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Learning Objectives On completion of this topic you will be able to understand:

1. Density if energy states and carrier concentration

Density of states

In statistical and condensed matter physics , the density of states (DOS) of a system describes the number of states at each energy level that are available to be occupied. A high DOS at a specific energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level. Explanation

Waves, or wave-like particles, can only exist within quantum mechanical (QM) systems if the properties of the system allow the wave to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of

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certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels. For example, the density of states for electrons in a semiconductor is shown in red in Fig. 2. For electrons at the conduction band edge, very few states are available for the electron to occupy. As the electron increases in energy, the electron density of states increases and more states become available for occupation. However, because there are no states available for electrons to occupy within the bandgap, electrons at the conduction band edge must lose at least Eg of energy in order to transition to another available mode. The density of states can be calculated for electron , photon , or phonon in QM systems. The DOS is usually represented by one of the symbols g, ρ, D, n, or N, and can be given as a function of either energy or wavevector k. To convert between energy and wavevector, the specific relation between E and k must be known. For example, the formula for electrons in free space is

and for photons in free space the formula is

where c is the speed of light in free space, is the reduced Planck's constant and m is the electron mass

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Calculation of the density of states

The density of states in a semiconductor equals the density per unit volume and energy of the number of solutions to Schrödinger's equation. We will assume that the semiconductor can be modeled as an infinite quantum well in which electrons with effective mass, m*, are free to move. The energy in the well is set to zero. The semiconductor is assumed a cube with side L. This assumption does not affect the result since the density of states per unit volume should not depend on the actual size or shape of the semiconductor. The solutions to the wave equation where V(x) = 0 are sine and cosine functions:

Where A and B are to be determined. The wavefunction must be zero at the infinite barriers of the well. At x = 0 the wavefunction must be zero so that only sine functions can be valid solutions or B must equal zero. At x = L, the wavefunction must also be zero yielding the following possible values for the wavenumber, kx.

This analysis can now be repeated in the y and z direction. Each possible solution then corresponds to a cube in k-space with size nπ/L as indicated on Figure.

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.The total number of solutions with a different value for kx, ky and kz and with a magnitude of the wavevector less than k is obtained by calculating the volume of one eighth of a sphere with

radius k and dividing it by the volume corresponding to a single solution, , yielding:

A factor of two is added to account for the two possible spins of each solution. The density per unit energy is then obtained using the chain rule:

The kinetic energy E of a particle with mass m* is related to the wavenumber, k, by:

And the density of states per unit volume and per unit energy, g(E), becomes:

The density of states is zero at the bottom of the well as well as for negative energies

The same analysis also applies to electrons in a semiconductor. The effective mass takes into account the effect of the periodic potential on the electron. The minimum energy of the

electron is the energy at the bottom of the conduction band, Ec, so that the density of states for electrons in the conduction band is given by:

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Calculation of the density of states in 3 dimensions We will here postulate that the density of electrons in k–space is constant and equals the physical length of the sample divided by 2 π and that for each dimension. The number of states between k and k + dk in 3 dimension.

Applications

The density of states appears in many areas of physics, and helps to explain a number of quantum mechanical phenomena.

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Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced. For quantum wires, the DOS for certain energies actually becomes higher than the DOS for bulk semiconductors, and for quantum dots the electrons become quantized to certain energies.

Photonic Crystals The photon density of states can be manipulated by using periodic structures with length scales on the order of the wavelength of light. Some structures can completely inhibit the propagation of light with certain wavelengths, causing the creation of a photonic bandgap. Other structures can inhibit the propagation of light in certain directions, creating photonic waveguides. These devices are known as photonic crystals .

Check your understanding

1. No of energy states in F(E) = 0? (a) 0 (b) 1 (c) 3 2. Calculate the number of states per unit energy in a 100 by 100 by 10 nm piece of silicon ( m* -1 = 1.08 m0) 100 meV above the conduction band edge. Write the result in units of eV .

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Summary

1. Density of energy states and carrier concentration in metals

Suggested Reading 1. Palanisamy P.K, ‘Engineering Physics – II’ Scitech Publications (India) Pvt. LTd., Chennai – 17. (2009).

Answers to CYU: 1. (a) 2. The density of states equals:

So that the total number of states per unit energy equals

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