Section 4.1 - 1

4.1 More on Band Gaps and Semiconductors Semiconductors • Semiconductors are distinguished from insulators by small band gaps that allow

excitation of electrons from filled valence bands to empty conduction bands, thereby creating charge carriers (holes and / or electrons).

• Semiconductors are generally classified by their electrical resistivity at room

temperature, with values in the range of 10-2 to 109 ohm-cm, and strongly dependent on temperature.

Comparing semiconductors to metals:

Metals

Semiconductors

- Conductivity decreases with increasing temperature due to electron-phonon interactions

- Conductivity increases with increasing temperature due to increased number of charge carriers (Boltzmann population distribution!)

σ = n |e| µ - n is large and effectively constant - µ is affected by the interaction of charge carriers with lattice vibrations

- n is small; conductivity is governed primarily by n - effects of µ are obscured by effects of changes in n

• At absolute zero, a pure, perfect crystal of most semiconductors will be an insulator,

if we arbitrarily define an insulator as having a resistivity above 1014 ohm-cm. • A highly purified semiconductor exhibits intrinsic conductivity, as distinguished

from the impurity conductivity of less pure specimens. In the intrinsic temperature range the electrical properties of a semiconductor are not essentially modified by impurities in the crystal.

Section 4.1 - 2

Section 4.1 - 3

Direct vs. Indirect Band Gaps

• The band gap is the difference between the lowest point of the conduction band (the

conduction band edge) and the highest point in the valence band (the valence band edge).

• The intrinsic conductivity and intrinsic carrier concentrations are largely controlled

by Eg / kBT, the ratio of the band gap to the temperature. • The best values of the band gap are obtained by optical absorption.

Section 4.1 - 4

• The threshold of continuous optical absorption at frequency ωg determines the band gap Eg = ħωg.

• In the direct absorption process, a photon is absorbed by the crystal with the

creation of an electron and a hole. • In the indirect absorption process, the minimum energy gap of the band structure

involves electrons and holes separated by a substantial wavevector kc.

In this process, a direct photon transition at the energy of the minimum gap cannot satisfy the requirement of conservation of wavevector, because photon wavevectors are negligible at the energy range of interest. BUT, if a phonon of wavevector K and frequency Ω is created in the process, then we can have: k(photon) = kc + K ≈ 0 and ħω = Eg + ħΩ

• The phonon energy ħΩ will generally be much less than Eg. • A phonon, even of high wavevector, is an easily accessible source of crystal

momentum because the phonon energies are characteristically small (~ 0.01 to 0.03 eV) in comparison to the energy gap (≤ 1 eV).

• If the temperature is high enough that the necessary phonon is already thermally

excited in the crystal, it is possible also to have a photon absorption process in which the phonon is absorbed!

Section 4.1 - 5

• The band gap may also be deduced from the temperature dependence of the

conductivity or of the carrier concentration in the intrinsic range (from measurements of Hall voltage), sometimes supplemented by conductivity measurements.

• It is the optical

measurements, however, that determine whether the gap is direct or indirect.

Section 4.1 - 6