Andrey Katanin, Doctor of Physical and Mathematical Sciences, Chief Researcher at the Institute of Metal Physics named after. M. N. Mikheeva Ural Branch of the Russian Academy of Sciences, Professor of the Russian Academy of Sciences
"Kommersant Science" No. 6, September 2017
The Earth's magnetism is determined by the iron and nickel in its core. But it is not yet entirely clear how exactly. Photo: Science Photo Library / AFP
People have known about terrestrial magnetism for several millennia, but its cause has not yet been clarified. Theoretical research by Ural physicists can answer this question.
The magnetic properties of iron were discovered several millennia BC. e. Thus, in China, pieces of magnetic materials were used to create a compass. In 1269, “The Book of the Magnet” was written by Peter Peregrin, and in 1600, William Gilbert wrote a treatise “On the Magnet,” describing the basic properties of magnets and analyzing the magnetism of the Earth. Today, iron, including its magnetic properties, has many different technological applications. Iron is not the only magnetic substance; nickel and cobalt can be noted, which became of interest to humanity much later and are also widely used today.
Despite such a long period of study of magnetism, this phenomenon still raises new questions. In everyday life, we feel magnetism as attraction or repulsion between bodies. In physics, magnetism is understood as the ability of a body to maintain residual magnetization (that is, its own magnetic field) in the absence of an external magnetic field. And this own field can influence other magnetic bodies.
Two concepts of magnetism
A common property of most magnetic substances is that their magnetism is due to atoms of so-called transition metals containing d
-electrons (index
d
refers to a certain type of symmetry of the electronic states of the atom). Transition metals are not only iron, cobalt and nickel, there are several dozen of them.
Localized ( top
) and band (
bottom
) patterns of ferromagnetism
With the advent of the concept of electron spin and its corresponding magnetic moment, two different quantum mechanical pictures of magnetism were proposed - localized and band.
The localized picture formulated by Heisenberg assumed that electrons in a crystal do not jump from one atom to the next one, but there is an exchange interaction between electrons from neighboring atoms. This is a purely quantum effect, caused by the difference in the energies of parallel and antiparallel spin ordering. Stoner's band picture, on the contrary, implied the possibility of electron movement, and their interaction was mainly carried out within a single atom.
At first glance, the band picture seemed more applicable to transition metals. But it could not explain some phenomena, for example, the Curie-Weiss law, which describes the linear dependence of inverse susceptibility on temperature (susceptibility is the response of a system to a weak external magnetic field). At the same time, it was not at all obvious why the picture of localized electrons, which, as it seemed, could not be applied to transition metals (in particular, iron), described the experiment much better.
In the late 1950s and early 1960s, Neville Mott, followed by John Goodenough, proposed that some of the electrons in iron (namely, electrons corresponding to the so-called eg
-states, two of the five possible
d
-states on an atom) are characterized by “non-conducting wave functions”, that is, they do not jump, they are localized.
Although by that time the concept of the transition of electrons from a band, conducting state to a localized one had already arisen (thanks to the work of Mott), the Mott-Goodenough assumption was far beyond the boundaries of the then existing theoretical approaches. -selective Mott transition introduced much later
).
Methods developed later (in the 1980s) for calculating exchange interactions in metals on the basis of band theory made it possible to obtain certain theoretical indications of the existence of localized moments in iron, but in the very method of these calculations, nevertheless, the conducting, band nature of electrons was inherent .
Petrus
Peregrinus - the 13th century French physicist Pierre Pelerin de Maricourt signed his works with this Latin pseudonym. The exact dates of his life are unknown. Peregrine is the author of the first experimental study and the first detailed scientific work on magnetism.William Gilbert
(William Gilbert), 1544–1603 - English physicist and court physician, researcher of electricity and magnetism, author of the first theory of magnetic phenomena.
Sir Neville Francis Mott
(Nevill Francis Mott), 1905–1996 - English physicist, winner of the Nobel Prize in Physics in 1977, together with Philip Anderson and John van Vleck, “for fundamental theoretical studies of the electronic structure of magnetic and disordered systems”, author of the theory of interaction-induced transitions from metallic to insulating state.
John Goodenough
(John Goodenough), b. 1922 - American scientist, specialist in the field of physics and materials science.
What is magnetic force?
Magnetic force is the force created by electrons that occurs between electrically charged particles. Applied by magnets to magnetic objects, this force creates and controls magnetism and electricity.
In fact, we cannot see the forces at work; they are invisible to the human eye, but we can observe their influence on various objects when conducting an experiment. The region where a magnetic force acts on a magnetic material is called a magnetic field.
Three types of metals interact with magnetic fields: ferromagnetic, paramagnetic and diamagnetic metals. Ferromagnetic metals are strongly attracted to magnets, others are not. Magnets also attract paramagnetic metals, but very weakly. Diamagnetic metals repel a magnet, although the force is usually very small.
How is a magnet made?
Inside a piece of iron or other magnetic metal are millions of tiny particles mixed together. When a magnet is placed next to a piece of metal, the particles line up and the piece of metal itself becomes a magnet. This is why a string of paper clips will hang from the end of the magnet. The stronger the magnet, the greater the strength of the magnetism and the longer the string of paper clips can be.
The most common materials used to make permanent magnets are iron, nickel, cobalt and some alloys of rare earth metals.
First steps towards a unified theory
The situation changed only in the late 1990s and early 2000s with the advent and development of the so-called dynamical mean field theory. This theory approximately reduces the complex problem of the movement of electrons in a crystal to the consideration of changes in their state over time on one selected atom. The theory made it possible to describe metal-insulator transitions in a number of substances, which naturally led to the question of its ability to explain the magnetism of transition metals.
In particular, iron and nickel were studied within the framework of this theory by Mikhail Katsnelson, Alexander Lichtenstein together with the American physicist Gabriel Kotlyar in 2001.
They were the first to obtain from a fully microscopic (that is, first-principles equations) calculation within the band picture the linear behavior of inverse susceptibility with temperature (Curie-Weiss law), which is usually interpreted as indicating the presence of local moments. They also found a weak dependence of local susceptibility on time (on the imaginary time axis, which is easier to study from a theoretical point of view), indicating the presence of local moments. At some point it seemed that the problem of iron and other transition metals was almost solved.
Energy zones
In an atom, electron energy levels are discrete. In a crystalline solid, entire ranges of allowed energies (allowed zones) and forbidden energies (forbidden zones) are formed. Simplifying somewhat, we can say that allowed bands are formed from atomic levels when atoms combine into a crystal, and the remaining space is occupied by forbidden bands.
History of discovery and application
Magnetism for people in the deep past most likely must have seemed like magic. Even the ancient Greeks and Chinese discovered fragments of meteorite iron or natural materials, which they used as a compass needle to determine direction. However, the first artificial magnets were not made until the 18th century, and even further progress in creating materials with highly pronounced properties was slow. The main opening dates look like this:
- 1740 - Gowin Knight developed a process for producing magnetized steel.
- 1750 - John Michell published eighty pages of his treatise in Cambridge on the theoretical basis for the creation of materials with structured domains.
- 1820 - André Marie Ampère's description of the phenomenon of electromagnetism.
- 1855 - Michael Faraday publishes his theory of electromagnetic induction.
- 1920s — principles for creating highly magnetic alloys were formulated.
- Mid-20th century — modern technologies for the production of ferrites have been developed.
- 1970s — start of production of rare earth magnets.
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Development of classical ideas with new methods
However, the emergence in the mid-2000s of the concept of orbital-dependent metal-insulator transitions once again forced us to reconsider and supplement the previously obtained results. Here I turn to my research, together with my colleagues. My interest in the problem of iron arose in 2007 as a result of discussions at the recently created Institute of Quantum Materials Science in Yekaterinburg, but then went beyond the scope of this institute. In particular, I was interested in the question of how the ideas of Mott and Goodenough could be further developed using modern methods for analyzing electronic correlations.
Diagram of the electronic configuration of an iron atom. Concentric circles
correspond to different energy levels of the atom.
Green dots
represent electrons in
s
(circular) orbitals,
blue dots
in
p
(elongated) orbitals,
orange dots
in
d
orbitals with a more complex distribution in space
In this regard, the idea arose to consider iron within the framework of the dynamic mean field theory, paying attention to the contribution of various electron orbitals to the observed properties. Already from the band structure it followed that the contributions t2g
and
eg
- electronic states in iron must be different.
(Here the designations t2g
and
eg
again refer to the symmetry of electronic
d
states on a cubic lattice; on each atom of the five possible
d
states there are three
t2g
states and two
eg
states on each of the two possible projections of the electron spin. In a solid these states form, respectively,
t2g
- and
eg
-zones).
Indeed, the two electronic bands - t2g
and
eg
- are structured completely differently (more precisely, they have a different dependence of the electron energy on the momentum). And the remaining less significant, so-called nonlocal effects could be considered using perturbation theory.
My collaboration with Vladimir Anisimov’s group at the Institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences began with consideration of this problem. As shown by calculations using the method of dynamic mean field theory, the behavior of t2g
- and
eg
-electrons are completely different.
In particular, the so-called intrinsic energy of electrons, which describes the influence of the interaction of electrons on their motion, has a different energy dependence for the t2g
and
eg
states.
Moreover, the dependence obtained for eg
states actually indicated the possibility of their localization.
In addition, the orbital contributions to the local (that is, corresponding to the reaction of one selected atom to an external magnetic field applied to it) susceptibility were also calculated. It turned out that the contribution of eg
states to local susceptibility is well described by the Curie law (a special case of the Curie-Weiss law), which again indicated the strong localization of these states.
At the same time, the contribution t2g
states exhibits a more complex temperature dependence, but due to the mixed
t2g
-
eg
contributions, the total local susceptibility also satisfies Curie’s law. Dynamic local susceptibility, defined as the response to a time-dependent external magnetic field, exhibits a narrow peak characteristic of systems with local moments.
Strictly speaking, there is also a small correction in the above-described Curie law for local susceptibility. It indicates the existence of a low temperature, below which local moments cease to exist, being, as they say, screened by mobile conduction electrons (this is called the Kondo effect, named after the Japanese physicist who discovered the effect).
Scheme of the occurrence of indirect exchange of RKKI between local moments ( long arrows
) through conduction electrons (
short arrows
).
i
,
j
correspond to two different atoms,
I
is the Hund interaction,
thin arrows
indicate the direction of the jump
The real consideration of nonlocal effects (including the nature of magnetic exchange) within the framework of perturbation theory was carried out much later, in 2015–2017, in collaboration with Peter Igoshev, Alexander Belozerov and Vladimir Anisimov. To calculate magnetic exchange, one can follow the old idea that it is due to indirect exchange through conduction electrons. This is the so-called RKKI mechanism: Ruderman - Kittel - Kasuya - Yosida; in the Soviet Union, similar ideas were developed by Semyon Shubin and Sergei Vonsovsky. Indirect exchange is the interaction of local moments through mobile, free electrons. The problem, however, is that there is no clear separation between these states in transition metals, since local moments have a finite lifetime, and the same electron can be localized or delocalized at different times. This problem, however, can be circumvented (at least for iron) with the help of mathematical transformations - by rewriting the susceptibility of the electronic system in a form where the interaction between effective magnetic moments becomes explicitly highlighted. Moreover, it just has the shape of the Red Army. The resulting "quasi-local" moments can then be related to the observed local moments. This approach gives good results, comparable to those obtained previously within the framework of purely band theories, in which, as already mentioned, the concept of a local moment is absent.
Thus, in iron there are well-defined local moments that appear as a result of exchange interaction. At the same time, nonlocal corrections were calculated, which made it possible to achieve good agreement with experimental data.
Iron, Nickel and Earth's Magnetism
As recent studies have shown, together with the group of G. Sangiovanni at the University of Würzburg (Germany), the magnetic properties of nickel exhibit both similarities and differences from iron. The difference between the nickel atom and the iron atom is that it has eight rather than six d
-electrons. Although, similar to iron, the local susceptibility of nickel obeys the Curie-Weiss law, in nickel it has a completely different origin, namely, it is largely due to band structure rather than interaction. In addition, in nickel the local moment is relatively small and, moreover, is already partially shielded at high temperatures. The only role of the interaction is to sharply reduce the Kondo temperature, above which the local moments are well defined, from several thousand to several hundred degrees.
Under high pressure, iron becomes non-magnetic. But adding a little nickel brings back the magnetism
These (and some other not described here) interesting features of nickel have recently been further developed in pressurized iron-nickel alloys. Under pressure, iron appears in a special, so-called epsilon phase, which is radically different from the “ordinary” alpha phase of iron. In particular, epsilon iron has absolutely no local magnetic moments. However, adding nickel atoms to epsilon iron, even in a small concentration, qualitatively changes the situation.
The peculiarities of the electronic structure of nickel and epsilon-iron are such that the magnetic properties of their alloy, containing even a small amount of nickel, turn out to be close to the properties of “ordinary” nickel. This fact may have important consequences for explaining terrestrial geomagnetism. The Earth's inner core is believed to contain iron and nickel in a concentration of approximately 4:1. Although iron is liquid at high intraterrestrial temperatures, the mobility of iron atoms is low, and their state can be correlated with one of the crystalline phases. In this case, the non-magnetic epsilon phase is apparently the most energetically favorable. And then it is the presence of nickel in the Earth’s core that can provide the appearance of the Earth’s magnetic field due to the so-called geodynamo effect (rotation of the liquid core), supported by the resulting low thermal conductivity of the iron-nickel alloy. Thus, absolutely theoretical studies of iron-nickel alloys can clarify one of the as yet unsolved problems - the problem of the origin of the Earth's magnetic field.
