Empirical tight-binding model for titanium phase transformations

For a previously published study of the titanium hexagonal close packed $\left(\alpha \right)$ to omega $\left(\omega \right)$ transformation, a tight-binding model was developed for titanium that accurately reproduces the structural energies and electron eigenvalues from all-electron density-functional calculations. We use a fitting method that matches the correctly symmetrized wave functions of the tight-binding model to those of the density-functional calculations at high symmetry points. The structural energies, elastic constants, phonon spectra, and point-defect energies predicted by our tight-binding model agree with density-functional calculations and experiment. In addition, a modification to the functional form is implemented to overcome the “collapse problem” of tight binding, necessary for phase transformation studies and molecular dynamics simulations. The accuracy, transferability, and efficiency of the model makes it particularly well suited to understanding structural transformations in titanium.

Figure 1

Interpolated intersite function with short-range spline. The parametrized function $h\left(r\right)$ grows exponentially as $r$ approaches zero, though the function is only sampled in the fitting database down to $R_{\min }$. At $r=R_{\min }$, we replace the function with a quartic spline that matches the value, first, and second derivatives at $R_{\min }$; the dashed curve shows the growth of the original function. The spline smoothly goes to a constant value in a width of $\sigma$. Only one adjustable parameter $\sigma$ is added to the entire fitting database, as $\sigma$ is the same for all functions.

Figure 2

Energy of Ti dimer calculated with tight binding using short-range splining. Without any short-range splining, the overlap matrix becomes artificially large, creating a bonding state with very low energy at small distances; at $1.92\mathrm{\AA }$, the tight-binding dimer Hamiltonian problem becomes unsolvable. As described in the text, by short-range splining of the Hamiltonian and overlap functions, the model is stable and becomes repulsive at small distances.

Figure 3

Tight-binding intersite Hamiltonian and overlap functions. The parametrized hopping integrals are shown for distances from 2.4 to $3.5\mathrm{\AA }$. The $R_{\min }$ in the fit is $2.35\mathrm{\AA }$; below this, these functions are smoothly interpolated to a constant value. Circles represent $\sigma$ integrals, squares $\pi$ integrals, and triangles $\delta$ integrals; black is for $s$, dark gray for $p$, and light gray for $d$.

Figure 4

Tight-binding onsite energy terms for hcp structure. The onsite energies are environment dependent in our model; we show the variation with respect to the volume of an hcp crystal with $c∕a=1.588$. The low volume of $10{\mathrm{\AA }}^3$ has a lattice constant of $2.44\mathrm{\AA }$, and the high volume of $25{\mathrm{\AA }}^3$ has a lattice constant of $3.31\mathrm{\AA }$. The equilibrium hcp volume is $17.56{\mathrm{\AA }}^3$.

Figure 5

(Color online) Comparison of tight-binding energy as a function of volume for $\alpha$ and $\omega$ with first principles data. The two filled points were included in the fit; the lines are FLAPW total energies. Our tight-binding model reproduces the fit data—slightly lower ground-state energy and equilibrium volume for $\omega$—and the equation of state of the full-potential calculations.

Figure 6

Comparison of tight-binding phonons for the $\alpha$ phase with experimental phonon data. The crosses are the experimental phonon frequencies at $295\mathrm{K}$ (Ref. 34). The deviation from the experimental values at small $q$ corresponds to the mismatch in the $\alpha$ elastic constants. Our tight-binding model does well for the high-energy optical and acoustic branches which are important for modeling the $\alpha \to \omega$ transformation.

Figure 7

Predicted $\omega$ phonons from tight binding. As expected from the $c∕a$ ratio of 0.620, the phonon modes are stiffer along the $\left[00\xi \right]$ direction than the basal plane directions $\left[\xi 00\right]$ and $\left[0\xi 0\right]$.

Figure 8

Comparison of tight-binding bcc phonons with calculated spectra using GGA. At $T=0$, the bcc phase in Ti is unstable, as shown by the imaginary phonon frequencies. The agreement between the phonons calculated using the current model and those with GGA is good, for both stable and unstable phonons. The deviation at the $P$ point indicates too much stiffness in the TB model for motion of [111] chains in bcc compared to GGA. The dip in the $\left[\xi \xi \xi \right]$ branch is near the L-$\frac{2}{3}\left[111\right]$ phonon, which corresponds to the $\mathrm{bcc}\to \omega$ transformation pathway. The imaginary phonon for T-$\left[011\right]$ corresponds to the $\mathrm{bcc}\to \alpha$ transformation mechanism (Ref. 35).