Entropy Explains Metal-Insulator Transition of the Si(111)-In Nanowire Array

Density functional theory calculations are performed to determine the mechanism and origin of the intensively debated ($4\times 1$)–($8\times 2$) phase transition of the Si(111)-In nanowire array. The calculations (i) show the existence of soft phonon modes that transform the nanowire structure between the metallic In zigzag chains of the room-temperature phase and the insulating In hexagons formed at low temperature and (ii) demonstrate that the subtle balance between the energy lowering due to the hexagon formation and the larger vibrational entropy of the zigzag chains causes the phase transition.

Figure 1

Schematic top views of (a) room-temperature ($4\times 1$) and (b) the low-temperature ($8\times 2$) hexagon structure of the Si(111)-In nanowire array. Red (dark gray) balls indicate In atoms.

Figure 2

Calculated eigenvectors for three prominent phonons modes (notation as in Table ) of the Si(111)-($4\times 1$)-In (a),(b) and Si(111)-($8\times 2$)-In phase (c). The mode shown in (b)—occuring at the $X$ point of the ($4\times 1$) BZ—is twofold degenerate due to the existence of an equivalent mode at the neighboring In chain.

Figure 3

Difference of the free energy $F(T)$ calculated for the $4\times 1$) and $8\times 2$) phase of the Si(111)-In nanowire array. The stable phase is indicated. The inset (enlarged) shows the entropy difference calculated by neglecting the electronic contributions and by restricting the BZ sampling to the $\Gamma$ point.

Figure 4

Phonon density of states calculated for the ($4\times 1$) and ($8\times 2$) phases of the Si(111)-In nanowire array ($4{\mathrm{cm}}^{-1}$ broadening). The inset shows a specific displacement pattern that hardly changes upon the phase transition but shifts in frequency. Circles/crosses indicate up/down movements.