Three independent contributions to the magnetically induced spontaneous polarization of multiferroic TbMn 2 O 5 are uniquely separated by optical second harmonic generation and an analysis in terms of Landau theory. Two of the contributions are related to the magnetic Mn 3+/4+ order and are independent of applied fields µ 0 H x of up to ±7 T. The third contribution is related to the long-range antiferromagnetic Tb 3+ order. It shows a drastic decrease upon the application of a magnetic field and mediates the change of sign of the spontaneous electric polarization in TbMn 2 O 5 . The close relationship between the rare-earth long-range order and the non-linear optical properties points to isotropic Tb 3+ -Tb 3+ exchange and O 2− spin polarization as mechanism for this rare-earth induced ferroelectric contributions.
4-6A characteristic feature of the magnetically induced ferroelectrics is their complexity.Various mechanisms, demanding in themselves, cooperate for promoting the spontaneous polarization: geometric frustration, incommensurate spin order, low crystallographic symmetry, and electronic 3d − 4f interactions may all play a role. This inevitably results in rich magnetoelectric phase diagrams with a multiplicity of contributions to the ferroelectric polarization P . The situation culminates in compounds like TbMn 2 O 5 which hosts three magnetic subsystems and five different magnetically ordered phases. This leads to a complex temperature-and field-dependence of the magnetically induced polarization, including a sign change of P induced by a moderate magnetic field.3 Phenomenological two-and threepolarization models were proposed to explain such features and the 3d − 4f interplay of the manganese and the terbium sublattices was scrutinized. The four-dimensional irreducible representation associated with k leads to a ferroelectric polarization P y net along the y axis when δ x becomes zero in a second-order phase transition at T C ≡ T 2 = 38 K. Landau theory predicts a temperature dependence of the polarization according toIn a subsequent second-order phase transition at T 3 = 33 K commensurate magnetic lockin with δ x = δ z = 0 occurs. The predicted temperature dependence of the ferroelectric polarization is now described by the expressionHere, ε is a constant responsible for a change of slope of P y net (T ) at T 3 . In a first-order phase transition at T 4 = 22 K the magnetic order of TbMn 2 O 5 becomes incommensurate again with δ x = 0, δ z = 0. The first-order phase transition is parametrized by the emergence of now two four-dimensional order parameters whose coupling gives rise to an additional polarization contribution P y II withBecause of the pronounced decrease of P y net observed at T 4 the net polarization is expressed aswith ε 1,2 as proportionality factors. Finally, a second-order magnetic phase transition is observed at T 5 ≈ 10 K. Optical SHG describes the induction of a light wave at frequency 2ω by an incident light wave at frequency ω.16,17 This is expressed as S i (2ω) = ǫ 0 χ ij...