Intrinsic Spin Hall Conductivity Platform in Triply Degenerate Semimetal
Zhengchun Zou, Pan Zhou, Rui Tan, Wenqi Li, Zengsheng Ma, Lizhong Sun
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n APS/123-QED
Intrinsic Spin Hall Conductivity Platform in Triply DegenerateSemimetal
Zhengchun Zou ‡ , Pan Zhou ‡ , ∗ Rui Tan, Wenqi Li, Zengsheng Ma, and Lizhong Sun † Hunan Provincial Key laboratory of Thin Film Materials and Devices,Xiangtan University, Xiangtan 411105, China School of Material Sciences and Engineering,Xiangtan University, Xiangtan 411105, China (Dated: January 6, 2021)
Abstract
It is generally believed that conductivity platform can only exist in insulator with topologicalnontrivial bulk occupied states. Such rule exhibits in two dimensional quantum (anomalous)Hall effect, quantum spin Hall effect, and three dimensional topological insulator. In this letter,we propose a spin Hall conductivity (SHC) platform in a kind of three dimensional metallicmaterials with triply degenerate points around the Fermi level. With the help of a four bands k · p model, we prove that SHC platform can form between | , ± i and | , ± i states of metallicsystem. Our further ab initio calculations predict that a nearly ideal SHC platform exhibits in anexperimentally synthesized TaN. The width of the SHC platform reaches up to 0.56 eV, hoping towork under high temperature. The electrical conductivity tensor of TaN indicates that its spinHall angle reaches -0.62, which is larger than many previous reported materials and make it anexcellent candidate for producing stable spin current.
1n recent years, spin Hall effect (SHE) receives much attention due to its potential toproduce pure spin current for future high speed spintronics devices[1]. The transverse spincurrents mainly derive from the atomic spin-orbit coupling, and intrinsic or extrinsic mech-anisms may contribute the spin Hall conductivity (SHC) of real materials[2]. Among them,the large SHC with intrinsic mechanism attracts much attention recently[3–7]. A well-knownmaterials with intrinsic spin Hall effect is 2D topological insulator, in which the SHC canbe quantized when the Fermi level is tuned into the band gap and the width of the con-ductivity platform matches with the value of the gap[8–11]. The similar spin Hall platformalso appears in 3D topological insulator[12] but the conductivity is not necessary to bequantized. For device application, however, the low carrier density in topological insulatormake it improper to produce large spin current in concrete electronic devices. We wonderis there any metallic material can produce stable SHC platform independent of the dopingconcentration.Finding materials with large SHC is also an important topic in the area of spin Halleffect. Latest works proposed some material families with large SHC, such as A supercon-ductors [5, 13] and Weyl semimetal[6]. However, the SHC of the materials strongly dependson the concentration of doping and their SHC value changes rapidly with the change of theFermi level (for example the change produced by the thermal fluctuations in real device).It is a big barrier for the SHE to be applied in future device to produce stable spin currentunder high temperature.In this letter, we propose that a SHC platform can form in triply degeneratesemimetals[14–17]. The results obtained from k · p model and first-principles calculationsindicate that the SHC platform is mainly contributed by the spin Berry curvature (SBC)between | , ± i and | , ± i states. The SBC distribution around the triply degenerate point(TDP) shows strong localization and characteristics of strong polarized p orbitals. Further-more, the contribution of positive and negative SBC around TDP to the conductivity isseverely imbalance forming a platform. To realize the SHC platform in concrete materials,an experimentally synthesized TaN is predicted. The first-principles results confirm that anearly ideal SHC platform as width as 0.56 eV exhibits in the material and its spin Hallangle reaches up to -0.62.The intrinsic SHC can be obtained from k · p model and Wannier Hamiltonian by the2 IG. 1: (color online) (a) is the first Brilloune zone for crystal with space group of P¯6m2. Highsymmetry points and lines are label with the corresponding letter of P¯6m2. The C rotation axis, σ h , and one of the σ v mirror are also presented in the figure. Side (b) and top view (c) of crystalstructure of TaN. Kubo-like formula: σ lij ( ω ) = − e ~ X k Ω lij ( k ) (1)where σ lij represents the spin current along the i direction generated by an charge currentor electric field along the j direction and the spin current is polarized along the l direction.Here ω is used to consider possible electron or hole doping. The k-resolved SBC can bewritten as Ω lij ( k ) = P n f n k Ω ln,ij ( k ). f n k is the Fermi-Dirac distribution function for theband n at k and the band-projected SBC term is:Ω ln,ij ( k ) = ~ X m = n − h n k | ˆ J li | m k ih m k | ˆ v j | n k i ( E n ( k ) − E m ( k )) (2)where ˆ J li = (ˆ s l ˆ v i + ˆ v i ˆ s l ), ˆ v j = (1 / ~ )( ∂ ˆ H/∂k j ).In present letter, we concentrate on the SHC in a hexagonal materials with the existence3 E n e r g y ( e V ) AG H -1-0.500.51 E n e r g y ( e V ) AG H (a) (d)(c)(f)(b)(e) (g) (h)
FIG. 2: (color online) Band structures of four-band k · p model (without and with triply degeneratepoint) as well as the SBC and SHC distribution. The momentum zero point is set at A. (a) is theband structure without triply degenerate point. Two blue dashed horizontal lines represent E = E F - 0.1 eV and E = E F + 0.1 eV. The k-resolved SBC on a log scale in a slice of the 2D BZ at k z = 0 (b) and k y = 0 (c) at E F , respectively. (d) is the SHC σ zxy relative to the Fermi energy. Twoblue dashed vertical lines represent E = E F - 0.1 eV and E = E F + 0.1 eV, respectively. (e) is theband structure with triply degenerate point. The red dashed horizontal line cross triply degeneratepoint represent E = E F + 0.027 eV. The k-resolved SBC on a log scale in a slice of the 2D BZ at k z = 0 (f) and k y = 0 (g) at E = E F + 0.027 eV , respectively. Where red and blue areas representpositive and negative regions. (h) is the SHC σ zxy relative to the position of Fermi energy. of TDP, which may appear in a C rotation invariant axis in reciprocal space and come fromthe crossing between non-degenerate band and double degenerate bands. The typical spacegroup of this type of material is P ¯6 m h and the typically electronic states for triply degeneratefermions possibly come from double degenerate J z = ± and ± . On high symmetry line∆ ( Γ-A), the wave vector point group reduces to C v . The bands with J z = ± mustbe double degenerate restricted by vertical mirror symmetry and J z = ± must separatewith each other and form two non-degenerate band. When the two groups of bands withthe J z = ± and ± cross each other, TDPs may form in C rotation invariant axis. Intime-reversal symmetry ( T ) system, space inversion symmetry ( P ) must be broken because4 T operator force each band double degenerate, which makes TDP impossible to appear inany k points.To describe the electronic states around TDP, a four-band minimal k · p model was con-structed from the point group D h including C rotation operator, one perpendicular mirror( σ h ), and three vertical mirror operator( σ v ), as shown in Fig. 1 (a). Time reversal symmetrymust be considered for the time reversal invariant points Γ and A. With the basis of | , ± i and | , ± i [19], we can construct the following k · p Hamiltonian: H ( k ) = M ( k ) A k z N − ( k ) A k + k z A k z M ( k ) − A k − k z N + ( k ) N + ( k ) − A k + k z M ( k ) 0 A k − k z N − ( k ) 0 M ( k ) (3)where k ± = k y ± ik x , N ± ( k ) = ± A k ± + A k ∓ , M ( k ) = ε + B k z + B k + k − , M ( k ) = ε + C k z + C k + k − . The detailed process to construct the model can be found insupplemental materials[19].Although following the above symmetry and state conditions, the TDP only possiblyappear. Firstly, we consider the two group of bands | , ± i and | , ± i [19] form topologicalinsulator, namely the two group of bands do not cross each other. With proper coefficientsin the above Hamiltonian as shown in the supplemental materials[19], the two groups ofbands separate with each other and form global band gap, as shown in Fig. 2 (a). Toensure the system topological non-trivial, through adjusting the on-site energies and thequadratic term coefficients, we set the energy of | , ± i lower than that of | , ± i , whichis the typical case of semimetals with TDP. The energy band structure, SBC, and SHC arepresented in Fig. 2 (a)-(d). We find that the SBC mainly localizes around symmetry lineΓ-A, especially around A point as shown in Fig. 1 (b) and (c). When Fermi level locatein the valence bands or conduction bands, the SBC drops dramatically due to the severedispersion of band around the A point. When the Fermi level is located in the band gap,a electronic conductivity platform forms for SHC as shown in Fig. 2 (d) due to the stronglocalization of the SBC.Secondly, we investigate the properties when the system is triply degenerate semimetal.According to the Hamiltonian (3), the energy band structure with triply fermion can formon the A-Γ high symmetry line if proper quadratic coefficients are used. An example is5 s zxy s zyx S HC (( / e ) S / c m ) Energy(eV) -0.03-0.59 -1 -0.5 0 0.5 1-600-400-2000200400600 S HC (( / e ) S / c m ) Energy(eV) s yzx s xzy -1 -0.5 0 0.5 1-600-400-2000200400600 S HC (( / e ) S / c m ) Energy(eV) s yxz s xyz (c) (b)(a) (d) FIG. 3: (color online) (a) Band structure of TaN along high symmetry lines with SOC. Two bandcrossings T and T are signed by red dots. (b)-(d) SHC tensor elements in function of the Fermienergy of TaN. Two black dashed vertical lines represent E = E F - 0.03 eV and E = E F - 0.59 eV. presented in Fig. 2 (e)-(h) in which two TDPs appear (the coefficients can be found in thesupplemental materials[19]). In this case, the Fermi level locates between the energies of thetwo TDPs. As shown in Fig. 2 (f) and (g), the distributions of SBC in k z = 0 and k y = 0planes are strong localization. However, positive SBC appear around TDP (Fig. 2 (g)) andit is similar with the distribution of SBC around Weyl point[6]. The sign of SBC is oppositeon the opposite sides of TDP, just like the characteristics of p orbital. Furthermore, it ismuch like strong polarized p orbital due to the large difference of band dispersions on theopposite sides of the TDP. The results also indicate that the negative contribution of SBC(blue zone in Fig. 2 (g)) is greatly larger than that of positive one (red zone in Fig. 2 (g)).Amazingly, the strong localized SBC also produce SHC platform as shown in Fig. 2 (h)whose width is equal to the energy difference of the two group bands at A point.The conditions to produce SHC platform in metallic system can be briefly summarizedas: (i) symmetry protection similar with that of triply degenerate semimetals; (ii) special6tates such as | , ± i and | , ± i [19] around the Fermi level; (iii) strong localization ofSBC. Although such conditions can be easily fulfilled in a simplified k · p model, to realizethe SHC platform in a real metallic material is still challenging. In present letter, we findthat the experimentally synthesized θ -TaN (we call it as TaN in following paper), a typicaltriply degenerate semimetal[14, 18], can achieve such SHC platform. We take TaN as aprototype to study the SHC using first-principles method[19]. The crystal structure of TaN[14, 20–23] is shown in Fig. 1 (b) and (c). It is WC-type[21] hexagonal crystal structurewith space group P ¯6 m ′ )and A points. There are two triply degenerate points on high symmetry line Γ-A belowand above Fermi level, respectively. When the SOC is ignored, the states around Fermilevel composed of one degenerate d z and double degenerate e g states ( d xy and d x − y ) ofTa, the results are in good agreement with previous report[14]. The SOC effect will inducethe d z and e g states to | , ± i and | , ± i [19] around the Fermi level, meeting the secondrequirement above. The band structures as shown in Fig. 3 (a) indicate that the localextreme point along A-H for the two bands around Fermi level deviates from A point, whichwill greatly impact on the distribution of SBC around A point, the details will be discussedbelow. Fig. 3 (b)-(d) show the different components of SHC with different concentrations ofelectron/hole doping. It is fascinating that platforms of SHC form for the four components σ zxy , σ zyx , σ xzy and σ yzx , as shown in Fig. 3 (b) and (d). The energy range of the platformreach up to 0.56 eV and the width is larger than quantized platform width in most previousreport on 3D or 2D topological insulator. Such wide SHC platform is favorable for thesystem producing stable spin polarized current counter act with temperature fluctuation.The maximum energy of the platform is 0.03 eV lower than the intrinsic Fermi level, whichindicates hole doping is necessary to constrain the Fermi level in the energy range of theplatform. Considering the height of the platform, the σ zxy and σ zyx are more valuable forfurther application. Therefore, we will focus on the origin mechanism of σ zxy platform below.As discussed above, the SHC platform should be related to the localization of theSBC of the system. We present the SBC along high symmetry lines and some sections offirst Brilloune zone in Fig. 4. The results indicate that the SBC mainly concentrates onhigh symmetry line Γ-A, A, and K points, especially around A point. Moreover, the SBC7 D › æ| D fl æ (a)(b) (f)(e) (d)(c) A GD U LMP DG MK U A LH
E-E F = -0.03eV -0.251 FIG. 4: (color online) (a) Band structures with the projection of SBC on a log scale. The reddashed horizontal lines represent E = E F - 0.03 eV. (b) The k-resolved SBC at E = E F - 0.03 eV.The k-resolved SBC on a log scale in a slice of the 2D BZ at k z = 0 (c), k z = π /c (d), k z = 0.52 π /c(e), and k y = 0 (f) for σ zxy SHC of TaN at E = E F - 0.03 eV, respectively. is isotropic in the plane perpendicular to the c axis around A point. As shown in Fig. 4(d), the extreme of SBC form a circular ring around A point, which is different from theresults obtained from k · p model whose maximum of SBC just locate at A point. SuchSBC distribution feature derives from the deviation of the band extreme around the Fermi8evel from A point, as shown in Fig. 3 (a). The circular ring around A point can be wellillustrated with formula (2) where the SBC inversely proportional to the square of theenergy difference. As illustrated in Fig. 3 (b), the SHC platform of σ zxy is not absolutely flatin the energy range of [-0.59, -0.03] for TaN. The absolute height of the platform increaseswith the increasing of the Fermi level. The difference between absolute maximum (456( ~ /e) S/cm) and minimum (411 ( ~ /e) S/cm) of the SHC platform is 45 ( ~ /e) S/cm. Theinclination of the SHC platform mainly comes from the complication of the band dispersionaround TDPs. Along with the increasing of the Fermi level in the energy range of [-0.59,-0.03], the SBC between A-L as shown in Fig. 4 (b) is nearly unchanged. However, theaverage SBC on the high symmetry line of Γ-A increases with the increase in the Fermilevel due to the complicated energy band dispersion around the TDPs. As shown in Fig. 4(a), the color for two split | , ± i bands on Γ-A are blue (negative). As the Fermi levelmove from -0.59 to -0.03 eV, more and more negative SBC contribute the total SHC.Especially at the energy of -0.03, two peaks of SBC can be observed on high symmetry lineΓ-A. The evolution of the SBC around the TDPs directly produce the inclination of theSHC platform which is different from the absolute flat platform obtained from simplified k · p model. The results as shown in Fig. 4 (b) and (c) indicate that besides around theTDPs there are large peaks of SBC around K and K ′ . However the SBC peaks are severelyrestricted in a small region confined by the crossing line between the energy band aroundthe K (K ′ ) and the Fermi level. The distribution feature can be found in Fig. 4 (c). Thecontribution of the SBC around the K and K ′ points to the SHC is negligible. It is worthnoting that the spin of the bands around K and K ′ point will split with each other due tothe k-points are not time-reversal invariant. However, the time reversal and C rotationsymmetry will force the sign of SBC around K and K ′ to be the same[24]. Generally, thepoint group of SBC is the subgroup of Berry curvature (BC)[25]. However, the time reversalsymmetry correlation between SBCs around K and K ′ produces the original C symmetryof BC into C for SBC, as shown in Fig. 4 (c). Moreover, the results as shown in Fig. 4 (a)reveal that both valence and conduction bands are spin-splitting for TaN, which is differentfrom the case of monolayer MoS [24, 26] that only valence bands are spin-splitting inducedby spin-orbit coupling. Such spin-splitting of the low energy electronic states around K forTaN can be explained by a similar k · p Hamiltonian, and the results can be found in thesupplemental materials[19]. 9pin Hall angle (SHA) is another important parameter for SHE to characterize theconversion efficiency from the charge current to spin current. It is defined as the ratio ofSHC to the charge conductivity, namely SHA=( e/ ~ ) σ zxy /σ xx where σ xx is the longitudinalcharge conductivity, σ zxy is the transverse SHC. Obviously, if we want to obtain a largeSHA, large SHC as well as low charge conductivity are equally important. For pure 5 d metal Pt, although its SHC can reach 2400 ( ~ /e ) S/cm , its SHA is small with the value of-0.07 because of large charge conductivity[27]. Considering the value of SHC platform ofTaN is -456 ( ~ /e ) S/cm at the energy of -0.03 eV and the longitudinal charge conductivityis estimate to 736
S/cm [19], the SHA of TaN reaches up to -0.62. The absolute SHA valueof TaN is much larger than that of Pt and other reported values[13, 28], which indicatesTaN is a high-performance material to produce pure spin current.In summary, with the four bands effective k · p model, we theoretically predict a SHCplatform in the triply degenerate semimetals. The large SHC platform mainly come fromthe localized SBC produced between | , ± i and | , ± i states. It is worth to mentionthat the platform is robust against the TDP around Fermi level. With the example ofTaN, we find, although the low energy electronic states of concrete materials maybe morecomplicated, the SHC platform can be well kept. The large SHC platform make TaN tobe excellent material to produce stable spin current in a large of doping range. Moreover,we find that in the TaN material family, TaP, TaAs, TaBi, and TaSb show similar SHCplatform, the results can be found in the supplemental materials[19]. The wide SHCplatform of TaN material family is potential to spintronics devices with stable spin currentunder high temperature.This work is supported by the National Natural Science Foundation of China (Grant No.11804287, 11574260), Hunan Provincial Natural Science Foundation of China (2019JJ50577)and the Scientific Research Fund of Hunan Provincial Education Department (18A051). ‡ These authors contributed equally to this work. ∗ Electronic address: [email protected] † Electronic address: [email protected]
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