TUNING TOPOLOGICAL PROPERTIES: ONE STEP TOWARDS 2D QUANTUM COMPUTING

Karunya Shirali, William Shelton and Ilya Vekhter

Louisiana State University

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MOTIVATION

• Topological insulators (TIs): �insulators in the bulk, topologically protected metallic surface states
• Topological quantum computing:�information encoded in braiding of anyons
• Class of systems predicted to host anyons:�Topological superconductors, etc

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• Hard to observe experimentally
• Alternative: TI-superconductor heterostructure�Proximity induced superconductivity

��resistant to decoherence

Topological insulator

Surface states^[1]

SPIN-ORBIT COUPLING AND HELICAL SURFACE STATES

Helical state² (k space)

Helical state¹ (real space)

Surface states observed with ARPES³

LEVEL SPLITTING AND SPIN-ORBIT COUPLING

• Bi₂Se₃: Start from atomic p orbitals¹ of Bi, Se

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• Consider chemistry of the system

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• Find Bi pz, Se pz levels near chemical potential

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• Include SOC:�Obtain band inversion �

¹: PRB 82, 045122 (2010)

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Band inversion

(i) (ii) (iii) (iv)

WHY DO WE USE DFT?

BI₂SE₃ AND BI₂TE₃: EXISTING WORK AND PROBLEMS

• Most calculations keep lattice parameters fixed without optimizing them
• Reason: some lattice parameters, especially d_int, are significantly different from experiment if they aren’t fixed�
• But DFT is a ground-state theory: could obtain metastable states�
• We fill this gap and set up systematic means¹ of performing first-principles calculations on 3D TIs
• Obtain bulk lattice constants, inter-QL distances,�properties of topological surface states accurately�
• Important to include SOC�� ¹: K Shirali, W A Shelton and I Vekhter 2021 Journal of Physics: Condensed Matter 33 035702

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REPRODUCE BULK, SURFACE PROPERTIES

Hexagonal warping

METHODOLOGY TO INCLUDE VDW FOR TIS

• Set up systematic way¹ to calculate structural & electronic properties self-consistently
• Bulk: lattice parameters and band gap, inter-QL distance�Surface: Dirac velocities and spin textures �
• Have methodology for: bulk ✅� surface ✅

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• Moving forward: how does strain affect topological properties?�

¹: Shirali K, Shelton W A and Vekhter I 2020 Journal of Physics: Condensed Matter 33 035702

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TOPOLOGY FROM BAND INVERSION

• Bi₂Se₃ under strain, fix c/a, P_hydr = 0
• QL remains intact; inter-QL distance expands as c increases

Inverted bands

No inversion

Bi₂Se₃ bulk bands

COULOMB AND VDW DRIVE TRANSITION

Figures:

(a) Bi₂Se₃ band gap

(b) Bi pz, Se pz levels

as a fn. of c/a

Se1 pz (outer atomic layers of QL) levels change with c/a:

Mainly inter-QL physics

Transition

Level switching

at transition

COULOMB AND VDW DRIVE TRANSITION

Figures:

(a) Bi₂Se₃ band gap

(b) Bi pz, Se pz levels

as a fn. of c/a

Se1 pz (outer atomic layers of QL) levels change with c/a:

Mainly inter-QL physics

Transition

Topological

Trivial

COULOMB VERSUS VDW

• Emergent picture: Se-Se (Te-Te) Coulomb repulsion �between QLs compete with vdW attraction �as c/a is varied �Drives transition
• Subtract ionic radii:�universal value for critical atomic spacing

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Can be measured in experiment

arXiV 2202.12867

TOPOLOGICAL PHASE TRANSITIONS IN DISORDERED TIS

• Topological phase transition in (Bi_1-xSb_x)₂Se₃: Sb substitution reduces strength of SOC
• Range of values (78%-83%) predicted¹ for critical impurity concentration
• Do Bi and Sb contribute to the same bands or distinct bands?

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• Simulating disorder in DFT: large unit cells required to mimic randomness
• SCRAPS² generate supercells for Bi/Sb alloying, atomic pair correlations �zero up to third nearest neighbor: no configurational averaging required�

In collaboration with Ames Laboratory

¹: Liu J and Vanderbilt D 2013 Phys. Rev. B 88 224202

²: Nature Computational Science 1, 54–61 (2021)

Figure: (Bi_1-xSb_x)₂Se₃ crystal structure, x = 0.5

d_int

Sb

Se

Bi

QL

CONCLUDING REMARKS, OUTLOOK

• 50%, 62.5%, 75% concentrations: obtain trivial insulator ground-states from SCRAPS supercells
• But end-point materials both topological: �Two topological transitions, one at low x and other �at large x

Future work

• Bi₂(Se_1-yTe_y)₃ alloy: �- Se and Te atoms have two sublattices to choose from

x = 0.5, 0.625, 0.75

x = 0.02

CONCLUSION

• Set up comprehensive framework for ab initio calculations of topological insulators�
• Reproduce bulk, surface properties accurately�
• Explained essential physics of topological phase transitions under strain�Due to competing Coulomb and van der Waals interactions between the quintuple layers �
• Foundation to study and address other problems involving topological insulators

ACKNOWLEDGEMENTS

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This research was supported by NSF via Grant No. DMR-1410741 and by the U.S. Department of Energy under EPSCoR Grant No. DE-SC0012432 with additional support from the Louisiana Board of Regents. Portions of this research were conducted with high performance computing resources provided by Louisiana State University (http://www.hpc.lsu.edu).

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Thank you!