Doping-induced topological phase transition in Bi: The role of quantum electronic stress

Kyung-Hwan Jin, Han Woong Yeom, and Feng Liu

Phys. Rev. B 101, 035111 – Published 8 January 2020

Abstract

Charge doping is an essential means to tailor a materials' properties. However, besides moving the Fermi level, charge doping is generally not expected to induce topological phase transition (TPT). Here, using first-principles calculations, we demonstrate an electron doping-induced TPT in bulk Bi from a higher-order topological insulator (HOTI) to a TI. The underlying mechanism is revealed to be driven by an electron doping-induced quantum electronic stress (QES), which in turn induces a highly anisotropic lattice expansion to close/reopen the small energy gap in Bi band structure. Our finding significantly resolves an outstanding controversy concerning the topological characterization of bulk Bi among existing experiments and theories, and explains the physical origin of the topological order in Bi (111) thin films. It sheds new light on the fundamental understanding of topological properties of small band gap materials in relation to doping and QES.

Received 9 July 2019
Revised 10 December 2019

DOI:https://doi.org/10.1103/PhysRevB.101.035111

Physics Subject Headings (PhySH)

Topological materials Topological phase transition

First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kyung-Hwan Jin ¹, Han Woong Yeom^2,3, and Feng Liu ^1,*

¹Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA
²Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea
³Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea

^*Corresponding author: fliu@eng.utah.edu

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Vol. 101, Iss. 3 — 15 January 2020

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Images

Figure 1
(a) Rhombohedral (cyan) unit cell and hexagonal crystal structure (black) of Bi bulk. (b) Brillouin zone of the Bi bulk and surface in the [111] and [110] directions. Top and side view of (c) Bi (111) and (d) Bi (110) surfaces. The unit cell of the Bi (111) and Bi (110) surfaces is indicated by a yellow rhombus and a blue rectangle, respectably.
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Figure 2
(a) The QES as a function of the variation of carrier density $Δ n$ . (b) The change of lattice volume, the (111) in-plane lattice constant $a$ , the bilayer buckling height $b$ , and interbilayer spacing $c$ , as a function of $Δ n$ . (c) Band structure of Bi bulk near the $L$ point varying with carrier density $Δ n$ and (d) corresponding Fermi level shift ( $Δ E_{F}$ ). The shift of $Δ E_{F} = 0.05 eV$ as marked by star corresponds to electron density $Δ n \sim 1.1 \times 10^{- 4} / Å^{3}$ .
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Figure 3
(a) Calculated band structure for intrinsic Bi, doped with $Δ n = 0, 0.49 \times 10^{- 4} / Å^{3}$ , and $0.98 \times 10^{- 4} / Å^{3}$ ( $Δ V = 0 %$ , 1.4%, and 2.8%), respectively. Parity ( ${\hat{C}}_{3}$ ) eigenvalues are indicated by $+$ and $-$ ( $π$ and $\pm π / 3$ ) signs. (b) Enlarged band structures of (a) near the $L$ point. (c) Band gap at the $L$ point as a function of $Δ n$ (or optimized volume) using mBJ and HSE06 functionals.
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Figure 4
(a) Calculated surface band structures of 10 BL, 40 BL, and 100 BL films and semi-infinite surface for intrinsic Bi indicating a HOTI. (b) Calculated surface band structures of 10 BL, 40 BL, and 100 BL films and semi-infinite surface for $Δ n = 0.98 \times 10^{- 4} / Å^{3}$ ( $Δ V = 2.8 %$ ) indicating a TI.
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Figure 5
(a)–(c) Comparison of DFT surface state of Bi (111) thin films ( $Δ n = 0.98 \times 10^{- 4} / Å^{3}$ and $Δ V = 2.8 %$ ) for 14 BL, 18 BL, and 79 BL with experimental ARPES results (left panel), respectively. The experimental images were taken from Ref. [32]. The experimental Fermi level was set to zero. The calculated surface states have been shifted downward (i.e., Fermi level shift upward) by 0.05 eV for electron doping to match with ARPES data within the rigid-band approximation. The nontrivial character of surface states is well reproduced. (d) Experimental ARPES and Fermi surface for Bi (110) surface states which were taken from Refs. [52, 53]. (e) Calculated surface states and Fermi surface of Bi (110) semi-infinite surface. For Bi (110) surface states, we used Wannier fitting parameters of the equilibrium structure of Bi bulk ( $Δ n = 0$ ). To match with experimental ARPES data, no need to shift Fermi level for Bi (110) surface.
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Figure 6
(a) and (b) The Fermi surface of semi-infinite Bi (111) films ( $Δ n = 0.98 \times 10^{- 4} / Å^{3}$ and $Δ V = 2.8 %$ ) and corresponding spin textures, respectively. The arrows (with red/blue background) indicate the in-plane (out-of-plane) spin component. (c) and (d) The calculated spin texture of Bi (110) semi-infinite surface ( $Δ n = 0$ ) and the QPI pattern at the Fermi level, respectively.
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