## 2 Ferromagnetic Semiconductor EuO  
The target material for this study is europium oxide (EuO) which is known as a representative ferromagnetic semiconductor. EuO, more generally, europium chalcogenides Eu\(X\) (\(X\) = O, S, Se, and Te), have long been attracting attention since its discovery in 1960s as magnetic semiconductors that possess both the electrical properties of semiconductors and the magnetic properties of magnets [14; 15]. Recently, a multitude of extreme properties in EuO (or slightly carrier doped EuO) such as nearly 100% spin polarization in the ferromagnetic state [16; 17; 18], drastic insulator-metal transition [19], colossal magnetoresistance [20; 21], pronounced linear and nonlinear magneto-optical effects [22; 23; 24; 25; 26],  
Figure 1: Characteristic energy and timescales of various magnetic phenomena. Timescales \(t\) are determined by the interaction energies \(E\) via the Heisenberg relation \(t=h/E\) (\(t\)(fs) \(\sim 4.13/E\)(eV)). Here, \(h=4.13\times 10^{-15}\) eV\(\cdot\)s is [PERSON]’s constant. The slowest dynamics is the domain-wall motion whose typical timescale is few ns to hundred \(\mu\)s. Precession of magnetization occurs within 10–100 ps and gets damped in sub-ns to tens of ns. Thus, the spin waves in ferromagnetic materials can propagate within few hundreds of ps to tens of ns before it gets damped out. Elementary interactions in magnetism such as the spin-orbit interaction (\(\sim\)10 fs–1 ps) and the exchange interaction (\(\sim\)10 fs) are related to faster dynamics. theoretical prediction of strain-induced ferroelectricity [27] have been attracting much attention from the viewpoint of basic science and application. In fact, its structural and electronic compatibility with Si, GaN, and GaAs make EuO a very attractive candidate for future semiconductor-based spintronics applications [28; 29; 30]. In the past, it was difficult to precisely control the sample quality due to undesirable oxidation and defects. With the recent developments of thin film fabrication technology, however, it is now possible to prepare high quality EuO films with well-controlled carrier density [31]. Undoped EuO is a typical Heisenberg ferromagnet with the Curie temperature \(T_{\mathrm{C}}\) of 69 K. The Eu\({}^{2+}\) ions on a centrosymmetric cubic rock salt structure (Figure 2a) have strongly localized 4\(f\)[7] electrons with \({}^{8}S_{7/2}\) as the ground state (\(S=7/2,L=0\)) and the saturation magnetic moment as large as 7 \(\mu_{\mathrm{B}}\) arises from the 4\(f\)[7] spins. Due to the strong localization of the 4\(f\) shell, the direct 4\(f\)[-4\(f\) exchange interaction with the nearest Eu\({}^{2+}\) neighbors is not enough to explain the \(T_{\mathrm{C}}\) of EuO. Instead, it has been predicted that the ferromagnetic order is driven by a virtual exchange mechanism [32]. According to this model, localized 4\(f\) electrons are virtually excited into the empty 5\(d\) orbital and create virtual magnetic excitons, and the excited 5\(d\) electron undergoes the intra-atomic 4\(f\)[-5\(d\) exchange interaction with 4\(f\) spins at the adjacent sites so that the magnetic exchange coupling \(J_{\mathrm{ex}}\) can be mediated via the spatially more extended 5\(d\) orbitals (Figure 2b). This indirect exchange interaction is considered to be stronger than the direct one. Therefore, it should be able to enhance the exchange interaction by truly populating the 5\(d\) orbitals. A typical way to enhance the exchange interaction is the introduction of carriers by chemical doping [31; 33]. For example, by substituting a part of Eu\({}^{2+}\) ions (Eu\({}^{2+}\): [Xe] 4\(f\)[7]) with magnetically equivalent Gd\({}^{3+}\) ions (Gd\({}^{3+}\): [Xe] 4\(f\)[7]), spatially spread excess 5\(d\) electrons are introduced and expected to enhance the exchange interaction (Figure 2c).