Quantum mechanical effects controlling the magnetic properties of transition metal based nanoglass

Nanoglasses consist of nanometer sized amorphous grains that are connected to one another by interfaces.¹ Amorphous nanoparticle grains are held by interfaces. The features of nanoglasses are the high volumetric ratio between interfaces and nanograins. An aspect of nanoglasses is that the interfacial regions open the way to atomic arrangements and physical properties that were found to be different from the properties of melt-quenched glasses. It is, therefore, of central significance to investigate the physical properties of an amorphous sample with high proportion of interfaces to grains. An example of a sample with high proportion of interfaces to grains is the metallic Fe₉₀Sc₁₀ nanoglass. Fe₉₀Sc₁₀ nanoglasses have been prepared by the consolidation of nanometer-sized amorphous clusters at pressures ranging from 1.5 to 4.5 GPa. The interfaces between adjacent nanometer-sized clusters are regions in which Fe atoms form distorted nanometer-sized bcc clusters that contain small cavities, Fig. 1.

The movement of electrons from one cluster to the adjacent cluster through the interface between both was found to result in remarkable physical properties of nanoglasses. The interface of nanoparticles shows physical properties that are discussed in this contribution.

The physical properties at the interface are the results of a quantum mechanical effect called “indistinguishability principle” discovered by Heisenberg.^2–6 The indistinguishability principle indicates that, for example, the position of two moving electrons given at one point of time cannot be predicted at any later point of time unless both electrons are strongly coupled. However, for weakly coupled electrons, the positions cannot be predicted and, hence, all properties of solids that depend critically on the positions of these two electrons cannot be predicted according to the “indistinguishability principle.”^2–6 The indistinguishability of electrons has an influence on the magnetic properties of nanoglasses.

Generally, the magnetic moment of a metal is linked to the magnetic moment of 3d-electrons and itinerant moments.^7–9 For example, the magnetic moment of bcc-Fe measured using SQUID is mainly the sum of 3d-moments, known as local moments μ_3d = μ_local, and the itinerant magnetic moment, μ_i, between atoms. The magnetic moment of the 3d-electrons is called local magnetic moment, μ_local = μ_3d, since the electron probability density of the 3d-electrons is concentrated around the atoms.¹⁰ The Mössbauer spectroscopy and magnetic Compton scattering measurements⁷ permit to specify the contribution of μ_local and μ_i to the total magnetic moment, μ_sum, in Bohr magnetron, μ_B: μ_sum (bcc-Fe) = μ_local + μ_i = 2.57–0.49 µ_B = 2.1 µ_B. The metallic Fe₉₀Sc₁₀ nanoglass has the largest itinerant magnetic moment among transition metals, μ_{i_interface} = −0.95 µ_B.⁷ The significant difference of the physical properties of a melt cooled glass and a nanoglass is caused by the itinerant magnetic moment of the interface. On the basis of these facts, we shall discuss the reason for the large magnetic moment of the interfaces. In order to do this, let us start by recalling that the itinerant magnetic moments of metals, such as bcc-Fe, fcc-Ni, and hex-Co, and amorphous alloys prepared by rapid quenching are distributed between −011 µ_B $≤$ μ_i $≤$ −0.49 µ_B.⁷ The band theoretical computations¹⁰ indicate that the magnetic moment of Fe and Co is the sum of the local 3d-moment and a negative value, μ_i = −0.02 µ_B, of spin polarization circulating between atoms. The value of the theoretically calculated magnetic moment of bcc-Fe and Co is μ_i = −0.02 µ_B, which is less than the value of μ_{i_interface} = −0.95 µB at the interface of the nanoglass. The comparison of μ_{i_interface} = −0.95 µB with μ_{i_grain} = −0.24 µB of amorphous Fe₉₀Sc₁₀ alloys suggests that μ_{i_grain} and μ_{i_interface} have different origins. This difference can be understood by applying the indistinguishability principle of quantum mechanics.

The alloy used in this study was prepared in an arc-furnace under Ar atmosphere and was subsequently heated for 3 days at 1100 K. Amorphous Fe₉₀Sc₁₀ ribbons were prepared using a conventional single roller melt spinning method with a wheel velocity 40 m/s under an argon atmosphere. In this process, a viscous melt of the Fe₉₀Sc₁₀ alloy is rapidly quenched to room temperature. The amorphous state of a Fe₉₀Sc₁₀ sample with a thickness of ∼30 µm and a width of ∼2 mm has been checked by the X-ray diffraction method.

The nanoglass Fe₉₀Sc₁₀ was prepared by the Inert Gas Condensation (IGC) method.¹ The preparation of nanoglasses consists of two steps. In the first step, nanometer-sized particles are generated by evaporating the Fe₉₀Sc₁₀ alloy from a source in flowing He gas. He atoms act as a nucleation source. The evaporated Fe₉₀Sc₁₀ alloy condenses in the form of nanometer-sized amorphous alloys and accumulates on the surface of a cold-finger available in the UHV chamber. The produced amorphous nanoparticles on the surface of the cold finger have average sizes ranging from 2 to 12 nm. In the second step, the nanometer sized amorphous particles are collected and transferred under UHV conditions to a pressure chamber. Under high exerted pressures, amorphous nanoparticles condense in a pellet-shape. Pellets used in this experiment were produced under 2 GPa. The resulting pellet, known as nanoglass, has a relatively high proportion of grains to interfaces. The amorphous nanoparticle grains are held by interfaces. The interface of nanoparticles shows physical properties that are discussed in this contribution. Energy Dispersive X-ray (EDX) spectroscopy (Oxford Instruments) was used for the analysis of the composition of the alloys. Magnetization measurements were obtained using a Superconducting Quantum Interference Device (SQUID).

The transition metals exhibit a linear relationship between μ_sum and μ_i. An exception from this relationship is the value of μ_i at interfaces, Fig. 2.

Using the indistinguishability principle, it is possible to distinguish the individual itinerant p-electrons inside the interface originating from dv₁ or dv₂. This results in an exchange interaction between the moments of two p-electrons as it is given in Eqs. (1) and (3). The indistinguishability principle causes the spins of two itinerant elctrons inside interfaces couple togtherand generate a large magnetic moment of −0.95 µB, which represents the sum of two, individual parallel spins in a triple state. A measure for the strength of the coupling between two electrons is the distance between electrons as shown in Fig. 3. The coupling of two moving electrons inside the interface is mainly negative. Maximum exchange coupling is achieved at small distances between electrons. For the reasons of antisymmetry, the linear combinations of the wave functions of electron 1 and electron 2 have to obey the antisymmetric principle. Inside the interface, the antisymmetric requirements⁶ for μ_{i_interface} = −0.95 µ_B are fulfilled when the A is negative as presented in Fig. 3.

Further investigation work is necessary to understand the physical phenomenon. It seems, however, important at this state to recognize that the indistinguishability principle results in the formation electronic states that have not been considered in solids and that may result in new effects that can only be recognized if quantum mechanical effects are taken into account. This effect opens new doors for the understanding of the properties of materials.

In conclusion, it can be stated that the macroscopic measured magnetic moment of transition metals and alloys is the sum of the delocalized magnetic moment between atoms and the magnetic moment of 3d-electrons that are concentrated around atoms.

The magnetic moment of localized 3d-electrons and delocalized electrons are strongly related by the indistinguishability principle, which plays an important role, since this effect is the main cause for the existence of the delocalized magnetic moment.

We acknowledge the support from the Open Access Publication Fund of the University of Münster.

The authors have no conflicts to disclose.

This work is the result of teamwork and discussion between all authors. All authors have read and agreed to publish the manuscript.

Mohammad Ghafari: Conceptualization (equal); Investigation (equal); Visualization (equal). Herbert Gleiter: Conceptualization (equal); Supervision (equal); Writing – original draft (equal). Gerhard Wilde: Conceptualization (equal); Formal analysis (equal); Validation (equal); Writing – original draft (equal).

The data that support the findings of this study are available within the article.