Mesoscopic orbital paramagnetism: The role of zero-point energy

Highlights

• Brief overview of d-zero magnetism in oxides.

• Explanation of coherent electronic structure formation at a solid surface in response to zero-point energy of mixing localized ground and excited states of a bandgap UV transition.

• Modified mixing of the states in an applied magnetic field leads to saturating orbital paramagnetism, matching that found in d-zero oxides.

• Accessible step-by-step presentation of the quantum field theory.

• Discussion of possible future experimental and theoretical work.

Abstract

A model is presented to explain the temperature-independent saturating paramagnetic response of some transparent oxides with no unpaired d-electrons. It is based on an array of N surface defect-related electrons with bound states in the gap that can form a coherent mesoscopic many-electron state in response to fluctuations of the zero-point electromagnetic field. Individual electronic orbits expand by 0.083 pm, as in the theory of the Lamb shift, and these expansions add to produce a global volume change. This modifies the energy density in the zero-point electromagnetic field, thereby lowering the energy per electron sufficiently to stabilize a coherent multi-electron state of the two-dimensional system at room temperature. A net magnetic moment can be induced by an applied magnetic field, which mixes coherent ground and excited states, producing a paramagnetic orbital magnetization of magnitude $M=M_S\frac{x}{\sqrt{1+x^2}}$, where x is proportional to the applied field. Orbital saturation moments per coherent surface electron range from ${10}^{-3}$ to ${10}^{-1}$ Bohr magnetons.

Introduction

An unusual magnetic phenomenon is sometimes observed in oxides that contain none of the unpaired 3d electrons that are normally required for magnetic order at room temperature and above. The experimental syndrome is illustrated in Fig. 1 for a nanoporous amorphous alumina membrane [1], but similar effects have been observed in a variety of other oxide systems, including reduced strontium titanate powder [2], cerium dioxide nanoparticle clusters [3] and Sc-doped ZnO thin films [4].

After correcting raw magnetization data obtained by SQUID magnetometry for a linear paramagnetic or diamagnetic background, the residual signal illustrated in Fig. 1 is obtained. The following are the characteristic features:

• The signal is independent of temperature, from cryogenic temperatures to room temperature and above. The curves measured at different temperatures superpose directly, without any need to normalize the vertical axis.

• There is little or no hysteresis at any temperature.

• The value of the saturation magnetization M_s is from one to four orders of magnitude smaller than the saturation field $H_0$ $\approx$ 100 kAm^-1 extrapolated from the initial susceptibility. For clarity, M is plotted against H in Fig. 1, with both in multiples of the same SI unit of Am^-1.

It must be emphasised that the behaviour is completely different from superparamagnetism – the saturation magnetization and especially the initial susceptibility of an ideal superparamagnet is temperature-dependent, the susceptibility varying as T. The magnetization in the present case typically lies in the range from 10 Am^-1 to 10 kAm^-1. It is therefore vital in practice to avoid possible contamination of the samples by traces of ferromagnetic or ferrimagnetic impurities, and to be aware of various possible experimental artefacts [5], [6]. The data are frequently not reproducible from one laboratory to another, or even in the same laboratory from one run to the next. These difficulties can be circumvented with appropriate experimental precautions, but the most convincing evidence of an intrinsic origin in a particular sample is the ability to reversibly create or destroy the magnetism. It should be possible to turn it on or off by adopting a particular experimental protocol. This is the case for systems we have studied in detail [7], where it was established that the magnetism is associated with surface oxygen defects.

A simplified but unconventional model was able to account for the characteristic features listed above, and make testable predictions for the cerium dioxide nanoparticle system [3]. In that case, measurements on dispersed powder samples established that there was a characteristic length scale of order 100 nm for the appearance of the magnetism, in agreement with prediction. Furthermore, UV absorption at a specific frequency $\omega$ was predicted for magnetic samples of CeO₂ nanopowders, and it was subsequently observed [3].

The originality of the model is to consider effects of the zero-point electromagnetic field on the orbitals of bound electrons in the oxides. The results are explained in terms of saturating orbital paramagnetism. The main idea is that coherence may be possible within a mesoscopic region containing millions of atoms due to mixing of the ground and excited atomic orbital states. The stabilization energy of the coherent electrons is about 1 percent of their UV excitation energy. The coherent multi-electron state formed in this way is much more likely to be stable at the surface of the mesoscopic region than in the bulk. The effect of an applied magnetic field B is then to form a new mixture of the the coherent ground and excited states that exhibits a saturating paramagnetic response similar to the one observed experimentally. In view of the unfamiliarity of these concepts in magnetism, it is worthwhile to explain the underlying physics in more detail, and suggest further developments and experimental tests of the ideas.