Hourglasslike nodal net semimetal in ${\mathrm{Ag}}_{2}{\mathrm{BiO}}_{3}$

Hourglasslike nodal net semimetal in ${Ag}_{2} {BiO}_{3}$

Botao Fu, Xiaotong Fan, Dashuai Ma, Cheng-Cheng Liu, and Yugui Yao

Phys. Rev. B 98, 075146 – Published 27 August 2018

Abstract

Based on first-principles calculations and analysis of crystal symmetries, we propose a kind of hourglasslike nodal net (HNN) semimetal in centrosymmetric ${Ag}_{2} {BiO}_{3}$ that is constructed by two hourglasslike nodal chains at mutually orthogonal planes in the extended Brillouin zone (BZ) when the weak spin-orbit coupling (SOC) mainly from the $6 s$ orbital of Bi atoms is ignored. The joint point in the nodal net structure is a special double Dirac point located at the BZ corner. Different from previous HNNs [T. Bzdušek et al., Nature (London) 538, 75 (2016)] where the SOC and double group nonsymmorphic symmetries are necessary and also different from the accidental nodal net, this HNN structure is inevitably formed and guaranteed by spinless nonsymmorphic symmetries and thus robust against any symmetry-remaining perturbations. The Fermi surface in ${Ag}_{2} {BiO}_{3}$ consisting of a toruslike electron pocket and a toruslike hole pocket may lead to unusual transport properties. A simple four-band tight-binding model is built to reproduce the HNN structure. For a semi-infinite ${Ag}_{2} {BiO}_{3}$ , the “drumhead”-like surface states with nearly flat dispersions are demonstrated on (001) and (100) surfaces, respectively. If such a weak-SOC effect is taken into consideration, this HNN structure will be slightly broken, leaving a pair of hourglasslike Dirac points at the twofold screw axis. This type of hourglasslike Dirac semimetal is symmetry enforced and does not need band inversion anymore. Our discovery provides a platform to study novel topological semimetal states from nonsymmorphic symmetries.

Received 22 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.075146

Physics Subject Headings (PhySH)

First-principles calculations Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Botao Fu, Xiaotong Fan, Dashuai Ma, Cheng-Cheng Liu^*, and Yugui Yao^†

Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China

^*ccliu@bit.edu.cn
^†ygyao@bit.edu.cn

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Vol. 98, Iss. 7 — 15 August 2018

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Figure 1
(a) Left: the hourglasslike band structures protected by $g_{1} = {M_{z} | a / 2}$ for systems without $P$ symmetry including the SOC effect. Right: the corresponding nodal rings at the $k_{x} − k_{y}$ plane (red solid line) and the $k_{x} − k_{z}$ plane (green dashed line) constructing an HNC in the extended BZ. (b) Left: hourglasslike band structures without SOC protected by $g_{1}$ and $g_{2}$ in the absence of SOC effect. Right: the corresponding HNCs at the $k_{x} − k_{y}$ plane (red solid line) and the $k_{x} − k_{z}$ (green dashed line) constructing an HNN structure in the extended BZ. The eigenvalues of ${M_{z} | a / 2}$ are given. Notice that the HNC in (b) is formed by a large nodal line traversing the whole BZ, while the HNC in (a) is formed by two tangent nodal rings alternately in orthogonal planes.
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Figure 2
(a) The unit cell of ${Ag}_{2} {BiO}_{3}$ in the $P n n a$ phase. The violet, red, and gray balls represent bismuth, oxygen, and silver atoms, respectively. (b) The bulk Brillouin zone and (001)/(100) surface Brillouin zone. (c) The band structure of ${Ag}_{2} {BiO}_{3}$ without SOC. (d) The toruslike Fermi surface with electron (red) and hole (blue) pockets. (e) All the twofold-degenerate points (green) and two nodal lines (red and blue) in the BZ. (f) Display of the HNN structure in the extended BZ. The color stands for the energy dispersion of the HNN with respect to the Fermi level. The red dashed lines labeled $L_{1}, L_{2}$ represent two distinctive loops along which the Berry phase is calculated.
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Figure 3
(a) The band structure at the $X U R S$ plane. The eigenvalues of ${M_{x} | b / 2 + c / 2}$ are shown. (b) Three-dimensional view of the energy spectrum at the $X U R S$ plane. (c) The band structure at the $X Γ Y S$ plane. The eigenvalues of ${M_{z} | a / 2}$ are shown. (d) Three-dimensional view of the double Dirac cone around the $S$ point at the $R T Y S$ plane. The bands are doubly degenerate. (e) and (f) Calculated surface states of ${Ag}_{2} {BiO}_{3}$ on the (001) and (100) surfaces, respectively. The color represents the weight of the local density of states.
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Figure 4
(a) The lattice with four bismuth atoms for the TB model. $t_{1}$ is the NN hopping between sites 1 and 2 and 3 and 4. $t_{2}$ is the NN hopping between sites 2 and 4 and 1 and 3. $t_{3}$ is the NNN hopping between sites 1 and 4 and 2 and 3. (b) TB band structure $(t_{1} = - 0.294 eV, t_{2} = - 0.172 eV, t_{3} = 0.038 eV, ɛ_{i} = 0.0 eV)$ . (c) TB band structure with the same parameters as (b) except $t_{2} = 0.0 eV$ . (d) TB band structure with the same parameters as (b) except $t_{3} = 0.0 eV$ .
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Figure 5
(a) and (b) Band structures from DFT including SOC. There are gaps inside the dashed circles. (c) The enlarged band structure around the $S$ point near the Fermi level. (d) The hourglasslike band structure along $S X$ below the Fermi level. The eigenvalues of ${C_{2 y} | a / 2 + b / 2 + c / 2}$ are given.
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Physical Review B

covering condensed matter and materials physics

Hourglasslike nodal net semimetal in ${Ag}_{2} {BiO}_{3}$

Botao Fu, Xiaotong Fan, Dashuai Ma, Cheng-Cheng Liu, and Yugui Yao

Phys. Rev. B 98, 075146 – Published 27 August 2018

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Physical Review B

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Hourglasslike nodal net semimetal in Ag2BiO3

Botao Fu, Xiaotong Fan, Dashuai Ma, Cheng-Cheng Liu, and Yugui Yao

Phys. Rev. B 98, 075146 – Published 27 August 2018

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Hourglasslike nodal net semimetal in ${Ag}_{2} {BiO}_{3}$