Electron and hole trapping by grain boundaries and dislocations in polycrystalline materials is important for wide ranging technological applications such as solar cells, microelectronics, photo-catalysts and rechargeable batteries. In this article, we first give an overview of the computational and methodological challenges involved in modelling such effects. This is followed by a discussion of two recent studies we have made on electron/hole trapping in wide gap insulators. The results suggest that such effects can be important for many applications which we discuss. These computationally demanding calculations have made extensive use of both the HPCx and HECToR services.
The electronic properties of polycrystalline materials have long been known to be affected by the presence of grain boundaries (GBs), which in many cases are able to trap electrons and holes that can be introduced by doping, by excitation or by electrical injection. These effects impact on a wide range of technological applications, including high-Tc superconductors (Hilgenkamp & Mannhart 2002), metal oxide field effect transistors (Seto 1975; Walker et al. 2004; Asenov et al. 2008; Saha 2010), solar cell technologies (Yan et al. 2006), varistors (Clarke 1987, 1999) and sensors (Göpel & Schierbaum 1995). In nature, charge trapping at GBs is even thought to play a role in reactions on interstellar dust grains (Caruana & Holt 2010) and electromagnetic seismic phenomena in the Earth’s crust (Takeuchi et al. 2004). GBs may trap electrons and holes both owing to their intrinsic electronic structure, and because there are often enhanced concentrations of electronically active defects segregated there. Separating these inter-related effects and characterizing the electronic properties of GBs by experiment alone is extremely difficult and theoretical modelling has proved to be invaluable.
Theoretical modelling of electron/hole trapping at GBs has predominantly focused on semiconducting materials. In particular, elemental semiconductors like Si have been the most widely studied (Kohyama 2002) with compound semiconductors like GaAs, ZnO, SiC and CuInSe2 and metal-oxide semiconductors like TiO2, SnO2, BaTiO3 and SrTiO3 also receiving attention (Greutert & Blatter 1990). However, until recently there have been very few theoretical studies of the electron/hole trapping properties of wide gap insulators. This is surprising considering that electron/hole trapping in polycrystalline insulating materials, such as HfO2 or MgO, may be very important for applications in semiconductor electronics, spintronic devices, catalysts and resistive switching materials with applications as low power and high density non-volatile memories (Szot et al. 2006; Waser et al. 2009; Borghetti et al. 2010).
In this article, we highlight our groups’ recent activity in the first-principles modelling of electron/hole trapping in the wide gap metal oxide materials MgO and HfO2. The former is often considered as a model oxide but also has numerous important applications, while HfO2 is a more complex material with very important application as a gate dielectric in field effect transistors. First-principles calculations on the complex systems presented here are only now becoming possible thanks to the increasing power of high performance computing systems and these studies have made extensive use of both HPCx and HECToR. The article is organized in the following way. In §2, we describe the challenges involved in modelling electron/hole trapping at grain boundaries and describe the approaches which we employ. This is followed by a detailed discussion of results for MgO and HfO2, highlighting their relevance to various applications. Finally in §5, we summarize the current status and outlook.
The first question one is faced with when trying to model the electron/hole trapping properties of polycrystalline materials is ‘What is the structure of interfaces between grains?’. In general, this is a difficult question to answer as polycrystalline materials can contain a wide variety of interfaces that can be influenced by numerous factors, such as growth conditions, thermal treatment and the presence of impurities. In practical modelling, one is usually forced to focus attention on a smaller subset of these boundaries that have high site coincidence on the expectation that such interfaces will have the lowest energies1 and higher abundances in real polycrystalline materials. This expectation seems to be confirmed for the few systems on which statistics have been gathered (e.g. in MgO; Chaudhari & Matthews 1971; Saylor et al. 2003). The problem is then reduced to finding the lowest energy interface structure for a given grain misorientation which can be achieved using methods such as direct energy minimization (Harding et al. 1999), simulated annealing (von Alfthan et al. 2006) or genetic algorithms (Chua et al. 2010).
Due to the complex multi-dimensional potential energy landscapes of GBs, it is common to employ empirical potentials to screen out structures with the lowest energy, before recalculating them using more accurate quantum mechanical methods which also can be used to calculate their electronic properties. This is the approach we have employed in our calculations. At the simplest level of theory, we employ shell model potentials (e.g. Lewis & Catlow 1985) to predict candidate structures for the interface between two grains or particles oriented at prescribed angles. The GB is modelled as a bi-crystal, which has finite extent perpendicular to the GB plane but is periodic parallel to the interface. The energy minimization procedure is carried out using the metadise code (Harding et al. 1999), which minimizes the total energy with respect to the positions of ions near the interface and with respect to translation of one crystal relative to the other.
Quantum mechanical calculations of grain boundaries usually employ periodic boundary conditions in all three dimensions; therefore, supercells must be constructed containing two grain boundaries that are sufficiently separated so that they interact only very weakly. This often requires supercells containing many hundreds of atoms. Early quantum mechanical studies of GBs employed semi-empirical methods such as tight-binding as the complexity and size of GB structures prohibited application of more accurate methods. However, more recent studies have been able to employ first-principles methods, such as density functional theory (DFT), thanks to the increasing power of high performance computing systems and the development of more efficient methods. Often DFT is employed for modelling GBs using the localized density or generalized gradient approximations (LDA or GGA) which tend to overestimate the tendency for electrons and holes to be delocalized and also underestimate the band gap. The underlying cause for this is the self-interaction error and there are numerous approaches to rectify this problem such as self-interaction correction and hybrid functionals. The periodic calculations presented in this paper are carried out using the vasp code (Kresse & Furthmüller 1996a,b).
An alternative approach to periodic DFT is provided by the embedded cluster method, which combines quantum mechanical treatment of a small region of a system (e.g. near a GB) with classical treatment of more distant ions (e.g. using shell model potentials). As such, the approach is inherently non-periodic and this brings several other advantages. As the quantum region is finite it can be modelled using a range of quantum chemistry methods which surpass the accuracy available with standard periodic codes and also allow properties such as optical excitation spectra and electron paramagnetic resonance (EPR) spectra to be computed. It also eliminates the artificial image–charge interactions which must be corrected for in supercell calculations. The embedded cluster approach we employ in our calculations is implemented in the guess code (Sushko et al. 2000), which can be used in conjunction with either gaussian03 (Frisch et al. 2004) or nwchem (Govind et al. 2009; Valiev et al. 2010) for the quantum mechanical part of the calculation.
3. MgO tilt grain boundary
GBs in MgO have been well-studied experimentally, partly because it is possible to manufacture very well-defined bi-crystals which can be studied using techniques such as high-resolution transmission electron microscopy (Kizuka et al. 1998; Yan et al. 1998). The simple crystal structure and highly ionic nature of MgO also allows the structure of GBs to be interpreted in terms of simple coincidence lattice models. Theoretical calculations using both empirical and quantum mechanical methods (Duffy 1986; Watson et al. 1996; Harding et al. 1999) have been employed to elucidate the atomic scale structure of GBs and also the diffusion and segregation of defects near them. However, the electronic properties of GBs in MgO have received less attention despite numerous applications where electron trapping may play an important role, such as secondary electron emitters in flat panel displays (Vink et al. 2002), microwave dielectrics for wireless communications and gate dielectrics in transistors. Recently, we investigated the electronic properties of a simple tilt GB in MgO and discovered a new class of electron trapping that can take place in materials with negative electron affinity (NEA). NEA materials find a wide range of applications (Martinelli & Fisher 1974) and these effects should be important in many other polycrystalline ceramics.
We performed calculations on the (310) symmetric tilt GB in MgO using DFT both using periodic and embedded cluster methods (McKenna & Shluger 2008, 2009a,b; Shluger et al. 2009). The latter approach allowed us to employ the B3LYP hybrid functional which predicts a band gap in much better agreement with experiment for MgO. The supercell used for the periodic calculations, corresponding to the lowest energy GB configuration, contains 120 ions and has dimensions 13.35×8.33×28.96 Å (figure 1a). The projector-augmented wave (PAW) method and the Perdew–Burke–Ernzerhof (PBE) functional were employed as implemented in the vasp code and we used only a single k-point (the gamma point) and plane waves with energies up to 400 eV. Atomic coordinates and the cell dimension perpendicular to the GB plane were optimized to within a force tolerance of 0.01 eV Å−1. Figure 1a shows how the optimized (310) symmetric tilt GB structure consists of an array of fairly open one dimensional dislocation channels. The formation energy (defined with respect to the bulk crystal) of this GB is calculated to be 1.95 J m−2 and it is stable with respect to translation of one grain relative to the other and to insertion of additional Mg or O ions inside the dislocation channels.
To characterize the electron/hole trapping properties of this GB we analysed the electronic structure and found shallow interface states split from the MgO valence band by approximately 0.1 eV and deeper states split from the conduction band by approximately 1 eV. To analyse the nature of these states we calculated the electron density associated with an additional electron added to the GB supercell. A contour plot of this electron density on a plane cut through the supercell (figure 1a) shows how the additional electron is trapped at the GB and confined in the voids formed by the dislocation channels. Bader analysis shows that 80 per cent of the electronic charge is localized inside the dislocation channel, while the remaining 20 per cent is associated with nearby anions. This trapping is unusual as normally trapped electrons are associated with under coordinated or strained atoms at the interface, whereas here the electrons are confined in the empty space between the crystals. Although the GGA DFT-predicted gap is too small by nearly 3 eV, we found the nature of electron trapping to be similar when the B3LYP functional was employed within the embedded cluster method. In these calculations the GB is modelled as an interface between two large grains (about 50 Å edge length) each containing 16 128 ions. The quantum cluster contains 54 ions and is treated at an all-electron level using the B3LYP hybrid density functional and a 6-31G basis set (e.g. McKenna & Shluger 2009b).
To understand the origin of this unusual electron trapping one must recall that MgO is an NEA material (e.g. Cox & Williams 1986). As such, the bottom of the conduction band is positioned above the vacuum level by about 0.5–1.0 eV, as shown in figure 1b. One consequence of this is that two-dimensional electronic states are formed just above the MgO surface, bound by the attractive image interaction (Rohlfing et al. 2003). In an analogous way, the open dislocation channels in the MgO (310) tilt GB can also serve as traps for conduction band electrons. Whether trapping is favourable or not is determined by the balance between the kinetic energy cost of confinement, the depth of the potential well at the GB, and the energy gained by polarization of the dielectric. Our calculations showed that this balance is favourable for the MgO (310) tilt GB, yielding a deep electron trap using both GGA DFT and B3LYP with different amounts of Hartree–Fock exchange (McKenna & Shluger 2008). On the other hand, holes are only weakly trapped by this GB. Recent calculations (Wang et al. 2009) have demonstrated that dislocations in MgO are also able to trap electrons in an analogous way to the (310) tilt GB. Scanning tunnelling spectroscopy and electron paramagnetic resonance studies on thin MgO films have also provided some experimental evidence for this effect (Benia et al. 2010).
More generally, the electron trapping ability of MgO GBs depends upon their atomic structure. For example, dense GBs which involve narrower dislocations channels are likely to present much shallower electron traps than the (310) tilt GB as the kinetic energy cost to confine the electron will be increased. Electron trapping properties can also be affected by the segregation of defects and impurities to GBs. For example, we recently showed how oxygen vacancies and proton impurities segregate to the MgO (310) tilt GB and introduce localized electronic states in the MgO band gap (McKenna & Shluger 2009a,b). The intrinsic electron trapping ability of GBs, coupled with the enhanced concentration of electron trapping defects at GBs, provides an effective mechanism for the charging of GBs, for example, under irradiation or applied electrical voltage. These effects can be important for a very wide range of applications. For example, MgO is used as a tunnel barrier in magnetic random access memory devices (Parkin et al. 2004), and the trapping of electrons by GBs and dislocations can negatively affect its transport characteristics. MgO is also a widely used support for surface science studies of nanoparticles and molecules. The trapping of electrons which may be introduced by UV illumination, scanning probes or transmission electron microscopy is an important issue that is not usually considered but deserves further investigation.
4. HfO2 grain boundary
The wide gap oxide HfO2 has received considerable attention owing to its suitability as a high dielectric constant replacement for SiO2 in metal-oxide field effect transistors (MOSFETs). Theoretically, it has been predicted on the basis of hybrid DFT calculations that both electrons and holes can self-trap in HfO2 (Muñoz Ramo et al. 2007). This can be explained by the fact that HfO2 has such a large dielectric constant (about 20 for the monoclinic phase). Whether self-trapping of electrons or holes can occur is decided by a competition between kinetic and potential energies. The large dielectric constant shifts the balance in favour of self-trapping by increasing the potential energy that is gained following localization. Unlike SiO2, the previously preferred gate dielectric, which is amorphous, HfO2 is polycrystalline and GBs may serve as electron/hole traps. GBs have been suspected to affect the electronic properties of MOSFET devices significantly (Ranjan et al. 2004; Yanev et al. 2008; McKenna et al. 2009). For example defects that participate in electron tunnelling processes, such as oxygen vacancies, may segregate there with increased concentrations. On the other hand, charged defects and impurities may diffuse more readily along extended defects in response to an electric field.
In order to learn about the electronic properties of GBs in m-HfO2, we constructed a simple twin boundary as shown in figure 2a. Empirical shell model potentials were used to initially search for candidate structures of the (101) twin boundary before employing periodic DFT calculations to determine the most stable structure. For this, the GB was modelled using a supercell containing 240 ions with dimensions 34.67×10.35×7.87 Å. The PAW method and the PBE functional were employed as implemented in the vasp code and we used 3×3×1 k-points and plane waves with energies up to 400 eV. Atomic coordinates and the cell dimension perpendicular to the GB plane were optimized to within a force tolerance of 0.01 eV Å−1. In the optimized structure all O ions are at least three-coordinated and all Hf ions are seven-coordinated as is the case in the bulk. The main differences with respect to the bulk are confined within a region about 5 Å on either side of the GB plane. In this region, the electrostatic potential varies by about 0.3 V and bond lengths deviate from the bulk by about 2 per cent. As this boundary has very low energy and no under-coordinated ions we consider this to be a best case scenario and more general boundaries may have stronger perturbations near the interface.
Figure 2b shows the calculated electronic density of states (DOS) for the GB supercell with associated interface states identified by comparison to the bulk. There are interface states in a range of about 0.3 eV near the conduction band minima but corresponding hole states are much weaker. Figure 3a shows the band structure of the top of the valence band and bottom of the conduction band for the GB supercell in the interfacial Brillouin zone (X → Γ, Γ → Y and Y → M). The corresponding bulk bands projected along the (101) direction are also shown for comparison. The high dispersion near the bulk conduction band minima is responsible for the more noticeable effect of the interface on the DOS (figure 2b). Figure 3b shows an electron density isosurface for an electron trapped at this GB. The electron density is delocalized over Hf ions near the interface. The corresponding hole density decays much more slowly into the bulk.
We note that the result that electron trapping is stronger than hole trapping in HfO2 has also been found in other wide gap metal-oxides such as alumina (Ching et al. 2006). This is very important for electronic devices such as MOSFETs as it leads to a local variation in the tunnel barrier in the polycrystalline HfO2 gate dielectric. Using conducting atom force microscopy on a polycrystalline HfO2 film Yanev et al. (2008) found that increased tunnelling currents are associated with GBs providing some evidence for this effect. As discussed above it has also been predicted that electrons and holes can self-trap in HfO2, therefore GBs should also affect the properties of polarons. For example, the very low dispersion of the GB state near the bulk conduction band (figure 3b) should enhance the stability of electron polarons leading to preferential segregation at the GB. Polarons at the GB may face high barriers to escape into the bulk limiting their contribution to electrical conductivity at low temperatures. Hole polarons, on the other hand, should experience smaller barriers to escape from GBs as the corresponding GB state is much closer to the bulk valence band. Computationally modelling these effects remains extremely challenging as to model polarons one should employ approaches which do not suffer from the self-interaction error (or at least suffer from it less than DFT). Appropriate methods, such as hybrid functionals, are computationally very expensive to apply to the large systems needed to model GBs (e.g. containing hundreds of atoms) and push the capabilities of high performance supercomputers to their limits.
5. Summary and outlook
We now summarize the key differences between the electron trapping behaviour of the m-HfO2 and MgO GBs described above. In the former case, the trapped electron is delocalized over Hf ions near the GB plane which are electrostatically and elastically perturbed with respect to the bulk. It is this perturbation which is responsible for forming the shallow electron trap (0.3 eV below the bulk m-HfO2 conduction band). In the case of MgO, the trapped electron is confined inside the open channels at the GB rather than associated with ions near the GB plane. This happens because, unlike m-HfO2, MgO is a negative electron affinity material which leads to a qualitatively different form of electron trapping. In the case of MgO it is the presence of the open channels rather than the elastic and electrostatic perturbation that is responsible for forming the deeper electron trap (approx. 1 eV below the bulk MgO conduction band).
One can summarize the current understanding of electron trapping at GBs, which is based mainly upon DFT studies of narrow to medium gap insulators (Dawson et al. 1996; Kohyama 2002), in four main points: (i) GBs that have the lowest energy (i.e. those thermodynamically favoured) reconstruct to leave no dangling bonds or low coordinated atoms (von Alfthan et al. 2006). (ii) Such reconstructed GBs possess either very shallow electron traps (split from the bulk conduction band by tenths of an eV) or no traps at all. (iii) Disordered intergranualar phases (Clarke 1987), defects and impurities can be responsible for deep electron traps. (iv) Deep electron traps can be present in negative electron affinity materials even at well-ordered GBs.
GBs also serve as preferential sites for the segregation of defects and impurities. The interplay between defect and impurity segregation and electron and hole trapping is important for a wide range of applications such as solar cells, resistive switching memories, MOSFETs, batteries and fuel cells but is only beginning to be explored by computational methods. Key challenges are the development of more accurate and more efficient methods for modelling charge trapping and defects and also methods for the statistical characterization of GB structure in polycrystalline materials.
Computational time on HECToR and HPCx was provided by EPSRC through the UK Materials Chemistry Consortium. Some of the embedded cluster calculations were performed using NWChem/Guess on the Chinook supercomputer with time provided by EMSL. KPM also acknowledges support from MEXT KAKENHI project number 2274019.
One contribution of 16 to a Special feature ‘High-performance computing in the chemistry and physics of materials’.
↵1 As GBs are in fact a long-lived metastable state it is not sufficient to search for the lowest energy configuration of atoms, as this would correspond to a single bulk crystal. Instead one must find low energy boundary configurations that satisfy certain constraints, such as the relative orientation between adjacent grains.
- Received October 7, 2010.
- Accepted November 19, 2010.
- This journal is © 2011 The Royal Society