Introduction to Nanotechnology

28 INTRODUCTION TO PHYSICS OF THE SOLID STATE
Ge
(a)
Figure 2.19. Ellipsoidal constant-energy surfaces in the conduction band of germanium (left)
and silicon (right). The constant energy surfaces of Ge are aligned along symmetry direction A
and centered at symmetry point L. As a result, they lie half inside (solid lines) and half outside
(dashed lines) the first Brillouin zone, so this zone contains the equivalent of four complete
energy surfaces. The surfaces of Si lie along the six symmetry directions A (i.e., along
fk,, fky, fk,), and are centered 85% of the way from the center point r to symmetry point
X. All six of them lie entirely within the Brillouin zone, as shown. Figure 2.14 shows the positions
of symmetry points I?, L, and X, and of symmetry lines A and A, in the Brillouin zone. (From
G. Burns, Solid State Physics, Academic Press, Boston, 1985, p. 313.)
2.2.4. Effective Masses
On a simple one-dimensional model the energy E of a conduction electron has a
quadratic dependence on the wavevector k through the expression
h2k2
E=-
2m*
The first derivative of this expression provides the velocity u
1dE fik
fidk -m*
-0
and the second derivative provides the effective mass m*
--
1 d2E 1
- -
fi2 dk2 - m*
(2.10)
(2.11)
which differs, in general, from the free-electron mass. These equations are rather
trivial for the simple parabolic energy expression (2.9), but we see from the energy