Quantum Spin-Valley Hall Kink States: From Concept to Realization
Tong Zhou, Shuguang Cheng, Hua Jiang, Zhongqin Yang, Igor Zutic
QQuantum Spin-Valley Hall Kink States: From Concept to Realization
Tong Zhou, ∗ Shuguang Cheng, Hua Jiang, Zhongqin Yang, and Igor ˇZuti´c † Department of Physics, University at Bu ff alo, State University of New York, Bu ff alo, New York 14260, USA Department of Physics, Northwest University, Xi’an 710069, China School of Physical Science and Technology, Soochow University, Suzhou 215006, China State Key Laboratory of Surface Physics and Key Laboratory of Computational Physical Sciences (MOE) and Department ofPhysics and Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, China (Dated: February 12, 2021)We propose a general and tunable platform to realize high-density arrays of quantum spin-valley Hall kink(QSVHK) states with spin-valley-momentum locking based on a two-dimensional hexagonal topological in-sulator. Through the analysis of Berry curvature and topological charge, the QSVHK states are found to betopologically protected by the valley-inversion and time-reversal symmetries. Remarkably, the conductance ofQSVHK states remains quantized against either nonmagnetic or long-range magnetic disorder, verified by theGreen function calculations. Based on first-principles results, we show that QSVHK states, protected with a gapup to 287 meV, can be realized in bismuthene by alloy engineering, surface functionalization, or electric field,supporting non-volatile applications of spin-valley filters, valves, and waveguides even at room temperature.
Two-dimensional (2D) hexagonal lattices o ff er a versatileplatform to manipulate charge, spin, and valley degrees free-dom and implement di ff erent topological states. While pio-neering predictions for quantum anomalous and quantum spinHall (QSH) e ff ect [1, 2] were guided by graphene-like sys-tems, graphene poses inherent di ffi culties with its weak spin-orbit coupling (SOC) and a gap of only ∆ ∼ µ eV [3].The quest for di ff erent 2D hexagonal monolayers (MLs) witha stronger SOC on one hand reveals, as in transition metaldichalcogenides (TMDs), an improved control of valley-dependent phenomena [4], emulating extensive research inspintronics [5], while on the other hand, as in ML Bi on SiCsubstrate, topological states remain even above the room tem-perature, with a huge topological gap ∼ Ω ( k ) in di ff erent val-leys is responsible for a valley Hall e ff ect, where the carriersin di ff erent valleys turn into opposite directions transverse toan in-plane electric field [7, 13]. A striking example of sucha sign reversal in Ω ( k ) along an internal boundary of a film isrealized in quantum valley Hall kink (QVHK) states [14–26].The resulting 1D topological defect supports counterpropagat-ing 1D chiral electrons, topologically protected by the valley-inversion symmetry [14–18]. The underlying mechanism forthe formation of zero-energy states, expected from the indextheorem [27, 28], shares similarities with many other systemsin condensed mater and particle physics [28–32]. While theproposals for QVHK states mainly focus on bilayer graphene(BLG) systems [19–23], the required sign reversal in Ω ( k ) re-alized by either the random local stacking faults [19, 21], or adual-split-gate structure [20, 22, 23], is challenging to imple-ment to achieve high-density channels. With the required ap-plied electric field, the volatility of QVHK states limits theirenvisioned use in valleytronics. A small and trivial gap ofBLG ∼
20 meV [20] excludes high-temperature applications,
FIG. 1. (a) Schematic of quantum spin-valley Hall kink (QSVHK)states ( A , B ), at the valleys K and K (cid:48) , and quantum spin Hall (QSH)edge states ( C , D ) in a junction of quantum valley Hall (QVH) andQSH insulators. L is the nanoribbon width. The red and blue arrowsdenote the spin-up and spin-down channels. (b) and (c) the schematicof the bands and Berry curvatures (black lines) for QVH and QSHinsulators, distinguished by the relative strength of SOC, λ SO , andstaggered potential, U . In (c) QSH edge states are marked in blue. and QVHK states were limited to 5 K [19–23]. Crucially,disorder easily induces intervalley scattering, preventing theexpected ballistic transport in QVHK states [19, 20].Motivated by these challenges, we propose a robust plat-form to realize high-density arrays of spin-polarized QVHKstates at room temperature based on a ML 2D hexagonal topo-logical insulator, where the QVHK states are simultaneouslythe QSH edge states, formed in the QVH-QSH junction, asshown in Fig. 1. The QVH region is characterized by a quan-tized valley Chern number C V = C K − C K (cid:48) = Z = C K and C K (cid:48) are the Chernnumbers at the K and K (cid:48) valley. The QSH region is describedby Z = Z and C V change the sign, thus both QSH edge and QVHKstates emerge, giving a largely unexplored topological state,we term quantum spin-valley Hall kink (QSVHK) state. Un-like the studied QVHK state, our proposed QSVHK state isnot only protected by the valley-inversion symmetry against a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 2. (a) Bands and wave function distributions ( | Ψ ( x ) | ) for topological states A - D in the QSH-QVH junction with L =
60 unit cells. (b)Bands and schematic of QVH-QSH-QVH junction with pure QSVHK states E - H . (c) Junction conductance versus nonmagnetic Andersondisorder strength W NA at the Fermi level, E F =
0, for QSVHK states in (b) and QVHK states in a QVH ( C V = C V = −
1) junction,where C V is the valley Chern number. (d) Junction conductance versus long-range magnetic disorder strength W LM at E F = W MA ). The parameters U , λ SO for QVH and QSH regionsare taken from the BiAs / SiC and Bi / SiC, respectively. The hopping parameter t , = long-range magnetic disorder (larger than lattice spacing toexclude valley mixing) [35], but also by the time-reversalsymmetry against the nonmagnetic disorder [36], giving riseto robust ballistic spin-valley-momentum locking transport.QSH-QVH junction can be implemented by inducing a stag-gered potential, U , in a part of the 2D topological insulators.When the strength of the intrinsic SOC λ S O is smaller than U ,the QVH state is obtained, while λ S O > U (including U = U can be induced by alloy engineering or by chemical surfacefunctionalization and easily controlled by the electric filed.We first present our basic idea through the analysis of atight-binding model based on p x and p y orbitals, which iswidely used to describe the electronic and topological prop-erties of the MLs with honeycomb lattice including function-alized arsenene [37, 38], antimonene [39–42], bisthumene [6,43, 44], and binary element group-V MLs [45, 46] H = λ S O (cid:88) i c + i σ z ⊗ s z c i + (cid:88) i U i c + i σ ⊗ s c i + (cid:88) i (cid:88) j = , , c + i T δ j c i + δ j + H . C . . (1)Here, c i represents the annihilation operator on site i . σ and s indicate the Pauli matrices acting on orbital and spin spaces,respectively. The second term gives the staggered potentialwith U i = U ( − U ) for the A ( B ) sublattice. The hopping term T δ j = (cid:34) t z (3 − j ) t z j t t (cid:35) ⊗ s , (2)describes the nearest hopping from site i to i + δ j , where z = exp(2 i π/
3) and t / is the hopping coe ffi cient. In the ab-sence of the first two terms in Eq. (1), it is known that twogapless Dirac points exist at the two valleys [43, 44]. Thestaggered potential and intrinsic SOC term open a gap of2 | λ S O − U | at the Dirac points and their competition deter-mines the topology of the system. When λ S O < U , the systemis in a QVH state with Z = Ω ( k ) (cid:44) λ S O > U , the systemhas QSH helical edge states characterized by Z = U > Ω ( k )] is reversed as compared to the QVH state[Fig. 1(c)]. We consider a junction formed by the QVH andQSH insulators, as shown in Fig. 1(a). QSVHK states emergealong the interface, where both Z and Ω ( k ) change the sign.To identify such QSVHK states, we calculate the spec-trum of the QVH-QSH junction along the zigzag direction asshown in Fig. 2(a). In the bulk band gap there are four non-degenerate gapless states, A − D . The helical C and D statesare the common QSH edge states localized at the outer edge ofthe QSH region, verified by their wave function distributionsin Fig. 2(a). The A and B states, at the K and K (cid:48) valleys, showthe QSVHK states localized at the inner interface [Fig. 2(a)].Unlike the QVHK states in the BLG, the QSVHK states hereare fully spin-polarized. Specifically, the kink state A ( B ) at K ( K (cid:48) ) valley has spin-up (down) channel. Such spin-valley-momentum locking supports a perfect spin-valley filter.To better understand the emergence of the helical QSVHKstates, we focus on the low-energy physics of the tight-bindingmodel. Away from the interface, we expand the Hamiltonianaround the K ( K (cid:48) ) point and obtain a continuum model H = (cid:126) v F (cid:16) k x σ x + τ z k y σ y (cid:17) + λ S O s z τ z σ z + U σ z , (3)where ν F is the Fermi velocity, σ , s , and τ are Pauli ma-trix for orbital, spin, and valley degrees of freedom, respec-tively. From the index theorem [27], the number of the valley-dependent gapless kink channels is directly related to thesign change in the bulk topological charges across the inter-face [14, 18, 47]. The spin- and valley-projected topolog-ical charge C s z τ z is calculated by the integration of the spin-dependent Ω ( k ) around each valley [14, 18, 47], where Ω (k) ≡ ∇ k × A ( k ), A ( k ) ≡ (cid:104) u k | i ∇ k | u k (cid:105) is the Berry connection ofthe valence bands, and | u k (cid:105) is the periodic part of the Blocheigenstate. From the continuum model in Eq. (3), we obtain C s z τ z = τ z U − τ z s z λ S O ) . (4)In the QVH region, we get (cid:16) C ↑ K , C ↓ K , C ↑ K (cid:48) , C ↓ K (cid:48) (cid:17) = (0.5, 0.5, -0.5, -0.5), while in the QSH region, (cid:16) C ↑ K , C ↓ K , C ↑ K (cid:48) , C ↓ K (cid:48) (cid:17) = (-0.5,0.5, -0.5, 0.5). The number of the kink modes per spin / valley (cid:16) ν ↑ K , ν ↓ K , ν ↑ K (cid:48) , ν ↓ K (cid:48) (cid:17) is an integer evaluated from the di ff erencebetween the topological charges in two regions [14, 18], i.e., (cid:16) ν ↑ K , ν ↓ K , ν ↑ K (cid:48) , ν ↓ K (cid:48) (cid:17) = (1, 0, 0, -1). It is clear the spin-up topolog-ical charge change the sign at the K valley while spin-downtopological charge change the sign at the K (cid:48) valley, givingrise to the spin-valley polarized QSVHK states. This topolog-ical charge analysis is consistent with our discussion about theenergy spectrum in Fig. 2(a). In QSH-QVH junctions, thereare still QSH edge states along the outer edge. To eliminatethem and realize a pure QSVHK state transport, we propose inFig. 2(b) a QVH-QSH-QVH junction, where the two pairs ofQSVHK states are verified by the calculated bands. Multiplechannels can be expected with more QSH-QVH boundaries.To explore the influence of the disorder on QSVHK states,we calculate the junction conductance, G , using the Landauer-B¨uttiker formula [48] and the Green function method [49–52]in the presence of nonmagnetic Anderson disorder [36, 50]in the energy range (- W NA / W NA / W MA / W MA / W LM / W LM /
2) (see details in Sup-plemental Material [55]), where W NA , W MA , and W LM mea-sure their respective strengths. For comparison, we also cal-culate G ( W NA ) in QVHK states, G ( W LM ) and G ( W MA ) inQSH edge states, shown in Figs. 2(c)-(e). For the QVHKstate with valley-momentum locking, its conductance de-creases as W NA increases, consistent with previous calcula-tions and observations [19, 20], which can be readily under-stood since the nonmagnetic Anderson disorder breaks thevalley-inversion symmetry and leads to the backscattering forthe QVHK states. For QSH edge states with spin-momentumlocking, its conductance decreases with W LM ( W MA ) becausethe time-reversal symmetry is broken by the magnetic disor-der, in agreement with the experiments [56, 57]. In contrast,for QSVHK states with spin-valley-momentum locking pro-tected by both valley-inversion and time-reversal symmetries,its conductance remains quantized against either nonmagneticAnderson disorder or long-range magnetic disorder as shownin Figs. 2(c) and (d). The backscattering in QSVHK statescan only be induced by simultaneously breaking the valley-inversion and time-reversal symmetries, for example by mag- FIG. 3. (a) Top and side views of a ML BiSb or BiAs on a SiC sub-strate. (b)-(d) Bands (black) with SOC and Berry curvatures, Ω ( k )(blue), of the valence bands for the Bi / SiC, BiSb / SiC, and BiAs / SiC.(e)-(g) Calculated bands of the zigzag nanoribbons for the Bi / SiC,BiSb / SiC, and BiAs / SiC, respectively. The fitted tight-binding pa-rameters ( λ SO , U ) for Bi / SiC, BiSb / SiC, and BiAs / SiC are (0.44 eV,0 eV), (0.30 eV, 0.26 eV), and (0.24 eV, 0.38 eV), respectively. netic Anderson disorder as shown in Fig. 2(e). However, theconductance of the QSVHK state still remains almost quan-tized with small W MA (half of the gap) and decreases muchslower as W MA increases compared to that in QSH states. Suchdisorder-dependent conductance calculations further corrobo-rate the robust topological protection of the QSVHK states. Material design . The key factor to achieve QSVHK statesis creating an interface of the QVH and QSH insulators. Sincethere are large number of QSH insulators with hexagonal lat-tice [58], a natural way to obtain such an interface is to de-sign a part of QSH insulator to be a QVH region. To getsuch a QVH region, U > λ S O is required. In graphene-basedsystem, U is induced by the substrate or a vertical electricfield [19, 20]. However, the SOC in graphene is too small tosupport a detectable QSH state [3]. Recently, group-V MLsbismuthene, antimonene, and arsenene on the SiC(0001) sub-strate were predicted to be high-temperature 2D topologicalinsulators [59]. In particular, for Bi / SiC, depicted in Fig. 3(a),a huge nontrivial gap of 0.8 eV has been measured [6], origi-nating from the onsite intrinsic SOC of p x and p y orbitals ofBi atoms [55]. However, due to the presence of the inversionsymmetry, Bi / SiC fails to show valley-dependent e ff ects, asverified by Ω ( k ) = at all k [Fig. 3(b)].To break the inversion symmetry, we propose to use alloyengineering to induce U in bismuthene, a well-established ap-proach to tailor electronic and topological properties [60, 61].Specifically, we propose to grow binary group-V MLs BiSbor BiAs on the SiC substrate. We expect that the change in thebinary composition alters the strength of SOC (growing withthe atomic number Z) and U (growing with a relative di ff er-ence in Z of the two group-V elements), thus favoring eitherQVH or QSH insulators, as shown in Fig 1(a). FabricatedBiSb and BiAs films [37, 38], can be grown on a SiC substrateusing a similar method to that of Bi / SiC [6] or by selectivearea growth and stencil litography [62]. From first-principlescalculations, we see that the BiSb / SiC and BiAs / SiC bandsnear E F are dominated by p x and p y orbitals [55], which canbe accurately described by the Hamiltonian in Eq. (1).Without considering SOC, Bi / SiC has a gapless Dirac bandat the K and K (cid:48) valleys, while the trivial gaps of 0.52 eV and0.76 eV are opened in BiSb / SiC and BiAs / SiC [55], respec-tively. Such gaps, originating from the staggered potential,give U BiSb / SiC = U BiAs / SiC = / SiC and Ω ( k ) (cid:44) ,shown in Fig. 3(c), giving a QSH insulator with Z = ff erentsituation for BiAs / SiC. Due to U > λ S O , a gap of 287 meV at K and K (cid:48) is trivial, with Z = / SiC, the sign reversal of Ω ( k )for BiAs / SiC gives the desired QVH phase.The resulting QVH-QSH insulator junction, shown inFig. 1, then can be realized combining BiAs / SiC (QVH) withBi / SiC (QSH) or BiSb / SiC (QSH) systems, to host the emerg-ing QSVHK states along their interface. Alternatively, tosimplify the fabrication and yield QSH states with an evenlarger nontrivial gap, BiAs-Bi / SiC junction is desirable, wherethe verified QSVHK states are shown in Fig. 2(a). In thisanalysis we excluded Rashba SOC [5], since its strength ismuch smaller than λ S O and U , and the influence is negligi-ble in the BiAs-Bi / SiC system [55]. The BiAs-Bi / SiC junc-tion provides a robust platform for QSVHK states, protectedby a global gap of 287 meV, which is ∼
14 times larger thanin BLG [20], supporting ballistic transport at high tempera-tures. Even multiple QSH-QVH boundaries can be flexiblycreated by spatially-selective deposition [62, 63], i.e., by al-ternating depositing Bi and BiAs arrays on SiC substrate, en-abling transport of high-density channels. Unlike in BLG, theQSVHK states in bismuthene-based system are spin polarizedand require no external field. This o ff ers non-volatility in un-explored applications coupling spin and valley, going beyondlow-temperature BLG realization of valley filter, valve, or awaveguide [22]. For example, QSVHK states support fullyspin-polarized quantum valley currents, making spin-valleyfilters, valves, and waveguides possible, or extend the fuction-alities for spin interconnects [64–66].Another way to realize QSVHK states in bismuthene sys-tem is chemical surface functionalization, widely used tomodify electronic structure and topological states of the 2Dmaterials [67]. Particularly, hydrogenation and halogenationhave been a powerful tool to induce a large-gap QSH statesin group-IV and V MLs [43, 68]. Based on first-principlescalculations, we show in Fig. 4 that the λ S O and U in MLsBiAs can be tuned by the chemical surface functionalization,giving either a QSH or a QVH phase. The structure of thehydrogenated (BiAsH ) and halogenated (BiAsI ) BiAs MLs FIG. 4. (a) Top and side views of the MLs BiAsH or BiAsI . (b)and (c) Bands (black) with Berry curvature (blue) for MLs BiAsH and BiAsI . (d) Electric-field dependent gap in a ML BiAsI . are shown in Fig. 4(a). From the calculated electronic struc-ture and Ω ( k ) in Figs. 4(b) and (c), we see the desired di ff er-ences between MLs of BiAsH and BiAsI . While the firstis a QVH insulator with a trivial gap of 26 meV, Z =
0, and Ω ( k ) (cid:44) at K and K (cid:48) , the second BiAsI is a QSH insulatorwith a nontrivial gap of 49 meV, Z =
1, and reversed Ω ( k ).When such two MLs form a junction, as in Fig. 1, the QSVHKstates can emerge along the interface. With hydrogenated andhalogenated graphene routinely fabricated [69, 70], by usinga similar method, the BiAsH -BiAsI junction could likely beobtained from ML BiAs to support QSVHK states. Since theelectric field, ε , is used to directly change U in 2D materi-als [20, 22], we also explore the possibility of ε -controlledQSVHK states. Figure 4(d) shows that for ε applied alongthe z direction in ML BiAsI , U is increased and the gap isclosed when ε = . V / Å, the value achievable with ion-liquidgating [71, 72]. Such a gap closing indicates a topologicaltransition from QSH to QVH states. Thus, the electric fieldcan also be used to generate and control the QSVHK states.With experimental realization of QSVHK states it would bepossible to verify their inherent robustness of quantized con-ductance of spin-polarized channels, in contrast to QSH insu-lators, where this quantization is fragile even at He tempera-tures [56, 57]. Furthermore, QSVHK states o ff er an intriguingopportunity to study their manifestations of topological super-conductivity through proximity e ff ects [57, 65, 73] and test therelated role of disorder [74, 75].We thank Prof. Fan Zhang for fruitful discussions. Thiswork is supported by the U.S. DOE, O ffi ce of Science BES,Award No. DE-SC0004890 (T. Z. and I. ˇZ.), NSFC underGrant Nos. 11874298 (S. C.), 11822407 (H. J.), 11874117 (Z.Y.), and 11574051 (Z. Y.), and the UB Center for Computa-tional Research. ∗ tzhou8@bu ff alo.edu † zigor@bu ff alo.edu[1] F. D. M. 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