Giant orbital diamagnetism of three-dimensional Dirac electrons in ${\mathrm{Sr}}_{3}\mathrm{PbO}$ antiperovskite

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Giant orbital diamagnetism of three-dimensional Dirac electrons in ${Sr}_{3} PbO$ antiperovskite

S. Suetsugu, K. Kitagawa, T. Kariyado, A. W. Rost, J. Nuss, C. Mühle, M. Ogata, and H. Takagi

Phys. Rev. B 103, 115117 – Published 10 March 2021

Abstract

In Dirac semimetals, interband mixing has been known theoretically to give rise to a giant orbital diamagnetism when the Fermi level is close to the Dirac point. In ${Bi}_{1 - x} {Sb}_{x}$ and other Dirac semimetals, an enhanced diamagnetism in the magnetic susceptibility χ has been observed and interpreted as a manifestation of such giant orbital diamagnetism. Experimentally proving their orbital origin, however, has remained challenging. The cubic antiperovskite ${Sr}_{3} PbO$ is a three-dimensional Dirac electron system and shows the giant diamagnetism in χ as in the other Dirac semimetals. ${}^{207}{Pb}$ NMR measurements are conducted in this study to explore the microscopic origin of diamagnetism. From the analysis of the Knight shift $K$ as a function of χ and the relaxation rate ${T_{1}}^{- 1}$ for samples with different hole densities, the spin and the orbital components in $K$ are successfully separated. The results establish that the enhanced diamagnetism in ${Sr}_{3} PbO$ originates from the orbital contribution of Dirac electrons, which is fully consistent with the theory of giant orbital diamagnetism.

Received 30 December 2020
Accepted 22 February 2021

DOI:https://doi.org/10.1103/PhysRevB.103.115117

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Dirac semimetal Topological materials

Nuclear magnetic resonance

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

S. Suetsugu ^1,2,3, K. Kitagawa¹, T. Kariyado⁴, A. W. Rost^2,5,6, J. Nuss², C. Mühle², M. Ogata ¹, and H. Takagi^1,2,5

¹Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113–0033, Japan
²Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
³Department of Physics, Kyoto University, Kyoto 606–8502, Japan
⁴International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305–0044, Japan
⁵Institute for Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
⁶SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St. Andrews, Fife KY16 9SS, United Kingdom

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Vol. 103, Iss. 11 — 15 March 2021

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Figure 1
Giant orbital diamagnetism of Dirac electrons. (a) Schematic energy dispersion of a massive Dirac electron band (left) and the expected giant orbital diamagnetism as a function of the Fermi level $E_{F}$ (right). Δ is a Dirac mass gap. The orbital diamagnetism takes a maximum with $E_{F}$ in the Dirac mass gap. (b) Energy density $E_{⊥} D (E_{⊥})$ as a function of $E_{⊥}$ for a Dirac band (left) and an ordinary parabolic band (right). $E_{⊥}$ and $D (E_{⊥})$ represent the kinetic energy originating from the two-dimensional momentum perpendicular to the applied field and the density of states as a function of $E_{⊥}$ , respectively. The Landau levels in a magnetic field with spin up and down are indicated by the blue and the red lines, respectively. The average $E_{⊥}$ of the Landau levels in a magnetic field, to which the gray shaded area of the zero field $D (E_{⊥})$ condenses, is indicated by the gray dashed lines. They are larger than the average $E_{⊥}$ of the corresponding gray shaded area (black dashed line) for the Dirac band but identical to that for the normal parabolic band.
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Figure 2
Basic electronic structure and bulk magnetic susceptibility in ${Sr}_{3} PbO$ antiperovskite. (a) and (b) Crystal structure and the first Brillouin zone for ${Sr}_{3} PbO$ antiperovskite. Six Dirac points [red points in (b)] are protected by $C_{4}$ rotational symmetry along Γ-X lines. The blue line parallel to the (110) direction connects two Dirac points, between which a saddle point (SP) exists. (c) Field dependence of Hall resistivity $ρ_{x y}$ in the zero-field limit. The slopes yield the hole densities $1.6 \times 10^{18}, 5.0 \times 10^{19}, 2.0 \times 10^{20}$ and $2.2 \times 10^{20} c m^{- 3}$ for samples A, C, D, and E, respectively. (d) Total carrier density as a function of the Fermi energy ( $E_{F}$ ) obtained from a band calculation, which gives the estimates of $E_{F}$ (dashed lines) for samples A, C, D, and E as –45, –125, –235, and $- 250 meV$ , respectively, from the experimental hole densities obtained from (c). (e) Schematic band structure of ${Sr}_{3} PbO$ antiperovskite for a $k$ axis along the blue line in (b), which is divided into three regimes, the Dirac bands (red area), the saddle point (SP) and the multibands (blue area). The Dirac bands enclosed by the gray dashed rectangle give rise to the giant orbital diamagnetism as illustrated in Fig. 1. (f) Temperature dependence of magnetic susceptibility χ for samples A, C, and E (solid lines). The magnitude of diamagnetic susceptibility increases with decreasing the hole density $p$ , in particular very rapidly from sample C to A. By subtracting Curie-like contributions at low temperatures, the intrinsic behaviors of χ are estimated (dashed lines).
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Figure 3
NMR spectra, Knight shift $K$ and spin lattice relaxation rate ${T_{1}}^{- 1}$ in ${Sr}_{3} PbO$ antiperovskite. (a) NMR spectra for samples A–E at a temperature $T = 150 K$ . The peak positions systematically shift to the negative side upon decreasing the hole density $p$ . The hole density $p$ and the Fermi level $E_{F}$ for each sample are displayed in Figs. 2 and 2. (b) Temperature dependence of the NMR Knight shift $K (T)$ for samples A-E. Note the positive correlation between $K (T)$ and $χ (T)$ in Fig. 2 is indicative of the positive hyperfine coupling constants $A_{spin}$ and $A_{orb}$ . (c) Temperature dependencies of the spin-lattice relaxation rate divided by temperature ${(T_{1} T)}^{- 1} (T)$ for samples A-E. The Korringa law, ${(T_{1} T)}^{- 1} =$ constant (black dashed line) holds well. A crossover from $T$ independence to $T^{2}$ dependence (red dashed line) for sample A reflects the strongly energy dependent density of states of Dirac electrons. (d) Fermi energy $E_{F}$ dependence of ${(T_{1} T)}^{- 1 / 2}$ at 100 K (closed circles), which is scaled well with the calculated partial density of states for Pb $6 p$ orbitals at $E_{F}, D_{p - orb}$ (black line).
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Figure 4
Separation of Knight shift $K (T)$ into the spin and the orbital contributions. (a) Knight shift $K (T)$ vs magnetic susceptibility $χ (T)$ plot for samples A, C, and E. The hole density $p$ and the Fermi level $E_{F}$ for each sample are displayed in Figs. 2 and 2. Here the intrinsic $χ (T)$ after the subtraction of the Curie contribution [dashed line in Fig. 2] was used for the plot. The orbital hyperfine coupling constant $A_{orb} = 88 \pm 14 kOe / μ_{B}$ can be estimated from a linear fit to the $K (T) − χ (T)$ relationship for sample A (red line), where the spin contribution $K_{spin}$ is almost zero. An upward deviation for samples C and E from the red line can be attributed to $K_{spin}$ . The black dashed line indicates the estimated $K_{spin}$ with the assumption of $K_{orb} \sim 0$ . The crossing point between the red line and the black dashed line represents the chemical shift $K_{chem}$ and the core magnetic susceptibility $χ_{0}$ . (See the main text.) (b) Knight shift $K (T)$ as a function of the spin-lattice relaxation rate ${(T_{1} T)}^{- 1 / 2}$ for sample E to sample A. The spin contribution $K_{spin}$ expected from the Korringa relationship $T_{1} T {K_{spin}}^{2} = S$ is indicated by the gray dashed line. The black dashed line indicates a modified Korringa relationship with an enhanced Korringa constant $S^{*} \sim 6.3 S$ , which assumes a dominant spin contribution and hence almost zero orbital contribution, $K_{orb} \sim 0$ , for heavily doped samples B–E. Note that the extrapolation of the black dashed line to ${(T_{1} T)}^{- 1 / 2} = 0$ gives an estimate of the chemical shift $K_{chem}$ with the assumption of $K_{orb} \sim 0$ for samples B–E. The large and additional negative shift of sample A should be ascribed to the orbital contribution, $K_{orb}$ .
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Figure 5
Theoretical calculation of magnetic susceptibility as a function of the Fermi level $E_{F}$ for ${Sr}_{3} PbO$ antiperovskite. The Fermi level $E_{F}$ dependence of magnetic susceptibility $χ^{cal}$ at 232 K (0.2 eV) and 348 K (0.3 eV) (solid lines), calculated for tight-binding bands of ${Sr}_{3} PbO$ by incorporating the interband effect. The diamagnetic contributions from the core electrons $χ_{0}$ , which are temperature and hole density independent constants, are not included in the calculation. To compare with the experimental results, $χ^{cal} + χ_{0}$ should be used. A large diamagnetism grows with approaching the Dirac mass gap. The deconvoluted orbital contribution ${χ^{cal}}_{orb}$ at 232 and 348 K are shown by the dashed lines. The experimentally determined $E_{F}$ for samples A–E are indicated by the vertical dashed lines.
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Physical Review B

covering condensed matter and materials physics

Giant orbital diamagnetism of three-dimensional Dirac electrons in ${Sr}_{3} PbO$ antiperovskite

S. Suetsugu, K. Kitagawa, T. Kariyado, A. W. Rost, J. Nuss, C. Mühle, M. Ogata, and H. Takagi

Phys. Rev. B 103, 115117 – Published 10 March 2021

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Physical Review B

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Giant orbital diamagnetism of three-dimensional Dirac electrons in Sr3PbO antiperovskite

S. Suetsugu, K. Kitagawa, T. Kariyado, A. W. Rost, J. Nuss, C. Mühle, M. Ogata, and H. Takagi

Phys. Rev. B 103, 115117 – Published 10 March 2021

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Giant orbital diamagnetism of three-dimensional Dirac electrons in ${Sr}_{3} PbO$ antiperovskite