In the most simplified version of the free electron gas, the true three-dimensional potential was ignored and approximated with a constant potential conveniently put at 0 eV.
The true potential, however, e.g. for a Na crystal including some energy states, is periodic and looks more like this:
Semiconducting properties will not emerge without some consideration of the periodic potential - we therefore have to solve the Schrödinger equation for a suitable periodic potential. There are several (for real potentials always numerical) ways to do this, but as stated before, it can be shown that all solutions must have certain general properties. These properties can be used to make calculations easier and to obtain a general understanding of the the effects of a periodic potential on the behavior of electron waves.
The starting point is a potential V(r) determined by the crystal lattice that has the periodicity of the lattice, i.e.
V(r) = V(r + T)
With T = any translation vector of the lattice under consideration.
We then will obtain some wavefunctions yk(r) which are solutions of the Schrödinger equation for V(r). As before, we use a quantum number "k" (three numbers, actually) as an index to distinguish the various solutions.
The Bloch theorem in essence formulates a condition that all solutions yk(r), for any periodic potential V(r) whatsoever have to meet. In one version it ascertains
yk(r) = uk(r) · exp (i · k · r)
With k = any allowed wave vector for the electron that is obtained for a constant potential, and uk(r) = arbitrary functions (distinguished by the index k that marks the particular solution we are after), but always with the periodicity of the lattice, i.e.
uk(r + T) = uk(r)
Any wavefunction meeting this requirement we will henceforth call a Bloch wave.
The Bloch theorem is quite remarkable, because, as said before, it imposes very special conditions on any solution of the Schrödinger equation, no matter what the form of the periodic potential might be.
We notice that exactly as in the case of the constant potential, the wave vector k has a twofold role: It is still a wave vector in the plane wave part of the solution, but also an index to yk(r) and uk(r) because it contains all the quantum numbers, which ennumerate the individual solutions.
Blochs theorem is a proven theorem with perfectly general validity. We will first give some ideas about the prove of this theorem, and then discuss what it means for real crystals. As always with hindsight, Blochs theorem can be proved in many ways; the links give some examples.
Here we only look ageneral outlines of how to prove the theorem:
*It follows rather directly from applying group theory to crystals. In this case one looks at symmetry properties that are invariant under translation.
*It can easily be proved by working with operator algebra in the context of formal quantum theory mathematics.
*It can be directly proved in simple ways - but then only for special cases or with not quite kosher "tricks".
*It can be proved (and used for further calculations), by expanding V(r) and y(r) into a Fourier series and then rewriting the Schrödinger equation. This is a particularly useful way because it can also be used for obtaining specific results for the periodic potential. This proof is demonstrated in detail in the link, or in the book of Ibach and Lüth.
Blochs theorem can also be rewritten in a somewhat different form, giving us a second version:
*It follows rather directly from applying group theory to crystals. In this case one looks at symmetry properties that are invariant under translation.
*It can easily be proved by working with operator algebra in the context of formal quantum theory mathematics.
*It can be directly proved in simple ways - but then only for special cases or with not quite kosher "tricks".
*It can be proved (and used for further calculations), by expanding V(r) and y(r) into a Fourier series and then rewriting the Schrödinger equation. This is a particularly useful way because it can also be used for obtaining specific results for the periodic potential. This proof is demonstrated in detail in the link, or in the book of Ibach and Lüth.
Blochs theorem can also be rewritten in a somewhat different form, giving us a second version:
yk(r + T) = yk(r) · exp(ikT)
This means that any function yk(r) that is a solution to the Schrödinger equation of the problem, differs only by a phase factor exp(ikT) between equivalent positions in the lattice.
This implies immediately that the probability of finding an electron is the same at any equivalent position in the lattice since, exactly as we expected, because
[yk(r + T)]2 = [yk(r)]2·[exp(ikT)]2= [yk(r)]2
Since [exp(ikT)]2 = 1 for all k and T.
That this second version of Blochs theorem is equivalent to the first one may be seen as follows.
Implications of the Bloch Theorem
One way of looking at the Bloch theorem is to interprete the periodic function uk(r) as a kind of correction factor that is used to generate solutions for periodic potentials from the simple solutions for constant potentials.
We then have good reasons to assume that uk(r) for k vectors not close to a Brillouin zone will only be a minor correction, i.e. uk(r) should be close to 1.
But in any case, the quantity k, while still being the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = k, or of its wavelength l = 2p/k, since:
The momentum of the electron moving in a periodic potential is no longer constant (as we will see shortly); for the standing waves resulting from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while k is not.
One way of looking at the Bloch theorem is to interprete the periodic function uk(r) as a kind of correction factor that is used to generate solutions for periodic potentials from the simple solutions for constant potentials.
We then have good reasons to assume that uk(r) for k vectors not close to a Brillouin zone will only be a minor correction, i.e. uk(r) should be close to 1.
But in any case, the quantity k, while still being the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = k, or of its wavelength l = 2p/k, since:
The momentum of the electron moving in a periodic potential is no longer constant (as we will see shortly); for the standing waves resulting from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while k is not.
There is no unique wavelength to a plane wave modulated with some arbitrary (if periodic) function. Its Fourier decomposition can have any spectra of wavelengths, so which one is the one to associate with k?
To make this clear, sometimes the vector k for Bloch waves is called the "quasi wave vector".
Instead of associating k with the momentum of the electron, we may identify the quantity k, which is obviously still a constant, with the so-called crystal momentum P, something like the combined momentum of crystal and electron.
Whatever its name, k is a constant of motion related to the particular wave yk(r) with the index k. Only if V = 0, i.e. there is no periodic potential, is the electron momentum equal to the crystal momentum; i.e. the part of the crystal is zero.
To make this clear, sometimes the vector k for Bloch waves is called the "quasi wave vector".
Instead of associating k with the momentum of the electron, we may identify the quantity k, which is obviously still a constant, with the so-called crystal momentum P, something like the combined momentum of crystal and electron.
Whatever its name, k is a constant of motion related to the particular wave yk(r) with the index k. Only if V = 0, i.e. there is no periodic potential, is the electron momentum equal to the crystal momentum; i.e. the part of the crystal is zero.
The crystal momentum P, while not a "true" momentum which should be expressible as the product of a distinct mass and a velocity, still has many properties of momentums, in particular it is conserved during all kinds of processes.
This is a major feature for the understanding of semiconductors, as we will see soon enough!
One more difference to the constant potential case is crucial: If we know the wavefunction for one particular k-value, we also know the wavefunctions for infinitely may other k-values, too.
This follows from yet another formulation of Bloch's theorem:
If yk(r) = uk(r) · exp(ikr) is a particular Bloch wave solving the Schrödinger equation of the problem, then the following function is also a solution.
This is a major feature for the understanding of semiconductors, as we will see soon enough!
One more difference to the constant potential case is crucial: If we know the wavefunction for one particular k-value, we also know the wavefunctions for infinitely may other k-values, too.
This follows from yet another formulation of Bloch's theorem:
If yk(r) = uk(r) · exp(ikr) is a particular Bloch wave solving the Schrödinger equation of the problem, then the following function is also a solution.
3-D Crystal Lattice Images
Simple lattices and their unit cells
Simple lattices and their unit cells
Simple Cubic (SC) - There is one host atom ("lattice point") at each corner of a cubic unit cell. The unit cell is described by three edge lengths a = b = c = 2r (r is the host atom radius), and the angles between the edges, alpha = beta = gamma = 90 degrees. There is one atom wholly inside the cube (Z = 1). Unit cells in which there are host atoms (or lattice points) only at the eight corners are called primitive.
Body Centered Cubic (BCC) - There is one host atom at each corner of the cubic unit cell and one atom in the cell center. Each atom touches eight other host atoms along the body diagonal of the cube (a = 2.3094r, Z = 2).
FCC Primitive - It is also possible to choose a primitive unit cell to describe the FCC lattice. The cell is a rhombohedron, with a = b = c = 2r, and alpha = beta = gamma = 60 degrees. [A cube is a rhombohedron with alpha = beta = gamma = 90 degrees!]
Simple Hexagonal (SH) - Spheres of equal size are most densely packed (with the least amount of empty space) in a plane when each sphere touches six other spheres arranged in the form of a regular hexagon. When these hexagonally closest packed planes (the plane through the centers of all spheres) are stacked directly on top of one another, a simple hexagonal array results; this is not, however, a three-dimensional closest packed arrangement. The unit cell, outlined in black, is composed of one atom at each corner of a primitive unit cell (Z = 1), the edges of which are: a = b = c = 2r, where cell edges a and b lie in the hexagonal plane with angle a-b = gamma = 120 degrees, and edge c is the vertical stacking distance.
Closest Packing
Hexagonal Closest Packing (HCP) - To form a three-dimensional closest packed structure, the hexagonal closest packed planes must be stacked such that atoms in successive planes nestle in the triangular "grooves" of the preceeding plane. Note that there are six of these "grooves" surrounding each atom in the hexagonal plane, but only three of them can be covered by atoms in the adjacent plane. The first plane is labeled "A" and the second plane is labeled "B", and the perpendicular interplanar spacing between plane A and plane B is 1.633r (compared to 2.000r for simple hexagonal). If the third plane is again in the "A" orientation and succeeding planes are stacked in the repeating pattern ABABA... = (AB), the resulting closest packed structure is HCP.
HCP Coordination - Each host atom in an HCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal array (B layer), and six (three in the A layer above and three in the A layer below) form a trigonal prism around the central atom.
Cubic Closest Packing (CCP) - If the atoms in the third layer lie over the three grooves in the A layer which were not covered by the atoms in the B layer, then the third layer is different from either A or B and is labeled "C". If a fourth layer then repeats the A layer orientation, and succeeding layers repeat the pattern ABCABCA... = (ABC), the resulting closest packed structure is CCP = FCC. Again, the perpendicular spacing between any two successive layers is 1.633r.
CCP Coordination - Each host atom in a CCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal (B) plane, and six (three in the C layer above and three in the A layer below) form a trigonal anti-prism (also known as a distorted octahedron) around the central atom.
Rhombohedral (R) lattice - If, in the (ABC) layered lattice, the interplanar spacing is not the closest packed value (1.633r), then the primitive (Z = 1) unit cell is a rhombohedron with a = b = c <> 2r and alpha = beta = gamma <> 60 degrees. The non-primitive hexagonal unit cell (Z = 3).may also be chosen.
2- & 3-layer repeats - There is only one way to produce a repeat pattern (crystal lattice) in two layers of hexagonally closest packed planes: (AB) = HCP. Likewise, there is only one way to produce a repeat pattern in three layers of hexagonally closest packed planes: (ABC) = CCP.
4-layer repeats - However, there are two ways to produce a closest packed lattice in four layers: (ABAC) and (ABCB). By extension, there are increasing numbers of ways to produce closest packed lattices in five layers, six layers, etc., up to and including non-repeating random stacking. Thus, there are many closest (and pseudo-closest) packings in natural and artificial materials.
Holes ("Interstices") in Closest Packed Arrays
Tetrahedral Hole - Consider any two successive planes in a closest packed lattice. One atom in the A layer nestles in the triangular groove formed by three adjacent atoms in the B layer, and the four atoms touch along the edges (of length 2r) of a regular tetrahedron; the center of the tetrahedron is a cavity called the Tetrahedral (or Td) hole; a guest sphere will just fill this cavity (and touch the four host spheres) if its radius is 0.2247r.
Octahedral Hole - Adjacent to the Td hole, three atoms in the B layer touch three atoms in the A layer such that a trigonal antiprismatic polyhedron (a regular octahedron) is formed; the center of the octahedron is a cavity called the Octahedral (or Oh) hole. A guest sphere will just fill this cavity (and touch the six host spheres) if its radius is 0.4142r. It can be shown that there are twice as many Td as Oh holes in any closest packed bilayer.
Simple Crystal Structures
CsCl Structure - Each ion resides on a separate, interpenetrating SC lattice such that the cation is in the center of the anion unit cell and visa versa. The two lattices have the same unit cell dimension.
NaCl Structure - Each ion resides on a separate, interpenetrating FCC lattice. The two lattices have the same unit cell dimension.
Halite Structure - The sodium chloride structure may also be viewed as a CCP lattice of anions (Z = 4), with smaller cations occupying all Oh cavities (Z = 4).
Fluorite Structure - The structure of the mineral fluorite (calcium fluoride) may be viewed as a CCP lattice of cations (Z = 4), with the smaller anion occupying all of the Td holes (Z = 8). The Td cavities reside on a SC lattice which is half the dimension of the CCP lattice.
Zinc Blende Structure - The structure of cubic ZnS (mineral name "zinc blende") may be viewed as a CCP lattice of anions (Z = 4), with the smaller cations occupying every other Td hole (Z = 4). [Note: the other ZnS mineral, wurtzite, can be described as a HCP lattice of anions with cations in every other Td hole.]
Zinc Blende lattices - The lattice of cations in zinc blende is a FCC lattice of the same dimension as the anion lattice, so the structure can be described as interpenetrating FCC lattices of the same unit cell dimension. Note that the only difference between the halite and zinc blende structures is a simple shift in relative position of the two FCC lattices.
Orlaning Colmenares
C.I.V.- 18.991.089
Electronica del Estado Sólido
Referencias Bibliograficas:
Visitar mi BLOG:
Connect to the next generation of MSN Messenger Get it now!
No hay comentarios:
Publicar un comentario