Ordinary and intrinsic anomalous Hall effects in Nb${}_{1-y}$Fe${}_{2+y}$

The Hall effect of the dilution series Nb${}_{1-y}$Fe${}_{2+y}$ has been studied on selected samples spanning ferromagnetic (FM) and spin-density-wave (SDW) ground states, including a sample in the vicinity of the quantum critical point. Ordinary and anomalous contributions are observed, with positive ordinary Hall effect for all single crystals dominating at high temperatures. Consistent analysis of the anomalous contribution is possible for iron-rich Nb${}_{0.985}$Fe${}_{2.015}$ with ferromagnetic ground state and a SDW state at elevated temperatures and stoichiometric NbFe${}_2$ with a SDW ground state. The anomalous Hall coefficient is consistent with an intrinsic (Berry-phase) contribution, which is constant below the ordering temperature $T_N$ and gradually vanishes above $T_N$. The ordinary Hall coefficient is positive at all temperatures with a moderate temperature dependence. For stoichiometric NbFe${}_2$ and niobium-rich Nb${}_{1.01}$Fe${}_{1.99}$—both having a SDW ground state—an additional contribution to the Hall resistivity impedes a complete analysis and indicates the need for more sophisticated models of the anomalous Hall effect in itinerant antiferromagnets.

Figure 1

Phase diagram of Nb${}_{1-y}$Fe${}_{2+y}$ after Ref. 12. Arrows indicate the three samples selected for Hall effect measurements. The inset illustrates sample orientation for Hall effect and resistivity measurements.

Figure 2

(a) Magnetic ac susceptibility of the studied Nb${}_{1-y}$Fe${}_{2+y}$ single crystals. Arrows indicate magnetic transitions identified as peaks. (b) Temperature dependence of the resistivity for the selected single crystals. Normalization to the resistivity at $T=300\mathrm{K}$ allows for best comparison of the sample excluding uncertainties in contact geometry. With phononic scattering processes dominating at these high temperatures the resistivity of all single crystals may be converted into absolute resistivities using an approximate room temperature value of $\rho (T=300\mathrm{K})=100\mu \Omega \mathrm{c}\mathrm{m}$. Arrow marks the FM transition temperature $T_C$ of sample 1 Nb${}_{0.985}$Fe${}_{2.015}$.

Figure 3

Temperature dependence of the Hall coefficient for samples of Nb${}_{1-y}$Fe${}_{2+y}$ with $y=-0.01$, 0, and 0.015 at a fixed field of $1\mathrm{T}$ as defined in Eq. (2). Open symbols denote the ordinary Hall coefficient $R_0$ as extracted for iron-rich Nb${}_{0.985}$Fe${}_{2.015}$ and stoichiometric NbFe${}_2$ from fits of Eq. (4) to ${\rho }_H\left(H\right)$ as shown in Figs. 4a and 5a. Arrows mark the (zero-field) transition temperatures $T_C$ and $T_N$.

Figure 4

Hall resistivity of Nb${}_{0.985}$Fe${}_{2.015}$ as a function of magnetic field. For clarity, the curves have been shifted subsequently by $-5\times {10}^{-2}$ $\mu \Omega$cm (cf. colored bars on $y$ axis). Solid lines reflect least-square fits of Eq. (4) to ${\rho }_H\left(H\right)$ utilizing the magnetization and magnetoresistance measured simultaneously (cf. Fig. 6). Inset shows a magnified view of the data and fits at low fields and low temperatures. (b) Hysteresis of Hall resistivity and (c) magnetization in the ferromagnetic phase of iron-rich Nb${}_{0.985}$Fe${}_{2.015}$. In order to obtain the Hall resistivity at both positive and negative magnetic field, the Hall resistivity was calculated by subtracting the magnetoresistance contribution such that the hysteresis loops are centered around the origin.

Figure 5

Hall resistivity of Nb${}_{1-y}$Fe${}_{2+y}$ as a function of magnetic field for (a) $y=0$ and (b) $y=-0.01$. Curves in (a) have been offset subsequently by $-2.5\times {10}^{-2}$ $\mu \Omega$cm for clarity (cf. colored bars on ordinate). Lines represent fits of Eq. (4) to the data up to the field indicated by arrows.

Figure 6

Isotherms of the (a) magnetoresistance and (b) magnetization of the studied Nb${}_{1-y}$Fe${}_{2+y}$ single crystals at $7\mathrm{K}$.

Figure 7

Decomposition of ordinary and anomalous contributions for (a) Nb${}_{0.985}$Fe${}_{2.015}$ and (b) NbFe${}_2$. Plotting ${\rho }_H/{\mu }_{0}H$ versus $\rho M/{\mu }_{0}H$ reveals the ordinary Hall coefficient $R_0$ as the intercept with the ordinate, whereas the slope reflects the coefficient of the anomalous contribution $S_H$. Lines are the same fits to the data included in Fig. 4. (b) For NbFe${}_2$ the fits are limited to small magnetic fields [cf. arrows in Fig. 5a].

Figure 8

(a) Anomalous and (b) ordinary contribution of the Hall effect of Nb${}_{0.985}$Fe${}_{2.015}$ (black) and NbFe${}_2$ (red) as extracted from fits of Eq. (4) to ${\rho }_H\left(H\right)$ [cf. Figs. 4a and 4b]. Open squares and diamonds reproduce $R_H$ measured at $1\mathrm{T}$ as shown in Fig. 3. Lines are guides to the eye. Arrows denote (zero-field) transition temperatures $T_N$ and $T_C$.

Figure 9

Representation of ${\rho }_H/H$ versus ${\rho }^{2}M/H$ for niobium-rich Nb${}_{1.01}$Fe${}_{1.99}$. Solid lines highlight the low-field linear regime and represent the fits to ${\rho }_H\left(H\right)$ given in Fig. 5b.