Effective carrier mean-free path in confined geometries

The properties of transport in confined geometries has received substantial attention in recent years (see, e.g., Ref. 1). In this letter, the effects of boundary scattering on transport in small samples is examined. Our goal is to provide concise expressions for the effect of boundaries on transport without explicitly evaluating the Boltzmann equation. We present analytical expressions for the effective mean free path in samples where the bulk mean free path is determined by other scatterers present in the sample. Expressions are obtained for samples of circular and rectangular cross section. These results are applicable to samples which are small enough that the carrier mean free path is on the order of the sample dimensions but not so small that the carrier spectrum is substantially modified from the bulk. In other words, the sample dimensions will be assumed to be much greater than the carrier wavelength. We employ a method first proposed by Flik and Tier? for the calculation of the size effect in thin films. The method assumes that, for a carrier of a given frequency in a bulk sample, a characteristic mean free path, I, can be defined. The goal, then, is to examine how this bulk value of I is modified by the presence of boundaries in the sample. The calculation utilizes the concept of the exchange length I ex, 53 which is defined as the average distance normal to a plane that a carrier travels after having been scattered within that plane. Specifically, we consider a carrier that has undergone a scattering event within a plane that is perpendicular to the direction of net transport, which will be referred to from here on as the positive z direction. We now allow the carrier to propagate to the point of its next scattering event, which, in the bulk, is a distance I away. This propagation is assumed to proceed with equal likelihood in all directions. I,, is then defined as the average z component of all possible such propagation vectors, where the average is performed over the hemisphere in the positive z direction. The bulk value of this quantity, I, is l/2. In the following, we consider the scenario in which the mean free path is on the order of the sample dimensions. For this case, some carriers will strike the boundaries before traveling a full distance I and the exchange length will be correspondingly shorter. We will assume that scattering at the boundaries is diffuse, which will be valid when the carrier wavelength is smaller than the characteristic roughness features of the sample surface. This hypothesis needs to be examined within the context of a particular measurement, but holds for many materials. We note that a principle purpose of this work is to extend the most widely used form for the effects of boundary scattering on the thermal conductivity4 and in this form, perfect sample roughness is also assumed. We will also consider the sample to be free of grain boundaries, though the expression derived could perhaps also be applied to samples whose grains have characteristic geometries which match those investigated here. We first calculate the exchange length for axial transport in a cylindrical sample of infinite length. We initially assume that the excitation can originate with equal likelihood anywhere within a given circular cross section of the sample. The average value of the exchange length in the sample, I,, , is then obtained by averaging I,, (which is itself an average over a hemisphere of solid angle) over the entire cross section. The geometry to be considered is shown in Fig. 1. We consider an excitation originating at some point a distance p from the center of the cross section of radius R and propagating in a random direction within the hemisphere of solid angle whose base is normal to the positive z direction. The quantity 8 is defined as the angle between the propagation vector 1 and the z axis, and 4 is the angle between the radius along which the origination point is located and the projection of I into the plane of the cross section. Note that I may or may not have length Z, depending upon whether or not it is truncated by a boundary. The average exchange length is then given by