Ab initio study of displacive phase transformation paths in As, Sb and Bi

P. Lazar^{ab}, M. Friák^{ab}, D. Legut^{c}, M. Šob^{a}

a Institute of Physics of Materials, Academy of Sciences of the Czech

Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic

b Institute of Condensed Matter Physics, Faculty of Science, Masaryk

University, Kotlářská 2, CZ-611 37 Brno, Czech Republic

c University of Technology Brno, Faculty of Chemistry, Department of

Chemistry of Materials, Purkyňova 118, CZ-612 00 Brno, Czech Republic

\underline{Abstract:} Total energy along the trigonal deformation path was calculated ab initio for three elements with A7 structure (As, Sb, Bi) using the full-potential linearized augmented plane wave method (FLAPW) within the generalized gradient approximation (GGA). In As, the existence of a high-pressure simple cubic phase was theoretically confirmed and the transition pressure was determined. Also elastic moduli for materials studied were calculated and compared with available experimental data.

Introduction: Bi, As, Sb

• ground state: orthorombic A7 structure

• one internal degree of freedom, u:

distance of (111) planes

• two nonequivalent atoms at ±u(a, a, a) in the unit cell • applied pressure: A7 ⟹ simple cubic structure

• semimetallic under normal conditions (A7 phase), highpressure sc phase found to be metalic

The A7 structure may be derived from simple cubic (sc) by applying two distortions: trigonal deformation along [111] together with shrinking of internal parametr u:

sc: \frac{c}{a} = \sqrt{6} ≈ 2.449 ⟶⟶ A7: \frac{c}{a} = 2.608

u = 0.25 ⟶⟶ u = 0.23394

\underline{Objectives:}

• ab initio study of structural behaviour of three elements with A7-structure along one-parameter transformation paths connecting the simple cubic and A7 structures

• analysis of energetics of higher-symmetry metastable structures

• behavior of internal parameter u in trigonal defomation

• influence of spin-orbit coupling

Details of the calculations

\underline{Method:}

• all-electron full-potential linear augmented plane waves method (FLAPW)

• WIEN 97 code (P. Blaha et al., Technical University of Vienna)

• for exchange-correlation energy:

generalized gradient approximation – GGA

• calculations with/without spin-orbit coupling

• muffin-tin radii sufficiently small to guarantee non-overlapping at every point of the transformation path (R_{MT} = 1.9 au was used)

• all calculations at T = 0 K Calculations performed:

• total energy of arsenic, antimony and bismuth as a function of trigonal distortion; internal parametr u was linearly interpolated

• total energy of arsenic as a function of elementary cell volume connected with the analysis of the A7-sc transformation with calculation of transition pressure

• equilibrium lattice parameters calculated for bismuth and compared with experimental data

• trigonal shear modulus c_{44} estimated for all three elements and compared with available experimental values

• total energy of bismuth as function of trigonal distortion with spin-orbit coupling included Symmetry-dictated extrema:

• appear at the points of higher symmetry along the transformation paths

Other extrema (not dictated by symmetry) may exist; they are a property of the system. If the extrema (of both types) are minima they may correspond to

• ground-state structures

• metastable configurations found often in extended defects.

Examples:

• A7 structure of As, Sb, Bi

• 9R (α-Sm) structure found at grain boundaries (GB’s) in Cu and Ag

• bcc Cu found at some GB’s

• B19-type structure found in Ti–48at%Al Trigonal deformation of arsenic. Along this deformation path, one symmetry-dictated maximum (sc) and two minima not dictated by symmetry may be found.

We used 861 k-points in irreducible Brilouin zone (IBZ); calculations were performed at the experimental elementary cell volume without spinorbit coupling.

RESULTS

Trigonal shear modulus c_{44}[GPa]

– calculated on experimental unit cell volume

– no relaxation of internal parameter - it was linearly related to trigonal deformation

This work Experiment^{a}
Arsenic  33.9  –
Antimony  18.4  28.5
Bismuth  16.6  14.9

Bismuth: ground state lattice parameters

– experimental volume, internal parametr fully relaxed

This work Experiment^{b}
c/a  2.59  2.77
u  0.236  0.228

Arsenic: ground-state volume and A7→sc transition pressure

– A7 experimental lattice parameters c/a = 2.7717 and u = 0.2271

This work Experiment^{c}
Volume[a.u.^{3}/per atom]  150.7  143.8
Pressure[GPa]  11  25 (at 300K)

a Tichonov et al.: Struktura u svojstva metallov u splavov

b Donohue: Structure of the elements, New York, 1974

c Beister H.J., Strossner K., Syassen K.: Phys.Rev. B 41, 1990

CONCLUSION

• The existence of metastable high-pressure sc phase in arsenic theoretically confirmed. Computed transition pressure is about half of the experimental value corresponding to 300 K.

• The dependencies of the total energy on the trigonal distortion are similar for all three elements studied.

• Theoretical lattice parameters of Bi were found within 6% of their experimental values.

• GGA overstimated the ground-state cell volumes: Theoretical volume of As was found 4% higher than the experimental one.

• Nonconsistent results appeared in the calculation of elastic constants. Bismuth shear modulus c_{44} is overstimated by 10% while for antimony underestimated by 30%.

• The relaxation of internal parametr u is showed as important. It may also bring better numerical values of shear moduli.

• The effect of spin-orbit coupling was found to be negligible in our calculations for bismuth.