From a Normal Insulator to a Topological Insulator in Plumbene

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be 8.5. SOC is included in all the ﬁrst-principles calculations unless stated otherwise.

C. Tight-binding model

In most of the previous works about 2D group-IV materials, only the NN hopping is considered in tight-binding mod- els, because the itineracy of electrons is bad and they focus on the band structure near the Fermi level6,26,34–36. There is no definitive strategy to fit the bands. In this study, both the NN and next-nearest-neighbor (NNN) hoppings are intro- duced into the total Hamiltonian so that the two bands above E can be fitted better. Three NN and six NNN translation FIG. 1: (Color online) Lattice structure of the buckled plumbene F vectors are from (a) the top view and (b) the side view. (c) The angle between the Pb-Pb bond and the z direction is defined as φ. (d) The first Bril- a √3 1 d1 = φ , louin zone with hight-symmetry points. Green and blue spheres cor- √3 2 2 cot a √3 1 respond to A and B sublattices, respectively. a1 and a1 are primitive d2 = cot φ , (1) √3 − 2 2 vectors. b1 and b1 are reciprocal lattice vectors. d = a 0 1 cot φ , 3 √3 − we present two rich phase diagrams and a theoretical design v = a 1 √3 , v = a 1 0 0 , for a four-state spin-valley filter is proposed; substrate effects 1 2 2 0 2 v = a 1 √3 , v = a 1 √3 , (2) on plumbene are also discussed. Finally, we conclude this 3 2 2 0 4 2 2 0 − 1− √3− work with a brief summary in Sec. IV. v5 = a 1 0 0 , v6 = a 0 , − − 2 2 where a is defined as the NNN distance. The total Hamilto- II. COMPUTATIONAL DETAILS nian read

′ ′ A. Crystal structure H = H00 + H0 + HA0 + HB0 + Hso, (3) where the five terms correspond to on-site energy, NN hop- Plumbene has a buckled honeycomb structure, as shown in ping, NNN hopping in A and B sublattices, and effective on- Fig. 1. The lattice constant a = 4.93 Å and the nearest- 0 site SOC interaction, respectively. They reads neighbor (NN) Pb-Pb distance d = 3.00 Å are obtained through structural optimization, which agree with the previous H = ε c† c , (4) calculations26. Thespace groupis P3¯m1. Theangle φ, defined 00 X α Liασ Liασ as being between the Pb-Pb bond and the z direction in Fig. 1 L,i,α,σ (c), is 108.34◦. In comparison to graphene with φ = 90◦, the larger Pb-Pb bond length weakens the π π overlap. Mean- H = t c† c − 0 ABδαα′ A,i+δ,ασ Biα′σ σ π i,δ,α,αP ,σ while, the buckling can enhance the overlap between and ′ (5) + orbitals and stabilizes the system. The same mechanism also tBAηα′αc† + cAiασ, P B,i η,α′σ applies for other 2D materials in group IV which share the i,η,α,α′,σ similar buckled configuration.

H′ = t c† c , A0 X AAvαα′ A,i+v,ασ Aiα′σ (6) B. First-principles calculations i,v,σ

In this paper, our first-principles calculations for electronic ξ0 ξ0 ξ0 i 2 c†Aiz cAiy i 2 c†Aiy cAiz + i 2 c†Aiz cAiy  ↑ ↓ − ↓ ↑ ↓ ↑  structures of plumbene were carried out using the WIEN2K  ξ0 + ξ0 ξ0  29  i 2 c†Aiy cAiz i 2 c†Aiy cAix i 2 c†Aix cAiy  package with a full-potential augmented plane wave , based  − ↑ ↓ ↑ ↑ − ↑ ↑   ξ0 + ξ0 ξ0  on the Perdew-Burke-Ernzerhof generalized gradient approx-  i 2 c†Aiy cAix i 2 c†Aix cAiy 2 c†Aiz cAix   − ↓ ↓ ↓ ↓ − ↑ ↓  30  ξ0 ξ0 ξ0  imation (PBE-GGA) . Besides, the energy gaps of plumbene  c† cAiz + c† cAix + c† cAiz  H =  2 Aix ↑ 2 Aiz ↑ 2 Aix ↓  so X  − ξ0 ↓ ξ0 ↓ ξ0 ↑  have also been investigated with the local density approxima-  +i c† cBiy i c† cBiz + i c† cBiy  31 32 i  2 Biz 2 Biy 2 Biz   ξ ↑ ↓ − ξ ↓ ↑ ξ ↓ ↑  tion (LDA) , the Wu-Cohen GGA (WC-GGA) and a com-  0 + 0 0   i 2 c†Biy cBiz i 2 c†Biy cBix i 2 c†Bix cBiy  bination of the modified Becke-Johnson exchange potential  − ↑ ↓ ↑ ↑ − ↑ ↑   ξ0 + ξ0 ξ0  and the PBE-GGA correlation potential (mBJ-PBE)33. The  i 2 c†Biy cBix i 2 c†Bix cBiy 2 c†Biz cBix   − ↓ ↓ ↓ ↓ − ↑ ↓   ξ0 ξ0 ξ0  number of k points is 10000 in the Brillouin zone. The muffin-  2 c†Bix cBiz + 2 c†Biz cBix + 2 c†Bix cBiz   − ↓ ↑ ↓ ↑ ↑ ↓  tin radius is 2.5 Bohr radius for Pb and RmtKmax is chosen to (7) 3

where c†Liασ and cLiασ represent the electron creation and an- nihilation operators at site i of sublattice L for electrons with spin σ and orbital α, respectively. δ and η correspond to NN vectors of the honeycomb lattice and η = δ. tABδαα − ′ is the hopping integral along δ from orbital α′ of sublattice B to orbital α of sublattice A and is given by the Slater- Koster formula37. In Table I the eight Slater-Koster parameters Vssσ, Vspσ, Vppσ, Vppπ, V′ssσ, V′spσ, V′ppσ, V′ppπ correspond to the σ and π bonds formed by 6s and 6p orbitals of NN and NNN sites. The other three fitted parameters εs, εp and ξ0 are the energy levels of s and p orbitals and effective on-site SOC strength, respectively. For HSO the effective on-site SOC coefficients among atomic orbitals are determined by following 6 ′ earlier theoretical studies . HB0 can be obtained by replacing the index A with B in Eq. (6).

TABLE I: The NN and NNN hopping integrals between s and p orbitals are respectively listed in the left and right columns and are considered as functions of the direction cosine l (l′), m (m′) and n (n′) of the vector d (v). Other hopping integrals are found by permuting indices. NN NNN

tss Vssσ V′ssσ tsx lVspσ l′V′spσ FIG. 2: (Color online) (a) Band structures of plumbene obtained txs lVspσ l′V′spσ from the first-principles (thin blue lines) and tight-binding (thick red 2 − 2 2 − 2 t l V + (1 l )V l′ V′ + (1 l′ )V′ lines) calculations. Two gaps around E = 0 and 1.7 eV are marked xx ppσ − ppπ ppσ − ppπ t lm(V V ) l′m′(V′ V′ ) by G1 and G2, respectively. (b) Density of states projected into Pb-6s xy ppσ − ppπ ppσ − ppπ t mn(V V ) m′n′(V′ V′ ) and 6p orbitals. Wannier centers with (c) EF inside the gap G1 (the yz ppσ − ppπ ppσ − ppπ electron-doping concentration n = 0) and (d) EF inside the gap G2 (n = 2). The red dotted and green dashed lines are the evolution and reference lines, respectively. When external perpendicular electric and magnetic fields (Ez and Bz) are applied, the corresponding Hamiltonians can be respectively expressed as follows, and exhibits a normal insulating behavior17,24. Besides the gap = + at EF , we find another gap ( 200 meV) between the conduc- HE Ez zmcmiασcmiασ, (8) ∼ m,Pi,α,σ tion band and higher energy bands. The two gaps are denoted as G1 and G2 in Fig. 2 (a).

= + + HB Bz cmiα cmiα cmiα cmiα , (9) − mP,i,α ↑ ↑ − ↓ ↓ TABLE II: The parameters are ﬁtted to the ﬁrst-principles calcula- where z is the z coordinate of the sublattice m. m tions in the tight-binding model in units of eV.

εs εp Vssσ Vspσ Vppσ Vppπ III. RESULTS AND DISCUSSION 7.009 1.388 0.879 1.326 1.990 0.676 − − −

A. Electronic structures & topological characteristics V′ssσ V′spσ V′ppσ V′ppπ ξ0 0.015 0.109 0.253 0.103 0.800 Fig. 2 (a) and (b) show the band structure and density of − − − states (DOS) of plumbene from first-principles calculations, respectively. The states around the Fermi level are domi- nantly contributed by Pb-6p orbitals. Three isolated bands are formed near the Fermi level and an energy gap of 400meV ∼ is opened at EF. The gap is much larger than that of the other In order to study the properties of plumbene in detail, 3 group-IV 2D materials, which are 10− , 8, 23, 72 meV from including its topological characteristics and microscopic re- theoretical calculations for graphene, silicene, germanene and sponses to external fields, we have employed a sixteen-band stanene, respectively3,24,34. Although plumbene has stronger tight-binding model with NN and NNN hoppings to fit the SOC interaction, leading to a larger energy gap, it is totally band structure according to the first-principles result (see Ta- different from the other four topologically nontrivial systems ble II). The corresponding band structures are shown in Fig. 4

plumbene with electron doping can be topologically nontrivial, we have studied plumbene nanoribbon. The width of nanoribbonis selected to be large enough to avoid interactions between edge states. Without loss of generality, we consider a zigzag nanoribbon, which is periodic along x direction with open boundaries in y direction. There are Ny zigzag chains. Fig. 3 (a) shows the band structure of the zigzag plumbene ribbon with Ny = 20. Through comparing it with the bulk band structure, we find that two bands cross the bulk gap G2 in the ribbon model. Through a weight analysis, we can determine that the states within the bulk gap G2 are edge states, as shown in Fig. 3 (b). The largest contribution are from the zigzag chains at the boundaries, including ny = 1,2,19 and 20. The weight at inner sites can be negligible. In contrast, the gap G1 is fully opened. It corresponds to a normal band insulator, which is in agreement with previous studies17,24. Through further spin-resolved analysis, the edge states within the bulk gap G2 show a distinct helical property: two states with oppo- site spin-polarization counterpropagate at each edge, as illus- trated in the inset of Fig. 3 (b). We have also test other cases with different Ny (e.g. 10,30,50) and get qualitatively consis- tent results. In addition, the ribbon with armchair-type edges has also been studied and similar results are obtained. Here, we report the nontrivial topological characteristic of plumbene without any chemical functionalization. Such a large bulk gap also make plumbene be an ideal candidate for realizing the QSH effect at room temperature. To show the robustness of the topological properties in plumbene, we have also studied adiabatic evolution of the electronic structures under external in-plane strain ∆, which FIG. 3: (Color online) (a) The band structure of the zigzag plumbene is defined as ∆= (a a0)/a0 100%. The z coordinates of Pb nanoribbon with a width Ny = 20. (b) The spacial weight distribu- atoms are fully relaxed.− In× the range of 6% < ∆ < 6%, tions corresponding to the states that are on the red dotted line in (a). the energy bands near the Fermi level are− always in isola- The states inside the gap G2 are edge states. Inset: Schematic of the 39 spin-resolved edge channels in the plumbene nanoribbon. tion, as plotted in Fig. S1 of the Supplemental Material . So the topology of the system remains unchanged. Fig. 4 (a) shows the evolutions of the total energy and the angle 2 (a). By comparing the model bands with first-principles φ under different strains. The structure with ∆ = 0, cor- bands, we can see that they agree well with each other in the responding to the optimized lattice structure, is the lowest- energy range ( 10 eV 3 eV). energy one. φ decreases monotonously with increasing ∆. The − ∼ gaps G1 and G2 are investigated with four approximations to To determine whether plumbene is topologically trivial or 31 nontrivial, we have studied the Wannier center. Its evolution the exchange-correlation functionals, including LDA , PBE- GGA30, WC-GGA32 and mBJ-PBE33. For energy gaps of sp- with k2 can be easily obtained by looking at the phase factor θ (k )38, which is an equivalent expression for the Z topologi- semiconductors, the mBJ potential performs in general better 2 2 than others29. Fig. 4 (b)showsthat thegap G by mBJ-PBE is cal invariant. k is defined as k = k a . k = 0 0 2 2 2 2 2 2 about 100 meV larger than that by the other three approxima- π π · ∼ a along b2 direction. a2 and b2 are defined in tions. On the whole, their results are in qualitative agreement − 0 √3a0 Fig. 1. The Wannier centers are calculated based on the tight- with each other. The two gaps versus ∆ show totally different binding model and shown in Fig. 2 (c) and (d). When the behaviors. G1 increases with increasing ∆, while G2 remains Fermi level is inside the gap G1, through moving the green relatively stable. When the compressive strain is excessively reference line we find it always crosses the evolution lines of large, the system is transformed into a metallic phase due to Wannier centers zero or two (even) times, indicating that the two bands crossing the Fermi level. It also leads to a nega- system is topologically trivial. In contrast, for the case that tive gap (Supplementary Fig. S1 (a)39). With increasing the the Fermi level is inside the gap G2, the reference line always value of ∆, the global gap G1 changes from an indirect gap to crosses the evolution lines one or three (odd) times, indicating a direct gap, while the global gap G2 always keeps an indirect that the system is topologically nontrivial. gap. The Fermi level can be shifted to the top of the conduc- Topological insulators are characterized by topologically tion band using gating methods by electric fields27,28 and the protected metallic edge states with helical spin polarization corresponding doping concentration n = 2. For this doping residing inside an insulating bulk gap. To determine whether level, when the strain ∆ changes from 6% to 6%, there are − 5

FIG. 4: (Color online) (a) Total energy per unit cell and φ as functions of ∆. (b) Two gaps G1 and G2 as functions of ∆ calculated with LDA, PBE-GGA, WC-GGA and mBJ-PBE exchange-correlation functionals. Inset: The ∆ n phase diagram of plumbene. The horizontal and vertical axes correspond− to the strain ∆ and the electron doping concentration n. The red, orange and green regions represent topo- FIG. 5: (Color online) (a, b) The Ez Bz phase diagram of plumbene logical insulating (TI), band insulating (BI) and metallic (M) phases, without and with electron doping n−= 2, based on the tight-binding respectively. model. The orange, green, blue, red and purple regions represent band insulator (BI), spin valley polarized metal (SVPM), spin polarized metal (SPM), quantum spin Hall insulator (QSHI) and quantum Hall insulator (QHI), respectively. In a strict sense, only the QSHI not any band crossings near the Fermi level and the system region on the E axis is robust; the edge states are unstable for B > 0. keeps topologically nontrivial. So the topological insulator z z state of plumbene is a robust state with respect to a certain range of in-plane strain. Its indirect bulk gap also enhances the stability. Based on the above results, we can obtain a ∆ n and it can lead to a high spin polarization at K and K′ points 39 phase diagram, as plotted in the inset of Fig. 4 (b). And both− (see the Supplemental Material for details ). Even so, the the large gaps G1 and G2 are ample for applications at room system remains nonmagnetic due to their degeneracy. In or- temperature. der to realize the spin-polarized transport, we further consider an external perpendicular magnetic field Bz. When both electric and magnetic fields are applied, the degeneracy of bands is almost lifted in the entire Brillouin zone, including at the B. External-field responses & device design time-reversal invariant points. With increasing the strength of fields, the band splitting is further increased. Fig. 5 (a) shows 40,41 Plumbene, analogous to graphene , has extraordinary an Ez Bz phase diagram in a pure plumbene (n = 0). When electronic properties. There are two degenerate and inequiv- the external− fields are small, the system is a normal band insu- alent valleys at the corners of the Brillouin zone (K and K′ lator. With increasing the fields, the spin is polarized gradu- points). Intervalley scattering is strongly suppressed due to ally and the degeneracy between K and K′ points is also lifted. their large separation in momentum space. Moreover, because When one of the two valleys is occupied by electrons, the sys- of the buckled structure, the presence of an external perpen- tem becomes a spin valley polarized metal. It is worth men- dicular electric field can induce a voltage difference between tioning that the electrons in the occupied valley (e.g. K′ point) the two sublattices. An electric field Ez is assumed to be ho- are from the states around Γ and K points in the spin valley mogeneous along the z direction in the tight-binding model polarized metallic phase, while for graphene and silicene the 6 electrons in the occupied valley are only from the latter. With excessively large electric and magnetic fields, the valley polarization vanishes, resulting in a spin polarized metal. In the absence of Bz, the increasing Ez makes the gap G1 close and reopen. Simultaneously, the system changes from a band insulator to a QSH insulator. Furthermore, we have also investigated the response of plumbene to the external fields with electron doping n = 2. The corresponding Ez Bz phase diagram is plotted in Fig. 5 (b). In small external− fields the system remains topologically nontrivial as a QSH insulator. With increasing Bz, the spin of the edge states is polarized, leading to a quantum Hall insulator. For a small magnetic field, the increasing electric field can destroy the edge states and the system is transformed into a band insulator. Similar to the case without electron doping, when both Ez and Bz are excessively large, the system becomes a spin polarized metal. Since the above results are based on the tight-binding model fitted to the first-principles band structure, the two phase diagrams are qualitatively correct for describing quantum phase transitions. FIG. 6: (Color online) (a) Conduction bands near K and K′ points with Ez = 10 meV/Å, Bz = 1.16 meV. (b) Spin polarizations as Fig. 6 (a) shows the conduction bands near K and K′ points functions of chemical potential µ with Ez = 10 meV/Å and Bz = with Ez = 10 meV/Å and Bz = 1.16 meV, which respectively 0.58, 1.16, 1.74 meV. The horizontal axis corresponds to the vertical correspond to the electric field intensity 100 MV/m and the axis in (a). (c) Schematic diagram of a four-state spin-valley filter. magnetic induction intensity 20 T. The degeneracy between Left column: four combinations of Ez and Bz with different directions two valleys is lifted and the valley at K′ points is lower than as inputs. Middle column: spin-split conduction bands at K and K′ that at K point. The corresponding spin polarization remains points. The chemical potential is tuned to µ0. Right column: four large and reaches 96.6%, as shown in Fig. 6 (b). Moreover, spin-valley electric states as outputs. with increasing Bz the spin polarization at the bottom of the conduction band almost remains constant but the energy range of the polarization increases. The carrier concentration can film has also been predicted as a topological insulator with be controlled and tuned into the blue area in the band struc- a larger bulk gap, there are no conduction-band valleys at K 20–24 ture, and then the spin and valley polarizations can be realized and K′ points , so that Bi monolayer film can not be used simultaneously without other bands crossing the Fermi level. to fabricate the four-state spin-valley filter mentioned above. Therefore, plumbene with the perpendicular electric and magnetic fields can serve as a spin-valley filter, with spin sign and valley switchable by altering the field directions. C. Substrate effects Following this idea, we present a schematic diagram of a four-state spin-valley filter in Fig. 6 (c). For an experimental Normally, in experiments a 2D film needs to be suspended realization of the proposal, there are three preliminary steps to on a certain substrate which may induce some strain on the be made. (i) External perpendicular fields Ez and Bz are ap- film due to lattice mismatch and affect the electronic struc- plied on plumbene as a controller. (ii) The chemical potential ture. The study of the electronic structures under different is tuned to the energy µ0 which is marked in the middle col- in-plane strains has shown the robustness of topological prop- umn of Fig. 6 (c). This progress can be feasible using gating erties in plumbene. This means that it is insensitive to the methods in experiments27,28. Now the spin and valley polar- in-plane strain of the substrate. In reality, the substrate ef- izations are achieved simultaneously. (iii) Through fixing the fect is not usually restricted to stretching or compression of strengths of the external fields and just changing their direc- the 2D film by the substrate. The interaction between them, tions we can obtain four different output states. This device such as chemical bonding, may strongly modify the electronic can be of utility as a four-state spin-valley filter, which can structure and properties43–45. Assuming that plumbene can be applied in spintronics, valleytronics and even the hybrid be synthesized experimentally, there are several suggestions structures of them. for future experiments about the substrate effect. (i) Choose Besides, there exist topologically protected edge states in the substrate which induces as little effect as possible on the the bulk gap G2. As is well known, the edge states of topo- electronic structure of the 2D Pb film. Specially, it is bet- logical insulators with strong SOC interaction are usually re- ter to avoid the bonding between Pb atoms and the substrate. garded as a spin generator42. When the position of the Fermi (ii) When the band structure near the Fermi level is modified level is tuned into the gap G2, plumbene can become a spin slightly, we can adjust it by the perpendicular electric field, generator with a large bulk gap, which can help to screen the based on the responses of plumbene to external fields. (iii) If interference of thermally activated carriers and allow a large the properties of the film are changed dramatically (e. g., the controlling range of external fields. Although Bi monolayer system becomes a metal), one can perform a chemical func- 7 tionalization with appropriate adsorbates (such as CH3 that the IV. SUMMARY corresponding functionalization was observed to show much more moderate reaction kinetics than plasma fluorination and In summary, we employed first-principles and tight-binding hydrogenation46,47), in order to reduce the substrate effect and methods to investigate the electronic structures and topolog- keep the global gap open. But the topological characteristic ical properties of plumbene. Two large bulk gaps were pre- may be changed, which depends on the chemical functional- dicted. Both of them allow for viable applications at room ization. Meanwhile, both the valleys at K and K points may ′ temperature. The n φ and E B phase diagrams show novel vanish due to the saturation of p dangling bonds. z z z and rich phase transitions.− Moreover,− we find that the QSH Recently Zhou et al. theoretically showed that the Si(111) state is very robust with respect to external in-plane strain. surface functionalized with one-third monolayer of hydro- The large bulk gap and stable edge states make plumbene an gen or halogen atoms exhibiting a trigonal superstructure pro- ideal candidate for realizing the QSH effect at room temper- vides an ideal template for epitaxial growth of heavy metals, ature. According to the responses of plumbene in external such as Bi, Pb and Tl, which self-assemble into a hexago- perpendicular electric and magnetic fields, we propose a de- nal lattice with high kinetic and thermodynamic stability48,49. sign for a four-state spin-valley filter. The substrate effects The template of a functionalized Si surface has already been of plumbene have also been discussed. Plumbene may open widely studied and used in early surface science research50–56. a new and exciting avenue for realizing the QSH effect and Most remarkably, the hexagonal metal overlayer is atomically designing spintronic devices at room temperature. bonded to but electronically decoupled from the underlying Si substrate, exhibiting large-gap QSH states completely isolated from Si valence and conduction bands in Bi@X-Si(111) ( X = H, Cl, Br, I) and Pb@H-Si(111). Moreover, hexagonal lat- tices of indium overlayer have been successfully grown on the Acknowledgments Si(111) √3 √3 Au surface in experiments57. Therefore, it is highly− feasible× to− experimentally realize the large-gap QSH This work is supported by the National Natural Sci- states in the hexagonal Pb film, based on the existing related ence Foundation of China (Grants No. 11681240276, No. experiments and theoretical calculations. We also note that the 11674152 and No. 11604142), and the Shenzhen Peacock substrate may hence the topological properties of plumbene, Plan and Shenzhen Fundamental Research Foundation (Grant which are of interest and will be exploredin our future studies. No. JCYJ20150630145302225).

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Fig. S1 shows the evolution of bands near the Fermi level under different in-plane strains, and the result corresponds to Fig. 4 in the main text. We have employed a sixteen-band tight-binding model to fit the band structure according to the first-principles result in the main text. An electric field Ez is assumed to be

homogeneous along the z direction. The calculated band structure with Ez = 0.5 eV/A˚ is addressed in Fig. S2 (a). The perpendicular electric field results in a band splitting in the entire Brillouin zone except the time-reversal points Γ and M. With increasing Ez, the band splitting is increased. Meanwhile, the energy gap is decreased linearly, as plotted in Fig. S2 (d). To explore and understand the properties of plumbene under perpendicular electric field, we next do a detailed analysis near the Fermi level around K and K′ points. Fig. S2 (b) and (c) show spin-resolved band structures of plumbene and spin polarizations at K and K′ points. The polarization of conduction band is much larger than that of valence band and up to 96% for Ez = 0.5 eV/A.˚ Fig. S2 (e) shows the energy splittings of valence and conduction bands at K point as functions of Ez. Both of them increase linearly with increasing the electric field strength. This directly leads to a linear reduction of the global gap (Fig. S2 (d)). Fig. S2 (f) shows the evolution of the polarization of the conduction band at K point with the perpendicular electric field Ez. The value of the polarization is almost decreased linearly, but it is still more than 95% when the field is upto1.5 eV/A.˚ We

2 Fig. S 2: (Color online) (a-c) The band structures of plumbene in the presence of an perpendicular

electric ﬁeld Ez = 0.5 eV/A.˚ The color and the arrow represent the spin-polarized direction. (d)

The global band gap, (e) the energy splittings of the conduction (∆c) and valence (∆v) bands at K point and (f) the spin polarization ratio of the conduction band at K point as functions of the perpendicular electric ﬁeld Ez.

further consider an external magnetic ﬁeld Bz along the z direction. Fig. S3 (b) shows the

band structure with Ez =0.3 eV/A˚ and Bz =0.1 eV. The corresponding spin polarization reaches 96% at the bottom of the lower-energy valley. To theoretically investigate transport properties of the four-state spin-valley filter mentioned in the main text, we consider such a system which contains a plumbene nanoribbon with zigzag boundaries, left and right semi-infinite electrodes, as shown in Fig. S3 (a). The retarded Greens function is defined as follows,

1 GR (ε)= , (1) + − − − ε+iη0 I HP b ΣL ΣR

where HP b is the Hamiltonian of the plumbene nanoribbon with external perpendicular ﬁelds

Ez and Bz, and I is the identity matrix. ΣL/R are the self-energy operators and describe the eﬀect of two semi-inﬁnite electrodes. During the calculations, the wide-band approximation has been used and the real part of the self-energy is ignored[1]. The imaginary part is set to

3 Fig. S 3: (Color online) (a) Schematic model of a zigzag plumbene nanoribbon with two electrodes, perpendicular electric and magnetic ﬁelds. (b) The bulk band structure and (c) the spin-resolved

transmission spectrum of the zigzag nanoribbon with Ez = 0.3 eV/A˚ and Bz = 0.1 eV. The number of zigzag chains is Ny = 6 and the number of Pb atoms on each zigzag chain is Nx = 20 in (c). Inset of (b): the spin polarization corresponding to the blue area on the conduction band. Inset of (c): a partial enlargement in the energy range of −0.1 ∼ 0.3 eV.

R A 0.05 eV. The Fisher-Lee relation T (ε)= T race ΓLG (ε)ΓRG (ε) is used to calculate the transmission spectrum through the plumbene nanoribbon, where GA (ε) is advanced Greens function and ΓL/R = −2ImΣL/R[2].

Fig. S3 (c) presents a spin-resolved transmission spectrum of the filter with Ez = 0.3 eV/A˚ and Bz =0.1 eV. Actually, their values are very large. Here we choose them theoretically in order to display the external-field effects clearly and explicitly. The transmission coefficients are splitted due to the presence of external fields. In the range of E =0.06 ∼ 0.2 eV, the down-spin one is suppressed significantly and the up-spin one is relatively high. When the chemical potential is tuned into the energy range, it can achieve a spin-polarized transport. Moreover, according to the characteristic of the conduction band in Fig. S3 (b), the valley polarization is also realized in this energy range. Therefore, a four-state

4 Fig. S 4: (Color online) Spin polarizations as functions of chemical potential µ with Ez = 1, 10, 50 meV/A˚ and Bz = 1.16 meV. spin-valley filter can be obtained through changing the directions of the applied fields, as described in the main text. As a supplement to Fig. 6 (b) of the main text, the spin polarizations with different electric field strengths are investigated in Fig. S4. These field strengths can be realized in experiments. We find that both the spin polarization and the energy range of the polarization almost remain constant.

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