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
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 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 (18971962)
“In the woods in a winter afternoon one will see as readily the origin of the stained glass window, with which Gothic cathedrals are adorned, in the colors of the western sky seen through the bare and crossing branches of the forest.”
—Ralph Waldo Emerson (18031882)
“It was a pretty game, played on the smooth surface of the pond, a man against a loon.”
—Henry David Thoreau (18171862)
“When some one remarked that, with the addition of a chaplain, it would have been a perfect Cromwellian troop, he observed that he would have been glad to add a chaplain to the list, if he could have found one who could fill that office worthily. It is easy enough to find one for the United States Army. I believe that he had prayers in his camp morning and evening, nevertheless.”
—Henry David Thoreau (18171862)
“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 (18171862)
“Mankind always sets itself only such tasks as it can solve; since, looking at the matter more closely, we will always find that the task itself arises only when the material conditions necessary for its solution already exist or are at least in the process of formation.”
—Karl Marx (18181883)