Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P[Formula: see text], which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects

Canfield, Paul

Jo, Na Hyun

Kaminski, Adam

Kuthanazhi, Brinda

Lange, Gunnar

Lee, Kyungchan

Orth, Peter

Schrunk, Benjamin

Slager, Robert-Jan

Trevisan, Thais

Wang, Lin-Lin

English

arXiv

Lange, Gunnar F.

Trevisan, Thaís Victa

Orth, Peter P.

Canfield, Paul C

The University of Manchester - Institutional Repository

Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2.

Abstract: Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P1¯, which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects

Trevisan, Thaís V.

Canfield, Paul C.

Apollo (Cambridge)

Supplementary information: Discovery of a weak topologicalinsulating state and van Hove singularity in triclinic RhBi2Kyungchan Lee,1, 2 Gunnar F. Lange,3 Lin-Lin Wang,1, 2 Brinda Kuthanazhi,1, 2Tháıs V. Trevisan,1, 2 Na Hyun Jo,1, 2 Benjamin Schrunk,1, 2 Peter P.Orth,1, 2 Robert-Jan Slager,3, 4 Paul C. Canfield,1, 2 and Adam Kaminski1, 21Ames Laboratory, Ames, Iowa 50011, USA2Department of Physics and Astronomy,Iowa State University, Ames, Iowa 50011, USA3TCM Group, Cavendish Laboratory, University of Cambridge,Cambridge CB3 0HE, United Kingdom4Department of Physics, Harvard University, Cambridge MA 02138, USA(Dated: February 19, 2021)1SUPPLEMENTARY NOTE 1: CRYSTAL GROWTH AND CHARACTERIZA-TION0 50 100 150 2000.00.20.40.60.81.00 50 100 150 200 250-1.0-0.50.00.51.0 ( T )/ ( 235 K )T ( K ) 0 T 7 T(a) 0.1 T(b)M ( emu/mol )T ( K )aSupplementary Figure 1. Resistivity and magnetization data for RhBi2. a, Resistivity datameasured in zero field and 7 T. b, magnetization measured as a function of temperature in 0.1 Tfield.Resistivity as a function of temperature was measured with current in the bc-plane,along one of the crystallographic axis. ρ(T) shows clear metallic behavior with no featuresthat would suggest any sort of phase transition in the measured temperature range. Withan applied field of 7 T, ρ(T) do not show any significance changes, except slightly higherresistance values at low temperatures. This very small magnetoresistance is not inconsistentwith the relatively low RRR value ( ∼ 6) of the sample. Magnetization as a function oftemperature, M(T), was measured with field applied perpendicular to the a-axis, and showsnearly temperature independent behavior, with a small, negative value of M over most ofthe temperature range. This fundamentally diamagnetic response is consistent with a verysmall density of states at the Fermi energy resulting in a larger core diamagnetic component.2SUPPLEMENTARY NOTE 2: DETAILS OF THE EFFECTIVE MODELWe aim to build an effective model for the observed surface state close to the Z point. Aswe know the form of the symmetries in the bulk, it would perhaps be most transparent tobuild a bulk model, which we then explicitly break to the relevant surface. In this section,we discuss this approach and explain why we instead opt to build a surface Hamiltoniandirectly. We also discuss the construction of this surface Hamiltonian.A minimal bulk Hamiltonian requires four bands, due to the presence of doubly degeneratebulk Kramer’s pairs. A general four-band model can be written in terms of 5 Γ matrices,and their corresponding commutators. However, we can restrict our model by imposing thebulk symmetries. The bulk has inversion and time-reversal symmetry. Combining thesegives an operator A = ΘP which satisfies:AH(k)A−1 = H(k) (1)We can choose our Γ-matrices to commute with the A operator1. Requiring this givesthat the commutators are all zero, so the effective model can be written using only the 5Γ-matrices. We can represent our symmetry operations as1:P = σx ⊗ I, Θ = i(I ⊗ sy)K (2)Where K is complex conjugation. This gives the 5 Γ-matrices commuting with A andsatisfying the Clifford algebra as:Γ(1,2,3,4,5) = {σx ⊗ I, σy ⊗ I, σz ⊗ sx, σz ⊗ sy, σz ⊗ sz} (3)Note that Γ1 equals P . We can then expand the Hamiltonian as:H(k) = εI4×4 +MΓ1 +5∑i=2AiΓi = εI4×4 +MΓ1 + S (4)Now we note that:ΘΓaΘ−1 = PΓaP−1 =1 if a = 1−1 if a 6= 1 (5)From which it follows that ε and M are even in k → −k, whereas the {Ai} are odd.Furthermore, S anticommutes with Γ1, which which it follows that:PH(k)P−1 = Γ1H(k)Γ1 = εI4×4 +MΓ1 − S (6)3To get the surface states from this bulk Hamiltonian, we organize our Hamiltonian into aterm H0 which contains constants and kx dependent terms, and a perturbation H1 whichdoes not contain any kx dependence. As there is no preferred direction under the symmetryoperations, H0 contains all 5 Γ-matrices, and is therefore not block diagonal. We can thenin principle solve H0 on a finite slab geometry, which would give us four states, two of whichare regular on the boundary. These can be interpreted as the edge states. From equation 6,it follows that the opposite parity states will be associated with an opposite sign for matrixS. Explicitly diagonalizing H0 gives the eigenvectors:v1 =ab10 , v2 =cd01 , v3 =ef10 , v4 =gh01 (7)Where the states (v1, v2) and (v3, v4) have pairwise the same eigenvalues, and the coefficientsare complicated functions of the free parameters. These are not block diagonal as H0 isnot block diagonal. We could now in principle investigate which of these four states remainregular at the boundary, given some boundary conditions, and then project the perturbationH1 in this basis. Note, however, that the resulting expression will be rather convoluted, withmany free parameters. As we are only interested in the surface dispersion to compute theDOS, we choose to instead build a model for the surface directly.The surface close to the TRIM Z exhibits time-reversal symmetry. Inversion symmetry isbroken at the surface. However, there will be some residual effects of inversion symmetry, asotherwise H0 should contain more terms, coming from the commutators of the Γ-matrices.Thus, inversion symmetry is broken down to some effective symmetry on the surface. Theexact nature of this symmetry will depend on the orbital contents. Note in particular that ifinversion acted purely as an effective C2 symmetry, taking (ky, kz) → (−ky,−kz), then ourexpansion in equation 13 in the main text would be trivial. The exact nature of this effectivesymmetry can be elucidated by solving the full bulk Hamiltonian above. For our purposes,however, it suffices to ignore this residual symmetry, and only focus on the time-reversalsymmetry. The residual inversion symmetry will only constrain the form of our effectiveHamiltonian further, potentially resulting in less free parameters. Thus, we do not lose anyphysical solutions by only ignoring the residual inversion symmetry, though we may find anunphysical parameter combination. As we are only interested in the shape of the surface4dispersion, this is of no concern.The time-reversal symmetry Θ = iσyK acts in spin space on the surface by constrain-ing:H(k) = (iσy)H∗(−k)(iσy)−1 (8)We can write a two-band model for the surface states as:H(k) = d(k) · σ (9)Hermiticity then requires that d(k) should be real. Imposing time-reversal symmetry gives:d0(k) = d0(−k), d1(k) = −d1(−k), d2(k) = −d2(−k), d3(k) = −d3(−k) (10)With energy given by:E(k) = d0 ±√d21 + d22 + d23 (11)Expanding to second order then gives:H2(k) = (E0 +A1k2y+A2k2z+A3kykz)I+(B1ky+C1kz)σx+(B2ky+C2kz)σy+(B3ky+C3kz)σz(12)With energyE2±(k) = E0 + A1k2y + A2k2z + A3kykz ±√√√√ 3∑i=1(Biky + Cikz)2 (13)If we want to account for strain, we should expand to third order which gives an effectivemodel:H3(k) = (E0 + A1k2y + A2k2z + A3kykz)I + (B1ky + C1kz +D1k2ykz + E1k2zky)σx+(B2ky + C2kz +D2k2ykz + E2k2zky)σy + (B3ky + C3kz +D3k2ykz + E3k2zky)σz(14)With energy:E3±(k) = E0 + A1k2y + A2k2z + A3kykz ±√√√√ 3∑i=3(Biky + Cikz +Dik2ykz + Eik2zky)2 (15)Note that our energy is symmetric under (ky, kz) → (−ky,−kz) even though the Hamilto-nian is not. This explains the observed effective C2 symmetry in the FS.The free parameters are fit using simulated annealing. This gives the result shown insupplementary table I and II.5A1 A2 A3 B1 C1 B2 C2 B3 C31.779 3.827 0.886 -0.248 -1.225 -0.138 -0.252 0.185 0.273Supplementary Table I. Fit parameters for E2±, see equation 13A1 A2 A3 B1 C1 D1 E1 B2 C2 D2 E2 B3 C3 D3 E31.895 8.051 2.701 0.145 0.021 -41.264 -150.345 0.082 -0.154 -2.524 -34.948 0.330 1.731 33.890 160.426Supplementary Table II. Fit parameters for E3±, see equation 15SUPPLEMENTARY NOTE 3: DFT CALCULATION OF 2D CUTSAlthough the triclinic structure of RhBi2 only has inversion symmetry, k → −k, whenprojected on the (100) surface, the inversion symmetry in 2D gives an effective twofoldrotation symmetry. This can also be understood in the 2D cuts of the 3D FS as shown inFig. 22D Bulk FS Cuts along k1k1=0.0 k1=0.2k1=0.1k1=0.5k1=0.3 k1=0.4 G’Supplementary Figure 2. FS projection along k1 direction 2D bulk Fermi surface cuts forRhBi2 along k1=0.0, 0.1, 0.2, 0.3, 0.4 and 0.5 planes. The 2-fold rotation symmetry only appearson the k1=0.0 and 0.5 planes.6SUPPLEMENTARY NOTE 4: BAND STRUCTUREA saddle point in the dispersion is characterized by a vanishing gradient, ∇εk = 0, andby opposite curvatures along orthogonal directions away from the extremum. Our ARPESmeasurements show evidence of the presence of a saddle point around the Z̄ point in thesurface Brillouin zone (sBZ) of RhBi2. To precisely determine the position of the saddlepoint, we perform a close investigation of the band structure using an effective low energyk · p model valid close to Z̄.A first look into the band structure of the k · p model suggest the existence of two saddlepoints in the SBZ [see Fig.4 (a)]. Interestingly, as shown in Fig.3, the lower band ε−,k isalmost flat along the diagonal direction parametrized byky = kZ̄,y + α(kz − kZ̄,z) , (16)where α ≈ 4.15 and (kZ̄,y, kZ̄,z) ≈ (0, 0.5)πa , denotes the position the Z̄ point. Zooming intothe region around Z̄ clearly shows two local minima along this flat direction at (ky−kZ̄,y, kz−kZ̄,z) = ±(−0.042, 0.010)πa ≡ ±k0. These points, on the other hand, behave as local maximaof the dispersion along the direction orthogonal to Eq. (16). This behavior is evidenced inFigs. 4(a) and (b), respectively.In fact, the gradient of the lower band∇ε−,k vanishes at±k0 [see Figs.4(c) and (d)], whichmakes these points true saddle points of the lower band. They occur at energy E = −83meV, where the peak in the surface density of states occurs [see Fig.4 (a)].Below, we provide details on the analytic solution of the gap equation (5) in the maintext in the presence of a logarithmically divergent density of states ρ(ξ) = ρ0 log(D/ |ξ|).After the change of variables y = ξ/Tc, the gap equation can be rewritten as1 = gρ0Λ/Tc∫0dy log(D/Tcy)1ytanh(y2). (17)Using properties of the logarithm, this integral can be further divided into two contributions,1gρ0= log(DTc) Λ/Tc∫0dy1ytanh(y2)−Λ/Tc∫0dylog(y)ytanh(y2). (18)The first integral in the right-hand-side of Eq.(18) is identical to the one encountered inthe standard BCS gap equation when approximating the DoS by its (constant) value at the7Fermi level:Λ/Tc∫0dy1ytanh(y2)= log(κΛTc), (19)where κ = 2eγ/π ≈ 1.13, and γ is Euler’s constant. For the second integral in Eq.(18), asimple integration by parts yieldsΛ/Tc∫0dylog(y)ytanh(y2)=12log2(ΛTc)− 14Λ/Tc∫0dy log2 (y) sech2(y2). (20)Since Λ/Tc  1 and log2 (y) sech2(y/2) remains finite as y →∞, we can approximateΛ/Tc∫0dy log2 (y) sech2(y2)≈∞∫0dy log2 (y) sech2(y2)≈ C , (21)where C = 2.669 is a numerical constant. Substituting Eqs.(19) and (21) into Eq.(18), weobtain1gρ0= log(DTc)log(κΛTc)− 12log2(ΛTc)+C4. (22)Supplementary Figure 3. Gradient of the lowest energy band of the k · p model. Thedirection where the norm of the gradient is minimum corresponds to the diagonal ky − kZ̄,y =−4.15(kz − kZ̄,z). In particular, it vanishes at (ky0, kz0) = ±(−0.042, 0.010)πa with respect to Z̄.These are saddle points of the lowest energy band. The momenta ky and kz are given in units ofπ/a. Accordingly, |∇ε−,k| has dimensions of energy.8Supplementary Figure 4. Cuts of the band structure of the k · p model along the direction(a) ky − kZ̄,y = −4.15(kz − kZ̄,z)and the orthogonal direction (b) ky − kZ̄,y = 0.24(kz − kZ̄,z).Upper (lower) band is shown in yellow (blue). The dispersion is centered at the saddle pointk0 = (−0.042, 0.010)πa . Panels (c) and (d) show the norm of the gradient |∇εk| of the dispersionin a and b, respectively. The momenta ky and kz are given in units of π/a. Accordingly, |∇εk| hasdimensions of energy.Since typically we have D > Λ2, it is convenient to rewrite the logarithms in Eq.(22) interms of D/Tc rather than Λ/Tc. This simplifies Eq.(22) significantly, since the linear termin log(κD/Tc) vanishes identically. We thus find2gρ0= log2(κDTc)+C2− log2(ΛD)− log2(1κ). (23)The strongest divergence comes from the term log2(κD/Tc), so that we can neglect the threelast terms in the right-rand-side of Eq.(23), which yields Tc as given in Eq.(6) of the maintext.9REFERENCES1 Liang Fu and Charles L Kane, “Topological insulators with inversion symmetry,” Physical ReviewB 76, 045302 (2007).2 J Labbe and J Bok, “Superconductivity in alcaline-earth-substituted La2CuO4: a theoreticalmodel,” EPL (Europhysics Letters) 3, 1225 (1987).10

Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi 2

Lange, Gunnar F

Trevisan, Thaís V

Orth, Peter P

Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P1¯" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1¯1¯, which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.</p

Digital Repository @ Iowa State University (ISU)

Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2

Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected 'strong' TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P1, which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy. Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.This article is published as Lee, Kyungchan, Gunnar F. Lange, Lin-Lin Wang, Brinda Kuthanazhi, Thaís V. Trevisan, Na Hyun Jo, Benjamin Schrunk, Peter P. Orth, Robert-Jan Slager, Paul C. Canfield, and Adam Kaminski. "Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi 2." Nature Communications 12, no. 1 (2021): 1855. DOI: 10.1038/s41467-021-22136-w. Posted with permission.</p

Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected \u27strong\u27 TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P1, which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy. Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects

Trevisan, Thais V.

Iowa State University Computer Science Technical Reports

https://www.repository.cam.ac.uk/bitstream/handle/1810/322122/Discovery%20of%20a%20weak%20topological%20insulating%20state%20and%20van%20Hove%20singularity%20in%20triclinic%20RhBisub2sub.pdf?sequence=1&isAllowed=y

Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2.

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Digital Repository @ Iowa State University (ISU)

Iowa State University Computer Science Technical Reports