The Escape of Particles from a Confining Potential Well
- PDF / 313,100 Bytes
- 6 Pages / 420.48 x 639 pts Page_size
- 95 Downloads / 168 Views
THE ESCAPE OF PARTICLES FROM ACONFINING POTENTIAL WELL JAMES P. LAVINE-, EDMUND K. BANGHART%, and JOSEPH M. PIMBLEY** *Microelectronics Technology Division, Eastman Kodak Company, Rochester, NY 14650-2008
"**Department of
Mathematical Sciences, Renssalaer Polytechnic Institute, Troy, NY
12180-3590 ABSTRACT Many electron devices and chemical reactions depend on the escape rate of particles confined by potential wells. When the diffusion coefficient of the particle is small, the carrier continuity or the Smoluchowski equation is used to study the escape rate. This equation includes diffusion and field-aided drift. In this work solutions to the Smoluchowski equation are probed to show how the escape rate depends on the potential well shape and well depth. It is found that the escape rate varies by up to two orders of magnitude when the potential shape differs for a fixed well depth. 1. INTRODUCTION Interest continues to focus on the escape rate of particles from a potential well. Kramers [1] established that the escape rate depends in an exponential fashion on the well depth and that increases in the well depth lead to decreases in the escape rate. He also showed that the potential well shape enters to first-order through the well curvature at the well bottom and at the well top. The well depth, or alternatively the barrier height of the potential well, affects the operation of electron devices such as charge-coupled devices [2] or quantumwell devices [3] and the rate of chemical reactions [4-6]. When the diffusion coefficient of the particle trapped in the well is small, the carrier continuity equation, i.e., the Smoluchowski equation, is used to study the escape rate [7]. This equation includes diffusion and field-aided drift with the field the negative derivative of the potential well. Solutions to the Smoluchowski equation are probed here to learn how the escape rate depends on the potential well shape and well depth. Several techniques lead to the dominant time dependence of the solutions of the Smoluchowski equation, that is, the c in e- t ITwith t the time [2]. Two of these approaches, eigenvalue determination and mean first passage time evaluation [4-6], are described in the next two sections. Numerical solution, closed-form expressions, and random walks are also utilized. Linear, rectangular, and polynomial potentials are investigated. All of these techniques are used in Section 4, where it is shown that t varies by up to two orders of magnitude when the potential well shape differs for a fixed barrier height. In addition, the variation is found to grow with the barrier height. Section 5 is a summary of the present findings. 2. THE SMOLUCHOWSKI EQUATION AND ITS EIGENVALUES Diffusion with field-aided drift is described by the Smoluchowski equation
an(x,t) = a._ [ D(x) an(x,t) + P x q n(x,t) aW(x)] &t
a×x
ix
kT
(1)
ik
Here n(x,t) is the particle density as a function of the spatial variable x and the time t, and D(x) is the diffusion coefficient of the particle. kT/q is the thermal voltage, W(x) is th
Data Loading...
