Shine intense light on a crystal and a little of it comes back at exactly twice the frequency – half the wavelength, which is how a green laser pointer turns invisible infrared into green light. It happens only if the crystal has no centre of symmetry, which makes it a quick, contactless test of symmetry, layer number and crystal orientation.
Going deeper
Second-harmonic generation is forbidden where a crystal has a centre of symmetry, so in 2H TMDCs odd layer numbers shine and even ones stay dark. Rotating the polarisation traces a six-lobed pattern locked to the crystal axes, which reads orientation and, in a stack, the twist angle.
Why symmetry decides
A strong light field drives the electrons in a crystal slightly anharmonically, so the induced polarisation contains a term proportional to the square of the field, oscillating at twice the frequency. That term changes sign under inversion, while the crystal does not if it is – so in such a crystal the second-order response must vanish, at least in the electric-dipole approximation that dominates the signal.
In layered materials this makes the signal a symmetry meter. A has no inversion centre and generates strongly; a bilayer regains one and goes nearly dark; a trilayer shines again. Rhombohedral 3R stacking never regains a centre, so its signal keeps growing with thickness. Residual signals in centrosymmetric samples come from surfaces, edges and higher-order terms, so “dark” means much weaker rather than exactly zero.
What the pattern tells you
Rotating the polarisation of the incoming light, or the sample, and recording the signal parallel to it traces a six-lobed pattern whose maxima line up with the armchair directions of the lattice. That fixes the crystal orientation of a without touching it, which matters for materials, for aligning edges and for cutting devices along known directions.
In a stack of two layers the patterns of the two add as amplitudes with a phase difference set by the twist, so the combined signal reads the – the standard quick check for before more demanding measurements. Intensity also responds to and to layer number, and because it is a contactless optical measurement it works through and at speed over a whole flake.
Where else it is used
Anything that breaks inversion symmetry can in principle be seen. domains give signals of opposite phase, so second-harmonic imaging maps them and follows switching. Some break inversion combined with time reversal, and second-harmonic generation has been used to detect antiferromagnetic order in layered magnets where no net moment exists to measure.
Practical care: the signal scales with the square of the intensity, so pulsed lasers are used and damage is a real risk for thin, flakes. Absolute susceptibilities are hard to compare between laboratories because they depend on pulse length, focusing and reference standards; ratios measured on one setup are far more reliable than absolute values.
For specialists
A second-order nonlinear response, forbidden in centrosymmetric media, so in 2H TMDCs it appears for odd layer numbers and vanishes for even ones. The polarisation dependence gives crystal orientation, and so the twist angle in a stack; the intensity follows layer number and strain; and it detects broken inversion symmetry in ferroelectric and magnetic layers where no other quick probe exists.