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Hie degeneracies d, of the confine« (square or parabolic well) enetgy k-.xfa depend on the particular level. The Heaviaide Rttp function &(x) is zero fbri<0 and one for* > 0; the delta function ¿(x) k zero for .v ^iO, infinity for x=0, and integrates to a unit area. The values of the constants K,, K-z- miK} aie given in Table A.3 of Appendix A.

Hie degeneracies d, of the confine« (square or parabolic well) enetgy k-.xfa depend on the particular level. The Heaviaide Rttp function &(x) is zero fbri<0 and one for* > 0; the delta function ¿(x) k zero for .v ^iO, infinity for x=0, and integrates to a unit area. The values of the constants K,, K-z- miK} aie given in Table A.3 of Appendix A.

dimensionality and of the confinement associated with a particular nanostructure have a pronounced effect on its properties. These considerations can be used to predict properties of nanostructures, and one can also identify types of nanostruc-tures from their properties.

### 9.3.6. Properties Dependent on Density of States

We have discussed the density of states D{E) of conduction electrons, and have shown that it is strongly affected by the dimensionality of a material. Phonons or quantized lattice vibrations also have a density of states Dm{E) that depends on the dimensionality, and like its electronic counterpart, it influences some properties of solids, but our principal interest is in the density of states D(E) of the electrons. In this section we mention some of the properties of solids that depend on the density of states, and we describe some experiments for measuring it.

The specific heat of a solid C is the amount of heat that must be added to it to raise its temperature by one degree Celsius (centigrade). The main contribution to this heat is the amount that excites lattice vibrations, and this depends on the phonon density of states DpH(E). At low temperatures there is also a contribution to the specific heat Cd of a conductor arising from the conduction electrons, and this depends on the electronic density of states at the Fermi level: Cei = n2D(Ef)k^T/3, where kB is the Boltzmann constant.

The susceptibility x = M/H of a magnetic material is a measure of the magnetization M or magnetic moment per unit volume that is induced in die material by die application of an applied magnetic field H. The component of the susceptibility arising from die conduction electrons, called the Pauli susceptibility, is given by the expression xei — /k|Z)(£'f), where is die unit magnetic moment called the Bohr magneton, and is hence Xa characterized by its proportionality to the

Bulk

Conductor

Quantum Well

Quantum Wire

Figure 9.15. Number of electrons N(E) (left side) and density of states 0(£) (right side) plotted against the energy for four quantum structures in the square well-Fermi gas approximations.

Number of Electrons N<e)

Bulk

Conductor

Quantum Well

Quantum Wire

Quantum Dot

Figure 9.15. Number of electrons N(E) (left side) and density of states 0(£) (right side) plotted against the energy for four quantum structures in the square well-Fermi gas approximations.

electronic density of states IXE) at the Fermi level, and its lack of dependence on the temperature.

When a good conductor such as aluminum is bombarded by fast electrons with just enough energy to remove an electron from a particular A1 inner-core energy level, the vacant level left behind constitutes a hole in the inner-core band. An electron from the conduction band of the aluminum can fall into the vacant inner-core level to occupy it, with the simultaneous emission of an X ray in the process. The intensity of the emitted X radiation is proportional to the density of states of the conduction electrons because the number of electrons with each particular energy that jumps down to fill the hole is proportional to D(E). Therefore a plot of the emitted X-ray intensity versus the X-ray energy E has a shape very similar to a plot of D(E) versus E. These emitted X rays for aluminum are in the energy range from 56 to 77 eV.

Some other properties and experiments that depend on the density of states and can provide information on it are photoemission spectroscopy, Seebeck effect (thermopower) measurements, the concentrations of electrons and holes in semiconductors, optical absorption determinations of the dielectric constant, the Fermi contact term in nuclear magnetic resonance (NMR), the de Haas-van Alphen effect, the superconducting energy gap, and Josephson junction tunneling in superconductors. It would take us too far afield to discuss any of these topics. Experimental measurements of these various properties permit us to determine die form of the density of states D(E), both at the Fermi level £F and over a broad range of temperature.

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