Numerous multiparticle phenomena such as the Stoner ferromagnetic instability [1, 2], Fermi-liquid renormalizations of the mass and spin susceptibility of conductance electrons [3–5], the collapse of the exchange energy, and magnetoexcitons with unconventional properties [6] occur even in the simplest quantum Hall states with integer filling factors in two-dimensional electron systems (2DESs) with strong interaction. Nevertheless, the spin configuration of states of the integer quantum Hall effect is collinear and is determined by the filling of spin Landau levels (LLs) with electrons or quasiparticles. At deviation from ferromagnetic Hall states, the magnetic order becomes topologically nontrivial. It is known that the interplay between the exchange and Zeeman interactions in 2DESs can result in the formation of skyrmion spin textures [7]. Their existence in, e.g., high-quality GaAs 2DESs is experimentally confirmed both by the fast spin depolarization of 2DESs [8, 9] and by the appearance of an additional spin mode [10, 11], which indicates the breaking of spin rotation symmetry in the system, in the spectrum of low-frequency collective excitations. In ZnO and AlAs two-dimensional systems, which are almost similarly perfect, skyrmion textures are energetically unfavorable because of large Zeeman gaps and a suppressed scale of the exchange energy [12]. At the same time, due to enhanced Coulomb correlations, spin and orbital splittings between quasiparticle LLs are renormalized and they can depend on the filling factor, the compressibility of states, and the Coulomb mixing LLs. The convergence and crossing of different spin LLs can be accompanied by the transformation of the spin order between Hall ferromagnets whose conditions in strongly interacting 2DESs can be strongly distorted depending on the filling factor. The magnetotransport study of MgZnO/ZnO structures with the strongly interacting 2DES (the parameter rs ~ 8) revealed a nonmonotonic dependence of the critical tilt angle of the magnetic field on the filling factor [13]. It includes segments with both sharp switching of the spin polarization (e.g., at 1.7 < ν ≤ 2) and with a qualitatively different character manifested in the form of smeared regions on the resistance map at ν ~ 1.2–1.3. Later inelastic light scattering experiments [14] on similar samples under the same conditions ν ~ 1.2 ± 0.15 demonstrated extraordinary properties of the spectrum of low spin excitations with an additional low-frequency mode. This indicates the formation of a noncollinear spin order in the system, as previously shown for spin textures of the 2DES in GaAs [10, 11]. Thus, the transformation of the spin order at fractional ν values can occur through qualitatively different scenarios, which are not explained only by the renormalization of the Fermi-liquid parameters but depend on residual Coulomb correlations between quasiparticles. In this work, the evolution of the spin order in the strongly interacting 2DESs at 1 ≤ ν ≤ 2 is studied under the variation of the key parameters determining the energy scales and hierarchy of LLs—the density of the 2DES, the filling factor, and the tilt angle of the magnetic field—by the inelastic light scattering scanning of the spectrum and dispersion of lowest spin excitations.

Inelastic light scattering studies with controlled momentum transfer were performed on high-quality MgZnO/ZnO heterostructures grown by molecular beam epitaxy [5]. The carrier density and mobility in the 2DES were in the ranges of (1.14–2.85) × 1011 cm–2 and (4–7) × 105 cm2/(V s), respectively. A semiconductor heterostructure was photoexcited at photon energies of about 3.4 eV near direct optical band gap of ZnO by a wavelength-tunable Ti:sapphire laser with resonant frequency doubling [15]. Scattered light was analyzed by a spectrometer in combination with a liquid nitrogen cooled CCD camera. The momentum transfer in the process of inelastic light scattering was unambiguously specified by the angular configuration of waveguides approaching the sample and was in the range of (1.0–3.0) × 105 cm–2. A cryostat with the evacuation of He3 vapors and a working temperature of ~0.35 K was used for the low-temperature experiment. Samples were placed on a rotary table, which allowed the variation of the sample tilt angle in the magnetic field with an accuracy of about 0.5°, and the maximum magnetic field was 15 T. Spectral lines of inelastic light scattering were identified against the photoluminescence signal background by their fixed energy shift with respect to the laser energy position.

We focused on inelastic light scattering spectra on spin excitons (SEs) within partially filled LLs. Figure 1a presents characteristic spectra on SEs at a nonzero momentum k and at three filling factors with qualitatively different types of the spin order. The collinear Hall ferromagnet at ν = 1 supports a single Larmor SE (left spectrum) with the energy above Ez due to k dispersion. An additional low-energy spin mode with E < Ez (middle spectrum) appears in the spectrum in the range of \(1.1 \lesssim \nu \lesssim 1.3\), which indicates the formation of the noncollinear spin order. Figures 1b and 1c show the dependences of the differences between the spin mode energies and the single-particle Zeeman splitting on the filling factor in the range of 1 ≤ ν ≤ 2. Anticrossing is observed between spin modes denoted as SEhi and SElo. It was previously shown [14] that repulsion between modes has a Coulomb character: it increases with the momentum transfer and the density of the 2DES. Such a picture of excitations has a unidirectional character at the deviation of the filling factor from ν = 1, which is due to the specificity of filling of next spin LLs with electrons. Finally, when approaching ν = 2, the picture can be qualitatively diverse: either a smooth evolution of the SE with E = Ez up to its disappearance in the paramagnetic phase is observed (Fig. 1c) or the energy and intensity of the SE change sharply at a certain filling factor ν = νFMT (right part of Fig. 1a and Fig. 1b), indicating the transition of the 2DES to the ferromagnetic phase. The scheme of the filling of quasiparticle LLs in both phases is shown in the right parts of Figs. 1b and 1c (the 0↑ and 1↓ levels are inverted). The inversion of the 0↑ and 1↓ levels can be achieved both by increasing the tilt angle of the magnetic field (case shown in Fig. 1b) and by reducing the density of the 2DES (due to the renormalization of the spin susceptibility).

Fig. 1.
figure 1

(Color online) (a) Inelastic light scattering spectra on spin excitons at various filling factors and experimental parameters presented in panel (b). (b, c) Multiparticle contributions to the energy of the spin exciton (SE) versus the filling factor at various tilt angles of the magnetic field. The filling factor ν* marks the position of the minimum in the splitting Δ of two modes. The filling factor νFMT corresponds to the ferromagnetic transition. Spin orders in various phases are illustrated on diagrams in terms of filling of quasiparticle Landau levels.

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In the case of the strong mixing of LLs and at the filling factor ν = 1+, the hierarchy of spin levels is already inverted [13, 14], and the filling of the quasiparticle 1↓ LL begins (left part of Fig. 1c). In the intermediate filling factor range near ν ~ 1.2, the 0↑ and 1↓ spin levels are almost degenerate, the energy splitting between them δ can be considered as small compared to the exchange energy Σ, and spin textures are thereby energetically favorable (schematically shown in the middle diagram in Fig. 1c).

The picture of collective excitations changes significantly with an increase in the tilt angle of the magnetic field. The ferromagnetic phase transition is shifted towards the filling factors νFMT < 2 at tilt angles above a certain critical value (Θν = 2). The dependence of νFMT on the tilt angle for one of the samples is shown by circles in Fig. 2a. However, the anticrossing of the SEhi and SElo spin modes has the fixed central filling factor ν* ≈ 1.2 up to much larger tilt angles. The mode splitting Δ vanishes sharply beginning with the critical angle Θс1 (see inset of Fig. 3), and the central filling factor ν* is shifted beginning with the angle Θс2 > Θс1 (see squares in Fig. 2a). This result is in quantitative agreement with magnetotransport studies of a sample close in parameters [13]. The angular evolution of νFMT at Θ > Θν = 2 is manifested as a sharp boundary in the phase diagram (Fig. 2b) near 1.7 < ν < 2. Any sharp boundary is absent on the side ν < 3/2; a smeared vertical band is instead observed in a certain angular range, which means a smooth change in the spin polarization at ν ~ 1.2–1.3. the current experiment showed that this transformation occurs through the formation of spin textures. A similar picture was also observed for other samples. Figure 3 presents the electron density dependences of the critical angles Θс1, Θс2, and Θν = 2.

Fig. 2.
figure 2

(Color online) (a) Filling factors of (squares) the anticrossing point ν* and (squares) the ferromagnetic transition νFMT versus the tilt angle of the magnetic field. (b) Magnetotransport map for the ZnO sample with ns = \(2.3 \times {{10}^{{11}}}\) cm–2 taken from [13] and reproduced with the inverted spin directions (the g-factor with the opposite sign was used in [13]).

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Fig. 3.
figure 3

(Color online) Cosines of the critical angles Θс1 and Θс2 and the angle of the ferromagnetic transition at ν = 2 versus the density of the two-dimensional electron system. The angular range Θс1 < Θ < Θс2 is given in gray. The oval marks the parameter region where the incompressible state of the fractional quantum Hall effect ν = 3/2 was detected in [13]. The inset shows the parameter Δ/k versus the tilt angle of the magnetic field, where Δ is the splitting of spin modes and k = (black symbols) (2.2–2.62) × 105 and (red symbols) (1.5–1.96) × 105 cm–1, on one of the samples.

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The anticrossing of spin modes at ν < 3/2 and the sharp ferromagnetic transitions are no longer identified at angles larger than Θс2, but two corresponding inflection points on the dependence of the SE energy on the filling factor ν remain at not too large angles (e.g., 43° and 45° in Fig. 4). Even if the fine structure of SE lines exists in these regions, it is not resolved, and only some broadening of inelastic light scattering peaks is observed. In the filling factor region \(1 \leqslant \nu \lesssim 1.3\) expanding with increasing angle, the multiparticle contribution to the SE energy is positive, is almost constant, and corresponds to the stable collinear ferromagnetic phase. In the range \(1.7 \lesssim \nu < 2\), the SE energy also has a stable negative Coulomb shift corresponding to the two-component Hall ferromagnet ν = 2 (as in [6]). Two extreme filling factors in this angular dependence (Fig. 2a) are close to ν ≈ 4/3 and ν ≈ 5/3, which can indicate a specific role of these states of the fractional quantum Hall effect in the transformation of the spin order (indications also exist on the magnetoresistance map taken from [13] and reproduced in Fig. 2b). In contrast to the stable behavior in two extreme ferromagnetic phases, the SE is sharply rearranged near ν ≈ 3/2: its energy loses the exchange Coulomb component and becomes equal to the Zeeman energy (Fig. 4). This evidences that the paramagnetic order still holds near ν = 3/2 at these angles. Finally, with a further increase in the tilt angle of the magnetic field (Θ = 52° and 60° in Fig. 4), the SE energy has a positive exchange component in the gradually expanding range covering the region 1 < ν ≤ 3/2; i.e., the classical Hall ferromagnet of a Heisenberg type with a positive spin stiffness is energetically favorable. This is also manifested in the accompanying decrease in the magnitude of the negative dispersion shift of SE near ν = 2 (from the comparison of data for Θ = 52° and 60° at the same momentum). It was technically difficult to measure curves at larger angles, but the revealed trend to the positive spin stiffness of the Hall ferromagnet is obviously expectable at an increase in the tilt angle and the separation of 0↑ and 1↓ spin LLs from each other.

Fig. 4.
figure 4

(Color online) Multiparticle contribution to the energy of spin excitons versus the filling factor at angles exceeding Θс2. The momenta and densities of the two-dimensional electron system are indicated. The dashed rectangles mark the regions, where the spectral peaks of spin excitons are broadened but the fine structure is not resolved.

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The described signatures of spin transformations at fractional filling factors ν are inconsistent with the commonly accepted picture of crossing of quasiparticle LLs. First of all, the dependences of all discussed critical tilt angles of the magnetic field on the filling factor (e.g., on the magnetoresistance map with a strongly nonmonotonic boundary at 1 < ν < 2) are related to the corresponding nonmonotonic change in the renormalized spin susceptibility g*m*. Its values are different even for integer ν = 1 and 2, because the order of the 0↑ and 1↓ LLs for all studied samples in the limit of low tilt angles Θ is direct at ν = 2 but is inverted at ν = 1. Differences appear in details of the dielectric polarizability of the 2DES in these states of the quantum Hall effect, which affects the character of Fermi-liquid renormalizations [16]. In the case of partially filled LLs, there is an additional mechanism of the metal screening of the Coulomb interaction fur to intralevel density fluctuations and, as a result, the renormalization of the spin susceptibility of the Fermi liquid g*m* is weaker than that at integer filling factors ν. As a result, the crossing of LLs with half-integer filling factors ν requires a larger slope of the magnetic field than that in the case of integer filling factors.

Even taking into account the nonmonotonic dependence of g*m*, the simplified model of noninteracting Fermi-liquid quasiparticles does not describe the system at fractional filling factors ν. Residual correlations between quasiparticles are responsible for qualitatively different spin transformations at the crossing of LLs: a sharp avalanche spin-flip process occurs at ν = νFMT, whereas a smooth transformation of the spin order through the formation of textures proceeds near ν*. Furthermore, up to the critical angle Θс2, some mechanism pins the formation of the noncollinear spin order to the filling factor ν* ≈ 1.2 (Fig. 2a), and ν* begins to increase only after the collapse of textures. With a further increase in the angle, the size of spin textures apparently decreases significantly due to the difference between the energies of the 0↑ and 1↓ LLs. Detailed mechanisms of these spin transformations are unclear; therefore, it would be extremely relevant to theoretically describe the spin configuration of the ground state of the 2DES at least in terms of interacting quasiparticles on renormalized LLs.

For one of the measured heterostructures with the electron density n = 2.23 × 1011 cm–2, the angular range of the collapse of spin textures Θс1 < Θ < Θс2 (marked by the ellipse in Fig. 3) coincides with an accuracy of arc minutes with the region where the incompressible state of the quantum Hall effect at ν = 3/2 was detected in magnetotransport experiments (quantization of the resistance was observed in [13] at T ~ 20 mK).

To summarize, the evolution of the spin order at fractional filling factors 1 ≤ ν ≤ 2 in strongly interacting MgZnO/ZnO two-dimensional electron systems has been studied in terms of the behavior of low-energy spin excitations in inelastic light scattering spectra. A sharp ferromagnetic transition has been observed at \(\nu \gtrsim 1.7\) in the dependence of the tilt angle of the magnetic field and has been accompanied by the jump in the energy of the spin exciton. A qualitatively different transformation of the spin order has been observed at ν < 3/2: in the region \(1.1 \lesssim \nu \lesssim 1.3\), the excitation spectrum contains not only the Larmor spin excitons but also an additional spin mode with the energy below the Zeeman one, which indicates the breaking of the spin rotational symmetry in the system (formation of spin textures). This noncollinear spin configuration in the system exists up to a certain critical angle depending on the density of the two-dimensional electron system. The dependences of the critical angles of disappearance of spin textures on the electron density have been obtained. At larger angles, signatures of spin textures and the ferromagnetic transition disappear, and the system passes to the ferromagnetic state in a wide filling factor region. It has been shown that the magnetic order formed under these conditions is symmetric in the filling factor with respect to the value ν = 3/2, where the spin stiffness is zero.