Solid-Gas Interface

Polarons in Nanometer-Scale Thin Film

In a typical covalently bonded crystal such as Si, electrons and holes can be approximated to move through a crystal whose atoms are frozen into place; however, such an approximation is inadequate in ionic or highly polar crystals such as oxide perovskites, where the Coulomb interaction between conduction electrons and vibrating lattice ions results in a strong electron-phonon coupling. Under such a strong interaction, conduction electrons and holes coupled with vibrating lattice deformations can be treated as polarons.

For example, stoichiometric SrTiO₃ or BaTiO₃, which have a d⁰ electronic configuration, can be treated as band insulators with a band gap of about 3.2 eV [M. Cardona, Phys. Rev. 140 (1965) A651.]; however, in electron-doped SrTiO₃ or BaTiO₃ doped electrons enter the bottom of the empty Ti 3d conduction band, and then, the properties become different from the one of conventional semiconductors such as Si and Ge. For example, because of the strong electron-phonon coupling resulting from ionic displacements (the displacement of Ti⁴⁺ and O^2- in a TiO₆ octahedron), in SrTiO₃ and BaTiO₃ the electron effective mass (m*) is heavier than that of free electron (m_e = 9.1 * 10^-31 kg): m* in 0.85-at.%-electron-doped SrTiO₃was estimated to be 1.2 m_e  for light electrons and to be 7.0 m_e  for heavy electrons  by Chang et al. [Y. J. Chang, A. Bostwick, Y. S. Kim, K. Horn, E. Rotenberg, Phys. Rev. B 81 (2010) 235109. They observed that the Fermi surface of SrTiO₃ consists of three degenerate ellipsoids (d_xy, d_yz, and d_zx) above 105 K, and that d_xy band has lower minimum energy than the doubly degenerate d_yz and d_zx bands below 105 K]; the dynamic Jahn-Teller splitting of the 3d-t_2g states of Mo⁵⁺ in SrTiO₃ is 60 meV at 1.65 K [B. W. Faughnan, Phys. Rev. B 5 (1972) 4925.]. For another example, the splitting of the 3d-t_2g band in BaTiO₃ increases from 400 (paraelectric cubic phase) to 330 K (ferroelectric tetragonal phase) and to be about 30 meV [F. M. Michel-Calendini, R. N. Blumenthal, J. Am. Ceram. Soc. 54 (1971) 515, 577.].

The polaronic nature in SrTiO₃ [K. A. Muller, J. Supercond. 12 (1999) 3; H. P. R. Frederikse, G. Candella, Bull. Am. Phys. Soc. 11 (1966) 108; R. P. Feynman, R. W. Hellwarth, C. K. Iddings, P. M. Platzman, Phys. Rev. 127 (1962) 1004; J. L. M. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, I. I. Mazin, Phys. Rev. Lett. 100 (2008) 226403; D. M. Eagles, P. Lalousis, J. Phys. C 17 (1984) 655.] and BaTiO₃ [S. Lenjer, O. F. Schirmer, H. Hesse, T. W. Kool, Phys. Rev. B 66 (2002) 165106; P. Gerthsen, R. Groth, K. H. Hardtl, D. Heese, H. G. Reik, Solid State Commun. 3 (1965) 165; E. V. Bursian, Y. G. Girshberg, E. N. Starov, Phys. Status Solidi B 46 (1971) 529.] has been intensively studied: it has been observed that electron doping and chemical substitutions (e.g., Ba²⁺-substitution into Sr²⁺-site) give significant influences on the polaronic nature, and that perturbations, which can cause spacially fluctuating potentials, tend to induce small polarons.

As for the perturbations resulting in the potential fluctuation, oxygen adsorption can be indeed a case in point: at paraelectric SrTiO₃ surfaces adsorbed electronegative molecules such as oxygen (O₂) can induce local distortions of TiO₆ unit cells; as a result, carrier-electrons become frequently trapped near the oxygen-adsorbed surfaces of SrTiO₃.

For simplicity, Cochran's soft mode [W. Cochran, Adv. Phys. 9 (1960) 139.] is schematically drawn; however, it should be noted that I do not deny Bersuker's order-disorder mode [I. Bersuker, Phys. Lett. 20 (1966) 589; B. Ravel et al., Ferroelectr. 206-207 (1998) 407; B. Zalar et al., Phys. Rev. Lett. 90 (2003) 037601; G. Volkel, K. A. Muller, Phys. Rev. B 76 (2007) 094105; J. Hlinka et al., Phys. Rev. Lett. 101 (2008) 167402.].

Note that oxygen molecule O₂ becomes negatively charged O₂^- at an n-type semiconductor surface (n-SS) because of charge transfer between them. Negatively charged O₂^- and positively charged  (n-SS)⁺ become stabilized under an image potential, although the electron affinity of O₂ is not high. The adsorbate/adsorptive-medium stabilization due to an image force was described in one of Gurney's paper [R. W. Gurney, Phys. Rev. 47 (1935) 479.].

Details can be shown at http://iopscience.iop.org/1347-4065/50/6R/065807/