Bandgap Dependence and Related Features of Radiation Ionization Energies in Semiconductors
1.“High” compared to the bandgap energy
2.See, e.g., Fig. 1 in C. A. Klein, Phys. Letters 24A, 513 (1967).
3.The quantity ε is often referred to as “energy required to form an electron‐hole pair in a semiconductor.” I wish to stress that this is not only incorrect but highly misleading in view of the rapport with threshold energies for impact ionization (see Sec. II). It could well be that much of the confusion in the field of radiation‐ionization energies originated from frequent usage of such loose semantics.
4.This was clearly stated by Lappe, as early as 1961 [F. Lappe, J. Phys. Chem. Solids 20, 173 (1961)].
5.These secondaries, in turn, are the product of branching processes commonly associated with “δ Rays” originating along the primary’s path; they may also result from the decay of plasmons, which are expected to play a central role in the absorption of high‐energy radiation by semiconductors [in this context, see C. A. Klein, J. Phys. Soc. Japan Suppl. 21, 307 (1966)].
6.E. Baldinger and W. Czaja, Nucl. Inst. Methods 10, 237 (1961).
7.K. G. McKay, Phys. Rev. 84, 829 (1951).
8.W. Shockley, Solid‐State Electron. 2, 35 (1961).
9.W. Czaja, Helv. Phys. Acta 34, 760 (1961); Czaja writes where k is the Boltzmann constant and θ the Debye temperature, finds θ to be a linear function of for elemental semiconductors, and then sets in order to reproduce the Ge result.
10.P. J. Dean and J. C. Male, J. Phys. Chem. Solids 35, 311 (1964).
11.This comment also applies to the recent work of W. van Roosbroeck, Phys. Rev. 139, A1702 (1965).
12.R. B. Day, G. Dearnaley, and J. M. Palms, IEEE Trans. NS‐14:1, 487 (1967).
13.This contrasts with the situation in the immediate neighborhood of the absorption edge, where any pair‐producing event, of course, consumes an amount of energy at least equivalent to the width of the forbidden gap.
14.Yu. M. Popov, in Soviet Researches on Luminescence (Consultants Bureau, New York, 1964), p. 62.
15.Strictly speaking, it is by no means certain that optical phonon emission constitutes the dominant scattering mechanism in the energy range we are concerned with. In Ge and GaAs, e.g., Conwell considers it likely that intervalley acoustic scattering will be more important [see her comments following R. A. Logan and S. M. Sze, J. Phys. Soc. Japan 21, Suppl., 434 (1966)]. However, since the relevant phonons have wavevectors biased towards zone‐edge values they should be comparable in energy to so that it might be immaterial to consider “details” of this sort.
16.Following P. A. Wolff, Phys. Rev. 95, 1415 (1954).
17.Right at the threshold the cross section is evidently zero but rises rapidly as very fast carriers are presumed to dissipate their energy through impact‐ionization scattering rather than other inelastic encounters (see Sec. V).
18.The threshold is normally found to be larger than the bandgap because of the requirement to conserve k [see, e.g., the calculations of J. R. Hauser, J. Appl. Phys. 37, 507 (1966)]. It can be argued that some ambiguity arises in connection with phonon‐assisted transitions but they probably are of little consequence (Ref. 16).
18.On the other hand, it may well be that because of the strong energy dependence of impact‐ionization cross sections the effective threshold reflects the physical situation under consideration [E. O. Kane, J. Phys. Soc. Japan 21, Suppl., 37 (1966)].
19.C. R. Crowell and S. M. Sze, Appl. Phys. Letters 9, 242 (1966), where the reader will find suitable references.
20.Note that represents an average energy, that is, it involves an integration over the whole distribution and thus may not be too sensitive regarding exact shapes.
21.It is interesting to compare this to the frequently used £ value of that holds for uniform distributions in energy [see, e.g., J. Tauč, J. Phys. Chem. Solids 8, 219 (1959)].
22.See Ref. 11.
23.They require drastic idealizations; in particular, they ignore the critical role of the density of states in the frequency distribution of
24.Henceforth, the number r will be interpreted as a mean‐free‐path ratio for ionizing collisions and phonon emission.
25.U. Fano, Phys. Rev. 70, 44 (1946)
25.and U. Fano, 72, 26 (1947); though Fano’s arguments concern gaseous ionization chambers, they are highly relevant to the problem in hand since fluctuations in the number of electron‐hole pairs also govern the ultimate resolution of semiconductor counters., Phys. Rev.
26.Note that, in our case, N̄ is simply the quantum yield defined via Eq. (1); in essence, the Fano factor measures what is meant by that famous slogan… “l’indépendance dans l’interdépendance.”
27.We refer to Y as an “ionization efficiency” rather than a “relative yield” for the following reason: With the advent of electron‐beam pumped lasers the ratio Y takes on operational significance because it sets an upper limit for the conversion efficiency of beam power to coherent light [C. A. Klein, IEEE J. Quant. Electron. (to be published)].
28.G. D. Alkhazov, A. A. Vorob’ev, and A. P. Komar, Bull. Acad. Sci. USSR (Phys. Series) 29, 1231 (1965).
29.H. M. Mann, H. R. Bilger, and I. S. Sherman, IEEE Trans. NS‐13:3, 252 (1966).
29.Note added in proof: It now appears that in Ge at 77 °K, [H. R. Bilger, Phys. Rev. 163, 238 (1967)].
30.S. O. Antman, D. A. Landis, and R. H. Pehl, Nucl. Instr. Methods 40, 272 (1966). Their ε value is believed highly accurate (for a discussion, see Ref. 2). On the other hand, judging from the work reported in Ref. 29 their Fano‐factor estimate must be discarded.
31.Apropos of diamond it is felt that the wide range of ε values which have been obtained [cf. P. J. Kennedy, Proc. Roy. Soc. A253, 37 (1959)] renders quantitative considerations inappropriate in the context of the present paper; yet, see Fig. 3.
32.J. Tauč and A. Abraham, Czech. J. Phys. 9, 95 (1959).
33.A. Smith and D. Dutton, J. Opt. Soc. Am. 48, 1007 (1958).
34.See Ref. 2.
35.C. Bussolati, A. Fiorentini, and G. Fabri, Phys. Rev. 136, Al756 (1964);
35.F. E. Emery and T. A. Rabson, Phys. Rev. 140, A2089 (1965).
36.I am indebted to Dr. Walter L. Brown, of Bell Telephone Laboratories, for enlightening comments about this matter.
37.L. Koch, J. Messier, and J. Valin, Compt. Rend. 250, 74 (1961);
37.J. J. Smithrick and I. T. Myers, Document NASA‐TN‐D‐3694 (CFSTI, Springfield, Virginia, 1966). Note added in proof: Recent work done at the Lawrence Radiation Laboratory [K. H. Pehl, F. S. Goulding, D. A. Landis, and M. Lenzlinger, Nucl. Inst. Methods (to be published)] demonstrates that the difference between and in Si is considerably less than reported in Ref. 35.
38.D. B. Wittry and D. F. Kyser, J. Appl. Phys. 36, 1387 (1965).
39.H. Pfister, Z. Naturforsch. 12a, 217 (1957).
40.J. W. Mayer, J. Appl. Phys. 38, 296 (1967).
41.B. Goldstein, J. Appl. Phys. 36, 3853 (1965).
42.For references, see N. F. Malyuk, G. A. Fedorus, V. D. Fursenko, I. A. Shakh‐Melikova, and M. K. Sheinkman, Sov. Phys.‐Solid State 8, 2513 (1967).
43.P. J. van Heerden, Phys. Rev. 106, 468 (1957);
43.O. Dehoust, Z. Physik 172, 19 (1963).
44.See Rgf. 4
45.G. P. Golubev, V. S. Vavilov, and V. D. Egorov, Sov. Phys.‐Solid State 7, 3000 (1966).
46.For references, see R. A. Logan, J. M. Rowell, and F. A. Trumbore, Phys. Rev. 136, A1715 (1964).
47.Here, we simply wish to mention that an r of 20 accords with the cross‐section ratio recently derived from the avalanche‐breakdown dynamics in bulk InSb [J. C. McGroddy and M. I. Nathan, J. Phys. Soc. Japan 21, Suppl., 437 (1966)].
47.Their figure, however, could easily be in error by a factor of two, either way, because ionization rates are known to be quite insensitive to r [G. A. Baraff, Phys. Rev. 128, 2507 (1962)].
48.This may be a good opportunity to stress that, in our case just as in the case of diodes (see citation in Ref. 15), poorer agreement is obtained when taking as suggested by Shockley (Ref. 8), or as one might surmise from theoretical arguments (Ref. 16). Indeed, any agreement of a “quality” such as exhibited in Fig. 3 would be fortuitous unless
49.G. A. Baraff (reference cited in footnote 47).
50.W. E. Spicer, J. Phys. Chem. Solids 22, 365 (1961).
51.This follows standard practice but is subject to criticism because it ignores potential energy gains through phonon absorption (see, e.g., Ref. 19).
52.C. A. Lee, R. A. Logan, R. L. Batdorf, J. J. Kleimack, and W. Wiegmann, Phys. Rev. 134, A761 (1964).
53.E. O. Kane (reference cited in footnote 18).
54.D. L. Bartelink, J. L. Moll, and N. I. Meyer, Phys. Rev. 130, 972 (1963);
54.their figure was subsequently revised and upped to 270 Å by G. A. Baraff, Phys. Rev. 135, A528 (1964).
55.Which in turn supports the premise (see Sec. I) that most of the pairs originate either through plasmon decay or in the final stages of a branching process.
56.Note that a pattern of this type, namely a probability that rises from zero to one over a narrow range, was already anticipated by Wolff (Ref. 16); he speculated, however, that this would happen at a substantially lower energy, around 2.3 eV.
57.D. E. Gray, in Style Manual (American Institute of Physics, New York, 1959), p. 4.
58.There does not appear to be prior specific evidence about this in the literature, but existing secondary‐emission data further substantiate the overwhelming influence of low‐energy phenomena in the pair‐creation process.
59.In this context it is amusing to remark that Czaja’s equation (Ref. 9) actually boils down to which explains why it “fits.”
60.The reader is expressly referred to the summary of theoretical calculations and experimental results for Ge, Si, and GaAs given in Table I of Hauser’s paper cited in Ref. 18.
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