Page 123 - Materials Chemistry, Second Edition

110 2 Solid-State Chemistry

approach the positively-charged nuclei of the lattice atoms, their electrostatic
2
potential energy will decrease by e /4pe o r. Hence, the energy of an electron in c þ
will be lower than an electron in c ,orE þ < E
2
ð hkÞ
E þ ¼ V n
ð40Þ 2m e
2
ð hkÞ
E ¼ 2m e þ V n
The first terms of Eq. 40 correspond to the energy of a free electron as a traveling wave
(i.e.,awayfromk ¼ np/a), obtained from solving the familiar “particle in a box”
problem. The term V n corresponds to the electrostatic potential energy resulting from
electron-nuclei interactions. As one can see from Figure 2.74,the E vs. k plot for an
electron wave in the 1-D lattice will result in a parabolic increase in energy with k
until k ¼ p/a is reached, at which point a sharp discontinuity is found. Another
parabolic increase in energy is then followed until k ¼ 2p/a is reached, and so on.
The range of k-values between p/a < k < p/a is known as the first Brillouin zone
(BZ). The first BZ is also defined as the Wigner-Seitz primitive cell of the reciprocal
lattice, whose construction is illustrated in Figure 2.75. First, an arbitrary point in the
reciprocal lattice is chosen and vectors are drawn to all nearest-neighbor points.
Perpendicular bisector lines are then drawn to each of these vectors; the enclosed area
corresponds to the primitive unit cell, which is also referred to as the first Brillouin zone.
Extending the number of reciprocal lattice vectors and perpendicular bisectors
results in the 2nd, 3rd, ..., nth BZs, which become increasingly less useful to

Figure 2.74. The energy of an electron as a function of its wavevector, k, inside a 1-D crystal, showing
energy discontinuities at k ¼ np/a. Reproduced with permission from Kasap, S. O. Principles of
Electronic Materials and Devices, 3rd ed., McGraw-Hill: New York, 2006.