Conductance viewed as transmission Yoseph Imry Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel Rolf Landauer IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 Electric current flow, in transport theory, has usually been viewed as the response to an applied electric field. Alternatively, current flow can be viewed as a consequence of the injection of carriers at contacts and their probability of reaching the other end. This approach has proven to be particularly useful for the small samples made by modern microelectronic techniques. The approach, some of its results, and related issues are described, but without an attempt to cover all the active subtopics in this field. [S0034-6861(99)00102-6] CONTENTS essential to take the interfaces into account. Mesoscopic physics, concerned with samples that are intermediate in I. Conductance Calculated from Transmission S306 size between the atomic scale and the macroscopic one, II. Spatial Variation, Conductance Fluctuations, can now demonstrate in manufactured structures much Localization S308 of the quantum mechanics we associate with atoms and III. Electron Interactions S310 molecules. IV. Electron Coherence; Persistent Currents; Other When scattering by a randomly placed set of point Features S311 defects was under consideration, it quickly became cus- Acknowledgments S312 References S312 tomary in resistance calculations to evaluate the resis- tance after averaging over an ensemble of all possible defect placements. This removed the effects of I. CONDUCTANCE CALCULATED FROM TRANSMISSION quantum-mechanically coherent multiple scattering, which depends on the distance between the scatterers. Early quantum theories of electrical conduction were This approach also made the unwarranted assumption semiclassical. Electrons were accelerated according to that the variation of resistance between ensemble mem- Bloch’s theorem; this was balanced by back scattering bers was small. The approach made it impossible to ask due to phonons and lattice defects. Cross sections for about spatial variations of field and current within the scattering, and band structures, were calculated sample. Unfortunately, as a result, the very existence of quantum-mechanically, but the balancing process al- such questions, which distinguish between the ensemble lowed only for occupation probabilities, not permitting a average and the behavior of a particular sample, was totally coherent process. Also, in most instances, scatter- ignored. ers at separate locations were presumed to act incoher- Electron transport theory has typically viewed the ently. Totally quantum-mechanical theories stem from electric field as a cause and the current flow as a re- the 1950s, and have diverse sources. Particularly intense sponse. Circuit theory has had a broader approach, concern with the need for more quantum mechanical treating voltage sources and current sources on an equal approaches was manifested in Japan, and Kubo’s formu- footing. The approach to be emphasized in the following lation became the most widely accepted version. Quan- discussion is a generalization of the circuit theory alter- tum theory, as described by the Schro¨ dinger equation, is native: Transport is a result of the carrier flow incident a theory of conservative systems, and does not allow for on the sample boundaries. The voltage distribution dissipation. The Schro¨ dinger equation readily allows us within the sample results from the self-consistent pileup to calculate polarizability for atoms, molecules, or other of carriers. isolated systems that do not permit electrons to enter or The viewpoint stressed in this short note has been ex- leave. Kubo’s linear-response theory is essentially an ex- plained in much more detail in books and review papers. tended theory of polarizability. Some supplementary We can cite only a few: Beenakker and van Houten handwaving is needed to calculate a dissipative effect (1991), Datta (1995), Ferry and Goodnick (1997), Imry such as conductance, for a sample with boundaries (1997). There are also many conference proceedings and where electrons enter and leave (Anderson, 1997). After special theme volumes related to this subject, e.g., Sohn all, no theory that ignores the interfaces of a sample to et al., 1997; Datta, 1998. the rest of its circuit can possibly calculate the resistance It seems obvious that the ease with which carriers of such a sample of limited extent. Modern microelec- penetrate through a sample should be closely related to tronics has provided the techniques for fabricating very its conductance. But this is a viewpoint that, with the small samples. These permit us to study conductance in exception of some highly specialized limiting cases, cases where the carriers have a totally quantum me- found slow acceptance. That the conductance of a single chanically coherent history within the sample, making it localized tunneling barrier, with a very small transmis- S306 Reviews of Modern Physics, Vol. 71, No. 2, Centenary 1999 0034-6861/99/71(2)/306(7)/$16.40 ©1999 The American Physical Society Imry and Landauer: Conductance viewed as transmission S307 fied in Eq. (1.2) occurs at the connections to the reser- voir (Imry, 1986). Consider the left reservoir. Deep in- side that reservoir there is a thermal equilibrium population. In the 1D channel only the right-moving electrons are present. The effective Fermi-level, or ef- fective electrochemical potential, measures the level to which electrons are occupied. At point P in Fig. 1, deep FIG. 1. Two reservoirs on each side of a perfect tube, at dif- inside the left-hand reservoir, the electron distribution is ferent electrochemical potentials m1 and m2 . P is well inside that characteristic of thermal equilibrium with the Fermi reservoir 1; Q is in its entrance. level m1 . At point Q in Fig. 1, in the tapered part of the connection, the electron population shows some effect sion probability, is proportional to that probability was of the lowered density of electrons which have come out understood in the early 1930s (Landauer, 1994). In the of the right hand reservoir with electrochemical poten- case of a simple tunneling barrier it has always been tial m2 . Thus there is a potential difference between P apparent that the potential drop across the barrier is and Q. Along the ideal one-dimensional channel the localized to the immediate vicinity of the barrier and not electron population is equally controlled by both reser- distributed over a region of the order of the mean free voirs, and the electrochemical potential there must be path in the surrounding medium. The implicit accep- 1 2 (m11m2). Therefore the voltage drop specified by Eq. tance of a highly localized voltage drop across a single (1.2) is divided equally between the two tapered connec- barrier did not, however, readily lead to a broader ap- tors. The physics we have just discussed is essential. preciation of spatially inhomogeneous transport fields in Conductance can only be calculated after specifying the the presence of other types of scattering. The localized location where the potential is determined. The voltage voltage drop across a barrier has been demonstrated specification deep inside the reservoir and the geometri- with modern scanning tunneling microscopy (STM) cal spreading, are essential aspects of the derivation of probing methods (Briner et al., 1996). We return to spa- Eq. (1.2). Unfortunately, supposed derivations that ig- tial variations in Sec. II. nore these geometrical aspects are common in the litera- Figure 1 shows an ideal conducting channel with no ture. irregularities or scattering mechanisms along its length. If we insert an obstacle into the channel, which trans- A long perfect tube is tied to two large reservoirs via mits with probability T, the current will be reduced ac- adiabatically tapered nonreflecting connectors. Carriers cordingly, and we find approaching a reservoir pass into that reservoir with cer- 2 tainty. The reservoirs are the electronic equivalent of a G5~e /p\!T. (1.3) radiative blackbody; the electrons coming out of a res- Note that it does not matter whether the T in Eq. (1.3) is ervoir are occupied according to the Fermi distribution determined by a single highly localized barrier or by a that characterizes the deep interior of that reservoir. As- more extended and complex potential profile. Expres- sume, initially, that the tube is narrow enough so that sions for the conductance with this same current, but only the lowest of the transverse eigenstates in the chan- with the potential measured within the narrow channel, nel has its energy below the Fermi level. That makes the on the two sides of the obstacle, also exist (Sec. 2.2 of channel effectively one-dimensional. Take the zero tem- Datta, 1995; Sec. 1.2.1 of Ferry and Goodnick, 1997, perature case and let the left reservoir be filled to up to Chap. 5 of Imry, 1997). If, in that case, the potential is level m1 , higher than that of the right-hand reservoir, averaged over a region long enough to remove interfer- m2 . Then in the range between m1 and m2 we have fully ence oscillations, then T in Eq. (1.3) is replaced by occupied states pouring from left to right. Thus the cur- T/(12T). rent is The preceding discussion can easily be generalized to a channel that involves more than one transverse eigen- j52~m12m2!ev~dn/dm!, (1.1) state with energy below the Fermi level (Imry, 1986). In where dn/dm is the density of states (allowing for spin that case we utilize the transmission matrix t of the scat- degeneracy) and v is the velocity component along the tering obstacle, which specifies the transmitted wave tube at the Fermi surface. Now (m12m2)52e(V1 functions relative to the incident wave, utilizing the 2V2), where V is a voltage and e the magnitude of the transverse eigenstates of the channel as a basis.