Topological phase transition with nanoscale inhomogeneity in (Bi 1−x In x ) 2 Se 3
Wenhan Zhang, M. X. Chen, Jixia Dai, Xueyun Wang, Zhicheng Zhong, Sang-Wook Cheong, Weida Wu
TTopological phase transition with nanoscaleinhomogeneity in (Bi − x In x ) Se Wenhan Zhang, † M. X. Chen, ‡ Jixia Dai, † Xueyun Wang, † , ¶ Zhicheng Zhong, § Sang-Wook Cheong, † , (cid:107) and Weida Wu ∗ , † † Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey08854, USA ‡ College of Physics and Information Science, Hunan Normal university, Changsha, Hunan410081, China ¶ School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China § Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of MaterialsTechnology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China (cid:107)
Rutgers Center for Emergent Materials, Rutgers University, Piscataway, New Jersey08854, USA
E-mail: [email protected]
Phone: +1 848-445-8751. Fax: +1 732-445-43431 a r X i v : . [ c ond - m a t . s t r- e l ] M a r bstract Topological insulators are a class of band insulators with non-trivial topology, aresult of band inversion due to the strong spin-orbit coupling. The transition be-tween topological and normal insulator can be realized by tuning the spin-orbit cou-pling strength, and has been observed experimentally. However, the impact of chem-ical disorders on the topological phase transition was not addressed in previous stud-ies. Herein, we report a systematic scanning tunneling microscopy/spectroscopy andfirst-principles study of the topological phase transition in single crystals of In dopedBi Se . Surprisingly, no band gap closure was observed across the transition. Further-more, our spectroscopic-imaging results reveal that In defects are extremely effective“suppressors” of the band inversion, which leads to microscopic phase separation oftopological-insulator-like and normal-insulator-like nano regions across the “transition”.The observed topological electronic inhomogeneity demonstrates the significant impactof chemical disorders in topological materials, shedding new light on the fundamentalunderstanding of topological phase transition. Keywords:
Topological phase transition, nanoscale inhomogeneity, In defects, STM, first-principles calculation 2isorders are inevitably present in any functional materials or physical system. Often,they have significant impact on many areas of physics, including condensed-matter physics, photonics, and cold atoms. E.g., quench disorders induce both frustration and random-ness in doped magnets, the key ingredients of spin glass physics. Quenched disorders dueto chemical doping are associated with the observed nanoscale inhomogeneity in high T c cuprates and CMR manganites. Strong disorders can also localize itinerant electrons, which is crucial for many appealing phenomena such as metal insulator transition and quan-tum Hall effects. However, the realization of quantum Hall effect requires high magneticfield, a hurdle for its broader applications. The quest for quantum Hall state without externalmagnetic field leads to development of quantum anomalous Hall, quantum spin Hall, which eventually lead to the birth of 3D topological insulator (TI). In 3D TI, the non-trivial topology of band structure is a result of band inversion dueto strong spin-orbital coupling (SOC).
The change of topology class at the interfacebetween topological and normal insulators ensures the existence of metallic Dirac surfacestates.
Similarly, the topological distinction also enforces a zero band gap Dirac semi-metal state for spatially uniform topological phase transition (TPT). In theory, TPT can beinduced by gradual change of average SOC strength via chemical doping, and has been ex-perimentally established in TlBi(S − x Se x ) and (Bi − x In x ) Se via bulk measurements. Because of topological stability, the band inversion is robust against weak disorders. The im-pact of strong disorders is quite rich. On one hand, strong lattice disorder can drive a TI intoa normal insulator (NI). On the other hand, disorders could transform a metal with strongSOC into a topological Anderson insulator, though more elaborate theoretical analysissuggests that it is adiabatically connected to clean TI. Although TPT has been studiedin few systems, the impact of disorders due to chemical doping has not been addressed.Furthermore, Lou et al reported a mysterious sudden band gap closure across TPT in Indoped Bi Se using angle resolved photoemission spectroscopy (ARPES). In this letter, wereport systematic high-resolution scanning tunneling microscopy/spectroscopy (STM/STS)3tudies on single crystals (Bi − x In x ) Se , where x varies from 0.2% to 10.8%. To our sur-prise, no band gap closing and reopening was observed across the TPT ( x c ≈ i.e. suppressingthe topological surface states (TSS) and increasing band gap. The characteristic length ofsuppression is comparable with the decay length of TSS in Bi Se ( ∼ − x In x ) Se with various In concentrations ( x ) weregrown by self-flux method. An Omicron LT-STM with base pressure of 1 × − mbar wasused for STM/STS measurements. Electrochemically etched W tips were characterized onclean single crystal Au(111) surface before STM measurement. Samples were cleaved inUHV at room temperature then immediately loaded into the cold STM head. All STM datawere taken at 4.8 K. The differential conductance ( dI/dV ) measurements were performedwith standard lock-in technique with modulation frequency f = 455 Hz and amplitude V mod = 5 ∼ mV.The density functional theory (DFT) calculations were carried out using the Vienna abinitio Simulation Package.
A slab in 3 × ∼ and pseudopotentials were constructed by the projector augmented wavemethod. The 2D Brillouin zone is sampled by a 4 × × Local density of states4LDOS) are calculated by the layer projection method, which integrates wave functions inspatial windows in vacuum. d I / d V ( n S ) -0.6 -0.4 -0.2 0.0 0.2 0.4 V B (V) x = 10.8% x = 9.0% x = 6.1% x = 4.8% x = 2.1% x = 0.2%60pm d I / d V ( n S ) -0.3 -0.2 -0.1 0.0 V B (V) x = 10.8% x = 9.0% x = 4.8% x = 2.1% x = 0.2% x = 6.1%80pm -0.4-0.20.00.2 E ( e V ) x (%)0.550.500.450.40 G a p ( e V ) CBMVBM D P x C (a) (d) (e) (f)(g)(b)(c) (Bi In x ) Se c axisSe1InSe2Bi 0.2%9.0% Figure 1: (a) Schematic of crystal structure of 1 QL of (Bi − x In x ) Se . Se, Bi and In atomsare denoted by green/yellow, blue and red spheres, respectively. (b)-(c) STM topographicimages of x = 0 . % and 9.0 %. Tunneling condition: − . V, 1 nA. (d) dI/dV spectraof samples with different x (offset for clarity). Black lines indicate the estimation of bandedges. (e) Zoom-in dI/dV spectra inside the band gap of (d). Dashed lines indicate the zeroreference point and vertical bars mark the Dirac points. (f) VBM (blue), CBM (yellow) andDirac point (violet) vs. x . The size of Dirac point symbols represent the spectra weight atDirac point. (g) Estimated gap size at various x showing a gradual increase of average bandgaps. x c : the critical point of TPT.Figure 1(a) shows the side view schematic of one quintuple layer (QL) of (Bi − x In x ) Se . In atoms (red) preferentially occupy the Bi sites (blue). Atomically resolved topographic im-ages were obtained in all the samples. The representative STM images of x = 0 . % and9.0 % are shown in Fig. 1(b) and 1(c) (See section 1 in Supporting Information for othersamples). In atoms were observed in either the second or fourth atomic layer from surface.Fig. 1(d) shows the average dI/dV spectra ( − . The reference levels are marked by black dashedlines. Tunneling spectrum dI/dV is proportional to the LDOS. Due to the presence of TSS,5he LDOS within the band gap is not zero. Fig. 1(e) shows the zoom-in dI/dV spectra in-side the band gap. The Dirac point and the linear LDOS of TSS are clearly resolved in the dI/dV spectra for x ≤ x > . , indicating that the criticalIn concentration ( x c ) of TPT is ∼ Interestingly, the spectral weight of Dirac point (indicated by size of the sym-bol) gradually decreases as x increases. Although it is not straightforward to determine theband gap ∆ using STS in the presence of TSS, the slope change of dI/dV curves near bandedges provides a reasonable criterion for estimation of conduction band minimum (CBM)and valence band maximum (VBM). The band edges are defined by the intercepts with thehorizontal axis of the best linearly fitted dI/dV near band edges, as shown with the blacksolid lines in Fig. 1(d) (see detailed description in Supporting Information). Similar resultswere obtained using a different criterion ( i.e. “kinks” in the d I/dV curves, see SupportingInformation), corroborating the aforementioned method. Fig. 1(f) shows the the estimatedCBM and VBM with the Dirac energy E D as functions of x . The CBM remains almostunchanged as x increases while the VBM gradually shifts down. As shown in Fig. 1(g), thisresults in a monotonic increase of band gap toward that of In Se ( ∆ ∼ . eV). Such be-havior is in apparent conflict with the aforementioned picture of band closing and reopening,but is consistent with recent ARPES studies. Most of previous TPT studies assume a spa-tially uniform change of SOC with chemical doping. In reality, chemical doping inevitablyintroduces spatial inhomogeneity ( e.g. , SOC), so TPT might not happen in a spatially uni-form manner. To illustrate this, we performed systematic nanoscale spectroscopic mappingacross the TPT.Fig. 2 shows the spectroscopy mapping results on three representative samples acrossthe TPT ( . , . and . ). Fig. 2(a-c) show the topographic images where STS datawere collected. In defects distribute randomly in the crystals. The dI/dV maps of in-gapstates (presumably the TSS) at the same locations are shown in Fig. 2(d-f). The dI/dV maps of x ≤ . are taken at E = E D + 0 . eV to enhance the contrast. Note that the6 .1 nS0.02 0.1 nS0 0.12 nS043210 0.2-0.2 V B (V)0-0.4 43210 -0.4 0.0 V B (V)-0.22.52.01.51.00.50.0 d I / d V ( n S ) V B (V)0-0.4 TSS-intense TSS-weak -0.8-0.6-0.4-0.2 X . C o rr . x (%) dI/dV & In Density (2nd Layer) (a) (b) (c)(f)(e)(d) (i)(h)(g) (j)
Figure 2: (a-c) Topographic images of x = 2 . , . and . (Tunneling condition: − . V, 1 nA.). Scale bars are 5 nm. (d-f) dI/dV maps in the same area of (a-c). When x ≤ x C , E = E D + 0 . eV; for x > x C , E is inside the band gap, as indicated in Fig. 1(e).Red (green) arrows mark typical TSS-weak (intense) regions. (g-i) average dI/dV spectraof selective regions: Red (green) is from TSS-weak (intense) area. (j) x dependence of crosscorrelations between dI/dV ( r ) and local density n ( r ) of In in the second atomic layer.inhomogeneity of dI/dV intensity is robust within the band gap. For x > x C , the energy ofthe in-gap states (likely residue TSS) is marked by the arrow in Fig. 1(e). Clearly, the TSSis spatially inhomogeneous at nanometer scale. The average dI/dV spectra of regions withweak (red arrows) and intense (green arrows) spectro-weight of TSS are shown in Fig. 2(g-i).(See Supporting Information for details) In the TSS-intense area, the in-gap LDOS persistseven above x C samples. In contrast, the in-gap LDOS of TSS-weak area decreases rapidly tozero as x rises. The linear dispersion of TSS is visible only when x < x C . More interestingly,the TSS anti-correlates with the local In density n ( r ) (counting only those in second layer)as shown by the negative cross correlation coefficients ( X.Corr. ) in Fig. 2(j). Note that it7s practically impossible to count In defects at deeper layers. The observed anti-correlationsindicate that In defects are very effective “suppressor” of topological band inversion. Thus,it is imperative to reveal their impacts individual In defects on local topological properties. -0.4 -0.20.00.2 V B i a s ( V ) -1 -1 In -1 -1 Dimer N o r m d I / d V d (nm) d I / d V ( n S ) d (nm) L DO S ( a . u . ) (a) (b) (c) In Dimer (d) (e)(f) (g)
CBM
VBM E D Top View d (nm) d (nm) Figure 3: (a) Topographic image of a single In and an In dimer on the second layer. Inset:simulated STM image of In defect by DFT. (b) STS map at E D in the same area as (a). (c)The top view of atomic structure of the top 2 atomic layers: Se (green), Bi (blue) and In(red). (d,e) dI/dV intensity plots along the lines across single In (red) and In Dimer (blue).Tunneling conditions: − . V, 0.5 nA. (f) Suppression of TSS at E D by In defects: blue, dI/dV line profile across a In dimer; red, dI/dV line profile across a single In defect; green:calculated dI/dV of a single In defect. (g) Normalized Gaussian fitted of the dI/dV spectrain (f): blue for In dimer; red for single In defects; purple for simulated overlapping of dI/dV suppression by two single In defects.Fig. 3(a) shows the typical topographic image of a single In defect and an In dimer withnearest neighboring In defects on the . sample. Their configurations are illustrated inthe schematic in Fig. 3(c). Fig. 3(b) shows a dI/dV map measured at E D in the same area.Evidently the much lower spectro-weight on top of the In sites demonstrates that the TSS arestrongly “suppressed” by In defects. This is further illustrated by the 2D spectral map (biasvs. displacement) of a dI/dV line profile across the center of a single In defect (red arrow8n Fig. 3(b)) shown in Fig. 3(d). In addition, there is a slight increase of local band gap onthe In defect, illustrated by the band edges (dash lines). The dI/dV line profile at E D (redcurves in Fig. 3(f)) shows a bell-shape suppression ( ∼ ξ ≈ . ± . nm, indicating the influence range of single In defects extends toapproximately a ( a ≈ . nm is the in-plane lattice constant). Interestingly, the FWHM ξ is comparable with the decay length of TSS, ξ S = ¯ hv F ∆ ≈ . nm, where v F ≈ × m/s isthe Fermi velocity of Dirac surface states and ∆ ≈ . eV is the band gap. The significantsuppression of TSS indicates that a single In defect can be approximated as a point NIembedded in the TI matrix.More interestingly, the neighboring of In defects significantly enhances the suppressionof local topological band inversion. Fig. 3(e) displays the 2D spectral map of dI/dV spectrataken along the line across an In dimer (blue arrow in Fig. 3(b)). In addition to the furtherenhancement of local band gap, the suppression of TSS on In dimer is much stronger thanthat on single In defect, and even stronger than a simple superposition of the suppressionfrom two independent In defects. As shown in Fig. 3(g), the simple overlapping of two singleIn defects suppress the spectro-weight to 50%, while the observed spectro-weight of In dimeris reduced to ∼ . Systematic studies of In dimers with different spacing indicates thatsuch enhancement persists for In dimers with two In are separated by a , consistent withthe influence range ξ of single In defects (see Supporting Information).DFT calculations were carried out to corroborate STM observation. The inset of Fig. 3(a)shows a simulated STM image of single In defect, in good agreement with STM results.Furthermore, the simulated LDOS of TSS near In defects shows a similar suppression (greencurve in Fig. 3(f)), which is also in good agreement with experimental results. Because theTSS are protected by the band topology (the surface-bulk correspondence), they cannotbe “annihilated” without changing the band topology. Thus, the local suppression of TSSspectro-weight likely comes from an effective increase of the tunneling barrier width. BecauseIn defects convert nano-regions around them to NI, the interface between TI and NI (vacuum)9s shifted slightly into the bulk, effectively increasing the tunneling barrier width and thusreducing the tunneling spectro-weight of TSS. G a p s i ze ( e V ) x (%)TSS-intenseTSS-weakavg. 1.00.80.60.4 X . C o rr . x (%)0.65eV0.3 0.6eV0.40.65eV0.3 (a) (b) (c)(d) (e) Gap & In density
Figure 4: (a-c) Local band gap maps ∆( r ) of the same area as Fig. 1(a-c). (d) Average bandgaps of the whole (black), TSS-intense (green) and TSS-weak area (red) as functions of x .(e) x dependence of the cross correlation coefficients between local In density n ( r ) and bandgap ∆( r ) .The observation of enhanced band gap ∆ on In defects suggest a positive correlationbetween local In density and local band gap ∆( r ) (extracted from spectroscopy maps usingthe method mentioned in Fig. 1). Similar to the TSS, ∆( r ) is also spatially inhomogeneouson nanometer scale, as shown in Fig. 4(a-c) (see complete data set in Support Information).Fig. 4(d) shows the x dependence of average band gap (cid:104) ∆ (cid:105) of selected regions. ∆( r ) in theTSS-weak area (red arrows) is larger than that in the TSS-intense area (green arrows). (cid:104) ∆ (cid:105) of TSS-weak regions increases rapidly while that of TSS-intense area changes little. Thus,the rise of (cid:104) ∆ (cid:105) of the whole area (black curve) is mainly due to the increasing areal fractionof TSS-weak regions. Note that (cid:104) ∆ (cid:105) of the whole area agrees well with that extracted fromspatially averaged dI/dV spectra in Fig. 1(d) (see Supporting Information), corroboratingthe validity of the band edge estimation method. Furthermore, ∆( r ) is correlated with localIn density n ( r ) , as shown by the positive cross correlation coefficients ( X.Corr. ) in Fig. 4(e).The observed nanoscale electronic inhomogeneity of TSS and band gap in In doped Bi Se As x increases but < x c , more regions with higher local In density formNNRs with larger band gaps in the matrix of topological regions which remain topological.For x > x c , the topological regions form nano-size bubbles, denoted as topological nano-regions (TNRs) while the normal regions form the matrix. So the material is effectively anormal insulator (the right panel in Fig. 5). This phenomenological scenario qualitativelyexplains the nanoscale electronic inhomogeneity observed in our STM measurements. Furtherstudies by other experimental techniques may help to better understand the unconventionalTPT in In doped Bi Se . x c In%In
TINI
TINI
Side viewSS
Figure 5: Cartoon of the TPT due to nanoscale electronic inhomogeneity. The top andbottom are the boundaries of TI and vacuum. Left: In-dilute; Middle: underdoped, wheretopological regions dominate; Right: over-doped, where normal regions dominate.It is worth noting that this scenario does not contradict the band-closure scenario of TPT,which assumes spatially uniform SOC so the band gap must close at x c . The assumptionof spatially uniform SOC is invalid for (Bi − x In x ) Se . Prior DFT studies suggest thatIn 5 s orbitals are very effective on suprressing SOC, thus In defects can revert the bandinversion locally. Therefore, In doping would inevitably introduce spatial SOC disorders,likely resulting in a mixture of TNRs and NNRs, a proliferation of TSS inside the bulkcrystal, and a gradual increase of average band gap across the TPT. The proliferation of11SS inside the bulk crystal is also consistent with the large enhancement of optical absorptionaround x c . Note that the inhomogeneous TPT scenario present here is different from thetheoretical proposal of bypassing band gap closure via symmetry broken states. To conclude, our results provide compelling microscopic evidence of an inhomogeneousTPT in (Bi − x In x ) Se , which is driven by nanoscale mixture of NNRs and TNRs. As shownby our systematic STM and STS studies, the inhomogeneity of both TSS and local band gaporiginates from very effective suppression of local SOC (and band topology) by In defects,resulting in local NNRs. Our results demonstrate that strong disorders can play a significantrole in TPT, which is difficult to capture in spatially average measurements. The directobservation of nanoscale inhomogeneous TPT will motivate further studies on the impact ofdisorders in the topological materials and the associated quantum phase transitions. Acknowledgement
We are grateful to J. Liu, D. Vanvderbilt, P. Armitage and S. Oh for helpful discussions.The STM work was supported by NSF under Grand No. DMR-1506618. The synthesiswork was supported by the NSF under Grant No. DMR-1629059. The theoretical work issupported by the National Natural Science Foundation of China (grant No. 11774084). Z.Zhong acknowledges financial support by CAS Pioneer Hundred Talents Program.
Supporting Information Available
The following files are available free of charge. High-resolution Topographic Images of(Bi − x In x ) Se .Simulation of the Evolution of DOS with Band Closure.Estimation of Band Edges via Linear Curve fitting on Average dI/dV Spectrum.Estimated Band Gap via d I/dV Spectra.Definition of TSS-weak and TSS-intense Area and the Corresponding dI/dV
Spectra.12nticorrelation between the Spectral Weight of the Topological Surface States and Local InDensity n ( r ) .Suppression of Surface States by Single In Defect and In Dimers.Comparison of Estimated Band Edges from Averaged dI/dV Spectra and Averaged BandEdges from dI/dV
Maps.Band Structure Evolution across the Topological Phase Transition.
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