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7.4 Examples of Bandstructures
tion of thesp3hybrid (a mixture of 2sand 2pwavefunctions with tetrahedral bonding; Chap. 1), there is a modification of thes- andp-levels which mani- fests itself in a further splitting of thesp3hybrid band into two bands, each of which (including spin) can accommodate four electrons (Fig. 7.9).
The four electrons of the atomic 2s- and 2p-states thus fill the lower part of the sp3 band, leaving the upper part unoccupied. Between the two sp3subbands there is a forbidden energy gap of widthEg. This is the origin of the insulating property of diamond, as will be shown in Sects. 9.2 and 12.1. The semiconductors Si and Ge are similar cases (Chap. 12).
The form of the bandstructure shown in Fig. 7.9 cannot be derived using the simple approach outlined here. More complex methods are neces- sary for this calculation and these require the use of modern computing facilities. For further information about such calculations the reader is refer- red to theoretical reviews and more advanced text books.
symmetry directions and symmetry points in the first Brillouin zone of a face-centered cubic lattice are indicated in Figs. 3.8 and 7.11 b.
A striking feature of the Al bandstructure is that it can be described very well by the parabolic dependence of a free-electron gas (dotted lines).
The energy gaps at the Brillouin zone edges are relatively small and the Fig. 7.10.The four highest occu- pied energy bands of KCl calcu- lated as a function of the ionic separation in Bohr radii (a0= 5.2910±9cm). The energy levels in the free ions are indicated by arrows. (After [7.2])
Fig. 7.11.(a) Theoretical bandstructureE kfor Al along directions of high symmetry (C is the center of the Brillouin zone). The dotted lines are the energy bands that one would obtain if the s- andp-electrons in Al were completely free (``empty'' lattice). After [7.3].
(b) Cross section through the Brillouin zone of Al. The zone edges are indicated by the dashed lines. The Fermi ``sphere'' of Al (ÐÐ) extends beyond the edges of the first Bril- louin zone
complexity of the bandstructure stems largely from the fact that the energy parabolas are plotted in the reduced-zone scheme, i.e.,``folded'' back into the first Brillouin zone. This type of bandstructure is characteristic for sim- ple metals. The similarity to the free electron gas is particularly pronounced for the alkali metals Li, Na and K.
The filling of the bands with the available electrons continues up to the Fermi energyEF(indicated in Fig. 7.11). It can be seen that the correspond- ing constant energy surface, the so-called Fermi surface E(k) =EF, inter- sects several bands. Thus, even for Al, the Fermi surface is not a simple continuous surface: whereas the Fermi surfaces of the alkali metals are almost spherical and are contained wholly within the first Brillouin zone, the ``Fermi sphere'' of Al extends just beyond the edges of the first Brillouin zone. The Bragg reflections occurring at these edges cause a slight deviation from the spherical form in these regions. This is shown qualitatively in Fig. 7.11 b in a cross section through three-dimensionalk-space.
In comparison to the simple metals, the band structures of the transition metals are considerably more complicated, due to the significant influence
Fig. 7.12.BandstructureE(k) for copper along directions of high crystal symmetry (right).
The experimental data were measured by various authors and were presented collectively by Courths and HuÈfner [7.4]. The full lines showing the calculated energy bands and the density of states (left) are from [7.5]. The experimental data agree very well, not only among themselves, but also with the calculation
of the d-bands. Together with the bands that originate from s-levels and resemble the parabolic form of the free-electron gas, there are also very flat E(k) bands, whose small energy width (low dispersion) can be attributed to the strong localization of the d-electrons. This is readily seen for the exam- ple of copper, whose bandstructure is illustrated in Fig. 7.12. For transition metals such as Pt, W, etc., where the Fermi level intersects the complex d- bands, the Fermi surfaces possess particularly complicated forms.
Other interesting phenomena, such as semiconducting properties (Chap. 12), occur when the bandstructure possesses an absolute gap, i.e., a so-called forbidden region: in this particular energy range and for all k-directions in reciprocal space, there are no available electron states.
A typical bandstructure of this type is that of germanium (Fig. 7.13). Like diamond and silicon, germanium crystallizes in the diamond structure, whereby the tetrahedral bonding of the individual atoms is a consequence of the formation of sp3 hybrid orbitals. As was mentioned at the end of Sect. 7.3, the formation of sp3 hybrids leads to the existence of sp3 sub-
Fig. 7.13. Theoretically derived bandstructureE(k) for germanium along directions of high symmetry (right), and the corresponding electronic density of states (left). A number of criti- cal points, denoted according to their position in the Brillouin zone (C,X,L), can be seen to be associated with regions of the bandstructure where E k has a horizontal tangent. The shaded region of the density of states corresponds to the states occupied by electrons [7.6]
bands. The lower of these (below the forbidden gap) are fully occupied whereas the higher-lying sp3 subbands above the gap are unoccupied. The Fermi energy must therefore lie within the forbidden gap, a fact that will be important when we come to discuss the semiconducting properties of this crystal in Chap. 12.