Subscriber access provided by UB + Fachbibliothek Chemie | (FU-Bibliothekssystem)
Communication
Tunable Dirac electron and hole self-doping of topological insulators induced by stacking defects Hugo Aramberri, Jorge I. Cerda, and M Carmen Muñoz Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b00625 • Publication Date (Web): 08 May 2015 Downloaded from http://pubs.acs.org on May 11, 2015
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Tunable Dirac electron and hole self-doping of topological insulators induced by stacking defects Hugo Aramberri, Jorge I. Cerdá and M. Carmen Muñoz* Instituto de Ciencia de Materiales de Madrid (CSIC), Sor Juana Inés de la Cruz 3, 28049 KEYWORDS: topological insulators, twin boundaries, stacking faults, spontaneous polarization, doping.
ABSTRACT: Self-doping of Dirac cones in Bi2X3 (X=Se, Te) topological insulators can be tuned by controlling the sequence of stacking defects in the crystal. Twin boundaries inside Bi2X3 drive n- or p-type doping of the surface states, 12 13 2 originated by the defect induced spontaneous polarization. Dopings up to 10 -10 e/cm may be achieved depending on the defect distribution. Our findings open the route to fabrication of Bi2X3 surfaces with tailored intrinsic charge and spin densities.
Topological insulators (TI) represent an exciting quantum phase in condensed matter defined by a bulk insulator with Z2-invariant topology that guarantees metallic Dirac states at the surface. In three-dimensional (3D) TIs, the inverted ordering of the bulk electronic bands driven by the spin-orbit interaction dictates that an odd number of Dirac states exist at the surface. These topological surface states (TSSs) form Kramer pairs with linear dispersion, helical spin texture and are topologically protected as long as time-reversal symmetry is pre(1)-(4) served . Their linear dispersion gives rise to novel (5) phenomena such as the magnetoelectric effect , the (6) anomalous Hall-effect in magnetically doped TIs and to non-trivial elementary topological excitations such as (7) Majorana fermions or topological exciton condensates (8) with fractionally charged vortices. The most widely studied 3D TIs are the Bichalcogenides Bi2X3 (X=Se, Te), which exhibit a layered structure along the [0001] direction with X-Bi-X-Bi-X quintuple layers (QLs) as the basic unit –see Figure 1(a)–. Bi2X3 (0001) surfaces present a single nondegenerate Dirac cone around the Brillouin zone center. The electrons populating the TSSs, although free to move parallel to the surface, are mainly confined within the two surfacemost QLs, forming a two-dimensional massless Dirac fermion gas (2DFG) with a well-defined (9) spin helicity. Therefore, surfaces and nanostructures of Bi-chalcogenides are ideal candidates to realize the predicted exotic phenomena. In addition, they offer novel opportunities for applications; recent proposals for (13) spintronics(12) and quantum computation are founded on these materials. Advance towards the experimental realization of complex many-body excitations or TIbased devices requires manipulation of the TSSs, as well as tuning the chemical potential around the Dirac point (DP).
Whereas in the ideal bulk-truncated Bi2Se3 surfaces charge neutrality pins the Fermi level at the DP, TSSs, despite being protected against backscattering and localization, are susceptible to charge disorder caused by bulk defects. Experiments have shown that bulk doping with magnetic and non-magnetic impurities results in pronounced nanoscale spatial fluctuations of energy, (14) momentum and spin-helicity while adsorbates at the (15)(16) surface shift the DP to higher binding energies. In addition, local charging of the Dirac states at antiphase boundaries between neighboring grains has been re(17) cently reported in epitaxial (0001) films. In this letter we show that self-doping of the Bi2X3 (0001) TSSs can be achieved by introducing structural planar defects. More precisely a (0001) oriented slab of this material containing twin boundaries –reversal of the stacking sequence between QLs– drives a charge doping of the Dirac TSSs. Remarkably, depending on the precise orientation of the twin, either electron or hole gases with well defined spin-helicity can be optionally tuned. Therefore, electron and hole 2DFG with the same spin-helicity coexist at opposite slab surfaces. In addition, the surface charge density can be controlled by the number of planar defects and, to a lesser extent, by their relative location with respect to the surface. In this way the chemical potential can be fine-tuned to preselected electron or hole conductive regimes. We demonstrate that the surface doping is caused by the spontaneous polarization (SP) induced by the lattice defects in the bulk. Our findings pave the way to the fabrication of Bi2X3 surfaces and nanostructures with tailored surface charge and spin densities. The present study is relevant for both, understanding the fundamental properties of TSSs and also promoting an effective use of these states in applications.
ACS Paragon Plus Environment
Nano Letters (a) ideal
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(b) SF4
(c) TB4
(d) TB2 Se Bi
[001]
[100]
c0
4c0
4c0
2c0
Figure 1. Crystal structure for some of the Bi2Se3 polytypes considered. Se (Bi) atoms are shown in red (blue). (a) Ideal defect-free case, (b) SF4 structure with a SF every two QLs, (c) TB4 structure with a TB every two QLs and (d) TB2 structure with a TB every QL. Thin (thick) dashed lines indicate inter-QL van der Waals bonds for an ideal (defective) stacking. The arrows show the direction of the dipole moment associated to each TB (see text and SM #2). Lattice parameter c is indicated below each case.
All the calculations have been carried out at the ab initio level within the density functional theory (DFT) and (18) under the generalized gradient approximation (GGA) (19)-21 employing both the SIESTA-GREEN and (22)(23) packages. Spin-orbit interactions necesVASP sary to retrieve the TSSs were included as well as van (24)(25) der Waals semi-empirical corrections to account for the weak inter-QL coupling. Apart from the usual slab geometries we additionally constructed true semi-infinite (26) surfaces via Green’s functions matching techniques in order to obtain a precise description of each TSS and analyze its doping. Since most of the considered slabs are asymmetric, we could construct, from each calculation, an top and a bottom surface by retaining the matrix elements of either the upper or lower half of the slab and replacing the rest of elements below or above by those of the bulk. We assume throughout dopant-free and stoichiometric slabs, thus localized states arising from charged defects which may either trap or release electrons are discarded. Furthermore, we choose as boundary conditions vanishing electric fields in vacuum, which is achieved after applying the usual dipole-dipole correc(27) tions among image slabs. Full details of the calculation parameters are provided in the Supplementary Material (SM #1). We consider Bi2Se3 as a model system but the results are similar for Bi2Te3 and related TIs. Bi2Se3 crystallizes in the rhombohedral system with space group R3m ( ). Along the [0001] direction, the stacking pattern is fcc–like, -AbCaB-CaBcA-, where capital and small letters stand for Se and Bi, respectively. The Se-Bi bonds within the QLs are mainly covalent, while at adjacent QLs the Se-Se double-layer is only weakly bonded through van der Waals forces. Here we address (1×1) slabs contain-
Page 2 of 7
ing two-dimensional stacking defects (SDs) which may either be stacking-faults (SFs) or basal twin boundary domains (TBs). In a SF the fcc stacking changes to AbCaB-AbCaB-CaBcA-, where bold letters indicate the SF location –see Figure 1 (b) where a SF is inserted every two QLs–. At a TB, on the other hand, the two adjacent QLs are rotated by 180º about the surface normal leading to an -AbCaB-CbAcB-AcBaC- stacking. Figure 1 (c) and (d) sketch structures with a TB inserted every two (TB4) and one (TB2) QLs, resulting in polytypes with = 4 and 2 periodicities along the [0001] direction, where = 9.54 Å is the repeat vector in the ideal TI. A crucial property of almost all SDs, either interor intra-QL, is the rupture of the inversion symmetry of the ideal slabs despite the stoichiometry within each QL being preserved, the only exception being inter-QL SFs. As a result, defected slabs become asymmetric and exhibit inequivalent geometries at the top and bottom surfaces. In particular, TBs present opposite orientations with respect to each surface The layered structure facilitates the formation of extended planar defects. Indeed, SFs and TBs, analogous to those here investigated, have been frequently identified in thin films of Bi2Se3, Bi2Te3 and related materials. They are formed either through deformation of the crystal or directly during the growth and annealing process(28)-34 . The energy cost involved in creating different es SDs within a (0001) slab is analyzed in detail in SM #2. After considering both inter- and intra-QL SDs, we consistently found, as expected, a sensibly larger formation 2 energy for the latter, with differences in the 5-20 meV/Å range depending on the precise location of the SD. Inter2 QL TBs show the lowest formation energy, ≤ 3 meV/Å , while, surprisingly, SFs require an energy cost around three times larger. Furthermore, the energy cost for the occurrence of multiple TBs in a thin film is basically the sum of the single TB contributions. Notably, the calculated formation energy of (0001) TBs in Bi2Te3 is similar, 2 2 meV/Å , and agrees well with previously reported va(29),(33) lues. Therefore, based on energetic considerations we predict that (0001) TB boundaries between QLs are the predominant SDs in Bi2X3 materials which is in line with numerous experimental observations of single and multiple TBs in a large diversity of Bi2X3 based ma(28)(29)(33)(34) terials. In the absence of SDs and in (9) agreement with previous calculations, the Bi2Se3 (0001) surface presents a single TSS with helical spin texture spanning the bulk energy gap and with the DP pinned at the Fermi level. Since SDs are non-magnetic, involve zero net charge perturbations and exhibit an almost bulk-like structure, it would be expected that their impact on the TSSs is minor.
ACS Paragon Plus Environment
Page 3 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Figure 2. Electronic band structure for Bi2Se3 semi-infinite surfaces (a) TB4-top, (b) TB4-bottom, (c) TB2-top and (d) TB2-bottom. The maps are obtained from the k-resolved DOS projected on the 3 QLs closest to the surface. An 8 QL thick slab was employed in all cases to construct the surface, so that 4 (7) TBs are included in the TB4 (TB2) case. As a TB is not symmetric in , the top and bottom surfaces give raise to dopings of opposite sign (see Figure 1), with the left (right) column showing p-(n-)doped TSSs. Due to the smaller group velocity of the TSS in the valence band region, the DP shifts are smaller for p-doped surfaces.
Figure 3. (a) Charge density profiles, ∆ρ(z), averaged over the 2D unit cell for a 6 QL thick Bi2Se3 slab after substracting the defect free contribution. (b) Same as (a) but for the associated Hartree potential, ∆VH(z). Red, green and blue lines correspond to defective structures SF4, TB4 and TB2, respectively. The z origin is located at the bottom-surfaces. Vertical dashed lines indicate the two surfaces of the slab. Arrows in (a) indicate the direction of the dipole moment associated to the TBs.The drop in the potential at the vacuum seen in (b) is proportional to the total dipole moment of the system.
However, after analyzing around a hundred different surfaces constructed from slabs of different thicknesses and different interlayer SD arrangements, we have found that large dopings of the TSSs can be achieved. In Figure 2 we display the electronic structure around the DP for the top and bottom surfaces of structures TB4 (a-b) and TB2 (c-d), respectively. Clear shifts in the DPs with respect to the Fermi level can be observed in all of them, with the peculiarity that the sign of the doping can be tuned by inverting the orientation of the TB; top surfaces (a) and (c) present p-type doping while bottom surfaces (b) and (d) n-type. The shifts are sensibly larger for the TB2, indicating that the density of TBs is a key ingredient
in the doping mechanism. Remarkably, SFs (not shown) induce almost negligible DP shifts of less than 10 meV. To elucidate the origin of the doping and its dependence with the type and density of SDs, we examine their electronic properties. In Figure 3 we present 2D averaged profiles along the [0001] direction of the charge density, ∆ρ(z), and its associated Hartree potential, ∆VH(z), after substracting the defect-free contribution for 6 QL thick slabs with SF4, TB4 and TB2 defect arrangements. ∆ρ is nonzero only close to the SDs, where charge oscillations develop. Although at first sight the ∆ρ features seem similar for the TBs and the SFs, closer inspection reveals a marked asymmetry for the former, leading to a considerable dipole moment (in SM #2 we analyze in detail its origin). Differences among SFs and TBs are more clearly seen in the ∆VH(z) profiles. Whereas the formers show up as localized almost symmetric dips in the potential, the latter induce marked discontinuities with upwards jumps along the down-up direction and an overall potential difference across the slab that increases with the TB density.
Figure 4. (a) Dipole moments per unit area, , for Bi2Se3 slabs with different TB arrangements as a function of the slab thickness, . Blue, red and green lines correspond to best fits to the data points (circles) for structures TB2, TB4 and TB6 respectively, according to eq. 1. For the last two we used solid, dashed and dotted lines after groupping the slabs according to the location of the TB closest to the uppermost QL. SOC was not included in these calculations in order to remove the TSS and hence obtain an insulating (dielectric) slab. (b) Doping charge (left axis) and DP shifts in the TSS (right axis) for the top and bottom surfaces constructed from the same slabs as in (a), again as a function of . Horizontal lines at the right show the limiting charges for each polytype derived from their bulk SPs.
Therefore, and in analogy with ferro-electric materi(35) als , the existence of sizeable internal dipoles at the TBs yields a finite SP, , along the [0001] direction. The , across any of continuity of the displacement vector, the two slab’s surfaces dictates that, ∆ = ! " # = $ where %! " is the macroscopic field inside the TI (generally denoted as the depolarization field), σ is the surface charge density allocated at the TSSs and is a unitary vector normal to the surface that points out of the film. In the large thickness limit the depolarization field will vanish and, thus, a direct equivalence between the SP and the TSS doping is expected: (36) = $. Physically, this means that bound charges appearing at the surfaces of Bi2Se3 slabs containing SDs are fully compensated by the free TSS
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
charge. For finite films, however, the surface charge distribution causes a finite depolarization field that softens the SP discontinuity or, in other words, leaves the SP partially uncompensated. As a result, represents an upper limit to the TSS doping σ –which will depend on the slab thickness. Verifying that the polarization discontinuity causes the surface doping requires, on the one hand, the estimation of σ by integrating the TSS’s DOS between the Fermi level and the DP –which is not such a trivial task as shown in SM #4– and, on the other hand, the evaluation of the SP for a given periodic arrangement of the TBs in the crystal. Although in principle the SP is a bulk dielectric property, we have recently shown how it can be accurately determined from slab calculations without the (37) need of performing macroscopic averages . Here, we calculate the SP considering that no free charges are present at the surface –in the ab-initio calculations this condition is enforced by switching off the spin-orbit interactions thus preventing the appearance of the TSSs. In Figure 4 (a) we show the evolution of the slab’s dipole moment per unit area, , as a function of the slab’s thickness, , for the TB2, TB4 and TB6 polytypes. For the last two cases we consider different subsets –green or red solid, dashed and dotted lines– depending on the location, at each , of the TB closest to the uppermost QL. The data points can be accurately fitted by the ex(37) pression : = # & ⁄ ' ) = # &* ⁄ ' ) # +, -+, (eq.1) where is a surface localized contribution arising from the bond rearrangement at the surface, * the net polarization at the crystalline regions, -+, the number of TBs in the slab and +, the dipole moment per unit area associated to a single TB. The theoretically derived value of ' = 24.6 has been used here (see SM #3). * includes the depolarization field counteracting the SP as well as the electric field arising from the h/e doping at the top/bottom surface. It nevertheless attains small values * ≈ 0.2 1023 4⁄3 as can be already inferred from the almost flat segments in the TB4 and TB6 plots. On the contrary, the appearance of a new TB in the slab shows up as a downwards jump of average magnitude 〈 +, 〉 = 3.8 1089 4⁄Å which is hardly dependent on the precise defect configuration. The TB2 slabs, on the other hand, show an almost perfect linear behavior for &) due to the absence of crystalline regions and, hence, the slope directly provides +, . The bulk SP may now be estimated as: = 2 ' & +, ⁄ ), yielding values for our defected slabs of &:;2/4/6/8) = &9.6/4.6/3.0/2.3) 1023 4⁄3 . Notably, they correspond to substantial 12 2 doping charges of the same order (10 e/cm ) as those (38) measured for graphene on SiC polytypes, or oxide 12 14 2 interfaces (10 -10 e/cm ) and even larger than those 10 12 2 in doped semiconductors (10 -10 e/cm ).
Page 4 of 7
that all bottom (top) surfaces show a clear n-(p-)type doping, as dictated by the direction of the spontaneous polarization. Next, the doping levels increase with t and approach a limiting value, which is precisely the SP, associated to each polytype (indicated by horizontal lines in the figure). At the largest thickness we considered (10 QLs), the dopings reach up to 60% of this limit 12 2 (5.4 and -5.910 e/cm for the maximum hole and electron concentrations respectively), while we estimate by linear extrapolation that almost full compensation of the SP is reached at ∼ 18 QLs. In addition, there is a clear resemblance between these curves and the &) ones in Figure 4 (a), specially the ladder type behavior for TB4 and TB6. Again, sudden jumps in the doping are seen whenever a defective QL is added to the slab, whereas if the additional QL is non-defective, it tends to screen the SP causing a slight decrease in the doping. We may therefore conclude that the doping can be tuned via the actual number of TBs and its density, -+, ⁄ , as well as by the proximity of the first TB to the surface. Note that in the reported doped surfaces all TB's dipole moments are aligned –point along the same direction – and, hence, accumulate. Indeed, surfaces and slabs with anti-parallel configurations show no SP due to the cancellation of the dipole moments and, thus, a vanishing doping. Nevertheless, as shown in the SM #2, the anti-parallel TB arrangement results energetically less favorable than the parallel one. Having shown the relationship between the bulk TB derived SP and the surface doping, we conclude by examining the effect of the TBs in thin film geometries. In a finite slab with a single or multiple TBs, the Dirac states located at each side of the film experience energy shifts in opposite directions and equal to those of the corresponding semi-infinite surfaces. This yields a net accumulation of negative and positive charge of the same magnitude at the bottom and top surfaces, respectively. Figure 5 (a) and (b) represent the band dispersion around the DP of two 6 QL thick films, one without defects and the other containing 3 TBs (TB4) The p- and ndoped TSSs exhibit a left-handed spin helicity, equal to that corresponding to the ideal surface. As shown in Figure 5 (c), along the Γ−M direction the spin lies in the surface plane, as required by mirror symmetry, whereas away from this high symmetry direction and correlated with the hexagonal warping a perpendicular component develops, reaching its maximum along Γ−K. The out-ofplane component increases with increasing energy separation from the DP and consequently with increasing electron or hole doping. In addition, the rupture of the inversion symmetry allows for the development of a large Rashba spin-splitting in the bulk-like bands, which is otherwise absent –see for instance the conduction and valence bands in Figure 5 (b).
A summary of the dopings for all the bottom and top surfaces with TB defects is presented in Figure 4 (b), where the TSS charge (left axis) or, equivalently, the DP shifts (right axis in a non-linear scale) is plotted as a function of the slab thickness, . The first observation is
ACS Paragon Plus Environment
Page 5 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
ACKNOWLEDGMENT This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grant Nos. MAT2012-38045-C04-04 and MAT2013-47878-C2-R. We would also like to acknowledge the use of computational resources of CESGA and the i2BASQUE academic network. Figure 5. Band structure of 6 QL thick Bi2Se3 slabs, (a) with no SDs and (b) for the TB4 polytype. In (b), the TSS on the bottom surface (purple) shows n-doping, while the TSS on the top surface (orange) shows p-doping. Note the appearance of a Rashba splitting in the defective case. In (c) the spin texture of the right TSS is shown in reciprocal space for a TB2 system. The in-plane components of the spin are plotted as arrows, while the out of plane Sz is colorcoded as a percentage of the modulus of S (see inset). Spin-momentum lock of the helical TSSs is preserved when (39)(38) SDs are included.
Summarizing, the small density of states derived from the highly dispersive TSSs represents a highly precise gauge for measuring the impact of SDs on 3D topological insulators. By controlling the sequence of stacking defects close to a Bi2X3 (0001) surface, nanostructures with tailored surface charge and spin densities up to 13 2 10 e/cm , can be fabricated without the need of dopants, adsorbates or external fields. Our results provide a simple explanation for the local charging of the Dirac (17) states recently reported in epitaxial (0001) films. The origin of the surface self-doping relies on the spontaneous polarization generated by the lattice defects in the bulk. This mechanism allows to tune the Fermi level to preselected electron or hole conductive regimes. Furthermore, the SP effect persists even for large defect concentrations. Preliminary calculations for defected slabs indicate that the presence of a donor Se vacancy in the slab, despite being a source of additional doping, essentially preserves the difference between the TSS doping at the top and bottom surfaces. Thus, SDs may also be used to reduce undesired carrier densities which may screen the outstanding TSS spin-transport properties. In addition, because of the large Rashba splitting, high-efficient spin manipulation might be realized in thin films. Although we have considered Bi2Se3 as our model system, the results can be generalized to Bi2Te3 and related TI surfaces. Therefore, thin films of Bichalcogenides TI with stacking defects, provide a new playground where exotic excitations such as topological exciton condensates could be probed.
SUPPORTING INFORMATION AVAILABLE Additional information and figures. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION Corresponding Author *email: [email protected] Notes The authors declare no competing financial interest.
REFERENCES (1) Kane, C. L.; Mele, E. J. Phys. Rev. Lett. 2005, 95, 146802. (2) Bernevig, B. A.; Zhang, S.-C. Phys. Rev. Lett. 2006, 96, 106802. (3) Hasan, M. Z.; Kane, C. L. Rev. Mod. Phys. 2010, 82, 3045. (4) Qi, X.-L.; Zhang, S.-C. Rev. Mod. Phys. 2011, 83, 1057–1110. (5) Qi, X.-L.; Hughes, T. L.; Zhang, S.-C. Phys. Rev. B 2008, 78, 195424. (6) Chang, C.-Z. et al. Science 2013, 340, 167–170. (7) Fu, L.; Kane, C. L. Phys. Rev. Lett. 2008, 100, 096407. (8) Seradjeh, B.; Moore, J. E.; Franz, M. Phys. Rev. Lett. 2009, 103, 066402. (9) Zhang, H.; Liu, C.-X.; Qi, X.-L.; Dai, X.; Fang, Z.; Zhang, S.-C. Nature Physics 2009, 5, 438–442. (10) Xia, Y.; Qian, D.; Hsieh, D.; Wray, L.; Pal, A.; Lin, H.; Bansil, A.; Grauer, D.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. Nature Physics 2009, 5, 398–402. (11) Chen, Y.; Analytis, J.; Chu, J.-H.; Liu, Z.; Mo, S.-K.; Qi, X.-L.; Zhang, H.; Lu, D.; Dai, X.; Fang, Z.; Zhang, S.-C.; Fisher, I.; Hussain, Z.; Shen, Z.-X. Science 2009, 325, 178–181. (12) Pesin, D.; MacDonald, A. H. Nature materials 2012, 11, 409–416. (13) Nayak, C.; Simon, S. H.; Stern, A.; Freedman, M.; Das Sarma, S. Rev. Mod. Phys.2008, 80, 1083–1159. (14) Beidenkopf, H.; Roushan, P.; Seo, J.; Gorman, L.; Drozdov, I.; San Hor, Y.; Cava, R.; Yazdani, A. Nature Physics 2011, 7, 939–943. (15) Benia, H. M.; Lin, C.; Kern, K.; Ast, C. R. Phys. Rev. Lett. 2011, 107, 177602. (16) Bianchi, M.; Hatch, R. C.; Mi, J.; Iversen, B. B.; Hofmann, P. Phys. Rev. Lett. 2011,107, 086802. (17) Liu, Y.; Li, Y. Y.; Gilks, D.; Lazarov, V. K.; Weinert, M.; Li, L. Phys. Rev. Lett. 2013,110, 186804. (18) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868. (19) Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D. Journal of Physics: Condensed Matter 2002, 14, 2745. (20) Cerdá, J.; Van Hove, M. A.; Sautet, P.; Salmeron, M. Phys. Rev. B 1997, 56, 15885– 15899. (21) Cuadrado, R.; Cerdá, J. Journal of Physics: Condensed Matter 2012, 24, 086005. (22) Kresse, G.; Furthmüller, J. Computational Materials Science 1996, 6, 15 – 50. (23) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 48, 13115– 13118. (24) Grimme, S. Journal of computational chemistry 2006, 27, 1787–1799. (25) Ortmann, F.; Bechstedt, F.; Schmidt, W. G. Phys. Rev. B 2006, 73, 205101. (26) Rossen, E.; Flipse, C.; Cerdá, J. Phys. Rev. B 2013, 87, 235412. (27) Bengtsson, L. Phys. Rev. B 1999, 59, 12301. (28) Medlin, D.; Yang, N. Journal of electronic materials 2012, 41, 1456–1464. (29) Medlin, D.; Ramasse, Q.; Spataru, C.; Yang, N. Journal of Applied Physics 2010, 108, 043517–043517.
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(30) Li, H.; Wang, Z.; Kan, X.; Guo, X.; He, H.; Wang, Z.; Wang, J.; Wong, T.; Wang, N.; Xie, M. New Journal of Physics 2010, 12, 103038. (31) Jariwala, B.; Shah, D. Journal of Crystal Growth 2011, 318, 1179–1183. (32) Mao-Hai, X.; Xin, G.; Zhong-Jie, X.; Wing-Kin, H. Chinese Physics B 2013, 22,068101. (33) Shin, H. S.; Jeon, S. G.; Yu, J.; Kim, Y.-S.; Park, H. M.; Song, J. Y. Nanoscale 2014, 6, 6158–6165. (34) Tarakina, N. V.; Schreyeck, S.; Luysberg, M.; Grauer, S.; Schumacher, C.; Karczewski, G.; Brunner, K.; Gould, C.; Buhmann, H.; Dunin-Borkowski, R. E.; Molenkamp, L. W. Advanced Materials Interfaces 2014, 1, 1400134.
Page 6 of 7
(35) Hill, N. A. Journal of Physical Chemistry B 2000, 104, 6694–6709. (36) Meyer, B.; Vanderbilt, D. Phys. Rev. B 2001, 63, 205426. (37) Slawinska, J.; Aramberri, H.; Muñoz, M.C.; Cerdá, J. arXiv 2014, 1410.4551. (38) Mammadov, S.; Ristein, J.; Koch, R. J.; Ostler, M.; Raidel, C.; Wanke, M.; Vasiliauskas, R.; Yakimova, R.; Seyller, T. 2D Materials 2014, 1, 035003. (39) Basak, S.; Lin, H.; Wray, L. A.; Xu, S.-Y.; Fu, L.; Hasan, M. Z.; Bansil, A. Phys. Rev. B 2011, 84, 121401.
ACS Paragon Plus Environment
Page 7 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
TOC Graphic
Bi2Se3 slab containing a twin boundary (TB) every QL. The TSS on the left surface (purple) is n-doped, while the TSS on the right surface (orange) is p-doped. The spin-momentum lock of the helical TSS is preserved (green arrows represent spins).
ACS Paragon Plus Environment
Communication
Tunable Dirac electron and hole self-doping of topological insulators induced by stacking defects Hugo Aramberri, Jorge I. Cerda, and M Carmen Muñoz Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b00625 • Publication Date (Web): 08 May 2015 Downloaded from http://pubs.acs.org on May 11, 2015
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Tunable Dirac electron and hole self-doping of topological insulators induced by stacking defects Hugo Aramberri, Jorge I. Cerdá and M. Carmen Muñoz* Instituto de Ciencia de Materiales de Madrid (CSIC), Sor Juana Inés de la Cruz 3, 28049 KEYWORDS: topological insulators, twin boundaries, stacking faults, spontaneous polarization, doping.
ABSTRACT: Self-doping of Dirac cones in Bi2X3 (X=Se, Te) topological insulators can be tuned by controlling the sequence of stacking defects in the crystal. Twin boundaries inside Bi2X3 drive n- or p-type doping of the surface states, 12 13 2 originated by the defect induced spontaneous polarization. Dopings up to 10 -10 e/cm may be achieved depending on the defect distribution. Our findings open the route to fabrication of Bi2X3 surfaces with tailored intrinsic charge and spin densities.
Topological insulators (TI) represent an exciting quantum phase in condensed matter defined by a bulk insulator with Z2-invariant topology that guarantees metallic Dirac states at the surface. In three-dimensional (3D) TIs, the inverted ordering of the bulk electronic bands driven by the spin-orbit interaction dictates that an odd number of Dirac states exist at the surface. These topological surface states (TSSs) form Kramer pairs with linear dispersion, helical spin texture and are topologically protected as long as time-reversal symmetry is pre(1)-(4) served . Their linear dispersion gives rise to novel (5) phenomena such as the magnetoelectric effect , the (6) anomalous Hall-effect in magnetically doped TIs and to non-trivial elementary topological excitations such as (7) Majorana fermions or topological exciton condensates (8) with fractionally charged vortices. The most widely studied 3D TIs are the Bichalcogenides Bi2X3 (X=Se, Te), which exhibit a layered structure along the [0001] direction with X-Bi-X-Bi-X quintuple layers (QLs) as the basic unit –see Figure 1(a)–. Bi2X3 (0001) surfaces present a single nondegenerate Dirac cone around the Brillouin zone center. The electrons populating the TSSs, although free to move parallel to the surface, are mainly confined within the two surfacemost QLs, forming a two-dimensional massless Dirac fermion gas (2DFG) with a well-defined (9) spin helicity. Therefore, surfaces and nanostructures of Bi-chalcogenides are ideal candidates to realize the predicted exotic phenomena. In addition, they offer novel opportunities for applications; recent proposals for (13) spintronics(12) and quantum computation are founded on these materials. Advance towards the experimental realization of complex many-body excitations or TIbased devices requires manipulation of the TSSs, as well as tuning the chemical potential around the Dirac point (DP).
Whereas in the ideal bulk-truncated Bi2Se3 surfaces charge neutrality pins the Fermi level at the DP, TSSs, despite being protected against backscattering and localization, are susceptible to charge disorder caused by bulk defects. Experiments have shown that bulk doping with magnetic and non-magnetic impurities results in pronounced nanoscale spatial fluctuations of energy, (14) momentum and spin-helicity while adsorbates at the (15)(16) surface shift the DP to higher binding energies. In addition, local charging of the Dirac states at antiphase boundaries between neighboring grains has been re(17) cently reported in epitaxial (0001) films. In this letter we show that self-doping of the Bi2X3 (0001) TSSs can be achieved by introducing structural planar defects. More precisely a (0001) oriented slab of this material containing twin boundaries –reversal of the stacking sequence between QLs– drives a charge doping of the Dirac TSSs. Remarkably, depending on the precise orientation of the twin, either electron or hole gases with well defined spin-helicity can be optionally tuned. Therefore, electron and hole 2DFG with the same spin-helicity coexist at opposite slab surfaces. In addition, the surface charge density can be controlled by the number of planar defects and, to a lesser extent, by their relative location with respect to the surface. In this way the chemical potential can be fine-tuned to preselected electron or hole conductive regimes. We demonstrate that the surface doping is caused by the spontaneous polarization (SP) induced by the lattice defects in the bulk. Our findings pave the way to the fabrication of Bi2X3 surfaces and nanostructures with tailored surface charge and spin densities. The present study is relevant for both, understanding the fundamental properties of TSSs and also promoting an effective use of these states in applications.
ACS Paragon Plus Environment
Nano Letters (a) ideal
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(b) SF4
(c) TB4
(d) TB2 Se Bi
[001]
[100]
c0
4c0
4c0
2c0
Figure 1. Crystal structure for some of the Bi2Se3 polytypes considered. Se (Bi) atoms are shown in red (blue). (a) Ideal defect-free case, (b) SF4 structure with a SF every two QLs, (c) TB4 structure with a TB every two QLs and (d) TB2 structure with a TB every QL. Thin (thick) dashed lines indicate inter-QL van der Waals bonds for an ideal (defective) stacking. The arrows show the direction of the dipole moment associated to each TB (see text and SM #2). Lattice parameter c is indicated below each case.
All the calculations have been carried out at the ab initio level within the density functional theory (DFT) and (18) under the generalized gradient approximation (GGA) (19)-21 employing both the SIESTA-GREEN and (22)(23) packages. Spin-orbit interactions necesVASP sary to retrieve the TSSs were included as well as van (24)(25) der Waals semi-empirical corrections to account for the weak inter-QL coupling. Apart from the usual slab geometries we additionally constructed true semi-infinite (26) surfaces via Green’s functions matching techniques in order to obtain a precise description of each TSS and analyze its doping. Since most of the considered slabs are asymmetric, we could construct, from each calculation, an top and a bottom surface by retaining the matrix elements of either the upper or lower half of the slab and replacing the rest of elements below or above by those of the bulk. We assume throughout dopant-free and stoichiometric slabs, thus localized states arising from charged defects which may either trap or release electrons are discarded. Furthermore, we choose as boundary conditions vanishing electric fields in vacuum, which is achieved after applying the usual dipole-dipole correc(27) tions among image slabs. Full details of the calculation parameters are provided in the Supplementary Material (SM #1). We consider Bi2Se3 as a model system but the results are similar for Bi2Te3 and related TIs. Bi2Se3 crystallizes in the rhombohedral system with space group R3m ( ). Along the [0001] direction, the stacking pattern is fcc–like, -AbCaB-CaBcA-, where capital and small letters stand for Se and Bi, respectively. The Se-Bi bonds within the QLs are mainly covalent, while at adjacent QLs the Se-Se double-layer is only weakly bonded through van der Waals forces. Here we address (1×1) slabs contain-
Page 2 of 7
ing two-dimensional stacking defects (SDs) which may either be stacking-faults (SFs) or basal twin boundary domains (TBs). In a SF the fcc stacking changes to AbCaB-AbCaB-CaBcA-, where bold letters indicate the SF location –see Figure 1 (b) where a SF is inserted every two QLs–. At a TB, on the other hand, the two adjacent QLs are rotated by 180º about the surface normal leading to an -AbCaB-CbAcB-AcBaC- stacking. Figure 1 (c) and (d) sketch structures with a TB inserted every two (TB4) and one (TB2) QLs, resulting in polytypes with = 4 and 2 periodicities along the [0001] direction, where = 9.54 Å is the repeat vector in the ideal TI. A crucial property of almost all SDs, either interor intra-QL, is the rupture of the inversion symmetry of the ideal slabs despite the stoichiometry within each QL being preserved, the only exception being inter-QL SFs. As a result, defected slabs become asymmetric and exhibit inequivalent geometries at the top and bottom surfaces. In particular, TBs present opposite orientations with respect to each surface The layered structure facilitates the formation of extended planar defects. Indeed, SFs and TBs, analogous to those here investigated, have been frequently identified in thin films of Bi2Se3, Bi2Te3 and related materials. They are formed either through deformation of the crystal or directly during the growth and annealing process(28)-34 . The energy cost involved in creating different es SDs within a (0001) slab is analyzed in detail in SM #2. After considering both inter- and intra-QL SDs, we consistently found, as expected, a sensibly larger formation 2 energy for the latter, with differences in the 5-20 meV/Å range depending on the precise location of the SD. Inter2 QL TBs show the lowest formation energy, ≤ 3 meV/Å , while, surprisingly, SFs require an energy cost around three times larger. Furthermore, the energy cost for the occurrence of multiple TBs in a thin film is basically the sum of the single TB contributions. Notably, the calculated formation energy of (0001) TBs in Bi2Te3 is similar, 2 2 meV/Å , and agrees well with previously reported va(29),(33) lues. Therefore, based on energetic considerations we predict that (0001) TB boundaries between QLs are the predominant SDs in Bi2X3 materials which is in line with numerous experimental observations of single and multiple TBs in a large diversity of Bi2X3 based ma(28)(29)(33)(34) terials. In the absence of SDs and in (9) agreement with previous calculations, the Bi2Se3 (0001) surface presents a single TSS with helical spin texture spanning the bulk energy gap and with the DP pinned at the Fermi level. Since SDs are non-magnetic, involve zero net charge perturbations and exhibit an almost bulk-like structure, it would be expected that their impact on the TSSs is minor.
ACS Paragon Plus Environment
Page 3 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Figure 2. Electronic band structure for Bi2Se3 semi-infinite surfaces (a) TB4-top, (b) TB4-bottom, (c) TB2-top and (d) TB2-bottom. The maps are obtained from the k-resolved DOS projected on the 3 QLs closest to the surface. An 8 QL thick slab was employed in all cases to construct the surface, so that 4 (7) TBs are included in the TB4 (TB2) case. As a TB is not symmetric in , the top and bottom surfaces give raise to dopings of opposite sign (see Figure 1), with the left (right) column showing p-(n-)doped TSSs. Due to the smaller group velocity of the TSS in the valence band region, the DP shifts are smaller for p-doped surfaces.
Figure 3. (a) Charge density profiles, ∆ρ(z), averaged over the 2D unit cell for a 6 QL thick Bi2Se3 slab after substracting the defect free contribution. (b) Same as (a) but for the associated Hartree potential, ∆VH(z). Red, green and blue lines correspond to defective structures SF4, TB4 and TB2, respectively. The z origin is located at the bottom-surfaces. Vertical dashed lines indicate the two surfaces of the slab. Arrows in (a) indicate the direction of the dipole moment associated to the TBs.The drop in the potential at the vacuum seen in (b) is proportional to the total dipole moment of the system.
However, after analyzing around a hundred different surfaces constructed from slabs of different thicknesses and different interlayer SD arrangements, we have found that large dopings of the TSSs can be achieved. In Figure 2 we display the electronic structure around the DP for the top and bottom surfaces of structures TB4 (a-b) and TB2 (c-d), respectively. Clear shifts in the DPs with respect to the Fermi level can be observed in all of them, with the peculiarity that the sign of the doping can be tuned by inverting the orientation of the TB; top surfaces (a) and (c) present p-type doping while bottom surfaces (b) and (d) n-type. The shifts are sensibly larger for the TB2, indicating that the density of TBs is a key ingredient
in the doping mechanism. Remarkably, SFs (not shown) induce almost negligible DP shifts of less than 10 meV. To elucidate the origin of the doping and its dependence with the type and density of SDs, we examine their electronic properties. In Figure 3 we present 2D averaged profiles along the [0001] direction of the charge density, ∆ρ(z), and its associated Hartree potential, ∆VH(z), after substracting the defect-free contribution for 6 QL thick slabs with SF4, TB4 and TB2 defect arrangements. ∆ρ is nonzero only close to the SDs, where charge oscillations develop. Although at first sight the ∆ρ features seem similar for the TBs and the SFs, closer inspection reveals a marked asymmetry for the former, leading to a considerable dipole moment (in SM #2 we analyze in detail its origin). Differences among SFs and TBs are more clearly seen in the ∆VH(z) profiles. Whereas the formers show up as localized almost symmetric dips in the potential, the latter induce marked discontinuities with upwards jumps along the down-up direction and an overall potential difference across the slab that increases with the TB density.
Figure 4. (a) Dipole moments per unit area, , for Bi2Se3 slabs with different TB arrangements as a function of the slab thickness, . Blue, red and green lines correspond to best fits to the data points (circles) for structures TB2, TB4 and TB6 respectively, according to eq. 1. For the last two we used solid, dashed and dotted lines after groupping the slabs according to the location of the TB closest to the uppermost QL. SOC was not included in these calculations in order to remove the TSS and hence obtain an insulating (dielectric) slab. (b) Doping charge (left axis) and DP shifts in the TSS (right axis) for the top and bottom surfaces constructed from the same slabs as in (a), again as a function of . Horizontal lines at the right show the limiting charges for each polytype derived from their bulk SPs.
Therefore, and in analogy with ferro-electric materi(35) als , the existence of sizeable internal dipoles at the TBs yields a finite SP, , along the [0001] direction. The , across any of continuity of the displacement vector, the two slab’s surfaces dictates that, ∆ = ! " # = $ where %! " is the macroscopic field inside the TI (generally denoted as the depolarization field), σ is the surface charge density allocated at the TSSs and is a unitary vector normal to the surface that points out of the film. In the large thickness limit the depolarization field will vanish and, thus, a direct equivalence between the SP and the TSS doping is expected: (36) = $. Physically, this means that bound charges appearing at the surfaces of Bi2Se3 slabs containing SDs are fully compensated by the free TSS
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
charge. For finite films, however, the surface charge distribution causes a finite depolarization field that softens the SP discontinuity or, in other words, leaves the SP partially uncompensated. As a result, represents an upper limit to the TSS doping σ –which will depend on the slab thickness. Verifying that the polarization discontinuity causes the surface doping requires, on the one hand, the estimation of σ by integrating the TSS’s DOS between the Fermi level and the DP –which is not such a trivial task as shown in SM #4– and, on the other hand, the evaluation of the SP for a given periodic arrangement of the TBs in the crystal. Although in principle the SP is a bulk dielectric property, we have recently shown how it can be accurately determined from slab calculations without the (37) need of performing macroscopic averages . Here, we calculate the SP considering that no free charges are present at the surface –in the ab-initio calculations this condition is enforced by switching off the spin-orbit interactions thus preventing the appearance of the TSSs. In Figure 4 (a) we show the evolution of the slab’s dipole moment per unit area, , as a function of the slab’s thickness, , for the TB2, TB4 and TB6 polytypes. For the last two cases we consider different subsets –green or red solid, dashed and dotted lines– depending on the location, at each , of the TB closest to the uppermost QL. The data points can be accurately fitted by the ex(37) pression : = # & ⁄ ' ) = # &* ⁄ ' ) # +, -+, (eq.1) where is a surface localized contribution arising from the bond rearrangement at the surface, * the net polarization at the crystalline regions, -+, the number of TBs in the slab and +, the dipole moment per unit area associated to a single TB. The theoretically derived value of ' = 24.6 has been used here (see SM #3). * includes the depolarization field counteracting the SP as well as the electric field arising from the h/e doping at the top/bottom surface. It nevertheless attains small values * ≈ 0.2 1023 4⁄3 as can be already inferred from the almost flat segments in the TB4 and TB6 plots. On the contrary, the appearance of a new TB in the slab shows up as a downwards jump of average magnitude 〈 +, 〉 = 3.8 1089 4⁄Å which is hardly dependent on the precise defect configuration. The TB2 slabs, on the other hand, show an almost perfect linear behavior for &) due to the absence of crystalline regions and, hence, the slope directly provides +, . The bulk SP may now be estimated as: = 2 ' & +, ⁄ ), yielding values for our defected slabs of &:;2/4/6/8) = &9.6/4.6/3.0/2.3) 1023 4⁄3 . Notably, they correspond to substantial 12 2 doping charges of the same order (10 e/cm ) as those (38) measured for graphene on SiC polytypes, or oxide 12 14 2 interfaces (10 -10 e/cm ) and even larger than those 10 12 2 in doped semiconductors (10 -10 e/cm ).
Page 4 of 7
that all bottom (top) surfaces show a clear n-(p-)type doping, as dictated by the direction of the spontaneous polarization. Next, the doping levels increase with t and approach a limiting value, which is precisely the SP, associated to each polytype (indicated by horizontal lines in the figure). At the largest thickness we considered (10 QLs), the dopings reach up to 60% of this limit 12 2 (5.4 and -5.910 e/cm for the maximum hole and electron concentrations respectively), while we estimate by linear extrapolation that almost full compensation of the SP is reached at ∼ 18 QLs. In addition, there is a clear resemblance between these curves and the &) ones in Figure 4 (a), specially the ladder type behavior for TB4 and TB6. Again, sudden jumps in the doping are seen whenever a defective QL is added to the slab, whereas if the additional QL is non-defective, it tends to screen the SP causing a slight decrease in the doping. We may therefore conclude that the doping can be tuned via the actual number of TBs and its density, -+, ⁄ , as well as by the proximity of the first TB to the surface. Note that in the reported doped surfaces all TB's dipole moments are aligned –point along the same direction – and, hence, accumulate. Indeed, surfaces and slabs with anti-parallel configurations show no SP due to the cancellation of the dipole moments and, thus, a vanishing doping. Nevertheless, as shown in the SM #2, the anti-parallel TB arrangement results energetically less favorable than the parallel one. Having shown the relationship between the bulk TB derived SP and the surface doping, we conclude by examining the effect of the TBs in thin film geometries. In a finite slab with a single or multiple TBs, the Dirac states located at each side of the film experience energy shifts in opposite directions and equal to those of the corresponding semi-infinite surfaces. This yields a net accumulation of negative and positive charge of the same magnitude at the bottom and top surfaces, respectively. Figure 5 (a) and (b) represent the band dispersion around the DP of two 6 QL thick films, one without defects and the other containing 3 TBs (TB4) The p- and ndoped TSSs exhibit a left-handed spin helicity, equal to that corresponding to the ideal surface. As shown in Figure 5 (c), along the Γ−M direction the spin lies in the surface plane, as required by mirror symmetry, whereas away from this high symmetry direction and correlated with the hexagonal warping a perpendicular component develops, reaching its maximum along Γ−K. The out-ofplane component increases with increasing energy separation from the DP and consequently with increasing electron or hole doping. In addition, the rupture of the inversion symmetry allows for the development of a large Rashba spin-splitting in the bulk-like bands, which is otherwise absent –see for instance the conduction and valence bands in Figure 5 (b).
A summary of the dopings for all the bottom and top surfaces with TB defects is presented in Figure 4 (b), where the TSS charge (left axis) or, equivalently, the DP shifts (right axis in a non-linear scale) is plotted as a function of the slab thickness, . The first observation is
ACS Paragon Plus Environment
Page 5 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
ACKNOWLEDGMENT This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grant Nos. MAT2012-38045-C04-04 and MAT2013-47878-C2-R. We would also like to acknowledge the use of computational resources of CESGA and the i2BASQUE academic network. Figure 5. Band structure of 6 QL thick Bi2Se3 slabs, (a) with no SDs and (b) for the TB4 polytype. In (b), the TSS on the bottom surface (purple) shows n-doping, while the TSS on the top surface (orange) shows p-doping. Note the appearance of a Rashba splitting in the defective case. In (c) the spin texture of the right TSS is shown in reciprocal space for a TB2 system. The in-plane components of the spin are plotted as arrows, while the out of plane Sz is colorcoded as a percentage of the modulus of S (see inset). Spin-momentum lock of the helical TSSs is preserved when (39)(38) SDs are included.
Summarizing, the small density of states derived from the highly dispersive TSSs represents a highly precise gauge for measuring the impact of SDs on 3D topological insulators. By controlling the sequence of stacking defects close to a Bi2X3 (0001) surface, nanostructures with tailored surface charge and spin densities up to 13 2 10 e/cm , can be fabricated without the need of dopants, adsorbates or external fields. Our results provide a simple explanation for the local charging of the Dirac (17) states recently reported in epitaxial (0001) films. The origin of the surface self-doping relies on the spontaneous polarization generated by the lattice defects in the bulk. This mechanism allows to tune the Fermi level to preselected electron or hole conductive regimes. Furthermore, the SP effect persists even for large defect concentrations. Preliminary calculations for defected slabs indicate that the presence of a donor Se vacancy in the slab, despite being a source of additional doping, essentially preserves the difference between the TSS doping at the top and bottom surfaces. Thus, SDs may also be used to reduce undesired carrier densities which may screen the outstanding TSS spin-transport properties. In addition, because of the large Rashba splitting, high-efficient spin manipulation might be realized in thin films. Although we have considered Bi2Se3 as our model system, the results can be generalized to Bi2Te3 and related TI surfaces. Therefore, thin films of Bichalcogenides TI with stacking defects, provide a new playground where exotic excitations such as topological exciton condensates could be probed.
SUPPORTING INFORMATION AVAILABLE Additional information and figures. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION Corresponding Author *email: [email protected] Notes The authors declare no competing financial interest.
REFERENCES (1) Kane, C. L.; Mele, E. J. Phys. Rev. Lett. 2005, 95, 146802. (2) Bernevig, B. A.; Zhang, S.-C. Phys. Rev. Lett. 2006, 96, 106802. (3) Hasan, M. Z.; Kane, C. L. Rev. Mod. Phys. 2010, 82, 3045. (4) Qi, X.-L.; Zhang, S.-C. Rev. Mod. Phys. 2011, 83, 1057–1110. (5) Qi, X.-L.; Hughes, T. L.; Zhang, S.-C. Phys. Rev. B 2008, 78, 195424. (6) Chang, C.-Z. et al. Science 2013, 340, 167–170. (7) Fu, L.; Kane, C. L. Phys. Rev. Lett. 2008, 100, 096407. (8) Seradjeh, B.; Moore, J. E.; Franz, M. Phys. Rev. Lett. 2009, 103, 066402. (9) Zhang, H.; Liu, C.-X.; Qi, X.-L.; Dai, X.; Fang, Z.; Zhang, S.-C. Nature Physics 2009, 5, 438–442. (10) Xia, Y.; Qian, D.; Hsieh, D.; Wray, L.; Pal, A.; Lin, H.; Bansil, A.; Grauer, D.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. Nature Physics 2009, 5, 398–402. (11) Chen, Y.; Analytis, J.; Chu, J.-H.; Liu, Z.; Mo, S.-K.; Qi, X.-L.; Zhang, H.; Lu, D.; Dai, X.; Fang, Z.; Zhang, S.-C.; Fisher, I.; Hussain, Z.; Shen, Z.-X. Science 2009, 325, 178–181. (12) Pesin, D.; MacDonald, A. H. Nature materials 2012, 11, 409–416. (13) Nayak, C.; Simon, S. H.; Stern, A.; Freedman, M.; Das Sarma, S. Rev. Mod. Phys.2008, 80, 1083–1159. (14) Beidenkopf, H.; Roushan, P.; Seo, J.; Gorman, L.; Drozdov, I.; San Hor, Y.; Cava, R.; Yazdani, A. Nature Physics 2011, 7, 939–943. (15) Benia, H. M.; Lin, C.; Kern, K.; Ast, C. R. Phys. Rev. Lett. 2011, 107, 177602. (16) Bianchi, M.; Hatch, R. C.; Mi, J.; Iversen, B. B.; Hofmann, P. Phys. Rev. Lett. 2011,107, 086802. (17) Liu, Y.; Li, Y. Y.; Gilks, D.; Lazarov, V. K.; Weinert, M.; Li, L. Phys. Rev. Lett. 2013,110, 186804. (18) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868. (19) Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D. Journal of Physics: Condensed Matter 2002, 14, 2745. (20) Cerdá, J.; Van Hove, M. A.; Sautet, P.; Salmeron, M. Phys. Rev. B 1997, 56, 15885– 15899. (21) Cuadrado, R.; Cerdá, J. Journal of Physics: Condensed Matter 2012, 24, 086005. (22) Kresse, G.; Furthmüller, J. Computational Materials Science 1996, 6, 15 – 50. (23) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 48, 13115– 13118. (24) Grimme, S. Journal of computational chemistry 2006, 27, 1787–1799. (25) Ortmann, F.; Bechstedt, F.; Schmidt, W. G. Phys. Rev. B 2006, 73, 205101. (26) Rossen, E.; Flipse, C.; Cerdá, J. Phys. Rev. B 2013, 87, 235412. (27) Bengtsson, L. Phys. Rev. B 1999, 59, 12301. (28) Medlin, D.; Yang, N. Journal of electronic materials 2012, 41, 1456–1464. (29) Medlin, D.; Ramasse, Q.; Spataru, C.; Yang, N. Journal of Applied Physics 2010, 108, 043517–043517.
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(30) Li, H.; Wang, Z.; Kan, X.; Guo, X.; He, H.; Wang, Z.; Wang, J.; Wong, T.; Wang, N.; Xie, M. New Journal of Physics 2010, 12, 103038. (31) Jariwala, B.; Shah, D. Journal of Crystal Growth 2011, 318, 1179–1183. (32) Mao-Hai, X.; Xin, G.; Zhong-Jie, X.; Wing-Kin, H. Chinese Physics B 2013, 22,068101. (33) Shin, H. S.; Jeon, S. G.; Yu, J.; Kim, Y.-S.; Park, H. M.; Song, J. Y. Nanoscale 2014, 6, 6158–6165. (34) Tarakina, N. V.; Schreyeck, S.; Luysberg, M.; Grauer, S.; Schumacher, C.; Karczewski, G.; Brunner, K.; Gould, C.; Buhmann, H.; Dunin-Borkowski, R. E.; Molenkamp, L. W. Advanced Materials Interfaces 2014, 1, 1400134.
Page 6 of 7
(35) Hill, N. A. Journal of Physical Chemistry B 2000, 104, 6694–6709. (36) Meyer, B.; Vanderbilt, D. Phys. Rev. B 2001, 63, 205426. (37) Slawinska, J.; Aramberri, H.; Muñoz, M.C.; Cerdá, J. arXiv 2014, 1410.4551. (38) Mammadov, S.; Ristein, J.; Koch, R. J.; Ostler, M.; Raidel, C.; Wanke, M.; Vasiliauskas, R.; Yakimova, R.; Seyller, T. 2D Materials 2014, 1, 035003. (39) Basak, S.; Lin, H.; Wray, L. A.; Xu, S.-Y.; Fu, L.; Hasan, M. Z.; Bansil, A. Phys. Rev. B 2011, 84, 121401.
ACS Paragon Plus Environment
Page 7 of 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
TOC Graphic
Bi2Se3 slab containing a twin boundary (TB) every QL. The TSS on the left surface (purple) is n-doped, while the TSS on the right surface (orange) is p-doped. The spin-momentum lock of the helical TSS is preserved (green arrows represent spins).
ACS Paragon Plus Environment
