Phonon anomalies predict superconducting T(c) for AlB2-type structures.

We show that the well-known Kohn anomaly predicts Tc for ordered AlB2-type structures. We use ab initio density functional theory to calculate phonon dispersions for Mg1-xAlxB2 compositions and identify a phonon anomaly with magnitude that predicts experimental values of Tc for all x. Key features of these anomalies correlate with the electronic structure of Mg1-xAlxB2. This approach predicts Tc for other known AlB2-type structures as well as new compositions. We predict that Mg0.5Ba0.5B2 will show Tc = 63.6 ± 6.6 K. Other forms of the Mg1-xBaxB2 series will also be superconductors when successfully synthesised. Our calculations predict that the end-member composition, BaB2, is likely to show a Tc significantly higher than currently achieved by other diborides although an applied pressure ∼16 GPa may be required to stabilise the structure.


Introduction
Magnesium diboride, with alternating layers of Mg and B atoms of hexagonal symmetry, is a superconductor below the transition temperature, 1 T c B 39 K. Within the MgB 2 structure, shown in Fig. 1a and c, the boron layer is critical to the superconducting properties of MgB 2 .Theory [2][3][4] and experiment 5,6 demonstrate close links, or coupling, between phonons and electrons. 7Phonons are collective excitations of atoms or molecules in a solid that oscillate at a single frequency and, by their nature, are temperature dependent.Boron vibration frequencies in MgB 2 vary with direction in real space.A dominant collective excitation of these atoms -a phonon mode -is designated the E 2g mode and is readily detected by Raman spectroscopy. 8,9The wave vectors and the quantized energies of all atom vibrations are depicted in a phonon dispersion (PD) plot as shown in Fig. 1d.This plot identifies primary reciprocal space directions along the abscissa and, along the ordinate axis, the phonon frequency (or energy).The two in-plane orthogonal E 2g modes near the reciprocal space point G are also shown schematically in Fig. 1.These phonon modes are a key to understanding the superconducting properties of MgB 2 and metal-substituted forms [10][11][12] and, as we will show, provide a simple unambiguous method to calculate T c .
The density functional theory (DFT), which describes the electronic structure of solids in terms of an electron density distribution rather than a many-electron wave function 13,14 underpins our current understanding of many compounds and, in principle, allows prediction of bulk physical properties. 14Analysis and prediction is facilitated by major improvements in computational capacity and use of approximations to address the physics of electronic structure in order to describe electron density for specific crystal structures. 14,15For superconductors, many computational methods have been used 2,3,16,17 to estimate physical properties such as T c .However, predictions of T c based on DFT band structure calculations alone typically involve estimates of adjustable parameters to ensure theory conforms with experiment. 2,7Floris et al. 16 have shown that superconducting DFT (SCDFT) 13 can be employed, post facto, to estimate from the electronic band structure a calculated T c that closely matches experimental data for MgB 2 .However, this notionally ab initio method employs modifications to standard DFT models 13,16,18 including new approximations and additional parameterization to close the gap between theory and experiment.
We outline in this paper a method to determine phononmediated physical properties using ab initio DFT calculations and interpretation of PDs for AlB 2 -type structures.We describe an approach to predict T c that adds no new functionals to standard DFT models, 19 does not invoke free or adjustable parameters 4,6,7 and accommodates metal atom substitutions.We use the Mg 1Àx Al x B 2 system to demonstrate that ab initio DFT calculation 19,20 predicts T c across the compositional range 0.0 o x o 1.0 based on the magnitude of a phonon anomaly defined by PDs associated with the E 2g mode around the G-point in reciprocal space.This phonon anomaly is described as a Kohn anomaly along G-M based on inelastic X-ray scattering (IXS) experiments 5 on MgB 2 , which show a softening and broadening of the E 2g mode close to G.

Computational methods
DFT calculations were undertaken using the CASTEP 19,20 module of Materials Studio 7.0 and a more recent version, Materials Studio 8.0.This module provides the functionality to calculate vibrational properties for a wide range of materials.The linear response within the local density approximation (LDA) and generalized gradient approximations (GGA) with a dense k-grid mesh is used as detailed in our earlier work. 9,21alculations are undertaken with an ultra-fine cut-off typically 4990 eV.Convergence criteria for most calculations are as follows: energy at 5 Â 10 À6 eV per atom; maximum force at 0.01 eV Å À1 ; maximum stress at 0.02 GPa and maximum atom displacement at 5 Â 10 À4 Å.For BaB 2 calculations, the maximum stress value is varied.
Optimal calculation conditions using CASTEP for the AlB 2type structure are given in earlier work. 9Ordered superlattice structures for non-end-member compositions within the Mg 1Àx Al x B 2 and Mg 1Àx Ba x B 2 series are based on geometry optimization of unit cell parameters with P6/mmm symmetry.These optimized parameters correlate with experimentally determined values for Al substitutions. 22Experimentally determined values for Ba substitutions in the MgB 2 structure are not available.For endmember compositions, optimized unit cell parameters are based on literature values for MgB 2 and AlB 2 as shown in earlier work. 9,21Schematic models of crystal structures shown in Fig. 1 are built using optimized cell parameters from CASTEP as input to the program VESTA. 23Each schematic shows a unit cell in the c-axis direction only.
Calculations are undertaken with the High Performance Computing facility at QUT using o200 cores that are multiples of the k-grid mesh in a and b reciprocal space directions.Within these conditions, PD calculations for the more complex structures may require 7-10 days to achieve dispersion bands with positive values and convergence of the calculation.This requirement may force a compromise in the choice of k-grid mesh density, the accuracy of results and identification of the anomaly in PD plots.
Electronic band structure calculations for all compositions are consistent with earlier work, in particular, those that invoke a dense k-grid mesh. 2,4,16Electronic structure calculations of substituted compositions also requires consideration of the This journal is © the Owner Societies 2015 k-grid mesh density 2 and, for MgB 2 , influences resolution of the tubular sections of Fermi surfaces.
For each composition, calculations on a range of k-grid mesh densities are undertaken to determine the lowest k-value to achieve convergence.For extended superlattice models such as Mg 4 AlB 10 or Mg 5 AlB 12 , computations for k = 0.02, 0.025, 0.027 and 0.03, respectively, are evaluated.The lowest k-value to achieve convergence is listed in Table 1 and, for all compositions, k r 0.03 Å À1 although the computational cost increases by a cube power as grid size is reduced.
The values for the k-grid mesh density used in this work are comparable to, or higher than, that used for many PD calculations due to earlier studies 4,8,9,24 that identified key changes in PD characteristics with this parameter.These changes include shifts in E 2g frequency values at specific reciprocal lattice points 8,9,24 and the appearance of vibration mode branches in PD plots. 9omputational DFT methods are limited by fundamental assumptions on delocalisation and static correlation embedded in functional approximations 25 as well as the degree of complexity of material composition and structure, 17 particularly structures containing transition metals. 15In this work, practical limits include the extent of superlattice models, extension of DFT calculations to include transition metal diborides with substituted compositions and, as noted above, the trade-off between k-grid mesh density, PD detail and convergence.Indications of computational limit for a particular composition in the Mg 1Àx Al x B 2 series are shown by (a) failure to converge, (b) inconsistent or irregular format of the anomaly in PD plots (e.g.multiple lows and highs of an E 2g mode within one branch) and (c) negative phonon frequency values.

Stoichiometry and superlattices
We use superlattices as a computational strategy to model compositions for which x is not an integer.CASTEP 19 allows fractional occupancy of specific atoms in a structure for electronic band structure calculations.This structural description is an alternative method to represent intermediate compositions in the Mg 1Àx Al x B 2 series, but is not possible for PD calculations using CASTEP.Random fractional occupancy of atoms in a solid solution is an alternative approach for CASTEP calculations of band structure.However, this approach induces significant changes in electronic band structures that do not match experimental data.As noted by Kortus, 2 the approach is suited to low levels of metal doping (e.g.x o 0.1) but is expected to fail at higher doping concentrations.
Use of fractional occupancies may invoke inconsistent or uninterpretable shifts in the calculated electronic band structure, particularly in the proximity of the Fermi level.Hence, our PD calculations on ordered compositions do not utilise fractional site occupancies but invoke a superlattice along the c-axis.For superlattice unit cell calculations, appropriate multiples of the end member parameters are used and weighted combinations of these parameters are used as input for mixed compositions containing both Al and Mg or Ba and Mg.A schematic of the ordered composition for x = 0.33, in which an Al layer is sandwiched between two Mg layers, resulting in a 3x superlattice along the c axis, is shown in Fig. 1b.For x = 0.125, the cell size for a DFT calculation can increase to a B 3.07 Å and c B 28.0 Å.This size cell is the maximum we are able to optimise to then calculate a PD with convergence within a reasonable computation time when limited to o200 cores.
For each superlattice, the degrees of freedom per atom within the sub-lattice unit cell is nine although the number of phonon branches in calculated PDs will depend on the multiples of the sub-lattice used to form the P6/mmm unit cell for a specific composition.For structures used in these calculations, Z = 2, 3, 4 or 5 for the superlattice constructs while for the basic P6/mmm structure, such as MgB 2 , Z = 1. 26The Fermi energy corresponds to the average electron density of a structure  and measures the highest energy (in the ground state) of valence electrons in the conduction band as free or nearly free electrons.Therefore, to compare phonon anomalies of end member and intermediate compositions, each superlattice is normalized to a single unit cell.

Error estimates
We have described intrinsic sources of error for PD calculations in earlier work. 21Estimates of error in the calculation of T d are obtained by measurement of d for both branches of the E 2g mode in the G-M and G-K reciprocal directions for each calculation (i.e. for LDA and GGA).Our analysis and estimate of d is dependent on measurement of calculated vibrational frequencies that show a mean relative error 27 of about AE5% for crystal structures of similar size and complexity to MgB 2 .These values of d are converted to T d using eqn (1) below.The error estimate is one standard deviation for the values of T d determined from all the measured values of d.For some compositions, the value of T d is an average of up to eight separate estimates of d obtained by measurement of the frequency difference on two branches of E 2g either side of G for both LDA and GGA models.The individual values for frequency measurements are not shown in Table 1.This journal is © the Owner Societies 2015

Results
Phonon anomaly for Mg 1Àx Al x B 2 An example of the phonon anomaly for MgB 2 calculated using the LDA model with P6/mmm symmetry is given in Fig. 1d and, in closer detail, in Fig. 2a.Fig. 2a shows a portion of the total PD across the G-M (right-hand side of the diagram) and G-K (left-hand side of the diagram) reciprocal lattice directions.The phonon bands or branches that contain the degenerate E 2g modes at the G point and define the double parabolic phonon anomaly are shown in red.The upper limit of the anomaly is defined in this case by the B 2g mode at higher energy.For substituted compositions, the E 2u or the B 2g mode defines the upper extent of the phonon anomaly for E 2g bands in the G-M and G-K directions, as shown in Fig. 2b.For MgB 2 , we show that these upper modes are important for energy conservation through conversion of phonon energies by coherent relaxation. 9,21n the vicinity of the G-point, the E 2g PD bands are degenerate and extend along the basal plane directions (i.e., G-K and G-M) with a characteristic inflection along these directions that is limited, or defined, by the B 2g mode.Note that outside the anomaly, along the G-K and G-M directions, the E 2g mode changes symmetry to E 2u for MgB 2 .The magnitude of the anomaly, d, is measured in frequency units (cm À1 ), as shown in Fig. 2.This anomaly is evident in other publications 9,21,28 when there is sufficient resolution of the k-grid and is referred to as a Kohn anomaly in earlier work. 5,29or high-Al compositions such as MgAl 2 B 6 , the frequency difference between E 2g and E 2u bands at the G-point is minimal or zero, and no anomaly occurs.In addition, for these high-Al compositions, the E 2g and E 2u bands do not show the inflection along the G-M and G-K directions as we show for MgB 2 . 9For AlB 2 , the E 2g mode is the highest frequency optical phonon and does not show a phonon anomaly. 21Similar PD plots for endmember compositions such as MgB 2 and AlB 2 are described in earlier work 9,21 and have been calculated for the compositions shown in Fig. 3

Compositional trends
A summary of the modelled parameters for end-member and intermediate compositions of the Mg 1Àx Al x B 2 suite is shown in Table 1.The E 2g mode for non-end-member compositions in the Mg-Al series increases in energy with increase in Al content, as noted previously 4 and, in each case, a higher phonon energy (e.g.B 2g or E 2u ) occurs until x = 1 where E 2g is the highest frequency mode. 21hese trends are evident with either the LDA or GGA model in the DFT calculation and are consistent with data calculated at sufficient k-grid mesh density. 9The calculated values for E 2g modes at G shown in Table 1 correspond with experimental values determined by spectroscopic techniques. 9,30Calculated partial densities of phonon states (data not shown) show that Mg and Al contribute almost exclusively to modes below 350 cm À1 .As noted in earlier work, 4 B atoms contribute to modes above 350 cm À1 with minor contributions in a small overlap region at 100-350 cm À1 .
The E 2g band is doubly degenerate around the G-point along the basal plane directions for non-end-member compositions in the Mg 1Àx Al x B 2 series.Fig. 2b shows a portion of the PD around the G-point for x = 0.33 calculated using the LDA with a k-grid mesh value of 0.02 Å À1 .The two E 2g modes, which are degenerate at G, are consistent with ordered alternation of boron layers with a metal layer (e.g.either Mg or Al) in the AlB 2 -type structure.In Fig. 2b, the E 2g modes at B570 cm À1 and at B851 cm À1 are highlighted in red, and the higher energy E 2u mode is indicated in orange.For this composition, measurement of the anomaly d for the lower energy E 2g mode shows that it is significantly less (see Table 1) than the anomaly in Fig. 2a.This lower value of d is consistent with a dampening of the dominant mode that influences superconductivity in Mg 1Àx Al x B 2 .

Phonon thermal energy and T c
We have calculated a thermal energy, T d , for each composition in Table 1 based on the equation: where d is the phonon anomaly (in cm À1 ), n is the degrees of freedom per atom, N is the number of atoms per unit cell, Z is the number of formula units per unit cell, k B is Boltzmann's constant, and k B T d /2 is the well-known relationship between thermal energy and degrees of freedom. 9Values for the phonon anomaly, d, when substituted in this equation determine the calculated temperature, T d , (in Kelvin) for each composition listed in Table 1.Fig. 3 shows the calculated temperature (open symbols), T d , associated with the phonon anomaly compared with the experimentally determined T c (solid symbols) for Al content in Mg 1Àx Al x B 2 .Experimental data for Fig. 3 have been collated from studies that used structure refinements 22 and/or microstructural and compositional analyses 26,31 to define stoichiometry and to account for the presence of second phases (e.g.MgB 4 ). 26,32These experimental T c values are for compositions determined on the as-synthesized product(s) made by an internally consistent method.The trend towards a lower T d with increased x value at intermediate compositions for 0.125 o x o 0.5 is consistent with experimental data 22,26,31 on well-characterized Mg 1Àx Al x B 2 .For reference, experimental T c values for x = 0.5 range from B4 K 33 to B13.5 K. 10,26,34 We have used this method to estimate T c on other AlB 2 -type structures.In these cases, the experimentally determined values for T c are much lower (B10 K) and thus, features ascribed to a phonon anomaly will be more difficult to detect.The base case for low T c compounds is provided for the high-Al compositions listed in Table 1 (e.g.Mg 0.5 Al 0.5 B 2 ).6][37] We use the approach outlined above to estimate the T c of the disilicide compounds BaSi 2 and Ca(Al 0.5 Si 0.5 ) 2 .For these disilicides, our estimate for BaSi 2 of T c = 9.3 AE 0.5 K compares with an experimental determination 36 of 8.9 K.An estimated T c = 7.5 AE 0.5 K for Ca(Al 0.5 Si 0.5 ) 2 is similar to the experimental value 35 of 7.8 K.

Phonon anomaly for Mg 1Àx Ba x B 2
We show in Fig. 2c and d partial PD plots for MgBaB 4 and MgBa 2 B 6 .For these DFT calculations, both compositions are constructed as AlB 2 -type structures with P6/mmm symmetry.The extent of the phonon anomaly, d, is shown for MgBaB 4 in Fig. 2c and is similar in form to MgB 2 shown in Fig. 2a.Determinations of d, as shown in Table 2 show that the magnitude of the anomaly for MgBaB 4 is significantly higher than for MgB 2 .Table 2 summarises the calculated parameters for compositions of the Mg 1Àx Ba x B 2 series similar to that shown in Table 1.We have calculated fewer compositions for this series compared with the Mg 1Àx Al x B 2 series because similar conclusions are evident from these calculations.
Comparison of Tables 1 and 2 shows that the Fermi energies for Mg 1Àx Ba x B 2 are lower than the Mg 1Àx Al x B 2 series by approximately 0.7 eV to 1.0 eV across the compositional range.In addition, the optical phonons for Mg 1Àx Ba x B 2 compositions are typically at lower frequencies than Mg 1Àx Al x B 2 with the critical E 2g modes at lower frequencies at the G point for equivalent values of x.

Discussion
The Mg-Al diboride system is characterized by a sharp superconducting transition at x = 0 that decreases in value and gradually broadens with higher Al content. 12Superconductivity in Mg 1Àx Al x B 2 is moderated, but not extinguished, by other influences such as order-disorder, 38 superlattice(s) [10][11][12] and the presence of other phases 32,33 for 0.0 o x o 0.5.Band structure calculations, 2,39,40 experimental data 41 and phonon calculations 8 have established that the vibrational properties of the E 2g mode for MgB 2 influence electron-phonon interactions and superconductivity. 4,7dering and superlattice models Superlattices are observed in Mg 1Àx Al x B 2 systems 10,11,26,34 with prominent diffraction spots at the reciprocal lattice dimension c*/2, which demonstrate a 2x c-axis superlattice 34 for Mg 0.5 Al 0.5 B 2 .Superlattice structures are also predicted 42 for vacancy-ordered MgB 2 but are not considered in this analysis.Detailed analysis of Al-substituted MgB 2 shows that superstructures occur for a range of compositions (x = 0.17; 34 x = 0.25; 10 x = 0.45, 0.5 and 0.55 11 ) and microstructures. 10,11Microstructural studies 10 suggest that intimate mixtures of MgB 2 and Mg 0.5 Al 0.5 B 2 may also occur for 0.1 o x o 0.5.However, the dominant motif involves alternation of Mg and Al layers for x = 0.5. 12Long range ordering of Al and Mg is observed in the a-b plane 12 with an extent B10 nm.Representation of this periodicity in the a-b plane is non-trivial for DFT calculations and is not considered here.Brutti and Gigli 38 used DFT calculations to show that for x 4 0.31, the formation of an Al-rich phase leads to formation of thermodynamically stable Mg 0.5 Al 0.5 B 2 at 50% Al concentration in MgB 2 .
Raman spectroscopy and inelastic neutron scattering studies 43 on Mg 1Àx Al x B 2 samples show evidence for alternate stacking of Mg and Al layers for x B 0.5.In our earlier work, we note that additional Raman and IR peaks not predicted by P6/mmm symmetry for MgB 2 are due to super-lattice modes that approximate a dynamic, phonon-distorted lower-symmetry crystal. 9We show that a 2x super-lattice in the c-direction allows a simple correlation of the pair breaking energy and the superconducting gap. 9 We have evaluated other c-axis ordered structures in which multiple adjacent Al-layers alternate with Mg-layers for key  compositions (e.g.Mg-Mg-Al-Al for x = 0.5 or Mg-Mg-Mg-Mg-Al-Al for x = 0.33).In these cases, while there are differences of B0.15 eV in calculated enthalpies favouring the stability of single Al-layer stacking, ordered motifs with adjacent Al-layers also show a phonon anomaly similar to that in Fig. 2a and b.Thus, a superlattice repeat along the c-axis utilising a simple alternation of Mg and Al layers to minimise adjacent Al layers is an optimal configuration.For simplicity of computation, we follow a superlattice formalism confirmed by detailed experimental studies 11,43 and implied by DFT models 38

Phonon anomalies
The magnitude of the phonon anomaly varies with composition, as do experimentally determined T c values.The average value of d for MgB 2 shown in Table 1 is comparable to the gap energy, 5 2d, of B15 meV.The variation in magnitude of the phonon anomaly reflects the predominant role of boron layer vibrations in many AlB 2 -type structures 44,45 and, in this case, the influence of Al substitution on these vibrations.For the AlB 2 -type structure, the D 6h point group symmetry results in two E 2g modes at the G-point and equivalent displacement modes, E 2u , at the A point of the hexagonal Brillouin zone. 8he doubly degenerate E 2g (G) band describes two distinct displacement patterns that are equivalent within the harmonic approximation. 8The E 2g and E 2u modes have the same movement pattern, albeit with different parity (gerade or ungerade), via a difference in relative phase. 8,46One of the E 2g modes reflects in-plane B-B bond stretching modes that are strongly coupled to the sigma bonded Fermi surfaces related to the p x and p y in-plane orbitals. 46able 1 summarizes the difference in frequency, or the phonon anomaly, d, between the low and high points of the E 2g mode inflection for both G-M and G-K directions for each calculated composition using both LDA and GGA in the Mg 1Àx Al x B 2 series.For both DFT calculation methods, the range of values for the phonon anomaly, d, of each composition is similar, but not equal, because of different assumptions in the methods to calculate the charge distribution in the LDA and GGA methods, 4 differences in optimized lattice parameters and the k-grid value required to attain convergence.
For each calculated composition in the Mg-Al series, the value of T d is slightly higher than the experimentally determined values for T c as shown in Fig. 3.A difference of 1.0 K to 2.5 K is evident for models with x o 0.25 and probably relates to factors such as (a) a higher k-grid value (particularly for x = 0.125) that allows convergence of the PD calculation, (b) systematic errors associated with extended superlattice construction and (c) our DFT calculations are for absolute zero, ground state properties without correction for higher temperatures.Nevertheless, this ab initio determination of T d is internally consistent and in close agreement with the experimentally determined T c trends 22,26,31 for Al-substituted MgB 2 .
The phonon anomaly can be described in terms of interconnected hyperboloid surfaces in different proportions showing origins at different energies and inverse directions along a fixed axis that intercepts G.We can describe this hyperboloid for the E 2g mode(s) as follows.For the E 2g dispersion shown in Fig. 1 and 2, the phonon band is in the k x À k y plane for which k z = 0.For one hyperboloid, the equation for k z = 0 becomes which is symmetric in the k x and k y directions and can be extended to the k x À k y plane.Considering only the k x direction, The asymptote of the hyperbola described in eqn ( 3) is o = ak x .This hyperbola can be approximated to a parabolic dependence as below and as commonly encountered in descriptions of the Jahn-Teller effect: 47 which equals Taking the partial derivative of eqn (3), we obtain Since the group velocity, n g = qo/qk x and the phase velocity, n p = o/k x , we obtain by substitution In the asymptotic region of the E 2g band, where the phonon dispersion changes abruptly from the lowest point of the anomaly towards the higher E 2u phonon band, the curve displays an approximate linear behaviour.This behaviour indicates that the group velocity, n g , is constant.From eqn (6), the phase velocity, n p , is also constant.
In the asymptotic region of the E 2g band, where the phonon dispersion changes abruptly from the lowest point of the anomaly towards the higher E 2u phonon band, the curve displays an approximate linear behaviour (in Fig. 2, the distance along the G-M direction between the green dotted lines a-a 0 and b-b 0 ).This behaviour indicates that the group velocity, n g , is constant and, as shown above, the phase velocity, n p , is also constant.Therefore, there is an interval of k-vectors, Dk, relating to phonon waves for which the wave is non-dispersive.That is, the group of waves moves at constant group velocity and each component of the interval also moves at constant phase velocity.In this instance, the wave packet retains shape and can be viewed as a coherent wave.The slope of the E 2g band in the linear section of the anomaly appears approximately parallel to the acoustic band of highest energy.This relationship suggests that the group velocity of the corresponding optical waves matches the sound velocity.

Phonons and Fermi surfaces
Comparison of electronic bands with PDs shows that the phonon anomalies in the Mg 1Àx Al x B 2 system originate from cusps of paraboloid bands across the Fermi level in the electronic band structure at the G-point.These anomalies are effected by a transfer of electronic charge from the vicinity of the cusps to and from adjacent unit cells and the flat bands in the G-A direction. 39,48ur calculations show that cusp size is directly proportional to the PD anomaly.For example, the PD anomaly is deep when the cusp size in the electronic band is large.If the cusp dips below the Fermi level, for example as with AlB 2 , the phonon anomaly does not occur; this is consistent with experimental data 49 that show no superconductivity for this composition.In addition, for superlattice constructs, multiple parallel cusps that intersect with the Fermi level occur.These intersections are reflected in a multiplicity of E 2g modes and of tubular sections in Fermi surface models.
Calculations of Fermi energies link electrons on or near the Fermi surface to strongly coupled phonons in MgB 2 . 39For example, Fig. 4 shows the Fermi surface for MgB 2 calculated with the GGA model for k = 0.02 Å À1 .For this model, the Fermi energy is 8.1087 eV.In a free electron approximation, the Fermi wave vector, k F , is determined from the equation where h is Planck's constant, m is the electron mass, and E F is the Fermi energy.For MgB 2 , k F = 1.458Å À1 .Using reciprocal cell dimensions from our CASTEP calculations, and limiting this analysis to the k y direction, the Fermi wave vector is B62% of the magnitude of the first reciprocal space vector |a 1 *|.As shown in Fig. 4, the vector resides just outside the first Brillouin zone (point a in the extended zone).Re-plotting this vector to the reduced zone results in a point at position a 0 on the Fermi surface, as shown in Fig. 4.
For two electrons to interact through a phonon, conservation of energy and momentum give the equation: where k e 1 and k e 2 are electron wave vectors, K ph is the phonon wave vector, and G is a reciprocal space point.For an interaction in the k y direction, the magnitude of the electron wave vector will be +k F or Àk F .For a pairing mechanism in which electrons with opposite momenta or wave vectors interact, 2k F = K ph + G for interaction along k y .Substituting values for MgB 2 from CASTEP calculations, the ratio of 2k F (after re-plotting to the reduced zone) to |a 1 *| is 0.239.This ratio is equivalent to the point in reciprocal space along the G-M direction where the E 2g vibration mode meets the B 2g mode, as shown in Fig. 2a (green dotted vertical line denoted b-b 0 ).Thus, a one-to-one correspondence between tubular elements of the Fermi surface and the phonon anomaly occurs for this composition.This reciprocal space point is similar for LDA and GGA models of MgB 2 with a value B0.24 along G-M (equivalent to 0.56 Å À1 ).These values are approximately twice the experimentally estimated radii 5 for cylindrical sigma surfaces parallel to G-A which show average values of 0.17 Å À1 and 0.25 Å À1 using IXS.
For Al-substituted compositions, this point shifts closer to the G-point in a PD plot, as shown in Fig. 2b (green dotted line denoted b-b 0 ; B0.13 along G-M; equivalent to 0.31 Å À1 ).This shift implies a reduction in size of the sigma sheets coupled to the E 2g phonon and is consistent with de Haas van Alfen effect measurements 50 of Al-substituted MgB 2 .Our calculations for other compositions (data not shown) also show a size reduction of the cylindrical sigma sheets in Fermi surface projections with increased Al substitution.Al substitution in MgB 2 results in a commensurate change in the number of tubular sections in Fermi surface projections in proportion to the multiplicity of E 2g modes using superlattice models.

Predicted superconducting compositions
Extrapolation of the methods described above for Mg 1Àx Al x B 2 to other compositions 12,51 of MgB 2 and a consideration of diboride thermodynamics 52 suggest particular atom substitutions to the type structure may also result in superconducting behaviour.For example, our LDA calculations on BaB 2 reveal an electronic band structure similar to MgB 2 but with a PD that shows unusual frequency variation and negative frequency values for modes in the A-H direction and around G. Subsequent calculations constrained to include hydrostatic pressure reduced the number and range of negative frequency values.At an applied hydrostatic pressure of 16 GPa convergence of the LDA and GGA models is achieved with phonon anomalies similar to that shown for MgB 2 in Fig. 2. The E 2g modes show a strong anisotropy in the PD plot particularly in the G-K direction.Nevertheless, these calculations indicate that a BaB 2 structure with an applied stress at 16 GPa is likely to show a phonon anomaly, and by inference, superconductivity.
BaB 2 is not a well-known compound and may be structurally unstable due to a larger Ba +2 ionic radius compared with Mg +2 .Cava et al. 12 notes that a variation of approximately 50% of the This journal is © the Owner Societies 2015 metal atom size can be accommodated by the diboride structure.However, the existence of BaB 2 is not readily confirmed.Early literature 53 on the synthesis of BaB 2 reports cell dimensions that are inconsistent with an AlB 2 -type structure and suggests that this compound is yet to be synthesised.In addition, the geometry optimised cell dimensions for BaB 2 shown in Table 2 are comparable to MgB 2 for the a axis.The c-axis dimension is B33% greater than the calculated values for MgB 2 .These attributes, and the unstable nature of DFT models without a hydrostatic pressure constraint, suggest that BaB 2 may not be thermodynamically stable except at high pressure.
Our computational method utilises a linear response for both LDA and GGA models and is effective for structures with low anharmonicity.Outcomes from these linear calculations as well as structural considerations (e.g.Ba +2 ionic radius), infer that anharmonicity is an important factor for BaB 2 PD calculations.An alternative calculation for BaB 2 using the Finite Displacement (FD) method at similar hydrostatic pressure shows a PD anomaly very similar to Fig. 2a.In this case, the extent of anomaly is significant (B250 cm À1 ) and, by similar analysis to that for DFPT calculations, suggests a T c B 79.1 AE 10.2 K.
The result from DFT calculations on BaB 2 under an applied stress indicates that substitution of an appropriate valence atom for Ba may also induce a similar shift in structural parameters or an improvement in the calculated PD.DFT calculations for compositions of Mg 1Àx Ba x B 2 , where 0 o x o 1, display phonon anomalies of varying magnitude as shown in Fig. 2c and d.Table 2 lists phonon anomalies for three compositions (x = 0.333; x = 0.5 and x = 0.666) and for the end-member BaB 2 at a hydrostatic pressure of 16 GPa.In all cases, calculated PDs for Mg 1Àx Ba x B 2 show a phonon anomaly with an extent significantly greater than calculated for MgB 2 .This analysis predicts that Mg 1Àx Ba x B 2 will show superconductivity at T c 4 60 K over a wide compositional range.We infer from the smaller average c-axis cell dimensions shown in Table 2 that Mg 0.66 Ba 0.33 B 2 is more likely to be a stable phase in this compositional series.This T c prediction for Mg 1Àx Ba x B 2 is B20 K higher than that for MgB 2 , currently the stand-out material in the diboride suite. 54

Coherence lengths
Using the width Dk in reciprocal space, we can derive a width Dx in real space that may be associated with the coherence length of the superconductor.Using the calculated PD for MgB 2 shown in Fig. 2a, we estimate the coherence length in the a-b plane as the distance along the G-M direction (i.e. the reciprocal a axis) between the approximately linear sections of the phonon anomaly manifest in the E 2g modes.This distance, measured in reciprocal lattice dimensions, can be converted to real space dimensions for the symmetry conditions of the unit cell.For MgB 2 , our estimate of the coherence length is B50 Å AE 5 Å.This value compares favourably with experimentally determined values 54 between 61 Å and 65 Å for coherence length in the a-b plane for single crystals; noting that experimental determination of coherence length is dependent on the applied magnetic field 55 and temperature. 56Similar estimates of coherence length for substituted MgB 2 can be inferred from these PD calculations.
For Al-substituted MgB 2 , these estimates of coherence length range from B70 Å to B80 Å with increasing Al content.For Mg 1Àx Ba x B 2 , estimates of coherence length range from B30 Å to B50 Å in the a-b plane.

Conclusions
We have utilised ab initio DFT calculations to show that PD plots of AlB 2 -type structures including Mg 1Àx M x B 2 (where M = Al or Ba) and MSi 2 (where M = Ca, Al or Ba) are key indicators of physical properties.For known compositions, the phonon anomaly predicts physical properties such as the presence (or absence) of superconductivity in this structure type.The extent, or size (in frequency units), of the anomaly provides an estimate of T c by the well-known relationship between thermal energy and degrees of freedom for a particular structure.Agreement between theory and experiment is strong even though DFT models ground state properties at absolute zero temperature.In addition, this approach does not use modified functionals or post facto corrections and is unable, at this time, to account for time dependent phenomena.The phonon anomaly for the AlB 2 -type structure is also known as a Kohn anomaly.If present, the Kohn anomaly provides a means to predict T c of unknown materials and to estimate other key parameters such as coherence length and Fermi surface structure that link electronic and magnetic properties of these materials.This work predicts that Mg-Ba compounds with the AlB 2 -type structure and BaB 2 will show superconducting properties.The approach used in this work amplifies the value of DFT computations as a predictive tool.

Fig. 1
Fig. 1 Schematic of the AlB 2 -type structure and relationship of key atom vibration modes to real and reciprocal space directions.(a) Alternating layers of Mg (gold spheres) and B (green spheres) for MgB 2 viewed at an angle to the a-axis direction.(b) Schematic of the Mg 2 AlB 6 superlattice structure showing alternating layers of Mg, B and Al.For both (a) and (b) the c-axis is elongated by B30% and the a-axis tilted toward the viewer to highlight the hexagonal arrangements of atoms.(c) Projection of MgB 2 down the c* direction showing the two orthogonal E 2g phonon modes.(d) PD plot in reciprocal space of phonon modes in the MgB 2 structure showing the frequencies (or energies) of vibration with principal direction.The principal direction denoted G is at the origin of the unit cell (i.e.[0, 0, 0]).Phonon branches that contain the E 2g phonon modes are highlighted in red.The dotted rectangle along the G direction is the location of the phonon (or Kohn) anomaly and shown in detail in Fig. 2 for MgB 2 , Mg 2 AlB 6 and Mg 1Àx Ba x B 2 compositions.

Fig. 2
Fig. 2 Partial PD plots based on DFT models with k = 0.02 Å À1 along the G-M and G-K directions of reciprocal space: (a) for MgB 2 and (b) for Mg 2 AlB 6 using the LDA functional; for the predicted compositions (c) MgBaB 4 using the GGA functional and (d) for MgBa 2 B 6 using the LDA functional.The phonon anomalies for the E 2g mode around the G-point are highlighted in red and the magnitude of the anomaly, d, is shown.The dotted red lines show regions with non-degenerate E 2g modes.The green dotted line b-b 0 corresponds to reciprocal lattice dimensions equivalent to the average diameter of the Fermi surface for the specific composition.The distance between a-a 0 and b-b 0 along the G-M direction is an indicator of coherence length in the a-b plane of an AlB 2 -type structure.

Fig. 4
Fig. 4 Fermi surface projection along g 3 calculated for MgB 2 using the GGA model with k = 0.02 Å À1 .The projection shows the equivalent magnitude of the Fermi vector, 2k F , along the k y direction and tubular sections on the Fermi surface.Green spheres are Mg atoms; off-white spheres (partially obscured) are B atoms.

Table 1
Calculated parameters for DFT models of Mg 1Àx Al x B 2 d (av.) s T (error)

Table 2
Calculated parameters for DFT models of Mg 1Àx Ba x B 2