Computing Elastic Tensors of Amorphous Materials from First-Principles (2024)

1 Introduction

Amorphous materials are abundantly integrated into modern semiconductor technologies, assuming a diverse array of roles, from ‘passive’ functions like substrates or electrically insulating layers, to ‘active’ ones, involving charge carrier transport. For example, hydrogenated amorphous silicon (a-Si:H) plays a pivotal role in both photovoltaic solar cells and the control transistors of thin-film liquid crystal displays. This is attributed to its direct band gap, moderate electron mobility, and excellent optical absorption capabilities, see for example Refs. [1, 2]. The anticipated successor, amorphous InGaZnO₄ (a-IGZO), employed in high-end display technologies, is reported to offer substantial advantages over traditional silicon-based thin-film transistor materials. Notably, its deposition temperature is compatible with polymeric substrates. These advantages encompass reduced rigidity, improved stability, and enhanced electron mobility, making it feasible to manufacture flexible thin-film transistors for large-area electronic applications. [3, 4]

Furthermore, in Si-based Complementary Metal-Oxide–Semiconductor (CMOS) technology, amorphous silicon dioxide and hafnium dioxide act as dielectric barriers to prevent charge carrier leakage.[5, 6] Amorphous metals such as Ti based alloys are also frequently employed in gate metals [7], and the effective work function of the device can be adjusted using thin layers of other metals, such as amorphous TiAlC [8, 9]. Additionally, amorphous bulk materials find extensive use as low-dielectric constant materials to reduce the parasitic capacitance of the CMOS addressing interconnection circuitry or as cost-effective fill materials. [10, 11]

As the transistor’s dimensional reduction continues to approach the physical limit, the electronic industry is being driven to transition to three-dimensional integration schemes to sustain performance gains (see Refs.[12, 13]). As a direct consequence the impact of the mechanical properties of material on both the device operability and performances are becoming of paramount interest. The convergence of requirements arising from the device’s intended operation and the decreasing thickness of functional material layers necessitates stringent control measures to ensure the structural integrity of these ultra-thin material layers and their associated interfaces. They are also introducing new (or enhanced) sources of stress or strain variability. The effect of strain engineering in layered materials is a well studied topic, with examples including the enhancement of the silicon charge carriers mobility in Si based nanooelectronics [14, 15, 16]. At the atomic level, the dynamics of interfacial effects (strain, defects, etc.) can strongly impact different aspects such as adhesion, charge carrier mobility, band alignment, or the pinning of the Fermi-level.[17, 18, 19, 20] Our earlier work illustrated that sub-5nm thick crystalline Ru layers exhibit an increased mechanical flexibility [21]. In general, a less stiff elastic tensor has different consequences in terms of device affected properties, which include adhesion of material layers [22] and enhanced diffusion [23, 24]. Amorphous materials, due to their intrinsic atomic disordered nature, offer the perspective to accommodate external strain sources differently than their crystalline counterpart would, and are potentially opening new engineering mitigation strategies. For this to happen, there is a need for understanding how the mechanical integrity of amorphous materials is being impacted by thinning. Unfortunately, measuring the elastic properties of a few nanometer thick film is a non-trivial task [25, 26, 27], but one that is quantifiable from ab initio. An important advantage of computing the elastic tensor through ab initio methods is its adaptability for use in a range of simulations and experiments, including applications such as Technology Computer-Aided Design (TCAD) and stress analysis with Raman spectroscopy [28]. In this paper, we report a methodology to compute the elastic tensor of both bulk and ultra-thin film amorphous materials from first-principles. We illustrate the practicality of our approach using amorphous silicon dioxide, a widely utilized dielectric material in Si CMOS technology.

The methodology for numerically evaluating the bulk elastic tensor from ab-initio for crystalline materials is widely known, with implementations such as ElaStic [29] and Pymatgen [30]. However, there is a large difference when treating an amorphous material model that is not accounted for with their ordered counterpart; the non-affine transformation under external strain and lack of periodicity. In addition, when compared to ordered crystalline systems, amorphous materials are atomically disordered materials; in which, the bond lengths, coordination numbers (CNs), and atomic densities vary wildly throughout the model used. The difficulty is then to build a representation that captures a statically relevant distribution of the atomic disorder, while relying on condensed matter physics formalism with periodic boundary conditions. This typically results in the use of large periodic unit cell models, which can often exceed over 100 atoms, in which atomic disorder is being built. Due to these differences in morphology with respect to ordered systems, different local energetic minima solutions can co-exist. Therefore, modeling amorphous systems requires sufficient sampling of atomic configurations large enough to provide statistically relevant results. In what follows, we give a brief review on the background of amorphous materials, followed by a discussion of the algorithm used to compute the elastic tensor from ab-initio, and finally the application to SiO₂ bulk and ultra thin films.

2 The Amorphous Elastic Tensor Method

As has been reviewed earlier, shear deformations are the linchpin that drive large variations of atomic displacement in amorphous materials. For instance, Figure1a) illustrates the evolution of the morphology of an amorphous SiO₂ model after the minimization of its atomic degrees of freedom with a shear deformation of 3%. A colour spectrum is provided to demonstrate the amount of normalized displacement occurring per atomic site, accompanied with the direction of displacement as a green vector. It is observed that independent sections of the material shift in differing directions. For example, in the top facet of the unit cell, there is a clockwise-like shift in atomic positions. The resulting elastic tensor for such a system yields varying results since a different applied strain would nudge the system into a different energetic minima. As a result, the elastic stability criteria of Born [45, 46] is typically not satisfied, leading to negative elastic tensor’s eigenvalues and hence to a non-physical result.

At first glance, one might assume that either i) the applied strain is too large, which perturbs the system out of the linear elastic regime into a plastic one, or ii) that due to amorphous nature of the system, the configuration undoes the applied strain. The consequences on the Voigt-stress components of the elastic tensor in a normally applied direction (Figure2a)). At this stage of the discussion, we are focusing on reproducing the linear elastic response of the material, therefore neglecting any non-linear, viscoelastic, or any other mechanical variant. In this regime, the stress-strain relationship must be strictly linear. This assumption is reasonable (but should be used on a per case basis) as for instance, many glasses, such as silica (a-SiO₂), silicates, and aluminates, have been reported to exhibit elastic properties.[37]

The linear elastic regime is clearly broken when observing $\sigma_{1}$ in Figure2a), the system is responding to the external strain with non-affine transformations yielding a new local structural minimum. Additionally, recall that the relative magnitudes of each curve do not matter, but the variation of the stress is used to evaluate the elastic tensor. In crystalline materials, 1% strain does not force the material into a plastic regime for the majority of pure ordered materials. Is the non-linearity due to anharmonicity or is it truly due to the plastic limit? Instead of determining either of these options, if case ii) is correct, then the application of elastic tensor methodology to ultra-thin films or surfaces can be applied to amorphous materials.

If, due to applying strain to the unit cell of an amorphous material, the system undoes the applied strain, then we cannot use the finite perturbation technique in the quasi-harmonic potential limit to determine the mechanics of the material. One of the key-defining features of an amorphous material is its inherent lack of translational symmetry or order, so any enforced deformation to a unit cell without symmetry is therefore capable of reverting to its original state. This is very much the exact same phenomena observed when developing the elastic tensor methodology for ultra-thin films. The solution is to artificially build a region of symmetry that maintains the applied strain. To this end, taking a small volume ($dV$) within the edges of the unit cell periodic boundaries (see Figure3), atoms can be selected to remain invariant under internal relaxation. The internal relaxation of such a scheme can be clearly observed in Figure1b), where the atoms along the unit cell boundary do not move. The remaining interior atoms still relax their positions, but to a much lesser degree than in Figure1a). The result is an elastic tensor which typically satisfies the Born criteria for stability. The constraint also yields in a linear response to the applied strain, with respect to stress, as observed comparing the non-held system in Figure2a) to that of the held system in Figure2b).

To compute the ultra-thin elastic tensor refer to our previous paper [21], where the position of the atoms at the surface are held still (which is given for free by the amorphous algorithm). Additionally, the stress tensor must be scaled accordingly using a standardized definition of the thickness of the film which accounts for the electron density extending past its surface.

3 Computational Details

3.1 Generation of Amorphous Structures

The a-SiO₂ models were generated with a decorate and relax scheme (see Ref.[49]). The procedure begins with an atomic structure built from a random distribution of silicon, the atomic position is then compressed and subsequently decorated by oxygen atoms to reproduced coordination numbers similar to the crystalline phase. These models are then optimized using DFT. This methodology has been heavily employed in the study of IGZOs and in general leads to structures with reasonable radial distribution functions and band gaps [50].

A total of ten 96-atom structures and fifteen 200-atom structures were generated to sample the variations in configurations available at densities ranging from 2.3–2.9$g\cdot cm^{-3}$ (see Figure4). Two different sizes of unit cells were studied, a 96-atom cell (Si₃₂O₆₄) and variants of a 200-atom type cell (ex. Si₆₆O₁₃₂). The radial distribution functions of the 200-atom type models for the nearest neighbours and different bond types can be found in Figure5, which reproduce values observed in literature [51, 52, 53].

4 Results & Discussion

4.1 Bulk a-SiO₂

The mechanical properties of a-SiO₂ are widely varied due to the plethora of methods in which to synthesize it, such as the deposition process, and varying temperatures or pressures in the growth in environment. Its characterization produces differing results, which is understood to be due to the differences in structural configuration natural to amorphous materials. Depending on the strength of the covalent bonds, the orientations, their number, the coordination of each atom, and the number and locations of voids, the mechanical properties can be expected to change. The variation in the density and its associated shear, bulk, and Young Hills’ moduli are shown in Figure6. For reference, densities close to 2.2 $g\cdot cm^{-3}$ are used in devices. The best-fit was computed for the data and tends towards the values previously computed with ab-initio from Ref.[62], without the elastic tensor. Experimental values of a-SiO₂ have found the Young’s modulus to be 70GPa [37] for bulk silica and 76.6$\pm$7.2GPa [63] for nano wires, measured at room temperature. Extrapolated to 0K, the experimental Young’s modulus for bulk silica is near 65GPa (note the density from this reference is not reported), placing the trends of our results and those of other DFT calculations within good accuracy of experimental data.

A linear best-fit line was computed to demonstrate the general trend of the data, as it is too sparse to fully determine if the relation is linear or non-linear. The R² values of each line of fit ranges between 0.55-0.56, but a $\chi_{red}^{2}$ cannot be computed due to the lack of a quantity of variance. Traditionally, the Young’s modulus and the atomic density are given on a log-log plot, typically called a material selection chart (see Ref.[64]), but this does not imply that any material is non-linear with respect to the density and modulus. For amorphous materials, there have been reports of linear scaling with the two properties, one example being Harms et al. 2003 [65], where different annealing temperatures generated amorphous structures of different densities and stiffness. Therefore, any best-of-fit lines reported in this amorphous work should be considered general trends and not necessarily validation of linear behaviour.

The values reported by Bondi et al. 2010 for a-SiO₂ consist of up to 100-atom amorphous configurations where the moduli were computed using the standard technique of straining in a particular direction (or isotropically) and the subsequent stress response is measured. Namely, the Young’s modulus is typically computed as

$$Y=\frac{\sigma_{xx}}{\epsilon}\hskip 2.84544pt,$$

(3)

with $Y$ being the Young’s modulus, $\sigma_{xx}$ the stress in the xx component (or 11), and $\epsilon$ the magnitude of strain in the x direction.In contrast, the Young’s modulus for this work uses the components of the elastic tensor,

$$Y=\frac{K_{VRH}\cdot G_{VRH}}{3K_{VRH}+G_{VRH}}\hskip 2.84544pt,$$

(4)

where $K_{VRH}$ and $G_{VRH}$ are the Voigt-Reuss-Hill average bulk modulus and shear modulus. The shear components of the elastic tensor are included in this formulation, suggesting that the methodology applied to compute the tensor is within the variation of existing literature. At the higher densities, we observe a variation of approximately $\pm 20$GPa for the Young’s modulus and $\pm 10$ GPa for the bulk modulus. This range of stiffness is also observed by Bondi et al. [62] (with a lower density). They suggested, when compared to a-Si, that the addition of H or O to the system weakens the bonds, creating a more deformable system, while reducing the coordination numbers, demonstrating another key-defining attribute of less stiff amorphous systems.

4.1.1 Analysis of Bonds and Coordination

In an effort to determine the source of the changing elastic properties for a given density, a coordination analysis was performed on each structure’s relaxed state. The variation of the average coordination number with the Young’s modulus in Figure7 suggests that as the density is increased, the coordination number typically increases. This is observed from the colours becoming lighter as the density increases. Except for one or two coordination pairs, the majority of the pairs are Si-O. The result coincides with the literature that suggests that given any atom, the more bonds it has (higher CN), then the more resistant it should be to external stresses, thereby stiffening the material system. [31] Much like a simple spring system or ball and chain model of atoms, the more connections added to an atom, the more difficult it is to perturb it. The result does assume that the chemical environment, or strength of the bonds, remains approximately the same throughout each configuration computed. The change in average CN is rather small, on the order of 0.15 at the ends of the spectra. Within a small unit of density, the average CN does indeed change non-uniformly in some cases, but the overall trend towards higher stiffness and increased average CN remains.

To determine whether bond direction enforces stiffness in SiO₂, the CN result was further post-processed to yield the pseudo-bonds or vectors to the nearest neighbours, each vector was unique to avoid double counting of the same pair (only one of Si-O or O-Si). After taking the element-wise absolute values of the vector, it can then be projected onto the Cartesian axis to determine the portion of the bond in the $\hat{x},\hat{y},\hat{z}$ directions. Enforcing that each component is positive, is necessary to ensure that each vector adds constructively and not destructively which would yield cancellation. If these vectors are normalized to ignore bond length bias, then averaged, the resulting gradient in Figure8 and 9 is observed (colour scale). In particular, the averaged and standard deviation normalized bond direction components are shown with the $c_{11},c_{22},c_{33}$ (xx, yy, zz) elastic tensor components over the atomic densities. We note that in a sufficiently statistically sampled and large enough amorphous material such as SiO₂, the mechanical properties are expected to be isotropic. Since the $c_{11}$ and $c_{22}$ components have similar best-fit trends, but the $c_{33}$ component is different, this highlights the difficulty in sampling enough topologies and varied densities to properly determine the mechanical properties of amorphous materials.

In Figure8, there does not appear to be correlation over the spectrum of densities computed with regards to the elastic tensor component and the pseudo-bond in the same direction. If the average bond strength remains similar across each structure over every atom, one would expect the respective component of the tensor to be stiffer with more bonds along the same direction. To observe this, for any small width of atomic density, the data below the best linear fit should be darker in colour than those above. This suggests that either insufficient structures have been used to properly gather the necessary statistics, or that the direction of the bonds has no effect on the macroscopic stiffness of the amorphous material. The former being the obvious choice since amorphous materials restructure and compensate for external sources of stress.

The variation in average bond direction along each component is very small among the calculated structures, inferring that the amorphous configurations is well disordered. The standard deviation of the bond direction within each structure is shown in Figure9. The values suggest standard deviations on the order of 2% of the projection of the pseudo-bond components, which is substantially small, further confirming that the amorphous structures are sufficiently disordered. The lack of variation does indeed suggest that the average pseudo-bond directions for each structure do not affect the final macroscopic stiffness of the material. To further support this observation, the average bond length variation over the densities computed is $10^{-3}$Å, which is within numerical precision. Meaning that variations for the Si-O bonds within a given structure are extremely small or negligible.

4.3 Results & Discussion of Ultra-Thin Amorphous Silicon Dioxide

What has been discerned thus far is the dependency of SiO₂ on both the bond strength and the coordination number. Extending such a study to ultra-thin silicon oxide is valuable due to its prevalence in CMOS transistors, where they are typically on the order of sub-2nm. Decoupling the dielectric layer to study individually as a free-standing film enables us to study its individual mechanical properties, which is presented here. At the time of this research, it is not clear if mechanical properties of ultra-thin SiO₂ have been computed or derived experimentally.

Combining the approach of ultra-thin film and the amorphous elastic tensor methodologies, the elastic components of the different amorphous configurations (from their corresponding bulk structure) are calculated in Figure10. The films after cell optimization range from 1-1.6nm in thickness (surface to surface distance). The statistical averages are shown as the large dots, with the values of the many configurations given by the smaller, lighter dots. A guiding line is drawn between these averages to demonstrate the expected decrease for any configuration, on average, of a given tensor component. It is evident that, like the metallic and crystalline ultra-thin film systems, the most affected component in magnitude is that of $c_{33}$ of 50%- the one related to stress or strain applied normally to the surface. However, both the $c_{13}$ (43%) and the $c_{23}$ (48%) components demonstrate large percentage deviations relative to the bulk average, as observed in Table1. Which are responsible for the in-plane stiffness due to a normally applied stress. Any deviations of the tensor components with smaller magnitudes should be considered carefully, as the precision of the calculation is on the order of 10GPa, representing a 20% potential error. Note that due to selecting the thickness to be the surface-to-surface distance, the values of the elastic tensor will migrate to a larger thickness (rightward on the figure) and decrease in stiffness by the equivalent amount of thickness added. This is due to the stress being inversely proportional to any elastic tensor component.

	Avg. Bulk (GPa)	Avg. Film (GPa)	% $\Delta$
$c_{11}$	158	134	-15
$c_{22}$	161	129	-20
$c_{33}$	162	81	-50
$c_{12}$	50	38	-24
$c_{13}$	50	28	-43
$c_{23}$	49	26	-48
$c_{44}$	50	34	-31
$c_{55}$	48	36	-26
$c_{66}$	52	42	-19

In ultra-thin film systems (see Ref.[21]), dangling bonds at the surface contribute to the change in stiffness; increased strain at the surface was observed relative to where the surface would be if it were fully coordinated. Such an analysis cannot be performed easily for amorphous thin films, since the atomic configuration is widely varied, and the atoms drastically reorder themselves to accommodate for dangling bonds. Studying the change in average coordination of the top-most and bottom-most atoms within 1Å from the average coordination of the entire film suggests that indeed this is also the case in amorphous ultra-thin films between 1-1.6nm. The change in coordination number was found to be on average -0.9 with an associated standard deviation of 0.3, meaning that compared to the entirety of the ultra-thin film, there are on average atoms with one less bond at the surface. Thus it can be considered that the surface has dangling bonds. Additionally, the films were found to have expanded on average 8% times (standard deviation of 6%) in thickness when compared to its initial starting position from the fully coordinated equivalent. Combined, the lowering of the coordination at the surface and the stretching of the system in the direction of vacuum, suggest that even in amorphous ultra-thin films (compared to crystalline ones [21]), the change in stiffness is due to the dangling bonds and to the relaxed degrees of freedom along the film confined direction. The introduction of the vacuum breaks the hom*ogeneity found in-plane throughout the bulk amorphous state, creating a more deformable system out-of-plane. We expect that at larger thicknesses, the surface effects will be overwritten by the contribution from the centre of the film body, outweighing the effect of the dangling bonds and surface relaxation, thereby increasing the stiffness of the system towards the bulk values.

5 Conclusion

As more amorphous materials are being included in semiconductor device stacks, particularly so as miniaturization continues, the need for multiscale properties increases to properly simulate and predict these systems. This paper has addressed the need to calculate the elastic tensor of amorphous solids, required to understand the effect of strain in various subjects such as thermally induced strain, strain-enhanced diffusion, or adhesion of layers. We have reviewed the origins and discussed the plastic and elastic regimes of amorphous materials. Given that a material can be described in the linear elastic limit, a methodology to compute the elastic tensor has been shown to be in agreement with literature for a-SiO₂. The methodology can be extended to any amorphous system in the linear elastic regime, but the efficacy should still be evaluated. Due to the nature of amorphous materials, small changes in external strain often result in a change the topological phase, often leading to plastic transformations. Since the ab-initio methodology requires internal relaxation after externally applied strain, this often leads to a sudden change in topology, breaking the affine transformation requirement of the linear elastic regime. To accommodate this issue, a thin shell of atoms around the boundaries of a computational unit cell are selected to remain stationary during internal atomic relaxation. It was found that for a small statistical sample of bulk a-SiO₂, the average coordinate number combined with the atomic density is correlated with the Young’s Hill modulus. Additionally, there is no nearest neighbour directional component to the corresponding direction of elastic tensor component (ex. $c_{11}$ with the Cartesian X direction). For ultra-thin film a-SiO₂ constructed from the bulk counterpart, it was confirmed that similarly to crystalline materials, the dangling bond or electron density at the surface contributes to a reduced stiffness in the direction perpendicular to the surface.

6 Data availability

Pre-density functional theory relaxed unit cells of the bulk a-SiO₂ have been deposited in the Cambridge Crystallographic Data Centre (CCDC) online database under the deposition numbers of 2313547-2313596, which can also be found searching for this publication.

7 Acknowledgements

The resources and services used in this work were partly provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government. The various members of our group for discussions and aid over the course of this research.

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