Analysis of Topological Transitions in Two-dimensional Materials by Compressed Sensing
Carlos Mera Acosta, Runhai Ouyang, Adalberto Fazzio, Matthias Scheffler, Luca M. Ghiringhelli, Christian Carbogno
AAnalysis of Topological Transitions inTwo-dimensional Materials by Compressed Sensing
Carlos Mera Acosta , Runhai Ouyang , Adalberto Fazzio , Matthias Scheffler , LucaM. Ghiringhelli , and Christian Carbogno Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin-Dahlem, Germany Institute of Physics, University of Sao Paulo, CP 66318, 05315-970, Sao Paulo, SP, Brazil Brazilian Nanotechnology National Laboratory, CP 6192, 13083-970, Campinas, SP, Brazil * [email protected] ABSTRACT
Quantum spin-Hall insulators (QSHIs), i.e., two-dimensional topological insulators (TIs) with a symmetry-protected bandinversion, have attracted considerable scientific interest in recent years. In this work, we have computed the topological Z invariant for 220 functionalized honeycomb lattices that are isoelectronic to functionalized graphene. Besides confirming theTI character of well-known materials such as functionalized stanene, our study identifies 45 yet unreported QSHIs. We applieda compressed-sensing approach to identify a physically meaningful descriptor for the Z invariant that only depends on theproperties of the material’s constituent atoms. This enables us to draw a “map of materials”, in which metals, trivial insulators,and QSHI form distinct regions. This analysis yields fundamental insights in the mechanisms driving topological transitions.The transferability of the identified model is explicitly demonstrated for an additional set of honeycomb lattices with differentfunctionalizations that are not part of the original set of 220 graphene-type materials used to identify the descriptor. In thisclass, we predict 74 more novel QSHIs that have not been reported in literature yet. In the last decade, the experimental realization of graphene and of topological insulators has fueled interest in thefundamental physics and possible applications that are hosted in linear band crossings. In fact, the crossing of energy bands incondensed matter has been investigated since the very early applications of formulation of quantum mechanics to periodicsolids . Already in 1985, Volkov and Pankratov showed that interfacing two semiconductors with mutually inverted bands canlead to massless Dirac fermions, i.e., linear electronic-dispersion relations that cross (leading to a phenomenon referred to as“band inversion”) and connect conduction and valence bands. If this inversion occurs at time reversed pairs of reciprocal spacepoints ± (cid:126) k , the respective boundary states associated to different spins must exhibit opposite momentum, which in turn forbidsbackscattering . Quantum spin-Hall insulators (QSHI) are two-dimensional topological insulators (TIs) that intrinsicallyexhibit this property . In graphene, for instance, this band inversion is driven by spin-orbit coupling (SOC), which formallyleads to a minute band-gap opening . In close analogy to charge pumping in the integer quantum-Hall effect , the spincharge pumped through the edge states is quantized in QSHIs . The respective integer quantum, i.e., the topological Z invariant, is 1 in QSHIs and 0 in trivial insulators. Formally, this Z invariant is defined via the integral over the half Brillouin a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y one (HBZ): Z = π (cid:20) (cid:73) ∂ HBZ
AAA ( kkk ) dl − (cid:90) HBZ
FFF ( kkk ) d τ (cid:21) mod ( ) (1)of the Berry connection AAA ( kkk ) and Berry curvature FFF ( kkk ) . This can be interpreted as the effective magnetic flux of aself-induced magnetic field, the Berry curvature, through the HBZ. Here, ∂ HBZ is the contour of HBZ. Although theorypredicted a wide variety of QSHIs , only few of them feature a large enough intrinsic bulk band gap at finite temperatures toallow for an experimental characterization, e.g., bilayer Bi as well as HgTe/CdTe and InAs/GaSb quantum wells .So far, the computational search for new QSHIs has been a numerically costly trial-and-error process that required tocompute the Z invariant for each individual compound . Since no simple rule of thumb exists that allows to a priori distinguishtrivial from topological insulators , an “exhaustive search procedure seems out of reach at the present time” . Naturally, thesearch for new QSHIs was thus guided by experience and intuition, e.g., by focusing on heavy elements with high SOC . Forinstance, 17 potential TIs could be identified by carrying out high-throughput electronic band structure calculations for 60,000materials . In the same spirit, high-throughput studies in this field have been performed using semi-empirical descriptors as aguidance, e.g., the derivative of the band gap with no SOC with respect to the lattice constant .In this work, we focus on functionalized 2D honeycomb-lattice materials , a material class in which various QSHIcandidates have been found already, e.g., functionalized stanene . By computing the Z -invariant for 220 of these compoundsfrom first principles we are able to identify 45 new QSHIs that have not yet been reported in literature. Using a recentlydeveloped statistical-learning approach based on compressed sensing, we then derive a two-dimensional “map” of thesematerials, in which metals, trivial insulators, and QSHIs are separated in different domains. The axes of this map are given bynonlinear analytic functions that only depend on the properties of the material’s constituent atoms, but not on the properties ofthe material itself. The identified descriptor captures the character of the electronic structure, revealing that orbital interactionscan drive a band inversion even in compounds featuring relatively light elements and low SOC. Furthermore, we are also able topredict the topological character of materials without performing any additional first-principles calculations, just by evaluatingtheir position on the “map”. By this means, we predict 74 additional novel QSHI candidates in a distinct material class, i.e., aset of different honeycomb lattice compounds with different functionalizations. Results
First-principles Classification of Functionalized 2D Honeycomb-Lattice Materials
In a first step, we investigate the topological character of 220 functionalized 2D honeycomb-lattice materials
ABX (seeFig. 1) by computing their Z invariant from first principles. We include all possible combinations AB that are isoelectronicwith graphene (group IV-IV, III-V, and II-VI). For each honeycomb-lattice AB , functionalization with four different groupVII elements ( X either Cl, Br, F, or I) is considered. The resulting first-principles classification of these 220 compounds inmetals (zero band gap), trivial insulators (nonzero band gap and Z = Z =
1) is shownin Fig. 1. 103 compounds are identified to be QSHIs in our calculations: In most cases (66%), the TI character is independentof the actual functionalization, as highlighted in blue in Fig. 1. These 68 functionalization-independent (FI) QSHIs consist ofrelatively heavy elements, feature topological band gaps between 5 meV and 2 eV, and include 15 new QSHIs and 53 QSHIsreported in literature before, e.g., functionalized stanene, germanene, Bi , GaBi, InBi, TlBi . Additionally, we also igure 1. Side and top view of the functionalized honeycomb-lattice system. Classification (trivial insulators: white; QSHIs:red; metals: cyan) of the 220 investigated
ABX compounds. The x and y axes denote the A and B atoms. For eachcombination AB , the four individual squares correspond to a different functionalization with a group VII element (see legend).Compounds for which the topological character is independent of X are surrounded by a blue line. See Supp. Mat. for atabulated list.identify 35 QSHIs (34% of all QSHIs), for which the TI character depends on the actual functionalization (mostly iodides).These functionalization-dependent (FD) QSHIs with topological band gaps between 5 meV and 1 eV include 30 compoundsthat have not yet been reported in literature, e.g., AlNBr and GaAsI . Quite surprisingly, these TIs consist of relatively lightelements with low SOC in the honeycomb lattice; they are however functionalized with relatively heavy atoms with strongSOC, which further substantiates that the topological transition is driven by the functionalization. Descriptor Identification via Compressed Sensing
To identify descriptors that can a priori classify (i.e, the descriptor depends only on properties of the — isolated — atomicspecies constituting the material) functionalized 2D honeycomb-lattice materials in metals, trivial insulators, and QSHIs, weemployed the SISSO (sure independence screening and sparsifying operator) approach recently developed by Ouyang et al. .First, a pool of about 10 different potential descriptors D n is constructed by analytically combining the properties of the freeatoms A , B and X computed with SOC (namely, the eigenvalues of the highest-occupied and lowest-unoccupied Kohn-Shamstates (cid:15) ho and (cid:15) lu , the atomic number Z , the electron affinity EA, the ionization potential IP, and the size r s , r p , and r d of the s , p , and d orbitals, i.e., the radii where the radial probability density of the valence s , p , and d orbitals are maximal. See Supp.Mat. for a full list including the values.). Second, this compressed-sensing-based technique identifies which low-dimensionalcombination of these descriptors represents the classification best, i.e., minimizes the overlap (or maximizes the separation)
Representation of DFT result for the training (filled) and test (unfilled symbols) set in the domain defined by thetwo-dimensional descriptors. A logarithmic scale is used for D . Compounds functionalized with F, Cl, Br, I are represented bydiamonds, squares, circles and triangles, respectively. The symbols’ color is used to distinguish between metalls (cyan),FD-QSHIs (red), FI-QSHIs (blue), and trivial insulators (white/grey). The same color-code is used to highlight the differentregions identified by the SISSO descriptors. The boundaries of the map of materials are defined by α ≈ α ≈ . β ≈
70, and β ≈
85. The gap in the data points observed for 865 < D < Z A and Z B whenswitching from the 5 th to the 6 th row of the periodic system. See Supp. Mat. for a tabulated list.among the convex hulls that envelope the individual classes (metals, trivial insulators, QSHIs). This procedure, which isperformed for a “training” set (176 compounds randomly chosen from the total set of 220), reveals that the best descriptor forthe classification of the investigated compounds is two-dimensional and features the components D = ( Z A + Z B ) (cid:15) hoB EA B (2) D = EA X IP X ( r s , A + r p , B ) . (3)For the remaining 20% of compounds, i.e., the so called “test” set used to validate the model, we find that all materials witha very well defined structural and topological character are correctly classified. Only ZnOCl and AlNBr , which are bothFD-QSHIs with minute band gaps ( ≤
15 meV after SOC) at the verge of a topological transition to a trivial insulator or metal,respectively, are not correctly classified. This shows that the found descriptor, which exhibits a predictive ability greater than95%, is robust and transferable, i.e., not limited to the original training set used to identify D and D As shown in Fig. 2, all compounds with D > α are FI-QSHIs (blue). For D < α , the descriptor D matters as well:Trivial insulators (white) occur for values of D larger than the line connecting β and β , while for materials lying below thatline we find metals (cyan) in the region D < α and FD-QSHIs (red) for α < D < α . Note that D and D do not onlyclearly discriminate between metals, insulators, and QSHIs, but also separate FI-QSHIs from FD-QSHIs. n( D )(b) ⇤ sp z
The descriptor of components D and D does not only numerically and graphically sort the functionalized graphene-likematerials, but also captures which atomic properties are relevant “actuators” for the topological phase transition. To understandthis, it is necessary to clarify the character of the electronic states involved in the band inversion at Γ , as schematically sketchedin Fig. 3a: These are the twofold degenerate σ p xy state (blue), to which the two in-plane valence p xy -orbitals from atoms B contribute, and the σ ∗ sp z state (red), to which the valence s states from atoms A and the out-of-plane p z states from atoms X contribute. Although all quantities entering Fig. 2 are computed with SOC, the descriptor ( D , D ) can be qualitativelyrationalized as the relative energetic positions of these two states ∆ sp = E σ ∗ spz − E σ pxy before SOC: If the twofold degenerate σ p xy state lies lower in energy ( ∆ sp > σ p xy state, but not invert the band order. Conversely,the σ ∗ sp z state lies lower in energy for ∆ sp < σ p xy state and so a band-inverted, semi-metal before SOC. The SOC itself is thus solely responsible for theband-gap opening, which leads to the QSHI state.How the individual actuators present in the descriptor ( D , D ) influence the topological transition, can be qualitatively un-derstood from their influence on ∆ sp . As shown in Fig. 3b, ∆ sp correlates linearly with ln ( D ) ( ∆ sp ≈ ( − . ± . ) ln ( D )+( . ± . ) ) for functionalization-independent QSHIs. In this case, the influence of the p z orbital of X on σ ∗ sp z is negligible,so that ∆ sp is dictated by atoms A and B . The role of the actuators in the component D of th descriptor can the be rationalizedusing the tight-binding models developed for tetrahedral group IV and III-V semiconductors AB . Here, (cid:15) hoB / EA B captures igure 4. Representation of proposed V-V-VI and IV-VI-VI compounds in the domain defined by the two-dimensionaldescriptor components. See Supp. Mat. for a full list of the materials and their classification.the energetic position of the hybridized p xy -orbitals from atom B , whereas ( Z A + Z B ) captures the size and thus the overlap ofthe respective orbitals. Just as in the case of tetrahedral semiconductors, heavier atoms lead to larger band widths and thus toreduced band gaps ∆ sp and metallic electronic structures.Conversely, the component D of the descriptor captures the influence of the functionalization. Due to its strong dependenceon the electron affinity and ionization potential of atom X , it groups the compounds by functionalization for D > α , asapparent from Fig. 2. For D < α , however, it describes the actual stable geometry of ABX and with that its electronic state:For light, strongly bound AB compounds such as BN and C (small values of Z A + Z B in D and r s , A + r p , B in D ), only stronglyelectronegative atoms X (large values of EA X IP X ) form a chemical bonding with AB , thus realizing a trivial insulators. Lesselectronegative functionalizations only physisorb via van-der-Waals interactions with AB , thus resulting in a metallic compoundeven after SOC. In this case, we also observe a structural transition from the so called low-buckled (LB) to the high-buckledstructure . The insulator/metal transition with D < β and D < α is thus essentially a LB-HB structural phase transition thatis energetically favorable only in these particular compounds, as we explicitly checked. In particular, this holds for compoundswith D > α that are relatively close to a topological transition from the start. Here, the additional degree of freedom providedby the functionalization allows to tune the σ ∗ sp z state. For strongly electronegative atoms X (large values of EA X IP X ), thisallows to close the gap ∆ sp and thus leads to a topological transition, as discussed in the context of Fig. 3a before.To showcase that the gained insights and identified descriptors are transferable, we have computed D and D for 140 lesscommon honeycomb ABX compounds ( AB from groups V-V and IV-VI functionalized with a group VI element for X ). Forthese compounds, the identified descriptor predicts 20 FI-QSHIs, 54 FD-QSHIs, 42 trivial insulators and 24 metals, as shown inFig. 4 and tabulated in the Supp. Mat.. Since EA x IP X for X belonging to the VI group is lower, we get that the functionalizationstabilizes the HB phase for non-QSHI materials. We have verified the prediction for selected compounds (As X , Sb X , SnSe X ,PSb X , Bi X and PO with X :O,S,Se,Te). It has been shown that oxide blue-phosphorene is a trivial insulator, which can become QSHI by applying tensile strain . Recently, one compound has been reported as an intrinsic QSHI, AsO . Different fromindividual systems proposed by trial-and-error calculations, the employed SISSO approach allows to identify a complete familyof 74 new QSHI candidates. Methods
First-principle calculations
For each of these systems, we have first determined the equilibrium lattice constant by relaxing both the atomic positions and theunit-cell shape until the residual forces on the atoms were smaller than 0.01 eV/ ˚A using the all electron, full potential numericatom centered orbitals based electronic structure code
FHI-aims . For the equilibrium configuration, the topologicalinvariant Z was computed from the evolution of the Wannier center of charge that we implemented in FHI-aimsusing the band structures and wavefunctions. For these latter properties, SOC was accounted for using a second-variational,second-order perturbation approach recently implemented in FHI-aims and first used in Ref. . For a qualitative analysisof the band inversion mechanism, projected band structures were computed. All calculations were performed using thePerdew-Burke-Ernzenhof (PBE) generalized gradient approximation , the Tkatchenko-Scheffler van der Waals correctionmethod (DFT-TS) , and with numerical settings that guarantee a convergence of < xy -plane, and a vacuum of 20 ˚A was used inthe z -direction to avoid the undesirable interaction between the periodic images of sheets. Futhermore, “really tight” numericalsettings and basis sets as well as a 40 × × (cid:126) k point grid for the Brillouin zone were used. Statistical approach for identifying descriptors: Compressed Sensing
To learn a descriptor for the Z -invariant material property, we employed the compressed-sensing approach recently developedby Ouyang et al. , which mainly consists of two steps: i ) construction of feature space (potential descriptors) by buildinganalytical functions of the input parameters (atomic properties with SOC, in the case studied here), by iteratively applyinga set of chosen algebraic operators, up to a certain complexity cutoff (number of applied operators). The used input atomicparameters are the eigenvalues of the highest occupied and lowest unoccupied Kohn-Sham states (cid:15) ho (cid:15) lu , the atomic number Z , the electron affinity EA, the ionization potential IP, and the size of the s , p , and d orbitals ( r s , r p , and r d ), i.e., the radiiwhere the radial probability density of the valence s , p , and d orbitals are maximal, for A , B and X . All these features werecomputed using perturbative SOC . Consequently, the feature space is formed by N vectors DDD n = ( D n , , D n , , ..., D n , M ) , where D n , m is the n th combination of atomic features, e.g., ( (cid:15) hoA + (cid:15) hoB + (cid:15) hoX ), evaluated on the constituent atoms of the m th ABX compound. For more details about the feature space construction please refer to Ref. ; ii ) descriptor identification by ascheme combining sure independence screening and sparsifying operator, SISSO. SIS selects features DDD n , highly correlatedwith the Z topological invariant property, which is formally written as a vector of the training values of Z -invariant. Startingfrom the features selected by SIS, the SO looks for the Ω -tuples of features that minimizes the overlap (or maximize theseparation) , among convex hulls enveloping subsets of data. The dimensionality Ω of the representation is set as the minimalthat yields perfect classification of all data in the “training” set. In this work Ω = cknowledgments We thank the financial support by FAPESP under Grant Agreement No. 2016/04496-9 and the European Union’s Horizon2020 research and innovation program under Grant Agreement No. 676580 with The Novel Materials Discovery (NOMAD)Laboratory, a European Center of Excellence. L.M.G. acknowledges funding from the Berlin Big-Data Center (Grant AgreementNo. 01IS14013E, BBDC). This project has received funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (Grant Agreement No. 740233, TEC1p).
Author contributions statement
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