Spin correlations and colossal magnetoresistance in ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$

This study aims to unravel the mechanism of colossal magnetoresistance (CMR) observed in $n\text{−}\mathrm{type} {\mathrm{HgCr}}_2{\mathrm{Se}}_4$, in which low-density conduction electrons are exchange-coupled to a three-dimensional Heisenberg ferromagnet with a Curie temperature $T_C\approx 105$ K. Near room temperature the electron transport exhibits an ordinary semiconducting behavior. As temperature drops below $T^{*}\simeq 2.1T_C$, the magnetic susceptibility deviates from the Curie-Weiss law, and concomitantly the transport enters an intermediate regime exhibiting a pronounced CMR effect before a transition to metallic conduction occurs at $T<T_C$. Our results suggest an important role of spin correlations not only near the critical point but also for a wide range of temperatures $\left(T_C<T<T^{*}\right)$ in the paramagnetic phase. In this intermediate temperature regime the transport undergoes a percolation type of transition from isolated magnetic polarons to a continuous network when temperature is lowered or magnetic field becomes stronger.

Figure 1

(a) Temperature dependences of longitudinal resistivity ${\rho }_{\mathrm{xx}}$ for an $n\text{−}\mathrm{type} {\mathrm{HgCr}}_2{\mathrm{Se}}_4$ sample with carrier density $n\approx 1\times {10}^{18} {\mathrm{cm}}^{-3}$ for magnetic fields ${\mu }_{0}H=0$, 1, 4, and 8 T. The corresponding Hall measurement results are shown in Fig. 7 in Appendix A1. (b) $T$ dependence of the magnetization $M$ with an applied field ${\mu }_{0}H=0.01$ T. The left (right) inset is a sketch of the position of the spin-split (spin-degenerate) conduction band for temperatures below (above) the Curie temperature. (c) ${\rho }_{\mathrm{xx}}$ plotted as a function of ${\mu }_{0}H$ for $T=110$ K and 120 K. (d) Temperature dependence of the MR ratio ${\rho }_{\mathrm{xx},0}/{\rho }_{\mathrm{xx}}\left(H\right)$ with ${\mu }_{0}H=8$ T. The maximum is located at $T=106.9$ K. The Curie temperature is marked with a vertical dashed line in panels (a), (b), and (d).

Figure 2

(a) Arrott plot of the magnetization isotherms ($M^2$ vs $H/M$) in a temperature region of 80–130 K. The temperature interval is 0.5 K for 100–110 K and 2 K for the others. (b) $T$-dependences of the spontaneous magnetization $M_s$ (green squares, left) and the zero-field limit of the inverse magnetic susceptibility 1/${\chi }_0$ (yellow squares, right). Also shown are fitting curves (solid lines) to the critical equations. (c) Scaling plot of ${M\left|\varepsilon \right|}^{-\beta }$ vs ${H\left|\varepsilon \right|}^{-(\beta +\gamma )}$ for $T=98\text{–}110$ K, where $\varepsilon =(T-T_C)/T_C$ is the reduced temperature. (d) $T$ dependence of the zero-field resistivity derivative $d{\rho }_{\mathrm{xx},0}/dT$. In panels (b) and (d), the critical region is indicated with a gray zone.

Figure 3

(a) Magnetic field dependence of longitudinal resistivity ${\rho }_{\mathrm{xx}}$, (b) Dimensionless magnetization $m\left(H\right)=M\left(H\right)/M_{\mathrm{sat}}$, where $M_{\mathrm{sat}}$ is the saturation magnetization (equivalent to $3 {\mu }_B/{\mathrm{Cr}}^{3+}$). (c) Normalized resistivity ${\rho }_{\mathrm{xx}}\left(H\right)/{\rho }_{\mathrm{xx},0}$ plotted as a function of $m\left(H\right)$. The data for $T\ge 110$ K can be roughly scaled onto a single curve. The same set of temperatures (100–180 K) are used in panels (a) and (b).

Figure 4

(a) Temperature dependence of the resistivity in the form of $\ln \left({\rho }_{\mathrm{xx}}\right)$ vs $1000/T$ for magnetic fields ${\mu }_{0}H=0$, 1, 4, 8 T. The linear fit (dotted line) for $T>220$ K yields a thermal activation gap $\mathrm{\Delta }=0.19$ eV. (b) $T$ dependence of the inverse susceptibility ${\chi }^{-1}$ recorded at ${\mu }_{0}H=0.01$ T (solid line), which starts to deviate from the Curie-Weiss law (dotted line) at $T^{*}\approx 220$ K (see Appendix B1 for details). (c) The upper panel schematically shows the evolution of the $s$-orbital states with decreasing temperature. In zone $\mathrm{I} \left(T>T^{*}\right)$ the conduction band is spin degenerate, whereas in zone $\mathrm{III} \left(T<T_C\right)$ the ferromagnetic order causes a large spin splitting of the conduction band. The middle panel shows that spin correlation length $\xi$ becomes longer with decreasing temperature. In zone II, $\xi$ enters the nanometer scale and can substantially increase the size of magnetic polarons, as illustrated in the bottom panel.

Figure 5

(a) Derivative of resistivity, $d{\rho }_{\mathrm{xx}}/dH$, as a function of magnetic field for $T=120\text{–}160$ K. The position of the minimum is denoted as $H_{\mathrm{dRmin}}$. (b) Magnetization corresponding to the $d{\rho }_{\mathrm{xx}}/dH$ minimum, in the units of saturation magnetization, $M_{\mathrm{sat}}$; (c) Phase diagram in the ($T,H$) plane. In zone II, the $\left(T_{\mathrm{Rmax}},H\right)$ (squares) and the $\left(d{\rho }_{\mathrm{xx}}/dH\right)$ (circles) data roughly fall onto a single line, which separates zone II into two regions: one with isolated magnetic polarons (MPs) and the other with percolated or merged MPs.

Figure 6

Current-voltage characteristics at $T=140$ K for several pairs of electrical contacts on a $n\text{−}{\mathrm{HgCr}}_2{\mathrm{Se}}_4$ sample. The linear $I\text{−}V$ curves suggest that the contacts are ohmic despite that the impedances are on the order of $100 \mathrm{M}\mathrm{\Omega }$.

Figure 7

(a) The Hall resistivity of sample A in the temperature range from 4.2 K to 60 K and in magnetic field up to 9 T. The inset shows the Hall effect data at $T=4.2$ K, in which the anomalous Hall effect (AHE) can also be observed. (b) Temperature dependence of electron density $n_e$ and mobility $\mu$.

Figure 8

(a) Spontaneous magnetization $M_s\left(T\right)$ (squares) and the fit to the Eq. (A3a) (solid line). (b) Zero field inverse magnetic susceptibility 1/${\chi }_0\left(T\right)$ (squares) and the fit to Eq. (A3b) (solid line). (c) $M\left(H\right)$ at $T=105$ K (triangles) and 105.5 K (squares) and the fits to Eq. (A3c). Both $M$ and $H$ are in the logarithmic scale.

Figure 9

Temperature dependence of the quantities $X,Y$ (blue and gray square symbols), defined in the text, and the corresponding linear fits (red lines) to Eqs. (A4a) and (A4b). The temperature range with the data points falling onto the linear fits can be regarded as the critical region ($T=98\text{–}117$ K).

Figure 10

(a) Temperature dependence of the inverse magnetic susceptibility 1/$\chi$ (solid line) of $n\text{−}{\mathrm{HgCr}}_2{\mathrm{Se}}_4$ and the linear fit in the high temperature range (dashed line). (b) The temperature $T^{*}$, at which the deviation from the Curie-Weiss law begins, can be seen more clearly after the linear part of the inverse susceptibility curve is removed, i.e., $\mathrm{\Delta }{\chi }^{-1}\left(T\right)={\chi }^{-1}\left(T\right)-c(T-T_{\theta })$. The temperature coefficient $c$ and the Curie-Weiss temperature $T_{\theta }$ can be extracted from the linear fit shown in panel (a).

Figure 11

Carrier density dependence of the Curie temperature $T_C$ for several batches of $n\text{−}\mathrm{type} {\mathrm{HgCr}}_2{\mathrm{Se}}_4$ single crystals. The electron densities are extracted from the Hall measurements at 4.2 K. The Curie temperature $T_C$ is taken to be the kink point of the $M(T,H)\text{vs}T$ curve with a small magnetic field (typically $H=100$ Oe) applied. This method gives reasonably accurate $T_C$ values in comparison to those deduced from the critical exponent analysis.

Figure 12

Transport properties of an $n\text{−}\mathrm{type} {\mathrm{HgCr}}_2{\mathrm{Se}}_4$ single crystal (sample B). (a) Temperature dependences of the longitudinal resistivity plotted as $\ln \left({\rho }_{\mathrm{xx}}\right)\text{vs}1/T$ for magnetic fields ${\mu }_{0}H=0$, 1, 4, 8 T. The linear fit (dashed line) for $T>220$ K yields a thermal activation gap $\mathrm{\Delta }=0.28$ eV. The transport characteristics in temperature zones (I) $T>T^{*}$, (II) $T_C<T<T^{*}$, and (III) $T<T_C$ resemble those of sample A [see Fig. 4 in the main text]. (b) $T$ dependence of the MR ratio ${\rho }_{\mathrm{xx},0}/{\rho }_{\mathrm{xx}}\left(H\right)$ with ${\mu }_{0}H=8$ T. The maximum value is $4.2\times {10}^4$, which is located at $T=110$ K.

Figure 13

(a) X-ray diffraction (XRD) spectra of ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ in a temperature range of 35–299 K. (b) Temperature dependence of the lattice constant extracted from the XRD measurements (solid circles), which is compared with the Debye-Gruneisen (D-G) theory (dotted line).