Tamm surface states

However, even in perfect ID structures, the mixing of Frenkel and CT excitons destroys this simple picture. Below, following (43), we show that this mixing is responsible for the appearance of new excitonic states, which are localized at the ends of a one-dimensional crystal chain and which are analogous to Tamm surface states of electrons. Their energy can be blue- or red-shifted in comparison with the bulk states. In the case of red-shift, these states can determine the fluorescence spectrum of a molecular chain. They can also play an important role in quantum confinement of the states in the molecular chain. For the description... 

Tamm Surface States, Surface Molecule Limit. 

The appearance of the Tamm surface state on the NbC(lOO) surface can be explained by... 

As described before, a Tamm surface state is observed in ARPE spectra for the NbC(lOO) surface. However, the Tamm surface state has not always been observed on compound crystal surfaces because the degree of the modification of the electrostatic potential at the surface is dependent on several factors, such as the degree of charge transfer and surface relaxation. As for the transition metal nitride and carbide (100) surfaces, similar surface states have been found only for TiN(lOO) (36), ZrC(lOO) (36,37), and VC(IOO) (26) in addition to NbC(lOO) (28-31). 

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. 

No "Jilt has so far been assumed that the semiconductor-electrolyte interphase does not contain either ions adsorbed specifically from the electrolyte or electrons corresponding to an additional system of electron levels. These surface states of electrons are formed either through adsorption (the Shockley levels) or through defects in the crystal lattice of the semiconductor (the Tamm levels). In this case—analogously as for specific adsorption on metal electrodes—three capacitors in series cannot be used to characterize the semiconductor-electrolyte interphase system and Eq. (4.5.6) must include a term describing the potential difference for surface states. 

Little theoretical work has been done on the electronic structure of a solid with a free surface. The main contributions are those of Tamm (4), Shockley (5), Goodwin 6), Artmann (7), and Kouteck (5), and the main conclusion is that, in certain circumstances, surface states may exist in the gaps between the normal bands of crystal states. In this section we investigate the problem in the simplest way. The solid is represented by a straight chain of similar atoms, and its two ends represent the free surfaces. This one-dimensional model exhibits the essential features of the problem, and the results are easily generalized to three dimensions. 

The existence of surface states is a consequence of the atomic structure of solids. In an infinite and uniform periodic potential, Bloch functions exist, which explains the band structures of different solids (Kittel, 1986). On solid surfaces, surface states exist at energy levels in the gap of the energy band (Tamm, 1932 Shockley, 1939 Heine, 1963). 

The concept of surface states was proposed by Tamm (1932) using a one-dimensional analytic model. We start with reviewing the proof of the Bloch theorem for a one-dimensional periodic potential U x) with periodicity a (Kittel, 1986) ... 

In the original paper of Tamm (1932), the concept of surface state is demonstrated with a Kronig-Penney potential (Kittel, 1986) with a boundary, as shown in Fig. 4.5. By solving the Schrodinger equation, exphcit expressions for the surface states and their energy levels can be obtained. In... 

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.

After the first theoretical work of Tamm (1932), a series of theoretical papers on surface states were published (for example, Shockley, 1939 Goodwin, 1939 Heine, 1963). However, there has been no experimental evidence of the surface states for more than three decades. In 1966, Swanson and Grouser (1966, 1967) found a substantial deviation of the observed fie Id-emission spectroscopy on W(IOO) and Mo(lOO) from the theoretical prediction based on the Sommerfeld theory of metals. This experimental discovery has motivated a large amount of theoretical and subsequent experimental work in an attempt to explain its nature. After a few years, it became clear that the observed deviation from free-electron behavior of the W and Mo surfaces is an unambiguous exhibition of the surface states, which were predicted some three decades earlier. 

TAMM LEVELS. Surface states the extra electron energy levels found at ctystal surfaces,... 

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. 

This will be illustrated here in the derivation of ( 7). As a result of the analysis we will discover the conditions where surface states appear. For the case to be considered here they have first been discussed by Tamm and to the chemist it is the condition that the chemisorption state can be analyz in terms of surface molecule formation. The roots of 00 are given by (g follows from Eq.(2.160)) ... 

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. 

Tamm [3] was first to show that the special states of electrons exist near crystal surface (so-called surface states), which have the discrete energy spectrum and their wave functions decay exponentially on both sides of the surface. Similarly, the vibrations of crystal surface atoms can be considered, which also decay on both sides of the surface. In the long wave limit, one of such surface phonon branches transforms into well-known surface Rayleigh waves, while the others yield special optic branches [4, 5]. 

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. 

In the adiabatic approximation the states of the electronic subsystem are calculated at fixed positions of nuclei which are considered as parameters. That way the arrangement of nuclei is reflected in the Schrodinger equation for electronic states. Near the surface, the electronic states should thus be modified compared with those in the bulk crystal. To find them, one has to take into account the surface crystal structure in the equation for the electronic motion, which is in general a formidable problem. As in the case of bulk electronic states, a one-dimensional (ID) model is a useful first approach. Tamm (1932) was the first to demonstrate that the presence of a crystal surface itself leads to the existence of electronic surface states which are inaccessible to electrons in infinite crystals. 

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). 

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