Surface States - Origin of Surface States at Condensed Matter Interfaces

Origin of Surface States At Condensed Matter Interfaces

As stated by Bloch's theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves

\begin{align} \Psi_{n\textbf{k}} &=& e^{i\textbf{k}\cdot\textbf{r}}u_{n\textbf{k}}(\textbf{r}). \end{align}

Here is a function with the same periodicity as the crystal, n is the band index and k is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions . The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected.

A simplified model of the crystal potential in one dimension can be sketched as shown in figure 1 . In the crystal, the potential has the periodicity, a, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in figure 1 is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image charges and the formation of surface dipoles and it rather looks as indicated by the dashed line.

Given the potential in figure 1, it can be shown that the one-dimensional single-electron Schrödinger equation gives two qualitatively different types of solutions.

  • The first type of states (see figure 2) extends into the crystal and has Bloch character there. These type of solutions correspond to bulk states which terminate in an exponentially decaying tail reaching into the vacuum.
  • The second type of states (see figure 3) decays exponentially both into the vacuum and the bulk crystal. These type of solutions correspond to states, with wave functions localized close to the crystal surface.

The first type of solution can be obtained for both metals and semiconductors. In semiconductors though, the associated eigenenergies have to belong to one of the allowed energy bands. The second type of solution exists in forbidden energy gap of semiconductors as well as in local gaps of the projected band structure of metals. It can be shown that the energies of these states all lie within the band gap. As a consequence, in the crystal these states are characterized by an imaginary wavenumber leading to an exponential decay into the bulk.

Read more about this topic:  Surface States

Famous quotes containing the words origin of, origin, surface, states, condensed and/or matter:

    “The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.”
    —Charles Sanders Peirce (1839–1914)

    “The essence of morality is a questioning about morality; and the decisive move of human life is to use ceaselessly all light to look for the origin of the opposition between good and evil.”
    —Georges Bataille (1897–1962)

    “Voluptuaries, consumed by their senses, always begin by flinging themselves with a great display of frenzy into an abyss. But they survive, they come to the surface again. And they develop a routine of the abyss: “It’s four o’clock ... At five I have my abyss.””
    —Colette [Sidonie Gabrielle Colette] (1873–1954)

    “An ... important antidote to American democracy is American gerontocracy. The positions of eminence and authority in Congress are allotted in accordance with length of service, regardless of quality. Superficial observers have long criticized the United States for making a fetish of youth. This is unfair. Uniquely among modern organs of public and private administration, its national legislature rewards senility.”
    —John Kenneth Galbraith (b. 1908)

    “There is no doubt that the loftiest written wisdom is either rhymed or in some way musically measured,—is, in form as well as substance, poetry; and a volume which should contain the condensed wisdom of mankind need not have one rhythmless line.”
    —Henry David Thoreau (1817–1862)

    “No matter what life you lead
    the virgin is a lovely number:
    cheeks as fragile as cigarette paper,
    arms and legs made of Limoges,
    lips like Vin Du Rhône....”
    —Anne Sexton (1928–1974)