Research articles
Polarized neutron diffraction on a diamagnetic bismuth single crystal

https://doi.org/10.1016/j.jmmm.2019.04.098Get rights and content

Highlights

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    Magnetization density maps of a diamagnetic single crystal of Bismuth determined.

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    At least two contributions identified.

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    Polarized neutron diffraction ability demonstrated.

Abstract

A large single crystal of bismuth has been studied between 2 K and 150 K by means of polarized neutron diffraction in fields up 6.2 T applied along the trigonal axis. The recorder flipping ratios are very small in agreement with the diamagnetic response of bismuth. Maximum entropy reconstruction shows that the leading contribution to the magnetic field induced magnetization is indeed associated with electron states centered at bismuth. However, small contribution stems from states placed outside bismuth atoms.

Introduction

Bismuth is at ambient pressure diamagnetic. The diamagnetism that is connected with changes in electron trajectories induced by applied magnetic field exists in all the materials. However, in majority of the cases it is much smaller with respect to other magnetic contributions. Nevertheless, several materials that include graphite and bismuth exhibit anomalously large diamagnetism [1], [2], [3]. Bismuth is reported to be anisotropic with the c-axis magnetic susceptibility being in absolute value approximately 30% smaller than with field directed perpendicular to it [3].

At low temperatures is the diamagnetic response to magnetic field in bismuth oscillating due to crossing of Landau and Fermi levels, phenomenon known as the de Haas-van Alphen effect [4]. Diamagnetism in bismuth has been shown that it originates from inter-band features of a specific band structure with a small, temperature dependent, energy gap [2], [5], [6], [7], [8], [9]. Similar effect has been found to be responsible for anomalous diamagnetism in graphene and other low-dimensional semiconductors [10].

Bismuth’s crystal structure (space group R 3¯ 2/m, No. 166) can be described in three different ways using either rhomboedral or hexagonal unit cells. In the present work we use, when addressing Bragg reflection indexes, the hexagonal notation with lattice parameters a ≈4.5 Åand c ≈11.8 Å. Within this lattice, Bi atoms occupy the 6c (0, 0, z) site with z position parameter of about 0.23. Bismuth is a semi-metal with extremely small Fermi surface, consisting from a hole pocket aligned along the trigonal axis and three Dirac electron ellipsoidal pockets that are tilted by about 6 degrees out of the bisectrix-binary plane. [11] The low concentration of itinerant electrons leads to confinement of electrons to the lowest Landau level already at rather low fields [12]. Various very interesting phenomena have been studied in bismuth. For instance, the electrical resistivity increases at low temperatures with application of a moderate magnetic field by several orders of magnitude with respect to its zero field value [13], [14] and Nernst, Seebeck and Hall effects show oscillatory dependencies [15], [16], [17], [18], [19], [20], [21]. Oscillations have been detected even in ultrasound measurements [22].

Our motivation for this work was to experimentally verify whether it is possible to determine the magnetization distribution in bismuth and compare it with the magnetic susceptibility measurements. In this work we compare experimental data obtained at two different fields, close to the maximum and minimum of the magnetization oscillations.

It is well known that the use of polarized neutron diffraction (PND) experiment enhance the sensitivity to magnetism with respect to the unpolarized neutron diffraction experiment enormously [24]. The PND technique has been frequently used in determinations of the electron redistribution caused by the applied magnetic field near the Fermi surface. The magnitude of the signal depends on the density of states and is capable to give the direct information on the distribution of the magnetization in the unit cell and allows for the identification of different contributions to the magnetic moments [23]. In a pioneering work of C.G. Shull and R.P. Ferrier [24] it has been demonstrated that it is possible to discriminate by means of this method between the nuclear and electronic contributions to the scattering intensity and by comparing the intensities to a calculated magnetic form factors to disclose the electronic configuration. Later on, Stassis argued that PND experiment should be sensitive enough to separate the magnetic contribution due to diamagnetic current induced by applied magnetic field [25].

Section snippets

Experimental

A 2.5 cm3 large bismuth single crystal of 99.999% purity has been prepared on a by Bridgeman method by MaTeck GmbH. The X-ray and neutron Laue technique has shown that it consists from a majority grain that comprises 90% of the sample volume and 10% volume grain about 0.7 degrees away. Such a quality (although not perfect) does not hamper main conclusions of this work. The crystal has been glued to an aluminium rod (used later to attach the crystal in the course of the diffraction experiment

Magnetic bulk measurements

In Fig. 1 we show the field dependence of the magnetization measured at 2 K in fields up to 14 T. As it was expected, it is negative and shows almost a linear decrease with increasing field. There is no hysteresis between the sweep-up and sweep-down branches. The magnetization at 2 K and 6.2 T amounts −0.0031 μB/Bi atom. At 4.5 T the magnetization amounts to −0.0022 μB/Bi atom. These values were adopted in the evaluation of the neutron diffraction data.

However, one can observe on the

Discussion and conclusions

Our PND experiment clearly shows negative magnetization clouds in a close vicinity of the Bi atomic positions with additional magnetization between them. This result suggests that electrons centered at bismuth atoms are responsible for the signal detected in our PND experiment only partially. Similar conclusion has been drawn for bismuth by Wilkinson [31] and by Wilkinson et al. [32] for graphite. The negative magnetization signal in our experiment has threefold symmetry imposed by the crystal

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