Physica D 52 (1991) 458-476 North-Holland Gunn instability in finite samples of GaAs I. Stationary states, stability and boundary conditions Luis L. Bonilla Departamento de Estructura y Constituyentes de la Materia, Unicersidad de Barcelona, Diagonal 647, 08028 Barcehma, Spain and Francisco J. Higuera E. TS.L Aeronduticos, Unicersidad Polit~cnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Received 10 October 1990 Revised manuscript received 14 March 1991 Accepted 14 March 1991 Communicated by R. Westervelt The stationary states of a well-known phenomenological model of n-GaAs are characterized and constructed. Their dependence on realistic boundary conditions and their stability properties are analyzed. Two mathematical problems are studied corresponding to different biases. Under current bias, the total current is a known control parameter. Under voltage bias, the current is an unknown to be determined so as to keep the voltage constant. Under current bias, coexistence of multiple steady states is found for large enough samples. A theorem on stability under different bias conditions establishes, under appropriate conditions, a correspondence between critical values of the current and the voltage at which the basic stationary solution ceases to be stable. While under current bias, bifurcations from the stationary solution are branches of stationary solutions, under voltage bias bifurcating branches may be oscillatory. The consequences of this result for the bifurcation diagram of the Gunn instability are discussed. 1. Introduction: general picture Semiconductors in which spontaneous bulk current instabilities occur have been shown to exhibit a wide range of temporal oscillatory and chaotic behavior under suitable bias conditions, including period doubling and frequency locking routes to chaos in Ge [15, 23], GaAs [1], and InSb [19]. These phenomena are observed by measur- ing the current in the external circuit connected to the semiconductor. They are caused by the dynamics of nonlinear waves of electric charge inside the one-dimensional semiconductor and their interaction with the Ohmic contacts at its boundary. The simplest case seems to be that of the Gunn instability: a periodic oscillation of the current through a purely resistive external circuit under dc voltage bias. The oscillations are caused by the periodic generation of charge domains (solitary waves) at one contact, their uniform motion inside the semiconductor, and their anni- hilation at the other contact. Although the physics of the Gunn instability in, say, n-GaAs is well understood, we still lack: clear criteria for insta- bility of steady states (i.e. quantitative, as opposed to simple order of magnitudc N-L criteria [20, 21]), understanding of the effect of the contacts in the generation and annihilation of 0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)