Physical Chemistry Third Edition

28.3 The Electronic Structure of Crystalline Solids 1173

800 640

Au 4s

Au 4p1/2

Au 4p3/2

Au 5p3/2

Au 4d3/2,5/2

C 1s

Au 4f5/2,7/2

480

Binding energy/eV

320 160 0

Figure 28.10 X-Ray Photoelectron Spectrum of Gold.Each peak is labeled with the

subshell and the value ofJ, the quantum number for the total angular momentum. Courtesy

of Dr. Kevin Ogle.

of an upper band that is empty in the ground state. If the band gap between the lower

band and the upper band is not large compared withkBT, some electrons from the

lower-energy band will occupy states of the upper band and the crystal will conduct

some electricity. Figure 28.11 schematically depicts the band occupations in insulators,

conductors, and semiconductors.

Conductor

Large band gap

Insulator

Semiconductor

Small band gap

5 Empty states

5 Occupied states

Figure 28.11 Bands of Orbital

Energies in a Hypothetical Insula-

tor, Conductor, and Semiconductor.

The top figure, for an insulator, shows

the band gap, which is large compared

withkBT. The middle figure, for a

conductor, shows a partly filled band,

so that there is no band gap. The

bottom figure, for a semiconductor,

shows a band gap that is not large

compared withkBT.

T 2 >T 1

T 1 >0

T 50

f(^

)

5 

1

Figure 28.12 The Fermion Probabil-

ity Distribution for 0 K and for Two

Nonzero Temperatures.

Electrons are fermions. If we neglect the interaction between the electrons the prob-

ability distribution of states is given by Eq. (26.5-4):

f(εi)

e(μ−εi)/kBT

1 +e(μ−εi)/kBT



1

e(εi−μ)/kBT+ 1

(28.3-1)

The value ofμ, the chemical potential of the electrons, is called theFermi level,

and is denoted byεF. At 0 K all of the states with energies at or below the Fermi

level will be fully occupied and those above the Fermi level will be vacant. If

the temperature is increased from 0 K, some of the states with energies just below

the Fermi level become unpopulated, and some of the states just above the Fermi

level become populated. The range of energy over which the probability distribution

ranges from roughly unity to nearly zero is approximately equal tokBT. Figure 28.12

schematically shows the fermion probability distribution for 0 K and for two nonzero

temperatures.

If the Fermi level lies at the top of a band or between two bands the crystal will be

an insulator at 0 K. If the band gap is not large compared withkBTfor some nonzero

temperature, some of the highest-energy states in the filled band will be vacant, some

of the low-lying states in the first vacant band will be occupied, and the crystal will