Predicting Electronic Structure of Realistic Amorphous Surfaces

Amorphous materials underlie the functionality of many devices in photocatalysis, electronics, and other technological applications. The performance of such devices is largely altered by the interfacial characteristics of the material motivating better understanding of their structure function relationships. However, open questions remain about how to generate atomistic representations of amorphous surfaces that are realistic while being amenable to computational study using electronic structure methods. Here, model parameters are explored to generate accurate amorphous surface models using prototypical amorphous titanium dioxide and a melt‐quench approach. In particular, non‐standard model considerations such as minimum sample size, unit cell size, and quench conditions are varied to efficiently explore amorphous structural space. The results indicate different modeling parameters have a substantial effect on surface morphology and electronic structure that significantly alters the interpretations gained by computational study of amorphous interfaces. Critically, it is shown that the structural motifs that contribute to such differences are not detectable by short range structural analysis that has traditionally been used to assess the quality of melt‐quench derived amorphous structures.


Introduction
A fundamental goal of materials chemistry is to understand how specific arrangements of atoms in a solid give rise to observed properties.This is particularly challenging for amorphous materials, which are characterized by high degrees of disorder and lack of long-range structure; opposed to crystalline materials that can be defined in terms of a single repeating unit.Despite the difficulties in defining their atomistic structure, amorphous materials are ubiquitous and have high technological relevance.For DOI: 10.1002/adts.202300292example, structurally amorphous materials can act as the dielectric layers in transistors, [1,2] surface passivation layers, [3,4] and photocatalysts. [5,6]n the basis of this high technological interest, there have been a number of computational studies that attempt to probe the structural and electronic characteristics of amorphous materials.[12][13] .It is difficult to run first principles calculations with the large number of atoms needed to define the amorphous structure, as well as, generally to generate atomistic structures that are representative of real materials.To address the latter, the "melt-quench" method, [7][8][9][10]14,15] which involves using classical or ab initio dynamics simulations to heat a sample to a liquid phase, and then quenching the melt in order to lock in the amorphous configuration, has emerged as a convenient way to generate amorphous unit cells.
Alternative methods to generating amorphous structures, range from Reverse Monte Carlo Simulations to directly match to experimentally obtained X-ray data, [16,17] to amorphous unit cells generated via deep learning approaches, [18,19] to actually modelling the synthetic processes of real amorphous surfaces [20] .These vary in complexity, and each of these methods have their pros and cons, in particular, they require a lot of judgement calls on how to set up the simulations and determine what is a "good" fit to experimental data.Consequently, they have not displaced the melt-quench method.
While these atomistic models have provided unparalleled structural insights, a complete picture of their structure-function relationships has remained elusive as most of the models available are too large for high quality electronic structure methods, even density functional theory (DFT).A particularly challenging area of amorphous theory is understanding the surface structures and interfacial chemistry of these materials. [21]][27] However, the suitability of this approach for generating amorphous surfaces has recently been called into question. [28,29]Specifically, this approach necessarily involves a high amount of bond breaking as the periodic boundary condition in the surface direction of the low energy quench is disrupted when creating the interface.This may ultimately result in structural artifacts that render the unit cell non-representative of experimental counterparts.A more intuitive approach to generate thermodynamically relevant surface bonding is to cleave in the high energy melt phase so that the surface can be quenched and optimized under non-periodic boundary conditions.This allows for a larger amount of surface reconstruction and topology. [29,30]Critically, the morphology of experimental amorphous surfaces can be modified by differing synthetic conditions or synthesis precursors to produce varying degrees of surface roughness, [31,32] and thus a robust computational model of amorphous materials would need to be able to produce a range of surface morphologies.
Initial implementations of the different cleaving protocols have focused on the differences in the structural morphologies that can be generated.However, the electronic structure of these amorphous surfaces that are critical to the properties, for example identification of catalytic active sites, have been largely overlooked.To address this issue, this paper presents the first direct comparison between periodic and surface quench derived slab models of titanium dioxide (TiO 2 ).
Many applications, ranging from the electrodes in photocatalytic reactions, to hydrogen storage and sensors, to batteries, take advantage of TiO 2 's low cost, stability, and low environmental toxicity, making it one of the most studied amorphous materials.Indeed, it is oftentimes amorphicity of TiO 2 and its associated effects on electric and mechanical properties that determine device performance.For example, amorphous TiO 2 (am-TiO 2 ) nanosheets have shown increased surface enhanced Raman scattering, [34] the lithium-storage capability of silicon nanoparticles is greatly improved when encapsulated with am-TiO 2 , [35] and am-TiO 2 protective coatings on water splitting electrodes have been shown to simultaneously protect against oxidation while facilitating hole conduction. [36]n this work, we explore the parameters that influence the experimental accuracy of an am-TiO 2 surface model.In particular, the electronic structure of periodic versus surface quenched am-TiO 2 reveal critical differences in the predicted properties of the two methods.In addition, there are other modeling considerations that have not been standard for amorphous materials, including the number of unit cells needed to achieve adequate sampling of structural space, and the minimum unit cell sizes needed to reproduce experimental data, which have a substantial effect on the understanding that can be learned from computationally derived amorphous interfaces.The results of this study not only provide new insights into this important material, but also demonstrate the urgent need for robust computational approaches for generating realistic amorphous surfaces.

Melt-Quench Procedures
Common methodology for generating amorphous surfaces utilizes the "melt-quench-cleave" approach, where the unit cell was cleaved after quenching and may then be subject to geomet-ric optimization.This method had been implemented in the current work using the Matsui-Akaogi potential for TiO 2 [37] in LAMMPS. [38]First, individual Ti and O atoms were randomly packed into a cubic unit cell corresponding a density of approximately 3.9 gcm -3 using the packmol program. [39]The number of TiO 2 units and box size was varied from (TiO 2 ) 18 to (TiO 2 ) 100 with lattice parameters ranging from 8 x 8 x 8 Å to 15 x 15 x 15 Å.To study the effects of quench procedure and quench rate, the largest of these unit cells with 100 TiO 2 units was selected.Larger cells would be more ideal for studying amorphous structures, however unit cells with ≳ 300 atoms were already at the edge of what was computationally affordable for subsequent DFT calculations.Each cell was then equilibrated using fully periodic boundary conditions in an NVT ensemble for 100 ps at 3000 K, which was above the melting temperature of TiO 2 of 2116 K.This was following by further equilibration in an NPT ensemble for an additional 200 ps, where the density was converged to approximately 3.5 gcm -3 .These equilibration times were sufficient to yield converged ensembles (Figures S1 and S2, Supporting Information).Diffusion analysis showed the average mobility for each Ti atom was D ≈ 4x10 9 m 2 s -1 , which corresponds to the expected value for the liquid state.The system was then allowed to evolve in the NPT ensemble at 3000 K, with structures selected for cooling every 500 ps.The long sampling time of the melt ensured that unique atom configurations were selected at each point.The structures were then cooled in the NPT ensemble to 300 K at the prescribed cooling rate, following by additional NPT equilibration for 100ps.A total of six different cooling rates were examined ranging from 0.574 to 100 K/ps.

Periodic Quench
Cleavage of the structures was then accomplished by the insertion of approximately 100 Å of vacuum space above the surface, and neglecting the slab-slab dipole contribution to the total energy.The resultant structures were then subject to preliminary optimization using the Matsui-Akaogi potential until the maximum force on each atom was less than 1.0 x 10 3 kcal x mol Å -1 .Cells prepared using this method were henceforth referred to as those derived from periodic quenches.

Surface Quench
To prepare the surface quenched structures, the cleavage step above was performed following initial NPT and NVT equilibration of the bulk fully periodic unit cell at 3000 K.The system was then allowed to evolve for an additional 100 ps at 3000 K in the NVT ensemble prior to selecting structures for quenching.Structures were sampled from the NVT ensemble at the same interval as the bulk quenched structures, and cooled at the corresponding rates in the NVT ensemble.As the vacuum space in the cell prevents determination of unit cell lattice parameters, the x-and y-lattice parameters of each quench was adjusted to match the average achieved by the bulk cells of the corresponding quench rate.In most cases, the deviation of the lattice parameter from the high energy melt was less than 0.3 Å.Important structural motifs from each distribution are shown as insets.

Density Functional Characterization
Following molecular dynamics preparation, full quantum mechanical optimization of the unit cells was performed in the Vienna Ab initio Simulation Program (VASP). [40,41]An energy cutoff of 400 eV was adopted to expand the electronic wave function, and valence configurations of the atoms: Ti:[Ne]/3s3p4s3d, O:[He]/2s2p.All calculations were performed with Γ-point sampling using the PBEsol functional. [42]Each structure was relaxed until the energy between successive optimization steps was less than 10 −3 eV.Optimizations were performed with a 15 Å vacuum between periodic slab images, and neglect of the dipole contribution to the total energy in the direction orthogonal to the slab surface.

Unit Cell Size
There are well known issues of artificially periodic models representing experimentally aperiodic systems in DFT calculations, which has motivated many previous am-TiO 2 studies to select larger unit cell sizes than are traditionally used for crystalline materials.[46] In order to maximize the reliability of the representation of amorphous TiO 2 in the current work, a comparison of different DFT tractable unit cell sizes was performed for the surface quench procedure that is less well explored.Initial, bulk unit cell sizes and cubic cell volumes were: (TiO 2 ) 18 , V = 8 3 Å; (TiO 2 ) 36 , V = 10 3 Å; (TiO 2 ) 63 , V = 12 3 Å; and (TiO 2 ) 100 V = 15 3 Å.
In general, all unit cells (Figure 1) are able to reproduce the experimental radial distribution functions (RDFs) for O-O and Ti-O separation distances for sputtered amorphous TiO 2 . [33]Specifically, peaks occur for the innermost coordination spheres at approximately: r Ti − −O = 1.9, and r O − −O = 2.6.There is some small splitting of the r O − −O peak into a doublet for the (TiO 2 ) 18 unit cell that is not present in either the larger unit cells or experimental data.In the O-O RDF, the two peaks correspond to the separation between two O atoms that comprise a TiO 6 octahedra edge and a single octahedra, respectively (labeled in Figure 1a inset).The intensity of the latter is highly diminished in the experimental data.However, it is prominent in the calculated RDFs of many other am-TiO 2 models prepared using the Matsui-Akaogi potential. [8,47]hile O-O and Ti-O interatomic separation distances are reasonably captured by all unit cell sizes examined here, there is a notable difference in the Ti-Ti interatomic separation distances.In the experimental data, this spectra is characterized by two peaks centered on r Ti − −Ti = 3.0 Å and r Ti − −Ti = 3.5 Å.These two peaks correspond to the separation between Ti atoms of edgesharing and corner-sharing octahedra, respectively (Figure 1b inset).However, the small (TiO 2 ) 18 only shows the latter of these peaks.Furthermore, intensity of the first peak is greatly diminished in the (TiO 2 ) 36 unit cell.Indeed, only the (TiO 2 ) 63 and (TiO 2 ) 100 unit cells are capable of reasonably reproducing the experimental spectra.This suggests that 63 TiO 2 units or at least 12 Å lattice parameters are necessary to capture experimental structures.

Sampling Parameters
Independent amorphous quenches result in slightly different structures.Previous studies on bulk amorphous materials have generally produced sample sizes ranging from one [9,27] to a few [14] independent quenches.Such small sample sizes may be insufficient to explore enough structural space for universally applicable conclusions.However, it is important to avoid wasteful recalculation of duplicate structures to limit the computational cost of the electronic structure calculations.To quantify the amount of new structural information returned by each quench sample, the smooth overlap of atomic positions (SOAP) method [48] is adopted to describe 560 different quenches.A brief review of the SOAP methodology and its implementation in the current work is available in the SI, however a rigorous mathematical derivation is provided in refs. [48,49].
The SOAP method provides a distance metric to compute the similarity between two atomic environments that is invariant with respect to rotation or translation.The appeal of such an approach is that all structural features of the unit cell are considered, and a single distance indicates the similarity between different unit cells.To quantify the amount of new structural information that is being returned by each additional quench, this distance was projected into 1D.The sample standard deviation (s) of a given number of replicates was then divided by the population standard deviation () for all 560 quenches for increasingly large sample sizes (Figure 2).Using this method, it is found that s/ is 90 % converged (i.e., s/ <1.1) with approximately 124 different samples.By 172 unique quenches, s/ was converged to approximately 95%.This number provides a reasonable target for amorphous studies aiming to efficiently explore a large region of amorphous phase space using similar sized unit cells.Significant improvements in convergence beyond 95% were not achieved when testing up to 560 independent quenches.

Surface Versus Periodic Quench
Quench rate and quench boundary conditions have the largest effect on the surface structure generated in each amorphous surface unit cell.Six different quench rates were examined: 0.574 , 1, 5.745, 10, 57.447, and 100 K/ps.A total of 20 (TiO 2 ) 100 unit cells were prepared for each quench rate, with ten prepared using the surface quench protocol and ten prepared using the periodic quench protocol.
RDFs for each quench rate and quench protocol show excellent agreement among past experimental and theoretical results for bulk amorphous TiO 2 (Figure S3, Supporting Information), with no large distinctions [8,33] .Bond lengths and angle distributions (Figures S4-S6, Supporting Information) also are very similar for the differently prepared systems.Ti-O bond lengths are represented by a single distribution centered at 1.9 Å.The distribution of angle Ti-O-Ti is largely bimodal with peaks at approximately 97 • and 130 • .These angles can be attributed to the angles between corner-sharing and edge-sharing TiO 6 polyhedra, respectively.Accordingly, two populations of O-Ti-O are formed by axial-equatorial and axial-axial O-Ti-O angles.These results indicate there is no difference between the close-range geometry of individual TiO 6 polyhedra composing amorphous surfaces prepared by different methods.
In real am-TiO 2 samples, experimental measurements have shown that various degrees of surface roughness are possible depending on experimental conditions. [51]This ability to tune sur-face properties is key for optimizing the performance of amorphous devices.To determine the ability of the proposed computational model to capture these features the Van der Walls surface area and surface ruggedness for each quench rate (Figures S7 and S8, Supporting Information) were plotted.These results show that the surface quenching predicts comparable, but slightly less rough surfaces than the periodic quenching protocol at the two slowest quench rates.However, at increasing quench rates, the roughness of structures prepared from the surface quenching approach is increased, while structures from periodic quenches remain fairly constant.These findings are consistent with those found for melt-quench study of amorphous silica, [29] and demonstrate that a combination of surface quenching and quench rate can be used to control the degree of surface roughness.Using surface quenching approaches allows models with increased surface roughness of experimental samples to be generated, which is critical for predicting realistic electronic structure.
Experimentally different surface topologies have a significant effect on the surface chemistry of amorphous materials [2,31,32] .Thus, the generated compositions of each different surface topology (Figure 3) are critical to predicting the electronic and catalytic properties.For the 10 K/ps quench rate, the average Ti coordination number ( [N] Ti) is significantly reduced in periodic quenched cells compared to the surface quenched cells with average N = 5.25 and N = 5.34, respectively.For comparison, the experimental Ti coordination number in approximately 2 nm colloidal TiO 2 particles was found to range from 3.9 to 5.8 [52] , while [N] Ti for bulk sputtered am-TiO 2 was found at N = 5.4. [33]Plotting [N] Ti probability as a function of depth from the surface (Figure S9, Supporting Information) reveals an increased number of [4] Ti on the surface of periodic quenched cells, and almost none present in surface quenched cells.Importantly, at all quench rates, the periodically quenched cells have a high number of [4] Ti, resulting in drastically different surface structures for the periodic and surface quenches (Figure 3).Specifically, creating the vacuum necessarily involves breaking Ti-O bonds across the periodic boundary, resulting in under coordinated Ti atoms that are unable to fully relax post-cleavage during geometric minimization in the low energy melt.
The significant differences in the surface topology and structure of amorphous samples prepared by the two different approaches also result in significant differences in their electronic structure.Bandgaps for each quench method show that periodically-quenched structures (Figures S10-S12, Supporting Information) predict a narrowed band gap of up to approximately 0.18 eV at all quench rates.
Quite notably, surface quenched structures predict more reduced Ti atoms at all distances below the surface than periodically quenched structures (Figure 4).In all cases, electron density is concentrated away from the surface, reaching the bulk value at a depth of approximately 2.5 Å.However, periodic quenched cells predict an average Ti oxidation state that is approximately 0.01 e -greater at all depths.Average O oxidation is correspondingly reduced for surface quenched cells, indicating a more covalent character in these cells.This is indicative of the very different chemistry predicted from the two surface preparation protocols.
Visualization of the charge density shows that the valence band of periodic quenched structures is the result of localized surface orbitals (Figure 5a; Figure S13, Supporting Information).Such  Bader charge [50] on Ti (a) and O (b) as function of depth for surface and periodic quenches at 10 K/ps quench rate averaged over ten structures each.
orbitals have been found to compose the valence band maximum of approximately 70% (Figure S13, Supporting Information) of the quenches studied here, and primarily consist of the oxygen 2p orbitals near a [5] Ti.For surface quenched cells, the character of these orbitals remains the same (i.e., localized clusters on O 2p orbitals).However, these clusters occur primarily in the subsurface layers on the O (or Ti) directly under a surface Ti (or O), or in bulk states (Figure 5b; Figure S14, Supporting Information).Such or-bitals also comprise approximately 70% of the surface quenched structures studied here.However, in approximately 10% of cells prepared by both methods, the valence band maximum occurs localized around a [7] Ti located in the bulk away from the surface, indicating that bulk [7] Ti should not be neglected in understanding the properties of am-TiO 2 surfaces.
The subtle differences in charge distribution between surface and periodic quenched structures would result in vastly different material interpretations if a single one of these models were used to predict the technologically relevant properties of am-TiO 2 surfaces.Indeed, the planar averaged electrostatic potential along the z-axis (Figure 6) shows peaks at the surface for each of the periodically quenched cells here that differ from surface quenched cells by approximately 1 eV.This is true for all quench rates studied here (Figure S15, Supporting Information), and shifts the predicted ΔG • of electron transfer processes or adsorption processes by 40 times k B T.

Conclusion
This study established minimum conditions necessary for achieving experimentally accurate melt-quench derived amorphous surfaces.Specifically, it was found that unit cells smaller than (TiO 2 ) 63 with 12 Å lattice parameters were unable to match the experimentally determined Ti-Ti RDFs.Additionally, approximately 172 replicate quenches captures 95% of the amorphous structural motifs.Surface quenched structures generated with these guidelines provide efficient rules of thumb for future studies attempting to understand other amorphous phase surfaces.
Furthermore, a detailed structural and electronic investigation into the effects of periodic boundary conditions when quenching am-TiO 2 has been presented.The results indicate significant differences in the predicted properties between the two methods, including more oxidized Ti atoms across the entire unit cell in surface quenched cells, alteration of the band edge character and location, and increased electrostatic potential at interfaces of surface quenched cells, resulting in significant differences in the properties predicted of the technologically relevant experimental interfaces.Critically, the structural origin of these effects are not detectable by short-range structural analysis that has been commonly used to assess the quality of an amorphous material model, such as bond length and angle distributions and RDFs.This demonstrates the need for increased scrutiny of melt-quench prepared surfaces, and improved structural descriptors for medium-and long-range order, including surface properties like ruggedness and coordination.Additionally, real materials possess a range of defects or dopants that likely alter structural and electronic properties.Such effects can be built into the models proposed here via atomic deletion or substitution in a quenched cell.However, more robust methods for understanding the structural relaxations that occur upon doping will likely coincide with the development of improved force fields that accurately capture the interaction between the dopant and host lattice.
The most realistic interfaces were generated via surface quenching.Interestingly, the different quench rates provide a way to generate a range of surface topologies that can be compared to experimental structures.Importantly, these models are a promising foundation to enable further study of amorphous surface chemistry, including for applications in molecular adsorption, catalysis, and interfacial electron transfer.

Figure 1 .
Figure 1.Radial distribution functions O-O a), Ti-Ti b), and Ti-O c) of various (TiO 2 ) n of surface quenched amorphous unit cells compared to the sputtered.am-TiO 2[33] Important structural motifs from each distribution are shown as insets.

Figure 2 .
Figure 2. Structural variability (s/) with increasing sample sizes.The lines at number of samples = 124 and 172 indicate the number of quenches needed to converge s/ to 90 and 95%, respectively.Shading indicates convergence threshold for 90, 95, and 99%, with increasing opacity.

Figure 3 .
Figure 3. Periodic (left) and surface quenched (right) unit cells from quench rate 10 K/ps, with the Ti coordination colored.

Figure 4 .
Figure 4.Bader charge[50] on Ti (a) and O (b) as function of depth for surface and periodic quenches at 10 K/ps quench rate averaged over ten structures each.

Figure 5 .
Figure 5. Representative orbitals of surface a) and bulk b) states from the periodic and surface quenches, respectively.

Figure 6 .
Figure 6.Electrostatic potential along the z-direction.Results are averaged over ten cells for each quench condition prepared at 10 K/ps quenched cell.