How strongly neighbouring sheets in a stack affect one another. The pull holding them together is weak, but electrons can still hop from one sheet to the next and whole sheets can vibrate against each other – which is why one, two and many layers of the same crystal behave differently, and why twisting or sliding one sheet changes the whole stack.
Going deeper
Left: electrons in orbitals that reach out of the plane can hop to the next layer, so a level of a single sheet splits in two when a second sheet is added – the reason the band structure changes with the number of layers. Right: whole layers also vibrate against each other, sliding (shear) or moving apart and together (breathing), at frequencies that measure how stiffly they are coupled.
Weak glue, real consequences
The layers of a crystal are held together by forces of a few tens of millielectronvolts per atom, around a hundredth of the energy of a bond inside a layer. That is why they peel apart. Weak is not the same as absent, though. Electrons in orbitals that point out of the plane – the p orbitals of carbon in graphene, those of sulfur and the out-of-plane d orbital of molybdenum in MoS2 – overlap with the next layer, and that overlap lets them hop between layers with an energy of a few tenths of an electronvolt, about 0.4 eV in graphite.
Hopping splits and shifts the states that take part, and those are often the ones at the . Adding layers to MoS2 pushes up the valence-band top at Γ, built from out-of-plane orbitals, until it overtakes the one at K, and the direct gap of the turns indirect. Two graphene layers in Bernal stacking have parabolic bands instead of . The number of layers is a knob on the precisely because of this coupling.
Stacking, twist and sliding
How strongly two layers couple depends on how their atoms sit over each other. In bilayer MoS2 the high-symmetry stackings bring the layers closest and couple them most, lowering the gap the most; at any other the layers sit slightly further apart and couple more weakly, with a gap that barely depends on the angle. A twist also makes the coupling vary from place to place, following the , and at small angles the lattices relax into domains of the most favourable stacking separated by narrow walls.
The same coupling carries magnetism and polarisation from layer to layer. Whether neighbouring magnetic layers align parallel or antiparallel depends on their stacking: thin CrI3 couples its layers antiferromagnetically although the bulk crystal is , because thin keep a different stacking. Sliding one layer by a fraction of a cell changes the coupling too, which is the basis of .
Measuring it
The most direct measure is mechanical. Whole layers vibrate against each other like masses joined by springs – sliding past each other in shear modes, moving apart and together in breathing modes – at a few tens of wavenumbers, low enough that spectrometers need special filters to reach them. Because the layers are simply masses on springs, the frequencies follow a chain model as layers are added and give the interlayer force constant directly: in graphene the shear mode rises from about 31 cm−1 in a bilayer to about 43 cm−1 in graphite, which also makes it a quick way to count layers.
The electronic side shows up as band splittings in , as the shift of the indirect peak in bilayers, and as in . In calculations the coupling depends steeply on the , and plain semi-local density functionals get that distance wrong, so a is needed before any computed coupling can be trusted.
For specialists
The electronic hybridisation, mechanical force constants and electrostatic interaction between adjacent layers of a van der Waals crystal. Electronically it is hopping between orbitals that reach out of the plane, a few tenths of an electronvolt in graphite, which splits and shifts bands and makes the band structure depend on thickness, stacking and twist. Mechanically it sets the interlayer shear and breathing modes, whose Raman frequencies of a few tens of cm−1 measure it directly. Far weaker than in-plane bonding, it still decides the direct–indirect gap crossover of , the of twisted bilayers and interlayer magnetic order.