Quantum Materials

WOMANIUM Global Quantum Media Project Initiative — Winner of Global Quantum Media Project

Published in

Quantum Engineering

11 min readJul 30, 2023

Introduction

Quantum Mechanics, originally developed to understand the structure of atoms, reveals its profound influence on materials even at macroscopic scales when atoms come together to form solids. These remarkable materials, known as quantum materials, exhibit extraordinary behaviors and exotic properties due to the presence of quantum effects. In this article, we delve into the world of electrons in solids, where their behavior differs significantly from free electrons in empty space. Central to our exploration are two fundamental concepts in the domain of quantum materials: emergence and topology. Through a lucid analysis, we will elucidate how these concepts redefine our understanding of materials, offering intriguing instances where electrons masquerade as distinct particles yet unknown in nature.

Electrons in Free Space

To comprehend electrons’ distinct behavior in solids, it is essential to grasp their characteristics when unconfined in solids, allowing them to move freely through empty space. Classically, electrons possess mass, and their kinetic energy relates to their momentum in a quadratic fashion, as described by Newtonian physics. However, when dealing with highly relativistic particles moving at speeds close to that of light, Einstein’s theory reveals a linear relationship between energy and momentum.

In 1928, Dirac introduced a groundbreaking equation that unified special relativity and quantum mechanics, leading to two significant discoveries. First, electrons exhibit a property known as spin, akin to a constrained rotation, where their spin can only assume forward or reverse directions. Secondly, Dirac’s equation revealed the existence of negative energy states, unveiling the concept of antimatter, with the positron as the antiparticle of the electron.

Dirac Equation [a]:

Electrons in Solids

In the context of metals, a simple classical picture emerges, where electrons flow through a metal bar when connected to a battery, resulting in a current. This motion involves collisions with vibrating atoms, randomizing the electron’s velocity. Consequently, they traverse the metal until reaching the other end. This simplistic model, known as Ohm’s law, explains how current is directly proportional to voltage, with resistance determining the constant of proportionality. Furthermore, this model correlates the behavior of metal resistance with temperature, where higher temperatures lead to increased resistivity due to heightened atomic vibrations. However, this classical picture neglects the profound impact of quantum mechanics on electron behavior.

Quantum Mechanics in Solids — Bloch’s Theory

The application of quantum mechanics to electrons in solids led Felix Bloch to a profound realization concerning wave-particle duality. Bloch’s solution to the wave equation for electrons in periodic crystals demonstrated that energy-momentum graphs initially resemble those of unbound electrons, exhibiting a parabolic shape. However, breaks in the graph occur at specific momentum values, resulting in the absence of energy states.

To visualize this phenomenon, one can draw parallels to water waves passing through a barrier with two small holes. The resulting circular waves exhibit constructive and destructive interference. Similarly, for multiple electrons, quantum mechanics mandates filling up states starting from the lowest energy levels, leading to crucial electron states determining material properties. Electrons occupying these states, akin to stacked cars, are the only ones free to carry current, while others remain trapped, influencing whether a material behaves as a metal or an insulator.

Superconducting Materials

Intriguingly, Bloch’s theory does not account for the phenomenon of superconductivity, wherein certain metals attain a state of zero resistance at extremely low temperatures. Take niobium, for instance, which displays resistivity dropping towards zero below 10 K.

Superconductors also exhibit a fascinating property — the expulsion of all magnetic flux when subjected to a magnetic field. As a result, they levitate above magnets due to the bending of magnetic field lines. While Bloch attempted to explain superconductivity, his efforts proved inconclusive, leading him to express the view that any theory of superconductivity is inherently refutable. Despite this challenge, the study of superconductors has remained guided by this principle, inspiring researchers to unveil the mysteries of these extraordinary quantum materials.

What are Quantum Materials?

Quantum materials encompass a class of substances in which electrons, influenced by quantum effects, deviate from the behavior of free electrons. Within the realm of quantum materials, we distinguish between two subclasses. The first includes materials where emergence plays a significant role, occurring when forces between electrons cannot be neglected. Unlike the classical picture, such materials exhibit emergent phenomena, where large-scale patterns of behavior emerge from the cooperative interactions of numerous particles. The second subclass comprises topological materials, where electrons possess distinct geometric properties. In particular, the electron wave function in these materials differs from that of free electrons in empty space, giving rise to unique electronic states.

Emergence

Emergence characterizes the collective behavior of many-particle systems, wherein large-scale patterns emerge beyond the predictions based solely on local short-range forces between the particles. A vivid example is found in ice snowflakes, seemingly distinct in shape despite being composed of water molecules. Although physicists understand the interactions and behaviors of water molecules, predicting the specific shape of a snowflake proves challenging due to emergent phenomena. In condensed matter physics, a multitude of emergent phenomena arises from cooperative interactions among particles. When particles act in unison, even small stimuli can elicit colossal responses from the system, a quality beneficial for practical materials. Notably, quantum materials often exhibit low-frequency oscillations of emergent patterns that resemble weakly interacting quasiparticles, resembling a gas of cooperative entities.

Typical Example of Emergent phenomena is readily found in ice snowflakes.

Spin Ice

In the fascinating domain of quantum materials, there exists an intriguing phenomenon called Spin Ice, where magnetic monopoles emerge and become confined within solid structures. One particularly intriguing material, Cadmium Cyanide (Cd (CN)2), demonstrates this phenomenon. In Cd (CN)2, the cadmium (Cd) atoms arrange themselves into a network of tetrahedra, with each atom behaving like a miniature bar magnet with its magnetic moment. The unique chemical environment surrounding these atoms influences their orientations, causing them to align either towards or away from the center of the tetrahedron, creating a distinct “two-in, two-out” arrangement of magnetic moments. This arrange showcases the intriguing interplay of quantum effects within the solid state.

The two-in, two-out arrangement bears a striking resemblance to the arrangement of water molecules in ice, adhering to the Bernal-Fowler ice rules. The concept of Spin Ice further explores the intriguing phenomenon of emergent magnetic monopoles within a material’s structure.

The ice rule describes the arrangement of protons in water ice, where two protons are bonded closely to the oxygen at the center of tetrahedra and two are farther away, close to neighboring oxygens. This rule can be analogous to spins in pyrochlore spin ices, rare earth titanates with frustrated magnetic moments. In this case, two spins point inward, and two point outward, similar to the configuration of protons in water ice. This similarity with water ice contributes to the absence of low-temperature magnetic order and provides non-zero entropy density in pyrochlore spin ices. Image Credits: [4]

Emergent Magnetic Monopoles

To comprehend the behavior of emergent magnetic monopoles within Spin Ice, one can envision creating a blemish in the stable arrangement, leading to a tetrahedron with three spins pointing inward and one pointing outward. This creates a net inward flux, akin to a north magnetic monopole. Neighboring tetrahedra become south magnetic monopoles, forming monopole pairs. Interestingly, these monopoles are confined within the solid, with their motion subject to continuous spin flips. The transformation of Spin Ice into a gas-like ensemble of monopoles confined within a solid is a remarkable aspect of quantum materials.

The transition from dipoles to dumbbells involves replacing each spin in the original configuration with a pair of opposing magnetic charges positioned on adjacent sites of the diamond lattice. When obeying the ice rule, neighboring tetrahedra display two spins pointing in and two spins pointing out, resulting in zero net charge on each site. However, inverting the shared spin creates a pair of magnetic monopoles, with diamond sites exhibiting net magnetic charge. This configuration yields a higher net magnetic moment, particularly favored when an upward-oriented magnetic field (corresponding to a [111] direction) is applied. The large panel depicts separated monopoles represented by large red and blue spheres, with a highlighted chain of inverted dipoles (referred to as the ‘Dirac string’) between them and sketched magnetic field lines. Image Credits: [5]

Experimental Confirmation of Monopoles

Experimental confirmation of the behavior of emergent magnetic monopoles is crucial to the study of quantum materials. The stable state of Spin Ice adhering to the two-in, two-out rule is established through scattering beams of neutrons on Cadmium Cyanide samples. By measuring the distribution of the scattered radiation in a plane, physicists can infer the spins’ arrangement. Notably, simulation results align closely with experimental data, confirming the stable state’s validity.

Diffuse scattering maps of Ho2Ti2O7 spin ice material are compared between experimental and theoretical results. The 1.7 K experimental small-angle scattering displays pinch points at specific coordinates, and non-spin-flip (NSF) scattering and total scattering are also observed. The theoretical simulations using a 12x12x12 supercell closely match the experimental data for spin-flip (SF) scattering. The NSF scattering from this model is Q-independent, influenced solely by the Ho3+ magnetic form factor. Overall, the comparison shows strong agreement between the near neighbor spin ice model and the experimental data. Image Credits:

Additionally, the heat capacity of Spin Ice has been measured, revealing the presence of monopole excitations, providing further evidence for the material’s behavior as a weakly interacting gas of monopole particles confined within a solid.

The system displays a gas of deconfined magnetic monopoles. Ising spins are constrained to align with the connection between tetrahedra centers, resulting in six configurations with net ferromagnetic moments along equivalent <100> directions. Frustration arises from non-collinearity among the Ising axes. In the presence of an external magnetic field along [001], the tetrahedral magnetization aligns with the field direction. The measured heat capacity per mole of Dy2Ti2O7 at zero field is compared to Debye-Huckel theory for the monopoles and a single-tetrahedron (Bethe lattice) approximation, with spin ice regimes indicated by blue and yellow backgrounds representing paramagnetic states. Image Credits: [7]

Magnetricity and Low Power Dissipation

The remarkable behavior of magnetic monopoles within Spin Ice holds significant practical potential. By applying a magnetic field, monopoles can be manipulated to form monopole currents, akin to electricity flow, creating circuits where magnetricity replaces conventional electricity. The advantage lies in the absence of mass flow and resistive heating, leading to significantly reduced power dissipation in electronic circuits, offering a promising avenue for low-energy consumption technology.

Topological Electronic Materials

Topology, a branch of geometry concerned with continuous deformations of shapes, plays a significant role in quantum materials. Certain materials display states that differ topologically from the host matrix due to defects introduced within the material. These topologically distinct states are locally stable, requiring a substantial amount of energy to revert to the surrounding matrix structure. As a result, stable states can be achieved against weak perturbations, providing a means to create unique and robust states in quantum materials.

Skyrmions

Skyrmions, first proposed in the 1960s, are topological defects found in ferromagnetic materials. A key aspect of these defects lies in the introduction of a spin configuration different from the surrounding ferromagnetic pattern. When examining a cross-section of this pattern, the atomic bar magnets display a characteristic tumbling and rotating behavior. The stable arrangement of the original ferromagnet exhibits parallel spins, whereas the skyrmion has higher energy due to the unfavorable configuration of spins at right angles to each other. The high energy barrier required for the skyrmion to relax back into the ferromagnet makes them highly stable defects, rendering them favorable for applications such as information recording.

The Bloch-type skyrmion lacks mirror-plane symmetry and is chiral. On the other hand, the N´eel-type skyrmion has mirror-plane symmetry when combined with time reversal for vertical planes through its center. Image Credits: [8]

Weyl Fermions

Weyl fermions are another type of topological quasiparticle found in certain quantum materials. Named after physicist Herman Weyl, who investigated a hypothetical particle with zero mass, Weyl fermions exhibit unique properties not observed in ordinary free electrons. One striking feature of Weyl fermions is their relativistic behavior, characterized by coupling between spin and the direction of momentum.

This coupling results in helical motion, where the spin aligns either parallel or antiparallel to the velocity direction. Weyl fermions behave as chiral fermions, meaning they have a definite handedness, either left-handed or right-handed. While neutrinos were initially considered as possible examples of Weyl fermions, their discovery of mass eliminated them as free fermions.

The spin of fundamental particles is an inherently quantum mechanical property. Although there’s no classical sense of a spinning sphere, it serves as a useful analogy. The red arrow indicates the particle’s spin direction, while the gray arrow shows its motion. The orientation or handedness is defined by the red and gray arrow directions. Right-handed particles have the same orientation as your right hand, while left- or right-handedness is the particle’s chirality, a quantum mechanical property. Fermions, like electrons, have a spin of one-half, meaning a 360-degree rotation results in the same state with a minus sign due to quantum interference. Chirality determines the phase of a fermion’s wavefunction, with left- and right-chiral fermions shifted in opposite directions during rotation, showcasing a purely quantum phenomenon. Image Credits: [9]

However, materials can be engineered to exhibit the energy-momentum relationship predicted by the Weyl equation, and tantalum arsenic stands as an example of such a material, where experimental evidence supports the existence of Weyl fermions as emergent particles.

Weyl Semimetals

Weyl semimetals are a class of quantum materials that offer intriguing opportunities for engineering their electronic structure to exhibit behaviors akin to Weyl fermions. In 2015, tantalum arsenic emerged as the first example of such a material, displaying an energy-momentum relationship precisely predicted by the Weyl equation. This revelation was substantiated through the use of angular-resolved photoemission spectroscopy, a technique that directly probes the energy dependence of electrons within the solid. The resulting graph exhibits an inverted V shape, demonstrating the presence of Weyl fermions.

The dispersion map along the cut 1 direction exhibits the E−ky relationship and distinctly reveals the presence of two linearly dispersive W2 Weyl cones. Image Credits: [10]

Notably, this technique only measures electrons in occupied states, making the graph non-symmetric; however, if unoccupied states were accessible, a symmetric V shape would manifest. The congruence between the experimentally observed energy-momentum graph and the theoretical Weyl equation affirms the unique handedness exhibited by specific electrons, providing experimental evidence for the existence of these emergent particles, which remain undiscovered as fundamental entities in free space.

Magnetically induced Weyl semimetals

Magnetically induced Weyl semimetals provide an alternative means of generating exotic particles in materials with inherent magnetism, such as Europium cadmium arsenic. At low temperatures, atomic-scale bar magnets in these materials align in alternating planes with opposite spin directions. When subjected to a magnetic field, these spins can be realigned to all point upwards, resulting in the closure of the gap in the electron band diagram. Consequently, a linear energy-momentum plot emerges, consistent with the behavior of Weyl particles. Remarkably, the application of a magnetic field allows for the controlled switching on and off of Weyl particles, presenting intriguing possibilities for practical applications.

Conclusion

In conclusion, quantum materials are characterized by the prominence of quantum effects and the departure of their electron behavior from that of free electrons. The concepts of emergence and topology have revolutionized our understanding of these materials, revealing how cooperative behaviors of many particles lead to large-scale structures with unique properties. Advanced experimental and theoretical techniques have enabled the discovery, identification, and characterization of intriguing quantum phases and patterns in quantum materials. Additionally, the newfound ability to predict and engineer materials with specific properties promises further advancements in the field, opening new avenues for future scientific exploration and technological development.

References

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