Magnetism in Nb${}_{1-y}$Fe${}_{2+y}$ – composition and magnetic field dependence

Most measurements in previous studies,shiga1987 (); yamada1988 (); crook1995 () were carried out on polycrystalline samples of Nb${}_{1-y}$Fe${}_{2+y}$ prepared by arc-melting, followed by annealing for one week at $1000{}^{\circ }C$. X-ray powder diffraction was used to check for phase purity and to determine the lattice parameters, from which the composition was estimated.

Since C14-Nb${}_{1-y}$Fe${}_{2+y}$ crystallizes in a broad homogeneity range from 27.4 to 36.3 at.$\%$ Nb at $1100{}^{\circ }C$, and because the magnetic properties are very sensitive to $y$, it is important to verify not only the structure but also the final composition. We decided, therefore, to prepare samples in the range from 26 to 40 at.% Nb, covering the whole homogeneity range of the C14 phase, with two similar methods (indicated as method A and B in table 1) accompanied by characterization with X-ray powder diffraction, metallographic analysis, and wavelength and energy dispersive X-ray spectroscopy (WDXS and EDXS).

X-ray powder diffraction data (Fig. 1) have been collected on a Huber G670 diffractometer equipped with a Ge monochromator. Co K${\alpha }_1$ radiation was used instead of standard Cu K${\alpha }_1$ because of the strong absorption of iron. Silicon (National Institute of Standards & Technology SRM 640c, $a=5.43119\left(1\right)$ Åwas used as the internal standard. The lattice parameters were refined by least-squares fits of the diffraction angles in the range ${20}^{\circ }\le 2\theta \le {100}^{\circ }$, where $2\theta$ is the diffraction angle, using the program PPLP.gabe1989 ()

Together with the expected hexagonal C14 Laves phase (space group $P{6}_3/mmc$), X-ray powder diffractograms show evidence of a second phase with Ti${}_2$Ni structure type (space group $Fd\overline{3}m$, $a\approx 11.3$ Å) in almost all investigated samples. The structure type of the second phase was confirmed for the sample at the composition Nb${}_{1.06}$Fe${}_{1.94}$ (see Tab. 1) by electron-back-scatter diffraction.

According to EDXS analysis this phase is richer in Nb compared to the Laves phase. The actual composition of this phase could not be accurately determined due to its small grain size. It is plausible that in a sample with a large amount of this Nb-rich second phase the C14 main phase will be Fe-rich. This argument may explain the difference in lattice parameters between the samples with $y=0$ and $y=0.003$, which were both extracted from the same ingot.

The main difference between the two methods emerged in the metallographic analysis: While the first method resulted in high homogeneity in all regions of the final pellet with very small isolated filaments of the Ti${}_2$Ni-like phase, the second method produced drastic differences in homogeneity between the upper and the lower zone inside the pellet. The zone with almost no extra phases appears to be the middle of the pellet.

After having cut the samples into horizontal slices, we ground and polished them carefully (in two steps) and observed the surface with a high resolution optical microscope. The inset in Fig. 2 shows the optical micrograph taken from one of the upper slices of the NbFe${}_2$ polycrystal prepared with method B. It is the same sample as that measured in Fig. 1, where we observed the richest amount of second phase. The phase can be seen clearly near the C14-NbFe${}_2$ grain boundaries, which have dimensions of about $50\mu m$.

Afterwards, we measured the AC-susceptibility at low magnetic fields ($H\le 250Oe$) of samples extracted from the upper and the middle slice (Fig. 2): In the upper slice, we observed a FM signal with a transition temperature $T_s\approx 80K$, in addition to the expected SDW peak at $T_N\approx 10K$. In the middle slice, only the SDW signature is present. Comparing these measurements with the metallographic analysis, we conclude that the second phase is responsible for the FM part of the signal. Repeating the same procedure with different samples, we observed that $T_s$ is different in every sample, varying from 250 K to 15 K. It probably depends on the real Nb concentration at the grain boundaries. Looking at the Nb${}_{1-y}$Fe${}_{2+y}$ phase diagram (Fig. 19), this hypothesis makes sense because the Nb-rich region is FM and the transition temperature varies strongly with the quantity of Nb. In magnetic fields higher than 250 Oe these FM features disappear. It is therefore necessary to evaluate the quality of any sample by not only measuring the residual resistance ratio (RRR), but also the AC-susceptibility in zero magnetic field.

The lattice parameters of all polycrystals are listed in Table 1, together with the initial nominal compositions, WDXS results and transition temperatures. The discrepancy between the values of our lattice parameters for NbFe${}_2$ and those reported in literature is less than 0.001 Å.shiga1987 (); bi1996 (); zhu2003 () The measurements that we are going to present have been performed on a selected number of polycrystals, which show none or at most very low presence of the second phase.

Fig. 3 shows the average volume per atom of the C14 structure versus the nominal (solid line) and WDXS (dashed line) Nb content. The final composition of most samples, as determined by WDXS, is slightly richer in Nb than the nominal composition. In the range from 28 to 34 at.$\%$ Nb, a linear dependence of the atomic volume on composition determined by WDXS according to Vegard’s volume is observed. Around the upper border of the homogeneity range ($\approx 35$ at.% Nb), the sample composition could not be determined accurately by WDXS, probably because of the large amount of the second phase. Using the relation

obtained by a least-squares fit of the data, we can determine the composition of a sample by measuring the lattice parameters. This is especially useful for detecting small differences in composition that are difficult to observe using WDXS. The accuracy in the WDXS experiments is of the order of $0.1\%$ limiting the accuracy on $y$ to $\sim 0.003$. We note that a small systematic error of about 0.2 at.$\%$ cannot be ruled out. However, this error would only shift the absolute value without changing the shape of the phase diagram.

Measurements of the resistivity, heat capacity, magnetization and magnetic susceptibility down to 1.8 K have been carried out in a 9 T Physical Properties Measurements System and in a 7 T Magnetic Properties Measurements System (Quantum Design). High-resolution measurements of the resistivity at temperatures down to 50 mK were obtained in an adiabatic demagnetization refrigerator (Cambridge Magnetic Refrigeration) by a standard 4-terminal lock-in technique. The low temperature heat capacity was determined in the same refrigerator by a relaxation-time technique.

Fig. 4 shows the DC magnetic susceptibility of the two samples measured at 50 Oe. Clear jumps in magnetization are observed at $T\approx 72$ K and $T\approx 14.5$ K for the Fe-rich and the Nb-rich sample respectively. When increasing the external field, the transition temperature remains practically constant, suggesting that the ferromagnetism is not a consequence of the second phase. In addition, hysteresis signatures are evident in both samples, as shown in the inset of the same figure. Although in both samples the remanent magnetizations are very small, of the order of ${10}^{-2}{\mu }_B$ per atom, the coercive fields are quite large: 200 Oe and 700 Oe.

With increasing Nb excess the coercive field of Nb-rich samples decreases to a value that is not detectable anymore by the measurement technique, as in the case of soft magnets: The Arrott plot for sample $y=-0.06$ ($T_c=30$ K) in Fig. 5 at 2 K does not show any hysteresis, although the finite $M^2$ intercept for zero $H/M$ is a clear indication of a remanent magnetization and hence of ferromagnetism. For very slight Nb doping, $y=-0.012$, no FM signal has been observed down to 2 K and also the Arrott plot suggests a PM groundstate.

On the Fe-rich side, all samples exhibited large hysteresis (see inset of Fig. 5). The FM transition in samples $y=+0.04$ and $y=-0.035$ is also inferred by the behavior of ${\chi }^{-1}$ versus $T$ shown in Fig. 6. All samples in the homogeneity range follow a Curie-Weiss law below 100 K with a slope $C=1/\left(T\chi \right)$ corresponding to a large fluctuating moment ${\mu }_{eff}$ compared to the ordered moment ($\sim 0.02{\mu }_B$), where $C^{-1}=\frac{1}{3}\frac{N}{V}{\mu }_0{\mu }_{eff}^2/k_B$ and $\frac{N}{V}$ is the atomic number density.

As the composition is varied from slightly Fe-rich ($y>0$) to slightly Nb-rich ($y<0$), ${\mu }_{eff}$ remains constant at $\simeq 0.9{\mu }_B$ per atom (inset of Fig. 6), whereas the Curie-Weiss temperature ${\theta }_{CW}$ changes sign, from ${\theta }_{CW}\approx 3$ K for $y=0$ to ${\theta }_{CW}\approx -3$ K for $y=-0.012$.

To study the ground state properties of the FM phase, we measured the electrical resistivity and the heat capacity of a FM sample with $y=+0.015$ down to 100 mK. Although the sample shows a small SDW signal (peak in the magnetic susceptibility) at $T_N=32$ K, its groundstate properties are dominated by the FM transition at $T_c=25$ K: the resistivity $\rho \left(T\right)$, shown in Fig. 7, suddenly decreases only at the FM transition temperature $T_c=25$ K, following a $T^{5/3}$ law above and below $T_c$ (Ref. 11), which indicates a strong influence of FM fluctuations on the electron scattering rate. Below $T_{FL}\simeq 2$ K, $\rho \left(T\right)$ recovers a Fermi-liquid $T^2$ dependence (lower inset). The magnetoresistance is almost flat at 100 mK up to 1 Tesla (upper inset).

The temperature dependence of the heat capacity divided by temperature, $C_p/T$, of the sample with $y=+0.015$ is shown in Fig. 8. No clear heat capacity anomalies are visible at the SDW and FM transition temperatures (arrows), which suggests small ordered moments. Below 2 K, $C_p/T$ levels off at a value of 42 mJ/K${}^2$mol, consistent with a Fermi-liquid ground state.

Below 300 mK, however, $C_p/T$ increases dramatically, probably because of the nuclear Schottky contribution of the Nb atoms subject to a the internal field of a ferromagnet (inset). By fitting the points below 2 K with a Schottky contribution, we obtain $C_p/T=(41.6+0.02(T/K{)}^{-3})mJ/K^{2}mol$. From this equation we can estimate the internal magnetic field $B_{eff}$ experienced by the Nb atoms:

where ${\gamma }_{NMR}$ is given by NMR experiments and $I$ is the nuclear spin of the Nb atoms. For ${}^{93}$Nb atoms with $I=9/2$ and ${\gamma }_{NMR}=10.4$ MHz/T, the resulting field is $B_{eff}=1.08$ T. Only in this FM sample (the only FM sample measured down to 50 mK), has the Schottky contribution to the heat capacity been clearly observed.

We conclude that the groundstate of the Fe-rich compounds is ferromagnetic and that the low temperature transport and thermodynamic properties are well explained by the Fermi-liquid theory, as in many other FM transition metal compounds.niklowitz2005 ()

We focus now on stoichiometric NbFe${}_2$. Based on the broadening of the NMR line width and on the “S”-like feature in the magnetization curves below $T_N=10$ K, NbFe${}_2$ had been reported to exhibit spin density wave order at low temperature.shiga1987 (); yamada1988 (); crook1995 () This interpretation is consistent with our measurements of the magnetic susceptibility, specific heat capacity and magnetization.brando2008 (); moroni2005 (); brando2006 () Based on these earlier results, we suggested that the low temperature magnetic order in stoichiometric NbFe${}_2$ may take the form of a SDW with an helical arrangement of the Fe spins. Direct evidence for SDW order in NbFe${}_2$ from neutron scattering is still outstanding,cywinski () but a renewed attempt, using large single crystals grown in an infrared mirror furnace, is scheduled.

Fig. 9 shows the real and imaginary components of the AC-susceptibility. The distinct SDW peak is visible at 10 K, cutting off the Curie-Weiss-like behavior with decreasing temperature (see Fig. 6). The low transition temperature, for a d-metal compound, as well as the geometrically frustrated Kagomé structure of the Fe atoms in the C14 phase, might suggest a spin-glass transition.

However, the measured Curie-Weiss temperature ${\theta }_{CW}$ is close to 1 K giving a frustration factor $f={\theta }_{CW}/T_N$ of only 0.1. By contrast, the hall-mark of frustration in local moment systems is a frustration factor $f>1$. Moreover, the lack of a loss peak in ${\chi }^{\prime \prime }$ at $T_N$, as well as the absence of a distinct frequency dependence of ${\chi }^{\prime }$ argue against a spin-glass transition.

The analysis of the irreversible field-cooled (FC) and zero-field-cooled (ZFC) measurements (not shown) indicates a slight splitting of the magnetization curves at $T_N$, but the effect is so small that it would be difficult to attribute it to the main phase. Only at much lower temperatures ($T\le 3$ K) and at very low fields does this splitting become more pronounced, suggesting the formation of magnetic domains.dennis-thesis2006 () This behavior will be analysed in the next section, where it resurfaces in a slightly Fe-doped sample.

An applied magnetic field $H$ shifts the ${\chi }^{\prime }$ peak to lower $T$ and suppresses the magnetic order at a critical field ${\mu }_0{H}_c\approx 0.6T$ at $100mK$, switching the system to a paramagnetic state. This can be seen more clearly in the magnetization and susceptibility plots in Fig. 10. Such a behavior could be interpreted as a metamagnetic transition from a paramagnetic (PM) state to another PM state with a higher magnetization in field. Such transitions are not uncommon in nearly FM metals, such as YCo${}_2$,sakakibara1990 () but here, in contrast to the metamagnetic PM-to-PM case, $H_c$ is shifted to lower values with increasing temperature (see color plot of Fig. 16).brando2008 ()

A further indication of a bulk SDW transition at $T_N$ is given by the Arrott plot in Fig. 11, in which the high field, high temperature linear dependence of $M^2$ on $H/M$ changes over to an arc with negative slope below $T_N$ and $H_c$.yamada1988 () Moreover, in the specific heat capacity (Fig. 12), the bulk SDW transition is visible as a small hump at $T_N$, when the data are compared with measurements in magnetic fields $H\ge H_c$.brando2006 () In Fig. 12, this hump is shown not only for $y=0$ but also for the slightly Fe-rich sample with $y=0.007$, which shows signatures of an SDW transition as well, at $T_N\simeq 18K$.

The entropy below the transition is very low, of the order of $4mJ/Kmol$. Fig. 12 also shows the heat capacity of a slightly Nb-rich sample ($y=-0.012$). At this composition, no indication of a transition has been observed in the heat capacity down to 2 K. The inset of the Fig. 12 shows how the high temperature data can be fitted with the electronic and phononic contribution:

where the Sommerfeld coefficient is ${\gamma }_{el}=14.2mJ/K^{2}mol$ and the Debye temperature is ${\theta }_D=348K$. This Sommerfeld coefficient agrees very well with that predicted by band-structure calculations.takayama1988 (); inoue1989 () However, $Cp/T$ increases towards low temperatures $T<20K$ and saturates below 1 K near a value of $\approx 48mJ/K^{2}mol$ (cf. Fig. 17),wada1990 () three times larger than the band structure value.

The most dramatic signature of the SDW state is the behavior of the magnetoresistance ${\rho }_M$: Although the SDW transition cannot be detected in the temperature dependence of the resistivity, the low temperature magnetoresistance ${\rho }_M$ shows a jump of about $10\%$ at the critical field in NbFe${}_2$, and even more ($\approx 17\%$) in the Fe-rich sample with $y=0.007$, as shown in Fig. 13. The magnitude of the jump thus appears to be related to $T_N$. At $T_N$ this feature disappears and ${\rho }_M$ follows a linear field dependence at high field, as in a paramagnetic metal.

In NbFe${}_2$, all magnetic signatures below $T_N$ are practically identical to those observed in the helically ordered compound MnAu${}_2$, although the latter has a much higher transition temperature $T_N\simeq 380K$.samata1998 () Furthermore, the large magnetic anisotropy observed in single crystals,kurisu1997 () and the the critical field ${\mu }_0{H}_c=0.6T$ required to destroy the SDW state, which is a surprisingly low value when compared to $T_N$, suggest a long wavelength incommensurate magnetic groundstate. Moreover, the hexagonal C14 structure may favor helical order with propagation wavevector along the $c$-axis, as in rare earth metals.blundell2001 () Finally, the large Stoner factor $S\simeq 180$ ($I\cdot N\left({ϵ}_F\right)=0.99\approx 1$), computed by comparing the value of $\chi$ at $T_N$ to the one expected from band-structure DOS,takayama1988 (); inoue1989 () and the high Wilson ratio of about 60,brando2008 () indicates that this compound is on the border of a ferromagnetic instability. All these observations led us to propose that the SDW state grows out of the $Q=0$ ferromagnetism existing in Fe-rich NbFe${}_2$, and develops into a $Q\ne 0$ helical state (as illustrated in Fig. 20).

To examine our hypothesis that the SDW state grows continuously out of the FM phase in Fe-rich samples, we measured a slightly Fe-doped sample with $y=0.007$, which is located in the phase diagram around the point of crossover between the pure FM phase and the SDW one. The AC-susceptibility, plotted in Fig. 14, shows the same sharp peak as in NbFe${}_2$, but at a higher temperature $T_N=18K$. The bulk transition is also visible in the specific heat measurement shown in Fig. 12.

Hysteresis phenomena appear only at temperatures below $10K$, as shown, for example, in the Arrott plot of Fig. 5 at 2 K: In this plot the sample seems to be SDW between 10 K and 18 K, but develops a remanent magnetization suggesting FM at lower temperatures. These features are illustrated more drastically in the ${\rho }_M$ isotherms of Fig. 13: Below $T_N$, the sample develops a clear negative magnetoresistance, and ${\rho }_M$ jumps towards lower values as the critical field $H_c$ is exceeded. The magnitude of the jump increases with decreasing temperature. Around $10K$, ${\rho }_M$ begins to shows hysteresis until at 2 K the “up” and “down” lines are well separated. The resulting curves suggest that the hysteresis phenomena detected in the stoichiometric samples are not a consequence of secondary FM phases, but are an intrinsic property of the SDW state.

Supporting evidence for this interpretation is provided in Fig. 15, which shows a sequence of magnetization and AC-susceptibility isotherms for the same sample. As already pointed out by Crook and Cywinski,crook1995 () the hysteresis lines separate firstly not at $H=0$ but at finite fields. Following the measurements from the top to the bottom frame of the figure, we recognize the double peak at $15K$ and ${\mu }_0{H}_c=0.15T$ as a signature of the SDW phase. As the temperature is lowered further, between $15K$ and $4K$ hysteresis develops, which straddles $H_c$ but does not yet reach $H=0$. Finally, close to $2K$, the size of the hysteretic field region includes zero field, so that the two hysteresis regions associated with $+H_c$ and $-H_c$ merge, giving rise to a FM remanent magnetization.

All investigated samples with Fe concentrations $0\le y\le 0.02$ display qualitatively the same phenomena, only the transition temperatures and the magnitudes of the effects change with increasing $y$. For $y\ge 0.02$ the samples seem to show only the FM character. The fact that in these very iron-rich, purely FM samples the magnetoresistance ${\rho }_M$ is much weaker and does not show any hysteresis suggests that the hysteretic signatures in $M\left(H\right)$, ${\rho }_M\left(H\right)$ and ${\chi }^{\prime }\left(H\right)$ are a characteristic property of this specific SDW state and indicates the presence of SDW domains, as recently discussed by Jaramillo et al. .jaramillo2007 ()

While Fe-excess increases the SDW ordering temperature $T_N$, and eventually induces low-ordered-moment ferromagnetism, slight Nb-excess weakens the SDW state and tunes the system towards paramagnetism and therefore to a QCP. This has been observed in the behavior of the magnetic susceptibility $\chi$ in two samples with $y=-0.006$ and $y=-0.012$. Fig. 16 shows the color plots of $\chi (H,T)$ for the two samples at $2K$: The ordered SDW phase for the $y=-0.006$ sample, defined by the bright zone in the $H-T$ diagram, shrinks to a point for $y=-0.012$. A similar plot for a stoichiometric sample ($y=0$) is shown in Ref. 11.

Additional evidence for this evolution towards a paramagnetic state close to $y=-0.015$ is given by the sequence of Arrott plots shown in Fig. 11. For $y=0$ and $y=-0.006$, the $M^2$ vs. $H/M$ curves change appearance at $T_N$: for $T>T_N$, the curves bend towards the left, whereas for $T<T_N$ they bend towards the right. By contrast, for $y=-0.012$, no such change in behavior occurs down to $2K$.

The heat capacity is less sensitive to the magnetic transition and so the weak anomaly that was observed in the stoichiometric sample at 10 K, has not yet been observed in either Nb-rich sample. The surprisingly high value of the Sommerfeld coefficient $\gamma$ at low temperature for all the samples close to the stoichiometric composition has been considered as indication of the presence of strong spin fluctuations and of the vicinity of the system to a QCP.brando2006 (); brando2007 () Approaching the QCP by increasing the Nb-content, $\gamma$ gradually increases and, for $y=-0.012$, it follows a logarithmic temperature dependence down to about $0.6K$ (Fig. 17).

Non-Fermi-liquid behavior has also been observed in the temperature dependence of the electrical resistivity. The inset of Fig. 18 shows the temperature dependence of the resistivity $\rho \left(T\right)$: its power-law exponent $n=d\left(log\Delta \rho \right)/d\left(logT\right)$ varies from $3/2$ to $5/3$ between $3K$ and $0.6K$ and crosses over to the FL exponent 2 below $0.6K$.brando2007 () Combining both observations, the $T$-dependence of the heat capacity and of the resistivity below 3 K, the sample Nb${}_{1.012}$Fe${}_{1.988}$ seems to be close to a ferromagnetic QCP, where $C/T\propto log\left(T\right)$ and $\rho \propto T^{5/3}$ have been predicted for a 3D FM metal.moriya1985 (); lonzarich1997 () Below $0.6K$, Nb${}_{1.012}$Fe${}_{1.988}$ experiences a crossover to the conventional FL state.

Magnetoresistance measurements below $2K$ (Fig. 18) suggest that Nb${}_{1-y}$Fe${}_{2+y}$ does not have a paramagnetic ground state at $y=-0.012$. A little drop (less than 1$\%$) in ${\rho }_M$ is observed at 2 K and clear hysteresis phenomena develop below $1K$. This indicates the presence of very weak SDW order below $T_N\simeq 2K$, which develops domains at $T\le 1K$.

These findings contrast with our observations on a high quality Nb-rich ($y\simeq -0.01$) single crystal, in which non-Fermi liquid phenomena have been observed down to the lowest temperature.brando2008 () Despite residual low-temperature order below $T_N\simeq 2.8K$, presumably with a very small ordered moment, the resistance of the single crystal follows a $T^{3/2}$ power law down to $50mK$, and its heat capacity coefficient $C_p/T$ increases logarithmically with decreasing temperature, without any indication of a crossover to a FL state. The return of our Nb-rich polycrystal with $y=-0.012$ to FL behavior at low temperature, when such a return to FL behavior is not observed in a single crystal with a similar composition, may be attributed to the lower quality of the polycrystal either in terms of its purity or its homogeneity.

We interpret these findings as the first clear indication of a logarithmic breakdown of the Fermi-liquid state in a transition metal antiferromagnet: The fact that the precise power-law exponent varies between 3/2 and 5/3 and the discrepancies between the phenomena observed in single and polycrystals suggest that the precise form of $\rho \left(T\right)$ may depend on stoichiometry and on the sample orientation and that present data cannot decide whether FM or SDW spin fluctuations determine the low temperature behavior.