Oscillations of the Hall Resistivity in Natural Single Crystals of Pyrrhotite Associated with the Orientation of the Elementary Magnetic Moments of Fe Atoms

The Hall resistivity of natural single crystals of pyrrhotite Fe7S8 exhibits oscillations with the inverse external magnetic field 1/B at temperature 77 K and B up to 0.73 Tesla. This resembles phenomena due to Landau quantization of the carriers demanding very pure samples, temperatures near 4 K and magnetic fields of several Tesla. However, none of these requirements is met in the experiments. The oscillations appear only when there is orientation of the elementary magnetic moments of Fe atoms, which happens when B is parallel to the c‐plane at 77 K. At room temperature with the orientation destroyed by the thermal agitation and for B parallel to the c‐axis along which the alignment of the Fe magnetic moments is negligible, the oscillations disappear. According to the s–d model proposed for heavily doped magnetic semiconductors, defects and impurities produce large local fluctuations of carrier concentrations. These through the strong s–d exchange interaction between the carriers and the lattice magnetic moments of Fe establish variations of local magnetization. These constitute scattering centers which are enforced for certain values of B, though for others weaken giving the oscillatory behavior of the Hall resistivity.


DOI: 10.1002/pssb.202300118
The Hall resistivity of natural single crystals of pyrrhotite Fe 7 S 8 exhibits oscillations with the inverse external magnetic field 1/B at temperature 77 K and B up to 0.73 Tesla.This resembles phenomena due to Landau quantization of the carriers demanding very pure samples, temperatures near 4 K and magnetic fields of several Tesla.However, none of these requirements is met in the experiments.The oscillations appear only when there is orientation of the elementary magnetic moments of Fe atoms, which happens when B is parallel to the c-plane at 77 K.At room temperature with the orientation destroyed by the thermal agitation and for B parallel to the c-axis along which the alignment of the Fe magnetic moments is negligible, the oscillations disappear.According to the s-d model proposed for heavily doped magnetic semiconductors, defects and impurities produce large local fluctuations of carrier concentrations.These through the strong s-d exchange interaction between the carriers and the lattice magnetic moments of Fe establish variations of local magnetization.These constitute scattering centers which are enforced for certain values of B, though for others weaken giving the oscillatory behavior of the Hall resistivity.
Landau quantization of carriers giving a Shubnikov-de Haas behavior. [16]The possibility of observing such oscillations will be discussed in the following and other probable sources of it will be considered.

Experimental Section
The measurements were performed on parallelepiped samples approximately 1.5 Â 0.4 Â 0.2 cm 3 cut from natural hexagonal crystals of pyrrhotite with a composition close to Fe 7 S 8 , as it was revealed from the density of the samples, the determination of the percentage of Fe atoms and XRD diffractograms [17,18] In Figure 1, it is shown that the way of cutting a sample suitable for measurements with B parallel to the c-axis and another one with B perpendicular to it, with the electric field E applied along the samples.Their homogeneity was tested by passing a direct current and measuring the distribution of voltage along them.
A four-probe method was used for the Hall resistivity measurements in a cryostat filled with inert He gas.The temperature was monitored by an Oxford intelligent temperature controller ITC4.A Keithley 2400 SourceMeter and a Keithley 182 sensitive digital voltmeter were used to control and measure automatically via a PC the current and the voltage, respectively.A Varian water-cooled 4 in.electromagnet provided the magnetic field which was measured by a Walker MG-5D magnetometer.Each measurement was repeated ten times, from which the average and the standard deviation of the mean were calculated.

Results and Discussion
In Figure 2, the Hall resistivity ρ H = f (1/B) is shown for a sample with B perpendicular to c-axis at 77 K.The oscillatory behavior is apparent and a fitting of the experimental data by a relation of the form is included, which describe Shubnikov-de Haas oscillations of the Hall resistivity ρ H for a magnetic material. [16]In Equation ( 1), R 1 is the extraordinary Hall coefficient, μ 0 is the permeability of the vacuum, M(B) is the magnetization of the material, r is a constant that is inversely proportional to the Fermi surface curvature ∂ 2 A 0 It can be seen that the sample approaches saturation for B higher than 0.4 T approximately.As magnetic saturation M s is achieved for high B, the corresponding linear part of the curve in Figure 3b is described by the relation with intercept R 1 μ 0 M s = (6.488AE 0.002) Â 10 À9 Ω m and slope C = (0.037AE 0.002) Â 10 À9 m 3 C À1 .Taking R 1 = 1.4 Â 10 À7 m 3 C À1 for pyrrhotite [3][4][5] the saturation magnetization M s = (3.688AE 0.003) Â 10 4 A m À1 at 77 K results.The values of M s for the six series of measurements vary in the range 3.6-4.6Â 10 4 A m À1 .In the study of Besnus et al., [7] saturation magnetization at 77 K approximates the value M s = 6.520Â 10 4 A m À1 , although this is achieved in samples with a magnetic field higher than 1-1.5 T perpendicular to c-axis.So, the lower value of M s may be attributed to the lower value %0.73 T of the magnetic field used in our measurements.Although the fitting of the experimental data by Equation ( 3) seems satisfactory for all the results obtained, the conditions under which this equation is valid are hardly met.Shubnikov-de Haas oscillations are due to Landau quantization of the energy of the cyclotronic motion of the carriers on a plane perpendicular to the magnetic field B. [19] The energy separation of the Landau levels increases with B according to the relation where and In the above equations, ω c and m c * are the cyclotronic frequency and the effective cyclotronic mass, respectively, A is the area enclosed by the carrier in the inverse space, and E F is the Fermi energy. [20]y increasing B, the separation of the Landau levels increases and as each of them exceeds the Fermi energy depopulates to the succeeding level of lower energy, causing a fluctuation of the carrier concentration near the Fermi energy.This generates an oscillatory behavior of thermomagnetic and transport effects with a period of where A 0 is the cross-section of the Fermi surface perpendicular to the magnetic field.This provides a reliable way to reveal details of the Fermi surface along different directions of the crystal lattice. [15,16]n practice, temperature broadens the Landau levels by kT, where k is the Boltzmann constant and the collisions of the carriers with impurities, lattice defects, phonons, and the sample surfaces introduce an additional broadening of the order ħ/τ, where τ is the relaxation time. [20]This broadening may be considered due not to collisions, but to an increase of the temperature in Equation (3) by T D , the Dingle temperature.
It is obvious that oscillations of the Shubnikov-de Haas type require samples of high purity, intense B of several Tesla, and very low temperatures of a few Kelvin degrees.None of these requirements is met in our case: the samples of natural pyrrhotite contain high concentrations of impurities and lattice defects, [3][4][5] 77 K is too high a temperature, and 0.73 T is too weak a magnetic field.However, Equation (3) fits well with the experimental results of our six series of measurements.A remarkable characteristic of them is that although the amplitude of the oscillations decrease with decreasing B, they persist for B as low as 0.15 T.Moreover, all the six oscillatory results have periods in the rather narrow range 0.67-0.97T À1 .
The semiperiod results from the slope of a plot of the values 1/B for which maxima and minima occur in Figure 2 versus their corresponding integers, as it is shown in Figure 4. From Equation ( 6), the cross-section A 0 of the Fermi surface perpendicular to the magnetic field results 6.4 Â 10 15 -6.8 Â 10 15 m À2 .Although the application of the theory of Shubnikov-de  Haas is doubtful, the value of area A 0 is not unlikely.The crosssection of the "cylinders" of the Fermi surface of arsenic is 7.6 Â 10 15 m À2 comparable with our result. [21]The same is true for the Fermi energy, which from our measurements lies in the range 0.2-0.4eV in accordance with the experimental value 0.14 eV, which comes up from thermoelectric power measurements on pyrrhotite [22] and the energy band diagrams of this sulfide mentioned before.However, an unlikely small value of m c */m e % 4 Â 10 À4 comes up from Shubnikov-de Haas processing of our measurements.As mentioned before with B parallel to the c-axis an n-type conductivity is observed up to about 160 K reversing to p-type for higher temperatures.This reversal point was found to decrease with increasing external magnetic field being 158 K for B = 0.4 T, 156 K for 0.7 T and 154 K for 1.115 T. A similar behavior for Ge has been attributed to a small concentration of holes with effective mass smaller than that of an electron. [23]he behavior of the sample which gave the oscillatory behavior appearing in Figure 2 is shown in Figure 5 at room temperature 300 K. From Figure 5a, it is obvious that at room temperature the oscillations disappear and from Figure 5b we see that the thermal agitation makes the magnetic field 0.73 T inadequate to approach magnetic saturation.
Figure 6 shows the experimental points ρ H = f(1/B) from a sample with B parallel to the c-axis at 77 K.In this case oscillations do not appear.The available magnetic field of 0.73 T is too weak to orient the elementary magnetic moments along this axis.It would be interesting to see if oscillations appear in samples with B parallel to the c-axis for magnetic fields approaching the saturation value B % 10 Tesla.
For the sample of Figure 6 at room temperature 300 K the same situation occurs.The difficulty of the elementary magnetic moments orientation along the c-axis is enforced by the additional effect of thermal agitation.Lack of orientation is accompanied by the disappearance of oscillations Figure 7.
Apart from de Shubnikov-de Haas oscillations other oscillatory phenomena have been observed.26] This hole band splits into Landau sub-bands and its highest extreme lies near the bottom of the conduction band.With increasing B, the density of the electron states below the bottom of the conduction band is reduced periodically to zero. [25]owever, a temperature of 77 K and such a low magnetic field as B = 0.73 T, as in our case, do not seem suitable to produce this sort of oscillation in natural pyrrhotite.From the above, it can be concluded that neither the Landau quantization of the carriers nor the HTO theory can account for the observed oscillatory behavior of ρ H .
[29] In these solids, large fluctuations of the carrier density occur due to impurities and other defects randomly distributed in them.The strong indirect exchange interaction via the conduction carriers tends to establish a ferromagnetic ordering of the lattice magnetic moments in regions around impurities which provide an excess of conduction carriers, though the disorder is less in regions with  In natural pyrrhotite, there are dislocations with dangling chemical bonds and impurity atoms like Ni, Co, Cu, Zn, and Mn in high concentrations between 10 18 and 10 21 cm À3 randomly distributed in the samples. [17]This produces energy levels and narrow bands with localized or nearly localized carriers yielding spatial fluctuations of carrier concentration and magnetization.As in pyrrhotite, the orientation of the Fe magnetic moments is easy on the c-plane and very difficult normally to it, any disorder due to conduction carriers' concentration should extend almost exclusively on this plane.
With increasing B for certain values of it the fluctuations of magnetization in the c-plane diminish, though for others new fluctuations appear.Actually, in a sample with regions with high carrier concentration where magnetization has approached its maximum values the external magnetic field may be able just to raise the magnetization of regions with lower carrier concentration diminishing the magnetic fluctuations and so the scattering of the carriers.However, as the external magnetic field B increases, the magnetization of partly magnetized regions may be raised to such an extent that more fluctuations occur promoting the scattering.
As the orientation of the Fe magnetic moments approaches saturation with the increasing of B parallel to the c-plane, the scattering of the carriers is more and more dominated by the local fluctuations of the magnetization induced by the s-d exchange interaction, raising the amplitude of the observed oscillations.In contrast, at room temperature, the thermal agitation of the Fe magnetic moments introduces such a disorder that the domination of the scattering from the magnetic fluctuations is diminished making oscillations to disappear.Moreover, when the magnetic field B is parallel to the hard c-axis, the orientation of the magnetization fluctuations cannot practically change as they lie on the easy c-plane.So, the scattering centers remain basically invariable and oscillations are not observed.
The magnetic moment's rotation out of the c-plane at 77 K is estimated at about 14-15 deg. [9,30]The component of B parallel to the new orientation is not strong enough to produce oscillations, as when B is parallel to the c-plane.It would be interesting to see if oscillations appear with more intense B along the c-axis.

Conclusion
The oscillations of the Hall resistivity ρ H periodic in 1/B have been observed on natural single crystals of pyrrhotite at 77 K and magnetic fields up to 0.73 T. Six such oscillatory behavior appeared in measurements on three different samples and showed periods in the range (0.67 AE 0.05)-(0.97AE 0.05) T À1 .These oscillations appear only when there is an orientation of the elementary magnetic moments by the applied magnetic field B, i.e., on samples with B in the c-plane at low temperature 77 K.At room temperature where thermal agitation impedes the alignment of the elementary magnetic moments of Fe atoms and with B parallel to the difficult c-axis, along which the alignment of Fe magnetic moments is negligible for the highest of our magnetic field, oscillations cease to exist.These oscillations cannot be explained by Landau quantization of carriers, as this requires very pure samples, temperatures near 4 K, and magnetic fields of several Tesla.None of these requirements is met in the above measurements.
A possible explanation is given by the s-d model proposed for heavily doped magnetic materials, according to which the intense exchange interaction between the carriers and the Fe magnetic moments disturbs the orientation of the latter.As impurities and defects induce variations of carrier concentration around them, they cause fluctuations of magnetic moments which become intense scattering centers of carriers.
In pyrrhotite, the orientation of the Fe magnetic moments is easy on the c-plane and much more difficult normally to it, along the c-axis, so it is expected that the magnetic fluctuations should occur preferably on the c-plane.For some intensities, the external magnetic field can decrease and for others increase the local magnetic scattering centers, causing oscillations of ρ H .At room temperature, the thermal agitation of the lattice magnetic moments predominates, making negligible the contribution of the local magnetic fluctuations.Moreover, the external magnetic

∂k 2 hi
À1=2 at the external section along B, β* = eħ/m c * where m c * the effective mass of the carrier and E F the Fermi energy.Six analogous oscillatory experimental results have been obtained from measurements on three different samples of natural single crystals of pyrrhotite and all of them fitted convincingly by Equation (1).The third term of this equation describes a sinusoidal oscillation with amplitude decreasing with 1/B, overlying on a background expressed by the two first terms of Equation (1).This background is depicted versus 1/B and versus B in Figure 3a,b respectively.

Figure 1 .
Figure 1.Samples cut from a hexagonal single crystal of pyrrhotite.The direction of the applied magnetic field B is indicated by arrows I) parallel to the c-axis and II) perpendicular to it.

Figure 2 .
Figure 2. The Hall resistivity ρ H versus 1/B of a sample with B perpendicular to c-axis at 77 K.The experimental points are fitted to Equation (1).

Figure 3 .
Figure 3. a) The background CB þ R 1 μ 0 M(B) of the oscillations shown in Figure 1 versus 1/B and b) versus B.

Figure 4 .
Figure 4. Values of 1/B for which maxima and minima of the quantum oscillation of Figure 2 occur are plotted against their corresponding integers.The semiperiod of the oscillations of Figure 2 is given by the slope of the straight line.

Figure 5 .
Figure 5.The Hall resistivity of the sample of Figure 2 at room temperature 300 K. a) There are no oscillations as the thermal agitation impedes the orientation of the elementary magnetic moments and b) the sample does not approach saturation; compare with Figure 3b.

Figure 6 .
Figure 6.Sample with B parallel to c-axis at 77 K.

Figure 7 .
Figure 7.The sample of Figure 6 at room temperature 300 K. a) There are no oscillations.The difficulty of orientation along the c-axis is enforced by the thermal agitation of the elementary magnetic moments of Fe atoms.b) The sample exhibits the typical behavior of a paramagnetic material, with a linear increase of magnetization with B.