Introduction to Nanotechnology

32 INTRODUCTION TO PHYSICS OF THE SOLID STATE
the same number pi of holes in the valence band, that is ni =pi. We see that the
expression (2.15) contains the product memh of the effective masses me and mh of the
electrons and holes, respectively, and the ratios me/mo and mh/mo of these effective
masses to the free-electron mass mo are presented in Table B.8. These effective
masses strongly influence the properties of excitons to be discussed next.
2.3.3. Excitons
An ordinary negative electron and a positive electron, called a positron, situated a
distance r apart in free space experience an attractive force called the Coulomb force,
which has the value -e2/4mo?, where e is their charge and co is the dielectric
constant of free space. A quantum-mechanical calculation shows that the electron
and positron interact to form an atom called positronium which has bound-state
energies given by the Rydberg formula introduced by Niels Bohr in 19 13 to explain
the hydrogen atom
(2.16)
where a. is the Bohr radius given by a. = 4ncofi2/moe2 = 0.0529 nm, mo is the freeelectron
(and positron) mass, and the quantum number n takes on the values
n = 1,2,3,. . . , m. For the lowest energy or ground state, which has n = 1, the
energy is 6.8 eV, which is exactly half the ground-state energy of a hydrogen atom,
since the effective mass of the bound electron-positron pair is half of that of the
bound electron-proton pair in the hydrogen atom. Figure 2.20 shows the energy
levels of positronium as a hnction of the quantum number n. This set of energy
levels is often referred to as a Rydberg series. The continuum at the top of the figure
is the region of positive energies where the electron and hole are so far away from
each other that the Coulomb interaction no longer has an appreciable effect, and the
energy is all of the kinetic type, $mu’ = p2/2m, or energy of motion, where u is the
velocity andp = mu is the momentum.
The analog of positronium in a solid such as a semiconductor is the bound state of
an electron-hole pair, called an exciton. For a semiconductor the electron is in the
conduction band, and the hole is in the valence band. The electron and hole both
have effective masses me and mh, respectively, which are less than that (mo) of a free
electron, so the effective mass m* is given by m* = memh/(me + mh). When the
electron effective mass is appreciably less than the hole effective mass, me