Magnetic field dependence of nonlinear magnetic response and tricritical point in the monoaxial chiral helimagnet ${\mathrm{Cr}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$

We present a comprehensive study of the magnetization dynamics and phase evolution in the chiral helimagnet ${\mathrm{Cr}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$, which realizes a chiral soliton lattice (CSL). The magnetic field dependence of the ac magnetic response is analyzed for the first five harmonic components, $M_{\mathrm{n}\omega }\left(H\right)\left(n=1–5\right)$, using a phase sensitive measurement over a frequency range, $f=11–10000\mathrm{Hz}$. At a critical field, the modulated CSL continuously evolves from a helicity-rich to a ferromagnetic domain-rich structure, where the crossover is revealed by the onset of an anomalous nonlinear magnetic response that coincides with extremely slow dynamics. The behavior is indicative of the formation of a spatially coherent array of large ferromagnetic domains, which relax on macroscopic time scales. The frequency dependence of the ac magnetic loss displays an asymmetric distribution of relaxation times across the highly nonlinear CSL regime, which shift to shorter time scales with increasing temperature. We experimentally resolve the tricritical point at $T_{\mathrm{TCP}}$ in a temperature regime above the ferromagnetic Curie temperature which separates the linear and nonlinear magnetic regimes of the CSL at the phase transition. A comprehensive phase diagram is constructed which summarizes the features of the field and temperature dependence of the magnetic crossovers and phase transitions in ${\mathrm{Cr}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$.

Figure 1

Evolution of the spatially modulated chiral spin structure in ${\mathrm{Cr}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ with applied magnetic field. (a) The CHM structure continuously evolves into a CSL. The period of the helical ground state is $L\left(0\right)=48\mathrm{nm}$. The CSL period, $L$($H$), continuously grows with applied magnetic field $H$, perpendicular to the $c$ axis. The forced ferromagnetic (IC-C) transition occurs at magnetic fields above $H_{\mathrm{FFM}}$. (b) The change in spin coherence or uniformity of the magnetic structure as a function of magnetic field, ξ($H$), as described in the text. The CHM is coherent over the entire crystal at $H=0\mathrm{Oe}$. In the CSL regime, the spin uniformity corresponds to ferromagnetic domain (commensurate) component of the magnetic structure. (c) Schematic H-T phase diagram. The green line at $H=0\mathrm{Oe}$ represents the pure CHM state where $\xi \left(0\right)=\infty$. The PM-CHM transition occurs at $T_0$. The CSL is divided into two regimes by a crossover boundary at $H_{\mathrm{C},1}$ (blue line) which separates the linear and HNL CSL. The chiral phase boundary extends past the Curie temperature, $T_{\mathrm{C}}$ and terminates at $T_0$. A precursor region of strong correlations is marked by $T^{*}$. The tricritical point, $T_{\mathrm{TCP}}$, separates a second-order HNL CSL-FFM transition with a first-order linear CSL-PM transition.

Figure 2

Magnetic field dependence of the real and imaginary parts of the linear ac magnetic response, $M_{1\omega }\left(H,T\right)$, measured with an ac magnetic field amplitude, $h=5\mathrm{Oe}$. (a) Real and (b) imaginary parts of $M_{1\omega }\left(H,T\right)$ measured as a function of magnetic field at fixed temperatures in the range $T=129–133\mathrm{K}$. (c) Real and (d) imaginary parts of $M_{1\omega }\left(H,T\right)$ measured for a frequency of $f=1111\mathrm{Hz}$.

Figure 3

Comparison of the magnetic field dependence of the dc differential susceptibility, $\mathit{dM}/\mathit{dH}$, the ac susceptibility, $M_{1\omega }^{\prime }/h={\chi }_{1\omega }^{\prime }$, at $f=11\mathrm{Hz}$ (right axis), and ac magnetic loss, $M_{1\omega }^{\prime \prime }/h={\chi }_{1\omega }^{\prime \prime }$, at $f=11$ and 111 Hz (left axis). The dashed lines mark the field regime of the deviation between the dc and ac susceptibilities and the corresponding onset and destruction of ac magnetic loss. The maximum in $\mathit{dM}/\mathit{dH}$ vs $H$, defined as ${H_{\mathrm{dM}/\mathrm{dH}}}^{\mathrm{peak}}$, is dependent on measurement temperature and occurs at $H\sim 500\mathrm{Oe}$ at $T=129\mathrm{K}$.

Figure 4

Magnetic field dependence of the linear and nonlinear components of the ac magnetic response for selected temperatures, $M_{\mathrm{n}\omega }$($H, T$), at $f=111\mathrm{Hz}$. The magnitude of all harmonic components are highly dependent on frequency and hence do not display maximum magnitude simultaneously. (a) The magnitude of the ac magnetic loss, $M_{1\omega }^{\prime \prime }$($H$). The large absolute magnitudes of (b) the third, $M_{3\omega }$, and (c) fifth, $M_{5\omega }$, harmonic response show a remarkably similar field dependence to the ac magnetic loss term. (d) $M_{1\omega }^{\prime }$. The even harmonics (e) $M_{2\omega }$ and (f) $M_{4\omega }$, have sizeable contributions to the total magnetic response and closely follow the inflection points of $M_{1\omega }^{\prime \prime }$.

Figure 5

The third harmonic of the ac magnetic response as a function of magnetic field, $M_{3\omega }$($H$), for $f=25–10000\mathrm{Hz}$ measured at $T=129\mathrm{K}$.

Figure 6

Magnetic field dependence of the second harmonic of the ac magnetic response measured with $f=111\mathrm{Hz}$. (a) $M_{2\omega }\left(H\right)$ measured at 130.5 K. (Inset) Real and imaginary components of $M_{2\omega }\left(H\right)$. (b) Surface plot of $M_{2\omega }^{\prime }\left(H,T\right)$.

Figure 7

Frequency dependence of the linear ac magnetic response at $T=129\mathrm{K}$. (a) Real and (b) imaginary parts of the linear ac magnetic response, $M_{1\omega }^{\prime }$ ($H$) and $M_{1\omega }^{\prime \prime }$ ($H$), measured as a function of magnetic field. In- and out-of-phase linear susceptibility (c) ${\chi }_{1\omega }^{\prime }=M_{1\omega }^{\prime }h$ and (d) ${\chi }_{1\omega }^{\prime \prime }=M_{1\omega }^{\prime \prime }h$ as a function of frequency for the magnetic field corresponding to the dip in $M_{1\omega }^{\prime }$ ($H$) and $M_{1\omega }^{\prime \prime }$ ($H$) at $H_{\mathrm{dM}/\mathrm{d}}{}_{\mathrm{H}}^{\mathrm{peak}}$ marked by a blue asterisk in (b). The green line represents a fit of (4) and (5) to the low-frequency side of the inflection point in ${\chi }_{1\omega }^{\prime }$ and the peak in ${\chi }_{1\omega }^{\prime \prime }$, respectively, which correspond to ${\tau }_0=1/\left(2\pi f_0\right)$. The blue lines represent fits on the high-temperature side of ${\tau }_0$.

Figure 8

Dynamic parameters as a function of temperature (a) ${\tau }_0$, the characteristic time, and (b) ${\chi }_0^{\prime \prime }$, frequency-independent term, extracted from fits of Eq. (5) to the low-frequency side of ${\chi }_{1\omega }^{\prime \prime }$. The corresponding magnetic fields are marked by asterisks in Fig. 7.

Figure 9

The third harmonic of the ac magnetic response as a function of temperature and magnetic field. (a) Frequency dependence of real component of $M_{3\omega }\left(H\right)$ for $f=25–10000\mathrm{Hz}$. (Inset) $M_{3\omega }^{\prime \prime }$. (b)–(d) Surface plots of the real part $M_{3\omega }^{\prime }$ ($H, T$) for various frequencies.

Figure 10

Temperature dependence of the characteristic frequency, $f_0$, corresponding to the magnetic fields marked in Fig. 7. (Inset) ${\chi }_{1\omega }^{\prime \prime }$ vs $f$ for temperatures ranging from $T=129–131.5\mathrm{K}$. The peak in ${\chi }_{1\omega }^{\prime \prime }$ at $f_0$ shifts to higher frequency as temperature increases, as indicated by the black arrow, toward the tricritical point.

Figure 11

Linear and nonlinear magnetic response of the unpolarized CHM state as a function of temperature at $H=0\mathrm{Oe}$ and $f=10000\mathrm{Hz}$. The kink point at $T_0\sim 132.25\mathrm{K}$ marks the PM-CHM phase transition. Nonzero values of $M_{2\omega }-M_{5\omega }$ appear at $T^{*}=133\mathrm{K}$ and correspond with the inflection point in $M_{1\omega }$ that marks the onset of chiral correlations in the fluctuation-disordered precursor region.

Figure 12

Temperature dependence of the ac magnetic response measured with an ac field amplitude, $h=5\mathrm{Oe}$. (a) Real and (b) imaginary parts of $M_{1\omega }\left(T\right)$ measured as a function of temperature with fixed dc fields in the range $H_{\mathrm{dc}}=50–1200\mathrm{Oe}$ and $f=111\mathrm{Hz}$. $T_{\mathrm{m}}$ marks the PM–FP transition, which shifts to higher temperature with increasing magnetic field. The inset shows $M_{2\omega }$ which demonstrates the onset of strong FM correlations at $T_{\mathrm{m}}$. (c) $M_{2\omega }$ and (d) $M_{3\omega }$ as a function of temperature for fixed dc fields in the range $H_{\mathrm{dc}}=50–400\mathrm{Oe}$ at $f=10000\mathrm{Hz}$.

Figure 13

Determination of the critical values of field and temperature. (a) First derivative of $M_{1\omega }^{\prime }, M_{3\omega }$, and $M_{2\omega }$ as a function of magnetic field. $H_{\mathrm{C},1}$ is defined where the onset of rapid changes of slope in the linear and nonlinear magnetic response coincide. $H_{\mathrm{C},1}$, represents the crossover field for the onset of the HNL CSL. $H_{\mathrm{C},2}$ defines the critical field for the FFM transition. $\mathrm{d}{M}_{3\omega }/\mathrm{d}H$ and $\mathrm{d}{M}_{2\omega }/\mathrm{d}H$ are multiplied by a factor of 10. (b) Magnetic field dependence of $M_{1\omega }\left(T\right)$ for $H_{\mathrm{dc}}=400–550\mathrm{Oe}$ measured for $f=10000\mathrm{Hz}$. As magnetic field decreases, the peak at $T_{\mathrm{m}}$ shifts to lower temperature and disappears at the tricritical point. (Inset) Example of Lorentzian peak fit used to determine an accurate value of $T_{\mathrm{m}}$. Standard error obtained from fits of each data set range between ± 0.012–0.045 K.

Figure 14

H-T phase diagram determined from the linear and nonlinear components of the ac magnetic response plotted onto (a) $M_{1\omega }^{\prime }\left(H,T\right)$, (b) $M_{1\omega }^{\prime }\left(H,T\right)$, [(c) and (d)] $M_{\mathrm{n}\omega }^{\prime }\left(H,T\right)$ for ($n=2-5$) at $f=10000\mathrm{Hz}$, where $M_{\mathrm{n}\omega }\left(H,T\right)$ refers to the field dependence at fixed temperature. In (a), the critical temperatures are labeled: $T^{*}=133\mathrm{K}, T_0=132.3\mathrm{K}, T_{\mathrm{TCP}}=131.35\mathrm{K}$ (red dashed line), and $T_{\mathrm{C}}=130.75\mathrm{K}$ (black dashed line). Field-dependent anomalies, marked by open symbols, in $M_{1\omega }^{\prime \prime }$ (diamonds), $M_{2\omega }$ (circles), and $M_{3\omega }$ (squares) locate the low field increase at $H_{\mathrm{C},1}$ (blue), dip in magnitude at ${H_{\mathrm{dM}/\mathrm{dH}}}^{\mathrm{peak}}$ (blue), and high-field destruction at $H_{\mathrm{C},2}$ (black) of the magnetic loss and higher harmonics corresponding to the HNL CSL. Equivalent temperature-dependent anomalies from $M_{\mathrm{n}\omega }\left(T,H_{\mathrm{dc}}\right)$ measurements are marked by closed symbols. The IC-C phase line is given by the fall off of $M_{1\omega }^{\prime }$($H$) (stars). Above $T_{\mathrm{TCP}}$, the red symbols mark the lower and upper bounds of the anomaly across the linear CSL-PM phase transition. The magnetic loss and nonlinear response abruptly fall off at $T_{\mathrm{TCP}}=131.35\mathrm{K}$ given by the intersection of the PM-FP line defined by $T_{\mathrm{m}}$ (green/pink squares) and the IC-C phase line determined from the power law fit of $H_{\mathrm{C},2}$ (solid black line).