Debye versus Einstein




Debye vs. Einstein. Predicted heat capacity as a function of temperature.

So how closely do the Debye and Einstein models correspond to experiment?

Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not.

How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a functional form for the heat capacity. They are models, and no model is without a scale. A scale relates the model to its real-world counterpart.

One can see that the scale of the Einstein model, which is given by

is ε / k. And the scale of the Debye model is TD, the Debye temperature. Both are usually found by fitting the models to the experimental data. (The Debye temperature can theoretically be calculated from the speed of sound and crystal dimensions.)

Because the two methods approach the problem from different directions and different geometries, Einstein and Debye scales are not the same, that is to say

which means that plotting them on the same set of axes makes no sense. They are two models of the same thing, but of different scales. If one defines Einstein temperature as

then one can say

and, to relate the two, we must seek the ratio

The Einstein solid is composed of single-frequency quantum harmonic oscillators,

.

That frequency, if it indeed existed, would be related to the speed of sound in the solid. If one imagines the propagation of sound as a sequence of atoms hitting one another, then it becomes obvious that the frequency of oscillation must correspond to the minimum wavelength sustainable by the atomic lattice, λmin.

which makes the Einstein temperature

and the sought ratio is therefore

Now both models can be plotted on the same graph. Note that this ratio is the cube root of the ratio of the volume of one octant of a 3-dimensional sphere to the volume of the cube that contains it, which is just the correction factor used by Debye when approximating the energy integral above

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Debye_Theory

Phonon dispersion

Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms.

This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system is

where is the mass of each atom, and and are the position and momentum operators for the th atom. A discussion of similar Hamiltonians may be found in the article on the quantum harmonic oscillator.

We introduce a set of "normal coordinates" , defined as the discrete Fourier transforms of the 's and "conjugate momenta" defined as the Fourier transforms of the 's:

The quantity will turn out to be the wave number of the phonon, i.e. divided by the wavelength. It takes on quantized values, because the number of atoms is finite.

The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

The upper bound to comes from the minimum wavelength imposed by the lattice spacing , as discussed above.

By inverting the discrete Fourier transforms to express the 's in terms of the 's and the 's in terms of the 's, and using the canonical commutation relations between the 's and 's, we can show that

In other words, the normal coordinates and their conjugate momenta obey the same commutation relations as position and momentum operators! Writing the Hamiltonian in terms of these quantities,

where

Notice that the couplings between the position variables have been transformed away; if the 's and 's were Hermitian (which they are not), the transformed Hamiltonian would describe uncoupled harmonic oscillators.

This may be generalized to a three-dimensional lattice. The wave number k is replaced by a three-dimensional wave vector k. Furthermore, each k is now associated with three normal coordinates.

The new indices s = 1, 2, 3 label the polarization[disambiguation needed] of the phonons. In the one dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Phonon

Fermi Level

The number of energy levels in the conduction band is given by[2] Nc:

where me is the effective mass of an electron.

The number of energy levels in the valence band is given by Nv

where mh is the effective mass of a hole.

For an intrinsic semi-conductor, the number of conduction electrons must equal the number of conduction holes. Such that

nc = nv

where nc is the number electrons in the conduction band and nv is the number of conduction holes in the valence band, given by:

for an n-type extrinsic semiconductor, the number of conduction electrons nc must equal the number of conduction holes plus the number of ionized donor atoms, nd.

nc = nv + nd

where:

Figure 8: Carrier density for doped semiconductor

Figure 8 shows this relationship against temperature. At the operating temperature, the electrons available for conduction is relatively constant, as most donor electrons exist in the conduction band. For high temperatures electrons from the valance band begin to populate the conduction band, significantly increasing the carrier density. The electrons in the conduction band are now dominated by electrons from the intrinsic semiconductor and it is said to be intrinsic. For very low temperatures, the donor electrons no longer populate the conduction band and the semi-conductor is said to freeze out.

Conduction

electron mobility: When an electric field is applied to a semiconductor, the electrons experience a force and are accelerated in the opposite direction of the electric field. This acceleration is inhibited by what we term 'collisions' [1].

When a collision occurs, the velocity of the electron drops to zero and it accelerates again. The average time between collisions is given by τc.

The effect is a constant drift velocity for an n-type semiconductor Vn given by:

where μ is the mobility. Its derivation is complicated as the velocities have a Maxwellian distribution.

The current density Jn is given by:

where n is the number of electrons per unit volume A and q their charge. One may also express the current density in terms of the conductivity σ:

Jn = σξ

σ = qnμ

where σ is the conductivity in siemens per meter and ξ the electric field.

Conduction is further complicated by additional diffusion of carriers. The voltage drop across the semiconductor is gradual and therefore sets up an electron density gradient. Electrons which exist at higher densities experience a force towards less dense region. Thus a Diffusion co-efficient Dn is defined along with electron density gradient .

where

The same equations also apply for a p-type semicondcutor with a few minor differences.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikibooks.org/wiki/Semiconductors/What_is_a_Semiconductor

Semiconductors/What is a Semiconductor

Semiconductors are materials that have properties in between normal conductors (materials that allow electric current to pass, e.g. aluminium) and insulators (which block electric current, e.g. sulphur).

Semiconductors fall into two broad categories. First, there are intrinsic semiconductors. These are composed of only one kind of material. Silicon and germanium are two examples. They are also called "undoped semiconductors" or "i-type semiconductors".

Extrinsic semiconductors are made of intrinsic semiconductors that have had other substances added to them to alter their properties.

Intrinsic Semiconductors

Every atom consists of a nucleus surrounded by a number of electrons. Only the electrons are involved in electronic processes. The electrons can exist only in certain electron shells around the atom. There are many shells in each atom.

It requires energy to get an electron from a shell close to the nucleus to one further away, and if an atom's electrons are in a position which is not the position with least energy (i.e. they are in a higher (further from nucleus) shell and there is space in a lower shell), energy is given up so the electrons "fall" into the inner shells. Thereby, the shells closest to the nucleus are filled first, and then the next closest and so on. It requires more energy for an electron in a shell that is close to its nucleus to fill an outer shell than it is for an electron on an outer shell to fill an inner shell, so the inner shell is filled first.

We will consider our material to be arranged in a lattice, which is a regular arrangement, like a crystal. This helps to describe and explain the principles. In the lattice, each electron can "see" every atom in the entire lattice, and therefore is not just affected by the presence of electrons in its own atom, but by all the other atoms in the material. The huge number of atoms (usually greater than one thousand billion billion in a cube 1mm on a side) means that the number of electrons in each shell of each atom is not important - the shells "merge" into bands. All that matters is that if that band is filled, partially filled or empty. The size of bands and the gaps between them is determined by the nature of the material.

Figure 2: Electronic band structure of an insulator or semiconductor.

Figure 3: Comparison of the band gaps for a metal, a semiconductor and an insulator.

In a lattice, there will be a set of filled bands, with a full complement of electrons and unfilled bands which have no electrons (because they are in the lower-energy filled bands). The highest energy band with electrons in it is called the valence band, from the chemists' term "valence electrons" which are the electrons on the outermost shell of the atom which are responsible for chemical reactions. The conduction band is the band above the valence band. Electrons in the conduction band are free to move about in the lattice, and can therefore conduct current. The energy gap between the valence and conduction band is called the band gap.

Every material has associated with it a Fermi energy. Imagine the bands "filling up" from the bottom up, like water poured into a container. The continuous nature of the filling arises from the fact that there are such are large number of electrons they are essentially infinite in number. This behavior does not happen in a single atom, as the small number of electrons means that the amount of energy is heavily quantized. The Fermi energy is the level of the top of the "sea" that is formed.

This is defined at absolute zero, when there is no thermal energy to allow the electronics to form "ripples" on the the sea.

In insulators, the Fermi level lies between the valence and conduction bands, in one of the "forbidden zones" where electrons cannot exist. Thus all electrons in the lattice are in the valence band or a band under that. To get to the conduction band, the electron has to gain enough energy to jump the band gap. Once this is done, it can conduct. However ,the band gap for insulators is large (over 3 eV) so very few electrons can jump the gap. Therefore, current does not flow easily in insulators.

In metals, the conduction band and the valence band overlap or the valence band is only partially full, both with the Fermi energy somewhere inside. This means that the metal always has electrons that can move freely and so can always carry current.

In semiconductors, the Fermi energy is between the valence and conduction band, but the band gap is smaller, allowing electrons to jump the gap fairly easily, given the energy to do it. At absolute zero, semiconductors are perfect insulators, but at room temperature, there is enough thermal energy to allow occasional electron jumps, given the semiconductor limited conductivity, even though, by rights, it should be an insulator.

If there are no electrons in the conduction band of a semi-conductor it won't conduct. To move electrons out of the valence band and into the conduction band, one needs to give them energy. This may be through heat, incident light or high electric field. As most semiconductors operate at non-zero temperature, there are generally some electrons in the conduction band. This also means that if the semi-conductor
gets too hot (125°C for silicon), excess electrons will exist in the conduction band, hence the semi-conductor will act more like a conductor.

Because intrinsic semiconductors contain no "extra" electrons from impurities like extrinsic semiconductors do, every time an electron jumps the band gap, it leaves a hole behind. This hole represents a positive charge as it is the lack of an electron. Intrinsic semiconductors have exactly equal numbers of holes and electrons, so, where n is the number of electrons and p is the number of holes,

Direct and Indirect Semiconductors

Figure 4: Band-gap for silicon

The total energy of an electron is given by its momentum and its potential energy. To move an electron from the conduction band to the valence band, it may need to undergo a change in potential energy and a change in momentum. There are two basic material types, in-direct and direct band gap materials. In an indirect band gap material, such as silicon, shown in figure 4, to move into the valence band, the electron must undergo a change in momentum and energy[1]. The chance of this event is small. Typically this process is achieved in several steps. The electron will first move to a trap site in the forbidden band before moving into the valence band.

A change in potential energy will result in the release of a photon, while a change in momentum will produce a phonon (a phonon being a mechanical vibration which heats the crystal lattice).

Figure 5: Band-gap for GaAs, a direct semiconductor

In a direct band-gap material such as GaAs, only a change in energy is required, as seen in figure 5. As such GaAs is very efficient at producing light, although in the infrared spectrum.


Extrinsic Semiconductors

One may also dope the semiconductor material. Semi-conductor materials are doped with impurities chosen to give the material special characteristics. One may want to add extra electrons or remove electrons.

Figure 6: N-type silicon, doped with phosphorous

Figure 7: P-type silicon, doped with boron

Doping atoms are chosen from elements in group III or V of the periodic table[1] which are similar in size to silicon atoms. Thus individual intrinsic semiconductor atoms may be replaced with dopant atoms to form an extrinsic semi-conductor.

The binding energy of the outer electron added by the impurity is weak. This is represented by placing the excess electrons just below the conduction band. Thus very little energy is required to move these electrons into the conduction band. Thus an extrinsic semi-conductor operating at room temperature will have most of these "extra" electrons existing in the conduction band. Thus at normal operating temperature,

Where nc is the number of conduction electrons and Nd is the number of dopant atoms.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikibooks.org/wiki/Semiconductors/What_is_a_Semiconductor