244
QUANTUM WELLS, WIRES, AND DOTS
electronic density of states D(E) 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 e\!
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 the form of the
density of states D(E), both at the Fermi level EF and over a broad range of
temperature.
9.4. EXCITONS
Excitons, which were introduced in Section 2.3.3, are a common occurrence in
semiconductors. When an atom at a lattice site loses an electron, the atom acquires a
positive charge that is called a hole. If the hole remains localized at the lattice site,
and the detached negative electron remains in its neighborhood, it will be attracted to
the positively charged hole through the Coulomb interaction, and can become bound
to form a hydrogen-type atom. Technically speaking, this is called a Mott-Wunnier
type of exciton. The Coulomb force of attraction between two charges Q, = - e
and Qh = +e separated by a distance r is given by F = -k2/&?, where e is the
electronic charge, k is a universal constant, and E is the dielectric constant of the
medium. The exciton has a Rydberg series of energies E sketched in Fig. 2.20 and a
radius given by Eq. (2.19): ueff = 0.0529(~/~,)/(m*/m,), where & / E O is the ratio of
the dielectric constant of the medium to that of free space, and m*/mo is the ratio of
the effective mass of the exciton to that of a free electron. Using the dielectric
constant and electron effective mass values from Tables B. 11 and B.8, respectively,
we obtain for GaAs
E = 5.2meV
ueff = 10.4nm
(9.10)
QUANTUM WELLS, WIRES, AND DOTS
electronic density of states D(E) 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 e\!
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 the form of the
density of states D(E), both at the Fermi level EF and over a broad range of
temperature.
9.4. EXCITONS
Excitons, which were introduced in Section 2.3.3, are a common occurrence in
semiconductors. When an atom at a lattice site loses an electron, the atom acquires a
positive charge that is called a hole. If the hole remains localized at the lattice site,
and the detached negative electron remains in its neighborhood, it will be attracted to
the positively charged hole through the Coulomb interaction, and can become bound
to form a hydrogen-type atom. Technically speaking, this is called a Mott-Wunnier
type of exciton. The Coulomb force of attraction between two charges Q, = - e
and Qh = +e separated by a distance r is given by F = -k2/&?, where e is the
electronic charge, k is a universal constant, and E is the dielectric constant of the
medium. The exciton has a Rydberg series of energies E sketched in Fig. 2.20 and a
radius given by Eq. (2.19): ueff = 0.0529(~/~,)/(m*/m,), where & / E O is the ratio of
the dielectric constant of the medium to that of free space, and m*/mo is the ratio of
the effective mass of the exciton to that of a free electron. Using the dielectric
constant and electron effective mass values from Tables B. 11 and B.8, respectively,
we obtain for GaAs
E = 5.2meV
ueff = 10.4nm
(9.10)