Nonsaturating large magnetoresistance in semimetals

Ian A. Leahya, Yu-Ping Lina, Peter E. Siegfrieda, Andrew C. Tregliaa, Justin C. W. Songb, Rahul M. Nandkishorea,c, and Minhyea Leea,d,1

aDepartment of Physics, University of Colorado, Boulder, CO 80309; bDivision of Physics and Applied Physics, Nanyang Technological University, Singapore 637371; cCenter for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309; and dCenter for Experiments on Quantum Materials, University of Colorado, Boulder, CO 80309

Edited by N. Phuan Ong, Princeton University, Princeton, NJ, and approved September 4, 2018 (received for review May 22, 2018)

The rapidly expanding class of quantum materials known as topo- the charge neutrality point in semimetals allows the generic logical semimetals (TSMs) displays unique transport properties, quadratic two-band model to describe the MR and the Hall including a striking dependence of resistivity on applied magnetic effect in reasonable levels (19–23). However, a two-band model field, that are of great interest for both scientific and technologi- of this form generically predicts a MR that is quadratic in applied cal reasons. So far, many possible sources of extraordinarily large fields, whereas the materials frequently exhibit a MR linear in nonsaturating magnetoresistance have been proposed. However, the applied field. While various theoretical proposals for H lin- experimental signatures that can identify or discern the dominant ear MR have been advanced (e.g., refs. 8, 24–26), the origins mechanism and connect to available theories are scarce. Here we of extreme MR in topological semimetals remain unclear. As present the magnetic susceptibility (χ), the tangent of the Hall nonsaturating, large MR becomes more ubiquitous, it becomes angle (tan θH), along with magnetoresistance in four different particularly urgent to identify a set of distinct attributes that nonmagnetic semimetals with high mobilities, NbP, TaP, NbSb2, enable the delineation of their origins. and TaSb2, all of which exhibit nonsaturating large magnetore- In this article, we systematically examine the low-field dia- sistance (MR). We find that the distinctly different temperature magnetic susceptibility (χ), the transverse MR, and the Hall dependences, χ(T), and the values of tan θH in phosphides and effect as a function of T and H in four different semimetals antimonates serve as empirical criteria to sort the MR from dif- with high mobility (ν ≥ 104 cm2/Vs) and very large nonsaturating ferent origins: NbP and TaP are uncompensated semimetals with MR—NbP, TaP (phosphides), NbSb2, and TaSb2 (antimonates). linear dispersion, in which the nonsaturating magnetoresistance Characteristic parameters related to magnetic transport are arises due to guiding center motion, while NbSb2 and TaSb2 summarized in Table 1. are compensated semimetals, with a magnetoresistance emerging We present two different types of nonsaturating large MR from nearly perfect charge compensation of two quadratic bands. identified by the temperature (T ) dependence of diamagnetic Our results illustrate how a combination of magnetotransport susceptibility, χ(T ), and the H dependence of the Hall angle, ρxy σxy and susceptibility measurements may be used to categorize the tan θH = = , where ρxx and ρxy are longitudinal and ρxx σxx increasingly ubiquitous nonsaturating large magnetoresistance in Hall resistivity, respectively, and σxx and σxy are corresponding TSMs. conductivities. One type of MR originates from the presence of smooth dis- nonsaturating magnetoresistance | magnetic susceptibility | order that governs guiding center motion of charge carriers. Weyl semimetals | topological semimetals The linear H dependence of this type arises from the squeezed

agnetoresistance (MR) and the Hall effect are versatile Significance Mexperimental probes in exploring electronic properties of materials, such as carrier density, mobility, and the nature of The intensive recent investigations of topological semimetals scattering and disorder. In typical nonmagnetic and semicon- have revealed a large number of semimetal compounds that ducting materials, the MR increases quadratically with applied exhibit very large nonsaturating magnetoresistance. Multi- transverse magnetic field and saturates to a constant value when ple mechanisms for this magnetoresistance phenomenon have the product of the applied field and the mobility (ν) approaches been theoretically proposed, but experimentally it is unclear unity. Nonsaturating MR is commonly attributed to the semiclas- how to identify which mechanism is responsible in a par- sical two-band model, where electron-like and hole-like carriers ticular sample or how to make a clean connection between are nearly compensated (1), resulting in rich magnetotransport experimental observations and theoretical models. Our results characteristics that are strongly temperature (T ) and applied show that the magnetic susceptibility and the tangent of the transverse magnetic field (H ) dependent in nonmagnetic com- Hall angle successfully capture the fundamental differences pounds. A flurry of interest in nonsaturating, H -linear MR in seemingly similar nonsaturating large magnetoresistance, (2–5) in narrow-gap semiconductors led to two main theoretical where charge compensation, energy dispersion, and the roles accounts: (i) a 2D simple four-terminal resistor network model, of disorder are markedly distinct, and provide empirical tem- where strong disorder or inhomogeneity of the sample manifests plates to characterize the origins of the extraordinary mag- as charge and mobility fluctuations (6, 7), and (ii) the so-called netotransport properties in the newly discovered topological “quantum linear MR” which emerges in systems with linear band semimetals and beyond. crossings when the lowest Landau level is occupied (8). The former approach has provided a basis to engineer large magneto- Author contributions: M.L. designed research; I.A.L., Y.-P.L., P.E.S., A.C.T., R.M.N., and M.L. transport responses via macroscopic inhomogeneities or disorder performed research; I.A.L., Y.-P.L., J.C.W.S., R.M.N., and M.L. analyzed data; and I.A.L., (9–11). Meanwhile, the latter has remained rather elusive until Y.-P.L., J.C.W.S., R.M.N., and M.L. wrote the paper. y recently. The authors declare no conflict of interest.y Interest in nonsaturating very large MR has exploded fol- This article is a PNAS Direct Submission.y lowing the discovery of topological semimetals. These materials Published under the PNAS license.y are regularly reported to exhibit record-high nonsaturating MR, 1 To whom correspondence should be addressed. Email: [email protected] known as extreme magnetoresistance (XMR), with unusually This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. high mobilities for bulk systems (12–18) and relatively low resid- 1073/pnas.1808747115/-/DCSupplemental.y ual resistivity ρ0. The proximity of the chemical potential to Published online October 3, 2018.

10570–10575 | PNAS | October 16, 2018 | vol. 115 | no. 42 www.pnas.org/cgi/doi/10.1073/pnas.1808747115 Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 eh tal. et Leahy Moreover, discussion). recent center a guiding for to 25 due ref. invoking phosphides arises χ(T e.g., without MR the (see, the even that motion that dispersion, assume and linear we compensation, with if semimetals explained are be Finally, can study. at this features minimum pronounced in a exhibit field phosphides applied highest the In rates. field, crossover eddpnec fteHl eitvt togydvae from deviates strongly resistivity Hall the of dependence in quadratic field to nearly is down (<10 MR zero The temperature study. to this close in remains fields angle antimonates, the Hall in Meanwhile, the (ii) parameters. fit as point trality (α quasi-linear Rdfie as defined MR peculiar a has ple pto up indicates applied bar Each compensation. quadratic persistent by as transition ear (Left (TaP) phosphides for (TaSb dispersions energy sentative 1. Fig. when temperatures low phos- at the In constant (i) of 1: magnitude Fig. the in depicted phides, be are can and results Our follows latter. as the summarized into into antimonates phosphides the the and categorize former of the to facets able are different we the magnetoresponse, interrogate to measurements totransport characteristics framework. magnetic conventional the and model within transport two-band other type the accompanies in other it compensation The and charge compensation. from comes charge involvement MR the the of for require bands not multiple does and of fields 1) Table magnetic e.g., large semimetals, semiclassically in ν carriers of trajectories (ρH tan NbP data Sample magnetotransport of Summary 1. Table NbSb TaP TaSb B sn obnto fmgei ucpiiiyadmagne- and susceptibility magnetic of combination a Using resistivity Residual ) ≥ 2 ) 2 − 2 (Right ) losu oetatdpn eesrltv otecag neu- charge the to relative levels doping extract to us allows 1 ρ ceaial eitdnnauaigM hnmn n repre- and phenomena MR nonsaturating depicted Schematically 0 )/ρ esl civdi ierydsesn topological dispersing linearly in achieved (easily H 0 .Tepopie’M scaatrzdb us-iert lin- to quasi-linear by characterized is MR phosphides’ The ). > µ n tan and 0 ≤10 ≤10 H H H H 7.6 . 3.5 5.8 ∼ ∆ρ/ρ = H S dpnetnnoooi om h measured The form. nonmonotonic -dependent 2 nrae,wieteatmnts Ri characterized is MR antimonates’ the while increases, Rrmislna in linear remains MR , −4 −4 θ S 1Tat T 31 1. H ρ 2 5 sstb h cl twhich at scale the by set is 0 θ H H ± 0 tzr edi eotdat reported is field zero at t03Kand K 0.3 at ∝ T eedne rsn rmsmcasclcharge semiclassical from arising dependence, olna (α linear to 0.1) T H = tan 0 = α K. 0.3 1.9–2.5 2.2–4.3 ν T , . tlow at θ 99 3 × ∆ H −1 ρ/ρ 10 ,oeigKhe’ ue The rule. Kohler’s obeying K, auae oan to saturates µ 3 0 0 H H = T = 5 > xiisacosvrfrom crossover a exhibits 5T. 15 × H ∼ 10 H ρ S 1. 0 5 −2 , ' antcfil was field Magnetic %. 0 pto up µΩ 0.1 0.1 0.2 0.5 T talaccessible all at ) ± T 8 = tan n antimonates and ) min ,while T, cm H ,weethe where 0.1), and K 2 χ(T H -independent µ l fthese of All . θ H 0 rmroom from H o the for )s (H ∆ 31 = ρ ) ρ/ρ 27,800 20,200 xx ∆ρ/ρ 5,560 satu- (T 561 0 T, = 0 ) T surfaces 34). Fermi 33, smaller (29, the the antimonates and to the fields due of higher smaller in much only are but quan- magnitudes well, antimonates, as the emerge In oscillations phosphides. tum the in features both spike-like in Schubnikov–de apparent Strong are field). oscillations zero Haas near (except antimonates the when 2.5 around to saturating H values large reaches TaP and nioae hwsapycnrsigbehavior: contrasting sharply show antimonates H Magnetotransport. Results were samples by the System of Measurements Design. Properties Quantum susceptibilities Magnetic Magnetic T the 31 by K. to measured up 0.3 plane the to on field current down of applied direction the the with to (Dupont perpendicular performed paint were Ag of measurements ranges using transport resistance contacts crystals, contact single electrical with on 4966) Standard known directly 30–32). a made following were (17, method procedure transport vapor synthesis chemical the using for available yet not are studies antimonates. photoemission the the semimet- yet compensated nearly (29), vicinity of als picture NbSb the our for in with calculations consistent identified The also energy. were Fermi nodes the Weyl of thoroughly multiple studied 27, where (17, been studies 28), photoemission have and TaP calculations first-principle and via NbP calculations: ture with bands. semimetals quadratic compensated usual the for archetypical are for antimonates susceptibility mostly diamagnetic is The antimonates the antimonates. the in linearity resolved. clearly are vs. etrs (C features. at safnto of function a as 2. Fig. C AB 2 S T h hshdsadatmntsas ipa contrasting display also antimonates and phosphides The igecytl fNP a,NbSb TaP, NbP, of crystals Single struc- electronic existing with consistent well is finding Our tan

T tan at eednei alage i.2C Fig. angle. Hall in dependence

-4 -2 H

' H = 0 2 4 0 2 4 6 8 lte nlg–saei the in scale – log in plotted 010102020300 250 200 150 100 50 0 3 1 530 15 0 -15 -30 T 8 .Srn unu siltosi hshdsrsl nspike-like in result phosphides in oscillations quantum Strong K. 0.3 TaSb (A ,wiei ean w reso antd mle for smaller magnitude of orders two remains it while T, NbSb 0 = TaP and tan ) NbP NbP 2 .3 2 h alagetan angle Hall The B) 2 .I h ihfil ii,tepopie n the and phosphides the limit, high-field the In K. H θ H NbSb PNAS o a mgna n TaSb and (magenta) TaP for (B) and esrdat measured 0 H [T] i.2 Fig. 2 | coe 6 2018 16, October T =0.3K A T and

T [K] 2 µ

y tan needn.Teefaue fthe of features These independent. 10 10 0 H 10 H xssae hr NbSb where scale, axis -3 -1 θ B 1 = 0 0 300 200 100 0 H TaP displays 3 1 530 15 0 -15 -30 o b rd n NbSb and (red) NbP for (A) safnto of function a as T 7 TaSb TaP NbP NbSb TaSb 2 2 ρ n TaSb and , | xx h magneto- The Ω. ≤1−2 hw h temperature the shows 2 2 o.115 vol. tan and T [K] θ 0 H ρ H [T] 2 2 xy tan | safnto of function a as bak,measured (black), n TaSb and 2 2 T htgenerate that o 42 no. n TaSb and (Inset . eegrown were θ H 0.3 K o NbP for 2 tan ) | (green) 2 2 10571 H data are 2 θ > -4 -2 0 2 4 H

APPLIED PHYSICAL SCIENCES 2 2 6 8 10 6 8 10 dependenceof tan θH measured at µ0H = 7 T. Strikingly, tan θH 10 10 10 10 10 10 rises rapidly above unity with decreasing T around 100 K and 106 AB H 6 60 K, for NbP and TaP, respectively. In contrast, the antimonates H 10 0.3 K behave in the opposite fashion: Upon decreasing temperature, 104 2 [%] 4 0 80 10 tan θH rapidly decreases, giving values two orders of magnitude 150 smaller than the phosphides (Fig. 2C, Inset). We note that small (H)/ 102 102 Hall angles are frequently found in conventional metals and 300 semimetals (35), as well as in a wide range of XMR materials with NbP TaP 100 100 high mobilities for both holes and electrons (19, 22). 6 6 10 The field dependence of tan θH plays a deciding role in deter- 10 CD 2 H2 mining MR. For example, a field-independent tan θH indicates H [%] the field dependence of ρxx and ρxy should have the same func- 0 102 ρ H 102

tional form. In uncompensated systems, xy has an -linear Hall (H)/ contribution (i.e., ρxy = µ0RH H , where RH is the normal Hall NbSb TaSb coefficient), allowing a field-independent tan θH and therefore 2 2 -2 -2 10 a nonsaturating H -linear ρxx . This is exactly what we observe in 10 H ρ H ρ 106 108 1010 106 108 1010 the phosphides which exhibit -linear xy as well as -linear xx H/ [T/ m] H/ [T/ m] (below). 0 0 0 0 Furthermore, large tan θH can act to suppress resistivity and Fig. 4. (A–D) Kohler’s plots of phosphides in (A) NbP and (B) TaP and morph its T dependence. To demonstrate this, we express ρxx in of antimonates in (C) NbSb2 and (D) TaSb2. Arrows in A and B indi- terms of tan θH and σxx , cate the locations of HS, where tan θH saturates and the MR switches to H linear.   σxx 0 1 ρxx = 2 2 = ρxx 2 , [1] σxx + σxy 1 + tan θH in Fig. 3 C and D, Inset. This is consistent with a small tan θH  0 1 0 where ρ ≡ . As evident in Eq. 1, when tan θH ≥ 1, the 1. For all samples, the rapid rise in ρ at low temperatures xx σxx xx inverse relation between σxx and ρxx no longer holds. corresponds to a plummeting σxx . The effect of large tan θH magnitude on ρxx (T ) is particu- We now turn to the field dependence of MR. We show α larly pronounced for NbP and TaP in Fig. 3 A and B, where Kohler’s plots in Fig. 4 and find that ∆ρ/ρ0 ∝ H , where ρ0 = the T dependence of ρxx is plotted. Monotonic metallic T ρxx (T , H = 0), collapses into a single curve over the large T dependence switches to nonmonotonic behavior as the field range. In the antimonates, we observe α ≈ 2 in the entire tem- increases, with a peak at a temperature that coincides with the perature range up to 31 T. In contrast, in the phosphides the 0 onset of rapid increase of tan θH (Fig. 1C). Crucially, ρxx (as exponent α deviates from 2 even at low H (α ≈ 1.4 ± 0.1) and defined in Eq. 1) plotted as dashed lines in Fig. 3 A and B, switches over to the linear-H dependence (α ≈ 1) in the µ0H ≥ Insets, deviates significantly from the measured ρxx , reflecting 8 T, where the tan θH approaches a constant value. the large values of tan θH and the dominant role tan θH has Finally, ρxy (H )’s of NbP and NbSb2 are compared in Fig. 5 A in ρxx (T ). and B. The field dependence of NbSb2, shown in Fig. 5B, is far NbSb2 and TaSb2, however, display contrasting behavior plot- from linear and a higher power of H becomes more visible with ted in Fig. 3 C and D. We first note that ρxx (T ) initially exhibits increasing H . ρxy in TaP and TaSb2 showed similar behavior as a slight decrease in ρxx as T is lowered until the sudden rise, presented in SI Appendix, Fig. S2. mimicking a metal–insulator-like transition. This behavior is commonly observed in many XMR materials and ρxx continues Magnetic Susceptibility. In Fig. 6, we plot the T dependence of increasing as T is lowered further. In contrast to phosphides in the magnetic susceptibility in the low-field limit. All four samples Fig. 3 A and B, NbSb2 and TaSb2, however, display ρxx ≈ 1/σxx show negative susceptibilities, corresponding to diamagnetism. as reflected by the near overlay of the dashed lines and solid lines Fig. 6 displays χ as a function of T for (Fig. 6A) NbP and NbSb2 and (Fig. 6B) TaP and TaSb2. Both NbP and TaP have broad yet pronounced minima emerging at Tmin = 203 K and Tmin = 68 K, respectively. For TaP the minimum susceptibility (Fig. 6) and AB the resistivity peak under the field (Fig. 3B) both occur at sim- ilar temperatures, which are also close to the temperature where 2 tan θH first becomes appreciable (Fig. 2C). For NbP, these tem- peratures are within a factor of 2, although the agreement is not as close as for TaP. On the other hand, χ(T ) for the antimonates remains featureless and mostly constant. Discussion CD We begin by discussing the magnetic susceptibility plots shown in Fig. 6. The absence of Pauli paramagnetism in all sam- ples indicates that we do not have spin degenerate bands and is strong evidence for spin–momentum locking, such that the magnetic susceptibility is dominated by orbital diamagnetism (36). As we now discuss, the additional features can be well explained if we postulate that the phosphides are uncompensated semimetals with a linear dispersion, whereas the antimonates are compensated semimetals.

Fig. 3. ρxx (T) at different Hs is shown for (A) NbP, (B) TaP, (C) NbSb2, and We discuss first the phosphides. In particular, we fit the χ(T )s 0 (D) TaSb2. ρxx (T) at µ0H = 7 T, defined in Eq. 1, is plotted in A–D, Insets. of phosphides to the result of orbital magnetism in a linear

10572 | www.pnas.org/cgi/doi/10.1073/pnas.1808747115 Leahy et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 eto 5C) section Appendix, (SI crossing for band linear minimization simple energy a of from case the obtained is which (37), dispersion panels. main the to correspond boxes small the Appendix Eq. to fit a shows eh tal. et Leahy data. our in clearly find we as teslas, several that b yields this limit, field where Weyl of locations points, and reported crossing meV two the (−57 Eq. of NbP by one in 6) points with with (Fig. point consistent fitted crossing is well linear one are which find TaP we and NbP, For NbP 2. for data The and unity. points, neutrality charge the energy, cutoff a where T 5. Fig. yi om with form, lytic Eq. to cubic an gives uncompen- also for disorder) only ρ smooth operative with is semimetals (which sated scenario this that note direction, field the a exhibits along guiding and squeezed the become enabling to fields), trajectories enough disorder center the large an than (at smaller length to is radius correlation lead cyclotron in the naturally when diffuse arises to can This in way centers that guiding unusual note disorder, an we smooth particular, of In presence phosphides. the the on data transport uncompensated suggest as strongly understood dispersion. thus be linear with should data semimetals phosphides susceptibility the magnetic that Weyl of The pair (28). a at data of photoemission location in reported the nodes with consistent remarkably again ˜ AB xy = 1 = [m cm] hsbsccnetr sas ossetwt l u observed our all with consistent also is conjecture basic This -0.4 -0.2 xy 0.2 0.4 ned writing Indeed, . .Nt h ifrneo h antd of magnitude the of difference the Note K. 0.3 0 1/H .5 H b η ρ 3 × 0 ρ 6- 6 3 0 -3 -6 xy eto 1). section , χ(T stesadr em–ia itiuinfunction, distribution Fermi–Dirac standard the is xy eedneaie.Tedse iei i.5A Fig. in line dashed The arises. dependence n nsgo gemn ewe aaadti ana- this and data between agreement good finds and lk eednedominates dependence -like and 10 (H = safnto of function a as ) 6 = ) b ˜ σ σ C·T/m [m cm] 3 xy xx xy 2 r ytmseicprmtr 2) ntelarge- the In (25). parameters system-specific are -2 -1 µ 0 1 2 n n ahdln in line dashed and C 3 1 530 15 0 -15 -30 σ 0 e + hthsadominant a has that steceia oeta esrdfrom measured potential chemical the is ρ xy H [T] Z 1 = xy σ ρ 0 µ 3 (H xy xx  2 H diinly hs aaeesconfirm parameters these Additionally, . 1 0 NbP .9 ’ pt 1Taesonin shown are T 31 to up )’s 0 = H [T] -linear ntrsof terms in = η × (µ −51 (ne 10 H i /k 6 NbSb (B) and NbP (A) in /H e and meV +5 B ρ C/m xy B T C ) hw tt w-admdl(SI model two-band a to fit a shows hl ntelwfil ii a limit low-field the in while , ) 2 6- 6 3 0 -3 -6 e)(4.FrTP efind we TaP, For (34). meV) σ − 3 sacntn nteodrof order the in constant a is (b + xx ,  1/H ne η b σ and (−µ 0 xx 0mVand meV −40 0 /H

µ [m cm] 5 = H /H xy -0.4 -0.2 2 0.2 0.4 eedne(5.We (25). dependence o ed agrthan larger fields for 0 -linear σ i +11 = 3 1 530 15 0 -15 -30 0 .6 /k xy H [T] + A ρ xy × B and b ˜ 2) eobtain we (25), ahdln in line Dashed . T NbSb /H 10 µ 0 ) B, H [T] 2 ρ e.Ti is This meV. = 5 esrdat measured , 2 d xx where Insets, ) C/m 2 −40 , 2 +20 H 2,25). (24, , -linear safit a is 3 E meV, and , meV  -1.0 -0.5 0 0.5 1.0 F 0 [2] [3] 10 to is A -3 .Arw niaetelctosof locations the indicate Arrows (B). TaP for and nodes (A) linear NbP for node linear one for at measured at length correlation tan disorder the (r radius as cyclotron the that condition the the out important temperature. elevated become separate K at scatterings cleanly activated 60 thermally reader to around other the hard because remind is is we however, scale it K); temperature that 120 expected observed the of the (instead TaP, In two 1C). these fact, In where 4A. Fig. in plots 3A Fig. in estimated the the with NbP sistent In a respectively. above of TaP, occur and to order expected or are scale These temperature centers. guiding of tories V obtain we carriers), and the of most provide will the val- of the larger Taking potential). for chemical ues local in fluctuation (typical where than smaller a estimates is 25 radius ref. length, cyclotron above correlation the disorder fields the when at Specifically, phosphides 2). the (Fig. on observations with i.6. Fig. (v velocities Fermi using respectively, TaP, and NbP of order n,sc httedniyo ttsa h em ee ssmall, is level Fermi the at states of dop- density small the with dispersion that to linear point such have key ing, systems The these framework. that this is within note of understood dependence be temperature also the can Finally, 4. Fig. with an imply automatically (38). TaP in the revealed that were note that to faults interesting stacking is It of 33). 27, values (17, values reported the AB

[ /FU T] 0 ecnas xrc h iodrcreainlnt (ξ length correlation disorder the extract also can We field-independent a is MR linear to Central ent htlna Ri xetdt iapa for disappear to expected is MR linear that note We B field-independent a Meanwhile, -5.4 -4.8 -4.2 2)we nlsi cteigdgae h qezdtrajec- squeezed the degrades scattering inelastic when (25) -6 θ V H 0 0 300 200 100 0 0 µ 10 (A tan µ eoe edidpnet At independent. field becomes ≈ -4 steceia oeta and potential chemical the is r rmtemgei ucpiiiyfi frTPtkn the taking TaP (for fit susceptibility magnetic the from and ξ C 10 n ihtetmeauedpnec fteKohler the of dependence temperature the with and θ µ r ossetwt h eghsae o eet and defects for scales length the with consistent are = H 0 VfrTaP. for mV µ H B) (T mv eB = aus ic h alywt agrFrisurface Fermi larger with valley the since values, NbP χ T PNAS F ) .Sldln sfi oEq. to fit is line Solid T. 1 T [K] vs. tan esrdat measured n ti siae ob 6n n 4n for nm 14 and nm 26 be to estimated is it and cl nwihnnoooiiyi observed is nonmonotonicity which on scale T NbSb θ | T ntepopie rd n nioae (blue), antimonates and (red) phosphides the in H min coe 6 2018 16, October H ≈ 2 lna Ra ihfils consistent fields, high at MR -linear √ 27π µ 2 1 V H k = 0 0 300 200 100 0 B 0 S  1mVand meV 51 ≈ tan T eist ierpdy(Fig. rapidly rise to begins eV 90 V µ T clsas orsodto correspond also scales TaSb | min 0 0 θ hc gives which 2, H H H  C n 2 o NbP for K 120 and K o.115 vol. stedsre strength disorder the is S = 2 so h aeorder same the of is ) 3/2 n an and ≈ V V H µ T [K] 8 , 0 0 T tan 2 S = ≈ the , cl rmMR, from scale ,aoewhich above T, T TaP | clsaecon- are scales min 1mVfrtwo for meV −11 7 θ tan o 42 no. H H VfrNbP for mV µ mi text). (main ρ consistent , ξ -linear xx = θ F H so the on is H s from ’s) 0meV −40 S k Fg 3) (Fig. | from ) B as ' T 10573 8 -4.8 -4.5 -4.2 -3.9 ρ [4] 10  xy T -4

APPLIED PHYSICAL SCIENCES 2 ∼µ . Increasing the temperature T allows the system to access The field dependence of ρxy is also informative. With small states within kB T of the chemical potential and (since the density deviation from perfect compensation, one expects (SI Appendix, of states grows rapidly with energy), greatly enhances the num- section 4) that ρxy ∼ H for systems with linear dispersion, but ber of states that can participate in transport. We thus conclude for quadratic dispersion one expects ρxy ∼ H at low fields, with a 3 that increasing temperature can increase σxx through this density crossover to ρxy ∼ H at higher fields. The data in Fig. 5 are more of states effect, consistent with the observed monotonic decline consistent with the latter behavior, suggesting that the NbSb2 0 in ρxx with increasing temperature (Fig. 3 A–D, Insets). should be thought of as compensated semimetals with effec- These, together, establish that all salient observed features tively quadratic dispersion (i.e., appreciable band curvature on of the phosphides can be explained by an uncompensated the scale of the doping level). This conclusion is also consistent spin–orbit locked semimetal with linear dispersion; this is cor- with the magnetic susceptibility data, which are reminiscent of roborated by our linear MR-type magnetotransport expected the Landau diamagnetism of quadratic bands. from guiding center diffusion that is particularly pronounced in semimetals with linear dispersion (25). Moreover, a systematic Summary combination of thermodynamic and magnetotransport measure- We investigated magnetotransport and magnetic susceptibility ments can allow us to extract parameters such as chemical of four different semimetals. We find that the combination of potential (or doping level) and typical disorder strength and cor- susceptibility and magnetotransport measurements allows us to relation length. These enable us to make a direct correspondence cleanly characterize the nonsaturating behavior of MR. The with a microscopic guiding center description, e.g., identification two phosphide materials that we study (NbP and TaP) are well of fields above which linear MR dominates and identification of described by a model of uncompensated semimetals with lin- temperature scales below which tan θH becomes large and H ear dispersion, wherein the MR is well described by guiding independent. center diffusion with tan θH  1 and field independent. The The antimonates also exhibit diamagnetism, indicating that combination of measurements that we have made also allows these materials are also spin–orbit coupled, but their magnetic us to extract the disorder strength and disorder correlation susceptibility is not well fitted by an expression of the type of length in these materials, as well as the doping level. Mean- Eq. 2. Instead, the susceptibility is mostly temperature indepen- while, the antimonates (NbSb2 and TaSb2) are well described dent, closer to the expectation for Landau diamagnetism for as compensated semimetals governed by a two-band model quadratic bands (36). These materials also exhibit a MR that is with effectively quadratic bands and tan θH  1. The criteria 2 ∼H . These facts, as well as the smallness of tan θH in the anti- reported here highlight a distinct set of traits for nonsatu- monates, are all well explained if we postulate that these systems rating MR and will serve as a primary touchstone to classify are compensated semimetals described by a two-band model MR phenomena in materials, which in turn will provide design (SI Appendix, section 1). In compensated semimetals, tan θH is principles for material platforms and devices for technological small, as long as mobilities of carriers remain in a similar range, application. and the MR is ∼H 2, consistent with observation. The MR is nonsaturating for perfect compensation, but will eventually sat- ACKNOWLEDGMENTS. We thank David Graf for technical assistance with ∼1/δn2 δn the high-field measurement. This work was supported by the University of urate at a value , where is the difference between Colorado Boulder Office of Research Innovation. High–magnetic-field data electron and hole densities. We ascribe the lack of saturation of were obtained at the National High Magnetic Field Laboratory, which is MR up to 31 T to the systems being close enough to compen- supported by National Science Foundation Cooperative Agreement DMR- sation that we do not hit the saturation value at experimentally 1157490 and the State of Florida. This research was sponsored in part by the Army Research Office and was accomplished under Grant W911NF-17- accessible fields. Here we note that, despite high mobilities of 1-0482 (to Y.-P.L. and R.M.N.). J.C.W.S. acknowledges the support of the both carriers in antimonates (SI Appendix, Table S1) that sat- Singapore National Research Foundation (NRF) under NRF Fellowship Award isfy the condition of νB  1, tan θH is found to be much less NRF-NRFF2016-05. The views and conclusions contained in this document than unity. This implies that the system effectively remains in the are those of the authors and should not be interpreted as representing the ω τ  1 official policies, either expressed or implied, of the Army Research Office limit of c and the MRs of the antimonates should not be or the US Government. The US Government is authorized to reproduce and saturated within the experimentally accessible field range of this distribute reprints for Government purposes notwithstanding any copyright work. notation herein.

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