Density-Functional Study of the Si/SiO₂ Interfaces in Short-Period Superlattices: Structures and Energies

Abstract

The oxide-semiconductor interface is a key element of MOS transistors, which are widely used in modern electronics. In silicon electronics, SiO₂ is predominantly used. The miniaturization requirement raises a problem regarding the growing of heterostructures with ultrathin oxide layers. Two structural models of interface between crystalline Si and cristobalite SiO₂ are studied by using DFT-based computer modelling. The structures of several Si/SiO₂ superlattices (SL), with layer thicknesses varied within 0.5–2 nm, were optimized and tested for stability. It was found that in both models the silicon lattice conserves its quasi-cubic structure, whereas the oxide lattice is markedly deformed by rotations of the SiO₄ tetrahedra around axes perpendicular to the interface plane. Based on the analysis of the calculated total energy of SLs with different thicknesses of the layers, an assessment of the interface formation energy was obtained. The formation energy is estimated to be approximately 3–5 eV per surface Si atom, which is close to the energies of various defects in silicon. Elastic strains in silicon layers are estimated at 5–10%, and their value rapidly decreases as the layer thickens. The elastic strains in the oxide layer vary widely, in a range of 1–15%, depending on the interface structure.

1. Introduction

The oxide–semiconductor interface is a key element of MOS transistors, which are widely used in modern electronics and optoelectronics [1,2]. A study of the electronic and optical properties, structure, and defects that formed as soon as the structure stability of SiO₂ and Si contained heterostructures, has been performed in a number of theoretical and experimental papers. Among them, there are papers devoted to practically important cases of HfO₂/Si and HfO₂/SiO₂ interfaces [3,4,5,6,7,8] as well as Si/SiO₂ structures [1,2,9,10], which play a decisive role in silicon-based electronics and optoelectronics. The miniaturization requirements raise problems for technologists who wish to grow ultrathin oxide layers, which leads to the necessity of understanding the laws governing the formation of interfaces at the atomistic level. It has been shown [11] that the Si-SiO₂ interface can be made very thin (in few oxide monolayers), smooth (with a minimum of steps), and with a minimum defect density. However, the question of the dependence of the heterostructure’s properties on the thickness of the layers and the quality of the interface remains the subject of extensive research (see [12] and references therein).

Understanding the structure–property relationship is impossible without theoretical studies based on high-precision calculations of the electronic structure. Today, the most reliable methods for modeling the electronic structure of condensed matter are based on calculations within the framework of the density functional theory (DFT). The application of such calculations to the study of the structure and properties of the heterostructures with Si/SiO₂ interfaces has a long history (see [13] and references therein). In most of these studies, superlattices (SLs)—periodic layered heterostructures—were used to model the planar Si-SiO₂ interface [2,14,15,16,17,18,19,20,21,22]. This approach made it possible to use methods developed for 3D periodic structures in modeling 2D interfaces. Note that all previous theoretical studies were devoted to the study of the spatial and electronic structure of the Si/SiO₂ SL.

It is well established that the oxide in a Si/SiO₂ system is amorphous and stoichiometric at distances greater than about 1 nm from the interface [23]. Due to its elasticity, the amorphous material has the ability to dampen the elastic stresses. Crystalline materials do not have such a property. The experimental data provides evidence that the oxide material in the Si/SiO₂ planar heterostructures is stressed within this transition layer and the stress exhibits an exponential decay behavior [24]. The microscopic nature of a transition layer is still unclear. Within this layer the structure changes from perfect order to disorder. It can be suggested that the Si/SiO₂ transition proceeds via an ordered, stable, and bulk phase of SiO₂. As the distance from the interface increases, the crystallinity decreases, and the oxide structure becomes amorphous and unstressed. Therefore, the study of heterostructures with relatively thin and crystalline oxide layers can shed light on the oxide structure in the immediate vicinity of the interface.

Most theoretical studies have assumed that, at least locally, the oxide structure in the vicinity of the of Si/SiO₂ interface is close to any crystalline phase of SiO₂. Theoretical treatments were mostly based on density-functional theory computer modelling [2,14,15,16,17,18,19,20,21,22,25,26,27]. The key question was the choice of an appropriate structural model for the oxide and interface layers. Due to difficulty in dealing with amorphous silica, various crystalline polymorphs were considered [18,19,25,26,27]. No final decision on relative preference of the models was performed. A similar situation occurs in studies of interfaces with other oxides, such as ZrO₂ and HfO₂ [3].

Most often, the Si/SiO₂ interface was modelled as a junction of (001) surfaces of the Si and β-cristobalite crystals (see Figure 1a). Such an approach seems straightforward because both crystals are similar in structural organization. Both are cubic with Si atoms in the same positions. A direct junction of the lattices would provide a heterosystem with a very thin interface layer containing only one Si plane. We denote this interface type as I1.

Possible formation of the I1 interfaces was called into question due to the discrepancy between the lattice dimensions. Indeed, the mismatch of the silicon and β-cristobalite cubic unit cell parameters a_c is more than 30%. At the same time, it was noted that their ratio is close to √2. Therefore, we decided to join the lattices by turning one of them by 45°, as is shown in Figure 1b [16,28]. We shall refer to this interface type as I2. Formation of the I2 interface allows for the avoidance of large strains in the joint lattices but provides a smooth structural arrangement for only half of the interface Si atoms. Another half of the atoms rest with two dangling bonds. Various mechanisms to relax these structural defects have been discussed in a number of papers (see [23] and references therein). A common feature of these works was the conclusion that with any method of overgrowing dangling bonds, the interface becomes blurry, i.e., containing several planes of Si atoms in different oxidation states.

Recall, however, that the continual decrease in the device size places ever higher demands on the dimensions and quality of the nanoscale structures with the silicon/oxide interfaces. Thus, the main goal of the computer modelling is a search of the systems with ultrathin interfaces containing a minimal number of defects. That is why we decided to return to the I1 interfaces and to investigate possible relaxed structures resulting from the geometry optimization procedure. Realizing that such a structure is obviously unstable (due to its strong lattice dimension mismatch) we have decided to explore the relaxation paths induced by different instable phonon modes. By testing different relaxation paths, a stable structure with a minimum energy of formation was found. A similar study was also carried out for SLs with the I2 interfaces.

Several SLs, with interfaces of I1 and I2 types with different layer thicknesses, are considered in this study. For each of them, stable structures were found and the values of the elastic strains in the materials of the layers were calculated. The energies of the formation of the heterostructures from bulk materials were estimated. The stability of the found structures was tested by calculating the phonon spectrum. The phonon spectra of the SLs will be discussed in a forthcoming paper. The relative probability of the formation of different structures was estimated by comparing the corresponding energies of formation.

2. Theoretical Methods and Models

Computational Details

The ab initio calculations using the plane–wave pseudopotential method were carried out within the framework of density functional theory in the generalized gradient approximation (GGA) using PBEsol functional [29] as realized in the ABINIT software package [30,31,32]. The norm-conserving pseudopotentials were constructed according to the scheme described in [33]. For Si and O atoms, the 3s3p and 2s2p electrons were treated as the valence ones, respectively. The plane–wave basis size was controlled by cutoff energy E_cut = 45 Ha. The Brillouin zone (BZ) sampling, according to the Monkhorst–Pack scheme [34], was chosen as 6 × 6 × 2. The full geometry optimization was performed by varying both the lattice parameters and atomic positions. The convergence tolerance for geometry optimization was selected within 10⁻⁵ Ha/Bohr and 0.01 GPa in maximal force and stress tensor, respectively. Each self-consistent step was controlled by energy convergence lower than 10⁻⁸ Ha. The phonon eigenvectors and frequencies were calculated within the density functional perturbation theory (DFPT) [35].

Electron population analysis was performed using a projection of the plane–wave states onto a localized basis using a technique described by Sanchez-Portal et al. [36]. Population analysis of the resulting projected states was then performed using the Mulliken formalism [37].

The approach used in this study was restricted to static consideration of theoretically possible structural models. Unfortunately, within the framework of this approach, it is impossible to take into account the temperature factor. An experimental confirmation of the existence of such structures can be the observation of characteristic lines in the vibrational spectra of real heterosystems. The search for such spectral features is the subject of our next work, which is being prepared for publication.

3. Results and Discussion

3.1. Structural Models

First, we optimized the structures of bulk silicon and β-cristabolite crystals and made sure that the chosen calculation scheme reproduced the experimental data well (see

Table 1. Optimized structural parameters of bulk crystals and short-period SL with I1 interface.

System		a	c (Si)	c (SiO2)	τ (°)
Bulk crystals	Si	3.840 (3.840 1)	5.430 (5.430 1)	-	-
	β-SiO2	5.248 (5.067 2)	-	7.422 (7.166 2)	0
	$\tilde{\mathsf{\beta }}$-SiO2	4.991 (5.029 3)	-	7.331 (7.131 3)	21 (18 3)
SL	#115	4.908	3.865	8.156	0
	#81	4.202	5.105	8.189	33

¹ Reference [38]. ² Reference [39]. ³ Reference [40].

). For modeling systems of the I1 type, we started with the simplest model system, which is a binary periodic SL with Si and SiO₂ layers. Each layer contained four monolayers of Si atoms and four bilayers of SiO₂ (see Figure 2a). Thus, the unit cell of this SL was composed of two unit cells of the silicon crystal and two unit cells of the β-cristobalite lattices. Such systems can be referred to as Si₄(SiO₂)₄ superlattices with a $\overline{4}m2$ point symmetry in the $P\overline{4}m2$ (#115) space symmetry group. The tetragonal unit cell parameters are determined as a = <a_c>/√2 and c = a_c(Si) + a_c(β-SiO₂), where <a_c> is an average of the cubic cell parameters a_c(Si) and a_c(β-SiO₂). We determined the initial cell dimensions as a = 4.547 Å and c = 12.86 Å. The optimized structural parameters are given in Table 1. On comparing them with the corresponding parameters of the bulk crystals (also shown in Table 1), one can see the large elastic strains appearing in both materials. In particular, the Si layer becomes stretched by 28% in the ab plane and compressed by 29% in c-direction. Such high strains should cause structural instabilities.

In fact, the phonon calculations revealed the existence of several unstable modes. All of them included the SiO₄ tetrahedron rotations within the oxide layers. This fact is in line with the well-known instability of the β-cristobalite lattice related to the presence of many Rigid Unit Modes [41]. In search of stable configurations, we distorted the #115 structure along the eigenvector of each of the unstable modes and performed the total geometry optimization. Geometry optimization along one of the unstable modes resulted in the finding of a new structure with the $P\overline{4}$(#81) space symmetry group. Geometry optimizations along other unstable modes resulted in the finding of several less symmetric structures with about the same total energies. Taking into account the insignificant difference in the energies of all of the found stable configurations, we decided to start the study with a detailed analysis of the most symmetric of them. Thus, the rest of the paper is devoted to discussing the properties of the #81 SLs shown in Figure 2b,d.

Comparing Figure 2a,b, one can see that the structural differences between the #115 and #81 structures include the rotations of the SiO₄ tetrahedra around the axis parallel to the SL axis, i.e., the c-direction. Similar distortion of the bulk β-cristobalite lattice transforms it into a tetragonal modification called $\tilde{\beta }$-cristobalite [42], with spatial symmetry group $I\overline{4}2d$(#122). Hence, one can consider the #81 SL as a planar heterostructure in which the oxide layers are structured as in $\tilde{\beta }$-cristobalite with a tetragonal axis along the SL axis. Tetrahedron rotation is characterized by the tilt angle τ. The data in Table 1 show that this angle markedly increases when the $\tilde{\beta }$-SiO₂ material is locked up in an SL structure. One can see that the tetrahedron rotations result in significant reduction in the elastic strains: the ab compression strain diminishes to 9% in the Si layer (versus 28% in the #115 SL). As a result, the structure becomes stable with respect to all phonon modes. This result shows the invalidity of all previous statements about the impossibility of the existence of stable structures with interfaces of II type.

Dependence of the elastic strains and cohesive energy of SL on the layer thickness is of particular interest. To study this issue, we have considered several SLs of the I1 type with variable layer thicknesses. We shall refer to them by using the double index m × n, where m is the numbers of Si monolayers in the silicon layer and n is the number of SiO₂ bilayers in the oxide layer. The optimized structural parameters and elastic strains of the considered SLs are shown in

Table 2. Optimized structural parameters and elastic strains in the I1 type SLs Si_n/(SiO₂)_m.

m × n	τ(°)	Layer Parameters (Å)			Strains (%)
		c (Si)	c (SiO2)	a	Si		SiO2
					Uzz	Uxx	Uzz	Uxx
8 × 4	35.8	5.257 × 2	8.336	4.033	−3.2	+5.0	+13.7	−19.2
4 × 4	33.1	5.105	8.189	4.202	−6.0	+9.4	+11.7	−15.8
4 × 8	28.8/31.5	4.810	7.849 × 2	4.445	−11.4	+15.8	+7.0	−10.9
8 × 8	33.3/35.5	5.073 × 2	8.072 × 2	4.185	−6.6	+9.0	+10.1	−16.1

.

For the SL with a thick oxide layer (with n = 8), the two τ values given in Table 2 as τ₁/τ₂ correspond to the tilt angles in interior and exterior parts of the oxide layer, respectively. It is remarkable that the tilt angle depends primarily on the magnitude of the U_xx(SiO₂) strain. Furthermore, looking at Table 2, one can see two tendencies. Firstly, the material in the Si layer was compressed in c-direction and stretched in the interface plane, whereas the material in the oxide layer was stretched in c-direction and compressed in the interface plane. Secondly, thickening of one layer reduced the strains existing in it but increased them in another layer. This tendency can be characterized numerically by equations relating the strain in one layer U₁ and the relative thickness of another layer d₂. Such an equation has one evident point: U₁ = 0 at d₂ = 0. If one determines the relative thicknesses as d(Si) = m/(n + m) and d(SiO₂) = n/(n + m), the results presented in Table 2 can be plotted as shown in Figure 3.

The obtained dependencies U₁(d₂) can be approximated by the following relations:

U_x(Si) = 4.0x + 29.4x² and U_z(Si) = −0.5x − 23.7x² x = d(SiO₂)

(1)

U_x(SiO₂) = −38.2y + 13.8y² and U_z(SiO₂) = 23.1y − 3.4y² y = d(Si)

(2)

These approximating relations are shown in Figure 3 as red lines. In practical terms, the most interesting possibility is to use Equation (2) to estimate deformations in a thin oxide layer interfaced with a thick silicon layer, that is, at y = 1. They are U_xx(SiO₂) = 24.4% and U_zz(SiO₂) = 19.7%.

Now we turn to the SLs with I2 interfaces. As noted above, in all previous papers discussing the interface between Si and cristobalite SiO_2, the preference was given to such systems. Therefore, it is reasonable to compare properties of the I1 and I2 interfaces. An example of the SL with I2 interface is shown in Figure 2c. On comparing the structures shown in Figure 2a,c, one can see two significant differences. The first difference is that, on the I1 interface plane, there is one Si atom for each group SiO₂, whereas the surface density of Si atoms is twice as high on the I2 interface plane. The second fundamental difference is that, in the I2 interface on the side of the Si layer, there are atomic positions that do not have corresponding positions from the side of the SiO₂ layer. If we assume these positions to be occupied by Si atoms, then these Si atoms with two dangling bonds will have severe non-stoichiometric structural defects which would cause structural instability. Some authors proposed to eliminate these defects by adding various terminal groups (O, H or OH) to these atoms [28]. Another way, consisting of replacing the defective Si atoms with O atoms, was suggested in [25]. This structural model has been used by many authors dealing with DFT modeling (see e.g., [21,26,27]).

We also used this approach for studying a series of SLs with I2 interfaces, in which the defect Si atoms are replaced by O atoms. An example of such SL is shown in Figure 2c, in which the black circles show the added O atoms. We have studied several such SLs containing m Si monolayers in the silicon lattice and n SiO₂ bilayers in the oxide lattice (m × n = 8 × 4, 4 × 4, 4 × 8 and 8 × 8). For each of these systems, a complete geometry optimization was carried out. It was found that, similarly to the I1 systems, the stable configuration of the I2 systems included tilt rotations of the SiO₄ tetrahedra in the oxide layer. The optimized structural parameters are given in

Table 3. Optimized structural parameters and elastic strains in the I2 type SLs.

m × n	τ (°)	Layer Parameters (Å)			Strains (%)
		c (Si)	c (SiO2)	a	Si		SiO2
					Uzz	Uxx	Uzz	Uxx
8 × 4	19.1/17.5	5.6523 × 2	7.2391	5.1710	4.1	−4.8	−1.3	3.6
4 × 4	23.0/21.9	5.8828	7.4039	4.9629	8.3	−8.6	1.0	−0.6
4 × 8	22.9/21.7	5.8796	7.3730 × 2	4.9666	8.3	−8.5	0.6	−0.48
8 × 8	19.2/17.3	5.6675 × 2	7.25723 × 2	5.1508	4.4	−5.1	−1.0	3.2

. One of the optimized I2 structures is shown in Figure 2d.

According to the data in Table 3, one can draw certain conclusions about the properties of the I2 heterostructures. Firstly, the Si material is stretched in c direction and compressed in the ab plane, whereas the SiO₂ material is compressed in the ab plane and almost not deformed in c direction. Comparing the data in Table 2 and Table 3, one can see that the elastic strains were lower and the tilt angle was lesser in the I2 structures than in the I1 structures. Secondly, the calculated strain values were not directly related to the relative layer thicknesses as in the I1 systems. Moreover, the U_xx (SiO₂) and U_zz(SiO₂) strains may have the same signs, which seems rather strange from the point of view of minimization of the elastic strain energy. All these features of the I2 type SLs are evidently related to particularities of the I2 interface, which, having a finite thickness, is a separate layer of the heterostructure—a layer characterized by its special elastic properties.

The structural fragments of the two interface types are shown in Figure 4. The Si atoms in different oxidation states are colored in different colors. The Si⁴⁺ state is inherent to the atoms within the oxide material (upper part of Figure 4) and the Si⁰ state occurs inside the silicon material (bottom part of Figure 4). One can see that the main difference concerns the thickness of the interface layer. It consists of one monolayer of the Si²⁺ moieties in the I1 structure and of two monolayers of Si²⁺ and Si¹⁺ moieties in the I2 structure.

To better understand the difference in the structure of the two interfaces, we will look at the local structural parameters of the two types of SLs (bond lengths and angles). They are listed in

Table 4. Structural parameters for two SL structures in comparison with those of the bulk crystals.

Parameter		System
		4 × 4 I1	4 × 4 I2	Bulk
Valence bonds (Å)	Si4+–O	1.623	1.619	1.617
	Si2+–O	1.661	1.655
	Si1+–Ob	-	1.698
	Si2+–Si0	2.469	-	-
	Si2+–Si1+	-	2.483
	Si1+–Si0	-	2.310
	Si0–Si0	2.458	2.305	2.351
Valence angles (°)	Si4+–O–Si4+	126	148	147
	Si4+–O–Si2+	131	146
	Si1+–Ob–Si1+	-	138
tilt angle (°)		34	18	21

in comparison with similar parameters for the bulk Si and $\tilde{\beta }$-cristobalite crystals.

It follows from Table 2 and Table 3 that the specific volume of the Si material in the I1 structures increased, while in the I2 structures it decreased, as compared to the volume of the bulk crystal. Accordingly, the Si⁰-Si⁰ bonds were longer in the first case and shorter in the second case. The situation is opposite with the SiO₂ material: its specific volume in the I1 structures was decreased, while in the I2 structures it increased, as compared to the volume of the bulk crystal. However, in both I1 and I2 systems, the Si⁴⁺-O bonds were longer than similar bonds in the bulk SiO₂ crystal. The anomalous stretching of the Si-O bonds in the compressed oxide material in type I1 structures can be related to large tilting rotations of the SiO₄ tetrahedra. A similar effect takes place in course of the structural phase transition in cristobalite: when β-cristobalite transforms into α-cristobalite, a noticeable decrease in volume is accompanied by an increase in tilt angle and, at the same time, by an elongation of Si-O bonds [40].

Both I1 and I2 interfaces contain the Si atoms in the S²⁺ oxidation state. These atoms form two Si-O bonds in the oxide layer and two Si-Si bonds in the silicon layer. Thus, the SiO₂Si2 tetrahedra are formed around these Si²⁺ atoms. In such tetrahedra, the Si²⁺–O bonds are noticeably longer than the Si⁴⁺–O bonds inside the oxide materials. This effect can be due to the weakening of the ionic binding. It should also be noted that there are unusually long Si¹⁺–O_b–Si¹⁺ bridges in the I2 interface. It can be assumed that the enhanced length of the Si¹⁺–O_b bonds is a consequence of the fact that the Si¹⁺–O_b–Si¹⁺ bridges “replace” the much longer Si-Si-Si bridges in the I2 structure.

The Si-Si bonds in the vicinity of the interfaces behave differently: the Si²⁺– Si⁰ bonds in the I1 structure are of the same length as the Si⁰–Si⁰ bonds inside the silicon material, while the Si²⁺–Si¹⁺ bonds in the I2 structure are noticeably longer. Apparently, this is due to the difference in the oxidation states of the Si⁰ and Si¹⁺ atoms. It can be assumed that a significant part of the electron density is transferred from the SI1⁺ to the O_b atoms, which leads to weakening of the Si²⁺–Si¹⁺ covalent binding. The presence of long Si¹⁺–O_b–Si¹⁺ bridges and SiOSi₃ tetrahedra around the Si¹⁺ atoms are characteristic features which allow the detection of the I2 interfaces.

3.2. Mulliken’s Population Analysis

In order to elucidate the relation between the coordination environment and the oxidation state of Si atoms in the considered SL, we used the Mulliken’s population analysis. This approach makes it possible to analyze the distribution of the electron density distribution over atoms and bonds. We discriminated the Si atoms by indicating their oxidation states Si⁰, Si¹⁺, Si^2+, and Si³⁺. To discriminate the O atoms in a different environment, we supplied them with index ij, which corresponds to the oxidation states of the nearest Si atoms. For example, the O atoms in the interior of the oxide material (i.e., between two Si⁴⁺ atoms) were labelled as O₄₄.

For the bulk Si crystal, the Mulliken’s population analysis predicted the Si-Si overlap population to be equal to 0.74 e, and the atomic Si charges were obviously zero. The Si-O overlap population was equal to 0.53 e and Si charges were equal to 2.36 e in the bulk $\tilde{\beta }$-cristobalite crystal; the O charges were obviously equal to −1.18 e. Calculated atomic charges and overlap populations for two SLs of different types are presented in

Table 5. Mulliken atomic charges (Z_a) and overlap populations (P_ab) in SLs of different types. All quantities are given in e units. Atoms are listed in the order of the layers of the heterostructure. N_a is the number of atoms per unit cell. The overlap population P_ab corresponds to the pair of atoms in two neighboring (above and below) lines.

4 × 4 I1 Type					4 × 4 I2 Type
Layer	Atom	Na	Za	Pab	Layer	Atom	Na	Za	Pab
SiO2	O42	2	−1.09	0.42	SiO2	O42	2	−1.11	0.42
				0.52					0.54
	Si4+	1	+2.25	-		Si4+	1	+2.32	-
	O44	2	−1.12	0.52		O44	2	−1.17	0.52
				0.51					0.52
	Si4+	1	+2.26	-		Si4+	1	+2.34	-
	O44	2	−1.12	0.51		O44	2	−1.17	0.52
				0.52					0.52
	Si4+	1	+2.25	-		Si4+	1	+2.32	-
	O42	2	−1.09	0.52		O42	2	−1.11	0.54
				0.42					0.43
I1	Si2+	1	+1.17	-	I2	Si2+	1	+1.22	-
Si	Si0	1	−0.10	0.67		Si1+	2	+0.56	0.69
				0.66					0.67
	Si0	1	−0.06	-	Si	Si0	2	−0.25	-
	Si0	1	−0.10	0.66					0.67
				0.67	I2	Si1+	2	+0.56	-
I1	Si2+	1	+1.17	-		O11	1	−1.02	0.48
						Si1+	2	+0.56	-

.

The data presented in Table 5 show that the ionicity of the Si-O bonds in SL decreased: both Si⁴⁺ and O₄₄ charges in SL were lower than in the bulk SiO₂ crystal. This effect was more prominent in the I1 SL. At the same time, the Si-O overlap population remained almost unchanged (0.52 e in SL versus 0.53 e in bulk SiO₂). On the whole, the Si-O bonding in silica material of SL was less ionic and more valence.

As for silicon material, all Si atoms in the interior of the silicon layers took negative atomic chargers (−0.10 e in I1 SL and −0.25 e in I2 SL). This fact indicates that the formation of a layered Si/SiO₂ heterostructure causes a transfer of electron density from the interface region towards the interior of the silicon layers. At the same time, the overlap Si-Si populations in the Si layers decreased (0.66 e in SLs versus 0.74 e in the bulk Si crystal). Moreover, this effect takes place in SLs of both types, regardless of whether the Si-Si bonds are compressed (case of I2 SL) or stretched (case of I1 SL) relative to the bulk crystal. This effect may indicate a weakening of the valence Si-Si bonds.

The most interesting question concerns the distribution of charges in the interface region. The atomic charges of the interface Si²⁺ atoms were about two times lower relative to the charges in the interior of the oxide layer. At the same time, the Si⁴⁺-O₄₂ overlap population was almost unchanged. The Si²⁺-O₄₂ overlap population was 20% lower relative to the values in the interior of the oxide layer. The atomic charge of the Si¹⁺ atom was about four times smaller than that of the Si⁴⁺ atoms. Thus, the results confirm the suggested classification of Si atoms based on coordination environment analysis.

The overlap populations within the Si¹⁺-O₁₁-Si¹⁺ bridge were 0.48 e, and the O₁₁ atomic charge was −1.02 e. Both values are lower than in the bulk SiO₂ crystal. These results are consistent with the fact that such bridges are noticeably longer than those in the interior of the oxide material.

One more interesting circumstance can be noted. The overlap density for non-bonding O-O pairs in bulk $\tilde{\beta }$-cristobalite was −0.16 e for the in-plane contacts and −0.17 e for the out-of-plane contacts (we mean the plane perpendicular to the tetragonal axis). A negative overlap population value corresponds to repulsion, and the difference in values was consistent with the difference in contact lengths: the former (2.6709 Å) longer than the latter (2.6347 Å). A similar relation between bond length and overlap population took place for the O-O contacts in the SL structures. In particular, the O₄₄-O₄₄ in-plane contacts in I1 SL, which are the shortest ones (2.5055 Å), had minimal population overlap (−0.24 e). From this, one can conclude that a large negative overlap indicates a significant repulsion.

3.3. Energy of Formation and Interface Energy

The above results prove the possibility of the existence of two types of interfaces between silicon and cristobalite crystals. How likely is the formation of such structures in real samples? The answer to this question requires estimating the energy of formation of such heterostructures.

When deriving such an estimate, it is reasonable to assume that the initial state is two pure materials, crystalline silicon and crystalline cristobalite, and the final state is an optimized superlattice structure. The difference between the energies of these two states will give us an estimate of the energy of formation of the heterostructure.

It is quite easy to obtain such an estimate for the SLs of I1 type. These structures are formed by the alternation of Si and SiO₂ layers. If the Si layer contains m Si monolayers and the SiO₂ layer contains n SiO₂ bilayers, then the chemical composition of the unit cell can be written as Si_m(SiO₂)_n. Let E(m, n) is the energy of such an SL. The calculated E(m, n) values are listed in

Table 6. Calculated total E(n,m) energies and energies of formation E_f (eV/cell) for SLs of I1 and I2 types.

I1 Type			I2 Type
m × n	E(m,n)	Ef	m × n	p × q	E(m,n)	Ef
4 × 8	−8594.31014	4.8619	8 × 4	14 × 5	−6692.80403	5.77941
4 × 4	−4526.12430	3.74014	4 × 4	6 × 5	−5772.31922	5.15054
8 × 4	−4986.32200	4.09928	4 × 8	6 × 9	−9841.64586	5.13150
8 × 8	−9054.28784	5.44104	8 × 8	14 × 9	−10,762.07500	5.81604

. Let E(Si) and E(SiO₂) are the energies per formula units (f.u.) of the Si and $\tilde{\beta }$-cristobalite crystals. Then the difference:

E_f = E(m, n) – [mE(Si) + nE(SiO₂)]

(3)

Can be considered as the energy of the formation per unit cell of the SL. Using this definition and the calculated values:

E(Si) = −115.13921 eV/f.u. and E(SiO₂) = −1017.32690 eV/f.u.

(4)

we estimated the E_f values. They are also shown in Table 6.

The use of (3) implies that these energies can be represented as a sum of three additive contributions: intrinsic energies of the Si, SiO₂ materials, and the interface formation energy. The intrinsic energies of the materials are suggested to be proportional to the material content. The E(Si) and E(SiO₂) values given in (4) can be considered as normalized energies.

The application of this approach to the I2 SLs requires some clarification related to the peculiarities of these structures. Firstly, one should take into account that the unit cell of the I2 SL contains two Si atoms per monolayer (and not one, as was the case in the I1 systems). Secondly, half of the Si atoms in the interface plane must be replaced by O atoms. Thus, the formula describing the chemical composition of the I2 unit cell containing m monolayers of Si and n bilayers of SiO₂ takes the form Si_{(2m – 1 + n)}O_{(2n + 2)}. This formula can be rewritten as follows:

Si_{(2m – 1 + n)}O_{(2n + 2)} = Si_p(SiO₂)_q (p = 2m − 2, q = n + 1)

(5)

This expression is equivalent to that describing composition of the I1 SLs. Thus, one can use Equation (3) and, by replacing m and n by p and q, calculate the E_f values for the I2 SLs. The calculated E(m, n) energies and the E_f values for I2 structures are also given in Table 6. Comparing the E_f values for the two types of SLs we have come to the conclusion that they have quite comparable formation energies. This result casts doubt on the claims of previous works that declared the I1 interface model to be deliberately unrealistic. The arguments presented in this article confirm that this type of interface is quite realistic.

The closeness of the energies of the formation of two types of interfaces requires closer consideration. It is noticeable that the E_f value includes both the energy of formation of the interface and the energy of elastic strains. Indeed, during the formation of interfaces, the materials of the layers experience elastic deformations, the values of which are much larger in the case of I1 systems compared to I2 systems. On the other hand, formation of the I2 interfaces is associated with the penetration of a part of an O atom into the Si layer and with the appearance of the suboxide Si¹⁺ moieties (which is not present in the I1 interface). The formation of any suboxide Si atoms costs some chemical energy. The “suboxide penalty” for the Si¹⁺ suboxide was estimated as 0.47 eV/atom [43]. Thus, the formation of the I2 interface requires less energy for the elastic strains but is associated with extra energy costs for the formation of suboxide Si¹⁺ moieties. As a whole, these two factors make the formation energies of the two types of interfaces comparable. Note that the obtained estimate of E_f (about 3–5 eV per surface Si atom) is close to the energies of the formation of various defects in silicon [44]

This conclusion is a consequence of the high mechanical compliance of the cristabolite lattice. Due to concordant rotations of the tetrahedra forming this structure, the lattice can undergo large deformations and easily adapt to the interface dimensions specified by the silicon lattice parameters. To reveal the I1 and I2 interface structures experimentally, it is necessary to find a way to detect the interfaces in real samples. One such method is a Raman spectroscopy. Our next article is devoted to a discussion of the vibrational spectra of the considered SLs and the search for spectral lines characteristic of different types of interfaces.

4. Conclusions

The structure of the interface between crystalline silicon and cristobalite SiO₂ have been studied by considering planar binary Si/SiO₂ superlattices. Two scenarios of the interface formation have been considered. The first one lead to the interface type I1, which is based on the direct correspondence of the two cubic lattices; the second one, I2, implies the rotation of one of the lattices by 45°. In both cases, the total geometry optimization made it possible to determine stable heterostructures. Stability of the optimized structures was tested by the phonon state calculations, which were not performed in any of the previous works.

It was found that, in both cases, the silicon lattice conserves its quasi-cubic structure, whereas the oxide lattice is markedly deformed by rotations of the SiO₄ tetrahedra around axes perpendicular to the interface plane. Such deformations cause the flexibility of the cristobalite lattice, which allows it to interface with the silicon lattice, despite the large lattice mismatch. Elastic strains in the layers of a I1 superlattice are regular. It has been shown that their dependence on the relative thicknesses of the layers can be described with good accuracy by simple relations.

The I1 interface is extremely thin: it consists of only one Si monolayer with atoms in the Si²⁺ oxidation state. The I2 interface includes half of the Si atoms in the Si²⁺ oxidation state; another half of the Si atoms turn out with two dangling bonds. These severe structural defects make the structure instable. The most appropriate way to eliminate these defects is to replace the Si atoms with dangling bonds with O atoms. Such a replacement ensures the structural stability and reserves the stoichiometric net formula. At the same time, it makes the interface region twice as large: in addition to the layer of atoms Si²⁺, a layer of atoms Si¹⁺ appears. In order to compare the thermodynamic probability of the formation of these two types of interfaces, their formation energy was analyzed. The elastic strains in the silicon layer comprise 5–10%, and their value decreases rapidly as the layer thickens. The elastic strains in the oxide layer vary widely, in a range of 1–15%, depending on the interface structure. The obtained estimate of the interface formation energy is about 3–5 eV per surface Si atom, which is close to the energies of the formation of various defects in silicon. It has been shown that the formation energies of the two interfaces are quite comparable with a slight advantage in favor of the I1 type.