1. Introduction

As the size of transistors continues to shrink, there is an urgent need to replace traditional electrical interconnects with Si-based optical interconnects that have the advantages of fast transmission speed, high bandwidth, and multiplexability to boost computer performances^[1-8]. Growing III–V semiconductors on silicon is a promising method for realizing Si-based light sources because III–V materials have excellent luminescence properties due to their direct bandgap^{[1, 6-8]}. Currently, room temperature excitation of a Si-based III–V laser has been achieved by Liu et al.^[9] and Bowers et al.^[10]. To further improve the luminescence efficiency, we can focus on band offset, which largely determines the interfacial charge transfer and quantum confinement. If the carriers in the heterojunction are confined to the III–V semiconductors, it is beneficial to give full play to their advantages in luminescence.

However, unlike isovalent superlattices and alloys^[11], the Si and III–V semiconductors belong to different group semiconductors, and have different valence electrons (heterovalent). Therefore, the electric field at the interfaces could exist when they touch each other^[12], which could significantly affect the band offset. On the other hand, there is large lattice mismatch between Si and some III–V semiconductors and inevitably a large strain at the interfaces. These reasons make calculating the band offset of Si and III–V semiconductors difficult and challenging, and sometimes lead to the inconsistencies between experimental observations and theoretical predictions^[13-16]. For example, the excellent luminescent performance of InAs makes it have great potential to become a candidate material for future optoelectronic applications. However, the band offset type of Si/InAs has always been controversial^[13-16]. Experimentally, Mano et al.^[13] believed that the band offset of Si/InAs is the type-I, while Heitz et al.^[14] found that it belongs to the type-II. Theoretically, Tersoff et al.^[15] believed that the band offset of Si/InAs is the type-I, while Bru-Chevallier et al.^[16] reported that it belongs to the type-II. The reason for this controversy is probably due to these two kinds of semiconductors having a large lattice mismatch and heterovalence-induced electric field, which should be particularly and reasonably considered for the band offset predictions for the large-lattice-mismatched and heterovalent Si and III–V semiconductors.

Over the years, some models have been proposed and try to overcome the above problems, mainly by adding deformation potential to remove the influence of strain on energy levels. For example, Van de Walle et al.^[11] modified the calculation of the band offset by including the effect of strain on the band edge state. This method focuses on iso-valent and isostructural cubic semiconductors, so caution for this method should be taken in the calculations of heterovalent materials. Li et al.^[17] calculated the deformation value of the valence band between experimental crystal and average crystal through absolute deformation potential. However, since the absolute deformation potential is related to volume, this method is only useful in certain cases with small lattice mismatch. Lang et al.^[18] proposed a three-step method for band offset calculation for lattice-mismatched and isovalent systems. By considering the interfacial reconstructions for eliminating the electric field in the superlattices, Deng et al. calculated band alignment of lattice-matched and heterovalent II–VI/III–V semiconductors^[12]. However, so far the direct calculation of the band offset for heterovalent Si/III–V systems with large lattice mismatch is lacking.

In this paper, we present a modified, and reliable method for calculating the band offset of large lattice mismatch and heterovalent semiconductors, and simultaneously apply this to the calculations of band offsets of Si/III–V systems. We found that our proposed approach has the same accuracy as the widely used modified core level method for systems with small lattice mismatch. Moreover, the calculated results are much closer to the experimental values of the large-lattice-mismatched and heterovalent systems.

2. The method of band offset calculation

The calculations were performed using density functional theory^{[19, 20]} with the local density approximation (LDA)^[21] for the exchange-correlation functionals as implemented in the Vienna ab-initio simulation package (VASP)^[22]. The cutoff energy for the plane-wave basis was 450 eV, and dense k-point sampling was used in each case to ensure fully converged results. All atoms were relaxed until the Hellman–Feynman forces on individual atoms were less than 0.01 eV/Å. All calculations were performed at the equilibrium (experimental) lattice constants^[23].

The valence band offset is usually calculated as follows^[24]. If the two semiconductors are represented by AX and BY respectively, the valence band offset is expressed as:

Here $\Delta E_{\rm{v,C}}^{\rm{AX}} = E_{\rm{v}}^{\rm{AX}} - E_{\rm{C}}^{\rm{AX}}$ represents the energy difference between the valence band maximum (VBM) and the core energy level of semiconductor AX at the equilibriumlattice, the as same for $\Delta E_{\rm{v,C}}^{\rm{BY}}$. $\Delta E_{\rm{C,C}}^{\rm{AX/BY}} = \Delta E_{\rm{C}}^{\rm{BY}} - \Delta E_{\rm{C}}^{\rm{AX}}$ is the energy difference between the core levels of BY and AX in a relaxed heterojunction with the same reference level. The above method is reasonable only for band offset calculations of homovalent semiconductors with a small lattice mismatch^[17]. We have improved the method to accommodate large-lattice-mismatched heterostructures. The specific steps for calculating the band offset of semiconductor AX and BY are given below.

(i) Bulk semiconductors of AX and BY. This step is to perform separate bulk calculations for two semiconductors AX and BY at their equilibrium lattice constant. The different $\Delta E_{\rm{v,C}}^{\rm{AX}}$ and $\Delta E_{\rm{v,C}}^{\rm{BY}}$ between ${\varGamma }_{8\rm{v}}$ VBM state and core energy level in the two bulk materials is obtained by energy band calculations. In our calculations, we use the average of the cation and anion 1s core levels^[17].

(ii) Alignment of core energy levels in heterojunctions. The purpose of this step is to align the core energy levels in isolated systems of AX and BY, and which is achieved through a five-step calculation, as proposed by the Lang et al.^[18]. It should be noted in advance that we use ($a_1$, $a_2$, $a_3$), (b₁, b₂, b₃) to denote the lattice constants of AX and BY, respectively, and (c₁, c₂, c₃) to denote the average of the lattice constants of both, ${{c}}_{{1}}={(}{{a}}_{{1}}+{{b}}_{{1}}{)}/{2}$, ${{c}}_{{2}}={(}{{a}}_{{2}}+{{b}}_{{2}}{)}/{2}$ and ${{c}}_{{3}}={(}{{a}}_{{3}}+{{b}}_{{3}}{)}/{2}$. AX, AX' and AX'' denote AX semiconductors at the experimental lattice, unidirectional strain lattice and bidirectional strain lattice, respectively, and the same for BY. Fig. 1 illustrates the five calculation steps using the [100] direction as an example. In the first step, a unidirectional strained semiconductor AX' with lattice constants ($a_1$, c₂, $a_3$) is constructed and connected with AX along the (010) plane (lattice-matched interface) to form an (AX)n(AX')n superlattice. In the second step, a bidirectional strained semiconductor AX'' with lattice constants ($a_1$, c₂, c₃) is constructed and connected with AX' along the (001) plane to form an (AX')n(AX'')n superlattice. In the third step, a bidirectional strained semiconductor BY'' with lattice constants (b₁, c₂, c₃) is constructed and connected with AX'' along the (100) plane to form an (AX'')n(BY'')n superlattice. In the fourth step, a unidirectional strained semiconductor BY' with lattice constants (b₁, c₂, b₃) is constructed and connected with BY'' along the (001) plane to form a (BY'')n(BY')n superlattice. In the fifth step, BY' is connected with BY along the (010) plane to form a (BY')n(BY)n superlattice. For all of these steps, the lattice constants in both directions forming the interface are lattice matched. It should be pointed out that the direction of unidirectional strain is not unique; it can be either of the two directions of the bidirectional strain. The calculation method for the heterojunction along [110] and [111] directions is the same. Finally, the core energy levels in the middle of the two sides of the superlattice^{[17, 24]} (similar to the bulk material) are calculated to obtain the difference in the core energy levels of the two semiconductors at the equilibrium crystal^[18].

Figure  1.  (Color online) Five steps for correcting the effect of lattice mismatch on valence band offset calculations using core energy levels as a reference.

DownLoad: Full-Size Img PowerPoint

where $\Delta E_{\rm{C,C}}^{\rm{AX/AX'}} = E_{\rm{C}}^{\rm{AX'}} - E_{\rm C}^{\rm{AX}}$ represents the difference between the core energy levels of the equilibrium and unidirectional strain structures in AX semiconductors, and definitions of $\Delta E_{\rm{C,C}}^{\rm{AX'/AX''}}$, $\Delta E_{\rm{C,C}}^{\rm{AX''/BY''}}$, $\Delta E_{\rm{C,C}}^{\rm{BY''/BY'}}$ and $\Delta E_{\rm{C,C}}^{\rm{BY'/BY}}$are similar.

(iii) Reconstructed polarized interface. The purpose of this step is to reconstruct the interface to eliminate the internal electric field caused by the heterovalence in the polarized interface structures. When Si and III–V semiconductors form the heterojunction along the direction of [100] or [111], because they do not belong to the same group and the valence electrons are different, there is often a built-in electric field in the system^{[12, 25]}, which causes the energy levels of the system shifting, as shown in Fig. 3(a). Once the energy levels are shifted by the electric field, the resulting core energy levels will be meaningless. According to our previous research^[12], the anion or cation atoms of the interface are mixed to form the mixed structures that are able to effectively elimante the internal electric field. In such structures, the excess charge carriers compensate each other at the interface, and there is no built-in electric field in the heterojunction. After calculating the difference between the core energy levels of the mixed cation and mixed anion structures, the valence band offset of these two bidirectional strain systems is obtained according to the Eqs. (1) and (2), and the average of the two is taken as the band offset in the requested direction. The superlattice formed along the [110] direction is not treated because it is nonpolar.

Figure  2.  (Color online) (a–e) show the five superlattice structures constructed. (a) Equilibrium-unidirectional superlattice of Si. (b) Unidirectional-bidirectional strain superlattice of Si. (c) Bidirectional strain superlattice of Si and InAs. (d) Bidirectional-unidirectional superlattice of InAs. (e) Unidirectional-equilibrium superlattice of InAs.

DownLoad: Full-Size Img PowerPoint

3. Applications to large-lattice-mismatched and heterovalent Si and III–V semiconductors

The valence band offsets of Si and III–V semiconductors with zincblende structure along [100], [110], and [111] directions have been calculated, respectively. We choose the Si/InAs superlattice as an example, which has a lattice mismatch of 11.60%. Fig. 2 shows the five constructed superlattice structures of Si and InAs along the [100] direction. They are the equilibrium-unidirectional strain superlattice of Si, the unidirectional-bidirectional strain superlattice of Si, the bidirectional strain superlattice of Si and InAs, the bidirectional-unidirectional strain superlattice of InAs, and the unidirectional strain-equilibrium superlattice of InAs, respectively.

Fig. 3 shows the elimination of the interface potential between Si and InAs. The abrupt interface refers to the interface directly connected by Si and InAs, and the mixed anion interface refers to the structure in which anions are mixed in the interface, and the mixed cation interface is the same. Fig. 3 clearly shows that the electrostatic potential at the abrupt interface is oblique, indicating the presence of a built-in electric field. When the interface atoms are mixed, the electrostatic potential becomes flat, indicating that the built-in electric field is eliminated successfully.

Figure  3.  (Color online) The abrupt interface, mixed cation interface and mixed anion interface formed by Si and InAs along the [100] direction, and the corresponding potential distribution along the growth orientation of the superlattices.

DownLoad: Full-Size Img PowerPoint

The calculated valence band offsets are shown in Table 1. First of all for the system with small lattice mismatch, we compared with previous theoretical calculations without considering strain and built-in electric field as well as with the experimental measurement and found almost no difference. For example, for Si/AlP systems with a lattice mismatch of 0.55%, the valence band offset in the [100] direction is calculated to be –0.88 eV in this paper, in agreement with the previous theoretical value of –0.91 eV^[26]. For the Si/GaP system with a lattice mismatch of 0.37%, the valence band offset in the [100] direction is –0.28 eV, which also agrees with the previous theoretical value of –0.27 eV^[26]. In addition, the value of –0.24 ± 0.12 eV^[27] was measured experimentally for the valence band offset of the Si/GaP heterojunction, whose confidence interval contains the results of this study. This is due to the fact that our method is mainly used to revise the band offset of the large-mismatched system, and the core energy level of the small-mismatched system is less affected, so that the band offsets are comparable to the experimental value as well as to the value calculated by other theory.

Table  1.  Valence band offsets (in eV) of Si/III–V semiconductors with lattice mismatch and heterovalence along the orientation of [100], [110], [111], respectively. The lattice constants (in Å) of III–V semiconductors and mismatches (referred to the lattice constant of Si, e.g. 5.43 Å) are also given. The positive value represents that the position of the valence band of the III–V semiconductor is higher than that of Si.

System	Lattice constant	Mismatch (%)	[100]			[110] abrupt	[111]			Exp.\Theo.
			Mixed anion	Mixed cation	Average		Mixed anion	Mixed cation	Average
Si/AlP	5.46	0.55	–0.56	–1.21	–0.88	–0.84	–0.59	–1.22	–0.91	–\–0.91i
Si/AlAs	5.66	4.24	–0.07	–0.70	–0.39	–0.26	0.05	–0.53	–0.24	–\–
Si/AlSb	6.14	13.08	0.31	0.26	0.28	0.52	0.37	0.41	0.39	–\–
Si/GaN	4.53	–16.57	–1.92	–1.99	–1.95	–2.00	–1.52	–1.68	–1.60	–1.90ii\–
Si/GaP	5.45	0.37	0.04	–0.59	–0.28	–0.28	0.02	–0.58	–0.28	–0.24±0.12iii \–0.27i
Si/GaAs	5.65	4.05	0.29	–0.09	0.10	0.28	0.51	0.11	0.31	0.23±0.10iv \0.12v
Si/GaSb	6.10	12.34	0.41	0.26	0.34	0.62	0.36	0.27	0.31	–\–
Si/InP	5.87	8.10	0.10	–0.27	–0.08	0.25	0.43	0.07	0.25	0.12vi\0.36vii
Si/InAs	6.06	11.60	0.29	0.11	0.20	0.42	0.42	0.37	0.40	0.31viii\0.50vii
Si/InSb	6.48	19.34	0.16	0.14	0.15	0.73	0.22	0.24	0.23	0.06ix\0.25vii
i from Ref. [26]; ii from Ref. [28]; iii from Ref. [27]; iv from Ref. [29]; v from Ref. [30]; vi from Ref. [32]; vii from Ref. [31]; Viii from Ref. [13]; ix from Ref. [33].

DownLoad: CSV  | Show Table

For systems with a large lattice mismatch, the results obtained by our method are in good agreement with the experimental measurements. For example, for the Si/GaN system with lattice mismatch up to –16.57%, our calculated values are –1.95, –2.00, and –1.60 eV for the [100], [110], [111] directions, respectively, which are close to the experimental value of –1.90 eV^[28]. For the Si/GaAs system with a lattice mismatch of 4.05%, our calculated value of 0.28 eV along the [110] direction is more consistent with the experimental value of 0.23 ± 0.10 eV^[29] than the previously calculated result of 0.12 eV^[30]. In the Si/InAs system with a lattice mismatch of about 11.60%, our calculated valence band offset along the [100] direction is 0.20 eV, which is closer to the experimental value of 0.31 eV^[13] than the previous theoretical value of 0.50 eV^[31]. In the Si/InP system with a lattice mismatch of about 8.10%, the previous theoretical work^[32] shows a large difference of 0.24 eV compared with experimental value (0.12 eV^[31]) along the [110] direction, while our result indicates this difference is only 0.13 eV, and has a smaller error. In summary, the valence band offsets calculated by the current method are significantly more consistent with the experimental measurements than the previous ones. For the Si/AlAs, Si/AlSb and Si/GaSb systems, since the valence band offsets have not been previously studied, we have performed calculations in the hope of providing some reference for the study of these materials.

Combining the valence band offset with the experimentally measured band gap^[23] gives a clear picture of the carriers in Si/III–V semiconductor. Fig. 4 shows the band offsets of the Si/GaSb, Si/InAs, and Si/InSb systems along [100], [110], and [111] directions. Regarding the controversy about the band offset of the Si/InAs system mentioned in the previous section^[13-16], the Si/InAs heterojunctions exhibit type I band offsets along [100], [110] and [111] directions because the energy minimum of the X-valley in the Si conduction band lies at a higher energy than the minimum of the Γ-valley in the InAs conduction band, as shown in Fig. 4. In such material systems, the recombination between electrons and holes is type I in both momentum and real space, and the photoluminescence efficiency is therefore expected to be high. This is also the situation for the band offsets along [100], [110] and [111] directions for the Si/InSb system and along [100] and [111] directions for the Si/GaSb system. Other band offsets of Si/III–V systems are either of the type I where the carriers are localized in Si, or of the type II where the electrons and holes are localized in Si and III–V semiconductors, respectively. These systems cannot exploit the great advantages of III–V materials in luminescence due to the involvement of phonons required for carrier compounding. Therefore, the Si/InAs and Si/InSb systems connected along three directions, and the Si/GaSb systems connected along [100] and [111] directions are the most promising systems for improving the luminescence efficiency of Si-based III–V semiconductor lasers from the perspective of the energy band structure.

Figure  4.  (Color online) Band offset diagram of the Si/GaSb, Si/InAs and Si/InSb systems with zinc-blende structure along the [100], [110], and [111] directions. The energy is relative to the vacuum level of Si.

DownLoad: Full-Size Img PowerPoint

4. Conclusions

We present a modified method for theoretically calculating the band offset of large-lattice-mismatched and heterovalent semiconductors for the different directions based on the first-principles calculations. Using this method, the effect of interface reconstruction and lattice mismatch on band offsets is corrected by considering the reconstruction of the heterovalent interface and the inclusion of unidirectional strain crystals that play a transitional role between the equilibrium crystal structure and the bidirectional strain structure. Furthermore, we apply the method to calculate the band offsets of heterovalent Si and III–V semiconductors with large lattice mismatches in the [100], [110] and [111] directions. This research will provide some guidance for the selection of suitable light-source materials on the Si substrate from the perspective of energy band structure, and will be beneficial for the understanding and analysis of the properties of heterojunctions between the Si substrate and zincblende structured III–V semiconductors.

Acknowledgments

This work was supported by the National Key Research and Development Program of China (Grant No. 2018YFB2200100), the Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB22), and the National Natural Science Foundation of China (Grant No. 118764347, 11614003, 11804333). H. X. D. was also supported by the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No. 2017154).