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अमीषां निम्नत्वं वृषगिरिपतेरुन्नतिमपि

प्रभूतैः स्रोतोभिः प्रसभमनुकंपे समयसि ||

- dayaa satakam of vedaanta dezika, 13^th century

The flow of water is a profound force in our world. It cuts through rock to forge canyons and furrows through plains to create deltas. In fact, water once flowed on Mars. We know this from scars that run along its surface, big enough to be visible from the earth. This power derives from the kinetic energy of moving water. As each wave strikes a rock, some of its kinetic energy goes into breaking the bonds that hold the rock together. In time, the rock loses its integrity and crumbles. The flowing water continues its onslaught until all obstructions in its path have been worn down. Leonardo da Vinci spent much of his life thinking about the power of water and of ways to harness it. His water-powered machines are on display at Clos de Luce, a palace in France where the master spent his final years. His profound fascination is captured in this quote, displayed on the walls of Clos de Luce:

Water gnaws at mountains and fills valleys. If it could, it would reduce the earth to a perfect sphere. -Leonardo da Vinci (Codex Atlanticus, 185v)

In recent work as part of an excellent team of collaborators, I have explored the power of flow at a microscopic scale. The flowing liquid here is not water, but a ‘fluid’ of electrons that carries electric currents. The landscape is not rocks, but is the arrangement of ‘spins’ in an oxide material. These spins can be viewed as tiny magnets that are stacked within the material.

Before discussing our findings, we take a detour into the origin of (ferro)magnetism. What makes a material magnetic? It is ultimately because the electrons inside carry spins; that is, they behave as tiny magnets. If each spin points in a random direction, the system as a whole carries no magnetism. However, if a majority of the spins point in the same direction, the entire material behaves as a single large magnet. This picture becomes particularly interesting in metals and metal-like materials that are capable of conducting electricity. Within such materials, electrons do not remain stationary. Rather, they behave as a liquid that sloshes around. In a manner of speaking, every droplet in this fluid is a tiny magnet. Even as the droplets slosh around, their spins may all orient in the same way. This makes the material a magnet.

Why does any metal become a magnet? Why should all the tiny electron-magnets point in the same direction? To answer this question, one must bring in the rules of quantum mechanics. They tell us that the spins of electrons are ‘quantized’. They come in two types: pointing up or down. The ‘up’ and ‘down’ labels are given with respect to a certain reference direction. Any direction can serve as the reference. However, once a choice is made, each electron can only point up or down. We can now think of the electrons as coming in two types: ‘up’ and ‘down’. The underlying cause of magnetism in metals is repulsion between up- and down-electrons. Crudely, we can imagine electrons as people in a world where every one belongs to one of two religions (or ethnicities or political parties or any other such division). The two religions repel; that is, the followers of each religion bear hatred for those of the other. Imagine a room with a hundred people, with fifty belonging to each religion. The people move about at random so that they bump into one another. Ever so often, a pair from opposite camps will collide. This is unpleasant for both and will leave both of them unhappy. As we can imagine, every one in the room is left unhappy due to frequent inter-religious collisions. Is there a way to make life more pleasant?

As you may have guessed already, there is indeed a simple solution. It lies in partitioning the room into two, with each religion being given one side. People will now be able to move less, however they will no longer run into some one of the opposite religion. Readers may think of numerous parallels in history when countries were partitioned for this very same reason. It is essentially the same phenomenon that gives rise to magnetism - a material prefers to have all electrons of the same type so that electrons with opposite spins do not come together.

Do all magnets owe their existence to repulsion? Surprisingly, the answer is no. There are exceptions where magnetism arises from the ‘power of flow’. We will first discuss two classic examples that have been known for decades. We will then discuss our findings that offer a third, somewhat different, example. These three examples occur in materials that are half metallic (technically, systems that contain both localized and itinerant electrons). That is, some of their electrons are stationary, attached to fixed positions. The remaining electrons are liquid-like, flowing freely within the material. A new feature emerges in such materials - an interaction between the stationary and mobile electrons. This kicks in when a moving electron approaches a stationary electron. The two electrons may repel or attract, depending on the orientations of their spins. In certain materials, the electrons attract when they have the same spin and repel when the spins are opposite. In certain others, they repel when the spins are parallel and attract when they are opposite.

We now consider the first example of the power of flow in magnetism. It was put forward by Thouless and Nagaoka in the mid 1960’s (for an interesting writeup in the context of a recent paper, see here). Within a purely theoretical model, Thouless and Nagaoka used rigorous mathematical arguments to show that ferromagnetism emerges. Here, the model consists of a single flowing electron that moves on a background of fixed spins. The latter are fixed on the sites of a lattice. If the mobile electron is to sit at a certain site, its spin must be opposite to that of the fixed electron there. This requirement follows from Pauli’s exclusion principle, a consequence of quantum mechanics. In order to lower energy, the mobile electron tries to spread itself as widely as possible. However, the mobile electron can move smoothly only if the background electrons are all pointing along the same direction. This simple mechanism leads to ferromagnetism. This can be understood in analogy with flowing water. Each background spin comes in two types: with up or down spin. These can be viewed as tiles with two possible heights, e.g., the down tile is at ground level while the up tile is one foot higher. For an irregular arrangement of ‘up’ and ‘down’ background electrons, we have a rugged terrain of plateaux (‘up’) and valleys (‘down’). The mobile electron plays the role of water that flows on this terrain. In order to flow widely, it requires a smooth background. That is, it requires all background electrons to be ‘up’ or all to be ‘down’. This simple picture leads to ferromagnetism.

While this model has mathematical rigour in its favour, it remains an abstract theoretical idea. It does not directly lend itself to materials. Our second example, however, was designed to explain the observed behaviour of manganites – a family of oxide materials. They naturally contain two types of electrons – one type that flows and another that remains fixed at the sites of lattice. The number of flowing electrons can be experimentally controlled through a process called ‘doping’, i.e., by replacing a certain number of Lanthanum atoms with Strontium atoms while synthesizing the material. If we have a small number of flowing electrons, the background electrons form an ‘antiferromagnet’. They alternate with one pointing up, the next pointing down and so on. As the spins alternate, they cancel each other out so that the material, as a whole, is not magnetic. However, if the density of flowing electrons reaches a certain level, their flow drastically alters the magnet. Flow requires a flat background – a uniform arrangement of background spins. This forces all background spins to point in the same direction. Indeed, the material becomes a ferromagnet. This picture is known as the ‘double exchange mechanism’. It underlies the phenomenon of ‘colossal magnetoresistance’ which may be useful for technological applications, e.g., in new types of hard disks.

We finally come to the third example that derives from our work. We present a qualitative description here. Readers interested in technical details can find them in our paper. This example differs from the previous two in certain fundamental ways. It is also easily testable. We consider heterostructures of two materials, LSMO (Lanthanum-Strontium Manganite) and SRO (Strontium Ruthenate). A heterostructure can be thought of as solid state Lasagna, with alternating layers of the two materials. Both materials share a common feature – they are composed of static background electrons as well as a mobile fluid. However, they differ in a crucial property. In LSMO, the mobile electrons are attracted to static electrons that have the same spin. In SRO, mobile electrons are attracted to background electrons of opposite spin. When these materials are brought together in layers, the flowing electrons can move from one layer to another. However, if they are to move freely, the layers should not represent an uneven surface with troughs and hills. Consider an ‘up’ electron that flows between the layers. When it is within the LSMO layer, it prefers to have the background spins pointing ‘up’. When it is within the SRO layer, it prefers to have the background pointing ‘down’. As the electron flows, it forces the background electrons in the two layers to point in opposite directions. This leads to flow-induced ‘antiferromagnetism’. As a consequence of flow, the entire heterostructure weakens in magnetism as the alternating layers cancel each other out. This picture suggests several interesting properties – by controlling the flow of electrons (e.g., by injecting more current), we can tune the overall magnetism in the heterostructure. Conversely, by forcing the heterostructure to be a magnet (e.g., by imposing a magnetic field), we can suppress the flow of electrons. This suppresses the ability of the system to conduct electricity. In other words, an external magnetic field functions as a ‘switch’ to cut off currents.

We have presented this picture based on several theoretical calculations. However, the true test will have to come from experiments. We propose a simple test that can show a dramatic signature. We consider bilayers of SRO and LSMO – unlike a heterostructure, a bilayer simply consists of two adjacent layers. We show that flow of electrons can be strongly controlled by changing the relative widths of the layers. For example, we could increase the width of SRO while decreasing that of LSMO. Due to a quantum phenomenon called confinement, this dramatically alters the density of flowing electrons. It follows that the magnetic property of the bilayer will change sharply upon changing the widths. That is, if we build a series of samples with different widths, we will an abrupt change at some point. The entire bilayer will flip between a strong magnetic and a weak magnetic state.

Our paper:

• Carrier-driven coupling in ferromagnetic oxide heterostructures

Physical Review B 96, 184408 (2017), arXiv version here.