Magnetic phase diagram of MnSi inferred from magnetization and ac susceptibility

A. Bauer and C. Pfleiderer

Phys. Rev. B 85, 214418 – Published 14 June 2012

Abstract

We report simultaneous measurements of the magnetization and the ac susceptibility across the magnetic phase diagram of single-crystal MnSi. In our study we explore the importance of the excitation frequency, excitation amplitude, sample shape, and crystallographic orientation. The susceptibility, $μ_{0} d M / d B$ , calculated from the magnetization, is dominated by pronounced maxima at the transition from the helical to the conical and the conical to the skyrmion lattice phase. The maxima in $μ_{0} d M / d B$ are not tracked by the ac susceptibility, which in addition varies sensitively with the excitation amplitude and frequency at the transition from the conical to the skyrmion lattice phase. The same differences between $μ_{0} d M / d B$ and the ac susceptibility exist for Mn $_{1 - x}$ Fe $_{x}$ Si ( $x = 0.04$ ) and Fe $_{1 - x}$ Co $_{x}$ Si ( $x = 0.20$ ). Taken together our study establishes consistently for all major crystallographic directions the existence of a single pocket of the skyrmion lattice phase in MnSi, suggestive of a universal characteristic of all B20 transition metal compounds with helimagnetic order.

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Received 8 March 2012

DOI:https://doi.org/10.1103/PhysRevB.85.214418

Authors & Affiliations

A. Bauer and C. Pfleiderer

Technische Universität München, Physik-Department E21, D-85748 Garching, Germany

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Vol. 85, Iss. 21 — 1 June 2012

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Figure 1
Schematic depiction of the magnetization as a function of temperature and magnetic field. In the regime of the skyrmion lattice phase the magnetization is characterized by a tilted plateaux. The boundary of this plateaux is a discontinuous (first order) step. The traces illustrate typical scans as a function of magnetic field or temperature. In the experimental data the discontinuous steps are smeared out.Reuse & Permissions
Figure 2
Typical magnetization and ac susceptibility data of sample 1 as a function of applied magnetic field. Data were recorded for various temperatures in the regime of the A phase. The ac susceptibility data were recorded for an excitation amplitude of 0.1 mT parallel to $〈 100 〉$ at 10 Hz. See also Fig. 3 for definitions of the transition fields. (a) Magnetization as a function of applied magnetic field. (b) Real part of the ac susceptibility as a function of applied magnetic field. (c) Imaginary part of the ac susceptibility as a function of applied magnetic field. Data shown in panel (c) have been shifted by constant value of 0.025 for clarity.Reuse & Permissions
Figure 3
Comparison of the magnetic field dependence of the magnetization, $M$ , the susceptibility calculated from the magnetization, $μ_{0} d M / d B$ , and the real part of the ac susceptibility, $Re χ_{ac}$ . Data shown here were recorded in sample 1 for various temperatures in the regime of the A phase. The ac susceptibility data were recorded for an excitation amplitude of 0.1 mT along $〈 100 〉$ at 10 Hz.Reuse & Permissions
Figure 4
Typical magnetic field dependence of the susceptibility inferred from the magnetization, $μ_{0} d M / d B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected temperatures for various excitation frequencies. Data were recorded for an excitation amplitude of 0.1 mT. Curves at different temperatures have been shifted by constant values for clarity. See Table for a summary of the definitions of characteristic field values.Reuse & Permissions
Figure 5
Typical magnetic field dependence of the susceptibility inferred from the magnetization, $μ_{0} d M / d B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected temperatures for various excitation amplitudes. Data were recorded for an excitation frequency of 10 Hz. Curves at different temperatures have been shifted by constant values for clarity. See Table for a summary of the definitions of characteristic field values.Reuse & Permissions
Figure 6
Typical temperature dependence of the magnetization and the ac susceptibility of sample 1 at various magnetic fields in the regime of the A phase. Data shown here were recorded in sample 1. The ac susceptibility data were recorded for an excitation amplitude of 0.1 mT along $〈 100 〉$ at 10 Hz. See Fig. 7 for definitions of the transition temperatures. (a) Magnetization as a function of temperature. (b) $Re χ_{ac}$ as a function of temperature. (c) $Im χ_{ac}$ as a function of temperature. Data in panel (c) have been shifted by constant values of 0.025 for clarity.Reuse & Permissions
Figure 7
Comparison of the temperature dependence of the magnetization, $M$ , the susceptibility inferred from the magnetization, $μ_{0} Δ M / Δ B$ , and the real part of the ac susceptibility, $Re χ_{ac}$ . Also shown is the ratio $μ_{0} M / B$ . Data shown here were recorded in sample 1 for various magnetic fields in the regime of the A phase. The ac susceptibility data were recorded for an excitation amplitude of 0.1 mT along $〈 100 〉$ at 10 Hz.Reuse & Permissions
Figure 8
Typical temperature dependence of the susceptibility inferred from the magnetization, $μ_{0} Δ M / Δ B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected fields for various excitation frequencies. Data were recorded for an excitation amplitude of 0.1 mT. $μ_{0} Δ M / Δ B$ was calculated from magnetization data recorded for a field difference of 5 mT. Curves at different magnetic fields have been shifted by constant values for clarity. See Table for a summary of the definitions of temperature values.Reuse & Permissions
Figure 9
Typical temperature dependence of the susceptibility inferred from the magnetization, $μ_{0} Δ M / Δ B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected fields for various excitation amplitudes. Data were recorded for an excitation frequency of 10 Hz. $μ_{0} Δ M / Δ B$ was calculated from magnetization data recorded for a field difference of 5 mT. Curves at different magnetic fields have been shifted by constant values for clarity. See Table for a summary of the definitions of temperature values.Reuse & Permissions
Figure 10
Comparison of the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, as recorded while field heating after zero-field cooling (zfc-fh) and while field cooling (fc). Data were recorded in sample 1 for an excitation amplitude of 0.1 mT along $〈 100 〉$ at 10 Hz. Sets of curves for each field value are shifted by constants for clarity. No differences are observed in MnSi between zfc-fh and fc. This is contrasted by the properties of doped systems such as Mn $_{1 - x}$ Fe $_{x}$ Si, Mn $_{1 - x}$ Co $_{x}$ Si, and Fe $_{1 - x}$ Co $_{x}$ Si, which display distinct differences as shown below.Reuse & Permissions
Figure 11
Typical ac susceptibility data for different sample shapes and orientations as a function of applied field. Data were recorded at an excitation amplitude of 1 mT and 10 Hz. (a) Real part of the ac susceptibility, $Re χ_{ac}$ , for three different sample geometries. The pronounced qualitative and quantitative differences for different sample shapes are purely due to demagnetizing fields. (b) Real part of the ac susceptibility, $Re χ_{ac}$ for a cubic sample shape and various crystallographic directions. Differences are purely due to differences of crystallographic orientation.Reuse & Permissions
Figure 12
Magnetic phase diagram of MnSi as a function applied magnetic field along $〈 100 〉$ for different sample shapes. Sample 1 is a bar oriented along the applied field; sample 2 is a cube and sample 4 a thin platelet oriented perpendicular to the applied field. With increasing demagnetization fields the anomalies at $H_{c 1}$ and for the A phase broaden considerably. Data points between the regime marked IM and the paramagnetic state at higher temperatures represent a crossover.Reuse & Permissions
Figure 13
Typical field dependence of the susceptibility inferred from the magnetization, $μ_{0} d M / d B$ or $μ_{0} Δ M / Δ B$ , respectively, and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively. Data shown here were recorded in sample 4 (a thin platelet) to track the role of large demagnetizing fields. The ac susceptibility data were recorded for an excitation amplitude of 1 mT along $〈 100 〉$ at various excitation frequencies. Sets of curves for each temperature have been shifted by a constant for clarity. Values of the transition fields and temperatures have been determined in panel (a). The same values are marked in panel (b).Reuse & Permissions
Figure 14
Comparison of the magnetic field dependence of the susceptibility calculated from the magnetization, $μ_{0} d M / d B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected frequencies. Data shown here were recorded for magnetic field applied along $〈 100 〉$ , $〈 110 〉$ , and $〈 111 〉$ in samples 2 and 3. Both samples are cubic for the field directions studied. Thus even though data are shown as a function of applied fields they may be readily compared. The ac susceptibility was measured for an excitation amplitude of 1 mT. Sets of curves for the same temperature are shifted by constants for clarity.Reuse & Permissions
Figure 15
Comparison of the temperature dependence of the susceptibility calculated from the magnetization, $μ_{0} Δ M / Δ B$ , and the real and imaginary parts of the ac susceptibility, $Re χ_{ac}$ and $Im χ_{ac}$ , respectively, at selected frequencies. Data shown here were recorded for magnetic field applied along $〈 100 〉$ , $〈 110 〉$ , and $〈 111 〉$ in samples 2 and 3. Both samples are cubic for the field directions studied. Thus even though data are shown as a function of applied fields they may be readily compared. The ac susceptibility was measured for an excitation amplitude of 1 mT. $μ_{0} Δ M / Δ B$ was calculated from magnetization data recorded for a field difference of 5 mT. Sets of curves for the same magnetic field values are shifted by constants for clarity.Reuse & Permissions
Figure 16
Comparison of the susceptibility calculated from the measured magnetization, $μ_{0} d M / d B$ , and the ac susceptibility as measured at a wide range of different frequencies. The maxima at the lower and upper boundary of the A phase decrease dramatically with increasing frequency and vanish altogether for very large frequencies. Likewise maxima in the imaginary part of the susceptibility at the phase boundaries also decrease substantially with increasing frequency.Reuse & Permissions
Figure 17
Frequency dependence of the height of the peak in the ac susceptibility at the lower and upper boundary of the A phase denoted as $B_{A 1}$ and $B_{A 2}$ , respectively. Both maxima decrease exponentially and vanish for frequencies above $\sim$ $1.7 kHz$ and reach the size of the susceptibility in the conical phase.Reuse & Permissions
Figure 18
Comparison of the susceptibility $μ_{0} d M / d B$ calculated from the magnetization with the ac susceptibility for Mn $_{1 - x}$ Fe $_{x}$ Si ( $x = 0.04$ ). (a) Data after zero-field cooling. (b) Data after field cooling. The same qualitative differences are observed as in MnSi with additional differences between the zfc and fc data in the helical state.Reuse & Permissions
Figure 19
Comparison of the susceptibility calculated from the magnetization with the ac susceptibility for Fe $_{1 - x}$ Co $_{x}$ Si ( $x = 0.20$ ). (a) Data after zero-field cooling. (b) Data after field cooling. The same qualitative differences are observed as in MnSi with additional differences between zfc and fc data.Reuse & Permissions
Figure 20
Magnetic phase diagram of MnSi for field along different crystallographic directions. Phase diagrams are shown as a function of internal field; the effects of demagnetizing fields have been corrected. Data shown here were inferred from samples 1 through 4. Data points between the regime marked IM and the paramagnetic state at higher temperatures represent a crossover.Reuse & Permissions

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