Abstract
The periodic potential distribution of an electron in a crystal shown in Fig.2.5 involves N discrete levels if the crystal contains N atoms as we have seen in Fig. 2.9. A discussion of these levels can be confined to the first Brillouin zone. We have seen in the last chapter that due to the crystal periodicity the electron wave functions, which in one dimension are ψ(x) = u(x) exp(ikx), have also to be periodic (“Bloch functions”). Hence from
and
we obtain
or
where a is the lattice constant. We notice that Eq.(3.1) is actually valid for a ring-shaped chain which means that we neglect surface states (see Chap. 14a). Since for the first Brillouin zone k has values between −π/a and +π/a, we find that the integer n is limited to the range between −N/2 and +N/2. In Fig.3.1 the discrete levels are given for a “crystal” consisting of N = 8 atoms.
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References
This calculation has been adapted from E.Spenke,: Electronic Semiconductors, chap. VIII. New York,: McGraw-Hill. 1958.
This calculation has been adapted from E.Spenke,: Electronic Semiconductors, chap. VIII. New York,: McGraw-Hill. 1958.
See e.g. E.Schrödinger,: Statistical Thermodynamics. Cambridge,: Univ.Press. 1948.
J.S.Blakemore,: Semiconductor Statistics, appendix B. Oxford,: Pergamon. 1962
H.Fritzsche, Phys.Rev. 120 (1960) 1120.
H.Fritzsche, Phys.Rev. 120 (1960) 1120.
S.H.Koenig, R.D.Brown III, and W.Schillinger, Phys.Rev.128 (1962) 1668, and literature cited there. See also D.Long, C.D.Motchenbacher, and J.Myers, J.Appl.Phys.30 (1959) 353; J.Blakemore,: Semiconductor Statistics, Chap.3.2.4. Oxford,: Pergamon. 1962.
J.S.Blakemore, Phil.Mag. 4 (1959) 560.
N.B.Hannay,: Semiconductors (N.B.Hannay,ed.) p. 31. New York,: Reinhold Publ.Co. 1959
For the occupation probabilility of double donors see e.g. E.Spenke,: Electronic Semiconductors, chap. VIII, I. New York,: McGraw-Hill. 1965.
For a review see e.g. E.M.Conwell, Proc.IRE 46 (1958) 1281; T.H.Geballe in ref.4,p.341 and 342.
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Seeger, K. (1973). Semiconductor Statistics. In: Semiconductor Physics. Springer Study Edition. Springer, Vienna. https://doi.org/10.1007/978-3-7091-4111-3_3
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