Tensile Strained Gray Tin: a New Dirac Semimetal for Observing Negative Magnetoresistance with Shubnikov-de-Haas Oscillation
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Tensile Strained Gray Tin: a New Dirac Semimetal for Observing NegativeMagnetoresistance with Shubnikov-de-Haas Oscillation
Huaqing Huang and Feng Liu
1, 2 Department of Materials Science and Engineering,University of Utah, Salt Lake City, Utah 84112, USA Collaborative Innovation Center of Quantum Matter, Beijing 100084, China (Dated: September 26, 2018)The extremely stringent requirement on material quality has hindered the investigation and po-tential applications of exotic chiral magnetic effect in Dirac semimetals. Here, we propose thatgray tin is a perfect candidate for observing the chiral anomaly effect and Shubnikov-de-Haas (SdH)oscillation at relatively low magnetic field. Based on effective k.p analysis and first-principles cal-culations, we discover that gray tin becomes a Dirac semimetal under tensile uniaxial strain, incontrast to a topological insulator under compressive uniaxial strain as known before. In this newlyfound Dirac semimetal state, two Dirac points which are tunable by tensile [001] strains, lie in the k z axis and Fermi arcs appear in the (100) surface. Duo the low carrier concentration and highmobility of gray tin, a large chiral anomaly induced negative magnetoresistance and a strong SdHoscillation are anticipated in this half of strain spectrum. Comparing to other Dirac semimetals, theproposed Dirac semimetal state in the nontoxic elemental gray tin can be more easily manipulatedand accurately controlled. We envision that gray tin provides a perfect platform for strain engineer-ing of chiral magnetic effects by sweeping through the strain spectrum from positive to negative andvice versa. The discovery of Dirac and Weyl semimetals with chi-ral quasiparticles [1–5] opens a new avenue to realiz-ing the long-anticipated high-energy-physics Adler-Bell-Jackiw chiral anomaly[6–8] in condensed matter systems.As a defining signature, a large negative longitudinalmagnetoresistance (MR) is expected to be observable inDirac semimetals. In the presence of parallel magneticand electric fields, each Dirac point split into two Weylnodes with opposite chirality. The Weyl fermions resid-ing at one Weyl node are pumped to the other, resultingin non-conserved chiral charges and prominent negativeMR. The chiral anomaly induced negative MR need tobe observed in Dirac semimetals with ultralow carrierconcentration and high mobility, which require a highsample quality. Apart from the negative MR effect, theShubnikov-de-Haas (SdH) oscillation is another interest-ing magnetotransport phenomenon in Dirac semimetals,where the MR oscillates periodically in reciprocal mag-netic field (1 /B ). Analysis of the SdH oscillations of MRgives a nontrivial π Berry phase, which is a distinguishedfeature of Dirac fermions.[9, 10] This quantum oscillationis attributed to the Dirac band structure and ultrahighcarrier mobility of Dirac materials.However, to date, the chiral anomaly induced nega-tive MR was observed only in a few Dirac semimetalmaterials[11–13], and none of which exhibits negativeMR associated with a SdH oscillation in the quantumlimit. This is due to the lack of a high-quality Diracsemimetal with low carrier concentration and high car-rier mobility at the same time. For example, the negativeMR without SdH oscillation was observed in Cd As andNa Bi with low carrier concentrations ( ∼ cm − ) andmobilities ( ∼ cm V − s − )[11]. Although an ultra-high carrier mobility of ∼ cm V − s − that accom- panied by strong SdH oscillations was reported in dif-ferent Cd As samples, the chiral anomaly induced neg-ative MR was not observed due to the relatively highcarrier concentration (10 − cm − )[14]. In addition,Na Bi is unstable and decomposes rapidly upon expo-sure to air. The delicate synthesis together with the hightoxicity of Cd and As also makes handling of Cd As dif-ficult. Given the above challenges, identifying new Diracsemimetals with low carrier concentration and high car-rier mobility is of great importance for both basic re-search and potential applications.In this Letter, we predict that tensile strained graytin is a Dirac semimetal, and more importantly pro-pose it to be a perfect candidate to observe the chiralanomaly effect and SdH oscillation at relatively low mag-netic field. So far it has been taken for granted thatexternal strains just drive gray tin into a topological in-sulator state [15, 16]. Here, we reveal the missing halfof the strain spectrum of gray tin, where a previouslyunknown Dirac semimetal phase is discovered. Based oneffective k · p analysis and first-principles calculations,we demonstrated the existence of a pair of controllableDirac points in the k z axial of the bulk Brillouin zone(BZ) and Fermi arcs connecting the two projected Diracpoints on the (100) surface when gray tin is under tensileuniaxial strain. Due to the relatively low carrier concen-tration [17, 18] and anomalously high mobility [19, 20],a large chiral anomaly induced negative MR accompa-nied by SdH oscillation is expected to be observable inthe tensile strained gray tin. In fact, some measurementsof gray tin many years ago have addressed the negativeMR effect associate with the SdH oscillation manifestingan unconventional Berry phase [17, 18], which supportsour proposal. Comparing to other Dirac semimetals, the FIG. 1. (Color online) (a) Crystal structure of α -Sn with F d ¯3 m (No. 227) symmetry in a cubic unit cell. The dashedlines indicate a tetragonal unit cell. (b) Tetragonal unit cellunder external strains. Substrate induced strains are appliedas in-plane compressive strains leading to a uniaxial z -axistensile strain. (c) Topological phase diagram v.s. uniaxial z -axis strain. For tensile strain, ε zz >
0, the system becomes aDirac semimetal with two Dirac points; while for compressivestrain, ε zz <
0, the system is a topological insulator. proposed Dirac semimetal state in the nontoxic elemen-tal gray tin promises ease of fabrication and tuning. Ourfindings provide a perfect platform for strain engineeringof chiral magnetic effect by scanning through the strainspectrum from positive to negative and vice versa.Gray tin, also known as the α phase of Sn crystal ( α -Sn), is a common zero-gap semiconductor, in which theconduction band minimum and valance band maximumare degenerate at the Γ point [21, 22]. Gray tin crys-tallizes in the diamond structure with F d ¯3 m symmetry(space group No. 227) as shown in Fig. 1(a). Unlikeother group-IV compounds with the same crystal struc-ture, α -Sn has an inverted band ordering where the p -orbital derived Γ +8 state located at the Fermi level ishigher than the s -orbital dominated Γ − state with op-posite parity [see middle panel of Fig.1(c)]. The bandinversion is critical for the existence of topologically non-trivial states including topological insulator and Diracsemimetal phases. Since the fourfold degeneracy of theΓ +8 state is a consequence of the O h symmetry of the di-amond lattice, applying an uniaxial strain will split thedegenerate states. However, we discover that the splitmanifests in different ways depending on the directionof the applied strain, which lead to different topologicalstates. We first perform a k · p analysis to investigate generalbehavior of electronic structures under strain. The Γ +8 bands at the Γ point are fourfold degenerate as J = 3 / k · p Hamiltonian[23–25]: H ( k ) = − ( γ + 52 γ ) k +2 γ X i k i J i +2 γ X i = j k i k j { J i J j } , (1)where i, j = ( x, y, z ), J i are 4 × { J i J j } = ( J i J j + J j J i ). The inverse-mass parametersare γ = − . , γ = − . γ = − . ~ / m e ), which have been determined by Booth et al. from an experimental study of the anisotropy of the MRoscillations in gray tin [26]. The main perturbations in-duced by strains are given by H ε = ( a + 54 b ) ε − b X i J i ε ii − √ d X i = j { J i J j } ε ij , (2)where ε = ε xx + ε yy + ε zz . The deformation potentialsof b = − . d = − . k · p Hamiltonian H k · p = H ( k ) + H ε for strained α -Sn can be derived analytically, which are given in theSupplemental Material [29].For simplicity, let’s first consider an uniaxial [001]strain, i.e., ε xx = ε yy = ε zz and ε xy = ε yz = ε xz = 0[30, 31]. In this case, the cubic F d ¯3 m (No. 227, O h ) sym-metry reduces to tetragonal I /amd (No. 141, D h ).The Γ +8 state splits into Γ +6 and Γ +7 , and the correspond-ing eigenvalues along k z axis become, E , ( k z ) = − γ k z + aε ± (cid:20) γ k z + 2 γ b ( ε − ε zz ) k z + b ε − ε zz ) (cid:21) = − γ k z + aε ± (cid:12)(cid:12)(cid:12)(cid:12) γ k z + b ε − ε zz ) (cid:12)(cid:12)(cid:12)(cid:12) . (3)Apparently, the criterion for the existence of band cross-ing points in k z axis is given by the following condition: γ b ( ε − ε zz ) < . (4)We can apply either a tensile z -axis strain or a biax-ial compressive in-plane ( x, y ) strain, ε zz >
0, whereas( ε − ε zz ) = ε xx + ε yy − ε zz <
0. Inserting the pa-rameters given above, we found that γ b >
0, and thecriterion of Eq. (4) is satisfied. As a consequence, thesystem becomes a Dirac semimetal with two symmetricDirac points at (0 , , k cz = ± p b (3 ε zz − ε ) / γ ), whichare obviously tunable by external strain ε zz . This Diracsemimetal phase, which was previously overlooked, re-veals the missing half of the strain spectrum on α -Sn andalso suggests exotic properties of other strained materials FIG. 2. (Color online) (a) The Brillouin zone of bulk α -Snin tetragonal unit cells and the projected surface Brillouinzones of (001) and (010) surfaces. (b) Calculated HSE bandstructure of α -Sn under in-plane compressed strain of − with similar electronic structures, such as HgTe [32, 33].In contrast, when applying a compressive [001] strain or atensile biaxial in-plane strain, ε zz < γ b ( ε − ε zz ) > k · p analysis, we ob-tain the phase diagram of α -Sn under a [001] strain, asschematically shown in Fig. 1(c). A topological phasetransition from topological insulator to Dirac semimetalcan be driven by tuning ε zz through zero. We also stud-ied the effect of [111] strain, which show similar topolog-ical phase transition [29].To verify the prediction from the effective k · p analysispresented above, we perform first-principles calculationson strained gray tin using Vienna ab initio simulationpackage[34]. More details of models and computationalmethods are presented in the Supplemental Material [29].For a biaxial in-plane compressive strain of − +8 state splits into Γ +7 and Γ +6 , which are pushed down andup, respectively. Since the Λ and Λ bands belong todifferent irreducible representations and disperse upwardand downward respectively along the Γ-A line, the twobands cross at two discrete points: (0, 0, ± . π/a ), consistent with the Dirac points predicted by theeffective k · p theory, as shown in Fig. 2. The Fermi level is exactly at the band crossing points which are fourfolddegenerate due to the coexistence of time-reversal andinversion symmetries. Thus the Fermi surface consistsof two isolated points, around which the bands disperselinearly, resulting in a 3D Dirac semimetal. The separa-tion of the two Dirac points in momentum space increaseswith the increasing external strain. If the tensilel strainis too large, the conduction band at the R point wouldshift concave downwards and become occupied. To sat-isfy the charge neutral condition, the Fermi level woulddiverge from the Dirac point. In contrast, by applying acompressive [001] strain, the Γ +7 and Γ +6 states are liftedin opposite direction, and a globe band gap opens in theentire BZ (not shown). As the band inversion retains inthe compressively strained α -Sn, this gapped system is atopological insulator[15].Because the splitting of Γ +8 does not change the bandinversion in α -Sn, the nontrivial topology of the Diracsemimetal state under tensile strain should be similar tothe topological insulator state under compressive strain.Also, the band structures are gapped in both the k z = 0and the k z = π planes when the system is under com-pressive or tensile strains, the Z topological invariantsin these planes are well-defined. In fact, as inversionsymmetry retains in the strained system, we can simplydetermine the Z index from the parities of all occupiedbands at time-reversal invariant momentum (TRIM) k -points[15]. The parity products of occupied bands is − Z = 1 for the k z = 0 plane, whereas Z = 0 for the k z = π plane.Therefore, the strained α -Sn is always topologically non-trivial. Thus, topological surface states or Fermi arcs areexpected to appear on side surfaces of the compressivelyor tensile strained gray tin.One of the most important consequences of Diracsemimetal is the existence of topological surface statesand Fermi arcs on the surface. We have calculated boththe (001) and (010) surface states of tensile strained α -Sn, as shown in Fig. 3. For the (001) surface, two Diracpoints are projected to the same point of the surface BZ.Bulk continuum superimposes nontrivial surface states,and the Fermi surface of the (001) surface is just a sin-gle point [Fig. 3(b)]. For the (010) surface, even thoughthere are some trivial surface bands due to the danglingbond states of the unsaturated surface Sn atoms, thenontrivial surface states, which originate from the gap-less point, are clearly visible [see Fig. 3(c)]. As shown inFig. 3(d), the Fermi surface, which has a shape of butter-fly, is composed of two pieces of Fermi arcs, which con-nect the two projections of bulk Dirac points. Howeverthe Fermi velocity is ill-defined at these projected Diracpoints [i.e., singular points, see Fig. 3(e)]. Althoughthe Fermi arc pattern may change upon varying surfacepotential, its existence, which stems from the bulk 3DDirac points, is robust against such perturbations. Theseunique features, absent for topological insulators, can be FIG. 3. (Color online) The projected surface states and cor-responding Fermi surface of semi-infinite α -Sn under a com-pressive in-plane strain of − measured by angle-resolved photoemission spectroscopytechniques.This newly discovered Dirac semimetal phase instrained α -Sn is expected to facilitate the realization ofthe Adler-Bell-Jackiw chiral anomaly[6, 8], which is ob-servable as a negative longitudinal MR. To do so, it isrequired that the carrier density is low enough so thatthe Fermi level is located close to the Dirac point. Thiscondition is clearly satisfied by α -Sn with a known lowcarrier concentration on the order of 10 cm − [17, 18],Moreover, as the mobility of α -Sn is anomalously high( ∼ cm V − s − , comparable to that of the high-mobility Dirac semimetal Cd As ) and increases dramat-ically with decreasing carrier concentration [19, 20], it iseasy to drive the system into the extreme quantum limitat relatively low magnetic field. In fact, some measure-ments many years ago have shown the negative MR effectand the SdH oscillations with an anomalous oscillatoryphase of − π/
2, which indicate strong signatures of theAdler-Bell-Jackiw chiral anomaly in gray tin [17, 18]. Inaddition, a giant non-saturating linear transverse MR isexpected in strained α -Sn, which can be useful to clar-ify the unclear mechanism for the linear MR in Diracmaterials.To further assess the chiral anomaly induced negativeMR and SdH oscillation with nontrivial Berry phase, thebehavior of longitudinal MR in strained gray tin is sim-ulated. When an external electric field E is applied inparallel with the magnetic field B , the chiral charges atone node are pumped to the other with opposite chiral-ity due to the chiral anomaly induced ± e π ~ E · B term.This charge pumping yields a positive magnetic conduc- FIG. 4. (Color online) The estimated longitudinal MR asa function of magnetic field at 1.2 K. The SdH oscillationterm ∆ ρ SdH /ρ , the chiral anomaly induced negative MR∆ ρ chiral /ρ and the total MR ∆ ρ k /ρ are shown in red, blackand blue, respectively tivity (correspond to a negative MR) given by[35],∆ σ chiral = e τ a π ~ g ( E F ) B (5)where g ( E F ) is the density of state (DOS) at the Fermienergy E F , τ a is the internode scattering time. Mean-while, due to the high mobility of strained gray tin, thequantum oscillation of the MR are expected to be ob-served at low temperature, which can be described bythe Lifshitz-Kosevich formula[36]:∆ ρ SdH ρ = A ( T, B ) cos (cid:20) π ( FB − γ ±
18 ) (cid:21) . (6)The oscillatory phase factor 2 πγ = π − ϕ B is directlyrelated to the Berry phase ϕ B . A nontrivial ± π Berryphase can be acquired by electrons in cyclotron orbits.We estimated the longitudinal MR curve of a strainedgray tin with the carrier concentration of n = 2 . × cm − and the mobility of µ = 2 . × cm V − s − ,which are in the experimentally accessible range[17, 18].As shown in Fig. 4, the oscillatory MR ∆ ρ k /ρ decreasesrapidly with the magnetic field as expected. The chi-ral anomaly induced negative MR ∆ ρ chiral /ρ can ap-proach to − ρ k /ρ from the chiral anomaly. Due to the small cross-sectionalarea A F of the Fermi surface, the estimated oscillationfrequency F is only about 2 . F = A F ~ / eπ , much smaller than otherDirac semimetals. These novel behaviors of MR, is rarein non-ferromagnetic materials, thus can serve as one ofthe most definite signatures of the Dirac semimetal statein strained α -Sn (More details about the estimation arepresented in Supplemental Materials[29]).In conclusion, we discover a Dirac semimetal state inthe other missing half of the tensile strain spectrum ofgray tin, which offers a perfect candidate for the realiza-tion of chiral magnetic effects, addressing a long-standingexperimental challenge. The exotic chiral anomaly in-duced large negative longitudinal MR associated withSdH oscillation is estimated. Furthermore, gray tin alsoprovides a new route to studying the interplay betweendifferent topological states and other novel phenomena.For example, Weyl semimetals are hopefully realized ingray tin by breaking either time-reversal or inversionsymmetries. 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