaller than that in Ge. These results are not unexpected as the order of the atomic size is C < Si < Ge < Sn.
3.2.2. Eif and Pi of Cs-Ci-th i For Ci at D-site around the Cs atom, there are 26 independent atomic configurations in the 64-atom supercell after geometry
i-th i-th Fig. 2. Eif of (a) Cs-Ci-th at each atomic configuration in s , (b) Cs-Sns , and (c) Sns-Sns Si (Ge) 64-atom cell.
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113
Fig. 3. Atomic configuration of most stable [0 0 1] oriented Ci-Cs pair in Ge.
optimization. Fig. 3 shows the atomic configuration of the most stable [0 0 1] oriented Ci-Cs pair in Ge [10]. Table 2 also summarizes the coordination number of Ci-th around the Cs atom. i In the case of Ci-th , the atomic configuration between Ci-th and Cs i i differs in Si and Ge because the structure of Ci at [0 0 1] D-site is affected by the Cs atom. A [0 0 1] oriented Ci-Cs pair is the C2nd in i both Si and Ge crystal. Fig. 4(a) and (b) shows the Ei-th of the Ci-th of Cs-Ci-th at each f i i atomic configuration in the Si and Ge 64-atom cell, respectively. In these figures, the Ei-th of the Ci-th of Cs-Ci-th are also shown. For f s s
Fig. 5. Thermal equilibrium ratio Pi of Cs-Ci-th and Cs-Ci-th in (a) Si and (b) Ge 64s i atom cell.
Si crystals, the Ef of the most stable Cs atom (3rd site from the other Cs atom) is about 1 eV smaller than the Ef of the Ci of the most stable [0 0 1] oriented Ci-Cs pair (2nd site from Cs atom). In contrast
Fig. 4. Ei-th of Ci-th of Cs-Ci-th at each atomic configuration in (a) Si and (b) Ge 64-atom cell. Ei-th of Ci-th of Cs-Ci-th is also shown. f i i f s s
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Fig. 6. 64-atom cubic cell with tensile strain in a zigzag-bond vertical to [0 0 1] oriented Ci-Cs pair (zigzag-bond 1) and a compressive strain in a zigzag bond along the other directions (zigzag-bond 2).
Fig. 5(a) and (b) shows the ratio Pi of Cs-Ci-th and Cs-Ci-th to the s i total C atoms in the Si and Ge 64-atom cell, respectively. In the Si crystal, the thermal equilibrium ratio of Ci to the total C is almost zero. In contrast, in the Ge crystal, the Pi of the [0 0 1] oriented Ci-Cs pair is about 7.7% at T = 450 °C and 1.7% at T = 27 °C. Here, we briefly discuss the previous experimental results with the present calculations. In the experiments on Ge thin film growth with C doping, about 50–80% of the C atoms stay at interstitial sites with a total C concentration of 1–3% [9–12]. One of the reasons for the underestimation of Ci concentration in the present work is the assumption of uniform distribution (i.e., that two C atoms exist in each Si 64-atom cell) in the calculation. The calculated results showed that there is a tensile strain in a zigzag-bond vertical to the [0 0 1] oriented Ci-Cs pair (zigzag-bond 1 in Fig. 6) and a compressive strain in a zigzag-bond along the other directions (zigzag-bond 2 in Fig. 6). This result indicates that once the [0 0 1] oriented Ci-Cs pair in Fig. 6 forms, the Ef of the other Ci-Cs pair will be decreased in both zigzag-bonds 1 and 2 by arranging the orientation of the Ci-Cs pair. The other reason for the underestimation is the formation of multiple Ci atoms at one Cs atom. The Ef of one Ci of the [0 0 1] oriented triplet Ci-Cs-Ci is more stable than that of the single Cs atom [10]. 3.3. Ci formation in Si crystal
Fig. 7. Four most stable Cs atoms in Si 64-atom cell. The Cs atoms stay at 3rd neighboring site to each other and all of the zigzag-bonds include only one Cs atom.
to Si, the Ef of the most stable Cs atom (3rd site from the other Cs atom) and the Ci of the [0 0 1] oriented Ci-Cs pair (2nd site from Cs atom) is almost the same in a Ge crystal. This result indicates that the interstitial Ci will form rather easily when the concentration of uniformly distributed C atoms in Ge is higher than about 3%; the concentration of C atoms in Ge is 3.12% when the Ge 64-atom cell includes two C atoms.
As mentioned in previous sections, the formation of Ci is rather difficult in Si crystal with a C concentration up to 3%. Here, we discuss the dependence of stable Cs sites on the Cs concentration in Si. Our calculations for Cs-Cs showed that the Ef of Cs-Cs is increased in the zigzag-bond including two Cs atoms. Therefore, Cs atoms favor the sites out of the zigzag-bonds that include the other Cs atoms. Fig. 7 shows the four most stable Cs atoms distributed uniformly (C concentration of 6.24%) in the Si 64-atom cell. The Cs atoms stay at the 3rd neighboring site to each other, and interestingly, all of the zigzag-bonds include only one Cs atom. That is, the fifth Cs atom is forced to be included in the zigzag-bond that is already occupied by the other Cs atoms. Fig. 8(a) and (b) shows the Ef of the last Cs and Ci atoms of the m Cs atoms in the Si 64-atom cell. Here, the Cs atoms occupy the 3rd neighboring sites to each other up to m = 4. The Ef of the last Cs atom in Fig. 8(a) decreased until Cs concentration at 6.2% (4 Cs atoms and 60 Si atoms) and increased at 7.8% (5 Cs atoms and 59 Si atoms), as estimated above. The Ef of the Ci atom also decreased with increasing Cs concentration (as shown in Fig. 8(b)), probably
Fig. 8. Ef of last (a) Cs and (b) Ci atoms of m Cs atoms in Si 64-atom cell, and (c) difference of Ef of Cs and Ci in (a) and (b).
H. Koyama, K. Sueoka / Journal of Crystal Growth 463 (2017) 110–115 Table 3 Ef of Ci atom at Cs-Sns pair in Si and Ge crystals compared with Ef of Ci of the [0 0 1] oriented Ci-Cs pair. Matrix
Ef (eV) Ci at Cs-Sns
Ci of Ci-Cs
Si Ge
2.49 2.27
2.11 2.04
115
decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%. Fourth, the co-doping of C and Sn suppresses the formation of Ci atoms in Si and Ge crystals. The results obtained here are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for application to solar cells. Acknowledgements
due to the strain compensation between the tensile strain of the Cs atom and the compressive strain of the [0 0 1] oriented Ci-Cs pair. We found that the difference of Ef between Cs and Ci in Fig. 8 (a) and (b) decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%, as shown in Fig. 8(c). The thermal equilibrium ratio of Ci to total C in Si was not evaluated due to the enormous number of atomic configurations with m P 4. Even so, the data in Fig. 8 is very useful in terms of clarifying the mechanism behind Ci formation in Si crystals. 3.4. Effect of C and Sn co-doping on the suppression of Ci formation In Table 3, the Ef of the Ci atom at the Cs-Sns pair in Si and Ge crystals is compared with the Ef of the Ci of the [0 0 1] oriented Ci-Cs pair. We found that, in the Si (Ge) crystal, the Ef of the Ci atom at the Cs-Sns pair is about 0.38 eV (0.23 eV) larger than that of the [0 0 1] oriented Ci-Cs pair. That is, the Ci formation is suppressed by co-doping the C and Sn atoms. This is probably due to the reduction of tensile strain around the Cs atom with the co-doped Sn atom. 4. Conclusion Group IV semiconductor compounds, e.g., Si and Ge containing substitutional C (Cs) and/or Sn (Sns) atoms (mono-doping and codoping) with contents of several %, are attracting attention for application to solar cells because they are good for the environment and have an affinity with Si materials. One of the technological problems is how to control the atomic configuration, namely, the substitutional and interstitial sites of doped C and/or Sn atoms in the host Si and Ge crystals. The substitutional C and Sn atoms help control the band gaps but interstitial C and Sn atoms form complexes with other impurities and/or intrinsic point defects. This results in the formation of carrier traps in the band gap. In this study, we evaluated the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial C (Ci) atoms by using DFT calculations. The Hakoniwa method was applied to a 64-atom supercell to obtain the thermal equilibrium ratio of Ci to the total C atoms. The results of our analysis are fourfold. First, the isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Second, a Ci atom is formed in a [0 0 1] oriented Ci-Cs pair in Ge crystals with the ratio of 7.7% to total C atoms at 450 °C when the uniformly distributed C concentration is about 3%. Third, the difference of formation energy of Ci and Cs in Si
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crys
Density functional theory study of stable configurations of substitutional and interstitial C and Sn atoms in Si and Ge crystals Hiroki Koyama ⇑, Koji Sueoka Department of Communication Engineering, Okayama Prefectural University, 111 Kuboki, Soja, Okayama 719–1197, Japan
a r t i c l e
i n f o
Article history: Received 24 December 2016 Received in revised form 28 January 2017 Accepted 30 January 2017 Available online 2 February 2017 Communicated by T.F. Kuech Keywords: Semiconductor materials Solar cells Density functional theory Formation energy Atomic configuration
a b s t r a c t Group IV semiconductor compounds, e.g., Si and Ge containing substitutional C (Cs) and/or Sn (Sns) atoms (mono-doping and co-doping) with contents of several % are attracting attention for application to solar cells because they are good for the environment and have an affinity with Si materials. In this study, we evaluate the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial C (Ci) atoms by means of density functional theory calculations. The Hakoniwa method proposed by Kamiyama et al. (2016) is applied to a 64-atom supercell to obtain the thermal equilibrium ratio of Ci to the total C atoms. The results of the analysis are fourfold. First, the isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Second, a Ci atom is formed in a [0 0 1] oriented Ci-Cs pair in Ge crystals with the ratio of 7.7% to total C atoms at 450 °C when the concentration of uniformly distributed C atoms is about 3%. Third, the difference of the formation energy of Ci and Cs in Si decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%. Fourth, the co-doping of C and Sn suppresses the formation of Ci atoms in Si and Ge crystals. The results obtained here are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for solar cell application. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Multi-crystalline Si crystals are relatively inexpensive, so currently they are the main crystals used in solar cells. However, to achieve higher engineering efficiency in solar cells, the sensitivity zones of wavelengths in sunlight need to be expanded. One alternative candidate is group III–V semiconductors with multijunction solar cells composed of films with different band gaps [1]. However, some III–V elements negatively affect the environment and lack affinity with Si materials. Group IV semiconductor compounds, e.g., Si and Ge containing C and/or Sn atoms (monodoping and co-doping) with contents of several %, are attracting attention for their potential to control band structures and band gaps [2,3]. One of the technological problems with this is how to control the atomic configuration, i.e., the substitutional and interstitial sites of doped C and/or Sn atoms in the host Si and Ge crystals [4–13]. The substitutional C and Sn atoms help control the band gaps, but interstitial C and Sn atoms form complexes with other impurities and/or intrinsic point defects, which results in the formation of carrier traps in the band gap. First principles anal⇑ Corresponding author. E-mail address: [email protected] (H. Koyama). http://dx.doi.org/10.1016/j.jcrysgro.2017.01.054 0022-0248/Ó 2017 Elsevier B.V. All rights reserved.
ysis based on density functional theory (DFT) have been performed to obtain the basic data on atomic configurations, such as bond strain and stable sites of substitutional C and Sn atoms in bulk Si or Ge crystals [5–8,10,13]. In this study, we examined the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial Ci atoms by means of DFT calculations. The Hakoniwa method proposed by Kamiyama et al. [14] was applied to a 64-atom supercell to obtain the ratio of Ci to total C atoms. A clear difference between Si and Ge was found. We considered not only the mono-doping but also the co-doping of C and Sn. The phase diagrams of Si-C or Si-Sn allow very little solubility (fractions of percent maximum) of the dopant and rather phase separate into Si and SiC or Si and Sn. One possibility of enhancing the solubilities of C and Sn is to take advantage of non-equilibrium and surface effects in thin films [6,15]. Therefore, most of the structures considered in the present work are probably metastable. However, the results we obtained are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for solar cell application. Previous experimental and theoretical works [16–18] have considered the effects of co-doping Sn and C in Si crystals in terms of the interactions of the dopants with vacancies, and suppression of
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A-center defect complexes. Such defect complexes should be the scope in a more comprehensive study in the future. 2. Calculation details 2.1. DFT calculation for Ef of C and Sn atoms in Si (Ge) crystals DFT calculations were performed within the generalized gradient approximation (GGA) for electron exchange and correlation using the CASTEP code [19]. The wave functions were expanded with plane waves, and the ultra-soft pseudo-potential method [20] was used to reduce the number of these waves. The cut-off energy was 340 eV. The expression proposed by Perdew et al. [21] was used for the exchange-correlation energy in the GGA. A density mixing method [22] and the Broyden-Fletcher-GoldfarbShanno (BFGS) geometry optimization method [23] were used to optimize the electronic structure and atomic configurations, respectively. A three-dimensional periodic boundary condition was used with cubic 64-atom supercells for Si and Ge crystals. We carried out k-point sampling at 2 2 2 and 4 4 4 special points of the Monkhorst–Pack grid [24] for the Si and Ge 64-atom cells, respectively. For mono-doping of C or Sn atoms in Si, formation energies (Ef) of substitutional m Cs atoms or n Sns atoms in the Si 64-atom cell are calculated with Eqs. (1a) and (1b) as
Ef ðCs;m Þ ¼ Etot ðSi64-m Cs;m Þ ½Etot ðSi64 Þ-m=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ;
111
one Cs and one Ci in the 64-atom cell including several C and Sn atoms. A ‘‘gap problem” in DFT calculations [25,26] occurs when an incomplete description of the electron self-interaction by the exchange-correlation potential causes the calculated band gap based on GGA to be less than half that of the experimental value [27,28]. A number of methods have been proposed to overcome this problem. GW approximation [29], which is based on manybody perturbation theory, can calculate accurate defect levels [30], but the computational resources required for even a 64atom supercell are too high. More practical methods, such as the scissors operator scheme, the marker method [31], the LDA + U method [32], the use of adapted pseudopotentials [33], and the application of ad hoc extrapolation schemes [34], have been proposed. More recently, hybrid DFT has become popular as an accurate method of calculation [35,36]. Compared with general DFT calculations, these methods and functionals improve the band gap with higher levels of accuracy [37,38]. However, this does not guarantee high levels of accuracy for the calculated defect levels, such as the formation energy of impurities. Chroneos et al. [39,40] found that conventional DFT calculations (e.g., GGA-PBE with ultrasoft pseudopotentials), used in conjunction with aggressive trials designed to improve the theoretical approach, yielded reliable results for some neutral impurities and complexes in Ge crystal. In this work, we use GGA-PBE with ultrasoft pseudopotentials so as to generate useful systematic data on the stable sites of C and Sn atoms in Si and Ge crystals.
ð1aÞ 2.2. Ratio of each configuration of two C atoms
Ef ðSns;n Þ ¼ Etot ðSi64-n Sns;n Þ ½Etot ðSi64 Þ n=64Etot ðSi64 Þ þ n=64Etot ðSn64 Þ;
ð1bÞ
where Etot(Si64-mCs,m) and Etot(Si64-nSns,n) are the total energies of a Si 64-atom supercell including m Cs atoms and n Sns atoms, respectively. Etot(Si64), Etot(C64), and Etot(Sn64) are the total energies of a Si 64-atom supercell, a C 64-atom supercell, and a Sn 64-atom supercell, respectively. We also considered the case of one interstitial Ci included in the total m C atoms. Formation energies (Ef) of one Ci and substitutional (m1) Cs atoms in the Si 64-atom cell are calculated with Eq. (2) as
Ef ðCi;1 Cs;m1 Þ ¼ Etot ðSi64-mþ1 Ci;1 Cs;m1 Þ ½fEtot ðSi64 Þ ðm 1Þ=64Etot ðSi64 Þg þ m=64Etot ðC64 Þ;
ð2Þ
where Etot(Si64-m+1Ci,1Cs,m1) is the total energy of a Si 64-atom supercell including one Ci atom and (m1) Cs atoms. For co-doping of C and Sn atoms in Si, we considered Cs and Sns atoms and/or one Ci atom. The Ef of substitutional m Cs atoms and n Sns atoms in a Si 64-atom cell are calculated with Eq. (3) as
Ef ðCs;m Sns;n Þ ¼ Etot ðSi64-mn Cs;m Sns;n Þ ½Etot ðSi64 Þ ð3Þ
where Etot(Si64-mnCs,mSns,n) is the total energy of a Si 64-atom supercell including m Cs atoms and n Sns atoms. The Ef of one interstitial Ci atom and (m1) Cs atoms and n Sns atoms in Si is calculated with Eq. (4) as
Ef ðCi;1 Cs;m1 Sns;n Þ ¼ Etot ðSi64-mþ1n Ci;1 Cs;m1 Sns;n Þ ½Etot ðSi64 Þ ðm þ n 1Þ=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ þ n=64Etot ðSn64 Þ;
Pi ¼
Z¼
ðm þ nÞ=64Etot ðSi64 Þ þ m=64Etot ðC64 Þ þ n=64Etot ðSn64 Þ;
As discussed later in Section 3.1, the Ef of the isolated Ci is rather high compared to that of the isolated Cs. Thus, for two C atoms in the 64-atom supercell, we neglected the formation of two Ci atoms and calculated the ratio of each atomic configuration of two C atoms ((Cs and Cs) and (Cs and Ci)) in Si and Ge 64-atom supercells by using the Hakoniwa method [14]. This is the case for the uniform distribution of C atoms at 3.12% in Si and Ge crystals. There are 35 configurations of two C atoms (9 of (Cs and Cs) and 26 of (Cs and Ci)) in the 64-atom cell. Unfortunately, no reliable data are available for the formation entropy of each atomic configuration of (Cs and Cs), (Cs and Ci). Therefore, it is assumed as first order approximation that the formation entropy is constant in the analysis that follows. The ratio Pi of i-th atomic configuration (i = 1–9 for Ci-th at 1st–9th from the other Cs, and i = 10–35 for s st th Ci-th at 1 –26 from Cs) can be calculated with Eq. (5): i
" # Eif 1 i w exp z kB T
X i
"
Eif exp kB T
ð5Þ
# ð6Þ
where wi and Eif are the coordination number and formation energy of the i-th configuration, respectively, Z is the partition function, kB is the Boltzmann’s constant, and T is the absolute temperature. In the present work, we used T = 723 K (450 °C), which is the typical temperature of Si film growth, and T = 300 K (27 °C). 3. Results and discussion
ð4Þ
where Etot(Si64-m+1nCi,1Cs,m1Sns,n) is the total energy of a Si 64-atom supercell including one Ci and (m1) Cs atoms and n Sns atoms. Similar calculations were performed for a Ge 64-atom supercell. By using the calculated Ef, we obtained the formation energy of
3.1. Ef of isolated atom in Si and Ge crystals For interstitial Ci and Sni atoms, we considered [0 0 1] dumbbell (D)-site, [1 1 0] D-site, tetrahedral (T)-site, and hexagonal (H)-site. The calculated results showed that Ci (Sni) is the most stable at
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Table 1 Calculated Ef of isolated Cs, Ci ([0 0 1] D-site), Sns, and Sni (T-site) atom in Si and Ge crystals. Matrix
Ef (eV) Cs
Ci
Sns
Sni
Si Ge
1.45 2.27
3.56 4.17
0.72 0.28
5.40 3.88
Table 2 and Ci-th around Cs atom in Si and Ge 64-atom cell. Coordination number of each Ci-th s i In the case of Ci-th , the atomic configuration between Ci-th and Cs differs in Si and Ge, i i as the structure of Ci at [0 0 1] D-site is affected by Cs atom. A [0 0 1] oriented Ci-Cs pair is the Ci2nd in both Si and Ge. i-th
Cs
Ci in Si
Ci in Ge
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
4 12 12 3 12 12 4 3 1
24 6 48 24 24 24 24 48 12 48 48 48 24 24 24 24 24 48 24 48 24 48 24 2 12 12
24 6 24 48 24 24 12 24 48 48 48 24 48 24 24 24 24 24 48 48 24 24 48 2 12 12
Fig. 1. Calculation model of 64-atom cubic cell including classified atomic configurations of Ci-th according to distance from other Cs atom. s
[0 0 1] D-site (T-site). Table 1 summarizes the calculated Ef of isolated Cs, Ci ([0 0 1] D-site), Sns, and Sni (T-site) atoms in Si and Ge crystals. The isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Due to the very large Ef difference between Sns and Sni, we consider only Sns atoms hereafter. 3.2. Eif and Pi of two interacting atoms in Si and Ge 64-atom supercell i-th i-th 3.2.1. Eif of Cs-Ci-th s , Cs-Sns , and Sns-Sns Fig. 1 shows the calculation model, namely, the 64-atom cubic cell including the classified atomic configurations of Ci-th according s to the distance from the other Cs atom. In the 64-atom model, there are 1st, 2nd, . . ., and 9th Cs sites from the other Cs atom. Table 2 summarizes the coordination number of each Ci-th in the s 64-atom Si (Ge) supercell. For Cs-Sni-th and Sns-Sni-th s s , the classifications are the same as Cs-Ci-th s . To avoid any adverse effect from the cell constraint, some of the pairs are moved to the central region of the 64-atom model. i-th Fig. 2(a), (b), and (c) shows the Eif of Cs-Ci-th s , Cs-Sns , i-th and Sns-Sns at each atomic configuration in the Si (Ge) 64-atom cell, respectively. The Ef of Cs-Ci-th and Sns-Sni-th is large in the s s zigzag-bond (1st, 2nd, 5th, and 8th) including both Cs (Sns) atoms, while the Ef of Cs-Sni-th is the lowest at the nearest neighbor (1st) s site. The stress superposition of Cs-Cs (Sns-Sns) in the zigzagbond and the stress compensation of Cs-Sns at the 1st site are what changes their formation energies [13]. Comparing Si and Ge, the Ef of Cs-Cs (Sns-Sns) in Si is smaller (larger) than that in Ge. The Ef of Cs-Sns in Si is slightly smaller than that in Ge. These results are not unexpected as the order of the atomic size is C < Si < Ge < Sn.
3.2.2. Eif and Pi of Cs-Ci-th i For Ci at D-site around the Cs atom, there are 26 independent atomic configurations in the 64-atom supercell after geometry
i-th i-th Fig. 2. Eif of (a) Cs-Ci-th at each atomic configuration in s , (b) Cs-Sns , and (c) Sns-Sns Si (Ge) 64-atom cell.
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113
Fig. 3. Atomic configuration of most stable [0 0 1] oriented Ci-Cs pair in Ge.
optimization. Fig. 3 shows the atomic configuration of the most stable [0 0 1] oriented Ci-Cs pair in Ge [10]. Table 2 also summarizes the coordination number of Ci-th around the Cs atom. i In the case of Ci-th , the atomic configuration between Ci-th and Cs i i differs in Si and Ge because the structure of Ci at [0 0 1] D-site is affected by the Cs atom. A [0 0 1] oriented Ci-Cs pair is the C2nd in i both Si and Ge crystal. Fig. 4(a) and (b) shows the Ei-th of the Ci-th of Cs-Ci-th at each f i i atomic configuration in the Si and Ge 64-atom cell, respectively. In these figures, the Ei-th of the Ci-th of Cs-Ci-th are also shown. For f s s
Fig. 5. Thermal equilibrium ratio Pi of Cs-Ci-th and Cs-Ci-th in (a) Si and (b) Ge 64s i atom cell.
Si crystals, the Ef of the most stable Cs atom (3rd site from the other Cs atom) is about 1 eV smaller than the Ef of the Ci of the most stable [0 0 1] oriented Ci-Cs pair (2nd site from Cs atom). In contrast
Fig. 4. Ei-th of Ci-th of Cs-Ci-th at each atomic configuration in (a) Si and (b) Ge 64-atom cell. Ei-th of Ci-th of Cs-Ci-th is also shown. f i i f s s
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Fig. 6. 64-atom cubic cell with tensile strain in a zigzag-bond vertical to [0 0 1] oriented Ci-Cs pair (zigzag-bond 1) and a compressive strain in a zigzag bond along the other directions (zigzag-bond 2).
Fig. 5(a) and (b) shows the ratio Pi of Cs-Ci-th and Cs-Ci-th to the s i total C atoms in the Si and Ge 64-atom cell, respectively. In the Si crystal, the thermal equilibrium ratio of Ci to the total C is almost zero. In contrast, in the Ge crystal, the Pi of the [0 0 1] oriented Ci-Cs pair is about 7.7% at T = 450 °C and 1.7% at T = 27 °C. Here, we briefly discuss the previous experimental results with the present calculations. In the experiments on Ge thin film growth with C doping, about 50–80% of the C atoms stay at interstitial sites with a total C concentration of 1–3% [9–12]. One of the reasons for the underestimation of Ci concentration in the present work is the assumption of uniform distribution (i.e., that two C atoms exist in each Si 64-atom cell) in the calculation. The calculated results showed that there is a tensile strain in a zigzag-bond vertical to the [0 0 1] oriented Ci-Cs pair (zigzag-bond 1 in Fig. 6) and a compressive strain in a zigzag-bond along the other directions (zigzag-bond 2 in Fig. 6). This result indicates that once the [0 0 1] oriented Ci-Cs pair in Fig. 6 forms, the Ef of the other Ci-Cs pair will be decreased in both zigzag-bonds 1 and 2 by arranging the orientation of the Ci-Cs pair. The other reason for the underestimation is the formation of multiple Ci atoms at one Cs atom. The Ef of one Ci of the [0 0 1] oriented triplet Ci-Cs-Ci is more stable than that of the single Cs atom [10]. 3.3. Ci formation in Si crystal
Fig. 7. Four most stable Cs atoms in Si 64-atom cell. The Cs atoms stay at 3rd neighboring site to each other and all of the zigzag-bonds include only one Cs atom.
to Si, the Ef of the most stable Cs atom (3rd site from the other Cs atom) and the Ci of the [0 0 1] oriented Ci-Cs pair (2nd site from Cs atom) is almost the same in a Ge crystal. This result indicates that the interstitial Ci will form rather easily when the concentration of uniformly distributed C atoms in Ge is higher than about 3%; the concentration of C atoms in Ge is 3.12% when the Ge 64-atom cell includes two C atoms.
As mentioned in previous sections, the formation of Ci is rather difficult in Si crystal with a C concentration up to 3%. Here, we discuss the dependence of stable Cs sites on the Cs concentration in Si. Our calculations for Cs-Cs showed that the Ef of Cs-Cs is increased in the zigzag-bond including two Cs atoms. Therefore, Cs atoms favor the sites out of the zigzag-bonds that include the other Cs atoms. Fig. 7 shows the four most stable Cs atoms distributed uniformly (C concentration of 6.24%) in the Si 64-atom cell. The Cs atoms stay at the 3rd neighboring site to each other, and interestingly, all of the zigzag-bonds include only one Cs atom. That is, the fifth Cs atom is forced to be included in the zigzag-bond that is already occupied by the other Cs atoms. Fig. 8(a) and (b) shows the Ef of the last Cs and Ci atoms of the m Cs atoms in the Si 64-atom cell. Here, the Cs atoms occupy the 3rd neighboring sites to each other up to m = 4. The Ef of the last Cs atom in Fig. 8(a) decreased until Cs concentration at 6.2% (4 Cs atoms and 60 Si atoms) and increased at 7.8% (5 Cs atoms and 59 Si atoms), as estimated above. The Ef of the Ci atom also decreased with increasing Cs concentration (as shown in Fig. 8(b)), probably
Fig. 8. Ef of last (a) Cs and (b) Ci atoms of m Cs atoms in Si 64-atom cell, and (c) difference of Ef of Cs and Ci in (a) and (b).
H. Koyama, K. Sueoka / Journal of Crystal Growth 463 (2017) 110–115 Table 3 Ef of Ci atom at Cs-Sns pair in Si and Ge crystals compared with Ef of Ci of the [0 0 1] oriented Ci-Cs pair. Matrix
Ef (eV) Ci at Cs-Sns
Ci of Ci-Cs
Si Ge
2.49 2.27
2.11 2.04
115
decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%. Fourth, the co-doping of C and Sn suppresses the formation of Ci atoms in Si and Ge crystals. The results obtained here are useful for the prediction of possible atomic configurations of C and/or Sn in Si and Ge for application to solar cells. Acknowledgements
due to the strain compensation between the tensile strain of the Cs atom and the compressive strain of the [0 0 1] oriented Ci-Cs pair. We found that the difference of Ef between Cs and Ci in Fig. 8 (a) and (b) decreases to about 0.3 eV with an increase in the concentration of uniformly distributed C atoms up to 6%, as shown in Fig. 8(c). The thermal equilibrium ratio of Ci to total C in Si was not evaluated due to the enormous number of atomic configurations with m P 4. Even so, the data in Fig. 8 is very useful in terms of clarifying the mechanism behind Ci formation in Si crystals. 3.4. Effect of C and Sn co-doping on the suppression of Ci formation In Table 3, the Ef of the Ci atom at the Cs-Sns pair in Si and Ge crystals is compared with the Ef of the Ci of the [0 0 1] oriented Ci-Cs pair. We found that, in the Si (Ge) crystal, the Ef of the Ci atom at the Cs-Sns pair is about 0.38 eV (0.23 eV) larger than that of the [0 0 1] oriented Ci-Cs pair. That is, the Ci formation is suppressed by co-doping the C and Sn atoms. This is probably due to the reduction of tensile strain around the Cs atom with the co-doped Sn atom. 4. Conclusion Group IV semiconductor compounds, e.g., Si and Ge containing substitutional C (Cs) and/or Sn (Sns) atoms (mono-doping and codoping) with contents of several %, are attracting attention for application to solar cells because they are good for the environment and have an affinity with Si materials. One of the technological problems is how to control the atomic configuration, namely, the substitutional and interstitial sites of doped C and/or Sn atoms in the host Si and Ge crystals. The substitutional C and Sn atoms help control the band gaps but interstitial C and Sn atoms form complexes with other impurities and/or intrinsic point defects. This results in the formation of carrier traps in the band gap. In this study, we evaluated the stable configurations of C and/or Sn atoms in Si (Ge) crystals with a focus on the formation of interstitial C (Ci) atoms by using DFT calculations. The Hakoniwa method was applied to a 64-atom supercell to obtain the thermal equilibrium ratio of Ci to the total C atoms. The results of our analysis are fourfold. First, the isolated Cs atom is stabler than the isolated Ci atom in both Si and Ge crystals, and it is stabler in Si than in Ge. The isolated Sns atom is much stabler that Sni as well, but it is stabler in Ge than Si. Second, a Ci atom is formed in a [0 0 1] oriented Ci-Cs pair in Ge crystals with the ratio of 7.7% to total C atoms at 450 °C when the uniformly distributed C concentration is about 3%. Third, the difference of formation energy of Ci and Cs in Si
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