Thermodynamic evidence of a second skyrmion lattice phase and tilted conical phase in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

Precision measurements of the magnetization and ac susceptibility of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ are reported for magnetic fields along different crystallographic directions, focusing on the border between the conical and the field-polarized state for a magnetic field along the $\langle 100\rangle$ axis, complemented by selected specific heat data. Clear signatures of the emergence of a second skyrmion phase and a tilted conical phase are observed, as recently identified by means of small angle neutron scattering. The low-temperature skyrmion phase displays strongly hysteretic phase boundaries, but no dissipative effects. In contrast, the tilted conical phase is accompanied by strong dissipation and higher-harmonic contributions, while the transition fields are essentially nonhysteretic. The formation of the second skyrmion phase and tilted conical phase are found to be insensitive to a vanishing demagnetization factor. A quantitative estimate of the temperature dependence of the magnetocrystalline anisotropy may be consistently inferred from the magnetization and the upper critical field and agrees well with a stabilization of the low-temperature skyrmion phase and tilted conical state by conventional cubic magnetic anisotropies.

Figure 1

Magnetic phase diagrams inferred from magnetization and susceptibility data measured in field sweeps after zero-field cooling (a)–(c) and high-field cooling (d) with fields along $\langle 111\rangle$ (a), $\langle 110\rangle$ (b), and $\langle 100\rangle$ [(c) and (d)]. The following phases may be distinguished: helical (H), conical (C), field-polarized (FP), tilted conical (TC), high-temperature skyrmion (HTS), and low-temperature skyrmion (LTS) phases.

Figure 2

Typical magnetization and susceptibility data as a function of internal field following zero-field cooling with fields aligned along $\langle 111\rangle$ (blue), $\langle 110\rangle$ (green), and $\langle 100\rangle$ (red). Data were recorded at $T=57.5\mathrm{K}$ (left column) and $T=2\mathrm{K}$ (right column). [(a) and (e)] Magnetization data. [(b)–(d) and (f)–(h)] Susceptibility calculated from the magnetization, $dM/dH$ (symbols), the real part of ac susceptibility ${\chi }^{\prime }$ (black line), and the imaginary part of ac susceptibility ${\chi }^{″}$ (colored line) for a field along $\langle 111\rangle$ [second row: (b), (f)], a field along $\langle 110\rangle$ [third row: (c), (g)], and a field along $\langle 100\rangle$ [fourth row: (d), (h)].

Figure 3

Typical magnetization and susceptibility data at $T=2$ K following high-field cooling. Data were recorded in a field sweep from left to right. Data are shown as a function of internal field along $\langle 111\rangle$ (blue), $\langle 110\rangle$ (green), and $\langle 100\rangle$ (red). (a) Magnetization as a function of magnetic field under HFC. (b)–(d) Susceptibility calculated from the magnetization, $dM/dH$ (dots), the real part of ac susceptibility ${\chi }^{\prime }$ (black line), and the imaginary part of ac susceptibility ${\chi }^{″}$ (colored line) for fields along $\langle 111\rangle ,\langle 110\rangle$, and $\langle 100\rangle$.

Figure 4

Comparison of small angle neutron scattering intensity with the susceptibility as a function of internal magnetic field. Data recorded under HFC and ZFC are shown on the left- and right-hand side, respectively. (a1)–(a5) Typical neutron scattering intensity distribution in reciprocal space for the five different modulated phases as denoted in each panel. The scattering plane is marked in gray shading. [(b1) and (c1)] Neutron scattering intensity of the helical and conical state. [(b2) and (c2)] Neutron scattering intensity of the tilted conical and LTS phase. [(b3) and (c3)] Susceptibility as calculated from the magnetization, $dM/dH$ (dots), and the real part of the ac susceptibility ${\chi }^{\prime }$ (black line). [(b4) and (c4)] Imaginary part of the ac susceptibility ${\chi }^{″}$.

Figure 5

Higher-harmonic contributions in the ac susceptibility as a function of field at selected temperatures for field along the $\langle 110\rangle$ and $\langle 100\rangle$ direction. The set of panels on the left corresponds to $\mathbf{H}\parallel \langle 110\rangle$; the set of panels on the right corresponds to $\mathbf{H}\parallel \langle 100\rangle$. Rows display the first, second, and third harmonic. Columns correspond to different temperatures as marked in the plots. Data have been recorded following high-field cooling in a field sweep from negative to positive field values. The negative branch of the $\langle 110\rangle$ data is not shown for clarity, as it provides no further information.

Figure 6

Magnetization $M$, calculated susceptibility, $dM/dH$, as well as real and imaginary part of the ac susceptibility, ${\chi }^{\prime }$ and ${\chi }^{″}$, respectively, for different sample shapes with different demagnetization factors and field along $\langle 100\rangle$. The demagnetization factors are denoted above each column (see Table 1 for further details). Data were recorded following high-field cooling in a single field sweep, where data recorded at negative fields are mirrored to positive field values. Data branches recorded under decreasing and increasing magnitude of the field are shown in blue and red shading, respectively.

Figure 7

Magnetic phase diagrams as a function of demagnetization factor $N$ and applied magnetic field $H$ applied along $\langle 100\rangle$. Note that different units are used for $H$ in the theoretical calculations as compared with experiment. (a) Theoretical phase diagram obtained from a Ginzburg-Landau model in the presence of the cubic anisotropy of Eq. (6), as described in Ref. [16]. Here we assumed that only single-domain states exist. We also display the metastable tilted conical state in the regime where its energy is higher than the LTS phase but lower than both the conical and ferromagnetic phase. It vanishes for small $N_z$. Parameters and units correspond to those used in Ref. [16]. (b) Phase diagram of (a) when including the exchange anisotropy $c{\sum }_i{\left({\nabla }_i{M}_i\right)}^2$ with $c=-0.06$, as described in the main text. The metastable tilted conical state exists even for $N_z\to 0$. (c) Thermodynamic phase diagram [parameters as in (a)] including regions where demagnetization effects stabilize phase coexistence, see main text. Note that the experimental regions of phase coexistence can be much larger due to hysteresis effects in the region where phases are metastable. (d) Summary of the experimental critical fields for samples with different demagnetization factor $N$, as detailed in Fig. 6, with data for negative fields mirrored to positive fields.

Figure 8

Magnetization (first row) and specific heat (second row) as a function of internal field at 2 K for all major directions. Data were recorded following high-field cooling in a single field sweep from negative to positive field values, where data recorded at negative fields are mirrored to positive field values. Data branches recorded under decreasing and increasing are indicated in blue and red shading, respectively.

Figure 9

Magnetization of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ ($x=0.2$), and MnSi, illustrating the different strength of the cubic magnetic anisotropies. Data are shown as a function of internal magnetic field in units of the upper critical field, $H_{\mathrm{c}2}$, determined in the direction exhibiting the largest value of $H_{\mathrm{c}2}$. The magnetization is presented in units of the magnetization at high fields. (a) Magnetization of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, exhibiting strong $\langle 100\rangle$ easy-axes anisotropy. (b) Magnetization of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ ($x=0.2$), exhibiting moderate $\langle 100\rangle$ easy-axes anisotropy. (c) Magnetization of MnSi, exhibiting very weak $\langle 111\rangle$ easy-axes anisotropy.

Figure 10

Magnetic work $W$, transition fields $H_{\mathrm{c}1}$ and $H_{\mathrm{c}2}$, and anisotropy constant $K$ as inferred from the magnetization and the transition fields as a function of temperature. (a) Temperature dependence of the magnetic work $W$ for all major crystallographic orientations under magnetic field saturating the magnetization. (b) Difference of the magnetic work $\mathrm{\Delta }W$ with respect to field along $\langle 100\rangle$. (c) Transition fields $H_{\mathrm{c}1}$ and $H_{\mathrm{c}2}$ for field along $\langle 111\rangle$ (blue), $\langle 110\rangle$ (green), and $\langle 100\rangle$ (red) as a function temperature. (d) Anisotropy constant $K$ as a function of temperature, as extracted from the magnetic work (green, blue), and the upper critical field $H_{\mathrm{c}2}$ (orange). Shown in gray shading is the regime in which an LTS phase becomes favorable ($K\le -0.07$).

Figure 11

Illustration of the effective anisotropy that arises when the conical helix, Eq (12), is embedded in the magnetocrystalline energy landscape of Eq. (6). Large energies are shown in red shading, whereas low energies are shown in blue shading. (a)–(c) explain the representation of the orientational dependence of the energetic cost with $K=0,K<0$, and $K>0$, respectively. For $K<0$, the magnetic easy axes are $\langle 100\rangle$, whereas they are the hard axes for $K>0$. (d)–(l) illustrate the orientations that are covered by the magnetic moments of the conical helix with a certain cone angle $\alpha$ for different field orientations of the pitch vector $\mathbf{k}$. (m)–(o) display the effective anisotropy potential of Eq. (13) that changes sign as function of cone angle $\alpha$.

Figure 12

Magnetization and magnetic work illustrating the field dependence of the effective anisotropy. (a) Internal magnetic field as a function of magnetization for different directions. (b) Magnetic work, inferred from magnetization data relative to the $\langle 100\rangle$ orientation as a function of the magnetization.