Theory of electric, magnetic, and toroidal polarizations in crystalline solids with applications to hexagonal lonsdaleite and cubic diamond

R. Winkler and U. Zülicke

Phys. Rev. B 107, 155201 – Published 4 April 2023

An article within the collection: Emmanuel Rashba: Breaking New Ground in Solid-State Exploration

Abstract

Multipolar order in bulk crystalline solids is characterized by multipole densities—denoted as polarizations in this work—that cannot be cleanly defined using the concepts of classical electromagnetism. Here we use group theory to overcome this difficulty and present a systematic study of electric, magnetic, and toroidal multipolar order in crystalline solids. Based on our symmetry analysis, we identify five categories of polarized matter, each of which is characterized by distinct features in the electronic band structure. For example, Rashba spin splitting in electropolar bulk materials like wurtzite represents the electric dipolarization in these materials. We also develop a general formalism of indicators for individual multipole densities that provide a physical interpretation and quantification of multipolar order. Our work clarifies the relation between patterns of localized multipoles and macroscopic multipole densities they give rise to. To illustrate the general theory, we discuss its application to polarized variants of hexagonal lonsdaleite and cubic diamond structures. Our work provides a general framework for classifying and expanding current understanding of multipolar order in complex materials.

Received 24 January 2023
Accepted 17 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.155201

Physics Subject Headings (PhySH)

Antiferromagnetism Dresselhaus coupling Electric polarization Ferroelectricity Ferromagnetism Magnetic order Rashba coupling Spin-orbit coupling

Diamond structure Electronically polarized systems Magnetic systems Wurtzite

Discrete symmetries in condensed matter Tight-binding model k dot p method

Condensed Matter, Materials & Applied Physics

Collections

This article appears in the following collection:

Emmanuel Rashba: Breaking New Ground in Solid-State Exploration

Physical Review B is pleased to present the “Collection in Honor of Emmanuel I. Rashba and His Fundamental Contributions to Solid-State Physics” in the year of his 95th birthday, highlighting the many ways in which his work has changed the landscape of modern condensed matter physics. Papers belonging to this collection will be published through mid-2023. The contributed articles, and an editorial by Guest Editors Mark Dykman, Alexander Efros, Bertrand Halperin, Leonid Levitov, and Charles Marcus, are linked below.

Authors & Affiliations

R. Winkler ^1,2 and U. Zülicke ³

¹Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA
²Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
³MacDiarmid Institute, School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 15 — 15 April 2023

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Typical examples for spinful band dispersions $E_{σ} (k)$ associated with the five categories of polarized matter. The bands for the parapolar and antimagnetopolar categories are at least twofold spin-degenerate in the entire Brillouin zone. The upper (lower) row corresponds to $n \geq 1$ ( $n = 0$ ), where $n$ is defined in Table 4. The expressions in the lower left of the panels use the simplified notation of Table 4 to represent the indicators for the presence of multipolar order. Different colors represent opposite spin orientations. In the multipolar case, the spin-split bands have more complicated spin textures such that it is generally not possible to assign a spin index to these bands.
Reuse & Permissions
Figure 2
Multipole densities in the lonsdaleite family. Top row [(a)–(d)]: pristine lonsdaleite (c) and variants of the lonsdaleite crystal structure including wurtzite (d). In (c), the four atoms in a unit cell are highlighted in blue. (g) The local electric octupole moments $M_{0}$ on the identical atoms in pristine lonsdaleite give rise to an electric quadrupolarization ( $ℓ = 2$ ). Remaining panels in the central row [(e), (f), and (h)]: For the structures (a), (b), and (d) consisting of two distinct types of atoms, panels (e), (f), and (h) show the deviation $Δ M$ of the local octupole moments compared with the moments $M_{0}$ when all atoms are identical (g). These local moments $Δ M$ give rise to (e) an electric quadrupolarization ( $ℓ = 2$ ), (f) an octupolarization ( $ℓ = 3$ ), and (h) a dipolarization ( $ℓ = 1$ ). Bottom row [(i), (j), (k), and (l)]: local magnetic dipole moments give rise to (i) a hexadecapolarization ( $ℓ = 4$ ), (j) an octupolarization ( $ℓ = 3$ ), (k) a quadrupolarization ( $ℓ = 2$ ), and (l) a magnetization ( $ℓ = 1$ ). (i)–(l) show the local magnetic dipole moments with different shades of the same color because all sites are equivalent by symmetry (i.e., they have the same Wyckoff letter) so that the local moments are likewise symmetry-equivalent. The same situation arises for the electric octupole moments (g) of the nonmagnetic pristine lonsdaleite structure (c).
Reuse & Permissions
Figure 3
Magnetic lonsdaleite with oppositely oriented local magnetic moments pointing perpendicular to the lonsdaleite main axis, compare Fig 2. The local moments give rise to a magnetic quadrupolarization ( $ℓ = 2$ ).
Reuse & Permissions
Figure 4
Multipole densities in the diamond family. Crystal structure of (a) pristine diamond and (b) zincblende. In (a), the two atoms in a unit cell are highlighted in blue. Local octupole moments give rise to (c) a hexadecapolarization ( $ℓ = 4$ ) and (d) an octupolarization ( $ℓ = 3$ ). Local dipole moments give rise to (e) a quadrupolarization ( $ℓ = 2$ ) and (f) a dipolarization ( $ℓ = 1$ ).
Reuse & Permissions
Figure 5
Quadrupole densities in tetragonally distorted diamond. (a) The distorted electric octupole moments due to the $s p^{3}$ hybrid orbitals give rise to local electric quadrupole moments which, in turn, give rise to an electric quadrupole density. Alternating patterns of magnetic dipole moments oriented (b) parallel and (c) perpendicular to the tetragonal axis give rise to magnetic quadrupole densities.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics