When a material behaves differently depending on direction – conducting better along one line than across it, for example, much as wood splits easily along the grain but hardly across it. Some , such as black phosphorus, are strongly direction-dependent within the sheet itself.
Going deeper
Black phosphorus is puckered, so its two in-plane directions differ. In few-layer flakes, current flows and polarised light is absorbed most readily along the armchair direction, which lets the crystal axes be found optically.
Two kinds of direction dependence
Every layered crystal is strongly anisotropic between the in-plane and out-of-plane directions. Graphite conducts heat a few hundred times better along its layers than across them, and the same holds, less dramatically, for electrical conduction, stiffness and optical response.
Within the plane, symmetry decides. In a crystal with three-, four- or sixfold rotation symmetry – graphene, hBN, the – properties such as conductivity or linear absorption come out the same in every in-plane direction, although nonlinear signals like still show the rotational pattern. Lattices with lower symmetry, such as black phosphorus, ReS2, GeSe or CrSBr, have no such protection, and their in-plane properties differ along different crystal axes.
Black phosphorus as the example
Each black phosphorus layer is puckered into ridges. Along the armchair direction, across the ridges, carriers are lighter, so in conductance is higher that way; near the , light polarised along armchair is absorbed strongly and light polarised along zigzag hardly at all. That , together with polarised spectra, is how the axes of a flake are found.
The preferred directions need not agree. Heat flows more easily along zigzag than along armchair, so the best direction for current is not the best for removing heat. Calculations even suggest that in a single layer the anisotropy of can reverse compared with thicker flakes.
Measuring it without artefacts
In-plane anisotropy is measured by comparing directions on one flake: contacts placed around the edge at regular angles, polarisation-resolved absorption, Raman or , and angle-resolved second-harmonic generation. The crystal axes have to be identified first, because flakes cleave along arbitrary edges.
Device geometry can fake anisotropy. Contacts of different size or quality, non-uniform thickness and irregular flake shapes change the measured resistance along different paths, and from the can make an otherwise isotropic crystal slightly anisotropic. A convincing measurement shows the expected angular dependence across several directions on the same flake, not a single pair of numbers. – a preferred direction for magnetisation – is a related but separate idea with its own entry.
For specialists
Direction dependence of a property. All layered crystals are anisotropic between in-plane and out-of-plane directions; low-symmetry lattices such as black phosphorus, ReS2 and CrSBr are also anisotropic within the plane, with , optical absorption or differing along crystal axes.