The ease with which electrons move through some metals depends on which direction they take. Watching electrons move along metal planes gives new insight, curiously, into how they move between them.
One of the fundamental issues in condensed-matter physics is conductivity, in other words how charge is transported through a material in response to an applied electric field. The essence of the conduction phenomenon in most materials is understood, thanks to work done from the 1930s to the 1950s by Bloch, Peierls and others. Over the past decade, however, the accepted wisdom has been challenged by measurements made on a wide range of 'strongly correlated metals', such as high-temperature copper-oxide superconductors, in which interactions between the conduction electrons dominate the physics.
On page 627 of this issue, T. Valla and colleagues (Nature 417, 627–630; 2002) use a pleasingly indirect method to obtain a new insight into the conductivity of strongly correlated metals. Their work concerns electrically anisotropic materials (high-temperature superconductors are again an example) in which current can flow more easily in some directions than in others. Such materials pose a particular challenge: the temperature dependence of the conductivity varies markedly depending on the direction of current flow. Above a certain temperature — typically in the range 50–200 K, depending on the material — one finds 'metallic' behaviour (that is, conductivity decreases as the temperature rises) for the highly conducting directions and 'insulating' behaviour (conductivity increases as the temperature rises) for the poorly conducting directions. At lower temperatures the temperature dependence is the same for both directions of current flow, although the magnitudes of the conductivities differ greatly.
The existence of different temperature dependences for different directions of current flow is difficult to understand within the conventional Bloch–Peierls picture, in which conductivity is expressed as the product of electron velocity and the 'transport time' (roughly, the mean time between electron-scattering events, which significantly degrade the current). Electron velocity is generically anisotropic but not temperature dependent, whereas the scattering time is generically temperature dependent but is not expected to depend strongly on the direction of current flow. The difficulty has stimulated much creative theoretical work but no generally accepted solution has yet emerged, in part because conductivity results were the only data that theorists had to work with.
Valla et al. have performed a photoemission experiment that sheds new light on this phenomenon: they studied two different layered materials whose crystal structure forms planes along which current can easily flow but in which the conductivity between planes is much smaller. The photoemission measurements involve shooting a beam of high-energy photons at a solid, then measuring the energy and in-plane momentum of emitted electrons. It seems remarkable that an experiment in which electrons are violently ejected from a solid can tell us something about the motion of electrons inside that solid. It is even more remarkable that Valla et al. have extracted important information about the ability of electrons to move between planes by studying the dependence of the photoelectron spectrum on in-plane electron momentum: Polonius's phrase from Hamlet — “by indirection find directions out” — comes to mind.
To understand what Valla et al. have shown, it is helpful to recall the basic theory of conductivity in metals. The quantum-mechanical effect of wavefunction hybridization enables electrons to move from one atom in a solid to another. In most circumstances, the timescale associated with this hybridization is very short, so that one can think of electrons as moving ballistically through the crystal. Metallic conductivity is then associated with the presence of reasonably well-defined electron quasi-particle excitations. These quasi-particles carry unit charge and move ballistically through the crystal until they are scattered into another state, and can for all practical purposes be thought of just as normal electrons.
The mean time during which a quasi-particle propagates ballistically defines the quasi-particle lifetime. In the absence of imperfections in the crystal lattice, this lifetime is set by scattering off thermally excited electron-density fluctuations or off phonons (fluctuations of the lattice), and usually it diverges as the temperature decreases towards absolute zero. The quasi-particle lifetime is in fact shorter than the transport time (although Valla et al. conflate the two) for two reasons. First, a quasi-particle might be scattered out of one state and into another that is moving in substantially the same direction. Such forward-scattering processes do not degrade the current much. Second, a scattering process might cause a quasi-particle to decay into a combination of states that is not a quasi-particle but carries the same, or almost the same, current. The one-dimensional electron gas provides an extreme example of this: it possesses no stable excitation with quantum numbers that match those of the electron (so the quasi-particle lifetime is very short), but the possible stable excitations include states known as charged bosons that can move over very large distances without scattering, leading to a transport time that diverges strongly as the temperature approaches zero.
In quasi-two-dimensional materials, the presence of reasonably well-defined quasi-particle states can be inferred from the appearance of dispersing peaks in photoemission spectra (see for example in Fig. 1 on page 628). By comparing photoemission spectra with the interplane conductivity, Valla et al. demonstrate a striking correlation: quasi-particle peaks were present only in the low-temperature regime in which the interplane conductivity has a metallic temperature dependence. As the temperature is raised, the quasi-particle peak disappears at approximately the temperature at which the interplane conductivity crossed over from metallic to insulating behaviour.
At one level this is not surprising. The disappearance of the quasi-particle peak means that the quasi-particle lifetime has become very short. So a carrier is scattered many times in one plane before it can make a transition to the next — this is known as the 'incoherent transport regime'. It seems plausible that this strong scattering could lead to a non-metallic interplane conductivity, although a quantitative comparison of the data with the various theories of this effect has not yet been undertaken.
But if, thanks to the work of Valla et al., the interplane conductivity now seems less mysterious, the in-plane conductivity has become more so. The presence or absence of the quasi-particle peak is seemingly irrelevant to the in-plane conductivity, whose metallic temperature dependence extends smoothly through the crossover regime at which the quasi-particle peak vanishes. It may be that the in-plane conductivity is dominated by quasi-particles not observed in the photoemission experiment. But it is more likely either that the scattering that destroys the quasi-particle is strongly peaked in the forward direction or, intriguingly, that the in-plane current is not carried by quasi-particles at all. Perhaps, to paraphrase Polonius, this measurement of incoherence has found a new type of coherence out.