Topological Phase Diagram of BiTeX--Graphene Hybrid Structures
Zoltán Tajkov, Dávid Visontai, László Oroszlány, János Koltai
aapplied sciences
Article
Topological Phase Diagram of BiTeX–Graphene HybridStructures
Zoltán Tajkov * , Dávid Visontai , László Oroszlány and János Koltai Department of Biological Physics, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A,1117 Budapest, Hungary; [email protected] Department of Material Physics, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A,1117 Budapest, Hungary; [email protected] Department of Physics of Complex Systems, ELTE Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117Budapest, Hungary; [email protected] * Correspondence: [email protected]† MTA-BME Lendület Topology and Correlation Research Group, Budafoki út 8., H-1111 Budapest, Hungary.Received: 26 September 2019 ; Accepted: 2 October 2019 ; Published: date
Abstract:
Combining graphene with other novel layered materials is a possible way for engineeringthe band structure of charge carriers. Strong spin-orbit coupling in BiTeX compounds and the recentfabrication of a single layer of BiTeI points towards a feasible experimental realization of a Kane–Melephase in graphene-based heterostructures. Here, we theoretically demonstrate the tunability of thetopological phase of hybrid systems built from graphene and BiTeX (X = I, Br, Cl) layers by uniaxialin-plane tensile and out-of plane compressive strain. We show that structural stress inherently presentin fabricated samples could induce a topological phase transition, thus turning the sample in a novelexperimental realization of a time reversal invariant topological insulator.
Keywords: graphene; DFT; topological insulators; spin-orbit coupling
1. Introduction
After the first successful isolation of graphene in 2004, graphene has proved to be an excellenttemplate material with outstanding mechanical properties for revolutionary applications [1]. Due tothe small atomic weight of the carbon atoms building up the graphene lattice, the effect of spin-orbitcoupling (SOC) on the electronic band structure is negligible [2]. This makes graphene a semimetal.There has recently been a strong push to find ways to enhance spin-orbit coupling in graphene [3] inorder to enable spintronics applications [4]. Theoretical investigations showed that one can overcomethis limitation in several ways. Introducing curvature in the graphene sheet enhances the effect of thespin-orbit interaction up to ≈
17 meV in realistic situations [5]. It has been also shown that impuritiesinfluence the SOC in a positive way and can amplify its strength up to ≈ Appl. Sci. , xx a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t ppl. Sci. , xx , 5 2 of 7 structural asymmetry combined with a large in-plane gradient of the crystal field in this lattice resultsin giant Rashba spin-splitting semiconductors [7–11]. A single trilayer of BiTeI, that shows the Rashbaspin-split band structure can be exfoliated from the bulk BiTeI as it was recently shown by Fülöp et al.,using a novel exfoliation technique [12]. It was also shown theoretically that centrosymmetric thin filmscomposed from topologically trivial BiTeI trilayers are quantum spin Hall insulators and properly stackedbulk compounds of BiTeX results in topological insulating phase [13,14]. In monolayer BiTeI the bandgap is increased, compared to the bulk material, still the basic characteristics are preserved, hence it is agood candidate as a SOC inducing component in graphene-based heterostructures, as it was previouslyproposed by earlier works [15,16].The fact that fabricated devices are usually subject to mechanical stress requires the investigationof the impact of such effects on the electronic properties of the systems. In experimental setups aconsiderable effort has been made to control and manipulate the electronic and optical properties ofnovel heterostructures by strain fields [17–20].To realize mechanically driven 2D Z topological insulators, several theoretical proposals havebeen made. For example, in two-dimensional dumbbell stanene a topological phase can be tuned bycompression and in hydrogenated ultra-thin tin films a topological gap can be opened by mechanicaldistortions. [21]As we claimed in our previous work, the single sided heterostructures of BiTeX and grapheneunder appropriate conditions are topological insulators, however we showed evidence for only theBiTeBr–graphene structure [22]. In this manuscript we investigate in details the phase diagram of theBiTeCl–graphene and BiTeI–graphene systems in the level of density functional theory calculations, aswell as the BiTeBr–graphene structure. We analyze and discuss the differences of the electronic propertiesof the heterostructures.
2. Results
To set up the investigated systems, the experimental characteristics of the BiTeX crystals were used.The in-plane lattice constants are a BiTeX = 4.34 Å, 4.24 Å, 4.27 Å for X = I, Br, Cl respectively [23]. Inasmuchas these values are close to a √ × a Gr , a 30 ◦ rotated, √ × √ a BiTeX / a √ ×√ = − ppl. Sci. , xx , 5 3 of 7 Figure 2 we show the corresponding phase diagrams. In relevant cases we include the band structures aswell. The sign of the band gap indicates the topological invariant, it is negative if the system is topologicaland positive otherwise.The first subfigure (a) presents the calculations on the BiTeCl–graphene heterostructure. At theoptimal geometry the system is semi-metallic due to the touching altered Dirac cones of graphene as itcan be seen in subfigure (I.). As the tensile strain increases, a topologically non-trivial band gap opensin graphene (subfigure (II.)). Turning on the compressive out-of-plane strain, it widens the band gap inthe beginning, but later closes it (see the corresponding band structure in subfigure (III.)) and changesits topological favor to trivial. In this scenario, a 50 meV topological and a 120 meV trivial gap can bereached. The second subfigure (b) corresponds to the BiTeBr–graphene system. The same statementscan be made with a slightly different energy scale. The behavior of the band gap is similar to the caseof the BiTeCl–graphene structure. At fixed tensile strain, applying pressure widens the topological gapin the beginning up to 70 meV, but later closes the gap and the reopened gap is trivial, which can be ashigh as 140 meV. The third subfigure (c) shows the BiTeI–graphene structure. The effect of in-plane strainis the same as above: it promotes the topological phase. The consequence of the compressive strain issomewhat different. This system turns metallic instead of a trivial insulator due to some non graphenebands reaching the Fermi level as pressure is increased (see subfigure (IV.)).
TeBiXC (a) (b)
Figure 1.
Top ( a ) and side ( b ) view of the BiTeX–graphene (X = I, Br, Cl) heterostructure. The black dashedline indicates the unit cell. (color online). The deviating behavior of the BiTeI–graphene structure can be explained by the different workfunction of the BiTeX materials. In the case of the Te terminated BiTeBr and BiTeCl, the work function isaround 4.7–4.5 eV as it was calculcated by Fiedler et al. [24]. This is close to the theoretically calculatedvalue for graphene (4.6 eV Yu et al. [25]). The work function of Te terminated BiTeI is larger by almost0.5 eV. The altered Dirac cones of graphene can be distinguished in the band structure of the hybridsystems because the cones are placed in the band gap of the BiTeX systems due to the similar workfunctions. As we stated in our previous work, the hybrid structure consisting graphene and X terminatedBiTeX does not provide this nature [22], which is in agreement with the significantly larger work functionof X terminated BiTeCl and BiTeBr (6.2–6.0 eV [24]). In the case of the BiTeI–graphene structure the alteredDirac cones are placed closer to the occupied bands of BiTeI due to the larger value of the work function.As we apply pressure those bands populate the band gap at a higher rate as in the case of BiTeBr or BiTeCl,resulting in a metallic phase.Based on the presented results we conclude that both type of mechanical distortions have a strikingeffect on the band gap, however they favor different topological phases. Out-of-plane strain opens a trivialband gap, while in-plane strain drives the system into the topological phase.We would like to point out that the largest compressive strain we applied during our calculationswould correspond to a nominal pressure of 20 GPa. We estimated this value as the derivation of the total ppl. Sci. , xx , 5 4 of 7 energy per unit area over the reduced distance. In modern experimental setups these mechanical stressescan be routinely achieved [26–30]. -500 E - E F [ m e V ] K MK M K M K M -20 -15 -10 -5 0
Out-of-plane compressive strain [%] I n - p l a n e t e n s il e s t r a i n [ % ] -20 -15 -10 -5 0 Out-of-plane compressive strain [%] -80-60-40-20020406080100120140 -20 -15 -10 -5 0
Out-of-plane compressive strain [%] E n e r g y g a p [ m e V ] (c)(IV.) (I.) (II.) (III.) (IV.)(b)(a)(II.)(III.) (I.) Figure 2.
The calculated in-plane tensile–out-of-plane compressive strain phase diagram of theBiTeCl/BiTeBr/BiTeI–graphene structures ( a – c ) respectively. The blue colors indicate that the gap istopological, while the red hue means that is trivial. The metallic shade marks the phase boundary.Subfigures (I.)–(IV.) are the corresponding band structures of the marked phases near the Γ point inmomentum space. (color online).
3. Methods
The optimized geometry and ground state Hamiltonian and overlap matrix elements of each structurewere self consistently obtained by the SIESTA implementation of density functional theory (DFT) [31,32].For every calculation, the spin-orbit interaction was included as it was implemented in SIESTA by Dr.Ramón Cuadrado based on the original on-site SO formalism and implementation developed by Prof.Jaime Ferrer [33]. SIESTA employs norm-conserving pseudopotentials to account for the core electronsand linear combination of atomic orbitals to construct the valence states. We used the pseudopotentialsoptimized by Rivero et al. [34]. For all cases, the considered samples were separated with a minimumof 18.5 Å thick vacuum in the perpendicular direction. The generalized gradient approximation of theexchange and the correlation functional was used with Perdew–Burke–Ernzerhof parametrization [35],as the pseudopotentials were created to the PBE functionals, with a double- ζ polarized basis set and areal-space grid defined with an equivalent energy cutoff of 1000 Ry. The Brillouin zone integration wassampled by a 24 × × k -grid [36]. The geometry optimizations were performed untilthe forces were smaller than 0.01 eV/Å.The choice of pseudopotentials optimized by Rivero et al. ensures that both the obtained geometricalstructures and the electronic band properties are reliable. As a benchmark we validated our method bycomparing the electronic properties of the bulk BiTeI with the experimental data. This approach gave us ppl. Sci. , xx , 5 5 of 7
130 meV band gap and 4.6 eVÅ as Rashba parameter for BiTeI bulk. The corresponding experimentalresults are 190 meV and 3.8 eVÅ respectively [37].We extracted the topological phase information of the systems by determining the Wannier centerflow. The center of the Wannier function can be expressed as the phase of eigenvalues of a matrix obtainedas the product of the Berry connection along the “Wilson loop” by our own post-processing tool. The usedreal space Hamiltonian was calculated by the SIESTA self-consistent cycle. The Z topological numberswere expressed as the number of times mod 2 of the partner switching of these phases during a completeperiod of the “time reversal pumping” process [38–40].
4. Conclusions
In summary, we have explored the intriguing topological phase diagrams of bismuthtellurohalide/graphene heterostructures by means of first principles calculations. We showed that thein-plane uniaxial tensile strain opens a topologically non-trivial band gap in graphene in the vicinity of amaterial features strong inherent spin-orbit coupling. On the other hand, the compressive stress on thedevice promotes a trivial band gap. The BiTeBr–graphene and BiTeCl–graphene systems can be tuned fromtopological to trivial state and vica versa by applying mechanical stress. Furthermore, the BiTeI–graphenestructure can be tuned from topological state to metallic. These systems lead to a novel realization ofthe time reversal invariant topological insulating phase, thus making these heterostructures potentialcandidates for quantum technology applications without the necessity of low temperature.
Author Contributions:
Conceptualization, J.K. and L.O.; methodology, Z.T.; software, Z.T. and D.V.; validation, D.V.;writing–original draft preparation, Z.T.; writing–review and editing, J.K., L.O. and Z.T.; supervision, J.K. and L.O.
Funding:
This research was supported by the National Research, Development and Innovation Fund of Hungarywithin the Quantum Technology National Excellence Program (Project Nr. 2017-1.2.1-NKP-2017-00001); grantsno. K112918, K115608, FK124723 and K115575. This work was completed in the ELTE Excellence Program(1783-3/2018/FEKUTSRAT) supported by the Hungarian Ministry of Human Capacities. JK acknowledge theBolyai and Bolyai+ program of the Hungarian Academy of Sciences.
Acknowledgments:
We acknowledge [NIIF] for awarding us access to resource based in Hungary at Debrecen.
Conflicts of Interest:
The authors declare no conflict of interest.
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