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Abstract
In the scandium-hyperdoped silicon, scandiums tend to form interstitial dimers due to their lowest formation energies. The interstitial dimers of Sc formed in silicon can introduce several intermediate-bands (IBs) in the band gap, which can lead to strong sub-band gap absorption. When the two interstitial Sc atoms get close to each other, the infrared response decreases and shifts to short wavelengths. The absorption wavelength range of the interstitial dimers covers the main solar spectrum and the two primary telecommunications wavelengths, which would make material become a high efficiency IB solar cell and promising silicon-based infrared photodetector.
© 2017 Optical Society of America
1. Introduction
Recently, intermediate-band (IB) photovoltaic materials have attracted extensive attention due to the possibility of overcoming the Schockley-Queisser limit and increasing the efficiency [1–6]. The IB solar cells which were firstly proposed by Luque and Martí could utilize almost the full solar spectrum with one or more IBs properly placed into the band gap of the host semiconductor [7, 8]. The sub-band gap energy photons can be absorbed in an IB material through a two-step process: from valence band (VB) to the IB and from the IB to conduction band (CB) [2–4]. Due to the significant advantages of the IB solar cells, several materials have been proposed as IB materials, such as deep-level-impurity hyperdoped semiconductors and short-period superlattice structures [2–4, 9–13].
Among these IB materials, deep-level impurity hyperdoped crystalline silicon is widely reported in recent years for the clear advantage of silicon: good quality, low cost, and thorough knowledge of it [3]. Due to such advantages, a large amount of theoretical and experimental researches have been devoted at the silicon-based IB materials [3, 6, 10, 13–17]. Previous researches indicated that crystalline silicon hyperdoped with chalcogens (S, Se, Te) can largely absorb the sub-band gap light from 250 nm to 2500 nm [14, 18–20]. The nitrogen-hyperdoped silicon can lead to a strong absorption in the mid-infrared wavelength range [21–23]. Besides these, some transition metals, such as cobalt and titanium, were also introduced into crystalline silicon to expand its absorption range [3, 24, 25]. In view of these facts, a great deal of quantum calculations were reported to explore the mechanisms of the unique properties of the hyperdoped silicon and forecast whether the compound could become a promising IB material [3, 6, 13, 15, 16]. Most of the researches indicate that the deep-level impurity can form a band instead of isolated levels in silicon band gap when the impurity concentration exceeds the so-called Mott limit [3, 6, 13]. Moreover, it is necessary to notice that the stability of the impurity configurations in silicon is also important to the properties of the material [15, 22].
In our previous work, we have studied the optical properties of scandium-hyperdoped silicon preliminarily through first-principles calculations [26]. We have constructed several isolate-Sc doped configurations and found that the four interstitial configurations have the lowest formation energies and can lead to the sub-band gap absorption [26]. Considering the complexity of the implantation in actual experiments and to accurately forecast the properties of the material, it still needs to be further investigated. In this work, we study paired-Sc doped configurations in crystalline silicon by implementing first-principles calculations. Our results indicate that scandiums tend to form dimers in silicon in the equilibrium state. To anticipate the possible properties of the material and examine whether the material could be a appropriate IB solar cell, the electronic band structures, density of states (DOS), and optical properties of these configurations are analyzed in detail.
2. Method
We perform the calculations by density functional theory (DFT) within the generalized gradient approximation (GGA) with the PBE functional [27], as implemented in the CASTEP [28] module of Materials Studio (MS). The plane wave energy cut-off established for the basis set is 400 eV. In the cases of geometry optimization, the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm are performed [29]. The convergence tolerance of energy is 1 × 10−5 eV/atom, and the atomic force, stress, and displacement are minimized to below 0.03 eV/Å, 0.05 GPa, and 0.001 Å, respectively. All of the configurations of the Sc-hyperdoped silicon are calculated in 3 × 3 × 3 supercell of the conventional Si8 cubic cell. We construct several paired-Sc doped configurations with different Sc-Sc distances and compare their band structures and optical properties. The atomic concentration of Sc in the paired configurations is about 0.92 at.%.
3. Results and discussion
As for the nearest Sc-Sc distance case, we consider fifteen possible configurations in our calculations. The configuration with two Sc atoms replacing the two adjacent Si atoms is denoted as ScS-S and shown in Fig. 1(a). One Sc atom sitting at the substitutional position and the second Sc atom sitting at the neighboring bond-center interstitial (BI) position between the substitutional Sc and the Si atoms, denoted as ScS-BI and shown in Fig. 1(b). One Sc atom sitting at the substitutional position and the second Sc atom sitting at the neighboring split <110> interstitial (SI) position, denoted as ScS-SI and shown in Fig. 1(c). One Sc atom sitting at the substitutional position and the second Sc atom sitting at the neighboring hexagon-center interstitial (HI) position, denoted as ScS-HI and shown in Fig. 1(d). One Sc atom sitting at the substitutional position and the second Sc atom sitting at the neighboring tetrahedron-center interstitial (TI) position, denoted as ScS-TI and shown in Fig. 1(e). Figures 1(f)-1(i) show the four configurations with two Sc atoms sitting at same neighboring BI positions (ScBI-BI), SI positions (ScSI-SI), HI positions (ScHI-HI), and TI positions (ScTI-TI), respectively. Figures 1(j)-1(o) shows the six configurations with two Sc atoms sitting at different neighboring position, which are denoted as ScBI-SI, ScBI-HI, ScBI-TI, ScSI-HI, ScSI-TI, ScHI-TI, respectively.
Fig. 1 Fifteen typical configurations of Sc-dimer doped silicon after geometry optimization: (a) ScS-S; (b) ScS-BI; (c) ScS-SI; (d) ScS-HI; (e) ScS-TI; (f) ScBI-BI; (g) ScSI-SI; (h) ScHI-HI; (i) ScTI-TI; (j) ScBI-SI; (k) ScBI-HI; (l) ScBI-TI; (m) ScSI-HI; (n) ScSI-TI; (o) ScHI-TI. Gray and orange balls represent the Si and Sc atoms, respectively.
The formation energies of the Sc-dimers implantation in 3 × 3 × 3 supercell are defined by the equation
where E[ScmSin] is the energy of the compound which contain m Sc atoms and n Si atoms, E[Sin] is the energy of the bulk silicon which contain n atoms, and E[Scm] is the energy of the hcp structure of Sc which contain m atoms. For the ScS-S configuration, the formation energy isThe formation energies of the ScS-BI, ScS-SI, ScS-HI, and ScS-TI correspond toand the results are −0.92, −1.46, −1.46, and −1.46 eV, respectively. In the case of ScBI-BI, ScSI-SI, ScHI-HI, ScTI-TI, ScBI-SI, ScBI-HI, ScBI-TI, ScSI-HI, ScSI-TI, and ScHI-TI configurations, the formation energies is calculated byand the results are −1.69, −0.46, −1.68, −1.71, −0.47, −1.71, −0.47, −1.71, −1.71, and −0.47 eV, respectively.According to these calculational results, the ScS-S, ScSI-SI, ScBI-SI, ScBI-TI, and ScHI-TI configurations show the highest formation energies, which are higher than that of ScS-BI configuration by about 0.46 eV and higher than that of ScS-SI, ScS-HI and ScS-TI configurations by 1 eV. In view of such high formation energies of the five dimers, we predict that they are considerably less stable than the other doped dimers and can be neglected in the equilibrium state. The ScS-BI configuration shows the second-highest formation energy among these dimers. While for the interstitial-dimer configurations of ScBI-BI, ScHI-HI, ScTI-TI, ScBI-HI, ScSI-HI, and ScSI-TI, their formation energies are almost the same and apparently small relative to that of other dimers. It means that the six interstitial-dimer configurations would have an overwhelming fraction among these dimers in the equilibrium state. The tiny differences of formation energy among the six configurations also imply their similar proportions in silicon. In addition, the ScS-SI, ScS-HI, and ScS-TI configurations show the second-lowest formation energies, which are higher than that of the six interstitial-dimer configurations by about 0.24 eV. Owing to the complexity of traditional hyperdoping method, e.g., ion implantation followed by pulse laser melting, the impurity atoms are doped in a non-equilibrium process in which dynamics factor may play an considerable role. Therefore, we mainly consider the nine dimer configurations with the lowest and second-lowest formation energies in this work.
In our previous research, the interstitial configurations of Sc show the lowest formation energies in the single implantation. To determine whether the doped-Sc atoms tend to form dimers, we also calculate the formation energies of the paired-Sc structures which have a distance between the two Sc atoms. Figure 2 shows the schematic representation of the ScS-HI configurations with different Sc-Sc distances. The formation energy increases from −1.46 to −0.48 eV when we separate the two Sc atoms by a little distance. When we further increase the Sc-Sc distance, the formation energy of the compound increases to −0.35 eV. The other dimer configurations exhibit the same feature. All of the results indicate that the Sc atoms tend to form dimers when they are doped in silicon lattice, especially in the equilibrium state. Even so, the larger multi-Sc configurations, such as trimers, tetramers, and other bigger clusters, are unlikely to widely exist in the hyper-doped silicon either in equilibrium state or in non-equilibrium state, which should be due to the kinetic barriers. Actually, researchers have attempted to find the impurity clusters in the chalcogen-hyperdoped silicon, but they have not detected any clusters so far [30, 31], the specific reason for which still needs further study. Therefore, the larger multi-Sc configurations are not considered in this work.
Fig. 2 ScS-HI structures with different atomic distances of Sc after geometry optimization: (a) the nearest Sc-Sc distance configuration; (b) the slightly separate Sc-Sc configuration; (c) the remotest Sc-Sc distance configuration. Gray and orange balls represent the Si and Sc atoms, respectively.
According to the calculations of the nine dimer configurations, we find that the ScS-SI, ScS-HI, and ScS-TI configurations exhibit similar electronic band structures and optical properties, and the properties of the other six interstitial-dimer configurations are also similar to each other. It should be due to the similar properties of the four interstitial positions of Sc which we have reported previously [26]. Therefore, we choose several representative configurations (ScS-HI, ScBI-BI, and ScHI-HI) to discuss their electronic band structures and optical properties in detail, and the results are shown in Fig. 3(a)-3(c). For comparison, the band structures of the hexagonal interstitial configuration of the single-Sc implantation (ScHI) and the undoped silicon are given in Fig. 3(d) and 3(e). Consistent with our previous calculations, the interstitial configuration of the single-Sc can introduce an impurity band in silicon band gap and the impurity band overlaps with the CB. The same feature also occurs in other single doped interstitial configurations. The atomic concentration of Sc for the single-Sc doped configuration in 3 × 3 × 3 supercell is 0.46 at.%, several times lower than that in 2 × 2 × 2 supercell in our previous report. According to Luque and Martí's reports, an IB material must have an impurity band isolated from the CB and VB [7, 8]. According to the above results, we can determine that the single-Sc implantation cannot introduce an appropriate IB in silicon band gap even at very low concentration. While for the band structure of ScS-HI, there are two separate impurity bands in the band gap and the two bands appear completely isolated from both the VB and CB, and the higher impurity band is near the Fermi energy while the lower one is below the Fermi energy. Therefore, the two impurity bands can be called IBs. For the ScS-HI configuration, the energy difference between the VB maximum (VBM) and the CB minimum (CBM) is 0.53 eV, which is lower than that of undoped silicon (0.64 eV) calculated by DFT method with GGA-PBE as shown in Fig. 3(e). For the dimer-interstitial configurations, such as ScBI-BI and ScHI-HI, the band structures are similar and there are three impurity bands in the band gap. Among these impurity bands, the higher two overlap with each other and close to the Fermi energy and the lower one is independent from the other bands. The energy difference between the VBM and the CBM is 0.59 eV for both of the dimer-interstitial configurations.
Fig. 3 Electronic band structures of the five configurations of ScS-HI (a), ScBI-BI (b), ScHI-HI (c), ScHI (d), undoped silicon (e), and the dielectric function (imaginary part) of these configurations (f) in 3 × 3 × 3 supercell.
It is known that the band structures are closely connected with the optical properties. Therefore, the formation of the IBs should generate some new optical properties for crystalline silicon. We investigate the optical properties of these configurations by calculating their dielectric functions and the results are shown in Fig. 3(f). Consistent with our previous calculational results of the single-Sc implantation in 2 × 2 × 2 supercell [26], the ScHI structure shows a low frequency peak at around 0.5 eV. While for the dimer-Sc implantation, the low frequency peak shifts to shorter wavelength, e.g., the ScS-HI configuration shows a peak at around 0.8 eV, the ScBI-BI and ScHI-HI configurations show a peak at around 0.7 eV. By comparison, it can be observed that the ScBI-BI and ScHI-HI configurations exhibit higher sub-band gap absorption compared with ScS-HI configuration. However, it cannot be directly compared to that of the ScHI configuration due to the different dopant concentration between the dimer- and single-configurations.
To investigate the changing processes of the band structures and optical properties of these dominant configurations when transforming the material from non-equilibrium to near-equilibrium state, we calculate the band structures and dielectric functions of the paired configurations which have a distance between the two Sc atoms. Figure 4(a) and 4(b) show the band structures of the ScS-HI and ScHI-HI configurations with the remotest Sc-Sc distance which are denoted as ScS-HIR and ScHI-HIR. As expected, the impurity band formed by ScHI-HIR configuration overlaps with the CB, which is consistent with that of single-doped configuration of ScHI. However, the band structure of ScHI-HIR is quite different from that of the nearest configuration of ScHI-HI as shown in Fig. 3(a). While for the band structures of ScS-HIR configuration, there is an isolated IB appears in the band gap and close to the Fermi energy. Through the comparison with Fig. 3(a), it can be found that the IB splits into two separate one when the two remotest Sc atoms form a dimer. Figure 4(c) and (d) show the dielectric functions of ScS-HI and ScHI-HI configurations with different Sc-Sc distances. For the ScS-HI configuration, the optical property shows no remarkable change when changing the relative position of the two Sc atoms from the remotest distance to the nearest one. While the optical property of the ScHI-HI configuration is greatly impacted by the Sc-Sc distance. The low frequency peak appears at around 0.22 eV and is especially strong for the ScHI-HIR configuration. When the two Sc atoms get close to each other, the sub-band gap absorption of the compound is reduced and it shows an obvious blue-shift. Combined with the calculational results of the formation energy, the infrared absorption wavelength of the material would have blue-shift when transforming the material from non-equilibrium to near-equilibrium state. As we know, the solar spectrum is mainly distributed at the visible and near-infrared wavelength range, and the light intensity decreases sharply with wavelength in the near-infrared. Therefore, the blue-shift of the sub-band gap absorption may be beneficial to the light-converting efficiency for the IB solar cells, even though the absorption intensity is reduced. In addition, the infrared response range of the ScHI-HI-dimer configuration covers the two primary telecommunications wavelengths, 1330 and 1550 nm [32], which would make material become a promising silicon-based infrared photodetector.
Fig. 4 Electronic band structures of the configurations of ScS-HI (a) and ScHI-HI (b) with the remotest Sc-Sc distance, and the dielectric functions (imaginary part) of ScS-HI (c) and ScHI-HI (d) with different Sc-Sc distances.
To explore the origin of the impurity bands formed in silicon band gap, we calculate the total and partial density of states (DOS) of the ScS-HI and ScHI-HI configurations in 3 × 3 × 3 supercell, and the results are shown in Fig. 5. From the total DOS of the two configurations, we can see clearly that the ScS-HI configuration forms two distinct isolated impurity bands in the band gap, while the ScHI-HI configuration forms two overlapping bands and one isolate band in the band gap. All of the impurity bands formed by the two configurations are isolated from both CB and VB. In addition, the Fermi energies of the two dimer configurations are all located near the higher impurity band. All of the DOS results exhibit highly consistent with the band structural results that are shown in Fig. 3. From the partial DOS of the two dimer models we can understand the compositions of the impurity bands easily. For the ScS-HI configuration, it is clearly shown that the higher impurity band mainly consist of the hybridization of the d states of Sc atoms with the p states of Si atoms, while the lower impurity band is mainly contributed by the p states of Si atoms only. While for the ScHI-HI configuration, the three impurity bands mainly originate from the contribution of the d states of Sc atoms and the p states of Si atoms. In addition, for the two configurations, the s and p states of Sc atoms and the s states of Si atoms also contribute to the impurity bands, but they only account for a very small proportion of the total DOS.
Fig. 5 Total and partial density of states (DOS) of the dimer configurations of ScS-HI (left) and ScHI-HI (right) in 3 × 3 × 3 supercell.
Our calculations and other reports all indicate that the band gap of silicon calculated by DFT-GGA methods (0.64 eV) is underestimated compared to the experimental value (1.12 eV) [3, 6]. Although the estimation of band gap would not affect some of the conclusions, e.g. which configurations can introduce an IB in the band gap and can lead to the sub-band gap absorption, it still need a more accurate calculations to make a correction. In this work, we employ the hybrid functional method proposed by Heyd et al. (HSE06) [33, 34] to calculate the electronic band structure of ScS-HI configuration in 2 × 2 × 2 supercell, and compare the results with that of the GGA-PBE calculation in the same supercell. From Fig. 6, the energy difference between the VBM and the CBM is 1.38 eV calculated by HSE06, about twice the value calculated by PBE (0.68 eV). The larger energy difference calculated by HSE06 is mainly due to the dramatic decrease of VBM relative to the Fermi energy, which is accordance with the previous report [16]. The impurity bandwidths calculated by HSE06 are 0.43 and 0.41 eV, which are also larger than that calculated by PBE (0.36 and 0.34 eV). In addition, the relative position between the Fermi energy and the impurity bands calculated by HSE06 is different from that calculated by PBE. For the HSE06 calculation, the lower impurity band is almost isolated from the higher one and the Fermi energy passes between the two impurity bands. While for the PBE calculation, the two impurity bands overlap each other and above the Fermi energy. As expected, the higher impurity band overlaps the CB for both of the two calculational methods, which is due to the excessive impurity concentration (3.1 at.%) in the 2 × 2 × 2 supercell. Based on the above comparisons of HSE06 and PBE methods calculating for the 2 × 2 × 2 supercell, we could speculate the more actual features of the band structures in the 3 × 3 × 3 supercell. For the band structures in Fig. 3 and Fig. 4, the VBM should be lowered and the impurity bandwidths should be increased in the HSE06 calculations. In addition, the HSE06 calculations would separate the impurity bands to a larger distance. All of the conclusions need to be verified by the further theoretical and experimental studies.
Fig. 6 Electronic band structures of the configurations of ScS-HI in 2 × 2 × 2 supercell calculated by HSE06 (up) and GGA-PBE (down) methods.
4. Conclusion
In summary, the calculational results of the formation energy indicate that the dimer-Sc doped configurations are more stable than the single-Sc implanted ones and would have an overwhelming fraction in the hyperdoped crystalline silicon in equilibrium state. Among these dimer-Sc doped structures, the interstitial dimers, such as ScBI-BI, ScHI-HI, ScTI-TI, ScBI-HI, ScSI-HI, and ScSI-TI, show the lowest formation energy and can form several IBs in the band gap. For the ScHI-HI configuration, the sub-band gap absorption decreases and shifts to short wavelengths (blue-shift) when the two Sc atoms get close to each other. The blue-shift of the sub-band gap absorption would be beneficial to the silicon-based IB solar cells and infrared photodetectors.
Funding
PhD early development program of Henan Normal University (qd14207); Key Research Program of Henan Province Office of Education (15A140007); Basic and Cutting-edge Technology Research Program of Henan Province (142300410162).
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