Large anomalous Hall and Nernst effects from nodal line symmetry breaking in ${\mathrm{Fe}}_{2}\mathrm{Mn}X$ ($X$ = P, As, Sb)

Large anomalous Hall and Nernst effects from nodal line symmetry breaking in ${Fe}_{2} Mn X$ ( $X$ = P, As, Sb)

Jonathan Noky, Qiunan Xu, Claudia Felser, and Yan Sun

Phys. Rev. B 99, 165117 – Published 12 April 2019

Abstract

A large Berry curvature in the vicinity of the Fermi energy is required in order to obtain a large anomalous Hall and Nernst effect. This Berry curvature can be induced by Weyl points and gapped nodal lines. One of the possible mechanisms takes place in systems with a symmetry group where mirror planes lead to protected nodal line structures. When these mirror symmetries are broken, e.g., via fixing a magnetization direction, a gap in the former nodal line can create Weyl points and a large Berry curvature along the gapped lines. In this work we study this effect in a minimal effective model and relate the results to the real regular Heusler compounds ${Fe}_{2} Mn X$ ( $X = P, As, Sb$ ). These materials have three mirror planes in the nonmagnetic case, leading to three nodal lines near the Fermi level. However, dependent on the orientation of the magnetization, some of the mirror planes are broken and the respective lines are gapped, creating large Berry curvature. Because the Fermi level is located in vicinity of the gapped lines, this leads to a large anomalous Hall and Nernst effect, which can be tuned to even higher values with a little bit of doping in the system.

Received 17 October 2018
Revised 18 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.165117

Physics Subject Headings (PhySH)

Anomalous Hall effect Magnetism Nernst effect

Heusler alloy Node-line semimetals

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jonathan Noky, Qiunan Xu, Claudia Felser, and Yan Sun^*

Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany

^*ysun@cpfs.mpg.de

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Vol. 99, Iss. 16 — 15 April 2019

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Figure 1
(a) Nodal line in the $k_{x} - k_{y}$ plane protected by the mirror symmetry in this plane. (b) Gapped line in the $k_{x} - k_{y}$ plane due to breaking the mirror symmetry with two Weyl points. (c) Band structure with and without the mirror symmetry. The insets show the splitting in the $Γ - X$ direction. (d) Berry curvature in the Brillouin zone after breaking the mirror symmetry. It is mostly located along the former nodal line (green dashed). (e) Anomalous Hall conductivity. (f) Anomalous Nernst conductivity at 300 K.
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Figure 2
(a) Crystal structure of ${Fe}_{2} Mn X$ with magnetic moments. (b) Brillouin zone with path for the band structure (green lines) and nodal lines (blue lines). The dashed lines become gapped due to the chosen magnetization direction. (c) Band structures with (black) and without (spin up: red; spin down: blue) spin-orbit coupling for ${Fe}_{2} MnP, {Fe}_{2} MnAs$ , and ${Fe}_{2} MnSb$ . The circles mark the positions of the nodal lines, where the green one belongs to the still protected mirror in the $x - y$ plane and the orange one is gapped out when SOC is taken into account. In the inset this behavior is shown enlarged and with the respective irreducible representations of the bands (red: without SOC; black: with SOC).
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Figure 3
Contributions to the anomalous Hall and Nernst effect in the three-dimensional Brillouin zone for ${Fe}_{2} Mn X$ ( $X = P, As, Sb$ ). The largest contributions stem from the gapped nodal lines in the $k_{x} - k_{z}$ and $k_{y} - k_{z}$ plane. The $k_{x} - k_{y}$ plane does not contribute because in this plane the mirror symmetry is still valid, resulting in a closed nodal line.
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Figure 4
(a) Band gap in the $k_{z} = 0$ plane with $M ∥ (001)$ . The closed nodal line is visible. (b) Berry curvature (BC) in the $k_{z} = 0$ plane with $M ∥ (001)$ . Only weak BC is visible. (c) Band gap in the $k_{y} = 0$ plane with $M ∥ (001)$ . The nodal line is gapped out. (d) Berry curvature (BC) in the $k_{y} = 0$ plane with $M ∥ (001)$ . Strong BC is visible. (e),(f) BC in the Brillouin zone for $M ∥ (110)$ and $M ∥ (111)$ , respectively. All three nodal lines are gapped out and create strong BC. The total BC and thus the AHC and ANC are the same as for $M ∥ (001)$ . The results are shown for ${Fe}_{2} MnSb$ (the other compounds show a similar behavior).
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Figure 5
(a) Anomalous Hall effect of ${Fe}_{2} Mn X$ . (b) Anomalous Nernst effect of ${Fe}_{2} Mn X$ at $T = 300$ K in dependence on energy level. (c) Anomalous Nernst effect in dependence on temperature at $E = E_{F}$ . (d) Anomalous Nernst effect of ${Fe}_{2} MnAs$ depending on temperature and energy level.
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Physical Review B

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Large anomalous Hall and Nernst effects from nodal line symmetry breaking in ${Fe}_{2} Mn X$ ( $X$ = P, As, Sb)

Jonathan Noky, Qiunan Xu, Claudia Felser, and Yan Sun

Phys. Rev. B 99, 165117 – Published 12 April 2019

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Physical Review B

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Large anomalous Hall and Nernst effects from nodal line symmetry breaking in Fe2MnX (X = P, As, Sb)

Jonathan Noky, Qiunan Xu, Claudia Felser, and Yan Sun

Phys. Rev. B 99, 165117 – Published 12 April 2019

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Large anomalous Hall and Nernst effects from nodal line symmetry breaking in ${Fe}_{2} Mn X$ ( $X$ = P, As, Sb)