BCS Theory of Superconductivity

The properties of Type I superconductors were modeled successfully by the efforts of John Bardeen, Leon Cooper, and Robert Schrieffer in what is commonly called the BCS theory. A key conceptual element in this theory is the pairing of electrons close to the Fermi level into Cooper pairs through interaction with the crystal lattice. This pairing results from a slight attraction between the electrons related to lattice vibrations; the coupling to the lattice is called a phonon interaction.

Pairs of electrons can behave very differently from single electrons which are fermions and must obey the Pauli exclusion principle. The pairs of electrons act more like bosons which can condense into the same energy level. The electron pairs have a slightly lower energy and leave an energy gap above them on the order of .001 eV which inhibits the kind of collision interactions which lead to ordinary resistivity. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity.
Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity.

Ideas Leading to the BCS Theory

The BCS theory of superconductivity has successfully described the measured properties of Type I superconductors. It envisions resistance-free conduction of coupled pairs of electrons called Cooper pairs. This theory is remarkable enough that it is interesting to look at the chain of ideas which led to it.

1) One of the first steps toward a theory of superconductivity was the realization that there must be a band gap separating the charge carriers from the state of normal conduction.

A band gap was implied by the very fact that the resistance is precisely zero. If charge carriers can move through a crystal lattice without interacting at all, it must be because their energies are quantized such that they do not have any available energy levels within reach of the energies of interaction with the lattice.
A band gap is suggested by specific heats of materials like vanadium. The fact that there is an exponentially increasing specific heat as the temperature approaches the critical temperature from below implies that thermal energy is being used to bridge some kind of gap in energy. As the temperature increases, there is an exponential increase in the number of particles which would have enough energy to cross the gap.

2) The critical temperature for superconductivity must be a measure of the band gap, since the material could lose superconductivity if thermal energy could get charge carriers across the gap.

3) The critical temperature was found to depend upon isotopic mass. It certainly would not if the conduction was by free electrons alone. This made it evident that the superconducting transition involved some kind of interaction with the crystal lattice.

4) Single electrons could be eliminated as the charge carriers in superconductivity since with a system of fermions you don't get energy gaps. All available levels up to the Fermi energy fill up.

5) The needed boson behavior was consistent with having coupled pairs of electrons with opposite spins. The isotope effect described above suggested that the coupling mechanism involved the crystal lattice, so this gave rise to the phonon model of coupling envisioned with Cooper pairs.

Experimental Support: BCS Theory

Electrons acting as pairs via lattice interaction? How did they come up with that idea for the BCS theory of superconductivity? The evidence for a small band gap at the Fermi level was a key piece in the puzzle. That evidence comes from the existence of a critical temperature, the existence of a critical magnetic field, and the exponential nature of the heat capacity variation in the Type I superconductors.
The evidence for interaction with the crystal lattice came first from the isotope effect on the critical temperature.
The band gap suggested a phase transition in which there was a kind of condensation, like a Bose-Einstein condensation, but electrons alone cannot condense into the same energy level (Pauli exclusion principle). Yet a drastic change in conductivity demanded a drastic change in electron behavior. Perhaps coupled pairs of electrons with antiparallel spins could act like bosons?

Measured Superconductor Bandgap

The measured bandgap in Type I superconductors is one of the pieces of experimental evidence which supports the BCS theory. The BCS theory predicts a bandgap of

where Tc is the critical temperature for the superconductor. The energy gap is related to the coherence length for the superconductor, one of the two characteristic lengths associated with superconductivity.

Referencias Biliográficas:

http://hyperphysics.phy-astr.gsu.edu/Hbase/Solids/bcs.html#c2

http://hyperphysics.phy-astr.gsu.edu/Hbase/Solids/bcs.html#c4

Phonons

What is a Phonon?

Considering the regular lattice of atoms in a uniform solid material, you would expect there to be energy associated with the vibrations of these atoms. But they are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Such propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material.
The vibrational energies of molecules, e.g., a diatomic molecule, are quantized and treated as quantum harmonic oscillators. Quantum harmonic oscillators have equally spaced energy levels with separation DE = hu. So the oscillators can accept or lose energy only in discrete units of energy hu.
The evidence on the behavior of vibrational energy in periodic solids is that the collective vibrational modes can accept energy only in discrete amounts, and these quanta of energy have been labeled "phonons". Like the photons of electromagnetic energy, they obey Bose-Einstein statistics.
Considering a solid to be a periodic array of mass points, there are constraints on both the minimum and maximum wavelength associated with a vibrational mode.

By associating a phonon energy with the modes and summing over the modes, Debye was able to find an expression for the energy as a function of temperature and derive an expression for the specific heat of the solid. In this expression, vs is the speed of sound in the solid.

Debye Specific Heat

By associating a phonon energy

with the vibrational modes of a solid, where vs is the speed of sound in the solid, Debye approached the subject of the specific heat of solids. Treating them with Einstein-Bose statistics, the total energy in the lattice vibrations is of the form

This can be expressed in terms of the phonon modes by expressing the integral in terms of the mode number n.

Here the factor 3p/2 comes from three considerations. First, there are 3 modes associated with each mode number n: one longitudinal mode and two transverse modes. Then you get a factor of 4p2 from integrating over the angular coordinates, treating the mode number n as the radius vector. Finally you constrain the integral to the quadrant in which all the components of n are positive, giving a factor of 1/8: the product of those is 3p/2.

The Debye specific heat expression is the derivative of this expression with respect to T. The integral cannot be evaluated in closed form, but numerical evaluation of the integral shows reasonably good agreement with the observed specific heats of solids for the full range of temperatures, approaching the Dulong-Petit Law at high temperatures and the characteristic T3 behavior at very low temperatures.
The specific heat expression which arises from Debye theory can be obtained by taking the derivative of the energy expression above.

This expression may be evaluated numerically for a given temperature by computer routines.

Since the Debye specific heat expression can be evaluated as a function of temperature and gives a theoretical curve which has a specific form as a funtion of T/TD, the specific heats of different substances should overlap if plotted as a function of this ratio. At left below, the specific heats of four substances are plotted as a function of temperature and they look very different. But if they are scaled to T/TD, they look very similar and are very close to the Debye theory.

High and Low Temperature LimitsDebye Specific Heat

The energy expression from the Debye theory of specific heat is of the form

Even though this integral cannot be evaluated in closed form, the low and high temperature limits can be assessed.
For the high temperature case where T>>TD, the value of x is very small throughtout the range of the integral. This justifies using the approximation to the exponential from the exponential series, ex = 1 + x. This reduces the energy expression to

which is the Dulong-Petit result from classical thermodynamics. For low temperatures where T<<>

Then energy is then

This T3 dependence of the specific heat at very low temperatures agrees with experiment for nonmetals. For metals the electron specific heat becomes significant at low temperatures and is combined with the above lattice specific heat in the Einstein-Debye specific heat.

Referencias Bibliográficas:

http://hyperphysics.phy-astr.gsu.edu/Hbase/Solids/phonon.html#c2

Bragg's Law and more

When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions:

The angle of incidence = angle of scattering.
The pathlength difference is equal to an integer number of wavelengths.

The condition for maximum intensity contained in Bragg's law above allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the x-rays incident upon the crystal.

A continuación un pequeño programa para calculos de longitud:

Para mayor información seguir el siguiente enlace, es muy práctico.

Bragg Spectrometer

Much of our knowledge about crystal structure and the structure of molecules as complex as DNA in crystalline form comes from the use of x-rays in x-ray diffraction studies. A basic instrument for such study is the Bragg spectrometer.

To obtain nearly monochromatic x-rays, an x-ray tube is used to produce characteristic x-rays. In order to eliminate as much of the brehmsstrahlung continuum radiation as possible, matched filters are used in the x-ray beam to optimize the fraction of the energy which is in the K-alpha line. Such filters use elements just above and just below the metal in the x-ray target, making use of the strong "absorption edges" just above and below the K-alpha energy of the target metal.
The x-rays are collimated with apertures in a strong x-ray absorber (usually lead) and the narrow resulting x-ray beam is allowed to strike the crystal to be studied. The spectrometer arrangement couples the rotation of the crystal with the rotation of the detector so that the angle of rotation of the detector is twice that of the crystal. This satisfies the conditions of Bragg's law for diffraction of the x-rays from the crystal lattice planes.

Characteristic X-Rays

Characteristic x-rays are emitted from heavy elements when their electrons make transitions between the lower atomic energy levels. The characteristic x-rays emission which shown as two sharp peaks in the illustration at left occur when vacancies are produced in the n=1 or K-shell of the atom and electrons drop down from above to fill the gap. The x-rays produced by transitions from the n=2 to n=1 levels are called K-alpha x-rays, and those for the n=3->1 transiton are called K-beta x-rays.
Transitions to the n=2 or L-shell are designated as L x-rays (n=3->2 is L-alpha, n=4->2 is L-beta, etc. ). The continuous distribution of x-rays which forms the base for the two sharp peaks at left is called "bremsstrahlung" radiation.
X-ray production typically involves bombarding a metal target in an x-ray tube with high speed electrons which have been accelerated by tens to hundreds of kilovolts of potential. The bombarding electrons can eject electrons from the inner shells of the atoms of the metal target. Those vacancies will be quickly filled by electrons dropping down from higher levels, emitting x-rays with sharply defined frequencies associated with the difference between the atomic energy levels of the target atoms.
The frequencies of the characteristic x-rays can be predicted from the Bohr model . Moseley measured the frequencies of the characteristic x-rays from a large fraction of the elements of the periodic table and produces a plot of them which is now called a "Moseley plot".
Characteristic x-rays are used for the investigation of crystal structure by x-ray diffraction. Crystal lattice dimensions may be determined with the use of Bragg's law in a Bragg spectrometer.

Brillouin Scattering

Scattering of light from acoustic modes is called Brillouin scattering. From a strictly classical point of view, the compression of the medium will change the index of refraction and therefore lead to some reflection or scattering at any point where the index changes. From a quantum point of view, the process can be considered one of interaction of light photons with acoustic or vibrational quanta (phonons). Two examples will provide some context for this phenomenon.

Fisica Cuántica

Referencia Bibliografica:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuum. The French mathematician Augustin Louis Cauchy was the first to formulate such models in the 19th century, but research in the area continues today.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.
Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.

The concept of a continuum

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.
The validity of continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a Representative Volume Element (RVE) and 'separation of scales' based on the Hill-Mandel condition. [Sometimes, in place of RVE, the term Representative Elementary Volume (REV) is used.] This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure. When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the RVE size, one employs a Statistical Volume Element (SVE), which, in turn, leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
In fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.

Formulation of Model

Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time is labeled. A particular particle within the body in a particular configuration is characterized by a position vector, where are the coordinate vectors in some frame of reference chosen for the problem. This vector can be expressed as a function of the particle position in some reference configuration, for example the configuration at the initial time, so that. This function needs to have various properties so that the model makes physical sense. needs to be: continuous in time, so that the body changes in a way which is realistic, globally invertible at all times, so that the body cannot intersect itself, orientation-preserving, as transformations which produce mirror reflections are not possible in nature. For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.

Kinematics: deformation and motion

A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration. The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a pathline. There is continuity during deformation or motion of a continuum body in the sense that: The material points forming a closed curve at any instant will always form a closed curve at any subsequent time. The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at is considered the reference configuration. The components of the position vector of a particle, taken with respect to the reference configuration, are called the material or reference coordinates. When analyzing the deformation or motion of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

Lagrangian description
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference configuration is the configuration at . An observer standing in the referential frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, . This description is normally used in solid mechanics.

Referencias Bibliográficas:

http://en.wikipedia.org/wiki/Continuum_mechanics

Lattice Vibrations

So far we have been discussing equilibrium properties of crystal lattices. When the lattice is at equilibrium each atom is positioned exactly at its lattice site. Now suppose that an atom displaced from its equilibrium site by a small amount. Due to force acting on this atom, it will tend to return to its equilibrium position. This results in lattice vibrations. Due to interactions between atoms, various atoms move simultaneously, so we have to consider the motion of the entire lattice.

►The motion of atoms in a linear chain is coupled, giving rise topropagating waves

►The frequency of oscillation depends on the wavelength (i.e. the wave vector) of the propagating wave

►For an infinite chain, the possible frequency of oscillations is a continuous

► For a finite chain of quantum oscillator, only a discrete set of frequencies is possible

►Each propagating wave with a certain frequency and hence a certain group velocity is called a phonon phonon

►The frequency of atomic vibrations in a phonon depends on the phonon wave vector – k: This defines the dispersion relation.

►Phonon wave vectors for a 1D chain of length L are n·2π/L, where n is integer. Number of phonons is

►All phonon wave vectors lie between –π/a and π/a. Thereforethe number of phonons in a 1D chain is 2π/a / 2π/L = L/a

►Each phonon can be treated itself as a quantum oscillation. For low temperatures every atom can be approximated by an harmonic oscillator, the energy of the oscillation is

One-dimensional lattice

For simplicity we consider, first, a one-dimensional crystal lattice and assume that the forces between the atoms in this lattice are proportional to relative displacements from the equilibrium positions.

This is known as the harmonic approximation, which holds well provided that the displacements are small. One might think about the atoms in the lattice as interconnected by elastic springs.

Diatomic 1D lattice

Now we consider a one-dimensional lattice with two non-equivalent atoms in a unit cell. It appears that the diatomic lattice exhibit important features different from the monoatomic case. Shows a diatomic lattice with the unit cell composed of two atoms of masses M1 and M2 with the distance between two neighboring atoms a.

Three dimensional lattice

These considerations can be extended to the three-dimensional lattice. To avoid mathematical details we shall present only a qualitative discussion. Consider, first, the monatomic Bravais lattice, in which each unit cell has a single atom. The equation of motion of each atom can be written. The solution of this equation in three dimensions can be represented in terms of normal modes where the wave vector q specifies both the wavelength and direction of propagation. The vector A determines the amplitude as well as the direction of vibration of the atoms. Thus this vector specifies the polarization of the wave, i.e., whether the wave is longitudinal (A parallel to q) or transverse (A perpendicular to q). When we substitute into the equation of motion, we obtain three simultaneous equations involving Ax, Ay. and Az, the components of A. These equations are coupled together and are equivalent to a 3 x 3 matrix equation. The roots of this equation lead to three different dispersion relations, or three dispersion curves. All three branches pass through the origin, which means all the branches are acoustic. This is of course to be expected, since we are dealing with a monatomic Bravais lattice.

The three branches in differ in their polarization. When q lies along a direction of high symmetry - for example, the[100] or [110] directions − these waves may be classified as either pure longitudinal or pure transverse waves. In that case, two of the branches are transverse and one is longitudinal. One usually refers to these as the TA - transverse acoustic and LA − longitudinal acoustic branches, respectively. However, along non-symmetry directions the waves may not be pure longitudinal or pure transverse, but have a mixed character.

Shows the dispersion curves for Al in the [100] and [110] directions. Note that in certain high-symmetry directions, such as the [100] in Al, the two transverse branches coincide. The branches are then said to be degenerate. We turn our attention now to the non-Bravais three-dimensional lattice. Here the unit cell contains two or more atoms. If there are s atoms per cell, then on the basis of our previous experience we conclude that there are 3s dispersion curves. Of these, three branches are acoustic, and the remaining (3s −3) are optical. The mathematical justification for this assertion is as follows: We write the equation of motion for each atom in the cell, which results in s equations. Since these are vector equations, they are equivalent to 3s scalar equations, which have 3s roots. It can be shown that three of these roots always vanish at q = 0, which results in three acoustic branches. The remaining (3s −3) roots, therefore, belong to the optical branches, as stated above.

The acoustic branches may be classified, as before, by their polarizations as TA1, TA2, and LA. The optical branches can also be classified as longitudinal or transverse when q lies along a high-symmetry direction, and one speaks of LO and TO branches. As in the one-dimensional case, one can also show that, for an optical branch, the atoms in the unit cell vibrate out of phase relative to each other. As an example of a non-Bravais lattice, the dispersion curves. Since there are two atoms per unit cell in germanium, there are six branches: three acoustic and three optical. Note that the two transverse branches are degenerate along the [100] direction, as indicated earlier.

Phonons

So far we discussed a classical approach to the lattice vibrations. As we know from quantum mechanics the energy levels of the harmonic oscillator are quantized. Similarly the energy levels of lattice vibrations are quantized. The quantum of vibration is called a phonon in analogy with the photon, which is the quantum of the electromagnetic wave.

We see that there is a relationship between the amplitude of vibration and the frequency and the phonon occupation of the mode. In classical mechanics any amplitude of vibration is possible, whereas in quantum mechanics on discrete values are allowed. This is shown.

The lattice with s atoms in a unit cell is described by 3s independent oscillators. The frequencies of normal modes of these oscillators will be given by the solution of 3s linear equations as we discussed before.

Phonons can interact with other particles such as photons, neutrons and electrons. This interaction occurs such as if photon had a momentum qh. However, a phonon does not carry real physical momentum. The reason is that the center of mass of the crystal does not change it position under vibrations (except q=0). In crystals there exist selection rules for allowed transitions between quantum states. We saw that the elastic scattering of an x-ray photon by a crystal is governed by the wavevector selection rule, where G is a vector in the vector in the reciprocal lattice, k is the wavevector of the incident photon and ′ k is the wavevector of the scattered photon. This equation can be considered as condition for the conservation of the momentum of the whole system, in which the lattice acquires a momentum .

Referencias Bibliográficas:

http://physics.unl.edu/~tsymbal/tsymbal_files/Teaching/SSP-927/Section%2005_Lattice_Vibrations.pdf


http://users.physik.fu-berlin.de/~pascual/Vorlesung/SS09/slides/EPIV-09SS-SolSt_K3-Lattice%20vibrations.pdf

Debye theory

Referencias Bibliográficas:
mercury.chem.pitt.edu/~rob/chem2440/present244/ngo_an.ppt

The Bravais lattices

The Bravais lattice are the distinct lattice types which when repeated can fill the whole space. The lattice can therefore be generated by three unit vectors, a1, a2 and a3 and a set of integers k, l and m so that each lattice point, identified by a vector r, can be obtained from:

r = k a1 + l a2 + m a3

In two dimensions there are five distinct Bravais lattices, while in three dimensions there are fourteen. These fourteen lattices are further classified as shown in the table below where a1, a2 and a3 are the magnitudes of the unit vectors and a, b and g are the angles between the unit vectors.

Cubic lattices

Cubic lattices are of interest since a large number of materials have a cubic lattice. There are only three cubic Bravais lattices. All other cubic crystal structures (for instance the diamond lattice) can be formed by adding an appropriate base at each lattice point to one of those three lattices. The three cubic Bravais lattices are the simple cubic lattice, the body centered cubic lattice and the face centered cubic lattice. A summary of some properties of cubic latices is listed in the table below:

Cubic lattices have the highest degree of symmetry of any Bravais lattice. They belong to the (m3m) symmetry group which contains the following symmetry groups and operations:

Note that the (m3m) symmetry group is the highest possible symmetry group associated with a cubic crystal. A limited symmetry of the basis (the arrangment of atoms associated with each lattice point) can yield a lower overall symmetry group of the crystal.

Simple cubic lattice
The simple cubic lattice consists of the lattice points identified by the corners of closely packed cubes.

The simple cubic lattice contains 1 lattice point per unit cell. The unit cell is the cube connecting the individual lattice points. The atoms in the picture are shown as an example and to indicate the location of the lattice points. The maximum packing density occurs when the atoms have a radius which equals half of the side of the unit cell. The corresponding maximum packing density is 52 %.

Body centered cubic lattice
The body centered lattice equals the simple cubic lattice with the addition of a lattice point in the center of each cube.

The body centered cubic lattice contains 2 lattice point per unit cell. The maximum packing density occurs when the atoms have a radius which equals one quarter of the body diagonal of the unit cell. The corresponding maximum packing density is 68 %.

Face centered cubic lattice
The face centered lattice equals the simple cubic lattice with the addition of a lattice point in the center of each of the six faces of each cube.

The face centered cubic lattice contains 4 lattice point per unit cell. The maximum packing density occurs when the atoms have a radius which equals one quarter of the diagonal of one face of the unit cell. The corresponding maximum packing density is 74 %. This is the highest possible packing density of any crystal structure as calculated using the assumption that atoms can be treated as rigid spheres.

Diamond lattice

The diamond lattice consist of a face centered cubic Bravais point lattice which contains two identical atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of diamond, germanium and silicon.

The diamond lattice contains also 4 lattice point per unit cell but contains 8 atoms per unit cell. The maximum packing density occurs when the atoms have a radius which equals one eighth of the body diagonal of the unit cell. The corresponding maximum packing density is 34 %.

Zincblende lattice

The zincblende lattice consist of a face centered cubic Bravais point lattice which contains two different atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of zincblende (ZnS), gallium arsenide, indium phosphide, cubic silicon carbide and cubic gallium nitride.

Cristal Momentum

The Bloch Theorem

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:

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.

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.

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.

3-D Crystal Lattice Images
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.

Referencias Bibliograficas:

http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_2_3.html

http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_1_4.html

http://www.ece.rutgers.edu/~maparker/classes/581-chapters/Ch06-Structure-Phonons/Ch06Sec10XalMomentum.pdf