INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTERJ. Phys.: Condens. Matter 15 (2003) 855–861 PII: S0953-8984(03)54310-7Vacancy formation enthalpy at high pressures intantalumSonali Mukherjee1,RECohen1and O˘guz G¨ulseren2,31Carnegie Institution of Washington, 5251 Broad Branch Road, NW Washington,DC 20015, USA2NIST Center for Neutron Research, NationalInstitute of Standards and Technology,Gaithersburg, MD 20899, USA3Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia,PA 19104, USAReceived 3 October 2002Published 3 February 2003Online at stacks.iop.org/JPhysCM/15/855AbstractUsingamixed basis pseudopotential method, total energy calculations wereperformed to obtain the enthalpy of vacancy formation in Ta as a functionof pressure, which is important for understanding the effects of pressure onmechanical properties. The vacancy formation enthalpy is found to increasefrom 2.95 eV at ambient pressures to 12.86 eV at 300 GPa, and the vacancyformation volume decreasesfrom being 53±5% of the bulk volumeper atom atambient pressure to 20± 2% at 300 GPa, for a 54-atomsupercell. We also showthat there is a strong correspondence between the vacancy formation enthalpyand the melting temperature in Ta.The effects of compression on mechanical properties of transition metals are not easilyaddressed experimentally, yet are important in modelling dynamical behaviour of materialsunder compressive loads. This paper addressesfor the first time the pressure dependenceof the enthalpy of vacancy formation in tantalum. Ta remains stable in the simple body-centred cubic (bcc) structure over a wide pressure range [1] allowing study of the effects ofcompressionon vacancy formationinatransition metal overa wide compression range withoutthe complications of structural changes. High thermal, mechanical and chemical stability alsomakes Ta an important technological material [2].The role of vacancies in controlling plasticity through the mobility of dislocations has notbeen fully explored. The vacancies allow the dislocations to overcome the interstitials andimpurities by facilitating their climb to a plane normal to the glide plane by self-diffusion ofatoms [3]. Moreover, dislocation glide depends upon double-kink formation [4], and thoughpresent calculations neglect vacancies in the calculation of double-kink formation energy [5],it is greatly modified by vacancies [6]. Computations of the vacancy formation enthalpy andits pressure dependence provide us with the vacancy concentration, which in turn determinesthe effects of vacancies on the dislocation motion. Small changes in the vacancy formation0953-8984/03/060855+07$30.00 © 2003 IOP Publishing Ltd Printed in the UK 855856 SMukherjee et alenthalpy with pressure lead to large changes invacancy concentration due to the exponentialdependence of the vacancy concentration on the enthalpy.The vacancy formation enthalpy [7], Hvac,isgivenbyHvac= Evac(P) + Pfvac, (1)where Evacis the vacancy formation energy and fvacis the vacancy formation volume atpressure P.Wefind the vacancy formation energy by performing total energy calculations forsupercells with one atom removed, and compare the energy of the N -atom supercell with the(N − 1)-atom defective supercell. Theatomicpositions after the atom is removed are relaxedto minimize the total energy using analytic forces. Expressing Evacand fvacin terms of thesystem internal energy and volume one obtainsEvac(P) = E(N − 1, P) −N − 1NE(N, P) (2)andfvac= V (N − 1, P) −N − 1NV (N, P). (3)E(N, P) and V (N, P) are the internal energy and volume of an N-atom ideal system atpressure P,while E(N − 1, P) and V (N − 1, P) are the internal energy and volume of asystem with N − 1atoms and a vacancy at pressure P.Thus one needs the internal energyand volume of the ideal system and of the system with the vacancy, where the systems are atthe same pressure, P.We used the mixed basis pseudopotential method [8] within density functional theory tocompute E(N, V ) and E(N − 1, V ).Weusedthelocal density approximation (LDA) [9],the Perdew–Wang [10] and PBE [11] generalized gradient approximations (GGA) for thetreatment of the exchange–correlation potential. We found that thevacancy formationenergy at a given volume varies little with the exchange–correlation potential chosen. Wereport the PBE GGA results below unless otherwise stated. Calculations have shown thatcomputationally efficient mixed basis pseudopotential results are in close agreement to the all-electron, full potential linear-muffin-tin-orbital results for bcc transition elements at ambientpressure [12]. We generated a non-local, norm-conserving Troullier–Martins [13] semi-relativistic pseudopotential. The pseudopotential was generated from 5d36s26p0atomicconfigurationswith cut-offradii 1.46, 2.6 and 3.4 bohr for 5d, 6s and 6p potentials respectively,with non-linear core corrections. The basis set consists of pseudo-atomic orbitals and somelow-energy plane waves with acut-offof 60 eV. The cut-off for the plane waves used toexpand the potential and charge density is 550 eV. The equation of state of Ta obtained fromthepseudopotential method is in very good agreement with the LAPW calculations and theexperiments[14].The ambient pressure equilibrium volume obtained for Ta is 18.15 Å3whichis 1.5% higher than the experimental value and the bulk modulus is 199 GPa which is in veryclose agreementwith the experimental value of 195 GPa [1]. A special k-point mesh [15] gives35 k-points for the 16-atom supercell, and 10 k-points for the 54-atom supercell. Gaussianbroadening with a width of 0.172 eV was used to optimize convergence with respect to the k-point sampling. Convergence of the vacancy formation energy of up to 0.001 eV was achievedwith respect to k-point sampling.The total (internal)energies of the ideal system and the system with the vacancy at constantpressure are obtained by fitting the volume dependence of the total energies for both systemsto the Vinet equation of state [16]. Calculation of the internal energy of the system withthe vacancy included structural relaxation, which allowed the ions to readjust their positionsaround the vacancy. Relaxation was considered complete when forces on the ions were lessVacancy formation enthalpy at high pressures in tantalum 857Table 1. Comparison of available Ta vacancy formation energies (eV) at ambient pressure for the16- and 54-atom