Results

First-Principles Simulations of V-X Alloys

5.3 Geometry Relaxation

5.3.2 Results

The results of supercell geometry optimizations are summarized in table 5.1. Since the supercells have O_h point symmetry, the directions of impurity-host bonds are constrained to 111, 200 and 220 directions for the 1-st, 2-nd, and 3-rd nearest-neighbor bonds, respectively. The free parameters are the bond-lengths and lattice parameters. Our results are given in table 5.1.

A comparison with the experimental results listed in table 4.1 shows that the calculated lattice parameters are too small by about 1%. Additional calculations on pure V within the local density approximation (LDA) in VASP predicted an even smaller lattice parameter, about 4% smaller than the experimental value. Discrepancies of 1% are considered small with current implementations of density functional theory. Besides, this error is systematic

Table 5.1: Relaxed supercell geometries from first-principles calculations. L1, L2 and L3

are the 1-st, 2-nd, and 3-rd nearest-neighbor bond lengths;ais the equivalent BCC lattice parameter.

system VASP Wien2k Wien2k

2×2×2 2×2×2 3×3×3

a(˚A) L₁(˚A) a(˚A) L₁(˚A) a(˚A) L₁(˚A) L₂(˚A) L₃(˚A) V-Ti 3.015 2.653 3.013 2.653 3.004 2.676 3.003 4.247 pure V 2.996 2.594 2.996 2.595 2.996 2.595 2.996 4.237 V-Cr 2.989 2.561 2.986 2.557 2.994 2.544 3.009 4.225

V-Mn 2.980 2.543 2.993 2.538 3.010 4.223

V-Fe 2.978 2.547 2.973 2.546 2.992 2.547 3.002 4.219 V-Co 2.978 2.565 2.973 2.562 2.991 2.568 2.988 4.220 V-Ni 2.982 2.592 2.980 2.588 2.993 2.601 2.969 4.229

V-Zr 3.036 2.722

V-Nb 3.020 2.659

V-Pd 3.002 2.650 2.998 2.641 2.998 2.664 3.008 4.230

V-Hf 3.036 2.714

V-Ta 3.023 2.666

V-Pt 3.002 2.643 2.998 2.634 2.998 2.654 3.007 4.239

1.0 0.8 0.6 0.4 0.2 0.0

Energy (eV)

6.2 6.1

6.0 5.9

5.8

a (Å)

Figure 5.2: Comparison of volume relaxation curves from Wien2k and VASP for V₁₅Ti₁. The lines are fits to the Murnaghan equation of state.

and can be overcome by looking at the relative change in lattice parameter Δa/a between pure vanadium and the alloys. This quantity is plotted in figure 5.3. As seen on this figure, there is good agreement between the experimental and calculated change in lattice parame- ter. This indicates that the relaxation effect upon introduction of the impurities is captured well by the DFT simulations. The deviations are not much larger than the experimental uncertainties, except in the case of V-6.25%Ti, for which Δa/a is overestimated by about 50% in both the FP-LAPW and PAW calculations. Both types of simulations reproduce well the overall trend across the 3d-series, with first a linear contraction, followed by an upward curvature of Δa/a.

This trend is due mostly to the change in 1NN bond length, as we will now discuss.

Figure 5.4 shows the calculated change in 1NN bond length ΔL₁/L₁ for the 3d-series.

The values of ΔL₁/L₁ obtained from FP-LAPW and PAW calculations are in very good agreement and present a similar behavior as Δa/a. The magnitude of the change is much larger however, as physically expected. The upward curvature for later transition metal impurities is also stronger in ΔL₁/L₁ than in the lattice parameter, with almost no net change in 1NN bond length in the case of Ni impurities. This indicates that for Ni, the

1.2

0.8

0.4

0.0

Δ a/a (%)

8 7

6 5

4 3

2

N

_d

4d 5d XRD 4d 5d density 4d 5d Wien2k -0.8

-0.4 0.0 0.4

Δ a/a (%)

8 7

6 5

4 3

2

N

_d

XRD density

Wien2k-2x2x2 VASP-2x2x2

(a)

(b)

Figure 5.3: Relative change in lattice parameter between V-6.25%X alloys and pure V. (a):

impurities in the 3d-series, (b): impurities in the 4dand 5d-series. N_dis the formal number ofdelectrons of the impurity.

structure. The positions of 2NN and 3NN atoms around the impurity in the 2×2×2 supercell are constrained by symmetry, so we used 3×3×3 supercells to investigate the relaxation of these shells. The results are plotted in figure 5.5. This figure clearly shows that the relaxation predominantly occurs in the 1NN bonds, at least for earlier transition metal impurities up to Fe. For these elements, the relaxation in L_1NN is more than four times larger than in L_2NN orL_3NN. For Co impurities, this ratio decreases to about three and in the case of Ni the relaxation affects mostly L_2NN. For the other impurities of the Ni-column, Pd and Pt, the relaxation in the 1NN shell is sizeable, with 2.6% and 2.3%

increase in 1NN bond lengths for Pd and Pt impurities, respectively. The strain is also positive for 2NN bonds in the case of Pd and Pt impurities (about 0.4% increase). In all cases, the strains decay to negligible levels beyond the 3NN shell.

An interesting aspect is the alternating sign in the strain as the distance from the impurity increases, for impurities in the 3d-series. This feature may be related to Friedel oscillations in the electronic density. However, Singh et al. have also reported predictions of such oscillations in the strain around impurities in V [106], based on purely mechanical considerations of lattice forces. Singh et al. used an analytical potential for interionic interactions in transition metals developed by Wills and Harrison [107] and performed calculations of the strain induced by transition metal impurities (Ti, Cr, Mn, Fe, Nb, Mo, Ta, W) in vanadium with the Kanzaki lattice statics method [108]. The strains they calculated show trends similar to our results, but are much smaller in magnitude, by about a factor of three. Also, the calculations of Singhet al. predict a maximum strain at the 2NN shell for all the impurities they considered (except W), which disagrees with our results.

It is likely that their results suffer from the stringent approximations that were used, in particular the neglect of charge transfer between the solute and host atoms as well as the central potential approximation and the inclusion of ion-ion interactions only up to 2NN.

Also, the electric field gradient around the impurities calculated by the same authors in

3 2 1 0 -1 -2 Δ L

1NN

/L

1NN

(%)

8 7

6 5

4 3

2

Nd

LAPW-2x2x2 (Wien2k) LAPW-3x3x3 (Wien2k) PAW-2x2x2 (VASP)

Figure 5.4: Impurity-host 1NN bond-length calculated for different supercells.

a subsequent study are not in very good agreement with experimental results [109]. This discrepancy also illustrates the difficulty of making reliable predictions for transition metals using simple analytical potentials.