Chapter 4 Monte Carlo Calculation of Electron Transport in Solids
10.1016/s0080-8784(08)60267-7
Crossref book-chapter
Elsevier
Semiconductors and Semimetals (78)
Bibliography

Price, P. J. (1979). Chapter 4 Monte Carlo Calculation of Electron Transport in Solids. Lasers, Junctions, Transport, 249–308.

Authors 1
  1. Peter J. Price (first)
References 72 Referenced 90
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  3. For a good survey of Monte Carlo see J. M. Hammersley and D. C. Handscomb, “Monte Carlo Methods,” Wiley, New York, 1965. A more detailed work is Yu. A. Shrieder (ed.), “The Monte Carlo Method.” Pergamon, Oxford, 1966. The Monte Carlo method was introduced into modern physics by Ulam (1946) and von Neumann. See: correspondence between von Neumann and Richtmyer (1974) reproduced in J. von Neumann, “Collected Works” (A. H. Taub, ed.). Vol. 5. pp. 751–764. Macmillan, New York, 1963, and S. M. Ulam and J. von Neumann, Bull. Am. Math. Soc. 53, (1947) 1120. A personal account will be found in Chapter 10 of Ulam's autobiography “Adventures of a Mathematician” (1976). Their immediate concern was with nuclear reactor design, but there soon was work on a wide range of applications. The scope and flavor of this activity is indicated by the papers at a 1949 conference – “Monte Carlo Method.” National Bureau of Standards, Applied Mathematics Series, no. 12(1951), and by the bibliography with abstracts in “Symposium on Monte Carlo Methods” (H. A. Meyer, ed.), Wiley, New York, 1956. An interesting early paper with content related to that of the present work is M. L. Goldberger, Phys. Rev. 74, (1948) 1269. Pioneer papers on electron transport in solids were Lüthi and Wyder39, and T. Kurosawa, J. Phys. Soc. Jpn. Supple. 21, (1966) 424. An earlier use of Monte Carlo in physics, in which results of the kinetic theory of gases were tested by simulating the molecular motions, was Lord Kelvin, Phil, Mag. (6th ser.) 2, 1(1901).
  4. In an early discussion of the subject, von Neumann17 remarks: we could build a physical instrument to feed random digits directly into a high-speed computing machine. The real objection to this procedure is the practical need for checking computations. If we suspect that a calculation is wrong, almost any reasonable check involves repeating something done before. At that point the introduction of new random numbers would be intolerable.” The sequence of values of a physical variable produced in a particular computer “run” depends on the initial value of the seed integer in The pseudorandom number generator5; but the estimator values given by the computation are useful results to the extent that they are independent of the initial seed integer.:
  5. The pseudorandom number generator that was used in the unpublished calculations describedhere, and in Refs. 25, 26, 31, and 45, is an implementation of the Lehmer method for IBM 360 machines. See: D. W. Hutchinson, Commun. ACM 9, 432(1966); (10.1145/365696.365712)
  6. P. A. W. Lewis, A. S. Goodman, and J. M. Miller, IBM Syst. J. 8, (1969) 136. (10.1147/sj.82.0136)
  7. From another point of view, the fluctuations themselves can be what is calculated in a Monte Carlo simulation. An instance of this, use of a “fluctuation—dissipation theorem” to calculate a linear response coefficient, appears in Section 8. A fluctuation is, of course, given by an estimator that has its own variance.
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  9. The linearization of (8) and (9) to describe Ohmic conduction gives an inhomogeneous equation, for the deviation of/from the thermal equilibrium function, in which the scattering term is not algebraically the same as (10). See Part V.
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  26. The tensor-mass case reduces to that of (24), by a linear transformation of p space. See. for example, Ref. 10.
  27. Fawcett et al.8 The formula for acoustic-mode phonon scattering given in this paper is incorrect in the “nonparabolic” case because the coupling matrix element should be that of a plane-wave factor times a deformation-potential operator.
  28. For the physics referred to here see Section 2.9 in Mott and Davis, “Electronic Processes in Non-Crystalline Materials.” Oxford Univ. Press, London and New York, 1971.
  29. See Section 7.8 in Mott and Davis, “Electronic Processes in Non-Crystalline Materials,” Oxford Univ. Press, London and New York, 1971.
  30. With this model (having constant density of states, step function mobility, etc.) the overall mobility decreases with increasing field, because of an inversion effect at the lower energies above Ec, The interest in these unpublished computation results is limited by the evident divergence between the model and physical systems of present interest.
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  57. Corresponding to the sum in Eq. (136) is the series in the scattering operator, for vector mean free path, given in Section 5 of Ref. 41. It is equivalent to the direct scattering-series solution for the Boltzmann equation, which was utilized by D.L. RodePhys. Rev. B219701012
  58. Since no path duration actually is going to exceed Δt, we could replace the argument of thelogarithm in Eq. (21) by 1/(1 - a R), where a is a fixed number chosen so that none or few of the pseudorandom numbers R result in s values greater than Δt and hence are wasted. If. γ Δt is small we might, with small and known maximum error, replace the logarithm by asimple polynomial: a R, or a R - 1/2 (a R2. or.
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  63. In small bars of n-Ga As, the negative-differential-mobility feature of the variation of electron drift velocity with field, together with time lag in the dynamical response to a varying field, results for suitable macroscopic conditions in high-field domains travelling fromcathode to anode: Gunn domains. See, for example, J. E. Caroll, “Hot Electron Microwave Generators.” American Elsevier, New York, 1970.
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  66. The many-particle ensemble is obtained with a single-particle Monte Carlo procedure, bysaving the N preceding states in the history (where in practice N was 1500). In A. Matulionis, J. Pozela, and A. Reklaitis, Solid State Commun. 16, (1975) 1133. (10.1016/0038-1098(75)90131-3)
  67. For a general reference on such processes, see (for example) Chapter 6 of J. L. Moll, “Physics of Semiconductors.” McGraw-Hill, New York, 1964. Where electron-hole recombination through traps is significant it may be necessary to include both mobile electrons and mobile holes in the Monte Carlo scheme.
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  69. See J. E. Carroll, Ref. 48, Chapters 8 and 9.
  70. Such a displaced-Maxwellian function, fitted to moments obtained from the preceding part of a single-electron Monte Carlo history, has been used to generate electron-electron scattering in the computation in Ref. 53.
  71. 10.1088/0022-3719/9/2/017 / J. Phys. C / This has been done by by Bosi (1976)
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Created 14 years, 7 months ago (Jan. 5, 2011, 1:21 a.m.)
Deposited 6 years, 2 months ago (June 7, 2019, 11:32 a.m.)
Indexed 4 months ago (April 15, 2025, 1:36 a.m.)
Issued 46 years, 7 months ago (Jan. 1, 1979)
Published 46 years, 7 months ago (Jan. 1, 1979)
Published Print 46 years, 7 months ago (Jan. 1, 1979)
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@inbook{Price_1979, title={Chapter 4 Monte Carlo Calculation of Electron Transport in Solids}, ISSN={0080-8784}, url={http://dx.doi.org/10.1016/s0080-8784(08)60267-7}, DOI={10.1016/s0080-8784(08)60267-7}, booktitle={Lasers, Junctions, Transport}, publisher={Elsevier}, author={Price, Peter J.}, year={1979}, pages={249–308} }