First principles feasibility assessment of a topological insulator at the InAs/GaSb interface
Shuyang Yang, Derek Dardzinski, Andrea Hwang, Dmitry I. Pikulin, Georg W. Winkler, Noa Marom
FFirst principles feasibility assessment of a topological insulator at the InAs/GaSbinterface
Shuyang Yang, Derek Dardzinski, Andrea Hwang, Dmitry I. Pikulin,
2, 3
Georg W. Winkler, and Noa Marom
1, 4, 5, ∗ Department of Materials Science and Engineering,Carnegie Mellon University, Pittsburgh, PA 15213, USA Microsoft Quantum, Redmond, WA 98052, USA Microsoft Quantum, Microsoft Station Q, University of California, Santa Barbara, California 93106-6105, USA Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA (Dated: January 21, 2021)First principles simulations are conducted to shed light on the question of whether a two-dimensional topological insulator (2DTI) phase may be obtained at the interface between InAsand GaSb. To this end, the InAs/GaSb interface is compared and contrasted with the HgTe/CdTeinterface. Density functional theory (DFT) simulations of these interfaces are performed using amachine-learned Hubbard U correction [npj Comput. Mater. 6, 180 (2020)]. For the HgTe/CdTeinterface our simulations show that band crossing is achieved and an inverted gap is obtained at acritical thickness of 5.1 nm of HgTe, in agreement with experiment and previous DFT calculations.In contrast, for InAs/GaSb the gap narrows with increasing thickness of InAs; however the gapdoes not close for interfaces with up to 50 layers (about 15 nm) of each material. When an externalelectric field is applied across the InAs/GaSb interface, the GaSb-derived valence band maximumis shifted up in energy with respect to the InAs-derived conduction band minimum until eventuallythe bands cross and an inverted gap opens. Our results show that it may be possible to reach thetopological regime at the InAs/GaSb interface under the right conditions. However, it may be chal-lenging to realize these conditions experimentally, which explains the difficulty of experimentallydemonstrating an inverted gap in InAs/GaSb.
I. INTRODUCTION
Two-dimensional topological insulators (2DTIs) haveattracted increasing attention in recent years owing tothe emergence of helical edge states and backscattering-free edge currents relevant for applications in spintronicsand quantum computing [1–3]. 2DTIs were first pro-posed based on a theoretical model of graphene incorpo-rating spin-orbit interactions [4]. However, the requiredtype of spin-orbit coupling in graphene is too weak to ob-serve the quantum spin Hall effect (QSHE) experimen-tally [5]. Later, a proposal for 2DTI was made basedon a HgTe/CdTe quantum well (QW) [6] and the signa-tures of the QSHE were experimentally demonstrated [7].When the thickness of the HgTe in the QWs is varied,the band structure changes from a trivial insulator to a2DTI with an inverted gap when a critical thickness isreached [6–9]. 2DTIs have been proposed in additionalmaterials systems, some of which have shown promisingsigns [10–22].In the present work we focus on another QW struc-ture, InAs/GaSb. It has been proposed that a 2DTImay be realized in InAs/GaSb QWs because the bandlineup of coupled InAs/GaSb QWs could lead to the co-existence of electrons and holes at the charge neutralitypoint [23, 24]. The topological insulator phase wouldarise if the band ordering were inverted and coupling be-tween electron and hole states opened a hybridization ∗ [email protected] gap which is necessarily topological due to the orbitalstructure of the hybridized bands [25]. Such band order-ing could potentially be achieved by choosing appropri-ate QW thickness and by applying an external electricfield [5]. InAs/GaSb QWs are in the family of well-studied III-V compounds and have thus attracted con-siderable experimental interest [26–34]. The experimentshave provided some encouraging signs of edge conduc-tance in the material. However, a phase diagram showinga clear topological transition accompanied by edge stateformation has yet to be demonstrated. Here, we use firstprinciples simulations to investigate whether it would bepossible to realize a 2DTI at the InAs/GaSb interfaceand under what conditions.The HgTe/CdTe and InAs/GaSb interfaces have beenstudied theoretically using a variety of methods. This in-cludes the k · p method [9, 35–39], pseudopotential mod-els [40, 41], and tight-binding [42, 43]. The drawback ofthese semi-empirical methods is that the fitting to exper-imental data largely determines the extent of their pre-dictive capability. Atomistic ab initio simulations mayprovide a more accurate representation of the electronicproperties and their dependence on the structure of theinterface. First principles studies based on density func-tional theory (DFT) have investigated the influence ofthickness on the edge states of HgTe/CdTe(100) [44, 45].Using different exchange-correlation functionals and dif-ferent thicknesses of CdTe, Ref. [44] predicted a criticalthickness of 4.6 nm of HgTe, whilst Ref. [45] predicted acritical thickness 6.5 nm of HgTe. Both results are closeto the experimental critical thickness of 6.3 nm [6]. a r X i v : . [ phy s i c s . c o m p - ph ] J a n For InAs and GaSb, local and semi-local exchange-correlation functionals severely underestimate the bandgaps to the point that they reduce to zero [46], due tothe self-interaction error (SIE). Some DFT studies ofInAs/GaSb have applied an empirical correction to theDFT band gaps [47, 48]. Others have used hybrid func-tionals, which mitigate the effect of SIE by including afraction of exact exchange [49]. An alternative approach,which has been used to obtain more accurate band gapsfor InAs/GaSb is many-body perturbation theory withinthe GW approximation, where G stands for the one-particle Green’s function and W stand for the screenedCoulomb interaction [50]. Although hybrid DFT func-tionals and the GW approximation produce significantlyimproved band gaps, their high computational cost limitstheir applicability to relatively small system sizes. There-fore, these methods have been used only for periodic het-erostructures of InAs/GaSb with very few layers [49, 50].DFT studies of large interface slab models with vacuumregions have not been conducted. All previous ab ini-tio studies of InAs/GaSb have not reported the bandstructure and band alignment at the interface and havenot shown an inverted band gap. Furthermore, previousstudies have not considered the effect of applying an elec-tric field, which plays an important role in experiments,and therefore should be considered computationally.Recently, we have introduced a new method of DFTwith a machine-learned Hubbard U correction, which canprovide a solution for accurate and efficient simulationsof InAs and GaSb [51]. Within the Dudarev formulationof DFT+U [52] the effective Hubbard U is defined as U eff = U − J , where U represents the on-site Coulombrepulsion, and J represents the exchange interaction. Fora given material, the U eff parameters of each element aremachine-learned using Bayesian optimization (BO). TheBO algorithm finds the optimal U eff values that max-imize an objective function formulated to reproduce asclosely as possible the band gap and the qualitative fea-tures of the band structure obtained with a hybrid func-tional. The DFT+U(BO) method allows for negative U eff values. Negative U eff values are theoretically per-missible when the exchange term, J , is larger than theon-site Coulomb repulsion, U [53–57]. We have foundthat negative U eff values are necessary to produce bandgaps for narrow-gap semiconductors, such as InAs andGaSb. Because the reference hybrid functional calcula-tion is performed only once for the bulk material to de-termine the optimal U eff values, the computational costof DFT+U(BO) calculations for interfaces is comparableto semi-local DFT.In this work, we use the DFT+U(BO) method tostudy the HgTe/CdTe and InAs/GaSb interfaces. Forthe HgTe/CdTe interface, we obtain band crossing at acritical thickness of 5.1 nm of HgTe, and subsequently aninverted gap is observed. Our results are in agreementwith experiment and previous DFT studies, thus vali-dating the DFT+U(BO) method. For the InAs/GaSbinterface, we find that increasing the thickness of InAs leads to gap narrowing. However, band crossing is notobtained up to the largest number of layers calculatedhere. When an external electric field is applied acrossthe InAs/GaSb interface, the GaSb-derived valence bandmaximum is shifted up in energy compared to the InAs-derived conduction band minimum. Band crossing isachieved at a critical field, followed by an inverted gapwhich widens and shifts higher above the Fermi level asthe field is increased. Our results indicate that it may bepossible to reach the topological regime in InAs/GaSbQWs. However, doing so would require a combination ofcareful interface engineering, a considerable electric fieldacross the interface, and gating to tune the position of theFermi level. This explains the difficulty of experimentallydemonstrating an inverted gap in InAs/GaSb. II. METHODSA. Computational details
DFT calculations were performed using the Vienna abinitio simulation package (VASP) [58] with the projectoraugmented wave method (PAW) [59, 60]. The general-ized gradient approximation (GGA) of Perdew, Burke,and Ernzerhof (PBE) [61, 62] was used with a HubbardU correction [52] determined by Bayesian optimization[51], as detailed below. Spin-orbit coupling (SOC) [63]was included throughout and the energy cutoff was setto 400 eV. For bulk band structure calculations a 8 × × × × B. Performance of PBE+U(BO)
FIG. 1. Performance of different DFT functionals for CdTeand HgTe: PBE band structures of (a) CdTe and (b) HgTe;HSE band structures of (c) CdTe and (d) HgTe; PBE+U(BO)band structures of (e) CdTe and (f) HgTe; The contributionsof the Cd/Hg s , Cd/Hg d , and Te p states are indicated bythe red, green, and yellow dots, respectively. A is the pointalong X − Γ in the bulk’s Brillouin zone with the coordinates(0.1, 0.1, 0). A − Γ − A is mapped to A − Γ − A in the (001)direction. The PBE functional fails to provide an adequate de-scription of the band structures of the materials studiedhere. The cases of InAs and GaSb have been discussedin detail in [51]. For CdTe, Fig. 1a shows that PBEseverely underestimates the bad gap compared to the ex-perimental value of 1.60 eV [66]. This is because the Cd4 d states, which contribute significantly to the top of thevalence band, are pushed up in energy due to the SIE[67]. For HgTe, Fig. 1b shows that PBE produces an in-correct band shape and band ordering at the Γ point withthe Hg s orbitals and Te p orbitals inverted around 1 eVbelow the Fermi level [45]. These issues are rectified by the Heyd-Scuseria-Ernzerhof (HSE) [68, 69] hybrid func-tional, as shown in Fig. 1c for CdTe and Fig. 1d forHgTe. However, the computational cost of HSE is toohigh for simulations of large interface models.To achieve a balance between accuracy and efficiency,a Hubbard U correction was applied to the p orbitalsof In, As, Ga and Sb and the d orbitals of Hg and Cdwithin the Dudarev approach [52]. For each orbital, theoptimal value of U eff was machine learned by Bayesianoptimization [51]. The objective function was formulatedto reproduce as closely as possible the band structureproduced by HSE: f ( (cid:126)U ) = − α (E HSEg − E PBE+Ug ) − α (∆Band) (1)Here, (cid:126)U = [ U , U ,..., U n ] is the vector of U eff valuesapplied to different atomic species and U i ∈ [ − , (cid:118)(cid:117)(cid:117)(cid:116) N E N k (cid:88) i =1 N b (cid:88) j =1 ( (cid:15) jHSE [ k i ] − (cid:15) jP BE + U [ k i ]) (2) N E represents the total number of eigenvalues, (cid:15) , in-cluded in the comparison, N k is the number of k -points,and N b is the number of bands selected for compari-son. To avoid double counting the band gap differencein the calculation of ∆Band, the valence band maxi-mum (VBM) and conduction band minimum (CBM) areshifted to zero for both the PBE+U and HSE band struc-tures. Hence, ∆Band captures differences in the qualita-tive features of the band structures produced by PBE+Uvs. HSE, independently of the difference in the band gap.The coefficients α and α may be used to assign differentweights to the band gap vs. the band structure. The de-fault values are 0.25 and 0.75, respectively. For CdTe, weset α = α = 0 . α = 0 and α = 1.For InAs and GaSb the optimal values of U eff havebeen found to be: U In,peff = -0.5 eV, U
As,peff = -7.5 eV,U
Ga,peff = 0.8 eV, U
Sb,peff = -6.9 eV, as reported in [51].With these parameters, DFT+U(BO) yields a band gapof 0.31 eV for InAs, in good agreement with the experi-mental value of 0.41 eV [70], and a band gap of 0.45 eVfor GaSb, which is somewhat underestimated comparedto the experimental value of 0.81 eV [70]. For CdTe, BOproduces an optimal value of U
Cd,deff = 8.3 eV, somewhathigher than the value of 7 eV used in [67]. This resultsin a band gap of 0.87 eV, which is closer to experimentthan previous ab initio calculations [71, 72]. The qual-itative features of the PBE+U(BO) band structure arein agreement with HSE, as shown in Fig. 1e, howeverthe gap and the band width are still somewhat under-estimated. For HgTe, BO produces a value of U
Hg,deff =8.4 eV, somewhat lower than the value of 9.4 eV used inRef. [45]. The band structure, shown in Fig. 1d, hasthe correct band shape, comparable to the HSE bandstructure, and is in agreement with Ref. [45]. To demon-strate the transferability of the U e f f values obtainedby BO from bulk materials to interfaces, we comparethe band structures produced by PBE+(BO) and HSEfor an InAs/GaSb interface with 5 layers of InAs and5 layers of GaSb, constructed as detailed below. Fig 2shows that overall good agreement is obtained betweenPBE+U(BO) and HSE, however PBE+U(BO) somewhatunderestimates the band gap and the band width. Wenote that the U eff values obtained here are based onthe implementation of the Dudarev formalism in VASP.Different DFT+U implementations may yield differentresults [73, 74]. FIG. 2. The band structure of an InAs/GaSb interface with5 layers of InAs and 5 layers of GaSb obtained with (a) HSEand (b) DFT+U(BO). Orange and green dots indicate thecontributions of InAs and GaSb, respectively.
C. Interface model construction
For the HgTe/CdTe(100) interface, we constructed pe-riodic heterostructures, similar to Ref. [45]. However,we used a larger number of CdTe layers to ensure con-vergence, as detailed below. The thickness of HgTe wasvaried to study the evolution of the electronic structure.The experimental lattice constants of 6.45 (cid:6)
A for HgTeand 6.48 (cid:6)
A for CdTe are closely matched [75]. We as-sumed that an epitaxially matched HgTe film would growon top of a CdTe substrate with the experimental latticeconstant of 6.48 (cid:6)
A.For the InAs/GaSb interface, we constructed two typesof interface slab models: The InSb-type interface has Inand Sb as the terminal atoms at the surfaces and inter-face. The GaAs-type interface has Ga and As as theterminal atoms. The experimental lattice constants of6.058 (cid:6)
A for InAs and 6.096 (cid:6)
A for GaSb [70] are closelymatched. We assumed that an epitaxially matched InAsfilm would grow on top of GaSb with the lattice constantof 6.096 (cid:6)
A, based on the experiment in Ref. [34]. Tostudy the effect of the InAs and GaSb thickness, inter-face models were constructed with the number of layersof each material varying from 10 to 50. The notation
FIG. 3. The band gap obtained with PBE+U(BO) as afunction of the number of layers for InAs(100), GaSb(100),and CdTe(100) surface slabs. ”A/B” is used to describe an InAs/GaSb interface withA layers of InAs and B layers of GaSb. A vacuum regionof about 40 (cid:6)
A was added to the interface model to pre-vent spurious interactions between periodic images (forthe purpose of band unfolding the closest integer num-ber of primitive cells to 40 (cid:6)
A was used [65]). In orderto terminate dangling bonds, In and Ga atoms on thesurface were passivated by pseudo hydrogen atoms with1.25 fractional electrons, whereas As and Sb atoms onthe surface were passivated by pseudo hydrogen atomswith 0.75 fractional electrons. Structural relaxation wasperformed for the surface atoms and passivating pseudo-hydrogen atoms until the change of the all forces wasbelow 10 − eV/ (cid:6) A.The number of layers included in slab models needs tobe converged to the bulk limit to avoid quantum size ef-fects. For semiconductors the band gap is typically usedas a the convergence criterion [65, 76]. Fig. 3 showsthe band gap as a function of the number of layers forInAs(100), GaSb(100), and CdTe(100). We note thathere ”layer” is defined as one atomic layer. In each iter-ation, the number of layers was increased by 8 for InAsand GaSb and by 6 for CdTe. If the band gap differencebetween the current iteration and the previous iterationwas within 1 × − eV, the current number of layers wasregarded as converged. For InAs and GaSb surfaces, 50layers are required, whereas for CdTe 40 layers are re-quired to converge the band gap. The converged bandgap values are close to the bulk values. The size of theinterface models used to simulate the effect of an electricfield was limited to 10 layers of InAs with 10 layers ofGaSb due to convergence issues, as detailed in the SI. III. RESULTS AND DISCUSSIONA. HgTe/CdTe
To validate the DFT+U(BO) method, we begin by ap-plying it to the well-studied HgTe/CdTe interface. Bulk-unfolded band structures of HgTe/CdTe heterostructureswith 40 layers of CdTe and a varying number of HgTelayers are shown in Fig. 4. The red dots indicate thecontributions from Hg s orbitals and the blue dots indi-cate the contributions from Te p orbitals. The band gapvalue as a function of the number of HgTe layers is shownin Fig. 5. Negative values indicate an inverted band gap.A drastic change is observed with the thickness of HgTe.When the number of layers is below 16, the interface be-haves as a trivial insulator, with the Hg s orbitals formingthe bottom of the conduction band and the Te p orbitalsforming the top of the valence band. When the number ofHgTe layers reaches 16, a transition point from a trivialinsulator to a topological insulator occurs. At this tran-sition point, both the CBM and VBM show a hybridized sp character. When the number of HgTe layers exceeds16, an inverted gap opens, leading to the occurrence of atopologically nontrivial phase, in which the VBM is dom-inated by Hg s states and the CBM is dominated by Te p states. The critical thickness of 16 layers, corresponds to5.1 nm in good agreement with the experimental resultof 6.3 nm (around 19 layers) [6]. Our result is compa-rable to previous DFT calculations, which used differentfunctionals and considered structures with fewer layersof CdTe. Ref. [44] obtained a critical thickness of 4.6nm of HgTe on top of 4 layers of CdTe using the modi-fied Becke-Johnson (MBJ) functional.Ref. [45] obtaineda critical thickness of 6.5 nm of HgTe on top of 10 layers ofCdTe using GGA+U for HgTe and GGA for CdTe. Thus,the DFT+U(BO) method successfully describes the elec-tronic structure of the HgTe/CdTe interface and capturesthe transition from trivial to topological behavior. B. InAs/GaSb
1. Effect of layer thickness
FIG. 6. Band gap values as a function of number of layers for50-layer InAs/X-layer GaSb and X-layer InAs/50-layer GaSbof InSb-type and GaAs-type interface.
To investigate the influence of the thickness of InAsand GaSb on the band gap, we conducted two series ofcalculations for InSb-type and GaAs-type interfaces. Inone series, the thickness of InAs was fixed at 50 layersInAs and the number of GaSb layers (X) was varied. Inthe other series, the thickness of GaSb was fixed at 50 lay-ers and the number of InAs layers (X) was varied. Theresults are shown in Fig. 6. For the InAs(50)/GaSb(X)series, the band gap of the InSb-type interface increaseswith increasing GaSb thickness, whereas the band gapof the GaAs-type interface does not change significantly.For the InAs(X)/GaSb(50) series, the band gap decreaseswith increasing InAs thickness for both interface types,although the gap of the InSb-type interface remainssmaller than that of the GaAs-type interface through-out. The trend of the gap decreasing with the increase inInAs thickness is in agreement with experimental obser-vations [77]. The thickest interface we were able to calcu-late comprises 50 layers, which corresponds to about 15nm of each material. The bulk-unfolded band structureof a 50/50 InSb-type interface is shown in the SI. Be-cause this interface still has a gap of over 0.2 eV, and therate of the gap narrowing decreases with increasing InAsthickness, as shown in Fig. 6, we estimate that it wouldeither require a significantly thicker film of InAs for thegap to completely close or the gap would approach a fi-nite asymptotic limit rather than close. In addition toincreasing the QW thickness, strain engineering, whichis not taken into account here, may also help modulatethe gap. [78–80].We note that an analysis based on the empirical 8-band Kane model found band inversion and the quan-
FIG. 4. Band structures of a HgTe/CdTe interface with 40 layers of CdTe and (a) 4 layers, (b) 16 layers, and (c) 20 layers ofHgTe. The red dots indicate the contributions of Hg s states and the blue dots indicate the contributions of the Te p states.FIG. 5. The band gap of a HgTe/CdTe interface with 40layers of CdTe as a function of the number of HgTe layers.Negative values indicate an inverted band gap. tum spin Hall phase for an InAs thickness above 9 nm atfixed 10 nm GaSb thickness [25]. However, this analysiswas based on empirical parameters for the material andinterface properties and did not take the atomic detailsof the interface structure into account. For example, inRef. [25] the band alignment at the interface was cho-sen such that the GaSb valence band is 150 meV higherthan the InAs conduction band leading to a band inver-sion even for relatively thin layers. In contrast, withinour first principles approach, we find that band inver-sion is not achieved up to an InAs thickness of 15 nm fora range of GaSb thicknesses including 10 nm. Further-more, we found that the atomic details of the interface,like the type of bonds formed at the interface (InSb orGaAs), are relevant, which was neglected in the effectivetheory of Ref. [25]. Finally, it should be noted that exper-iments seem to indicate that an electric field is requiredto achieve an inverted regime in InAs/GaSb heterostruc-tures [81].
2. Effect of electric field
The band alignment at the interface of InAs/GaSbcan be manipulated by applying external gate voltages.Ref.[81] has presented strong experimental evidence thatthe gap closes when the external gate voltages reach acritical value. Therefore, we performed DFT simulationsfor interface slabs in presence of electric field. In theVASP code, an external electric field is simulated byadding an artificial dipole sheet in the vacuum regionof the unit cell [64]. Due to screening effects and theelectric susceptibility inside the materials, the effectiveelectric field at the interface may be significantly smallerthan the input electric field [64, 82]. To estimate theeffective electric field, we calculated the gradient of thepotential in the InAs and in GaSb, based on the elec-trostatic potential averaged over xy plane. The averagedgradient is taken as the effective electric field. The fullaccount of the effective field estimation is provided inthe SI. The electric field is applied perpendicular to theplane of the interface and points from the GaSb side tothe InAs side. We note that in VASP only an externalelectric field can be set, whereas in experiments the po-sition of the Fermi level can be independently controlledby applying front-gate and back-gate voltages. Owingto convergence issues in DFT calculations with externalelectric fields (see SI), the largest interfaces we were ableto calculate comprise 10 layers of InAs and 10 layers ofGaSb.Fig. 7 shows the band structure of a 10/10 InSb-typeinterface. When no electric field is applied, the inter-face is in the trivial insulator state. The CBM is domi-nated by the interface InAs layer (orange), whereas theinterface GaSb layer (green) contributes predominantlyto the VBM. As the electric field increases, the bandscontributed by the GaSb shift upwards with respect tothe bands contributed by the InAs and the gap narrows.When the input electric field reaches 0.25 V/ (cid:6)
A, whichcorresponds to an effective field of 0.014 V/ (cid:6)
A, the GaSbVBM overlaps with the InAs CBM, the gap closes, andband crossing occurs. Our results are qualitatively inagreement with previous studies [25, 81, 83], which indi-cated that the band gap in InAs/GaSb could be closed
FIG. 7. Electronic structure of a 10/10 InSb-type interface with different external electric fields. a-c) Bulk unfolded bandstructures with the contributions of the interface layers of InAs and GaSb colored in orange and green, respectively. via an external electric field. When the electric field isincreased further, an inverted gap opens. As the elec-tric field is increased, the inverted gap expands, but alsoshifts higher above the Fermi level. Fig 8 shows the po-sition of the inverted gap above the Fermi level at theΓ point as a function of the electric field. With an in-put electric field of 0.35 V/ (cid:6)
A, which corresponds to aneffective field of 0.018 V/ (cid:6)
A, the gap at the Γ point is65 meV and the bottom of the inverted gap is found 66meV above the Fermi level. For the GaAs-type inter-face, shown in the SI, the band gap also decreases as theelectric field increases. However, because the GaAs-typeinterface has a larger band gap and the effect of the elec-tric field is weaker than for the InSb-type interface, thegap does not close even for an input electric field as highas 0.55 V/ (cid:6)
A.Fig 9 shows the change in the band gap, ∆, as a func-tion of the input electric field, E in , for 10/10 GaAs-typeand InSb-type interfaces:∆ = Gap ( E in ) − Gap ( E in = 0) (3)The blue and orange dashed lines indicate the band gapsof the 10/10 GaAs-type and InSb-type interfaces, respec-tively. The gap closes when the dashed line is crossed. Toestimate the input electric field that would be requiredfor the gap to close for a 50/50 interface, we assume thatthe change in the gap would behave similarly to a 10/10interface. The green and red dashed lines indicate theband gaps of the 50/50 GaAs-type and InSb-type inter-faces, respectively. Based on this, we estimate that aninput electric field of 0.19 V/ (cid:6) A, which corresponds to aneffective electric field of 0.012 V/ (cid:6)
A, would be needed toclose the gap for a 50/50 InSb-type interface, as indi-cated by the red solid line. For the GaAs-type interfacean input electric field of 0.55 V/ (cid:6)
A, which corresponds toan effective electric field of 0.017 V/ (cid:6)
A, would be neededto close the gap, as indicated by the green solid line.We highlight that the effective electric field of 0.017 V/ (cid:6)
Acorresponds to a potential drop of 2 .
55 V over the 15nm thickness of the QW in this case, which is likely tomake the material conducting well before the topologicaltransition. Our results indicate that while it may be pos-
FIG. 8. The inverted band gap at Γ and its position abovethe Fermi level as a function of the input electric field, wherethe shift is defined as the energy difference between the po-sition of the bottom of the inverted band gap and the Fermilevel at the Γ point. sible to tune the InAs/GaSb interface into the topologicalregime, it would not be trivial.
IV. CONCLUSION
In summary, we have studied the HgTe/CdTe andInAs/GaSb quantum wells using DFT with a Hub-bard U correction determined by Bayesian optimization.DFT+U(BO) produces band structures of comparableaccuracy to a hybrid functional at the computational costof a semi-local functional. This enables us to conductsimulations of large interface models with hundreds ofatoms.For the HgTe/CdTe interface we find that an invertedgap opens at a critical thickness of 5.1 nm of HgTe, inagreement with experimental observations and previoustheoretical studies. For InAs/GaSb QWs with 50 layers(about 15 nm) of GaSb we find that the gap narrowswith increasing thickness of InAs in agreement with the
FIG. 9. The band gap reduction, ∆, as a function of electricfield for 10/10 GaAs-type and InSb-type interfaces. previous theory estimations. However, the gap does notcompletely close with up to 50 layers (about 15 nm) ofInAs. Based on the rate of gap narrowing, we estimatethat it would either require a significantly thicker InAsfilm to close the gap or the gap would decay to a finiteasymptotic limit.Simulations with an external electric field applied per-pendicular to the interface, pointing from GaSb to InAs,have been conducted for models with 10 layers of eachmaterial. We find that with increasing field strength theGaSb VBM shifts upwards relative to the InAs CBM,leading to narrowing of the gap at the interface. For theInSb-type interface, band crossing is observed at a criticalfield and subsequently an inverted gap opens. As the elec-tric field increases the gap increases but also shifts higherin energy above the Fermi level. Because the 10/10 in-terface has a larger gap due to the quantum size effect, we estimate the reduced critical field that would be re-quired to achieve band inversion and reach the topologi-cal regime for thicker QWs comprising 50 layers of eachmaterial.Our results explain the difficulty of experimentallyreaching the topological regime in InAs/GaSb QWs. Inprinciple, under the right conditions, an inverted gapcould be produced in this system. However, achievingthis requires a delicate balance between several parame-ters. To tune the initial gap, the structure of the QWsmust be precisely controlled, including the layer thick-ness, the bonding configuration at the interface, and pos-sibly also the lattice strain. Even if a smaller zero-fieldgap is obtained by interface engineering, a considerableelectric field may still be required to obtain band crossingand drive the system into the topological regime. Finally,gating or doping may be required to tune the Fermi levelposition inside the inverted gap.The HgTe/CdTe QW does not suffer from this diffi-culty because HgTe has an inverted band structure in-trinsically. Therefore, no electric field is necessary toachieve band inversion at the HgTe/CdTe interface andit is easier to reach the topological regime. Thus, our re-sults make a case for limited applicability of InAs/GaSbquantum wells for 2DTI production and suggest that al-ternative, more promising materials should be sought.
ACKNOWLEDGMENTS
We would like to thank Sergey Frolov from the Uni-versity of Pittsburgh, Chris Palmstrøm from the Univer-sity of California, Santa Barbara, Vlad Pribiag from theUniversity of Minnesota, and Michael Wimmer from TUDelft for helpful discussions. Work at CMU was fundedby the National Science Foundation (NSF) through grantOISE-1743717. This research used resources of theNational Energy Research Scientific Computing Center(NERSC), a DOE Office of Science User Facility sup-ported by the Office of Science of the U.S. Departmentof Energy under contract no. DE-AC02-05CH11231. [1] M. Z. Hasan and C. L. Kane, Colloquium: topologicalinsulators, Reviews of modern physics , 3045 (2010).[2] Y. Ando, Topological insulator materials, Journal of thePhysical Society of Japan , 102001 (2013).[3] L. Kou, Y. Ma, Z. Sun, T. Heine, and C. 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