Tamm state

S. Davison and G. Levin, Surface (Tamm) States, Mir Publ., Moscow, 1973... 

Fig. 4.5. Surface states. By solving the Schrodinger equation for a cut-off Kronig-Penney potential, it is found that in the energy gaps of the corresponding Kronig-Penney solid, there are surface states that decay exponentially into the vacuum and into the solid (Tamm, 1932). The explicit wavefiinction of a Tamm state with P = 15 and a = 3Aat = 5eV below the vacuum level is shown. The shaded areas represent allowed energy bands in the bulk.

Surface structure determination 325—331 Suspension springs 244—248 Takayanagi model See DAS model Tamm states... 

Tafel equation, 1054,1066,1106,1115,1133, 1249,1404,1440, 1456,1507,1528 applications, 1508 and distribution of electronic states, importance, 1466 importance, 1508 in quantum calculations, 1495 in semiconductors, 1085 tunneling, 1495 Tafel, Julius, 1106 Tafel lines, oxygen reduction, 1207 Tamm states, 1082 Tarasevich, 1495 Taylor, electrodeposition, 1303 Temkin isotherm, 927, 938, 1195... 

As mentioned earlier, the existence of surface shifted core levels has been questioned.6 Calculated results for TiC(lOO) using the full potential linearized augmented plane wave method (FLAPW) predicted6 no surface core level shift in the C Is level but a surface shift of about +0.05 eV for the Tis levels. The absence of a shift in the C Is level was attributed to a similar electrostatic potential for the surface and bulk atoms in TiC. The same result was predicted for TiN because its ionicity is close to that of TiC. This cast doubts on earlier interpretations of the surface states observed on the (100) surface of TiN and ZrN which were thought to be Tamm states (see references given in Reference 4), i.e. states pulled out of the bulk band by a shift in the surface layer potential. High resolution core level studies could possibly resolve this issue, since the presence of surface shifted C Is and N Is levels could imply an overall electrostatic shift in the surface potential, as suggested for the formation of the surface states. 

The elementary surface excited states of electrons in crystals are called surface excitons. Their existence is due solely to the presence of crystal boundaries. Surface excitons, in this sense, are quite analogous to Rayleigh surface waves in elasticity theory and to Tamm states of electrons in a bounded crystal. Increasing interest in surface excitons is provided by the new methods for the experimental investigation of excited states of the surfaces of metals, semiconductors and dielectrics, of thin films on substrates and other laminated media, and by the extensive potentialities of surface physics in scientific instrument making and technology. 

However, if Re re 0, we obtain a new type of solutions. We will see below that for these solutions the function un is localized near the ends of the chain. Below, such type of states will be called edge states. They are similar to Tamm states of electrons in 3D crystals. It is important to note that as with Tamm states, the edge states arise even in ideal chains which have neither diagonal nor nondiagonal disorder. 

Among surface states, there are some that originate simply from the sudden discontinuity in the ciystal lattice these are intrinsic surface states. They are sorted, depending on their source, into two categories Tamm states, which are caused by lattice deformation, and Schockley states, caused by the unsaturated bonds on the surface. There also appears on the real surfaces extrinsic surface states due to the presence of foreign species on the surface of the solid, namely adsorbed atoms or molecules originating from a gaseous phase. 

Now it is well known that the surfaces of crystals may contain electron traps, hole traps, and/or recombination centers. Clearly the electronic processes occurring at these surface sites or surface states can release sufficient energy to produce reactions at the crystal surface. Some states ( ) can be associated with defects or impurities in the crystal lattice or one or more types of atoms chemisorbed on the crystal surface. Other surface states, the Tamm states (" ), occur in or on perfect crystals and are a consequence of the quantum mechanical nature of the electronic properties of crystals. Clearly if the surface of a crystal is being eroded by photolytic decomposition there could be ever-present Tamm states on the surface. The more important carriers, surface states, and internal states or traps which are important for photolytic decomposition are summarized in Table I. 

Tamm states Electronic surface sfafes. Sometimes one refers to Tamm states as those electronic surface sfafes which are obtained in the model of tightly bound electrons. Such states correspond to the case of a direct forbidden energy gap. 

As discussed in Section 5.2, the very origin of this behavior is the electron-electron interaction. Zo denotes the position of the image plane. The precise position of Zo is not known a priori, although it is clear that it is located quite close to the surface. Attempts have been made to derive its position from the spectroscopy of surface states, which - as we will see - exist for such a long-range potential in addition to the Shockley states discussed before and the Tamm states discussed in Section 5.3.6 the image-potential states (see also Chapter 3.2.4). 

Since surface states with free-electron-like dispersion (Shockley type) have a low occupancy per surface unit cell and a low DOS at Ey, they are in general not considered to dominate the energetics of the surface, although - as discussed in Section 5.4.3 - situations exist where they can afiect the properties of the surface. On the contrary, metallic surface states derived from weakly dispersing bands (Tamm states) may have a high DOS at Ey and thus may influence the surface phase diagram considerably. As mentioned already in the case of quasi-2D states, transition metal surfaces are interesting in this respect, the question... 

The more localized Tamm surface states show less dispersion, and the angular resolution is less important. As an example, a spectrum and the corresponding fit for the M Tamm state on Cu(lOO) is shown Figure 6.19a [108]. The state shows a narrow intrinsic linewidth of T = 7 meV even at an energy of — 1.8 eV below the Fermi energy. This is attributed to the localized character of the d states and the small overlap with the sp bands, which provide the main decay channel for inelastic decay. 

While striving to reach the best surface quality systematic, quantitative studies of the influence of defects on photoemission spectra have been somewhat neglected. As an example, we show the sensitivity of the M Tamm state of Cu(lOO) to the surface quality in Figure 6.19b [71]. The surface quality was varied by argon-ion... 

Surface states are usually classified as Shockley [178] and Tamm states [179], and we now briefly discuss these two types of surface state in turn. However, we caution in advance that while useful, the distinction is somewhat arbitrary since both types of state describe the same physical phenomenon of a wave function that is localized at the surface and decays exponentially into the bulk (also see Chapter 1). 

Tamm states These are characteristic of more tightly bound systems such as the transition metals in which the valence electrons are d states. Tamm states are spht-off states because of the reduced atomic coordination of the surface... 

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