Temperature dependence and stability of surface structures

Several years ago, Burton and Jura' reported a calculation that was based on an Einstein model and that predicted a structural phase transition for the (100) surface of Ar. On the basis of this calculation, one might expect to be able to observe such a surface phase transition in noble-gas solids and other fcc materials, such as Al and Ni. We show in Fig. 1(a) the unrearranged fcc (100) surface, and in Fig. 1(b) the rearranged surface that is predicted at high temperatures in the calculation of Ref. 1. In the rearranged structure, every other row of atoms has been displaced by haU the edge of a square in the x direction. In order to study the possibility of this structural transition, we performed a lattice-dynamics calculation of the vibrational frequencies for the rearranged structure. The calculation involved the usual procedure, with the only complication being that there are two particles per unit cell. We used a Lennard-Jones potential of interaction between the atoms, as in Ref. 1, and we did the calculation for an ll-layer slab. (As a check, we also used smaller thicknesses and found that the results reported here are not much affected by the finite thic!~ness of the slab. ) Preliminary to calculating the vibrational frequencies, we allowed the rearranged and unrearranged atoms in the surface [represented, respectively, by the solid and open circles of Fig. 1(b)] to relax outward, taking the atoms in the second, third, etc. , layers to remain in their bulk positions. The outward relaxation was found to be 0. 213a and, 0.036a for rearranged and unrearraoged atoms, respectively, where v 2a is the nearest-neighbor spacing in the bulk. In Fig. 2, we show the dispersion curves from the origin to the edge of the surface Brillouin zone in the q„direction. Near the origin, for a pair of modes localized near the surface, the vibrational frequency ~ becomes imaginary; i.e. , u &0 for sufficiently small values of q„. Imaginary frequencies correspond to an instability, so this result implies that the rearranged structure is mechanically unstable. In Fig. 3, we show the eigenvector component E„ for the modes with u2& 0 at q„=0. We have X X FIG. 1. (a& Unrearranged (100) surface of fcc crystal. (b) Rearranged surface. Open circles represent unrearranged surface atoms, solid circles represent rearranged surface atoms, and x's represent atoms in the next plane beneath the surface. The arranged atoms in (b) are above the unrearranged atoms (see text). plotted the average value of ReE„as a function of layer number for the pair of modes associated with the instability. (These modes are almost identical; as usual, there are two surface modes of each type because the slab has two identical surfaces. ) For these modes, g, = ),=Im)„=0. It can be seen that the instability is associated primarily with motion of the rearranged particles at the surface in the x direction. We thus have the result that at long wavelengths (smali q„) the rearranged atoms move back to their unrearranged positions, i. e. , they "slide off" the atoms in the second layer back into the "holes" corresponding to the unrearranged positions. As short wavelengths (larger values of q,), the instability is removed and ur &0, presumably because the atoms are not vibrating in phase and an atom which tries to "slide off" into the "hole" that it came from will encounter another atom. Qnly for wavelengths bigger than about ten