Abstract
Current approaches of discovering new topological quantum materials, such as (magnetic) Weyl materials, by screening material databases or by rational design suffer from experimental challenges in growing the material in the desired geometry, hampering their potential applications. We take an approach of defect-engineering a topology starting from systems with experimentally well-controlled growth. Using SnTe, an experimentally viable topological crystalline insulator, as our model system, we elucidate the different effects of breaking time-reversal, inversion, and mirror symmetries using magnetic dopant defects, to controllably induce a topological phase transition. To this effect, we compute the thermodynamics, magnetism, and band-structure topology of magnetically doped SnTe using a full ab initio approach. In this process, we have discovered that Cr-doped SnTe is a Weyl semimetal. We have computed its classic Fermi-arc experimental signature on the (001) surface and also showed that it has a large anamolous Hall conductivity (AHC) of ∼250 ω-1cm-1. We predict a large Curie temperature (Tc) of 62 K even at 3.3% doping, with both Tc and AHC being tunable with Cr concentration, suggesting potential for room-temperature applications. Our study in general paves the way for defect-designing room-temperature topological quantum phases.
| Original language | English |
|---|---|
| Article number | 241110 |
| Journal | Physical Review B |
| Volume | 10 |
| Issue number | 24 |
| DOIs | |
| State | Published - Dec 16 2019 |
Funding
Pham Anh * Ganesh P. † Center for Nanophase Materials Sciences , Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA * [email protected] † [email protected] 16 December 2019 December 2019 100 24 241110 10 May 2019 ©2019 American Physical Society 2019 American Physical Society Current approaches of discovering new topological quantum materials, such as (magnetic) Weyl materials, by screening material databases or by rational design suffer from experimental challenges in growing the material in the desired geometry, hampering their potential applications. We take an approach of defect-engineering a topology starting from systems with experimentally well-controlled growth . Using SnTe, an experimentally viable topological crystalline insulator, as our model system, we elucidate the different effects of breaking time-reversal, inversion, and mirror symmetries using magnetic dopant defects, to controllably induce a topological phase transition. To this effect, we compute the thermodynamics, magnetism, and band-structure topology of magnetically doped SnTe using a full ab initio approach. In this process, we have discovered that Cr-doped SnTe is a Weyl semimetal. We have computed its classic Fermi-arc experimental signature on the (001) surface and also showed that it has a large anamolous Hall conductivity (AHC) of ∼ 250 Ω − 1 cm − 1 . We predict a large Curie temperature ( T c ) of 62 K even at 3.3 % doping, with both T c and AHC being tunable with Cr concentration, suggesting potential for room-temperature applications. Our study in general paves the way for defect-designing room-temperature topological quantum phases. Oak Ridge National Laboratory 10.13039/100006228 Laboratory Directed Research and Development 10.13039/100007000 U.S. Department of Energy 10.13039/100000015 Office of Science 10.13039/100006132 DE-AC02-05CH11231 DE-AC05-00OR22725 Motivation and summary . Topological materials have attracted great interest due to their potential application in low-energy nanoelectronic devices and quantum computing [1,2] . Since the first discovery of the inversion-symmetric and time-reversal-invariant topological insulators [3–5] , new materials with topological properties that break time-reversal/inversion symmetry like Chern and Weyl semimetals [6–8] have been predicted and realized experimentally. These topological semimetals exhibit various unique physical properties due to nonzero net Berry curvature in momentum space like open Fermi surface and chiral anomalous magnetoresistance [9–25] . The biggest technological promise of a Weyl semimetal is its ability to host quantized electronic conductivity without any dissipation when they are made sufficiently thin, due to the so-called quantum anomalous Hall effect (QAHE). However, current materials can only realize the QAHE at millikelvin temperature [26–28] , thus limiting the application of nontrivial topological materials in functional devices. Generally, the search for new Weyl semimetals and other topological quantum materials has lately been carried out by scourging through materials databases and looking for topological markers or by rational design [29–31] . This type of random search, even if fruitful in identifying interesting materials, may pose tremendous challenges in processing these new materials in the desired geometry to realize the predicted topological state. We instead take the approach of starting with an experimentally well controlled and studied material with nontrivial band-structure topology, i.e., SnTe, and break time-reversal (TR), inversion ( I ), and/or mirror ( M ) symmetries using magnetic dopants. SnTe belongs to a class of IV-VI rocksalt semiconducting topological materials that has attracted attention due to their novel topology [32–36] and functional properties like ferroelectricity [37] . Distinguished from the Bi/Sb-based Z 2 topological insulators, SnTe is classified as a topological crystalline insulator (TCI), in which, due to crystalline mirror symmetry ( M ), the dissipationless chiral spin current on the surface state is protected by a nontrivial mirror Chern number. Specifically, SnTe contains a mirror Chern number of 2; thus, based on the bulk-boundary correspondence the topological surface state will contain two Dirac band crossings at different k points depending on the different surface orientations [38] . Since the nontrivial topological property in SnTe-related TCI materials arises in the absence of time-reversal and inversion symmetry, different types of symmetry-breaking operations in these materials might result in emergent quantum phenomena like QAHE and topological superconductivity [39–41] . A rigorous understanding of how symmetry breaking will influence the topology based on full ab initio methods is still lacking, a necessary step for understanding and experimental control of these emergent phases. Since defect-doping allows us direct control of the Fermi level, this could be an ideal route to engineering experimentally viable topological quantum materials, starting with a TCI such as SnTe. In addition, experimental realization requires knowledge of defect thermodynamics as well as its long-range magnetic behavior, that can only be computed with full ab initio approaches. This motivates us to investigate the interplay of symmetry breaking, magnetic interactions, defect thermodynamics, and topology in magnetically doped SnTe to explore the emerging magnetic topological phases. The methodology can generally be applied to screening dopants, both in bulk and interfacial geometries, starting from materials with well controlled growth. In summary, we have elucidated how magnetic dopants can lead to emergent topological phases in a topological crystalline insulator, such as experimentally synthesizable (doped) SnTe [42–44] , using ab initio Wannier-based approaches [45] . Due to broken time-reversal symmetry (TRS) in Cr-doped SnTe (Cr:SnTe), a magnetic Weyl phase is realized, even at very low Cr concentrations. Most importantly, an increasing doping concentration is shown to not only tune the Fermi level to the Weyl nodes, but also enhance the quantized anomalous Hall conductivity (AHC). Combined with the fact that Cr:SnTe shows a mean-field Curie temperature of 62 K even at low defect concentrations of 3.3 % , consistent with experiments [42] , we expect this system to be a strong candidate to exhibit quantized conductance at high temperatures in both the bulk and thin-film limits. Breaking only the I symmetry preserves the Weyl phase, and its doping concentration dependence, but breaking only the M symmetries of the TCI leads to a nodal-line semimetal. In general, our study also provides a strategy of discovering topological quantum materials by selecting appropriate dopants for defect-engineering targeted topological properties starting from an experimentally viable topologically insulating quantum material. Methods . The geometry and electronic structure calculations were performed using the vasp code with the projected augmented wave method [46] and the Perdew-Burke-Ernzerhof (PBE) functional [47] , with an on-site Hubbard-type Coulomb interaction [48,49] . To study the magnetic and topological properties of the doped structure, the conventional unit cell comprising eight atoms as well as a 2 × 2 × 2 supercell with 64 atoms were used as shown in Fig. 1 . For each of the supercells and the unit cell, one atom of Sn was replaced with one atom of the dopant ( D M = Cr or Mn ), resulting in an overall doping concentration of 25% and 3.3%, respectively. The doping site preference of D M was also tested by comparing the formation energy of D M at the Te and Sn site in the 2 × 2 × 2 supercell. To characterize the topological properties of defect-doped SnTe, the tight-binding method with Wannier basis [50] was used as implemented in the wanniertools [45] code to calculate the surface Green's function [51] through an iterative scheme and the intrinsic AHC. Further details of the thermodynamics, magnetic and Wannier interpolation are presented in Ref. [52] . 10.1103/PhysRevB.100.241110.f1 1 FIG. 1. (a) Crystal structure of SnTe, schematically showing (time-reversal) symmetry breaking by replacing Sn with a (magnetic) point defect, D M . (b) Schematic showing the transformation of SnTe from a TCI to new topological phases by selectively breaking specific symmetries such as time-reversal (TR)/inversion ( I )/crystalline ( M ) symmetries, via defect-engineering. The band inversion between Sn (blue) and Te (red) is included in the illustration. Results and discussions . We will limit our study to point defects, specifically focusing on substitutional magnetic dopants. Before discussing the new topological phase arising out of magnetic dopants, it is important that we identify the general criteria of a magnetic dopant to induce a topological semimetal state in topological insulators (TIs) or (TCIs). Different from the QAHE, it is important that the magnetic dopant closes the bulk band gap of the original TCI or TI. Chemically, this means that the dopant needs to hybridize with the valence band, while contributing minimally to the density of state at the bulk Fermi level to result in accidental band crossings closer to the bulk Fermi level. Thus, dopants that can generate itinerant magnetism which interact ferromagnetically via a mechanism like Ruderman-Kittel-Kasuya-Yosida or p-d double exchange interaction would be preferable. Specifically, in the p-d exchange interaction, the d bands of the magnetic dopants would be located predominantly below the p -valence band, narrowing the overall band gap to induce a half-metallic state. In addition, it is also important that the dopants remain in the neutral state after being introduced into the host (i.e., the dopant prefers to be in the same charge state as the host metal atom). As shown in Fig. S1, Cr satisfies this thermodynamic requirement since it prefers the 2 + state when it is doped in SnTe at the Sn site. In addition, the ferromagnetic exchange interaction between Cr ions is favorable even at large distances, which is mediated via the p-d mechanism which results in a half-metallic state in SnTe. Furthermore, based on total energy calculations, we find that a configuration with two Cr atoms as nearest neighbor is 41.9 (8.85) meV lower than one with these two Cr atoms farthest apart, using PBE ( PBE + U ) methods, suggesting Cr dopants possibly cluster. In the case of Mn (Fig. S1), the dopant preserves the bulk band gap of SnTe while maintaining the neutral 2 + state, making it more suitable for realizing a QAHE. However, the Curie temperature of ferromagnetic interaction is predicted to be 4.3 K in agreement with the experimental result [46] , making it less of a desirable dopant for high-temperature QAHE. To quantify the type of semimetallic phase emerging from doping SnTe with Cr, its topological character, and the role played by chromium moments in breaking time-reversal symmetry, we replace one of the Sn atoms by a Cr atom in the unit cell configuration (corresponding to 25 % doping) in the rocksalt structure [as illustrated in Fig. 1(a) ], and investigate the electronic structure and its underlying topology with and without inclusion of the magnetic exchange interaction by setting the local moment of Cr to zero. It should be noted that for the conventional unit cell, it is not possible to break inversion/mirror symmetry, so when the local moments of Cr are zero, the underlying structure essentially preserves the mirror symmetry of the original SnTe. This should enable us to purely quantify the TRS breaking effect with the magnetic dopant. Indeed, when Cr is in the nonmagnetic state, Cr modifies the bulk band structure, resulting in a new band inversion between Cr's d and Sn's p bands at the R ( 0.5 , 0.5 , 0.5 ) point [Fig. S2(b)], but the surface projected bands on the (001) surface show a Dirac surface located along the high-symmetry line Γ ¯ - M ¯ perpendicular to the mirror plane, suggesting the preservation of the TCI state. When the magnetic moment of Cr is nonzero, the bulk band structure SnTe:Cr is significantly modified with the combined effect of the SOC and the exchange interaction. Here we only consider the magnetic orientation along the z axis. The band inversion at the R point is now dominated by p orbitals from Sn and Te, and the ground state of the system becomes semimetallic [Fig. S2(c)]. A further examination of the surface energy spectrum on the (001) surface shows an absence of the Dirac band crossing along the Γ ¯ - M ¯ projected high-symmetry lines [Fig. S2(f)], suggesting that these crossings are only preserved by the crystalline symmetry in the presence of TRS. An important feature of the Weyl semimetal is a topological Fermi arc, a result of a dispersive surface energy spectrum connecting the two Weyl points (also called Weyl nodes). Due to the semimetallic nature of the bulk bands, we search for the Weyl points above the Fermi level. Two pairs of Weyl points on opposite planes along the [001] direction at | k z | ∼ 0.4 were identified at ( − 0.501 816 18 , − 0.449 057 39 , − 0.403 965 23 ) - ( − 0.447755 95 , − 0.502 538 75 , 0.404 212 40 ) which act as a “sink” and “source” (i.e., “negative” and “positive” centers of Berry curvature), as shown by plotting the Berry curvature around these points [Figs. 2(a) and 2(b) ]. The Weyl points connect together at E F + 0.34 [Fig. 2(c) ]. Thus, heavy electron doping is required to move the Fermi level to these Weyl nodes. Due to the high doping concentration, these Weyl points are buried inside the bulk band which obscures the Fermi arc [Fig. S8(a)]. As seen from our defect phase diagram [Fig. S1(a)], the Fermi level is a function of Cr-doping concentration. We hence hypothesize that we can tune the Fermi level by lowering the Cr-doping concentration to bring it closer to the Weyl nodes. We also hypothesize that changing the amount of D M should change the strength of magnetic interaction in the bulk, thereby influencing the distance between the Weyl nodes, thereby allowing us to control the magnitude of the AHC. 10.1103/PhysRevB.100.241110.f2 2 FIG. 2. Topological property of the 25% magnetically doped SnTe. (a) and (b) The Berry curvature of the ( − 0.501 816 18 , − 0.449 057 39 , − 0.403 965 23 ) and ( − 0.447 755 95 , − 0.502 538 75 , 0.404 212 40 ) Weyl points on the k z = 0.4 and k z = -0.4 planes, respectively. (c) Band dispersion connecting the Weyl points projected on the (001) surface. To investigate these aspects, we study the topological property of the SnTe:Cr system at a diluted doping concentration of 3.3 % and 6.6 % . As seen from our magnetic-exchange curves [Fig. S1(c)], the Cr moments are ferromagnetically coupled even at separation distances corresponding to these low concentrations. As such, at these lower concentrations, we still break TRS. Note that we also break mirror and inversion symmetries, depending on the doping concentration. Similar to the case of 25 % doping though, doping Cr at 3.3 % in the host SnTe also modifies the bulk band structure. Specifically, the inclusion of the magnetic exchange and the SOC result in a band inversion between the p orbitals of Sn and Te, while the Cr d orbital hybridizes with Te- p inside the conduction band, resulting in a semimetallic state [Fig. S3(b)]. As we hypothesized, the Weyl points have moved closer to the Fermi level to E F + 0.161 at 3.3 % concentration, as seen in Fig. 2(d) . Interestingly, the Dirac surface states are absent in the ferromagnetic configuration [Figs. S3(d) and S4(d)]. To further confirm the existence of the Weyl node in the larger 2 × 2 × 2 supercell, we unfolded the bulk band structure into the primitive cell Brillouin zone using the effective band-structure approach [53] . As shown in Fig. S4, here again we see the Weyl point at 0.167 eV. This result confirms that the Weyl point is not an artifact of the band folding in the supercell approach. We also studied what would happen if instead of TRS, we only broke inversion ( I ) or mirror ( M ) symmetries. To do this, calculations of SnTe:Cr were performed using one or two nonmagnetic Cr dopants in the SnTe supercell, breaking just the I or M symmetry, as shown in Fig. S9. As shown in Fig. 3 , breaking I symmetry still gives Weyl nodes. Breaking only the M symmetry gives a nodal-line semimetallic phase. The nodal line is formed between the Sn- p and Cr- d orbitals and it is protected by inversion and time-reversal symmetries, in the absence of spin-orbit coupling (SOC). With SOC the system transforms to a Z 2 topological insulator due to the absence of the mirror symmetry. 10.1103/PhysRevB.100.241110.f3 3 FIG. 3. Different topological phases due to symmetry breaking in nonmagnetic Cr-doped SnTe. (a) Berry curvature of the Weyl nodes due to only inversion symmetry breaking. (b) Nodal line phase due to the absence of only the mirror symmetry. In addition to concentration, the position of the Weyl nodes may also be affected by local atomic distortions, due to the introduction of the dopant. Such an effect cannot be captured in standard k · p methods, but we can explicitly take it into account in our full ab initio study. As a result, we also studied the topology in an unrelaxed supercell with one Cr doping in the conventional and 2 × 2 × 2 supercell. Without structural distortion, the breaking of time-reversal symmetry results in the Weyl nodes with opposite chirality symmetric around the time-reversal-invariant momentum points. However, the positions of the Weyl nodes differ significantly on the energy scale in comparison to the relaxed structure which takes into account the role of local bond distortion due to Cr doping. Specifically, for the high doping concentration the Weyl points move further up inside the conduction band at E F + 0.486 eV while for the diluted doping of 3.3% the Weyl points move closer to the valence band at E F + 0.035 eV (Fig. S6). Consequently, these results emphasize the importance of capturing the effect of local lattice distortion due to impurity dopant which is absent in model Hamiltonian methods such as k · p in understanding the Weyl physics in doped topological material. An important property in Weyl semimetal is the large intrinsic AHC. The AHC is shown to be strongly dependent on the doping concentration. It reduces from 268 Ω − 1 cm − 1 at E F + 0.364 eV for a high doping concentration of 25 % to 54.4 Ω − 1 cm − 1 at the Weyl point energy level of E F + 0.156 eV for 3.3 % doping (Fig. S7). These values are in the same range as the recent AHC measurements of AHC in magnetic Weyl semimetals like half-Heusler materials [53] , CoNb 3 S 6 [54] , La-doped EuTiO 3 [55] , kagome metal [56] , and Fe 3 Ge Te 2 [57] . In an ideal Weyl semimetal, the AHC is proportional to the separation of the Weyl points [8] . As a result, the smaller separation between the Weyl points will result in a smaller AHC. In our doped TCI, the Weyl semimetal is far from an ideal system due to the minimal contribution of the Cr bands close to the Fermi level, but it is still instructive to show the separation between the Weyl points at the energy closest to the peak energy in the AHC curve at different concentration. As shown in Fig. 4(a) , the Δ k values decrease significantly from 25% to 3%, while their energy positions move closer to the bulk Fermi level with decreasing concentration, making it more accessible for transport study. As a result, by tuning doping concentrations, the spatial separation and energetic position of the Weyl points can be controlled, thus resulting in controllable AHC. 10.1103/PhysRevB.100.241110.f4 4 FIG. 4. (a) Energy position and momentum separation of the Weyl points at 25% and 3.3% Cr doping obtained from the supercell method. (b) Energy position of Weyl points for x = 0 to x = 0.25 from the VCA method. (c) Chirality of the Weyl point ( 0.496 , 0.506 , 0.495 ) for x = 0.03 in Sn 1 − x Cr x Te . (d) Zoom-in of the Fermi arc for x = 0.03 . In addition to the supercell approach, we perform a virtual crystal approximation (VCA) calculation of Sn 1 − x Cr x Te at different Cr doping levels. As shown in Fig. 4(b) , a Weyl phase emerges for x ≥ 0.03 , in agreement with the earlier supercell result, which results in four pairs of Weyl nodes around the bulk L point (0.5,0.5,0.5) as shown in the (001) surface projection in Fig. S11, with a chirality of 1 [Fig. 4(c) ]. We also confirm the Weyl phase at x = 0.03 by the existence of the Fermi arc [Fig. 4(d) ]. Overall, the VCA method confirms the persistence of a magnetic Weyl phase at a diluted doping concentration of 3 % with visible Fermi arc in spite of disorder, thus making it more feasible for experimental fabrication and confirmation. We would like to note that the introduction of dopants can potentially introduce quenched disorder. Such an effect can broaden the Weyl nodes/nodal lines, limiting their direct experimental observations and warrants a separate detailed theoretical investigation. Conclusion and outlook . In conclusion, by systematically investigating the thermodynamics, magnetic, and electronic properties of Cr- and Mn-doped SnTe, we have demonstrated that depending on which symmetries are broken, new topological phase transitions can be realized. Since defects are intrinsically disordered, models of disorder should take into account these possible symmetry-broken phases to interpret experiments, using an effective band-structure approach as shown in Ref. [58] . We identified SnTe:Cr as a magnetic Weyl semimetallic system with a large AHC even at room temperatures, tunable by changing the doping concentration. A possible strategy to potentially tune the energy level of the Weyl nodes is to codope Pb and Cr into SnTe since the content of Pb can be tuned to result in a Dirac semimetal phase with the Dirac point at the Fermi level and Cr can be used to break the time-reversal symmetry to produce a new ferromagnetic Weyl semimetal phase. Alternatively, strain engineering along the [001] direction can also narrow the band gap of SnTe [59] , and doping strained SnTe with dopant like Mn which has no in-gap states can allow for the formation of ideal Weyl nodes closer to the bulk Fermi level. In light of the ongoing debate on the cause of topological superconductivity in doped-topological insulators [37,38] our work emphasizes the need to systematically understand the role of different broken symmetries when interpreting such systems. Also, with advances in control over single-dopant positioning, there is a possibility of defect-designing dopants in thin films to realize specific topologies [60,61] . Acknowledgments. A.P. and P.G. were financially supported by the Oak Ridge National Laboratory's, Laboratory Directed Research and Development project. This research was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. Part of the research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. Part of this research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC05-00OR22725.
| Funders | Funder number |
|---|---|
| DOE Office of Science User Facility supported | |
| NERSC | |
| National Energy Research Scientific Computing Center | |
| Oak | |
| U.S. Department of Energy Office of Science User Facility operated | |
| U.S. Department of Energy | 10.13039/100006132 DE-AC02-05CH11231 DE-AC05-00OR22725 |
| Research and Development | |
| Oak Ridge National Laboratory | |
| Laboratory Directed Research and Development |