PHYSICAL REVIE%' B VOLUME 49, NUMBER 24 15 JUNE 1994-II Landauer formula for transmission across an interface B. Laikhtman Racah Institute of Physics, Hebrew University, Jerusalem 9190$, Israel S. Luryi ATHT Bell Laboratories, Murray Hill, ¹mJersey 0797$ (Received 14 June 1993; revised manuscript received 21 January 1994) We reconsider the classical problem of the quantum-mechanical resistance due to a quantum- mechanical reBection off a heterostructure interface. In the presence of a current, the electron distribution in the vicinity of the interface is different from that in the bulk due to the angular dependence of the re6ection coefBcient. The interface also modifies the electron concentration, leading to a violation of the local neutrality. This creates a self-consistent electric field and affects the angular distribution. The interface resistance depends on the actual form of the electron distribution. Incorporating all these factors in a kinetic transport model, we have reduced evaluation of the resistance to an integral equation. The equation is solved analyticaQy in the limits of either strong or weak renection. A simple model shows an appreciable difference between our results and the conventional Landauer approach. I. INTRODUCTION The Landauer resistance formula has been very helpful in describing mesoscopic devices where only a few scatter- ers are active and the resulting resistance is conveniently described by a scattering matrix. This approach is very difFerent from that employed in calculations of the re- sistivity of a medium with many random scatterers (e. g. , impurities or phonons). That approach, usually involving an average with respect to various possible arrangements of the scatterers, cannot be applied to the case of a small number of scatterers. It is also attractive to use the Landauer formula in heterostructure-interface-transmission problems. Indeed, a planar interface can be characterized by the trans- mission and refIection coefficients and one is naturally tempted to express the interface resistance in terms of these coefficients. Such an expression was originally ob- tained by Landauer. Assuming the zero-temperature Fermi distribution for electrons, he derived the follow- ing formula for the conductance per unit area of a planar scatterer: e~k2 1 — R(cos 8) cos 8 dO, 8vrs 5 R(cos 8) where R(cos8) is the reflection coefficient depending on the electron incident angle 0 and k is the Fermi wave vector. In the subsequent discussion Landauer considered the possibility that the electron distribution function was modified due to an interference between the scattering plane and the thermal scattering, leading to a deviation from Eq. (1). Landauer stated~ that such a deviation would imply a violation of Matthiessen's rule for resis- tivities. Precisely this situation takes place in heterostructure devices. The distance between the interface and the leads is often much larger than the electron mean &ee path. The distribution of electrons incident to the interface is formed by elastic scattering far kom it. Near the inter- face this distribution is changed for two reasons. First, the distribution is affected by the angular dependence of the re8ection and transmission coefficients. Second, an electric current across the interface leads to a violation of local neutrality. Due to re6ections, the electron con- centration is higher on the side of the incoming flux and lower on the other side. The resulting self-consistent field affects the electron distribution function. The latter ef- fect is linear in the electric 6eld and contributes to the Ohmic resistance. ~ 2 Therefore the Landauer formula, in which the re8ection and transmission coefficients of an interface are independent of any scattering in its vicin- ity, cannot be used in devices of the length larger than the electron mean free path. It needs some modi6cation. Such a modi6cation can be considered as the calculation of eH'ective reQection and transmission coefficients taking into account scattering in the vicinity of the interface. The purpose of the present paper is to discuss the nec- essary modi6cation of Landauer's formula. As an ex- ample, we calculate the resistance of a planar potential barrier, similar to that in a GaAs/Al Gaq As/GaAs heterostructure. In this case calculations are a bit sim- pler because the efFective masses and electron affinities on both sides of the interface are equal. For the one- dimensional case, such a problem has been studied by Eranen and Sinkkonen. The interface resistance prob1em has some simi1arity to a xnultichannel mesoscopic device. There are several def- initions of the multichannel device resistance which lead to different results (see Refs. 4 — 6 and references therein). In a degenerate electron gas only the angular distribution 0163-1829/94/49(24)/17177{8)/$06. 00 49 17 177 Q~ 1994 The American Physical Society