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Figure 2.18. Band structure plots of the energy bands of the indirect-gap semiconductors Si and Ge. The bandgap (absence of bands) Res slightly above the Fermi energy E = 0 on both figures, with the conduction band above and the valence band below the gap. The figure shows that the lowest point or bottom of the conduction band of Ge is at the energy E = 0.6 at the symmetry point L (labeled by and for Si it is 85% of the way along the direction A, from r15 to X,. It is clear from Fig. 2.15 that the bottom of the conduction band of the direct-gap semiconductor GaAs is at the symmetry point r6. The top of the valence band is at the center point r of the Brillouin zone for both materials.

Wavevector k

Figure 2.18. Band structure plots of the energy bands of the indirect-gap semiconductors Si and Ge. The bandgap (absence of bands) Res slightly above the Fermi energy E = 0 on both figures, with the conduction band above and the valence band below the gap. The figure shows that the lowest point or bottom of the conduction band of Ge is at the energy E = 0.6 at the symmetry point L (labeled by and for Si it is 85% of the way along the direction A, from r15 to X,. It is clear from Fig. 2.15 that the bottom of the conduction band of the direct-gap semiconductor GaAs is at the symmetry point r6. The top of the valence band is at the center point r of the Brillouin zone for both materials.

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 Briflouin 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 ±kx, ±ky, ±k2), and are centered 65% of the way from the center point r to symmetry point X. AH six of them lie entirely within the Brillouin zone, as shown. Figure 2.14 shows the positions of symmetry points r, L, and X, and of symmetry lines A and A, in the Brillouin zone. (From G. Bums, Solid State Physics, Academic Press, Boston, 1985, p. 313.)

2.2.4. Effective Masses

On a simple one-dimensional model the energy £ of a conduction electron has a quadratic dependence on the wavevector k through the expression

The first derivative of this expression provides the velocity v

1 dE fik

and the second derivative provides the effective mass m*

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 bands of Figs. 2.15 and 2.18 that the actual dependence of the energy E on k is much more complex than Eq. (2.9) indicates. Equation (2.11) provides a general definition of (he effective mass, designated by the symbol m", and the wavevector k dependence of m* can be evaluated from the band structure plots by carrying out the differentiations. We see from a comparison of the slopes near the conduction band minimum and the valence tend maximum of GaAs at the F point on Fig. 2.15 (see also Fig. 2.16) that the upper electron bands have steeper slopes and hence lighter masses than do the lower hole bands with more gradual slopes.

2.2.5. Fermi Surfaces

At very low temperatures electrons fill the energy bands of solids up to an energy called the Fermi energy Er, and the bands are empty for energies that exceed E?. In three-dimensional k space the set of values of kx,ky, and kz, which satisfy the equation %2(kl + k* + k})/2m = Er, form a surface called the Fermi surface. All kx,ky,kz energy states that lie below this surface are full, and the states above the svirfece are empty. The Fermi surface, encloses all the electrons in the conduction band that carry electric current. In the good conductors copper and silver the conduction electron density is 8.5 x 1022 and 5.86 x 1022 electroas/cm3, respectively. From another viewpoint, die Fermi surface of a good conductor can fill the entire Brillouin zone. In the intrinsic semiconductors GaAs, Si, and Ge the carrier density at room temperature from Fig. 2.17 is approximately 106,1010, and 10!3 carriers/cm3, respectively, many orders of magnitude below that of metals, and semiconductors are seldom doped to concentrations above 1019 centers/cm3. An intrinsic semiconductor is one with a full valence band and an empty conduction band at absolute zero of temperature. As we saw above, at ambient temperatures some electrons are thermally excited to title bottom of the conduction band, and an equal number of empty sites or holes are left behind near the top of the valence band. This means that only a small percentage of the Brillouin zone contains electrons in the conduction band, and the number of holes in the valence band is correspondingly small. In a one-dimensional representation this reflects the electron and hole occupancies depicted in Fig. 2.16.

If the conduction band minimum is at the T point in the center of the Brillouin zone, as is the case with GaAs, then it will be very close to a sphere since the symmetry is . cubic, and to a good approximation we can assume a quadratic dependence of the energy on the wavevector k, corresponding to Eq. (2.9). Therefore the Fermi surface in k space is a small sphere given by the standard equation for a sphere

where Ev is the Fermi energy, and the electron mass me relative to the tree-electron mass has the values given in Table B.8 for various direct-gap semiconductors.

Equations (2.12) are valid for direct-gap materials where the conduction band minimum is at point T. All the semiconductor materials under consideration have their valence band maxima at the Brillouin zone center point T.

The situation for the conduction electron Fermi surface is more complicated for the indirect-gap semiconductors. We mentioned above that Si and GaP have their conduction band minima along the A direction of the Brillouin zone, and the corresponding six ellipsoidal energy surfaces are sketched in Fig. 2.19b. The longitudinal and transverse effective masses, mL and mr respectively, have the following values for Si (2.13a)

for these two indirect-gap semiconductors. Germanium has its band minimum at points L of the Brillouin zone sketched in Fig. 2.14, and its Fermi surface is the set of ellipsoids centered at the L points with their axis along the A or (111) directions, as shown in Fig. 2.19a. The longitudinal and transverse effective masses for these ellipsoids in Ge are mh/ma = 1.58 and mT/m0 = 0.081, respectively. Cyclotron resonance techniques together with the application of stress to the samples can be used to determine these effective masses for the indirect-bandgap semiconductors.

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