Raman study of the temperature and magnetic-field dependence of the electronic and lattice properties of MnSi

The temperature and magnetic-field dependence of lattice and carrier excitations in MnSi is studied in detail using inelastic light scattering. The pure symmetry components of the electronic response are derived from the polarization-dependent spectra. The $E$ and $T_2$ responses agree by and large with longitudinal and optical transport data. However, an anomaly is observed right above the magnetic ordering temperature $T_{\mathrm{C}}=29$ K that is associated with the fluctuations that drive the transition into the helimagnetic phase first order. The $T_1$ spectra, reflecting mostly chiral spin excitations, have a temperature dependence similar to that of the $E$ and $T_2$ symmetries. The response in the fully symmetric $A_1$ representation has a considerably weaker temperature dependence than that in the other symmetries. All nine Raman active phonon lines can be resolved at low temperature. The positions and linewidths of the strongest four lines in $E$ and $T_2$ symmetry are analyzed in the temperature range $4<T<310$ K. Above 50 K, the temperature dependence is found to be conventional and given by anharmonic phonon decay and the lattice expansion. Distinct anomalies are observed in the range of the helimagnetic transition and in the ordered phase. Applying a magnetic field of 4 T, well above the critical field, removes all anomalies and restores a conventional behavior highlighting the relationship between the anomalies and magnetism. The anomaly directly above $T_{\mathrm{C}}$ in the fluctuation range goes along with an anomaly in the thermal expansion. While the lattice constant changes continuously and has only a kink at $T_{\mathrm{C}}$, all optical phonons soften abruptly, suggesting a direct microscopic coupling between spin order and optical phonons rather than a reaction to magnetostriction effects.

Figure 1

Raman spectra of MnSi at (a) 288 and (b) 17 $\mathrm{K}$. Plotted is the Raman susceptibility $R{\chi }_{\mathrm{i},\mathrm{s}}^{\prime \prime }(\omega ,T)$ as a function of the energy shift $\omega$. The spectra are measured with different sets of light polarizations with respect to the crystallographic axes as explained in the text. For each polarization combination, the symmetry components are indicated. For clarity, the spectra are consecutively offset vertically by 0.7 counts${\left(\mathrm{s}\mathrm{m}\mathrm{W}\right)}^{-1}$. To point out the small frequency differences between $E$ and $T_2$ phonons, the positions of the two $E$ phonons in the $xx$ spectra are marked by dashed lines. Whenever both $E$ and $T_2$ excitations contribute to a spectrum, double-peak structures appear. Upon lowering the temperature, all phonons harden. The black horizontal arrows indicate these frequency shifts for the $E$ phonons.

Figure 2

Symmetry-resolved spectra of MnSi at 17 $\mathrm{K}$. (a) Pure symmetries can be obtained via linear combinations of Raman spectra measured with different polarization settings according to Eq. (2). For clarity, the spectra are offset by 0.35 counts${\left(\mathrm{s}\mathrm{m}\mathrm{W}\right)}^{-1}$. All Raman active phonons $\left(2E+5T_2+2A\right)$ predicted by factor group analysis [35] are observed (marked by arrows). In the $A_1$ and $T_1$ spectra, there are additional peaks due to polarization leakage (marked by stars). The inset shows the weaker phonons of $T_2$ and $A_1$ symmetry on an expanded intensity scale (spectra offset by 0.1 counts${\left(\mathrm{s}\mathrm{m}\mathrm{W}\right)}^{-1})$. The $T_1$ spectrum contains the chiral excitations. It is featureless except for polarization leakage (stars), but the intensity of the continuum is in the same order of magnitude as the other symmetries. (b) Sums of spectra having orthogonal polarizations of the scattered photons. Two orthogonal measurements cover the full response of the sample, thus their sums must be invariant. This is used to check the consistency of the measurements.

Figure 3

Temperature and symmetry dependence of the electronic continua in all pure symmetries $\mu =A_1,E,T_1,T_2$. The response is obtained via linear combinations of $xy,x^{\prime }{y}^{\prime },rr$, and $rl$ spectra. Phonons were fitted with Voigt profiles and subtracted.

Figure 4

Temperature dependence of the $E$ phonons. Shown are the frequencies (left scale) and linewidths (right scale) of the two strongest lines with representative error bars (labeled by $E^{\left(318\right)}$ and $E^{\left(194\right)}$ according to their room-temperature positions). (a) and (b) show the analysis for the full temperature range, (c) and (d) zoom in on low temperatures. For both phonons, $E^{\left(318\right)}$ and $E^{\left(194\right)}$, the linewidth can be described by the model of anharmonic decay [51] (black lines). The main contribution to the frequency change is due to thermal expansion and can be described by a constant Grüneisen parameter ${\gamma }_i=2.5$ above 35 K (orange line). Right above $T_{\mathrm{C}}$ (dashed vertical line) in the fluctuation disordered regime [11] (shaded) there is a dip in the phonon frequency. If the helical order is suppressed by a magnetic field of $4\mathrm{T}$ (triangles and circles), the anomalies in the phonon frequencies disappear.

Figure 5

Temperature dependence of $T_2$ phonons. Shown are the frequencies (left scale) and linewidths (right scale) of the two strongest lines (labeled by $T_2^{\left(314\right)}$ and $T_2^{\left(197\right)}$ according to their room-temperature positions). (a) and (b) show data for the full temperature range, (c) and (d) zoom in on low temperatures. The phonon width deviates from the predictions of the Klemens model [51] (black lines) only for the $T_2^{\left(197\right)}$ phonon below $T_{\mathrm{C}}$ (dashed vertical line). Above 35 K, the frequency change of both phonons can be explained in terms of a thermal expansion shift ${\Delta }_i^{\left(1\right)}\left(T\right)$ (orange lines) assuming a constant ${\gamma }_i=2.5$ for all phonon modes $i$. There is a dip in the phonon frequency right above $T_{\mathrm{C}}$ in the fluctuation-disordered region [10, 11] (shaded). The dip can be reproduced qualitatively if the macroscopic Grüneisen parameter ${\gamma }_{\mathrm{macro}}\left(T\right)$ is inserted into Eq. (9) as described in the text (dashed magenta line). In the helimagnetic phase the phonon frequencies are lower than those predicted by the thermal expansion.

Figure 6

Difference of the experimental phonon energies ${\omega }_{\mathrm{ph}}\left(T\right)$ and those calculated via Eqs. (5) and (9) using ${\gamma }_{\mathrm{macro}}\left(T\right)$. Note the logarithmic temperature scale. The anomalies above $T_{\mathrm{C}}$ vanish almost completely while those at $T_{\mathrm{C}}$ have no correspondence in the thermodynamical properties.

Figure 7

Temperature and symmetry dependence of the relaxation rates ${\Gamma }_{\mu }(\omega ,T)$. The rates were obtained from the electronic continua as shown in Fig. 3. To calculate ${\Gamma }_{\mu }(\omega ,T)$, the procedure described by Opel and co-workers [49] was used. The smooth lines are phenomenological fits to the data according to Eq. (11).

Figure 8

Temperature and symmetry dependence of the optical masses $1+{\lambda }_{\mu }\left(T\right)$ obtained via the same formalism as the relaxation rates. The smooth lines are derived from the fits to the relaxation rates via Kramers-Kronig transformation, hence obey causality (see text). They are in reasonable agreement to the data except for a constant offset.

Figure 9

Static Raman relaxation rates and transport data. Panel (a) shows the Raman relaxation rates ${\Gamma }_{\mu }(\omega =0,T)$ (points, left axis) as a function of symmetry $\mu$ as derived from ${\Gamma }_{\mu }(\omega ,T)$ (see text). If the longitudinal dc resistivity $\rho \left(T\right)$ (black line, right axis) [80] is converted into a relaxation rate ${\Gamma }_{\rho }\left(T\right)$ using a Drude model with the experimental plasma frequency ${\omega }_{\mathrm{pl}}=2.3$ eV [42], an extra factor of 0.73 is needed to match transport and Raman data. Above $T_{\mathrm{C}}$ (dashed vertical line), the Raman data in $E,T_1$, and $T_2$ symmetry agree with $\rho \left(T\right)$, while the data in $A_1$ symmetry do not. (b)–(e) Zoom in on low temperatures. The phase transition has only a minor effect on ${\Gamma }_{\mu }(\omega =0,T)$. Above $T_{\mathrm{C}}$ close to the fluctuation-disordered regime (shaded), there may be a dip in $A_1$ and $T_2$ symmetry. Below the phase transition ${\Gamma }_{\mu }(\omega =0,T)$ decreases slower than ${\Gamma }_{\rho }\left(T\right)$.

Figure 10

Sketch of the light path close to the sample. With a polarizer (1) and a Soleil-Babinet compensator (2), the polarization and phase of the incoming laser light (dark green) can be adjusted. The laser light is directed via a prism (3) to an achromat (4) focusing on the sample surface (6). Since separate optical components are used for excitation and collection, the direct reflex (7) from the mirrorlike sample surface does not enter the collection optics (8) $(f\approx 30\mathrm{m}\mathrm{m},\mathrm{N}.\mathrm{A}.=0.34)$ for the scattered light (light green). Reflexes from the cryostat windows (5) are blocked separately. This considerably reduces the amount of elastically scattered light and fluorescence reaching the spectrometer. The residual background signal is further reduced by a spatial filter (9) in the path of the scattered light consisting of two confocal achromatic lenses $\left(f=75\mathrm{m}\mathrm{m}\right)$ and a circular aperture of 150 $\mu \mathrm{m}$ in the common focus. A quarter wave plate (10) and an analyzer (11) select the polarization of the scattered light. The sample is located in a He-flow cryostat in the center of a superconducting solenoid (12).

Figure 11

Temperature determination from thermal conductivity. (a) Thermal conductivity measurement; (b) integrated thermal conductivity. According to Eq. (B2), the integral $\int \lambda dT=:\alpha$ from $T_{\mathrm{h}}$ to $T_{\mathrm{h}}+\Delta T$ is constant as long as the laser power and the focus size are not changed. For the experimental setup used here, $\alpha$ is 6.4 W/m for an absorbed laser power of 4 mW. The inset of (b) illustrates how $\Delta T$ can be obtained from $\alpha$ and the integrated conductivity.

Figure 12

Temperature dependence of the laser heating $\Delta T$ for an absorbed laser power of 4 mW. Points correspond to the laser heating for various holder temperatures $T_{\mathrm{h}}$ and are determined from the integrated thermal conductivity as described in Fig. 11. An exponential fit was used to obtain $\Delta T$ in the full temperature range.

Figure 13

Fit of the data on the example of an $xx$ spectrum at 17 K. Panel (a) shows the measured data (green) together with the fit function (orange) consisting of a fifth-order polynomial and four Voigt shaped peaks. (b) Shows the difference (blue) between measurement and fit. A smoothed curve (red) points out how much and at which frequencies the measurement and the fit differ beyond the noise level. To separate the phononic part from the electronic continuum, only the fits of the peaks are subtracted from the spectra.

Figure 14

Grüneisen parameters in MnSi. The macroscopic Grüneisen parameter ${\gamma }_{\mathrm{macro}}\left(T\right)$ (green) as derived from thermodynamic properties in comparison with the constant phonon Grüneisen parameter ${\gamma }_i=2.5$ (orange) which describes the thermal expansion shift of the Raman mode $i$.

Figure 15

(a) Majority-spin and (b) minority-spin Fermi surfaces of MnSi (without spin-orbit coupling) as derived by Jeong and Pickett [72]. (c) Nodal hypersurface of the lowest-order $A_1$ Raman vertex.