Abstract
Atomicscale understanding and control of dislocation cores is of great technological importance, because they act as recombination centers for charge carriers in optoelectronic devices. Using hybrid densityfunctional calculations, we present perioddoubling reconstructions of a 90° partial dislocation in GaAs, for which the periodicity of likeatom dimers along the dislocation line varies from one to two, to four dimers. The electronic properties of a dislocation change drastically with each period doubling. The dimers in the singleperiod dislocation are able to interact, to form a dispersive onedimensional band with deepgap states. However, the interdimer interaction for the doubleperiod dislocation becomes significantly reduced; hence, it is free of midgap states. The Ga core undergoes a further perioddoubling transition to a quadrupleperiod reconstruction induced by the formation of small hole polarons. The competition between these dislocation phases suggests a new passivation strategy via population manipulation of the detrimental singleperiod phase.
Introduction
Dislocations generally degrade the optoelectronic properties of III–V optoelectronic devices such as lightemitting diodes and solar cells. III–V multijunction solar cells have attained efficiencies in excess of 44%,^{1, 2, 3, 4} and have the potential to reach efficiencies in excess of 50%.^{5} However, many of these device architectures rely on latticemismatched epitaxy, for which threading dislocations are inevitable. Because these dislocations act as recombination centers for charge carriers^{6, 7, 8} and reduce the device efficiency, the density of dislocations in the active region of the device must be minimized, generally by carefully varying the lattice constant in a thick graded layer.^{9, 10, 11} It may also be possible to mitigate the impact of these dislocations via passivation with impurities,^{12, 13, 14, 15} or by controlling the type^{16} or direction of the remaining dislocations.
A full dislocation may dissociate into two partial dislocations to lower its formation energy.^{17, 18, 19} All possible structural permutations for glissile <110> dislocations in a III–V crystal are illustrated by the dislocation loop shown in Figure 1.^{20} Dissociation into partial dislocations creates two partial dislocation loops separated by a stacking fault. The angle between the partial Burgers vectors (dashed arrows) and the line direction for all segments is either 30° or 90°, and the vector sum of the partial dislocation pairs always equals the Burgers vector of the full dislocation (solid arrow). By definition, an unreconstructed dislocation disrupts the bulk crystal structure and any edge component creates dangling bonds. A second descriptor, ‘α’ or ‘β’, in Figure 1 indicates whether these dangling bonds are for group V or group III atoms, respectively. Because these dangling bonds are energetically unfavorable, a dislocation core generally reconstructs to form likeatom dimer bonds. A 90° partial dislocation reconstructs with dimers angled away from the dislocation line direction. There are two wellestablished reconstructions with similar formation energies, referred to as ‘single period’ (SP) and ‘double period’ (DP) based on their periodicity along the dislocation line.^{20, 21, 22, 23, 24, 25} A 30° partial dislocation reconstructs with dimers aligned parallel to the dislocation line direction.^{20, 24, 26} An atomic model for each of these reconstructions is shown in Figure 2.
A reconstructed dislocation along a highsymmetry crystal direction can be regarded as a onedimensional extended defect consisting of likeatom dimers repeated along the dislocation line. For some dislocations (for example, 90° SP), the distance between the dimers is not much longer than the dimer bond length itself. Thus, such a onedimensional extended defect differs from a point defect in that the constituent likeatom dimers may strongly interact with each other. Nonetheless, previous theoretical work^{15} neglected the interaction between the dimers. Here we show that these interactions have a fundamental role in determining a dislocation’s electronic and atomic structure.
We performed hybrid densityfunctional theory (DFT) calculations of a 90° partial dislocation in GaAs, to investigate how the interdimer interaction affects its structural and electronic properties. For comparison, the calculation results for a 30° partial dislocation are also presented. Despite previous DFT studies of the SP and DP reconstructions of a 90° partial dislocation, there still remains large uncertainty in the resulting defectlevel positions with respect to the band edges because of the underestimated DFT bandgap. To overcome this, we used hybrid density functional proposed by Heyd, Scuseria and Ernzerhof (HSE) for the exchange correlation.^{27} In addition to the SP and DP reconstructions, we propose a ‘quadrupleperiod’ (QP) reconstruction as a new ground state for a neutral 90°βdislocation in GaAs (Figures 2d and e), which contains small polarons of holes that are uniformly distributed along the dislocation line. As we will show later, the interdimer interactions in the SP, DP and QP reconstructions are quite different and the electronic properties of a dislocation therefore depend strongly on which reconstruction it adopts.
Materials and methods
Firstprinciples calculations of partial dislocations
We performed firstprinciples density functional calculations to investigate the stability and electronic structure of partial dislocation cores in GaAs, using the Vienna abinitio Simulation Package.^{28} We used the HSE hybrid density functional^{27} for the exchange correlation and the projectoraugmented wave method.^{29} The atomic structures were optimized until the residual forces were <0.05 eV Å^{−1}. The optimized lattice constant and the band gap are 5.67 Å and 1.37 eV for bulk GaAs, respectively, close to the experiment values of 5.65 Å and 1.42 eV, respectively.
Our supercells contain two different partial dislocation cores, α and β, to make the sum of the two partial Burger’s vectors equal to zero, so as to give the periodic boundary conditions used in our supercell calculations. The two partial dislocations are separated by a stacking fault. To prevent charge transfer between the cores in a supercell, we passivated one of the cores by attaching pseudohydrogen atoms. Ga and As dangling bonds are passivated by pseudo hydrogen with 5/4 and 3/4 e, respectively.
Results
We first discuss the interdimer interaction in the 90° SP α and βcores. The 90° SP dislocation is a line defect embedded in GaAs, in which the likeatom dimers are ‘polymerized’ through electron hopping between the adjacent dimers. The two hopping parameters, t_{b} and t_{c}, describe the interdimer interaction, while the intradimer hopping parameter t_{a} describes the strength of the dimer bond (Figure 2a). For crystal momentum k, the Bloch Hamiltonian H(k) is given by
where d is the lattice constant along the dislocation line and ɛ_{db} is the energy of the dangling bond (db) basis state at the core. The defect band structure ɛ_{±}(k) is given by . Hereafter, the two bands, ɛ_{+}(k) and ɛ_{−}(k), which are well separated in energy by roughly 2t_{a}, are referred to as the antibonding dimer band and the bonding dimer band, respectively.
The signs of the hopping parameters in H(k), as well as their sizes, essentially determine the dispersion of a dislocation band. The signs of t_{a}, t_{b} and t_{c} are determined by the following considerations.
Figure 3a depicts the plike db orbitals of the 90°β SP dislocation for the dimer bands at k=Γ and X. It is interesting to note that the db orbitals are tilted relative to the dimer bond direction, as confirmed by the chargedensity plots in Supplementary Figure S2. The sign on each lobe of the db orbitals denotes the relative phase between the lobes. Depending on the hopping parameters involved in the db coupling, there are three different types of bonds, a, b and ctypes, as denoted in Figure 3a (upper left). The solid (dotted) line represents a p–p bonding (antibonding) interaction between a selected pair of db orbitals. When two lobes connected by a bond line are in phase, the corresponding bonding state has a p–p bonding character. On the other hand, for the lobes with the opposite phases, the bonding state has a p–p antibonding character. By definition, the antibonding dimer band is characterized by the atype antibonding state (blue dotted lines, upper panels), while the bonding dimer band is in the atype bonding state (lower panels). The sign of t_{a} depends on the choice of the db basis states. We select the db orbitals in the upper left panel as our basis states, so that t_{a} is positive. It is worth noting that for the antibonding dimer band at k=Γ, there are two ctype antibonding bonds and one btype antibonding bond per unit cell (upper left), whereas at k=X, all of these interdimer bonds have a bonding character (upper right). From the dispersion relation, the energies of the corresponding states are ɛ_{+}(Γ)=ɛ_{db}+t_{a}+2t_{c}+t_{b} and ɛ_{+}(X)=ɛ_{db}+t_{a}−2t_{c}−t_{b}. Therefore, the t_{b} and t_{c} should be positive to obtain the right band dispersion.
The tightbinding band structure of 90°β SP is shown in Figure 3b, with the hopping parameters t_{a}=1.22 eV, t_{b}=0.19 eV and t_{c}=0.29 eV, which are fitted to the HSE energy values. The db energy ɛ_{db} is 1.27 eV with respect to the valence band maximum (VBM) of GaAs. Although the bonding dimer band is less dispersive than the antibonding dimer band, owing to the mixed nature of the interdimer bonds, the interdimer interactions, parameterized by t_{b} and t_{c}, are responsible for the deepgap states of the 90°β SP near the Γ point.
Similarly, for the 90°α SP dislocation, we found that the interdimer interaction leads to a dispersive antibonding dimer band, creating deepgap states near the X point (Figure 4). In this case, the hopping parameters are t_{a}=2.54 eV, t_{b}=0.18 eV and t_{c}=0.22 eV. The db energy ɛ_{db} is −1.39 eV with respect to the VBM. Unlike in the case of the β core, the plike db orbitals in the αcore are conventional in the sense that they are oriented along the dimer bond direction (Figure 4 and Supplementary Figure S3). Consequently, the dimerization energy, which is proportional to t_{a}, is much larger for the αcore than for the βcore. In addition to the ‘conventional’ db states, we note that there exist other plike db states in the αcore (Supplementary Figure S4), for which the plike orbitals are tilted relative to the dimer bond direction. However, these db states are ‘hidden’ inside the GaAs bands and thus are not detrimental to the optoelectronic properties of GaAs.
Figures 5a and b show the projected density of states (PDOS) for the 90° SP β and αcores, respectively. The PDOS of the βcore close to the VBM is derived from the Ga–Ga bonding dimer band, whereas the PDOS of the αcore close to the conduction band minimum is from the As–As antibonding dimer band. The interaction among the dimers is responsible for the broad PDOS. The two peaks in the PDOS are due to the dispersion of the defect band, which is relatively flat at k=Γ and X. It is noteworthy that the dispersive defect band is partially occupied by residual electrons or holes (Figures 5a and b). For the neutral βcore, each Ga db contributes 3/4 electrons and the bonding dimer band is occupied by 0.5 hole per dimer. For the neutral αcore, each As db contributes 5/4 electrons and the antibonding dimer band is thus occupied by 0.5 electron per dimer.
For the 90° DP reconstruction in Figure 2b, dimers along the dislocation line are spatially separated and no midgap states are expected due to significantly weakened interdimer interaction. Indeed, the PDOS of 90° DP in Figures 5c and d exhibit shallower, lessdispersive defect bands than those of the SP dislocations. Likewise, we obtained the nearly dispersionless PDOS for the 30° β and αcores (Figures 5g and h), for which the interdimer interaction is negligible due to the large separation between the dimers (Figure 2c). In the 90° DP and 30° reconstructions, the dimers behave more like isolated dimers. As in the case of SP, each Ga–Ga bond has 0.5 residual holes in the bonding dimer state, whereas each As–As dimer has 0.5 residual electrons in the antibonding dimer state.
Our HSE calculations show that the formation energies of the 90° SP and DP of GaAs are very similar, partly because the density of dimers along the dislocation line is similar. For the neutral 90°βcore, the formation energy of the SP is 0.034 eV per dimer lower than that of the DP. For the neutral 90°αcore, their stability is reversed and the DP is 0.044 eV per dimer more stable than the SP. Here we point out two factors that determine their stability. First, the residual carriers in the defect states have a critical role in the competition between SP and DP. For the 90°β SP, the residual holes occupy the top of the bonding dimer band near the Γ point (Figure 3b), at which the ctype interdimer bonds have an antibonding character. For the 90°α SP, the residual electrons occupy the bottom of the antibonding dimer band near the X point (Figure 4b), which has the interdimer bonding character. Therefore, in both cases the interdimer interaction mediated by residual carriers increases the relative stability of SP. On the other hand, when the DP is formed, the strain accumulated in the SP core is released through the bond rearrangement, enhancing the relative stability of DP. These two effects are nearly balanced, so that the formation energies of SP and DP are similar.
Next, we show that despite containing broken dimer bonds, a novel QP reconstruction is even more stable than the SP and DP phases for a neutral 90°β partial dislocation in GaAs. There are two types of QP that can be derived from SP and DP (Figures 2d and e). These two QP phases are referred to as QP_{S} and QP_{D}. The QP_{D} is 0.149 eV more stable than the SP per dimer, and it is slightly more stable than the QP_{S} by 0.016 eV per dimer. The key to explaining the stability of QP is to understand the different hole distributions in the DP (or SP) and the QP. As discussed earlier, for 90° β DP or SP, the holes in the bonding dimer states are delocalized over the dimers along the dislocation line with 0.5 hole per dimer on average. However, the delocalized hole state turns out to be unstable in the presence of the hole–lattice interactions. For 90°β QP, the hole polarizes or deforms its surrounding lattice in a way that the two holes in the unit cell become localized around a single Ga–Ga dimer, so that the rest of the shallow bonding dimer states near the VBM can be fully occupied, thus lowering the total energy. In this case, one of the four dimer bonds in the QP_{D} unit cell is elongated by 50% to trap two holes, forming a small polaron of holes in the dislocation core (Figure 2f). In contrast, the remaining dimer bonds are shortened by 2%. We found that the small polaron state appears in the conduction band of GaAs. The fully occupied bonding dimer states of the remaining dimer bonds appear near the VBM (Figures 5e and f).
We also checked the possibility of the formation of small electron polarons in the 90°α QP core. The excess residual electrons in the antibonding dimer states may be trapped at a single broken As–As dimer in the QP unit cell, forming small electron polarons.^{30} However, our HSE calculation shows that the 90°α QP is not stable even as a local minimum structure, which is consistent with the large dimerization energy and the large strain energy associated with the lattice distortion in the As core.
Discussion
The above results contribute to the understanding of recombination at midgap states in GaAs. One thing to note is that the results discussed here are limited to perfectly straight dislocations, and that kink sites^{22, 31, 32, 33, 34} or any other imperfections^{35, 36, 37, 38} neglected here may host midgap states. Nonetheless, even perfectly straight, impurityfree, dislocations can exhibit additional structural complexity. Here we will show that a mixture of reconstructions is likely to form at elevated (growth) temperatures, and that this can affect the existence and passivation of midgap states.
We have learned that neither of the groundstate reconstructions, 90°α DP and 90°β QP, supports midgap states (Figure 5). Deepgap states appear only when the dimers are coupled to form a dispersive defect band as in the SP. The 90°α SP supports midgap states and its energy is only ΔE=0.044 eV per dimer higher than the groundstate DP. At temperature T, the population of the SP is estimated to be ρ_{SP}=1/[1+exp(2ΔE/k_{B}T)] within the crude approximation neglecting the boundary energy of the SP–DP mixed phases. In the equation, k_{B} is the Boltzmann constant, and the factor 2 in front of ΔE is necessary because the DPtoSP conversion creates two SP dimers per unit cell. From the thermodynamic consideration, the population of the SP phase is ρ_{SP}=3.1% at room temperature. At a growth temperature T=700 °C, the SP population would increase to 25.9 %. Because of the small SP population, the chance of having a long SP segment in between the DP regions is small. Then, an important question is whether a relatively short SP segment can support midgap states. Using the tightbinding Hamiltonian of a SP segment, we found that a very short SP segment containing just four SP dimers gives midgap states nearly as deep as for the pure SP phase (Figure 6). Therefore, the short SP segment in the mixed phase can create deep gap levels and act as a recombination center.
For improved optoelectronic properties, the formation of the SP segments in the 90°αcore should be suppressed by enhancing the stability of DP with respect to SP. Although previous studies only considered neutral dislocations in GaAs,^{15, 20, 24, 26} their stability depends on the charge state as shown in Si.^{39} For the neutral 90°α, as discussed earlier, the residual electrons occupying the antibonding dimer states are responsible for the enhanced relative stability of SP. Therefore, the energy difference, ΔE=E_{SP}−E_{DP}, can be tuned by changing the electron occupation of the defect states. Indeed, our hybrid DFT calculations of charged dislocation cores show that ΔE increases from 0.044 eV per dimer to 0.247 eV per dimer, as the residual electron of 0.5 electron per dimer is removed. Therefore, the population of the harmful SP minority phase should be substantially reduced by shallow ptype doping of the 90°αcore.
Conclusion
Our hybrid DFT calculations establish the SP, DP and QPreconstructed atomic structures of the 90° partial dislocation in GaAs. The electronic properties of the SP and DP are revisited, revealing the nature of the interaction between the constituent dimers of the dislocations. We found that the defect states of the DP are shallow, whereas the interdimer interaction in the SP leads to detrimental midgap states. Furthermore, for the first time, we report on the formation of small polarons in the 90°β dislocation, which leads to the groundstate QP phase for the neutral 90°βcore. The fundamental understanding of the SP, DP and QP reconstructions suggests a new passivation strategy by biasing the competition between these phases through the tuning of the carrier density in the dislocation. The insights obtained from this study will also help to understand and control other onedimensional extended defects such as grain boundaries in MoS_{2}^{40} and other twodimensional semiconductors.
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Acknowledgements
The work at NREL was supported by the US Department of Energy, EERE, under contract number DEAC3608GO28308. The work at DGIST was supported by the DGIST MIREBraiN Program. We thank Andrew Norman and Ryan France for their helpful discussions.
Author contributions
JSP performed and analyzed theoretical calculations, and contributed to the writing of the manuscript. JK performed and analyzed theoretical calculations, contributed to the writing of the manuscript and the direction of the project. BH performed and analyzed theoretical calculations. SHW developed theoretical methods, analyzed the theoretical calculations, contributed to the writing of the manuscript and the direction of the project. WEM contributed to the atomic structural modeling, the direction of the project and the writing of the manuscript.
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Park, JS., Huang, B., Wei, SH. et al. Perioddoubling reconstructions of semiconductor partial dislocations. NPG Asia Mater 7, e216 (2015). https://doi.org/10.1038/am.2015.102
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