Pressure-induced quantum phase transition in Fe${}_{1-x}$Co${}_x$Si ($x=0.1,0.2$)

We have investigated the effect of pressure on electrical transport and magnetic properties of ferromagnetic Fe${}_{1-x}$Co${}_x$Si alloys for $x=0.1$ ($T_{\mathrm{C}}~11\hspace{0.5em}\mathrm{K}$) and $x=0.2$ ($T_{\mathrm{C}}~32\hspace{0.5em}\mathrm{K}$) using electrical resistivity measurements up to about 30 GPa. We have also studied the magnetic properties of these samples ($x=0.1$ and $x=0.2$) and a sample with $x=0.3$ ($T_{\mathrm{C}}~43\hspace{0.5em}\mathrm{K}$) at ambient pressure using ${}^{57}\mathrm{Fe}$ Mössbauer-effect (ME) spectroscopy. The ME results indicate that the effective magnetic hyperfine field $B_{\mathrm{eff}}$ at the ${}^{57}\mathrm{Fe}$ nucleus exhibits the same concentration dependence as the macroscopic magnetic moment and confirm that the onset of magnetic order is above $x~0.02$. The analysis of the high-pressure results reveals in both samples a gradual suppression of the ferromagnetic state to a quantum phase transition (QPT) at pressures of $p~11\hspace{0.5em}\mathrm{GPa}$ and $p~12\hspace{0.5em}\mathrm{GPa}$ for $x=0.1$ and $x=0.2$, respectively. High-pressure x-ray diffraction measurements on the three samples indicate very similar change of the volume of the cubic unit cell with pressure and exclude that the observed QPTs are connected with a structural phase transition. We discuss the observed instability of the ferromagnetic state with increasing Co concentration in the context of increasing local crystallographic disorder, which causes a change of the distribution of the helix wave vector as well as a modification of the ferromagnetic half-metallic state. We further show that, in the pressure-induced nonmagnetic metallic state, all samples, regardless of their different local crystallographic disorder, exhibit similar non-Fermi-liquid behavior [$\rho (T)\propto T$]. Finally, we find a small but positive magnetoresistance in the high-pressure metallic state well beyond the QPT of Fe${}_{0.9}$Co${}_{0.1}$Si. This can be attributed to a slight field-induced modification of the spin majority and minority band which leads to a very small magnetic moment.

Figure 1

Left panel: Mössbauer spectra of Fe${}_{1-x}$Co${}_x$Si ($x=0.1$, 0.2, 0.3) at 4.2 K. Solid line corresponds to the histogram fit of the data involving a distribution of $B_{\mathrm{eff}}$ including mixed hyperfine interactions using the Blaes et al. routine.[34] Small linear variations of center shift $S$ and quadrupole splitting $\mathrm{QS}$ were allowed. Right panel: $B_{\mathrm{eff}}$ distribution $P(B_{\mathrm{eff}}$) obtained from fitting the spectra at 4.2 K.

Figure 2

Average magnetic hyperfine field ${\stackrel{̲}{B}}_{\mathrm{eff}}$ of Fe${}_{1-x}$Co${}_x$Si as a function of concentration $x$. Dotted line is only a guide to the eye. The inset shows that ${\stackrel{̲}{B}}_{\mathrm{eff}}$ is proportional to the magnetization $M$ obtained by macroscopic measurements. The values of $M$ are taken from Ref. 5. The solid red line is a linear fit to the data, the dotted part is an extrapolation of this linear fit toward zero field.

Figure 3

Temperature dependence of the resistivity of (a) Fe${}_{0.9}$Co${}_{0.1}$Si and (b) Fe${}_{0.8}$Co${}_{0.2}$Si for different pressures from ambient pressure up to $29.9\hspace{0.5em}$ and $15.3\hspace{0.5em}\mathrm{GPa}$, respectively, as obtained by four-point electrical resistivity measurements using a diamond anvil cell setup. The arrows at ambient pressure indicate the maxima in $\rho (T)$ at $T_{\max }$, below which the systems show a metallic-like temperature dependence of $\rho (T)$, and the anomalies at $T_{\mathrm{M}}$ which are connected with the onset of ferromagnetic order in Fe${}_{0.9}$Co${}_{0.1}$Si ($T_{\mathrm{C}}~11\hspace{0.5em}\mathrm{K}$) and Fe${}_{0.8}$Co${}_{0.2}$Si ($T_{\mathrm{C}}~32\hspace{0.5em}\mathrm{K}$).[5]

Figure 4

X-ray diffraction patterns of Fe${}_{1-x}$Co${}_x$Si ($x=0.1$, 0.2, 0.3) at room temperature as obtained from energy dispersive x-ray diffraction at beamline F3 (HASYLAB, Hamburg) using a diamond anvil cell setup. (a) Patterns of Fe${}_{0.9}$Co${}_{0.1}$Si for selected pressures up to $30\hspace{0.5em}\mathrm{GPa}$ recorded with ${\mathrm{Ed}}_{\mathrm{hkl}}=72.750\hspace{0.5em}\mathrm{keV}\hspace{0.5em}\AA$. (b) Patterns of Fe${}_{0.8}$Co${}_{0.2}$Si for selected pressures up to $23.1\hspace{0.5em}\mathrm{GPa}$ recorded with ${\mathrm{Ed}}_{\mathrm{hkl}}=73.247\hspace{0.5em}\mathrm{keV}\hspace{0.5em}\AA$. The arrow marks the peak from the detector electronics, which remains pressure independent as demonstrated by the dotted line. The asterisks mark the peaks from the gold pressure marker. (c) Patterns of Fe${}_{0.7}$Co${}_{0.3}$Si for selected pressures up to $9.0\hspace{0.5em}\mathrm{GPa}$ recorded with ${\mathrm{Ed}}_{\mathrm{hkl}}=69.862\hspace{0.5em}\mathrm{keV}\hspace{0.5em}\AA$. The arrows mark the peaks from solid nitrogen in the cubic phase above $p~5$ GPa, the asterisks mark the peaks from the gold pressure marker. The high intensity of the peak at about 37 keV for 7.6 GPa results from the texture of the sample and is not observed at other pressures.

Figure 5

Pressure dependence of the volume $V$ of the cubic unit cell at room temperature of (a) Fe${}_{0.9}$Co${}_{0.1}$Si up to $30.4\hspace{0.5em}\mathrm{GPa}$, (b) Fe${}_{0.8}$Co${}_{0.2}$Si up to $23.1\hspace{0.5em}\mathrm{GPa}$, and (c) Fe${}_{0.7}$Co${}_{0.3}$Si up to $9.0\hspace{0.5em}\mathrm{GPa}$. Red line are fits to the data using the Murnaghan equation of state (see text). The insets show the pressure dependence of the corresponding $a$ axis of Fe${}_{1-x}$Co${}_x$Si ($x=0.1$, 0.2, 0.3).

Figure 6

Temperature dependence of the normalized electrical resistivity $\rho (T)/\rho (20\hspace{0.5em}\mathrm{K})$ of Fe${}_{0.9}$Co${}_{0.1}$Si in the low-temperature region ($1.6\hspace{0.5em}\mathrm{K}⩽T⩽20\hspace{0.5em}\mathrm{K}$) for different pressures from ambient pressure up to $p=9.0\hspace{0.5em}\mathrm{GPa}$. Arrows mark $T_{\mathrm{M}}$, which is connected with the onset of magnetic ordering.

Figure 7

Pressure dependence of (a) the coefficient $A$ and (b) the exponent $n$ as obtained from the temperature dependence of the electrical resistivity ($\rho ={\rho }_0+A{T}^n$) in the nonmagnetic metallic state of Fe${}_{0.9}$Co${}_{0.1}$Si for $p⩾12.5\hspace{0.5em}\mathrm{GPa}$. The dark areas mark the region where magnetic order exists.

Figure 8

Temperature dependence of the electrical resistivity of Fe${}_{0.9}$Co${}_{0.1}$Si at $30\hspace{0.5em}\mathrm{GPa}$ for $T⩽20\hspace{0.5em}\mathrm{K}$ in external magnetic fields up to $8\hspace{0.5em}\mathrm{T}$. The inset shows the magnetoresistance $\mathrm{MR}$ at $2.5\hspace{0.5em}\mathrm{K}$ in percent as a function of external magnetic field for Fe${}_{0.9}$Co${}_{0.1}$Si at $30\hspace{0.5em}\mathrm{GPa}$ (see text).

Figure 9

Temperature dependence of the normalized electrical resistivity $\rho (T)/\rho (60\hspace{0.5em}\mathrm{K})$ of Fe${}_{0.8}$Co${}_{0.2}$Si for $1.6\hspace{0.5em}\mathrm{K}⩽T⩽60\hspace{0.5em}\mathrm{K}$ for different pressures from ambient pressure up to $p=9.1\hspace{0.5em}\mathrm{GPa}$. Arrows mark $T_{\mathrm{M}}$, which is connected with the onset of magnetic ordering. (b) Temperature dependence of the normalized electrical resistivity $\rho (T)/\rho (20\hspace{0.5em}\mathrm{K})$ of Fe${}_{0.8}$Co${}_{0.2}$Si for $1.6\hspace{0.5em}\mathrm{K}⩽T⩽20\hspace{0.5em}\mathrm{K}$ for different pressures in the range $11.5\hspace{0.5em}\mathrm{GPa}⩽p⩽15.3\hspace{0.5em}\mathrm{GPa}$.

Figure 10

Pressure dependence of the upturn or kink at $T_{\mathrm{M}}$ of the electrical resistivity at low temperatures, which is connected to the onset of magnetic order in Fe${}_{1-x}$Co${}_x$Si ($x=0.1$, 0.2, 0.3). The data points of Fe${}_{0.9}$Co${}_{0.1}$Si and Fe${}_{0.8}$Co${}_{0.2}$Si were obtained in this study whereas the data points of Fe${}_{0.7}$Co${}_{0.3}$Si were reported by Onose et al. (Ref. 5). Dotted lines are linear fits to the experimental data points.