ABSTRACT
Chapter 3 treated theoretically the first-order phase transition and the related S-shaped I-V characteristics in the one-level impact-ionization model. A spatially inhomogeneous solution of the I-V curve was transformed into a spatially homogeneous solution of the I-V curve, leading to our understanding of the formation of a stationary current filament in semiconductors. This chapter covers spatially homogeneous solutions and investigates the nonlinear dynamics of the carrier transport in the impact-ionization avalanche. Dynamical behaviors in the S-shaped I-V characteristics are strongly related to the cusp catastrophe in sections 4.1 and 4.2. Linear stability analysis is introduced in section 4.3, which provides a powerful tool for finding a criterion for stability and instability in a given dynamical system. In order to illustrate the idea of linear stability analysis, some examples (pendulum, Duffing equation and Van der Pol oscillator) are also given in section 4.3. Linear stability analysis is applied to a mathematical model in section 4.4 and a physical model of the impactionization avalanche in section 4.5. Some fundamental aspects of chaos will be presented in sections 4.6-4.12. The following text describes the catastrophe theory. The structural
stability of a certain dynamical system can be discussed, under the static regime, in term of the catastrophe manifold M constructed by the potential function fðx; c1; c2; . . . ; cmÞ of an m-parameter family. Catastrophe was first named by a mathematician, Thom [1], and means a sudden change at a critical point in the parameter space. A wave breaking, the diadem of a splash, a swallow’s tail, morphogenesis in biology and various types of sudden changes in nature are described by the catastrophe theory [1-3]. The
first-order phase transition in semiconductor carrier transport [4] and laser optics [2, 5] are strongly related to an elementary catastrophe (cusp catastrophe), as discussed below. Let x ¼ ðx1; x2; . . . ; xnÞ be state variables (internal variables, x 2 RnÞ; ci ði ¼ 1; 2; . . .mÞ be control variables ðc1; c2; . . . cmÞRm;Rm: control space), then catastrophe manifold M is defined [2] as:
M ¼ fY 2 Rsj@f=@x1 ¼ @f=@x2 ¼ . . . ¼ @f=@xn ¼ 0g; ð4:1Þ where s ¼ nþm;Y ¼ ðx1; x2; . . . ; xn; c1; c2; . . . ; cmÞ. The map function f : Rn ! R is assumed to be a smooth function of class C1. The function f can be noted also as potential function V for dynamical systems. By definition, Y are critical points since @f=@x1 ¼ @f=@x2 ¼ . . . ¼ @f=@xn ¼ 0½2. Then, the catastrophe manifold M is a set of the critical points such that
@f=@xijY ¼ 0; i ¼ 1; 2; . . . : ð4:2Þ Among Y, some locate at local maxima, some at local minima and even at points of inflexion of f. The values of f at Y are called critical values. Several examples are shown in Fig. 4.1. The critical point Y is structurally stable on manifold M if the determinant of the Hessian matrix [2]
HfjY ¼ @2@2f
@xi@xj
ð4:3Þ
is a nonzero (nonsingular) value, namely, det Hf 6¼ 0, while it is structurally unstable when det Hf ¼ 0. The critical point is nondegenerate if it is structurally stable, and degenerate if structurally unstable. For the nondegenerate critical point, the structural stability is not broken by adding an arbitrary, but sufficiently small, function gðxÞ of at least C2topology in the subspaceR1. The determinant of the Hessian matrix satisfies [2]
detHðfþ gÞ 6¼ 0: ð4:4Þ In contrast, all degenerate critical points are structurally unstable on the manifold M Rs. As an example, consider f ¼ x4; g ¼ "x2. The critical
point x ¼ 0 is degenerate with Hessian determinant zero. By adding g, the Hessian determinant takes a nonzero value. With " < 0, the potential function fþ g exhibits two local minima and one local maximum, as shown in Fig. 4.2. Thus fðxÞ is structurally unstable. The critical point x ¼ 0 corresponds to the cusp point in the cusp
catastrophe, as seen later. If one chooses f ¼ x4 cx2 ðc > 0Þ, then there are three critical points x ¼ 0; x ¼ ffiffiffiffiffiffiffic=2p . By adding g such as x or x2, the structural stability is not broken at two local minima, where the local
minima x ¼ ffiffiffiffiffiffiffic=2p shift slightly against the small perturbation, while the Hessian determinant keeps nonzero values. Then the potential function is
structurally stable against a sufficiently small perturbation. If f ¼ x4 cx3, there is a local minimum at x ¼ 3c=4 (nondegenerate) and a point of inflexion at x ¼ 0 (degenerate). Apparently, x ¼ 0 is structurally unstable, forming a cliff, while x ¼ 3c=4 is stable. A particle on the cliff begins to roll down suddenly against a small perturbation.
