Quantum transport in a compensated semimetal W2As3 with nontrivial Z2 indices
Yupeng Li, Chenchao Xu, Mingsong Shen, Jinhua Wang, Xiaohui Yang, Xiaojun Yang, Zengwei Zhu, Chao Cao, Zhu-An Xu
QQuantum transport in a compensated semimetal W As with nontrivial Z indices Yupeng Li,
1, 2, ∗ Chenchao Xu, ∗ Mingsong Shen, Jinhua Wang, XiaohuiYang, Xiaojun Yang, Zengwei Zhu, Chao Cao, † and Zhu-An Xu
1, 2, 6, ‡ Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, People’s Republic of China Wuhan National High Magnetic Field Center, School of Physics,Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105, People’s Republic of China Department of Physics, Hangzhou Normal University, Hangzhou 310036, People’s Republic of China Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, People’s Republic of China (Dated: October 2, 2018)We report a topological semimetal W As with a space group C2/m. Based on the first-principlescalculations, band crossings are partially gapped when spin-orbit coupling is included. The Z indices at the electron filling are [1;111], characterizing a strong topological insulator and topologicalsurface states. From the magnetotransport measurements, nearly quadratic field dependence ofmagnetoresistance (MR) ( B (cid:107) [200]) at 3 K indicates an electron-hole compensated compound whoselongitudinal MR reaches 11500% at 3 K and 15 T. In addition, multiband features are detected fromthe high-magnetic-field Shubnikov-de Haas (SdH) oscillation, Hall resistivity, and band calculations.A nontrivial π Berry’s phase is obtained, suggesting the topological feature of this material. A two-band model can fit well the conductivity and Hall coefficient. Our experiments manifest that thetransport properties of W As are in good agreement with the theoretical calculations. INTRODUCTION
The research of new topological phases such as topo-logical insulator (TI), topological superconductor, andtopological semimetal (TSM) is a hot spot in recent yearsin condensed matter physics. Among them, topologicalsemimetals are widely studied because there are manytypes, including Dirac semimetals [1, 2], Weyl semimet-als [3], nodal-line semimetals [4–9], and semimetals withtriply degenerate nodal points [10–12]. In addition, theyare closely related to other topological phases and arethus believed to be intermediate states of topologicalphase transitions [13, 14]. These topological semimet-als usually exhibit large magnetoresistance and topolog-ical surface states. Apart from these topological phases,another kind of topological materials are Z topologi-cal metals [15], which are characterized by non-trivialZ topological invariants, topological surface states, andlack of a bulk energy gap, such as the La X ( X = P, As,Sb, Bi) [16] family. Among this family, LaBi is the onlyone with topological nontrivial band dispersion provedby angle-resolved photoemission spectroscopy (ARPES)measurements [15].Recently, transition-metal dipnictides MPn (M = Nbor Ta; Pn = As or Sb) with C2/m structure have at-tracted great attention because of extremely large mag-netoresistance (XMR), negative magnetoresistance andother interesting properties [17–23]. According to thetheoretical calculations, type-II Weyl points could be in-duced by magnetic field in MPn [24]. Moreover, a su- ∗ Equal contributions † [email protected] ‡ [email protected] perconducting transition has been observed in NbAs athigh pressure[25], and the maximal T c is 2.63 K under12.8 GPa. What is very interesting is that the C2/mstructure remains up to 30 GPa, and thus the topolog-ical phase could exist in the superconducting state andit may be a candidate of topological superconductors.MoAs is another compound with C2/m structure pos-sessing quadratic XMR [26, 27], whose XMR may orig-inate from the open-orbit topology [27] instead of theelectron-hole compensation in MPn .These interesting properties in the compounds withC2/m structure have attracted much research attention.Here we report on a topological material W As withC2/m structure and nontrivial Z indices [1:111] whichbelongs to a strong TI family [5, 9, 28–30]. Large MR=[ R ( B ) − R (0)] /R (0) of approximately 11 500% at 3 K and15 T is observed in this electron-hole compensated sys-tem. Evident quantum oscillations have been observedby using a high pulse magnetic field, from which nineintrinsic frequencies are obtained from the fast Fouriertransform (FFT) spectrum, and a nontrivial π Berry’sphase can be detected from F h , F h , and F h . Themultiband character is also revealed from both band cal-culations as well as Hall resistivity measurements. Neg-ative Hall resistivity indicates that this compound is anelectron-dominated semimetal. A simple two-band modelis proposed to explain the temperature dependence of theHall coefficient, with which the Hall conductivity andlongitudinal conductivity can be well fitted. The Hallcoefficient obtained from the fitting data is in agreementwith the experimental value of R H . All of these illustraterelatively consistent transport behavior and the reliabletwo-band fitting. a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t ( b )( a )( c ) ( d ) W A s C 2 / mR w p = 6 . 8 9 %
Intensity (arb.unit)
O b s e r v e d C a l c u l a t i o n D i f f e r e n c e B r r a g p o s i t i o n q ( d e g ) 0 2 4 6 8 1 0 1 2E ( k e V ) Counts (a.u.) ( 1 0 0 0 )
Intensity (a.u.) q ( d e g ) ( 2 0 0 ) ( 4 0 0 ) ( 6 0 0 ) ( 8 0 0 ) S 3
A s W Intensity (a.u.) q ( d e g ) R o c k i n g c u r v eF W H M ~ 0 . 0 1 5 ( 2 0 0 ) FIG. 1. (a) Crystalline structure of W As . (b) EDS spectrum of W As and the inset is a scanning electron microscopyfigure. (c) Powder XRD pattern of polycrystalline W As and Rietveld analysis profiles. (d) XRD pattern of high-qualitysingle crystal showing sharp diffraction peaks of the (200) plane. The left inset is the rocking curve of the (200) peak, and theright inset is a photograph of W As crystals. EXPERIMENT
Single crystals of W As were synthesized by an iodine-vapour transport method. Firstly, polycrystalline W As was prepared by heating the mixture of W powder andAs powder with a stoichiometric ratio of 2:3 at 1173 K for2 days. Then iodine with a concentration of 10mg/cm was mixed with the reground polycrystalline W As , andthey were sealed in an ampoule to grow single crystalsfor 7 days in a two-zone furnace, where the temperaturegradient was set as 1323-1223 K over a distance of 16 cm.All the experimental processes were carried out in a glovebox filled with pure Ar, except the heating processes.X-ray diffraction (XRD) data were collected by a PAN-alytical x-ray diffractometer (Empyrean) with a Cu K α radiation. We used energy-dispersive x-ray spectroscopy(EDS) to analyze the composition ratio of W and As el-ement. Longitudinal resistivity and Hall resistivity wasmeasured by a standard six-probe technique. Transportmeasurements below 15 T were performed on an Oxford- 15T cryostat, and a pulsed magnetic field up to 60 T wasemployed to obtain apparent quantum oscillations in theWuhan National High Magnetic Field Center (WHMFC- Wuhan).The first-principles calculations were performed withVienna Abinitio Simulation Package (VASP) [31, 32]. Aplane-wave basis up to 400 eV was employed in the calcu-lations. Throughout the calculation, the atom relaxationwas performed with the Perdew-Burke-Ernzerhof (PBE)exchange correlation functional, and the electronic bandstructures, Fermi surfaces, and surface states were ob-tained with the modified Becke-Johnson (mBJ) method[33]. A Γ-centred 15 × × - 1 . 0- 0 . 50 . 00 . 51 . 0 - 1 . 0- 0 . 50 . 00 . 51 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 ( g ) ( e )( d ) Energy (eV) ( f )
Energy (eV) X X Y e l e c t r o n p o c k e t s ‘ Y ‘ X ( a ) h o l e p o c k e t s - 1 . 0- 0 . 50 . 00 . 51 . 0 ‘ X ( c ) Energy (eV) G Y F L I | I ZX | X F w i t h S O C ( b ) Energy (eV) G Y F L I | I ZX | X F w i t h o u t S O C FIG. 2. (a) A schematic diagram of primitive Brillouin zone and the corresponding projection of the (001) surface Brillouinzone. (b) Band structure of W As without SOC. (c) The band structure with SOC. (d, e) The 3D Fermi surfaces of electronpockets (green ones) and hole pockets (blue ones) with SOC, respectively. (f) Expanded view of bulk band structure with SOCalong Y − X . (g) Calculated (001) surface band structure along the X − Y − X line, and the inset shows the topologicalsurface state. logical indices Z were calculated by the method of paritycheck[28]. Using the maximally localized Wannier func-tion method[36], the Fermi surfaces were obtained with atight-binding Hamiltonian fitted from density functionaltheory (DFT) bands. The surface states were calculatedusing surface a Green’s function [37]. RESULTS
The crystal structure (a unit cell) of W As is pre-sented in Fig. 1(a) with the monoclinic C2/m (spacegroup, no. 12), which has the same space group withNbAs [17–19, 22, 27, 38]. EDS data in Fig. 1(b) givethe chemical component ratio W:As = 2:2.85, consistentwith the nominal ratio of 2:3 within the experiment er-ror. Rietveld refinement of powder XRD is fairly reliablewith R wp =6.89%, as shown in Fig. 1(c). The refinedlattice parameters at room temperature are a = 13.322˚A, b = 3.277 ˚A, c = 9.593 ˚A and β = 124.704 ◦ , whichare almost the same as in the previous report [39]. Sub-sequently, a single-crystal XRD pattern of sample S3 isshown in Fig. 1(d), and the rocking curve in the leftinset indicates a high-quality sample (S3) with a smallFWHM = 0.015 ◦ [40]. The obtained single-crystal sam-ples are usually needle-shaped, as shown in the inset ofFig. 1(d).The band structures of W As without and with spin- TABLE I. Parities of bands at time-reversal invariant mo-menta (TRIM). Π n is the multiplication of the parities forbands 1 to n . The highest occupied band at each TRIM is in-dicated with ◦ . The corresponding Z classification is [1,111].TRIM (cid:81) (cid:81) (cid:81) (cid:81) (0,0,0) − ◦ − + − ( π ,0,0) + − ◦ − +(0, π ,0) + − ◦ − +( π , π ,0) − + ◦ − − (0,0, π ) − ◦ + + +( π ,0, π ) − + ◦ + +(0, π , π ) − + ◦ + +( π , π , π ) − + ◦ − − orbit coupling (SOC) are shown in Fig.2, respectively.Like the TaSb compound[38] , there is a one-electronband and a one-hole band crossing the Fermi level, in-dicating the two-band feature of this system. W As ,however, has large Fermi surfaces (FS) [Figs. 2(d) and(e)], and three band-crossing features can be identifiednear the Fermi level. They are along the X Y, FL andLI directions, respectively, which all resemble the tran-sition metal dipnictides XPn (X = Ta, Nb; Pn = P,As, Sb) [38]. Once the SOC is included, all the bandcrossings are gapped, leaving these two bands separatednear the Fermi level E f . Therefore, this system can beadiabatically deformed to an insulator without closing oropening gaps, thus allowing us to define the Z index e 1 ( c ) F e 2 F e 1 F h 3 F h 1 F h 6 F h 5 F e 3 F h 4 F h 2 Amplitude (a.u.)
F ( T ) r xx ( mW cm) T ( K )
S 1 R R R ~ 2 9 1 S 2 R R R ~ 3 1 1 S 3 R R R ~ 3 7 2 ( a ) MR B ( T ) 3 K F i t t i n g )M R = ( ‘ m
B ) n n = 1 . 9 4 ( b ) MR B ( T )
S 3 - 1 . 5- 1 . 0- 0 . 50 . 0 r yx ( mW cm) r xx ( mW cm) B ( T )
Amplitude (a.u.)
T ( K ) F h 1 F h 2 F h 3 F e 1 F h 4 F e 2 F h 5 F h 6 F e 3 m * h 1 = 1 . 4 2 m e m * h 2 = 1 . 3 7 m e m * h 3 = 1 . 3 1 m e m * e 1 = 0 . 9 4 m e m * h 4 = 0 . 7 0 m e m * e 2 = 1 . 0 m e m * h 5 = 1 . 1 9 m e m * h 6 = 1 . 1 8 m e m * e 3 = 1 . 3 2 m e ( d ) FIG. 3. (a) Temperature-dependent resistivity of several samples. The inset is a fitting result with MR = (¯ µB ) n . (b) Visiblequantum oscillations under high magnetic field. The inset displays ρ xx measured up to 15 T which is much larger than ρ yx .(c) FFT curves suggesting the nine independent intrinsic frequencies. (d) Fitting the SdH amplitudes by the LK formula. Theobtained effective masses of the nine FFT frequencies are shown. with the parity at the time-reversal invariant momenta(TRIM) multiplied up to the highest valence band, sim-ilar to the cases of LaBi [15] and CeSb [41]. As both thehighest valence band (band 27) and lowest conductionband (band 28) are crossing the Fermi level, we list theband parities up to four bands Π , Π , Π , and Π in Table I, where Π n is the multiplication of the pari-ties for bands 1 to n . There are 54 valence electrons inthe primitive cell, corresponding to 27 filled bands. Theproduct of parities at all these TRIMs up to the 27thband is -1, suggesting a strong topological property ofthis compound. In addition, we also performed the PBEcalculations and obtained the same topological propertiesas the mBJ calculations, in contrast to the divergence be-tween the PBE and mBJ calculations in LnPn (Ln = Ce,Pr, Gd, Sm, Yb; Pn = Sb, Bi) [42] and LaSb [43], inwhich the PBE calculations underestimate the band gapbetween conduction band and valance band. Comparedwith the bulk band structure along X − Y in Fig. 2(f),we calculate the (001) surface states along X − Y − X TABLE II. Physical parameters of nine extremal orbits arelisted when magnetic field is along the (200) facet of W As . m is the static mass of the electron. DFT results here arethe SdH frequencies from calculated band energies [44].FS SdH(T) m ∗ ( m ) DFT(T) F h
106 1.42 58 F h
232 1.37 140 F h
505 1.31 181 F e
574 0.94 577 F h
653 0.70 605 F e
778 1.00 623 F h F h F e as shown in Fig. 2(g). Although there are strong bulkstates due to the large hole and electron pockets in Fig.2(g), the Dirac type of surface states can be still observedat the Y point in the inset of Fig. 2(g). ( b ) LL Index n - 1 ) F h 1 F e 2 F h 2 F h 5 F h 3 F h 6 F e 1 F e 3 F h 4 g h 1 - d = 0 . 2 1 (cid:1) g h 2 - d = - 0 . 4 4 (cid:1) g h 3 - d = 0 . 0 4 (cid:1) g e 1 - d = - 0 . 3 4 (cid:1) g h 4 - d = - 0 . 0 9 (cid:1) g e 2 - d = 0 . 1 6 (cid:1) g h 5 - d = - 0 . 0 7 (cid:1) g h 6 - d = - 0 . 3 4 (cid:1) g e 3 - d = 0 . 3 7 (cid:1) F e 2 F h 4 F e 1 F h 3 F h 2 Dr xx ( mW cm) - 1 ) F h 1 ( a ) n = 4 ( d ) ( c ) F h 6 F e 3 F h 5 - 1 ) e x p e r i m e n t a l d a t a s u m o f n i n e f i l t e r e d f r e q u e n c i e s FIG. 4. (a) Oscillatory part of the resistivity ( ∆ ρ xx ) as afunction of 1/B at 1.7 K (the red line is experimental data).∆ ρ xx is obtained after subtracting the background of ρ xx .The blue line is the sum of nine filtered frequencies. (b, c)The nine filtered oscillatory parts of ∆ ρ xx . The Landau-levelindex is obtained from the peak position of the oscillationcomponent, for example, one of the oscillatory peaks about F h is n = 4 in (c). (d) Landau-level index plot of the ninefrequencies. The intercepts are between − F h , F h , and F h , respectively. Open circles indicate theinteger Landau-level index from the peaks of high-frequencyoscillatory components of ∆ ρ xx , and closed circles denote thehalf-integer index (∆ ρ xx valley). We now turn to the transport properties of W As .The temperature-dependent ρ xx of different samplesS1, S2, and S3 all exhibit a typical metallic behavior,and the corresponding residual resistance ratio [RRR = ρ xx (300 K ) /ρ xx (1 . K )] is 291, 311, and 372, respectively.Therefore, the sample S3 is chosen to perform furtherstudies. The giant MR of about 11 500% at 15 T and3 K is displayed in the inset of Fig. 3(a) without obvi-ous quantum oscillations. The MR can be well fitted by M R = (¯ µB ) n with n = 1 .
94, indicating good electron- hole compensation [45, 46]. The geometric mean of themobilities is ¯ µ = √ µ e µ h = 0 . T − when n = 2 is used.In order to extract more information about the Fermisurface of W As , a pulsed magnetic field experimentup to 60 T is performed. In Fig. 3(b), large quantumoscillations in resistance become visible at various tem-peratures, and no sign of saturation in MR is detectedup to 60 T. The inset of Fig. 3(b) shows | ρ yx | (cid:28) | ρ xx | , sowe can use ρ xx to analyze the quantum oscillation data.After removing the background, nine distinct frequenciesof extremal orbits are observed from a complicated FFTspectrum in Fig. 3(c). The calculated frequencies arein good agreement with the experimental observations,considering the error between experimental results andtheoretical calculations [44], as shown in Table II. Usingthe Lifshitz-Kosevich (LK) formula,∆ R xx ∝ R T × R D × cos (cid:20) π ( FB + γ − δ ) (cid:21) , (1)where R T = (2 π k B T /β ) /sinh (2 π k B T /β ), β = e ¯ B (cid:126) /m ∗ , k B is Boltzmann constant and ¯ B is the aver-age field value [47, 48]. The obtained effective mass m ∗ of each pocket is also indicated in Fig. 3(d), where thesmallest m ∗ h = 0 . m and largest m ∗ h = 1 . m . Dueto the complicated Fermi surface, as seen in Figs.2(d)and 2(e), nine filtered oscillatory parts in Figs. 4(b) and4(c) are detected by the decomposition of ∆ ρ xx in Fig.4(a), which is obtained by subtracting the backgroundof ρ xx at 1.7 K. Although both the mussy frequenciesand harmonic frequencies are eliminated, the sum of ninemain frequencies (blue line) matches the experimentaldata (red line) very well in Fig. 4(a), both in amplitudesand phases. Therefore, we assign the Landau level bythe peak position of the oscillation component of eachfrequency in Figs. 4(b) and 4(c) [49–51]. In the Lifshitz-Onsager (LO) formula, A n (cid:126) eB = 2 π ( n + γ − δ ) , (2)where A n is the FS cross section area of the Landau level(LL) n . γ − δ = − φ B π − δ is the phase factor, where φ B is the Berry phase and δ is a phase shift induced by di-mensionality [ δ = 0 for two dimensions (2D), or δ = ± for three dimensions (3D)]. The Landau-level index is fit-ted by the LO formula in Fig. 4(d), and we can obtainthe intercepts of nine frequencies. Furthermore, | γ − δ | of F h , F h , and F h are 0.04, 0.09 and 0.07, respectively,which are all in the range between 0 and 1/8, exhibit-ing a nontrivial Berry’s phase of π [52, 53]. Theoreti-cally, when a singularity in the energy band is enclosedby the cyclotron contour under magnetic field, we coulddetect a nontrivial π Berry’s phase from Shubnikov-deHaas (SdH) oscillation [54, 55]. It is worth noting thatthe observation of a π Berry phase from SdH oscillationis affected by the magnetic field directions and the crosssections of Fermi surfaces; thus some nontrivial Fermisurfaces may not yield exact a π Berry phase in SdH ( b ) r yx ( mW cm) B ( T ) ( e )
T ( K )
RH (10-10m-3/C) R H E x p . R H F i t t i n g ( f )
T ( K ) ne, nh (1020cm-3) n e n h ( a )
051 0 m e m h m e, m h (103cm2/Vs) MR B ( T ) ( c ) s xx (108 W -1m-1) B ( T ) ( d ) s xy(106 W -1m-1) B ( T )
FIG. 5. (a) Magnetoresistance at various temperatures; (b) Hall resistivity vs magnetic field; (c) conductivity σ xx vsmagnetic field; (d) Hall conductivity σ xy vs magnetic field; and (e) temperature dependence of Hall coefficient R H . R H (solidsquare) obtained from the two-band model fitting is compared with the experimental data of R H (diamond). (f) Temperaturedependence of n e , n h , µ e , and µ h obtained in the two-band model fitting. measurements [54–56]. For the materials with a complexFermi surface, some unexpected frequencies, such as lowfrequencies and harmonic frequencies, would affect thefiltered main frequency more or less. For example, theharmonic frequency 2F h (1306 T) may be mixed into thefiltered F h (1305 T) and thus modulates the oscillationdata of F h in Fig. 4(b), which could affect the precisedetermination of the Berry phase of F h in Fig. 4(d).The measurements of ρ xx ( B ) and ρ yx ( B ) provide fur-ther insight on the transport properties, as seen in Fig.5(a) and 5(b). The negative ρ yx ( B ) shown in Fig. 5(b)indicates that the electron-type carriers play a dominantrole in transport properties, and the nonlinear magneticfield dependence implies the multi-band feature. Fig-ures. 5(c) and 5(d) display that the curves of conductiv-ity σ xx = ρ xx / ( ρ xx + ρ yx ) and σ xy = ρ yx / ( ρ xx + ρ yx ) canbe well fitted by the two-band model: σ xx = e (cid:20) n h µ h µ h B ) + n e µ e µ e B ) (cid:21) , (3) σ xy = eB (cid:20) n h µ h µ h B ) − n e µ e µ e B ) (cid:21) , (4)where n e , n h , µ e , and µ h are electron-type carrier den-sity, hole-type carrier density, electron-type mobility, andhole-type mobility, respectively. Both the electron andhole mobility decrease monotonously as temperature in-creases, as seen in Fig. 5(f), which is a typical behav-ior for metals. Intriguingly, the electron-hole compen-sation only holds at low temperatures and becomes in-effective above 50 K. The Hall coefficient obtained by fitting with the two-band model R H = [ R hH ( σ hxx ) − R eH ( σ exx ) ] / ( σ hxx + σ exx ) [57], as shown in Fig. 5(e), isquite consistent with the directly measured R H which isacquired from the slope of ρ yx near the zero field, where R hH and R eH are the Hall coefficient for hole-type andelectron-type charge carriers respectively, σ hxx and σ exx are the hole conductivity and electron conductivity, re-spectively, and σ xx = enµ . All of above fitting resultssuggest that the two-band model fits the experimentaldata reliably. It is apparent that the absolute value of R H changes little below 14 K and becomes large withincreasing temperature, then becomes small again above100 K, at which temperature the slope of ρ yx ( B ) reachesthe maximum [Fig. 5(b)]. We propose that the shiftof Fermi level with increasing temperature may accountfor the loss of the electron-hole compensation above 50K. Similar behavior and explanations were reported inWTe [58] and ZrTe [59].In summary, we have discovered a topologicalsemimetal W As with Z indices [1;111] which has astrong TI feature. The electron-hole compensation leadsto a colossal MR which is as large as 11 500% at 3 Kand 15 T and is unsaturated even up to 60 T. A non-trivial Berry’s phase of π is obtained from F h , F h , and F h , suggesting the nontrivial topological characteristic.A two-band model is effective to fit the Hall conductiv-ity and Hall coefficient. The transport properties are ingood agreement with the band structure calculations. ACKNOWLEDGMENTS
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