Self-sustained current oscillations in the kinetic theory of semiconductor superlattices

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We present the first numerical solutions of a kinetic theory description of self-sustained current oscillations in n-doped semiconductor superlattices. The governing equation is a single-miniband Boltzmann-Poisson transport equation with a BGK (Bhatnagar-Gross-Krook) collision term. Appropriate boundary conditions for the distribution function describe electron injection in the contact regions. These conditions seamlessly become Ohm's law at the injecting contact and the zero charge boundary condition at the receiving contact when integrated over the wave vector. The time-dependent model is numerically solved for the distribution function by using the deterministic Weighted Particle Method. Numerical simulations are used to ascertain the convergence of the method. The numerical results confirm the validity of the Chapman-Enskog perturbation method used previously to derive generalized drift-diffusion equations for high electric fields because they agree very well with numerical solutions thereof.
[1] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511–525. [2] L.L. Bonilla, L. Barletti, M. Álvaro, Nonlinear electron and spin transport in semiconductor superlattices, SIAM J. Appl. Math. 69 (2008) 494–513. [3] L.L. Bonilla, R. Escobedo, A. Perales, Generalized drift-diffusion model for miniband superlattices, Phys. Rev. B 68 (2003) 241304(R) (4 pages). [4] L.L. Bonilla, H.T. Grahn, Nonlinear dynamics of semiconductor superlattices, Rep. Prog. Phys. 68 (2005) 577–683. [5] E. Cebrián, F.J. Mustieles, Deterministic particle simulation of multiheterojunction semiconductor devices: the semiclassical and quantum cases, Compel 13 (1994) 717–725. [6] P. Degond, B. Niclot, Numerical analysis of the weighted particle applied to the semiconductor Boltzmann equation, Numer. Math. 55 (1989) 599–618. [7] F. Delaurens, F.J. Mustieles, A deterministic particle method for solving kinetic transport equations: the semiconductor Boltzmann equation case, SIAM J. Appl. Math. 52 (1992) 973–988. [8] L. Esaki, R. Tsu, Superlattice and negative differential conductivity in semiconductors, IBM J. Res. Develop. 14 (1970) 61–65. [9] R. Escobedo, L.L. Bonilla, Numerical methods for a quantum drift-diffusion equation in semiconductor physics, J. Math. Chem. 40 (2006) 3–13. [10] Y. Farjoun, B. Seibold, An exactly conservative particle method for one dimensional scalar conservation laws, J. Comput. Phys. 228 (2009) 5298–5315. [11] J. Feldmann, K. Leo, J. Shah, D.A.B. Miller, J.E. Cunnigham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, S. Schmitt-Rink, Optical investigation of Bloch oscillations in a semiconductor superlattice, Phys. Rev. B 46 (1992) 7252–7255. [12] R.R. Gerhardts, Effect of elastic scattering on miniband transport in semiconductor superlattices, Phys. Rev. B 48 (1993) 9178–9181. [13] K. Hofbeck, J. Grenzer, E. Schomburg, A.A. Ignatov, K.F. Renk, D.G. Pavel’ev, Yu. Koschurinov, B. Melzer, S. Ivanov, S. Schaposchnikov, P.S. Kop’ev, Highfrequency self-sustained current oscillation in an Esaki–Tsu superlattice monitored via microwave emission, Phys. Lett. A 218 (1996) 349–353. [14] A.A. Ignatov, V.I. Shashkin, Bloch oscillations of electrons and instability of space-charge waves in semiconductor superlattices, Sov. Phys. JETP 66 (1987) 526–530 [Zh. Eksp. Teor. Fiz. 93 (1987) 935–943]. [15] A.A. Ignatov, E.P. Dodin, V.I. Shashkin, Transient response theory of semiconductor superlattices: connection with Bloch oscillations, Mod. Phys. Lett. B 5 (1991) 1087–1094. [16] D. Issautier, Convergence of a weighted particle method to solve the Boltzmann (B.G.K.) equation, SIAM J. Numer. Anal. 33 (1996) 2099–2119. [17] C. Jacoboni, L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55 (1983) 645–705. [18] H. Kroemer, Gunn effect – bulk instabilities, in: W.D. Hershberger (Ed.), Topics in Solid State and Quantum Electronics, Chapter 2, John Wiley, NY, 1972, pp. 20–98. [19] S.A. Ktitorov, G.S. Simin, V.Ya. Sindalovskii, Bragg reflections and the high-frequency conductivity of an electronic solid-state plasma, Sov. Phys. Solid State 13 (1972) 1872–1874 [Fiz. Tverd. Tela 13 (1971) 2230–2233]. [20] B. Niclot, P. Degond, F. Poupaud, Deterministic particle simulation of the Boltzmann transport equation of semiconductors, J. Comput. Phys. 78 (1988) 313–349. [21] C. Zener, A theory of the electrical breakdown of solid dielectrics, Proc. Roy. Soc. London, Ser. A 145 (1934) 523–529.