Scaling study and thermodynamic properties of the cubic helimagnet FeGe

The critical behavior of the cubic helimagnet FeGe was obtained from isothermal magnetization data in very close vicinity of the ordering temperature. A thorough and consistent scaling analysis of these data revealed the critical exponents $\beta =0.368,\gamma =1.382$, and $\delta =4.787$. The anomaly in the specific heat associated with the magnetic ordering can be well described by the critical exponent $\alpha =-0.133$. The values of these exponents corroborate that the magnetic phase transition in FeGe belongs to the isotropic 3D-Heisenberg universality class. The specific heat data are well described by ab initio phonon calculations and confirm the localized character of the magnetic moments.

Figure 1

(a) Isothermal magnetization ($M$ vs $H$) of FeGe at selected temperatures [35] for magnetic fields $H\parallel \left[100\right]$. (b) Rescaled isotherms using the critical exponents $\beta =0.368$ and $\gamma =1.382$. Only a subset of the measured data is shown for clarity [35]. The lines are linear fits to the data (large symbols) for fields above $H_{\mathrm{cutoff}}=9$ kOe. In both panels the critical isotherm is highlighted in red and in (b) the applied magnetic field was corrected for demagnetization effects.

Figure 2

(a) Temperature dependence of the spontaneous magnetization $M_s\left(T\right)$ (left) and inverse susceptibility $\chi {\left(T\right)}^{-1}$ (right) of FeGe. The lines represent fits of Eqs. (1) and (2) to the data with the exponents $\beta =0.368$ and $\gamma =1.382$. The inset shows the critical isotherm $M\left(H\right)$. Its slope yields the exponent $\delta =4.787$. (b) The Kouvel-Fisher plots of $M_s\left(T\right){(dM/dT)}^{-1}$ (left) and $\chi {\left(T\right)}^{-1}{(d\chi {\left(T\right)}^{-1}/dT)}^{-1}$ (right) yield the exponents $\beta =0.369$ and $\gamma =1.337$, respectively. The vertical dashed line indicates $T_c=278.6$ K.

Figure 3

Scaling plots of the $M\left(H\right)$ data of FeGe for temperatures $275\mathrm{K}\le T\le 283$ K using the values $\beta =0.368,\gamma =1.382$, and $T_c=279.0$ K. (a) All data fall on two universal curves. The inset shows the effective exponents ${\beta }_{\mathrm{eff}}$ and ${\gamma }_{\mathrm{eff}}$ according to Eq. (5). The vertical dashed line indicates $T_c$. (b) Scaling plot in a double-logarithmic scale.

Figure 4

Temperature dependence of the magnetic moment of FeGe at various magnetic fields. These data were obtained from the $M$ vs $H$ data shown in Fig. 1.

Figure 5

Specific heat $C_p\left(T\right)$ of FeGe in the vicinity of $T_c$. Two data sets show a cusp at 278.0 K (squares) and 278.7 K (circles). The solid line represents a fit of Eq. (7) with $\alpha =-0.133$ and $T_c=279.7$ K to those data points marked with a dot; its extrapolation is shown as dashed line. The inset shows the change of the entropy $[-\mathrm{\Delta }S(T,\mathrm{\Delta }H$)] derived from the isothermal magnetization data via Eq. (9) (see text).

Figure 6

(a) Specific heat $C_p\left(T\right)$ of FeGe (circles). Nonmagnetic contributions to the specific heat are well described by a Einstein-Debye model (dashed line) and by ab initio phonon DOS calculations (solid line). Inset: $C_p\left(T\right)$ in the vicinity of $T_c$ at various magnetic fields. (b) Magnetic contribution to the specific heat, $C_p^{\mathrm{mag}}\left(T\right)$, after subtraction of the electronic part and lattice contributions, based on a Debye-Einstein model (circles) or the phonon DOS calculation (squares). (c) Magnetic entropy $S_{\mathrm{mag}}$ determined from the integration $C_p^{\mathrm{mag}}\left(T\right)/T$ [circles in (b)].

Figure 7

Calculated vibrational density of states (VDOS) and phonon dispersion of FeGe along lines between selected high symmetry points in the Brillouin zone. The color coding refers to the elemental character of the phonon modes (blue for Fe and orange for Ge).

Figure 8

Linear thermal expansion ${\alpha }_L\left(T\right)$ of FeGe. The experimental data (line) show a small cusp at 278.8 K (arrow) that is caused by the magnetic transition. The lattice contribution (dashed line) to the thermal expansion was obtained from the specific heat data according to Eq. (10). The dotted line is based on the phonon DOS calculations and represents ${\alpha }_L\left(T\right)={\alpha }_V\left(T\right)/3$. Inset: experimental $\mathrm{\Delta }{\alpha }_L\left(T\right)$ data close to $T_c$ after the subtraction of a linear background from ${\alpha }_L\left(T\right)$.