Identifying the ground state structures of point defects in solids

1]Irea Mosquera-Lois
These authors contributed equally to this work.

[1,2]Seán R<EMAIL_ADDRESS>These authors contributed equally to this work.

2]Aron Walsh

[1]David O<EMAIL_ADDRESS>
*[1]Thomas Young Centre and Department of Chemistry, University College London, 20
Gordon Street, London, WC1H 0AJ, UK

*[2]Thomas Young Centre and Department of Materials, Imperial College London, Exhibition Road, London, SW7 2AZ, UK

Point defects are a universal feature of crystalline materials. Their identification is often addressed by combining experimental measurements with theoretical models. The standard approach of simulating defects is, however, prone to missing the ground state atomic configurations associated with energy-lowering reconstructions from the idealised crystallographic environment. Missed ground states compromise the accuracy of calculated properties. To address this issue, we report an approach to efficiently navigate the defect configurational landscape using targeted bond distortions and rattling. Application of our workflow to a range of materials (CdTe, GaAs, Sb2S3, Sb2Se3, CeO2, In2O3, ZnO, anatase-TiO2) reveals symmetry breaking in each host crystal that is not found via conventional local minimisation techniques. The point defect distortions are classified by the associated physico-chemical factors. We demonstrate the impact of these defect distortions on derived properties, including formation energies, concentrations and charge transition levels. Our work presents a step forward for quantitative modelling of imperfect solids.

§ INTRODUCTION

Defects control the properties and performance of most functional materials and devices.
Unravelling the identity and impact of these imperfections is, however, a challenging task. Their dilute concentrations hinder experimental identification, which is often tackled by combining characterisation measurements with ab-initio techniques.
The standard modelling approach, based on local optimisation of a defect containing crystal, is prone to miss the true ground state atomic arrangement, however.
The chosen initial configuration, which is often initiated as a vacancy/substitution/interstitial on a known crystal site (Wyckoff position) and all other atoms retaining their typical lattice positions, may lie within a local minimum or on a saddle point of the potential energy surface (PES), trapping a gradient-based optimisation algorithm in an unstable or metastable arrangement[1, 2, 3, 4, 5, 6, 7, 8, 9] as illustrated in <ref>.
By yielding incorrect geometries, the predicted defect properties, such as equilibrium concentrations, charge transition levels and recombination rates, are rendered inaccurate. This behaviour can severely impact theoretical predictions of material performance, including photovoltaic efficiency[10], catalytic activity[11], absorption spectra[12] and carrier doping[13], highlighting the pressing requirement for improved structure prediction techniques for defects in solids.

Potential energy surface for the neutral cadmium vacancy in CdTe, illustrating the global minimum (Te dimer) and local minima (Bipolaron and Tetrahedral configurations). A standard optimisation from the initial, high-symmetry configuration (black star) gets trapped in the metastable Tetrahedral configuration. Adapted from Matter, I. Mosquera-Lois and S.R. Kavanagh, In Search of Hidden Defects (2021) with permission from Elsevier.

Several approaches have recently been devised to navigate the defect configurational landscape.
Arrigoni and Madsen[1] used an evolutionary algorithm enhanced with a machine learning model to explore the defect PES and identify low energy structural configurations. While its robust performance makes it suitable for in-depth studies of specific defects, its complexity and computational cost hinders its applicability in standard defect studies, where all intrinsic defects in all plausible charge states (and relevant extrinsic defects) are modelled.
On the other hand, Pickard and Needs[14] employed random sampling of the PES.
Essentially, they remove the defect atom and its nearest neighbours and reintroduce them at random positions within a 5 Å cube centred on the defect, thus ensuring sampling of an important region of the PES. Despite successfully identifying the ground state structures for several systems[15, 16, 17, 14, 18], constrained random sampling in a high-dimensional space lowers efficiency and increases computational cost.

To improve PES sampling efficiency, domain knowledge can be employed to bias the search towards energy-lowering structural distortions.
Accordingly, we exploit the localised nature of defect distortions, as well as the key role of the defect valence electrons, to guide the exploration of the PES.
By combining these biases with lessons learnt from crystal structure prediction, we develop a practical and robust method to identify the defect ground and low-energy metastable states. Its application to eight host materials (CdTe, GaAs, Sb2S3, Sb2Se3, CeO2, In2O3, ZnO, TiO2) reveals a myriad of energy-lowering defect distortions, which are missed when relaxing the ideal high-symmetry defect configurations.
Notably, energy-lowering distortions that are missed by standard geometry relaxation were found in each system investigated.
Classifying these atomic rearrangements by distortion motif, we outline the main physico-chemical factors that underlie defect reconstructions. Moreover, we demonstrate the strong effect on defect formation energies and their charge transition levels, illustrating the
importance of exploring the defect configurational space for accurate predictions of defect properties.
Additionally, while defect properties are often determined by the ground state structure, there are cases when low energy metastable structures can significantly affect behaviour[10, 19, 20]. Accordingly, we tested our approach for low-energy metastable configurations by considering the DX centres in GaAs, which have been extensively studied due to their anomalous physical properties and technological importance. By finding the vast majority ($>90$%) of configurations identified by previous investigations, we demonstrate the applicability of our approach to locate both ground state and low-energy metastable defect structures, constituting an affordable and effective tool to explore the defect configurational landscape.

§ METHODOLOGY

§.§  Structure search strategy

I. Local bond distortion

The role of valence electrons in defect reconstructions has been demonstrated by the defect-molecule model developed by Watkins[21, 22, 23, 24, 25, 26] and Coulson[27].
It successfully explained the Jahn-Teller distortions observed for vacancy centres in silicon and diamond, and continues to be applied to rationalise defect reconstructions[28, 29, 5, 30, 31, 10, 12, 32, 33, 34, 35].
This highlights the suitability and utility of defect valence electrons as an approximate indicator of likely defect distortions. As such, our method incorporates this feature, as well as the localised nature of defect reconstructions, to generate a reasonable set of chemically-guided distortions to sample the PES.
Specifically, we use the number of missing/additional valence electrons of the defect species to determine the number of nearest neighbour atoms to distort. To generate a set of trial structures, the initial defect-neighbour distances $d_0$ are distorted by varying amounts, with the range set by the minimum and maximum distortion factors $k_{min}$ and $k_{max}$, such that $d_{min} = k_{min}d_0$ and $d_{max} = k_{max}d_0$, and the increment parameter $\delta$ which determines the sampling density of the trial distortion mesh.
Thus, the distorted defect-neighbour distances $d$ are given by

\begin{equation}
d_n =  k_{n}d_0
\end{equation}

\begin{equation}
k_{n} = k_{min} + n\delta \quad \textrm{with} \quad n=0,1,2,...,\Big(\frac{k_{max}-k_{min}}{\delta}\Big)
\end{equation}

While these parameters may be varied, in our work we found a distortion range of $\pm60\%$ (i.e. $k_{min} = 0.4$ and $k_{max} = 1.6$) and increment of $10\%$ ($\delta = 0.1$) to be an optimal choice, with a sufficiently wide and dense distortion mesh to identify all known energy-lowering distortions (see SI Section III for more details).
Similar bond distortions have previously been applied by Pham and Deskins[36] to locate the ground state of small polarons in three metal oxides. Analogously, these initial distortions aim to escape the local energy basin in which the ideal, undistorted structure may be trapped.
Further, by only distorting the defect nearest neighbours — and thus restricting to a lower-dimensional subspace, the method aims to sample the key regions of the PES that may comprise energy-lowering reconstructions, such as Jahn-Teller effects or rebonding between under-coordinated atoms.

The utility of electron count as an indicator for the number of defect-neighbour bonds to distort is demonstrated for the neutral cadmium vacancy in CdTe in <ref> (a).
Here, bond distortions were applied to differing numbers of defect neighbours. Best performance (i.e. widest range of distortions yielding the ground state) was obtained when distorting two neighbours, supporting the feature choice given that the vacancy site lacks two valence electrons relative to the original Cd atom.

II. Atomic rattle

Following the nearest neighbour distortion, the method applies random perturbations to the coordinates $r$ of all atoms $N$ in the supercell. This aids to escape from any metastable symmetry trapping and locate small, symmetry-breaking distortions (i.e. nearby lower energy basins).
This step has proven useful in other structure-searching approaches, such as the investigation of symmetry breaking distortions in perovskites[37] or point defects[38, 20].
The magnitudes of these randomly-directed atomic displacements $d$ are themselves randomly selected from a normal distribution of chosen standard deviation $\sigma$ (<ref>).

\begin{equation}\label{eq:rattle}
\begin{aligned}
r'_i = r_i + d \quad
& \textrm{with} \quad i = 0,1,...,3N
& \textrm{and} \quad d
\leftarrow \frac{1}{\sigma \sqrt{2\pi}}
\exp \big( -\frac{d^2}{2\sigma^2}  \big)
\end{aligned}
\end{equation}

To determine the optimum magnitude of the perturbation, several standard deviations were tested for a series of defects – shown in the SI Section III and exemplified in <ref> (b) for VCd0 in CdTe. In general, we found values between 0.05 - 0.25Å to give good performance, and adopted 0.25Å since it was the only distortion capable of finding the ground state of $\rm S_i^{-1}$ in Sb2S3 (SI Fig. S3).

Additionally, we also investigated a `localised' rattle, where the perturbations were only applied to atoms located within 5 Å of the defect, as recently employed in other defect studies[39, 40, 38].
However, this yielded inferior performance for several defects (less bond distortions leading to the ground state) without significant reduction in the number of ionic relaxation steps (SI Fig. S4). While the final structure is still only locally deformed, it appears that the external potential induced by the long-range symmetry of the surrounding host crystal biases the initial forces toward retaining the high-symmetry metastable defect structure. Therefore rattling was applied to all atoms in the simulation supercell.

(a) Schematic representation of our approach, as described in the Methods section, using the neutral cadmium vacancy (VCd0) in CdTe as an example. Cd in blue, Te in gold and VCd in black, with the distorted Te neighbours shown in a gold/white. (b) Relative energy of relaxed structures for VCd0 in CdTe, for different numbers of distorted defect neighbours and (c) rattle standard deviations. A localised rattle (where only atoms within 5 Å to the defect are distorted) is also shown ($\sigma=0.25_{loc}$ Å). The widest interval of bond distortions leading to the ground state is obtained by distorting two nearest neighbours.
Only negative bond distortions are shown since all positive ones yield the metastable `Tetrahedral' configuration.

§.§ Computational details

The bond distortion and rattling procedure for defect structure generation has been implemented in Python[41, 42].
The underlying total energy and force calculations were performed with plane-wave Density Functional Theory (DFT) within VASP[43, 44], using the projector augmented wave method.[45].
All calculations were spin-polarised and based on the HSE screened hybrid exchange-correlation functional[46].
When reported in previous studies, the bandgap-corrected Hartree-Fock exchange fraction ($\alpha$) was used, which is common practice in the field[47, 48, 49, 50].
Otherwise HSE06 ($\alpha = 25\%$) was employed. The basis set energy cut-off and $k$-point grid were converged to 3 meV/atom in each case. The converged parameters, valence electron configurations and fractions of Hartree-Fock exchange used for each system are tabulated in the SI Table I.

The conventional supercell approach for modelling defects in periodic solids[51] was used.
To reduce periodic image interactions, supercell dimensions of at least 10 Å in each direction[52] were employed (SI Table I).
The use of a large supercell justifies performing the PES exploration with $\Gamma$-point-only reciprocal space sampling, thus increasing speed while retaining qualitative accuracy. This makes the cost of these test geometry optimisations very small compared to the total cost of the fully-converged defect calculations.
The $\Gamma$-point approximation was further validated by optimising the defect ground state and metastable geometry (obtained by relaxing the high-symmetry, undistorted structure) with denser reciprocal space sampling (converged within 3 meV/atom). For materials containing heavy-atom elements (CdTe, Sb2S3, Sb2Se3, CeO2, In2O3), relativistic effects were accounted for by including spin-orbit coupling (SOC) interactions. To calculate the defect formation energy, we followed the approach described by Freysoldt et al[53], using the charge correction developed by Kumagai and Oba[54]. Madelung energies were calculated using Mulliken charges with the LOBSTER package[55, 56, 57, 58, 59]. To generate and analyse the defect distortions described in this work, we developed the  python and command-line package (<https://github.com/SMTG-UCL/ShakeNBreak>) which will be described in a forthcoming publication.

§.§ Defect suite

To test our method we chose a set of eight host crystals that provide diversity in crystal symmetry, coordination environments and electronic structure. CdTe and GaAs are tetrahedral semiconductors that adopt the zincblende structure (space group $F\bar43m$). Sb2Se3 and Sb2S3 are layered materials (space group $Pnma$), composed of one-dimensional [Sb4X6]$_n$ ribbons with covalent metal-chalcogen bonds within each ribbon and van der Waals interactions between ribbons[60, 61, 62, 63, 64, 65]. Beyond these covalently-bonded crystals, we also studied four metal oxides (In2O3, ZnO, CeO2, TiO2). ZnO crystallises in the hexagonal wurtzite structure (space group $P6_3mc$) with tetrahedral coordination, while In2O3 adopts a bixbyite structure (space group $Ia3$). CeO2 crystallises in the fluorite structure with a face-centred cubic unit cell (space group $Fm\bar3m$). The anatase phase of TiO2 was considered, which adopts a tetragonal structure composed of distorted TiO6 octahedra (space group $I4_1/amd$). For the covalently-bonded systems, all intrinsic defects were studied, while for the oxides we focused on defects where reconstructions had been reported or were likely (oxygen interstitials in ZnO, In2O3 and a-TiO2 and dopant defects in CeO2). For the materials that can host defects in several symmetry inequivalent sites, the lowest energy one was selected if the energy difference with the other sites was higher than 1 eV, otherwise all symmetry inequivalent sites were considered.

§ RESULTS

The application of our method to a series of semiconductors revealed a range of defect reconstructions, which are missed when performing a geometry optimisation of the high symmetry configuration. We categorise these reconstructions into distinct motifs according to the chemical origins of the energy-lowering distortion, as exemplified in  <ref> and discussed hereafter.

Types of symmetry-breaking reconstructions at point defect sites identified by our method.

§.§ Rebonding

For covalently-bonded materials, defect reconstructions identified by our method often entailed a change in the bonding arrangement at the defect site (`rebonding'), such as dimer formation or replacing cation-cation/anion-anion bonds with more favourable cation-anion ones.

Dimerisation. For many vacancies and interstitials, symmetry breaking was found to produce dimer bonds between under-coordinated atoms. This novel bond formation results in these distortions being highly-favourable, with energy decreases from 0.4eV to over 2.5eV (<ref>), between the ground state and metastable structure obtained from standard relaxations.
As expected given the large energy differences, the metastable and ground state configurations also display significant structural differences, which we quantify here by summing the atomic displacements between structures which are above a 0.1 Å threshold. The large structural rearrangements for dimer reconstructions are demonstrated by displacement values ranging between 2.0 and 24.3 Å, with the distortion mainly localised to the dimer-forming atoms (SI Figs. S8, S9, S10). For vacancies, the distortion typically entails two of the under-coordinated defect neighbours displacing toward each other to form a bond, while for interstitials it is the additional atom which displaces towards a nearby neighbour.

Dimerisation. Energy difference and summed atomic displacements ($\Sigma$Disp.) between the ground state identified by our method and the metastable configuration found when relaxing the ideal, undistorted defect structure (i.e. $\rm \Delta E = E_{\rm ground} - E_{\rm metastable}$).

Defect     Material     Charge      $\Delta$E (eV)     $\Sigma$Disp. (Å)

VTe     CdTe     0     -0.42     6.0

Sb_i     Sb2S3     -3     -0.46      12.8

Defect     Material     Charge     $\Delta $E (eV)     $\Sigma$Disp. (Å)

VCd     CdTe     0     -0.50     6.1

-1     -0.66     8.2

0     -0.78     8.7

+1     -0.84     8.9

+2     -2.02     12.8

-1     -0.80     7.2

0     -0.91     9.0

+1     -2.10     14.7

+2     -2.15     16.6

-1     -0.80     9.9

0     -1.03     9.0

+1     -2.31     11.8

+2     -2.03     24.3

-1     -0.99     8.9

0     -1.10     12.2

+1     -2.50     12.8

+2     -2.59     21.9

$\rm S_i$    Sb2S3     -1     -0.54     9.4

0     -1.47     1.6

+1     -2.44     3.4

0     -1.87     4.3

+1     -2.22     3.9

0     -2.23     3.1

+1     -1.99     3.2

[1]Metastable and ground state structures with bond distance labelling shown in SI Figs. S8, S9, S10.

Within the cases reported in <ref>, it is instructive to further consider the antimony vacancies (VSb) in the quasi-one dimensional materials Sb2(S/Se)3.
In contrast to other systems, the dimer reconstruction is favourable across several charge states, with its stability increasing with the magnitude of the charge state of the defect. As indicated by Crystal Orbital Hamilton Population analysis (COHP) [66, 67, 68], from the singly negative to the double positive state, we observe greater net bonding anion-anion interactions (SI: Table 3, Figure 11), thereby leading to stronger bonds and thus a more favourable reconstruction. Furthermore, for the neutral and positive charge states, the under-coordinated vacancy neighbours form two anion-anion bonds (yielding a S-S-S trimer), which is achieved by a large displacement of one of the vacancy neighbours towards the other two (<ref>). This remarkable ability to distort likely stems from their soft, quasi-1D structure[61], with van der Waals voids between atomic chains[69] (SI Fig. S5).
Despite the key role of anion dimerisation for VSb, which significantly impacts its energy and structure, this reconstruction has not been reported by previous studies on Sb2S3[70] or Sb2Se3[71, 72, 73, 74, 75, 76], due to the local minimum trapping which our method aims to tackle. While sulfur dimers were not identified in Sb2(S/Se)3, they were reported for VBi and $\rm S_i$ in the isostructural material Bi2S3[77], where their formation stabilises abnormal charge states for both species, rendering them donor defects — contradicting the typical acceptor nature of cation vacancies and anion interstitials.

Anion-anion dimer formation undergone by VSb,2-1,0,+1,+2 in Sb2S3, where the subscript differentiates between the two symmetry inequivalent sites of Sb (shown in SI Fig. S7). Similar reconstructions were observed for the other symmetry inequivalent site and in Sb2Se3. For each charge state, on the top we show the configuration obtained by relaxing the ideal, high-symmetry geometry and on the bottom the ground state found with our method. For the singly negative charge states, the distortion is driven by the formation of one sulfur dimer while for the neutral and positively charge states a trimer is formed.
VSb in black and S in yellow. Pseudo-bonds from the vacancy position to neighbouring S atoms are shown for the metastable configurations to illustrate the coordination environment and the arrows illustrate the key displacement from the high-symmetry structure.

Beyond covalently-bonded materials, anion-anion bond formation is also observed for anion interstitials in systems with a stronger ionic character (In2O3, ZnO, a-TiO2).
Rather than leading to hole localisation in the interstitial atom and/or neighbouring anions (metastable state), the defect can accommodate the charge deficiency by forming oxygen dimers, resulting in highly favourable reconstructions (<ref>). When only two electrons are missing ($\rm O_i^{0}$), the interstitial atom displaces towards a lattice oxygen forming a peroxide ($\rm O_2^{-2}$, d(O-O)=1.45Å, SI Fig. S10), which is 1.5-2.0 eV more favourable than the metastable double polaron state. If an additional electron is removed (Oi+1), a hole is trapped in one of the peroxide antibonding $\pi$* orbitals (e.g. $\rm h^+ + O_2^{-2}$, d(O-O)=1.31Å, SI Fig. S12). Remarkably, this peroxide hole trapping yields a stabilisation $>$2eV compared to the system with 3 holes localised on the defect region, and has previously been reported for bulk lithium peroxide (Li2O2)[2].

Overall, these anion-anion dimers lead to highly-favourable distortions and stabilise abnormal charge states for anion interstitials and cation vacancies. This unexpected behaviour highlights the key role of exploring the defect configurational space for accurate theoretical predictions of defect properties, while also demonstrating the importance of considering a wide range of charge states, as unforeseen chemical interactions (facilitated by defect reconstructions) may lead to unexpected stabilisations.
Finally, we note that dimer reconstructions at defects are not uncommon, and have been previously reported for numerous vacancies and interstitials, including VSe0 in ZnSe, CuInSe2 and CuGaSe2[12], VS0 in ZnS[12], VTi0,-1 and VZr0 in CaZrTi2O7 [18], Oi0 in In2O3[78], ZnO[79], Al2O3[6], MgO[80, 81], CdO[82], SnO2[83, 84], PbO_2[85], CeO2[86], BaSnO3[87] and
In2ZnO4[88], $\rm Ag_{i}^{0}$ in AgCl and AgBr[89], VI-, Ii0,  Pbi0, $\rm Pb_{CH_3NH_3}^{0}$ and $\rm I_{CH_3NH_3}^{0}$ in CH3NH3PbI3 [90, 91, 92, 93], $\rm Pb_i$ in CsPbBr3[94], (CH3NH3)3Pb2I7[95],
(CH3NH3)2Pb(SCN)2I2[96] and Sni in CH3NH3SnI3[97].

Cation-anion rebonding.
Beyond dimerisation, rebonding between defect neighbours can also include replacing cation-cation/anion-anion (homoionic) bonds with more favourable cation-anion (heteroionic) bonds. This reconstruction motif is primarily observed for antisite defects, where the antisite may displace from its original position towards an oppositely-charged ion to form a new bond, while breaking a cation-cation/anion-anion bond. This is illustrated in <ref> for $\rm As_{Ga}^{-2}$ in GaAs and was also found for $\rm Sb_{S}^{+3}$ and $\rm S_{Sb}^{+3}$ in Sb2S3. In the former ($\rm As_{Ga}^{-2}$), both the cation and anion displace towards each other (<ref>), from an original separation of 4.00Å to 2.49Å (close to the bulk Ga-As bond length of 2.45Å), breaking an As-$\rm As_{Ga}$ bond in the process.
Interestingly, this behaviour contrasts with that of $\rm Te_{Cd}^{-2}$ in the isostructural compound semiconductor; CdTe, which displaces to form a Te-Te dimer[5] rather than the cation-anion rebonding witnessed here. The differing reconstructions exhibited by these nominally similar anion-on-cation antisite defects – both in zinc blende compound semiconductors, may be rationalised by the greater bond energy of Te-Te (260kJ/mol) compared to As-As (200kJ/mol)[98], suggesting a greater proclivity toward anion dimerisation.

Cation-anion rebonding undergone by $\rm As_{Ga}^{-2}$ in GaAs. On the left is the metastable $T_d$ structure identified by standard defect relaxation, and to the right the ground state $C_s$ symmetry identified by our method. Ga in green and As in purple.

Cation–anion rebonding. Energy difference and summed atomic displacements ($\Sigma$Disp.) between the ground state identified by our method and the metastable configuration found when relaxing the ideal, undistorted defect structure (i.e. $\rm \Delta E = E_{\rm ground} - E_{\rm metastable}$).

Defect     Material     Charge     $\Delta$E (eV)     $\Sigma$Disp. (Å)

Sb_S     Sb2S3
+3     -0.39     15.8

S_Sb     Sb2S3
+3     -1.22     9.0

S_i     Sb2S3
+3     -0.45     11.5

As_Ga     GaAs     -2     -0.37     5.2

Cd_i     CdTe     +2     -0.46     5.8

[1]Metastable and ground state structures with bond distance labelling shown in SI Fig. S13

Cation-anion rebonding is also found in the antimony antisite in Sb2S3 ($\rm Sb_{S}^{+3}$), where the high-symmetry metastable structure and the ground state differ by the formation of an additional Sb-S bond (SI Figure S13).
This extra cation-anion bond has a distance of 2.5Å, again comparable to the bulk Sb-S bond length of 2.5-2.8 Å, and lowers the energy by 0.4eV (<ref>). Beyond antisites, this behaviour is also expected for extrinsic substitutional defects and dopants, where the same preference for heteroionic bonds can drive reconstructions to distorted, lower-energy arrangements.

This behaviour is not exclusive to substitution-type defects, however, with similar reconstructions undergone by interstitial defects. For instance, the sulfur interstitial ($\rm S_{i}^{+3}$) in Sb2S3 displaces to form a $\rm S_i-Sb$ bond which lowers the energy by 0.45 eV (<ref>, SI Fig. S13).
Similar interactions drive the reconstruction for the cadmium interstitial in CdTe ($\rm Cd_{i}^{+2}$), where $\rm Cd_{i}^{+2}$ moves to a Te (rather than Cd) tetrahedral coordination, again with bond lengths (2.85Å) similar to the bulk cation-anion bond length (2.84Å) and lowering the energy by 0.46eV. Beyond the explicit formation of these heteroionic bonds, the change in the interstitial position also results in more favourable electrostatic interactions, as reflected by the lower Madelung energy of the ground state ($\Delta E_{\textrm{Madelung}} =$ -2.0eV).
While this configuration is also found by considering a range of potential interstitial sites without distortion, the identified reconstruction demonstrates the ability of this approach to locate major structural distortions, where both the position of the defect atom and its coordination change significantly (summed atomic displacements of 5.8Å and $\rm Cd_i^{+2}$ moving by 0.86Å).

§.§ Crystal field & Jahn-Teller

For defects in materials with a stronger ionic character, crystal field or Jahn-Teller effects can also drive symmetry-breaking reconstructions. In this category, we focus on the distortions introduced by a series of divalent metal dopants (Be, Ni, Pd, Pt, Cu) in ceria (CeO2).
Previous studies had shown these dopants to favour the formation of a charge-compensating oxygen vacancy (VO+2) and undergo significant reconstructions, which were only found by testing a range of chemically-guided manual distortions for each dopant[11].

Accordingly, we tested our method by applying it to this system, finding it to successfully reproduce all previously-reported ground state structures, as well as identifying a novel lower-energy reconstruction for the Ni dopant. Here, Ni and Pd are found to distort from an originally cubic coordination to a more favourable square planar arrangement (SI Fig. S14). This is also the case for Pt, however unlike the other dopants this ground state configuration is also found by a standard relaxation.
For Ni and Pd, crystal field effects drive the displacement of the dopant by 1/4 of a unit cell (1.2Å) from the centre of the cube towards one of the faces (<ref>, SI Fig. S14). By lowering the energy of the occupied $d$ orbitals (as depicted in the electron energy diagram of SI Fig. S15), the reconstruction leads to an overall stabilisation of 0.30 and 0.87eV for Ni and Pd, respectively.
Notably, the nickel distortion was overlooked by the previous study[11], likely due to limited exploration of the PES and the shallower and narrower energy basin – stemming from reduced crystal field stabilisation energy (<ref>) due to weaker hybridisation between the anions and less diffuse 3$d$ orbitals (compared to 4$d$ and 5$d$ for Pd and Pt).

Finally, in the case of copper, a $d^9$ metal, it is the Jahn-Teller effect which drives a reconstruction towards a distorted square-planar arrangement (<ref>). By splitting the partially-occupied electron levels, it lowers the energy of the occupied $d$ orbitals, as shown in the orbital energy diagram of SI Fig. S16, and leads to an overall stabilisation of 0.67 eV.

Crystal field and Jahn-Teller driven distortions. Energy difference and summed atomic displacements ($\Sigma$Disp.) between the ground state identified by our method and the metastable configuration found when relaxing the ideal, undistorted defect structure (i.e. $\rm \Delta E = E_{\rm ground} - E_{\rm metastable}$).

Defect     Material      $\Delta$E (eV)     $\Sigma$Disp. (Å)

$\rm Ni_{Ce}^{-2}$ + VO2+     CeO2     -0.30       2.2

$\rm Pd_{Ce}^{-2}$ + VO2+     CeO2     -0.87      5.8

$\rm Cu_{Ce}^{-2}$ + VO2+     CeO2     -0.67      3.7

[1]Metastable and ground state structures with bond distance labelling shown in SI Fig. S14

§.§ Electrostatically-driven

Finally, for certain interstitials or substitutional defects, symmetry-breaking distortions can yield lower energy structures through optimising electrostatic interactions. Compared to cation-anion rebonding, which is driven by a combination of ionic and covalent bonding interactions, here no explicit new bonds are formed, but rather the lattice electrostatic (Madelung) energy is lowered by defect rearrangement. The recurrence of this distortion motif for size-mismatched substitutions is related to their disruption of the nearby lattice framework (and thus long-range electrostatics), making reconstructions to reduce the lattice strain favourable.
For instance, this behaviour was observed for the beryllium dopant in CeO2, which also forms a charge-compensating oxygen vacancy, i.e. $\rm Be_{Ce}^{-2}+$VO+2 (as witnessed for other divalent dopants[11, 99, 100]).
Surprisingly, instead of adopting the tetrahedral coordination of its parent oxide, beryllium distorts to a trigonal environment (SI Fig. S17 (a,b)). This behaviour appears to be driven by the significant size mismatch between beryllium and cerium (with ionic radii of $\rm r_{Be^{+2}}=0.59$ Å and  $\rm r_{Ce^{+4}}=1.01$ Å), requiring significant lattice distortion to attain the optimum Be-O distances in the tetrahedral arrangement. To adopt the tetrahedral coordination, three O atoms significantly displace from their lattice sites towards Be, straining the other Ce-O bonds by 0.2 Å.
In contrast, the trigonal configuration enables optimum Be-O separation while maintaining near ideal distances for the nearby Ce-O bonds, thereby optimising the overall ionic interactions. This is reflected by the Madelung energy, which is 3.2 eV lower in the ground state configuration, and the overall energy lowering of 0.34 eV.

Interstitials can also exhibit these electrostatically-driven reconstructions, with their higher mobility allowing for displacement to more favourable Madelung potential sites.
For example, the main difference between the metastable and ground state structures of the tellurium interstitial ($\rm Te_{i}^{-2}$) in cadmium telluride (CdTe) lies in its coordination environment. While in the metastable configuration $\rm Te_i$ is tetrahedrally coordinated by other Te anions (distances of 3.4Å), in the ground state the Te atoms rearrange to a split configuration, both occupying a hexagonal void and minimising interactions with neighbouring Te (distances of 3.5, 3.5, 3.6, 4.0Å) (SI Fig. S17 (c,d)). This reduction of unfavourable anion-anion interactions is again witnessed in the Madelung energy, which is 2 eV lower for the ground state, and drives an overall energy lowering of 0.20eV. Although these examples are primarily driven by optimising electrostatic interactions, the different distortion motifs in Fig. 3 are not mutually exclusive. Often several effects can contribute to a certain distortion, as observed for $\rm Cd_i^{+2}$ in CdTe, where the distortion leads to cation-anion rebonding (via explicit replacement of cation-cation bonds with cation-anion) but also lowering the lattice electrostatic energy (by surrounding the interstitial with oppositely charged ions).

Electrostatically-driven reconstructions. Energy difference and summed atomic displacements ($\Sigma$Disp.) between the ground state identified by our method and the metastable configuration found when relaxing the ideal, undistorted defect structure (i.e. $\rm \Delta E = E_{\rm ground} - E_{\rm

Defect     Material      $\Delta$E (eV)     $\Sigma$Disp. (Å)

$\rm Te_{i}^{-2}$      CdTe     -0.20      13.0

$\rm Cd_{i}^{+2}$     CdTe     -0.46     5.8

$\rm Be_{Ce}^{-2}$ + VO+2     CeO2     -0.34      19.7

[1]Metastable and ground state structures with bond distance labelling shown in SI Figs. S17

Beyond these motifs, our approach can also be employed to identify defect bound polarons, as exemplified in the supporting information for VO0 in CeO2, VO0 and Tii0 in rutile TiO2, and VIn-1 in In2O3 (SI Section VI). For ceria and rutile, the two electrons donated by the defect localise on lattice cations reducing them to Ce/Ti(III). This leads to many competing states depending on both the defect-polaron and polaron-polaron distances. While the different localisation arrangements are successfully identified by our approach, the complexity of the PES (hosting many competing low energy minima with small energy and structural differences) warrants further study.

§.§ Impact on defect properties

In this section we demonstrate how these reconstructions can affect calculated defect properties.
We take the charge transition level diagram for antimony vacancies in Sb2S3 as an exemplar, and compare the formation energies and defect levels calculated for the ground (VSb) and metastable states (VSb*), for both symmetry-inequivalent vacancy positions (VSb,1 and VSb,2; SI Fig. S7). As shown in <ref>, the highly favourable dimerisation reconstructions undergone by several charge states (+2, +1, 0, -1) significantly lower the formation energy of VSb, by 0.8 – 2.6 eV depending on charge state. As a result, the predicted concentration under typical growth conditions (T = 550K)[101] increases by $\exp(\frac{\Delta E}{k_BT}) \simeq$ 21 orders of magnitude for VSb,2+1 (SI Table IV). Furthermore, the reconstructions also affect the nature and position of the charge transition levels. The neutral state of VSb,2 is now predicted to be thermodynamically unfavourable at all Fermi levels, which leads to the disappearance of the (+1/0) and (0/-3) transition levels (<ref>).
Similarly, for VSb,1, the reconstructions render the singly-positive vacancy stable within the bandgap, resulting in two new transition levels ((+2/+1) and (+1/-1)) (<ref>, SI Table V).

Defect formation energy (left) and vertical energy level (right) diagram for the antimony vacancies in Sb2S3, under sulfur rich conditions[102] ($\mu_{\rm S}$ = 0, $\mu_{\rm Sb}$=-0.55eV).
The two symmetry inequivalent sites are depicted in purple (VSb,1) and gold (VSb,2). For each site, the ground state structures are shown with solid lines (VSb), while the metastable states are shown with faded dotted lines (VSb*). The valence band maximum (VBM) and conduction band minimum (CBM) are shown in blue and orange, respectively.

More significant, however, is the change in the energetic position of the levels. The greater stabilisation of the positive charge states, leads to the levels shifting deeper into the bandgap (SI Table V). For instance, the (-1/-3) transition of VSb,1 shifts 0.3 eV higher in the bandgap, revealing it to be a highly amphoteric defect with strong self-charge-compensating character (<ref>) – in fact now aligning with the behaviour of the other intrinsic defects in Sb2(S/Se)3[70, 103]. This change in the position of the transition levels can have important consequences when modelling defects in photovoltaic materials, as mid-gap states often act as carrier traps that promote non-radiative electron-hole recombination and decrease performance.
This deep, amphoteric character of the antimony vacancies was missed by previous theoretical studies on Sb2S3 defects,[70, 103] due to the local minimum trapping which our approach aims to combat, giving significantly underestimated concentrations and shallower transition levels. Overall, this demonstrates the potent sensitivity of predicted concentrations and transitions levels on the underlying defect structures, highlighting the crucial importance of correct defect structure identification for quantitative and reliable defect modelling.

Beyond thermodynamic properties, these reconstructions can also affect calculated rates of non-radiative electron/hole capture. Indeed, charge capture rates depend intimately on the structural PESs of the charge states involved, as well as the position of the transition level. This effect has been recently demonstrated for the neutral cadmium vacancy in CdTe[10] as well as for the neutral lead interstitial in CH3NH3PbI3[104] (MAPI). For the latter, the timescale of non-radiative charge recombination varies by an order of magnitude depending on whether the Pb interstitial forms a dimer (ground state structure) or remains in a non-bonded position (metastable configuration).
Moreover, even when defect reconstructions do not lower the energy significantly, this can still result in completely different capture behaviour, as observed for the tellurium interstitial in CdTe[20] and the sulfur substitution in silicon[105]. For $\rm S_{Si}$, a small structural distortion (with a negligible energy difference) significantly affects its capture behaviour, giving a capture rate now in agreement with experiment[105] and once again highlighting the critical role of exploring the defect configurational landscape.

§.§ Locating metastable structures

Thus far, we have focused on atomic reconstructions that lower the energy and lead to a new ground state structure, which is missed with a standard relaxation from the high symmetry geometry.
While defect properties such as doping are often determined by the ground state configuration, low energy metastable structures can also be important to performance in device applications. For instance, they often represent transition states in the ion migration process[106] (crucial for battery materials and semiconductor doping), and can impact non-radiative carrier recombination in solar cells and LEDs, behaving as intermediate species to the charge trapping process[10, 107, 20]. They can produce anomalous properties, such as persistent photo-conductivity and photo-induced capacitance quenching in semiconductors[19].
Accordingly, we investigated the ability of our approach to locate relevant low-energy metastable configurations by considering the DX centres in GaAs (SiGa-1, SnGa-1, SAs-1, TeAs-1).

For Si, Sn and Te impurities, our method successfully identifies all low-energy metastable structures reported by previous studies,[108, 109, 110, 111, 112, 113, 114] while for S, it identifies two of the three structures previously found.[115, 114]
For instance, in the case of Si, we correctly identify the broken bond configuration (DX-BB)[108, 109, 110, 111, 112, 113], which entails a $C_{3\textrm{v}}$ Jahn-Teller distortion with the dopant displacing along the [111] direction thereby breaking a Si-As bond (<ref> (b)). We find it to lie 0.34 eV higher in energy than the ground state $T_d$ configuration (<ref>), in agreement with earlier local DFT (LDA) calculations which found it 0.5eV higher in energy[108, 112].
A similar bond-breaking $C_{3\textrm{v}}$ distortion is also found for the other group IV dopant (Sn), though now with two possible low energy configurations (DX-BB-$\alpha$ and DX-BB-$\beta$). While the former corresponds to the conventional DX behaviour with major off-centring of the dopant atom, in the latter it is mainly a neighbouring As which displaces to an interstitial position (<ref> (d, e)). We find the $\alpha$ configuration to lie slightly lower in energy than $\beta$ (0.26eV vs 0.41eV above the $T_d$ ground state) (<ref>), agreeing with previous local DFT calculations which found DX-$\alpha$ to be 0.4eV higher in energy than $T_d$[108].

Ground state ($T_d$) and low energy metastable configurations identified for SiGa-1 and SnGa-1 in GaAs. The DX-BB (DX-BB-$\alpha$) configuration consists of a $C_{3\textrm{v}}$ distortion, where Si (Sn) displaces away from the original Ga position, breaking a dopant-As bond.  In the DX-BB-$\beta$ configuration, a Sn-As bond is also broken, but in this case As undergoes the largest displacement.
Si in blue, Sn in grey and As in purple. The arrows illustrate how the atoms reconstruct from the $T_d$ configuration and the black spheres depict the original position of the displaced atoms in the $T_d$ structure.

Regarding the sulfur dopant, there are several metastable configurations. Similar to the other species, it can adopt a broken bond arrangement (DX-BB, <ref> (b)), where one of the dopant neighbours displaces off-site and breaks a S-Ga bond. In addition, it can adopt two configurations with cation-cation bonding (CCB) – corresponding to dimerisation reconstructions. As depicted in <ref>, two of the neighbouring Ga can displace towards each other and form a Ga-Ga bond (2.5Å), with the dopant remaining either in the original position (DX-CCB-$\alpha$) or displacing slightly off-site (DX-CCB-$\beta$). The dimer (DX-CCB) configurations are found to be more favourable than the broken bond (DX-BB) one, with energies of 0.30eV (DX-CCB-$\alpha$, DX-CCB-$\beta$) and 0.44eV (DX-BB) relative to the ground state $T_d$ structure. Again, these results agree with previous local DFT (LDA) calculations, which found relative energies of 0.5eV for DX-CCB ($\alpha$ & $\beta$) and 0.6eV for DX-BB, relative to $T_d$[115, 114].
We note that our method did not locate the DX-CCB-$\beta$ arrangement, likely due to a soft PES (with an energy barrier of only 25meV between alpha and beta, SI Section VIII), the close similarity of structure and energies for DX-CCB-$\alpha/\beta$, and the bias toward ground state configurations. This highlights a limitation of our approach, where identifying several metastable defect structures on a complex PES may require a more exhaustive exploration (e.g. distorting different number of neighbours or using a denser grid of distortions).

Ground state ($T_d$) and low-energy metastable configurations (DX-BB, DX-CCB-$\alpha$ and DX-CCB-$\beta$) identified for SAs-1 and TeAs-1 in GaAs.  In the DX-BB configuration, a dopant-Ga bond is broken,
while the DX-CCB configurations correspond to cation-cation bond (CCB) formation.
S in yellow, Te in gold, Ga in green and As in purple.

Finally, for TeAs-1, we identify DX-BB as the lowest energy metastable structure (<ref> (f)), in agreement with previous theoretical[108, 114] and experimental[110] studies.
We calculate an energy difference of 0.25eV between the metastable $C_{3\textrm{v}}$ and ground state $T_d$ configurations, similar to the value of 0.38eV obtained with local DFT[108]. Overall, these results demonstrate the ability of the method to identify low-energy metastable configurations at negligible additional cost.

Energy difference and summed atomic displacements ($\Sigma$Disp.) between the identified metastable configurations and the ground state $T_d$ for the DX centres in GaAs. For $\rm S_{As}^{-1}$, we include a configuration not found by our method (DX-CCB-$\beta$*) but reported in a previous study[114].

Defect     Ground state     Metastable     $\Delta$E (eV)     $\Sigma$Disp. (Å)

$\rm Si_{Ga}^{-1}$      $T_d$     DX-BB     0.34     2.0

2*$\rm Sn_{Ga}^{-1}$      2*$T_d$
DX-BB-$\alpha$     0.26     2.7

DX-BB-$\beta$     0.41     2.0

3*$\rm S_{As}^{-1}$
3*$T_d$     DX-BB     0.44     1.4

DX-CCB-$\alpha$     0.30     4.6

DX-CCB-$\beta$*     0.30      5.5

$\rm Te_{As}^{-1}$      $T_d$     DX-BB     0.25     1.9

§ CONCLUSIONS

In summary, we present a method to explore the configurational space of point defects in solids and identify ground state and low-energy metastable structures. Based on the defects tested, if computational resources only allow limited calculations to be performed, we suggest structural perturbations of $\pm30\%$ using our method provides a higher chance of finding the true ground state defect configuration over starting from the unperturbed bulk structure. Beyond its simplicity and automated application, the low computational cost and the lack of system-dependent parameters make it more practical than current alternative structure-searching methods (comparisons are provided in SI Section IV). The applicability to both standard and high-throughput defect investigations is demonstrated by the range of materials studied herein.

By exploring a variety of defects and materials, the key physico-chemical factors that drive defect reconstructions were also highlighted.
For systems with mixed ionic-covalent bonding (such as those involving the Ge, Sn, Sb, Bi cations), we expect `rebonding' reconstructions to prevail, with the defect neighbours displacing to yield novel homoionic (dimer) or heteroionic (cation-anion) bonds. Furthermore, if the crystals also display low symmetry (e.g. one-dimensional connectivity in SbSeI, SbSI, and SbSBr), these reconstructions will likely be more significant and prevalent, as exemplified by the antimony chalcogenides (Sb2(S/Se)3).
Here, its open and flexible structure hosts several coordination environments, with the empty spaces between the chains enabling atomic displacements without major strain on neighbouring sites, yielding many energy-lowering reconstructions away from the high-symmetry local minimum.
Notably, through the introduction of new bonds, these reconstructions can stabilise unexpected charge states, highlighting the impact on the qualitative behaviour of defects.
For crystals with stronger ionic character, distortions can either be driven by novel homoionic bonds (peroxides in In2O3, ZnO and TiO2) or Jahn-Teller/crystal field stabilisation effects (CeO2). Finally, optimising ionic interactions can also result in energy lowering rearrangements, especially for cases of significant size mismatch between dopant and host atoms, or for interstitials, which distort to minimise Coulombic repulsion with neighbouring ions.

Regarding future improvements, the approach could be enhanced with a machine learning model, which could identify subsets of likely reconstructions to improve sampling efficiency. Similarly, local symmetry analysis could be incorporated to suggest potential subgroup distortions from the initial point group symmetry (e.g. $T_d \rightarrow C_{3\textrm{v}}$, $C_{2\textrm{v}}$, $D_{2\textrm{d}}$ ...) and then generate distorted structures based on this. This feature would likely be ill-suited for low-symmetry materials such as Sb2(S/Se)3, however, as the wide range of symmetry-inequivalent distortions could become prohibitively expensive to compute with accurate levels of theory.

Overall, our results demonstrate the prevalence of energy-lowering reconstructions for defects, that are often missed through local optimisation from a high symmetry configuration. The major structural and energetic differences highlight the crucial impact on defect properties.
Beyond thermodynamic properties, the modification of the structural PES will completely change recombination, absorption and luminescence behaviour. Consequently, navigating the configurational landscape is key for accurate predictions of defect properties and their impact on materials performance, ranging from energy technologies (thermoelectric/photovoltaic efficiency, battery conductivity and catalytic activity) to defect-enabled applications such as lasing and quantum computing.

Supplementary information.
Computational details; Parameter optimisation of the defect structure searching approach; Comparison with other structure searching approaches; Figures of the metastable and ground state structures for the identified energy lowering distortions; Application of the method to defect bound polarons; Concentrations and charge transition levels predicted for the ground state and metastable structures of antimony vacancies in Sb2S3; Energy barrier between the DX-CCB-$\alpha$ and DX-CCB-$\beta$ configurations of $\rm S_{As}^{-1}$ in GaAs.

§ DECLARATIONS

Data & code availability.
The code used to generate and analyse defect distortions is available from <https://github.com/SMTG-UCL/ShakeNBreak>.
The identified ground state and metastable structures are available from the Zenodo repository with DOI 10.5281/zenodo.6558244.

Author contributions.
Conceptualisation & Project Administration: Seán R. Kavanagh, David O.Scanlon. Investigation and methodology: Irea Mosquera-Lois, Seán R. Kavanagh. Supervision: Seán R. Kavanagh, Aron Walsh, David O. Scanlon. Writing - original draft: Irea Mosquera-Lois. Writing - review & editing: All authors. Resources and funding acquisition: Aron Walsh, David O. Scanlon. These author contributions are defined according to the CRediT contributor roles taxonomy.

I.M.L. thanks La Caixa Foundation for funding a postgraduate scholarship (ID 100010434,  fellowship code LCF/BQ/EU20/11810070).
S.R.K. acknowledges the EPSRC Centre for Doctoral Training in the Advanced Characterisation of Materials (CDT-ACM)(EP/S023259/1) for funding a PhD studentship. DOS acknowledges support from the EPSRC (EP/N01572X/1) and from the European Research Council, ERC (Grant No. 758345).
Via membership of the
UK’s HEC Materials Chemistry Consortium, which is funded
by the EPSRC (EP/L000202, EP/R029431, EP/T022213),
this work used the UK Materials and
Molecular Modelling (MMM) Hub (Thomas EP/P020194
and Young EP/T022213).

Competing interests.
The authors declare no competing interests.

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