With these potentials the new KS orbitals can be obtained. This closes the SCF cycle. At this stage one can look at forces acting on the atoms in the unit cell. If symmetry allows there can be forces on the atoms which are defined as the negative gradient of the total energy with respect to the position parameters.
Take for example the rutile TiO 2 structure Figure 2 , in which oxygen sits on Wyckoff position 4f which has the coordinates x, x, 0 where x is not specified by symmetry. In this case x can be varied to minimize the energy and thus a force can occur on the oxygen which vanishes at the equilibrium geometry.
When all atoms are essentially at their equilibrium positions with forces around 0 then one can change the volume of the unit cell and minimize the total energy E. This would correspond to the equilibrium geometry of the system in the given structure. After this minimization is completed one can, as the last step, calculate various properties for this optimized structure.
Quantum-Mechanical Ab-initio Calculation of the Properties of Crystalline Materials / Edition 1
The treatment of exchange and correlation effects has a long history and is still an active field of research. Some aspects were summarized in the review articles [ 1 - 5 ] but also in many other papers in this field. The reader is encouraged to look at recent developments. An excellent book [ 18 ] by Cottenier covers DFT and many aspects around the WIEN2k program package and thus is highly recommend to the reader for finding further details. It was designed to approximate Hartree-Fock, which by construction treats exchange exactly but neglects correlation effects completely.
This type of error cancellation is typical for many DFT functionals. Early applications of DFT were done by using results from quantum Monte Carlo calculations [ 21 ] for the homogeneous electron gas, for which the problem of exchange and correlation can be solved exactly. Although no real system has a constant electron density, one can at each point in space use the homogenous electron gas result to treat exchange and correlation, leading to the original local density approximation LDA.
Surprisingly LDA works reasonably well but has some shortcomings mostly due to the tendency to overbind atoms, which cause e. For several cases this GGA gave better results and thus for a long time PBE has been a standard for many solid state calculations. During recent years, however, several improvements of GGA were proposed, which fall in two categories, both with good justifications:.
Semi-empirical GGA, which contain parameters that are fitted to accurate e. One criterion for the quality of a calculation is the equilibrium lattice constant of a solid, which can be calculated by minimizing the total energy with respect to volume. By studying a large series of solids as shown in Figure 7 some general trends can be found [ 23 ] : LDA has the tendency of overbinding, leading to smaller lattice constants than the experiment.
GGA in the version of PBE [ 22 ] always yield larger lattice constants, which sometimes are above the experimental value. The more recently suggested modifications, as discussed in [ 23 ], lead to a clear improvement at least for the lattice parameters. In addition, there are other observables such as cohesive energy or magnetism, to mention just two , which depend on the functional.
The best agreement with experiment may require different functionals for various properties. So far no functional works equally well for all cases and all systems. Therefore one must acknowledge that an optimal DFT functional has not yet been found, which is the reason why this remains an active field of research. Comparison of several GGA functionals, showing the relative error in the equilibrium lattice constant of many solids between DFT calculations and experiment for further details see .
The calculations were done with WIEN2k. A systematic improvement of the exchange and correlation treatment as in quantum chemistry section 2. In rung 4 one goes from the simple dependence on the density alone, to an orbital description, which for occupied orbitals allows a correct description of exchange, like in Hartree-Fock. At this level one limits the computation space to the occupied orbitals but can extend it to the hybrid functions mixing a fraction of Hartree-Fock with a part in DFT.
In the highest rung also unoccupied orbitals are included, as for example in the scheme called random phase approximation RPA. There are well documented cases for which conventional DFT calculations LDA or GGA disagree even qualitatively with experimental data and lead, for instance, to predict a metal instead of an insulator. One of the reasons can be the presence of localized states often f-electrons or late transition metal d-orbitals for which correlation is very strong.
For these highly correlated systems one must go beyond simple DFT calculations. One simple form of improvement is to treat theses local correlations by means of a Hubbard U see [ 26 ] but use LDA or GGA for the rest of the electrons. With this parameter U the on-site Coulomb repulsion between the localized orbitals is included, but by introducing a parameter. In a simple picture, U stands for the energy penalty of moving a localized electron to the neighboring site that is already occupied.
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Nevertheless optical excitations are commonly described in the independent particle approximations, using these quasi particle states from DFT in the single-particle picture. One well known case is the energy gap of insulators, which in this crude single-particle picture is typically underestimated by about 50 per cent. This has been well known for some time see e. It is worth considering that in Hartree Fock the gap found would typically be too large.
This is one of the reasons, why hybrid functionals were suggested which mix Hartree Fock with DFT in order to produce the correct gap. Better estimates of the quasi-particle spectrum can be obtained by GW calculations employing many-body perturbation theory, which is significantly more computationally expensive. Recently a modified Becke Johnson mBJ potential was proposed [ 27 ], which is still a local potential and thus cheap but yields energy gaps close to experiment.. When the Coulomb potential is written in terms of the density third term in equation 2 it contains the unphysical self interaction of an electron with itself.
In Hartree-Fock this term is exactly canceled by the exchange term. Due to the approximation in DFT, this cancellation is not complete and thus in some functionals a self-interaction-correction SIC is added [ 28 ]. In HF based methods including CI and CC the Hamiltonian is well defined and can be solved almost exactly for small systems but for large cases only approximately i.
In DFT, however, one must first choose the functional that is used to represent the exchange and correlation effects or approximations to them but then one can solve this effective Hamiltonian almost exactly. Thus in both cases an approximation enters either in the first or second step. This perspective illustrates the importance of improving the functionals in DFT calculations, since they define the quality of the calculation.
The advantage for DFT is that it can treat relatively large systems. A schematic summary of the main choices one has to make for computations is shown in Figure 1 , where our selections are marked in red. We want to represent a solid with a unit cell or a Supercell as discussed in section 2 and thus invoke periodic boundary conditions. The system may contain all elements of the periodic table, from light to heavy, main group, transition metals, or rare earth atoms as for example shown in Figure 3. Its electronic configuration is 1s 2 2s 2 2p 6 3s 2 3p 6 3d 2 4s 2 Figure 9.
With respect to this sphere, the Ti electronic states can be classified in three categories:. Traditionally the electronic properties of a material due to the chemical bonding are associated with only the valence electrons and thus the core electrons are often ignored. A typical scheme is the so called frozen core approximation, in which the electron density from the core electrons does not change during the SCF cycle see Figure 6.
This is often justified but there are cases like hyperfine interactions, where the change of the core electrons can contribute significantly and even more so the semi-core states. An all-electron treatment has therefore the advantage of being able to explore the contribution from all electrons to certain experimental data e. As long as a solid contains only light elements, non-relativistic calculations are well justified, but as soon as a system of interest contains heavier elements, relativistic effects must be included. In the medium range of atomic numbers up to about 54 so called scalar relativistic schemes [ 30 ] are often used, which properly describe the main contraction or expansion of various orbitals due to the Darwin s-shift or the mass velocity term but omit spin-orbit coupling.
Such schemes are computationally relatively simple and thus recommended for a standard case. The inner electrons can reach a high velocity leading to a mass enhancement. This causes a stronger screening of the nuclear charge by the relativistic core electrons with respect to a non-relativistic treatment and affects the valence electrons. The spin-orbit contribution can be included in a second—variational treatment [ 31 ] and is needed for heavier elements. In the latter spin remains a good quantum number and thus spin-polarized calculations are valid to treat magnetic systems.
Figure 1 schematically shows the topics where one needs to make a choice. We want to represent a solid with a unit cell or a supercell. Relativistic and spin-polarization effects can be included as mentioned above. The next crucial point is the choice of the potential that is closely related to the basis sets. This aspect is extensively explained in [ 18 ] discussing the advantages and problems connected with pseudo potentials.
The main idea is to eliminate the core electrons and replace the real wave functions of the valence states by pseudo wave functions which are sufficiently smooth so that they can be expanded in a plane wave basis set. In the outer region of an atom, where the chemical bonding occurs they should agree with the real wave function. In principle — in mathematical terms — plane waves form a complete basis set and thus should be able to describe any wave function. However, the nodal structure for example of a 4s wave function close to the nucleus would need to be described by extremely many plane waves.
For the all-electron case within DFT the potential looks like the one shown in Figure In the muffin-tin approximation the potential is assumed to be spherically symmetric around the atom but constant in between. In this notation the muffin-tin case is the first term in both cases, namely the 0 0 component for LM i. In the th s the muffin-tin approximation was widely used because it made calculations feasible. For closely packed systems it was acceptable but for more covalently bonded systems like silicon or even surfaces it is a very poor approximation. Another drawback of the muffin-tin approximation was that the results depended on the choice of sphere radii, whereas in the full-potential case this dependence is drastically reduced.
Due to the muffin-tin approximation different computer codes obtained results that did not agree with each other. This has changed with the use of full-potential calculations. Nowadays different codes based on the full potential yield nearly identical results provided they are carried out to full convergence and use the same structure and DFT version. This has given theory a much higher credibility and predictability see Section 7. For solving the Kohn Sham equations see Figure 1 basis sets are needed. A linear combination of such basis functions shall describe the Kohn-Sham orbitals.
One can use analytic functions-such as Slater type orbitals STO or Gaussian type orbitals-or just plane waves for example in connection with pseudo potentials. Already in Slater [ 32 ] proposed the augmented plane wave APW method. An extensive description including many conceptual and mathematical details is given in [ 18 ].
Therefore only the main concepts will be summarized below. In the APW method one partition the unit cell into non-overlapping atomic spheres type I centered at the atomic sites and the remaining interstitial region II Figure Inside each atomic sphere region I the wave functions have nearly an atomic character and thus assuming a muffin-tin potential can be written as a radial function times spherical harmonics. It should be stressed that the muffin tin approximation MTA is used only for the construction of the APW basis functions and only for that. In region II the potential varies only slowly and thus the wave functions can be well expressed in a series of plane waves PW.
Each plane wave is augmented by the atomic partial waves inside each atomic sphere i. Here only a brief summary will be given. Andersen [ 33 ], suggested to linearize that is treat to linear order this energy dependence as illustrated in Figure The main advantage of the LAPW basis set is that it allows finding all needed eigenvalues with a single diagonalization, in contrast to APW, which has the non-linear eigenvalue problem.
The LAPW basis functions u and it derivative are recalculated in each iteration cycle see Figure 6 and thus can adjust to the chemical changes for example due to charge transfer requiring an expansion or contraction of the radial function. The LAPW method made it computationally attractive to go beyond the muffin-tin approximation and to treat both the crystal potential and the charge density without any shape approximation called full-potential as pioneered by the Freeman group [ 34 ].
In section 4. Let us focus on the p-type orbitals. The 2p core state is treated fully relativistic as an atomic core state while the valence 4p state is computed within LAPW using a linearization energy at the corresponding high energy.
The 3p states are separated in energy from the 4p states and thus the linearization with the linearization energy of the 4p state would not work here. For such a case Singh [ 35 ] proposed adding local orbitals LO to the LAPW basis set in order to accurately treat states with different principal quantum numbers e. In this example the 3p LOs look very similar to the 3p radial function but are constrained to have zero value and slope at the sphere radius R MT. These local orbitals are labeled in lower case to distinguish them from the semi-core LO.
The matching is again as in APW only made between values Figure This new scheme is significantly faster while maintaining the convergence of LAPW [ 37 ]. It was known that LAPW converges somewhat slower than APW due to the constraint of having differential basis functions and thus it is an improvement to return to APW but only for the orbitals involved in chemical bonding. The local orbitals provide the necessary variational flexibility to make this new scheme efficient but they are added only where needed to avoid any further increase in basis set.
The crystalline wave functions of Bloch type are expanded in these APWs leading to a general eigenvalue problem. The size of the matrix is mainly given by the number of plane waves PWs but is increased slightly by the additional local orbitals that are used. As a rule one can say that about PWs are needed for every atom in the unit cell in order to achieve good convergence.
The WIEN2k code is widely used and thus there is an enormous literature with many interesting results which cannot all be covered here. Many of the publications with WIEN2k can be found on the web page www. A selected list of results, that can be obtained with WIEN2k, is provided below, where references are specified either to the original literature or in some cases to review articles [ 4 , 5 ].
After the SCF cycle has been completed one can look at various standard results: the Kohn-Sham eigenvalues E n k can be shown along symmetry lines in the Brillouin zone giving the energy band structure. A symmetry analysis can determine the corresponding irreducible representation see Fig. For each of these states with E n k the wave function a complex function in three dimensions contains information about how much the various regions of the unit cell contribute.
The fraction of the charge that resides in the interstitial region is contained in q out. These numbers, which depend on the choice of sphere radii, help to interpret each state in terms of chemical bonding. This is an advantages of this type of basis set. There is a useful option to show the character of bands. As one example, three options of presenting the band structure are illustrated for the refractory metal titanium carbide TiC shown in Fig. The crystal field of TiC splits the fivefold degenerate Ti-d orbitals into t 2g and e g states with a degeneracy of 3 and 2 respectively.
Another example is the band structure of Cu shown in Fig. The Fermi surface in a metal is often crucial for an understanding of properties for example superconductivity. It can be calculated on a fine k -mesh and plotted for example with XCrysDen [ 38 ]. With a calculation for a sufficiently fine uniform mesh of k -points s in the irreducible Brillouin zone as discussed in connection with Figure 6 one can determine the density of states DOS , which gives a good description of the electronic structure.
This decomposition becomes even more important in complicated cases, for example if one wants to find which state originates from an impurity atom in a supercell. The electron density is the key quantity in DFT and thus contains the crucial information for chemical bonding but the latter causes only small changes.
Therefore it is often useful to look at difference densities, computed as difference between the SCF density of the crystal minus the superposed atomic densities of neutral atoms , because in this presentation the changes due to bonding become more apparent. Sometimes it is useful to look at the densities corresponding to states in a selected energy window using various graphical tools 2-or 3-dimensional plots. It allows among other details, one to uniquely define atomic charges within atomic basins, a relevant quantity for charge transfer.
See also chapter 6. The typical chemical bonds, like covalent, ionic or metallic bonds, can well be described within DFT. For their analysis the APW type basis is very useful because it can provide chemical interpretations in term of orbitals. Van der Waals vdW interactions, however, are not properly represented in conventional DFT: They can approximately be included by adding a Grimme correction, for example [ 29 ].
The total energy of a system is the main quantity within DFT. Especially for large systems this can be a rather large number, but nowadays it can be calculated with high precision. The interest is often in total energy differences for example to find out which of two structures is more stable.
In such cases the two calculations need to be done in a very similar fashion same functional, comparable k -mesh and basis set, same sphere sizes, etc. It is also possible to compare cohesive or atomization energies, where the atoms must be modeled in the same fashion as the crystal that is in a large supercell containing just the isolated atom.
The derivative of the total energy with respect to the nuclear coordinates yield the force acting on the atom, which is needed for structure optimization. This mismatch in lattice spacing requires that 13 x 13 unit cells of h-BN are needed to match 12 x 12 unit cells of the underlying Rh lattice to make it commensurate with periodic boundary conditions. In order to simulate this system with a supercell, the face-centered-cubic fcc metal layer is represented with three layers with a 12 x 12 Rh lattice which are covered on both sides on the metal slabs with a single layer of BN with a 13 x 13 BN lattice and then an empty region is added to simulate the surface.
Although this is still a crude model of the real situation, it illustrates which kind of large systems can be studied nowadays. This model system contained atoms per unit cell but could be computed with WIEN2k. This corrugated BN surface was found to agree with experimental data for further details see [ 41 - 43 ]. Another example is the investigation of so called misfit layer compounds [ 11 ], in which the bonding between the layers of TaS 2 and PbS required that some Pb atoms are replaced by Ta in an disordered fashion.
Relatively large supercells were needed in order to represent this cross substitution. After relaxing the atomic positions the more likely arrangements have been determined on the basis of total energy differences. In the case of magnetic systems spin-polarized calculations can provide the magnetic moments. In addition to collinear magnetic systems also non collinear magnetism can be handled, which was for example used in a study of UO 2 see [ 44 ]. Another example is the Verwey transition that was investigated for double perovskite BaFe 2 O 5.
DFT calculations made it possible to interpret this complicated situation, see [ 45 ] and section 7. In the latter it was mentioned that it is now possible to use such calculations to look for fine details such as the magneto-crystalline anisotropy. This is defined as the total energy difference between a case, where the magnetic moment is in the y direction with the lowest energy or the x direction.
In this case the difference in energy is found to be about 0. Therefore the quantity of interest is in the tenth decimal illustrating the numerical precision that is needed for such a quantity. Needless to say that extremely well converged calculations were required, in which both cases are treated practically the same. This is necessary to have a cancellation of errors. The electric field gradient EFG is a ground state property that is sensitive to the asymmetric charge distribution around a given nucleus.
By measuring the nuclear quadrupole interaction e. This is a local probe which often helps to clarify the local atomic arrangement. The EFG is a case where the semi-core states can significantly contribute as was shown for TiO 2 in the rutile structure [ 46 ]. On the basis of DFT calculations for the EFG of several iron compounds this quantity had to be adjusted by about a factor of two [ 47 ]. Recently also the NMR shielding chemical shifts can be obtained [ 48 ], where the all-electron treatment opens the possibility of analyzing the dominant contributions that determine the chemical shifts as has been illustrated for fluorides [ 49 ].
The calculation of various spectra X-ray emission or absorption , optical spectra or energy loss near edge structure ELNES spectra can be performed within the independent particle model.
Some structures in the excitation spectra of interacting electrons, called quasi-particle peaks, can be directly related to the excitation of independent electrons as they are treated within DFT. However, others for example satellite structures cannot be understood in such a simple way and require more sophisticated approaches. For example, including the electron core-hole interactions require the solution of the Bethe-Salpeter equation BSE as was illustrated for x-ray spectra [ 50 ]. Often such schemes are based on many-body perturbation theory. One of such approaches is the GW approximation [ 51 ].
This scheme allows calculating accurate band gaps or ionization potentials, which are not well determined by DFT eigenvalues. The interpretation of scanning tunneling spectroscopy STM data often require a simulation by theory, which can distinguish between proposed surface structures. It is based on the Tersoff-Hamann [ 53 ] approximation, in which the images can be obtained from the charge density originating form a set of eigenstates within a certain energy window around the Fermi energy consistent with the applied voltage used in the STM measurements see e. Phonons can be calculated based on the dynamical matrix, which is obtained by displacing one atom in a large unit cell or supercell in a certain direction and determining the forces on all the other atoms.
The necessary independent displacements are determined by the symmetry of the cell. By diagonalizing the dynamical matrix the phonon frequencies can be determined. Such information is also useful for example in connection with ferroelectrics, structural stability, thermodynamics or phase transitions. For the analysis of phase transitions a fundamental understanding requires a combination of concepts, namely group theory, DFT calculations, frozen phonons, soft modes or bilinear couplings, and Landau theory.
This was illustrated, for example for an Aurivillius compound [ 54 ], which shows multiple instabilities and has a phase transition to a ferroelectric state. For high pressure phase transitions a modified Landau theory was proposed and applied [ 55 ]. Maximally localized Wannier functions can be calculated with wien2wannier [ 56 ] and provide a good starting point for more sophisticated many body theory.
Dynamical mean field theory DMFT is one such example as is illustrated in [ 57 ]. Another extension of WIEN2k is the calculation of Berry phases with wien2kPI as modern theory of polarization in a solid for details see ref [ 58 ]. Computer graphics and visualization see [ 38 ]] can help to analyze the many intermediate results atomic structure, character of energy bands, Fermi surfaces, electron densities, partial density of states, etc. The more complex a case is the more support from computer graphics is needed. For an element one can plot all the energy bands, but for systems with over atoms one would be lost interpreting the band structure without the help of visualization.
From the experience of developing the WIEN2k code some general conclusions can be drawn. Some of the historical perspectives have been summarized in section 7 of [ 4 ]. During the last three to four decades it was often necessary to port the code to new architectures starting from main-frame computers, vector processors, PCs, PC-clusters, shared-memory machines, to multi-core parallel supercomputers.
The power of computers has increased in several areas by many orders of magnitude such as the available memory from kB to TB , the speed of communication e. An efficient implementation of a code made it necessary to closely collaborate with mathematicians and computer scientists in order to find the optimal algorithms, which perform well on the available hardware.
One example is the idea of using the scheme of iterative diagonalization [ 59 ]. A significant portion of the computational effort in the WIEN2k calculations is the solution of the general eigenvalue problem see Figure 6 which must be solved repeatedly within the SCF cycle. Changes from iteration to iteration are often small and thus one can use the information from the previous iteration to define a preconditioner for the next iteration and thus simplify the diagonalization and speed up the calculation.
Another aspect is the implementation of linear algebra libraries e. Simultaneously, increased computer power made it possible to treat much larger systems, especially using massive parallelization. The matrix size that we could handle on the available hardware has increased by about a factor over the last several decades.
Since solving the general eigenvalue problem scales as N 3 , the computer power needed to solve a times bigger system must be about a factor 10 9 higher, which is available now. Often our computational strategy had to be changed or extended. For example, to compute a metallic crystal with a small unit cell many k -points s in the Brillouin zone were needed to reach a good convergence. In this case k -points parallelization was optimal.
Nowadays we can treat large unit cells containing about atoms. In such a case, the reciprocal space is small and thus only few k -points s are needed for a good calculation. This requires new parallelization strategies, in which the large matrices must be distributed to many processors, where data locality and reduced communication is essential for achieving good parallel performance. Another aspect is the complexity of the code with the many tasks that need to be solved see Figure 6.
If only a small fraction is not parallelized, it may keep many processors waiting for the result that is calculated on only a single processor. This has often led to new bottle necks, which did not occur for smaller systems and thus were ignored but load balancing is important. Better computer power requires a continuous improvement of the code. There are completely different ways of distributing a code giving representative examples :.
From a commercial point to view it is understandable that a company wants to have strict rules and do not make the source code available. From a scientific perspective, the WIEN2k group favors, the source code is made available to the registered users, who pay a small license fee once. It has helped in many aspects, such as to find and fix bugs, but also to add new features which are made available to all the WIEN2k users.
In addition, several valuable suggestions were made, which allowed improving the documentation as well as implementing requested new features. We have organized more than 20 WIEN2k workshops worldwide, in which users are introduced to important concepts and learn how to run calculations and use kinds of associated tools. It has become a standard to help each other and thus contribute to the development of computations of solids and surfaces. In total this policy has had very positive impacts for WIEN2k and the field. The user friendliness of WIEN2k has been improved over the years.
A graphical user interface w2web was mainly developed by Luitz see [ 7 ] and is especially useful for novice users or in cases which are not done routinely. Later many default options were implemented, which were based on the experience of many previous calculations. This has made it much simpler to set up a calculation. For novice users or experimentalists this helps one to get started without being an expert.
In the old version the users were forced to think about how to run the calculation and thus had to look at details. This is a common problem, which all codes face. With all the possibilities mentioned in the previous section it is often useful to combine different theories according to their advantages but keeping in mind their disadvantages. About 20 years ago the fields of quantum chemistry, DFT and many-body theory were completely separated and there was hardly any cooperation between them: this has fortunately changed.
The strength and weaknesses of the different approaches are recognized and mutually appreciated. The solution of complex problems can only be found in close collaboration of the corresponding experts. Independent of which computer code is used for computations some general questions should be asked, when theory and experiment do not agree.
Some possible reasons for a disagreement are listed below Figure 15 :. Is the atomic structure model that was chosen for the computation adequate for the experimental situation, as already discussed in Section 2. An advantage of theory is that the structure is well defined, because it is taken as input. Experiments may have uncertainties stoichiometry, defects, impurities, disorder.
It can also be that the theory is based on an idealized structure such as infinite crystal, whereas in the experiment surface effects cannot be neglected. If the latter are included in a supercell calculation, one has periodic boundary conditions and thus still assumes an ordered structure, while in the experiment the sample is disordered or contains some defects or impurities.
A delicate question for the experimentalist is whether the sample that has been measured is at least close to the system that was assumed for the simulation. Is the chosen quantum mechanical treatment appropriate for the given system? Is a mean field DFT approach adequate? Are more sophisticated treatments especially for correlation needed or can the self-interaction within DFT cause the problem? One of the effects, which is usually associated with segregation of impurities on the GB, is intergranular embrittlement. It is accompanied by a profound reduction of the ductility and strength.
Materials, which significantly suffer from intergranular fracture and low ductility, are Ni-based Ni 3 X intermetallic compounds with the L1 2 crystal structure [ 31 , 32 , 33 ] although they exhibit a large potential for high temperature applications in corrosive atmospheres [ 34 , 35 ]. Another explanation derived from behavior of impurities in elemental metals [ 38 ] can be based on electronegativity of X atom.
The more electronegative atom X at the GB is the higher the tendency for it to pull charge out of the Ni—Ni bonds at the boundary, thereby reducing the cohesive strength and promoting intergranular fracture [ 37 ]. In case of Ni 3 Al, this problem can be solved by adding a small amount of boron, which improves GB cohesion [ 39 , 40 ] and changes intergranular character of fracture to transgranular. The same improvement of properties was, however, not observed after adding boron into Ni 3 Si intermetallic compound. Boron slightly increases ductility of Ni 3 Si, while leaving the fracture mode unaffected [ 31 , 41 ].
On the other hand, GB properties can be strongly enhanced by adding a large amount of Ti resulting in a highly ordered Ni 3 Si,Ti alloy [ 42 , 43 ]. The main goal of our present study is to determine and analyze tensorial, in particular elastic, properties of material regions affected by grain boundaries in Ni 3 Si and to connect interface-induced changes with properties of individual inter-atomic bonds.
These studies focused on two possible interface variants, which differ by local chemical composition. Both of them were found to have a significantly reduced shear elastic constant C 55 when compared with the bulk. This softening subsequently lowers a homogenized shear modulus and, using a classical approximative model of Slater [ 46 ], we predict that it may lead to lowering of the melting temperature [ 45 ].
Importantly, Si atoms added close to the GB interface plane of the elastically weaker variant were shown to radically alter the elastic properties and improve the softening. In order to do so, we compute and analyze phonon spectra, crystal orbital Hamilton population and densities of electronic states of the studied interface states. Atomic configurations of GBs are often highly distorted and, therefore, it is truly advantageous to employ reliable theoretical tools, in particular quantum-mechanical also called ab initio or first-principles calculations, when studying them [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 ].
Our ab initio calculations are based on the density functional theory [ 56 , 57 ] and we used the Vienna Ab initio Simulation Package VASP [ 58 , 59 ] including the projector augmented-wave potentials [ 60 ]. The electronic wave functions were expanded in plane waves and the expansion was cut-off at those with the kinetic energy of eV. Regarding the exchange and correlation energy, we employed the generalized gradient approximation as parametrized by Perdew, Burke, and Ernzerhof [ 61 ]. The two different GB chemical compositions corresponding to either solely Ni or both Ni and Si atoms at the GB plane are shown in Figure 1 a,b, respectively.
It is worth mentioning that the computational supercells are, in fact, periodic approximants of the real GB-associated interface states because the periodic boundary conditions are applied. Finally, when simulating the application of external strains to determine the elastic constants see Refs. The Ni atoms are visualized as the blue spheres while the Si atoms as the gray ones. Arrows indicate specific inter-atomic bonds discussed in the text see below. Please note that some atoms are shown together with their periodic images.
Lattice-dynamics calculations were performed with the Phonopy [ 69 ] package via the supercell finite-displacement method [ 70 ]. We found that these cells were sufficiently large to converge the shape of the density of phonon states DPS. Our quantum-mechanical calculations of the bulk Ni 3 Si with the L1 2 structure predict its lattice parameter to be 3. The studied GBs have also different volumes when compared with the bulk Ni 3 Si see also the lattice parameters in Table 1. This additional volume is an averaged value when the additional volume obtained for the whole computational cell is divided by the total area of the two GBs inside of the supercell it is thus expressed as a length parameter.
Its value is nearly identical for both GB variants, 0. It should be noted that alternative ways of analyzing the additional volume exist in literature see, e. The calculated lattice parameters within the interface plane of the studied GBs. Next, we determine the elastic properties. Calculated elastic constants of all studied GB systems are given in Table 2 and Table A1 in the Appendix together with bulk elastic constants in the same coordination system.
In addition to providing numerical values of individual elastic constants, we also visualize these elastic properties. Tensorial elastic properties provide the wealth of insight and understanding. For example, they allow us to rigorously assess the mechanical stability of the studied system employing so-called Born stability criteria [ 78 , 79 ].
The interface states are thus softer than the bulk. The most dramatic reduction is related to C 55 elastic constant which is equal only to 12 GPa. Here, we recall Born stability criteria see, e. The diagonal elements C 44 , C 55 and C 66 must be positive and the above-discussed drop predicted for C 55 thus identifies the weakest link. Also, the elastic anisotropy is significantly enhanced when compared with the bulk. The C 55 elastic constant related to shear deformations has turned out to be crucial for the stability of the studied GB states.necromancynancy.com/sitemap1.xml
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As this violates the Born stability criteria, this configuration is mechanically unstable. It is a sheared atomic configuration see Figure 1 c and its elastic constants are given in Table 2 and Table A1 mentioned in the Appendix. In particular, the value of C 55 is predicted to be 37 GPa. This is nevertheless still nearly three times lower than the corresponding bulk value. The above-discussed lower value of the shear elastic constant C 55 is also, in fact, in qualitative agreement with previous findings [ 84 , 85 ] reported for GBs in elemental face-centered cubic fcc metals.
In these studies, atomistic simulations were combined with a method that allows to decompose an overall elasticity into that of different atomic layers and, similarly as in our study, the shear elastic constants were found reduced close to the GB plane [ 85 ]. Small deviations can be found around the E F see the inset in Figure 4 a. To identify which particular atoms contribute to instability of GBs, we further plotted values of local DOS at the E F as a function of z -coordinates in the supercell in Figure 5 a.
Squares represent Ni atoms in the layer occupied only by Ni atoms and circles represent Ni atoms in the layer occupied by Ni and Si atoms. Substitution of Al atom results in a decrease of this value, which could indicate stabilization of GB. The highest contribution to soft modes for GB without Al arise from Ni atoms in the 2nd layer.
Also contributions of atoms in the 3rd and 4th layer are high. Surprisingly, Ni atom at GB does contribute so significantly and contribution of Si atom at GB is just slightly higher than contribution of other Si atoms in the supercell. Substitution of Si by Al significantly decreases imaginary phonon states for all atoms, with the strongest effect for atoms at the GB plane. The highest contribution to imaginary modes now can be found for atoms in the 4th layer.
To get a deeper insight into mutual chemical interaction between individual atoms we employed the analysis of crystal orbital Hamilton population COHP [ 63 ], which helps us to identify weaker chemical bonds in studied periodic approximants of GBs. Bands which do not participate in bonding between particular atoms do not appear in COHP curves.
Another way how to judge the strength of a chemical bond from the point of view of lattice dynamics is a projection of the force constant obtained from phonon calculation on the unit vector along each bonding direction. The Ni—Ni bond seems to be much stronger than Ni—Si bonds, which is in disagreement with previous analysis of bonds in Ni 3 Si based on charge-density plots [ 87 , 88 ].
Next-nearest neighbor interactions exhibit significantly smaller values of both quantities and probably have just a side effect on the stability of Ni 3 Si lattice. Several regions can be recognized in each trend. The weakest interactions between the next nearest neighbors are located in the upper-left corner of the each respective trend.
Black solid lines correspond to values for nearest neighbors Ni—Ni bond in bulk Ni 3 Si, whereas cyan solid lines represent Ni—Si bond. The strongest interaction can be found in the bottom-right corner, which corresponds to interactions across the GB plane between the atoms from the 2nd layers.
Black solid lines correspond to values for nearest neighbors Ni—Ni bond in the bulk Ni 3 Si, whereas cyan solid lines represent Ni—Si bond. Arrows mark inter-atomic bonds discussed in text. This enhancement is not affected by the substitution by the Al atoms.
This conclusion is in line with the values of C 33 elastic constants in Table 2 which are lower for the studied GB states than in the bulk but still high enough to guarantee the mechanical stability. It can be found between the Ni atoms in the 3rd layer and the Si atoms in the 5th layer marked by red arrows in Figure 1 and Figure 7 and shows also the longest inter-atomic distance out of all Ni—Si bonds, significantly longer than the Ni—Si bond in the bulk.
Strengthening of this interaction as well as shortening of the bond length can be seen after replacing the Si atom at the GB plane by an Al atom. Although strengthening does not reach the level found in the case of the Ni—Si interaction in bulk, it is sufficient to stabilize the GB. A comparing of COHP curves in Figure 8 c shows that the character of interactions in both GBs is very similar to that in the bulk and it is just suppressed or enhanced due to a shorter or longer bond length.
To find the explanation for the weakening of this bond, we have to look closer at the GB plane. In particular, interaction between the Si atoms in the GB plane and the Ni atoms in the 3rd layer marked by cyan arrows in Figure 1 and Figure 7 belongs to that Ni—Si interaction with enhanced strength.
As can be seen in Figure 8 d, this enhancement arises from a strong s—s interaction at the bottom of valence bands. This very strong interaction, on the other hand, results in a weakening of the previously discussed Ni—Si bond between the 3rd and the 5th layer. When the Si atom in the GB plane is replaced by the less electronegative Al atom, the corresponding Ni-Al bond exhibits a significantly lower strength due to the missing s—s interaction. Stronger bonds between the Si atoms at the GB plane and Ni atoms in the 3rd layer can be also recognized from inter-layer distances shown in Figure 9.
Similarly, the distance between the 3rd and 4th layer in the same plane is significantly larger than all the other inter-planar distances between the 3rd and 4th layers. This indicates that the Ni atoms in the 3rd layer are strongly pulled towards the Si atoms at the GB plane.
Inter-layer distances in Figure 9 b also show that the Ni atoms in the 3rd layer are pushed back to the position far from the GB plane, when the Si atom is replaced by the Al atom. The elastic constants are found to depend very sensitively on the GB plane chemical composition. In particular, the GB variant containing both Ni and Si atoms at the interface is shown to be unstable with respect to a shear deformation one of the elastic constants, C 55 , is negative.
This instability is found for a rectangular-parallelepiped supercell obtained when applying a standard coincidence-lattice construction. Our elastic-constant analysis allowed us to identify a shear-deformation mode reducing the energy and eventually to obtain a mechanically stable ground-state characterized by a shear-deformed parallelepiped supercell. Lattice-dynamics properties represented by projected force-constant matrices on the unit vector along each bonding direction were considered as well. Such complex analysis reveals a weak interaction far from the GB interface between the Ni atoms in the 3rd plane and the Si atoms in the 5th plane.
However, this bond weakening is a consequence of a very strong interaction between the Si atoms in the GB plane and Ni atoms in the 3rd plane off the GB interface. The same strong interaction was not observed when Si atom at the GB is replaced by Al. Thus the strong interaction near the GB plane makes this GB variant mechanically unstable. Our study thus demonstrates very clearly shows the importance of anisotropic elastic-constant analysis for next studies of interface states close to GBs when determining their mechanical in- stability.
Our analysis represents a complement to numerous previous studies of GBs which were focused predominantly on scalar characteristics i. Our small-deformation anisotropic-elasticity assessment should be ideally extended in future by simulations of larger deformations [ , , , ] which have been rather rarely studied so far in case of GBs see, e.
The found sensitivity of the elasto-chemical inter-relations, which was additionally exemplified by studying the impact caused by Al atoms substituting Si atoms at the GB interface plane, paves a new approach towards a solute-controlled design of interface states with on-demand tensorial elastic properties and stability. Conceptualization, M. The authors acknowledge the Czech Science Foundation for the financial support received under the Project No.
National Center for Biotechnology Information , U. Journal List Materials Basel v. Materials Basel. Published online Nov Author information Article notes Copyright and License information Disclaimer. Received Oct 16; Accepted Nov 5. Introduction Grain boundaries GBs represent one of the most important classes of extended defects. Methods Atomic configurations of GBs are often highly distorted and, therefore, it is truly advantageous to employ reliable theoretical tools, in particular quantum-mechanical also called ab initio or first-principles calculations, when studying them [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 ].
Open in a separate window. Figure 1. Results Our quantum-mechanical calculations of the bulk Ni 3 Si with the L1 2 structure predict its lattice parameter to be 3. Table 1 The calculated lattice parameters within the interface plane of the studied GBs. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Acknowledgments M. Figure A1. Author Contributions Conceptualization, M. Conflicts of Interest The authors declare no conflict of interest. References 1. Duscher G. Bismuth-induced embrittlement of copper grain boundaries.
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