Nonsaturating large magnetoresistance in semimetalsIan 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, Singapore637371; cCenter for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309; and dCenter for Experiments on Quantum Materials, Universityof 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-logical semimetals (TSMs) displays unique transport properties,including a striking dependence of resistivity on applied magneticfield, that are of great interest for both scientific and technologi-cal reasons. So far, many possible sources of extraordinarily largenonsaturating magnetoresistance have been proposed. However,experimental signatures that can identify or discern the dominantmechanism and connect to available theories are scarce. Here wepresent the magnetic susceptibility (χ), the tangent of the Hallangle (tan θH), along with magnetoresistance in four differentnonmagnetic semimetals with high mobilities, NbP, TaP, NbSb2,and TaSb2, all of which exhibit nonsaturating large magnetore-sistance (MR). We find that the distinctly different temperaturedependences, χ(T), and the values of tan θH in phosphides andantimonates serve as empirical criteria to sort the MR from dif-ferent origins: NbP and TaP are uncompensated semimetals withlinear dispersion, in which the nonsaturating magnetoresistancearises due to guiding center motion, while NbSb2 and TaSb2are compensated semimetals, with a magnetoresistance emergingfrom nearly perfect charge compensation of two quadratic bands.Our results illustrate how a combination of magnetotransportand susceptibility measurements may be used to categorize theincreasingly ubiquitous nonsaturating large magnetoresistance inTSMs.

nonsaturating magnetoresistance | magnetic susceptibility |Weyl semimetals | topological semimetals

Magnetoresistance (MR) and the Hall effect are versatileexperimental probes in exploring electronic properties of

materials, such as carrier density, mobility, and the nature ofscattering and disorder. In typical nonmagnetic and semicon-ducting materials, the MR increases quadratically with appliedtransverse magnetic field and saturates to a constant value whenthe product of the applied field and the mobility (ν) approachesunity. Nonsaturating MR is commonly attributed to the semiclas-sical two-band model, where electron-like and hole-like carriersare nearly compensated (1), resulting in rich magnetotransportcharacteristics that are strongly temperature (T ) and appliedtransverse magnetic field (H ) dependent in nonmagnetic com-pounds. A flurry of interest in nonsaturating, H -linear MR(2–5) in narrow-gap semiconductors led to two main theoreticalaccounts: (i) a 2D simple four-terminal resistor network model,where strong disorder or inhomogeneity of the sample manifestsas charge and mobility fluctuations (6, 7), and (ii) the so-called“quantum linear MR” which emerges in systems with linear bandcrossings when the lowest Landau level is occupied (8). Theformer approach has provided a basis to engineer large magneto-transport responses via macroscopic inhomogeneities or disorder(9–11). Meanwhile, the latter has remained rather elusive untilrecently.

Interest in nonsaturating very large MR has exploded fol-lowing the discovery of topological semimetals. These materialsare regularly reported to exhibit record-high nonsaturating MR,known as extreme magnetoresistance (XMR), with unusuallyhigh mobilities for bulk systems (12–18) and relatively low resid-ual resistivity ρ0. The proximity of the chemical potential to

the charge neutrality point in semimetals allows the genericquadratic two-band model to describe the MR and the Halleffect in reasonable levels (19–23). However, a two-band modelof this form generically predicts a MR that is quadratic in appliedfields, whereas the materials frequently exhibit a MR linear inthe applied field. While various theoretical proposals for H lin-ear MR have been advanced (e.g., refs. 8, 24–26), the originsof extreme MR in topological semimetals remain unclear. Asnonsaturating, large MR becomes more ubiquitous, it becomesparticularly urgent to identify a set of distinct attributes thatenable the delineation of their origins.

In this article, we systematically examine the low-field dia-magnetic susceptibility (χ), the transverse MR, and the Halleffect as a function of T and H in four different semimetalswith high mobility (ν≥ 104 cm2/Vs) and very large nonsaturatingMR—NbP, TaP (phosphides), NbSb2, and TaSb2 (antimonates).Characteristic parameters related to magnetic transport aresummarized in Table 1.

We present two different types of nonsaturating large MRidentified by the temperature (T ) dependence of diamagneticsusceptibility, χ(T ), and the H dependence of the Hall angle,tan θH =

ρxyρxx

=σxyσxx

, where ρxx and ρxy are longitudinal andHall resistivity, respectively, and σxx and σxy are correspondingconductivities.

One type of MR originates from the presence of smooth dis-order that governs guiding center motion of charge carriers.The linear H dependence of this type arises from the squeezed

Significance

The intensive recent investigations of topological semimetalshave revealed a large number of semimetal compounds thatexhibit very large nonsaturating magnetoresistance. Multi-ple mechanisms for this magnetoresistance phenomenon havebeen theoretically proposed, but experimentally it is unclearhow to identify which mechanism is responsible in a par-ticular sample or how to make a clean connection betweenexperimental observations and theoretical models. Our resultsshow that the magnetic susceptibility and the tangent of theHall angle successfully capture the fundamental differencesin seemingly similar nonsaturating large magnetoresistance,where charge compensation, energy dispersion, and the rolesof disorder are markedly distinct, and provide empirical tem-plates to characterize the origins of the extraordinary mag-netotransport properties in the newly discovered topologicalsemimetals and beyond.

Author contributions: M.L. designed research; I.A.L., Y.-P.L., P.E.S., A.C.T., R.M.N., and M.L.performed research; I.A.L., Y.-P.L., J.C.W.S., R.M.N., and M.L. analyzed data; and I.A.L.,Y.-P.L., J.C.W.S., R.M.N., and M.L. wrote the paper. y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: minhyea.lee@colorado.edu.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1808747115/-/DCSupplemental.y

Published online October 3, 2018.

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Table 1. Summary of magnetotransport data

Sample tan2θH ν, T−1 ρ0, µΩ cm ∆ρ/ρ0

NbP 7.6 99 0.5 561TaP 5.8 3.5× 103 0.2 20,200NbSb2 ≤10−4 1.9–2.5 0.1 27,800TaSb2 ≤10−4 2.2–4.3 0.1 5,560

Residual resistivity ρ0 at zero field is reported at T = 2 K and ∆ρ/ρ0 =

(ρ(H)− ρ0)/ρ0 and tan2θH at 0.3 K and µ0H = 15 T.

trajectories of carriers in semiclassically large magnetic fieldsνB ≥ 1 (easily achieved in linearly dispersing topologicalsemimetals, e.g., Table 1) and does not require the involvementof multiple bands for the charge compensation. The other typeof MR comes from charge compensation in the two-band modeland it accompanies other transport and magnetic characteristicswithin the conventional framework.

Using a combination of magnetic susceptibility and magne-totransport measurements to interrogate the different facets ofmagnetoresponse, we are able to categorize the phosphides intothe former and the antimonates into the latter. Our results can besummarized as follows and are depicted in Fig. 1: (i) In the phos-phides, the magnitude of tan θH saturates to an H -independentconstant at low temperatures when H >HS ' 8 T, while ρxx (T )has a peculiar H -dependent nonmonotonic form. The measuredMR defined as ∆ρ/ρ0∝Hα at low T exhibits a crossover fromquasi-linear (α∼ 1.5± 0.1) to linear (α∼ 1.0± 0.1), where thecrossover field, HS is set by the scale at which tan θH (H ) satu-rates. In H >HS , MR remains linear in H up to µ0H = 31 T,the highest applied field in this study. Finally, χ(T )s for thephosphides exhibit a pronounced minimum at Tmin. All of thesefeatures can be explained if we assume that the phosphidesare semimetals with linear dispersion, even without invokingcompensation, and that the MR arises due to guiding centermotion (see, e.g., ref. 25 for a recent discussion). Moreover,χ(T ) allows us to extract doping levels relative to the charge neu-trality point as fit parameters. (ii) Meanwhile, in the antimonates,the Hall angle remains close to zero (<10−2) at all accessiblefields in this study. The MR is nearly quadratic in H from roomtemperature down to T = 0.3 K, obeying Kohler’s rule. Thefield dependence of the Hall resistivity strongly deviates from

Fig. 1. Schematically depicted nonsaturating MR phenomena and repre-sentative energy dispersions for phosphides (TaP) (Left) and antimonates(TaSb2) (Right). The phosphides’ MR is characterized by quasi-linear to lin-ear transition as H increases, while the antimonates’ MR is characterizedby persistent quadratic H dependence, arising from semiclassical chargecompensation. Each bar indicates ∆ρ/ρ0 = 5× 105%. Magnetic field wasapplied up to µ0H = 31 T at T = 0.3 K.

-30 -15 0 15 300H [T]

-4

-2

0

2

4

tan

H

NbP

NbSb2

-30 -15 0 15 300H [T]

-4

-2

0

2

4

TaP

TaSb2

0 50 100 150 200 250 300T [K]

0

2

4

6

8

tan

2H

tan

2H

NbP

TaP

NbSb2

TaSb2

0 100 200 300T [K]

10-3

10-1

101

T = 0.3 K

NbP

TaP

NbSb2

TaSb2

0.3 K

A B

C

Fig. 2. (A and B) The Hall angle tan θH (A) for NbP (red) and NbSb2 (green)as a function of H and (B) for TaP (magenta) and TaSb2 (black), measuredat T = 0.3 K. Strong quantum oscillations in phosphides result in spike-likefeatures. (C) tan2θH measured at µ0H = 7 T as a function of T . (Inset) tan2θH

vs. T plotted in log – scale in the y axis scale, where NbSb2 and TaSb2 dataare clearly resolved.

linearity in the antimonates. The diamagnetic susceptibility forthe antimonates is mostly T independent. These features of theantimonates are archetypical for compensated semimetals withthe usual quadratic bands.

Our finding is well consistent with existing electronic struc-ture calculations: NbP and TaP have been studied thoroughlyvia first-principle calculations and photoemission studies (17, 27,28), where multiple Weyl nodes were identified in the vicinityof the Fermi energy. The calculations for NbSb2 and TaSb2 arealso consistent with our picture of nearly compensated semimet-als (29), yet the photoemission studies are not yet available forthe antimonates.

Single crystals of NbP, TaP, NbSb2, and TaSb2 were grownusing the chemical vapor transport method following a knownsynthesis procedure (17, 30–32). Standard electrical contactswere made directly on single crystals, using Ag paint (Dupont4966) with contact resistance ranges of ≤1−2 Ω. The magneto-transport measurements were performed with the applied fieldperpendicular to the direction of current on the plane up to 31 Tdown to 0.3 K. Magnetic susceptibilities of the samples weremeasured by the Magnetic Properties Measurements System byQuantum Design.

ResultsMagnetotransport. Fig. 2 A and B displays tan θH as a function ofH at T = 0.3 K. In the high-field limit, the phosphides and theantimonates show sharply contrasting behavior: tan θH for NbPand TaP reaches large values saturating to around 2.5 when H >HS ' 8 T, while it remains two orders of magnitude smaller forthe antimonates (except near zero field). Strong Schubnikov–deHaas oscillations are apparent in both ρxx and ρxy that generatespike-like features in the phosphides. In the antimonates, quan-tum oscillations emerge as well, but only in higher fields and themagnitudes are much smaller due to the smaller Fermi surfacesof the antimonates (29, 33, 34).

The phosphides and antimonates also display contrastingT dependence in Hall angle. Fig. 2C shows the temperature

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dependenceof tan2θH measured atµ0H = 7 T. Strikingly, tan2θHrises rapidly above unity with decreasing T around 100 K and60 K, for NbP and TaP, respectively. In contrast, the antimonatesbehave in the opposite fashion: Upon decreasing temperature,tan θH rapidly decreases, giving values two orders of magnitudesmaller than the phosphides (Fig. 2C, Inset). We note that smallHall angles are frequently found in conventional metals andsemimetals (35), as well as in a wide range of XMR materials withhigh mobilities for both holes and electrons (19, 22).

The field dependence of tan θH plays a deciding role in deter-mining MR. For example, a field-independent tan θH indicatesthe field dependence of ρxx and ρxy should have the same func-tional form. In uncompensated systems, ρxy has an H -linear Hallcontribution (i.e., ρxy =µ0RHH , where RH is the normal Hallcoefficient), allowing a field-independent tan θH and thereforea nonsaturating H -linear ρxx . This is exactly what we observe inthe phosphides which exhibit H -linear ρxy as well as H -linear ρxx(below).

Furthermore, large tan θH can act to suppress resistivity andmorph its T dependence. To demonstrate this, we express ρxx interms of tan θH and σxx ,

ρxx =σxx

σ2xx +σ2

xy

= ρ′xx

(1

1 + tan2θH

), [1]

where ρ′xx ≡ 1σxx

. As evident in Eq. 1, when tan θH ≥ 1, theinverse relation between σxx and ρxx no longer holds.

The effect of large tan θH magnitude on ρxx (T ) is particu-larly pronounced for NbP and TaP in Fig. 3 A and B, wherethe T dependence of ρxx is plotted. Monotonic metallic Tdependence switches to nonmonotonic behavior as the fieldincreases, with a peak at a temperature that coincides with theonset of rapid increase of tan θH (Fig. 1C). Crucially, ρ′xx (asdefined in Eq. 1) plotted as dashed lines in Fig. 3 A and B,Insets, deviates significantly from the measured ρxx , reflectingthe large values of tan θH and the dominant role tan θH hasin ρxx (T ).

NbSb2 and TaSb2, however, display contrasting behavior plot-ted in Fig. 3 C and D. We first note that ρxx (T ) initially exhibitsa slight decrease in ρxx as T is lowered until the sudden rise,mimicking a metal–insulator-like transition. This behavior iscommonly observed in many XMR materials and ρxx continuesincreasing as T is lowered further. In contrast to phosphides inFig. 3 A and B, NbSb2 and TaSb2, however, display ρxx ≈ 1/σxx

as reflected by the near overlay of the dashed lines and solid lines

A B

C D

Fig. 3. ρxx(T) at different Hs is shown for (A) NbP, (B) TaP, (C) NbSb2, and(D) TaSb2. ρ′xx(T) at µ0H = 7 T, defined in Eq. 1, is plotted in A–D, Insets.

106 108 1010

100

102

104

106

(H)/

0 [%]

300

150

80 2

0.3 K

NbP

106 108 1010

100

102

104

106

TaP

106 108 1010

0H/

0 [T/ m]

10-2

102

106

(H)/

0 [%]

NbSb2

106 108 1010

0H/

0 [T/ m]

10-2

102

106

TaSb2

H

H2 H2

HA B

C D

Fig. 4. (A–D) Kohler’s plots of phosphides in (A) NbP and (B) TaP andof antimonates in (C) NbSb2 and (D) TaSb2. Arrows in A and B indi-cate the locations of HS, where tan θH saturates and the MR switches toH linear.

in Fig. 3 C and D, Inset. This is consistent with a small tan θH 1. For all samples, the rapid rise in ρ′xx at low temperaturescorresponds to a plummeting σxx .

We now turn to the field dependence of MR. We showKohler’s plots in Fig. 4 and find that ∆ρ/ρ0∝Hα, where ρ0 =ρxx (T ,H = 0), collapses into a single curve over the large Trange. In the antimonates, we observe α≈ 2 in the entire tem-perature range up to 31 T. In contrast, in the phosphides theexponent α deviates from 2 even at low H (α≈ 1.4± 0.1) andswitches over to the linear-H dependence (α≈ 1) in the µ0H ≥8 T, where the tan θH approaches a constant value.

Finally, ρxy(H )’s of NbP and NbSb2 are compared in Fig. 5 Aand B. The field dependence of NbSb2, shown in Fig. 5B, is farfrom linear and a higher power of H becomes more visible withincreasing H . ρxy in TaP and TaSb2 showed similar behavior aspresented in SI Appendix, Fig. S2.

Magnetic Susceptibility. In Fig. 6, we plot the T dependence ofthe magnetic susceptibility in the low-field limit. All four samplesshow negative susceptibilities, corresponding to diamagnetism.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 yetpronounced minima emerging at Tmin = 203 K and Tmin = 68 K,respectively. For TaP the minimum susceptibility (Fig. 6) andthe resistivity peak under the field (Fig. 3B) both occur at sim-ilar temperatures, which are also close to the temperature wheretan2θH first becomes appreciable (Fig. 2C). For NbP, these tem-peratures are within a factor of 2, although the agreement is notas close as for TaP.

On the other hand, χ(T ) for the antimonates remainsfeatureless and mostly constant.

DiscussionWe begin by discussing the magnetic susceptibility plots shownin Fig. 6. The absence of Pauli paramagnetism in all sam-ples indicates that we do not have spin degenerate bands andis strong evidence for spin–momentum locking, such that themagnetic susceptibility is dominated by orbital diamagnetism(36). As we now discuss, the additional features can be wellexplained if we postulate that the phosphides are uncompensatedsemimetals with a linear dispersion, whereas the antimonates arecompensated semimetals.

We discuss first the phosphides. In particular, we fit the χ(T )sof phosphides to the result of orbital magnetism in a linear

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-6 -3 0 3 60H [T]

-0.4

-0.2

0

0.2

0.4

xy [m

cm

]

NbP

-6 -3 0 3 6

0H [T]

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-0.5

0

0.5

1.0

NbSb2

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-2

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2

xy [m

cm

]

-30 -15 0 15 30

0H [T]

-0.4

-0.2

0

0.2

0.4

xy [m

cm

]

10-3A B

Fig. 5. ρxy (H) as a function of H in (A) NbP and (B) NbSb2, measured atT = 0.3 K. Note the difference of the magnitude of ρxy . Dashed line in Ashows a fit to Eq. 3 and dashed line in B shows a fit to a two-band model (SIAppendix, section 1). ρxy (H)’s up to 31 T are shown in A and B, Insets, wherethe small boxes correspond to the main panels.

dispersion (37), which is obtained from energy minimization forthe case of a simple linear band crossing (SI Appendix, section 5C)

χ(T ) =C

∫ ε0

0

η(µi/kBT )− η(−µi/kBT )

εdε, [2]

where η is the standard Fermi–Dirac distribution function, ε0 isa cutoff energy, µ is the chemical potential measured from EF tothe charge neutrality points, and C is a constant in the order ofunity. The data for NbP and TaP are well fitted (Fig. 6) by Eq.2. For NbP, we find one linear crossing point with µ=−40 meV,which is consistent with one of the reported locations of Weylpoints in NbP (−57 meV and +5 meV) (34). For TaP, we findtwo crossing points, µ1 =−51 meV and µ2 = +11 meV. This isagain remarkably consistent with the location of a pair of Weylnodes reported in photoemission data at −40 meV and +20 meV(28). The magnetic susceptibility data thus strongly suggestthat the phosphides should be understood as uncompensatedsemimetals with linear dispersion.

This basic conjecture is also consistent with all our observedtransport data on the phosphides. In particular, we note that inthe presence of smooth disorder, guiding centers can diffuse inan unusual way to naturally lead to an H -linear ρxx (24, 25).This arises when the cyclotron radius is smaller than the disordercorrelation length (at large enough fields), enabling the guidingcenter trajectories to become squeezed along the field direction,and exhibits a σxx that has a dominant 1/H dependence (25). Wenote that this scenario (which is operative only for uncompen-sated semimetals with smooth disorder) also gives an H -linearρxy . Indeed, writing ρxy in terms of σxx and σxy (25), we obtain

ρxy =σxy

σ2xy +σ2

xx

=ne/H

(ne/H )2 + (b0/H + b/H 2)2, [3]

where b0 and b are system-specific parameters (25). In the large-field limit, this yields H -linear ρxy , while in the low-field limit acubic H dependence arises. The dashed line in Fig. 5A is a fitto Eq. 3 and finds good agreement between data and this ana-lytic form, with ne = 1.9× 106 C/m3, b0 = 5.6× 105 C/m3, andb = 1.5× 106 C·T/m3. Additionally, these parameters confirmthat 1/H -like dependence dominates σxx for fields larger thanseveral teslas, as we find clearly in our data.

Central to linear MR is a field-independent tan θH , consistentwith observations on the phosphides at fields above HS ' 8 T(Fig. 2). Specifically, when the cyclotron radius is smaller thanthe disorder correlation length, ref. 25 estimates a tan θH as

tan θH ≈2√27π

(µ

eV0

)3/2, [4]

where µ is the chemical potential and V0 is the disorder strength(typical fluctuation in local chemical potential). Taking the val-ues for µ from the magnetic susceptibility fit (for TaP taking thelarger of the µ values, since the valley with larger Fermi surfacewill provide most of the carriers), we obtain V0≈ 7 mV for NbPand V0≈ 10 mV for TaP.

We note that linear MR is expected to disappear for kBTV0 (25) when inelastic scattering degrades the squeezed trajec-tories of guiding centers. These are expected to occur above atemperature scale or order of V0

kB≈ 90 K and 120 K for NbP

and TaP, respectively. In NbP the estimated T scales are con-sistent with the T scale on which nonmonotonicity is observedin Fig. 3A and with the temperature dependence of the Kohlerplots in Fig. 4A. In fact, these two T scales also correspond towhere tan θH (T ) measured at HS begins to rise rapidly (Fig.1C). In TaP, the observed temperature scale is around 60 K(instead of the expected 120 K); however, we remind the readerthat it is hard to cleanly separate out the V0 scale from MR,because other thermally activated scatterings become importantat elevated temperature.

We can also extract the disorder correlation length (ξ) fromthe condition that the cyclotron radius (rC ) is of the same orderas the disorder correlation length at HS ≈ 8 T, above whichtan θH becomes field independent. At H =HS , the ξ is on theorder of rC = mvF

eBand it is estimated to be 26 nm and 14 nm for

NbP and TaP, respectively, using Fermi velocities (vF ’s) fromthe reported values (17, 27, 33). It is interesting to note that thevalues of ξ are consistent with the length scales for defects andstacking faults that were revealed in TaP (38).

Meanwhile, a field-independent tan θH and an H -linear ρxyautomatically imply an H -linear MR at high fields, consistentwith Fig. 4. Finally, the temperature dependence of ρxx (Fig. 3)can also be understood within this framework. The key point tonote is that these systems have linear dispersion with small dop-ing, such that the density of states at the Fermi level is small,

0 100 200 300T [K]

-6

-5.4

-4.8

-4.2

[B

/FU

T]

10-4

NbP

NbSb2

0 100 200 300T [K]

-4.8

-4.5

-4.2

-3.9

10-4

TaP

TaSb2

Tmin

Tmin

A B

Fig. 6. (A and B) χ vs. T in the phosphides (red) and antimonates (blue),measured at µ0H = 1 T. Solid line is fit to Eq. 2, which gives µ=−40 meVfor one linear node for NbP (A) and µ1 = 51 meV and µ2 =−11 meV for twolinear nodes for TaP (B). Arrows indicate the locations of Tmin (main text).

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∼µ2. Increasing the temperature T allows the system to accessstates within kBT of the chemical potential and (since the densityof states grows rapidly with energy), greatly enhances the num-ber of states that can participate in transport. We thus concludethat increasing temperature can increase σxx through this densityof states effect, consistent with the observed monotonic declinein ρ′xx with increasing temperature (Fig. 3 A–D, Insets).

These, together, establish that all salient observed featuresof the phosphides can be explained by an uncompensatedspin–orbit locked semimetal with linear dispersion; this is cor-roborated by our linear MR-type magnetotransport expectedfrom guiding center diffusion that is particularly pronounced insemimetals with linear dispersion (25). Moreover, a systematiccombination of thermodynamic and magnetotransport measure-ments can allow us to extract parameters such as chemicalpotential (or doping level) and typical disorder strength and cor-relation length. These enable us to make a direct correspondencewith a microscopic guiding center description, e.g., identificationof fields above which linear MR dominates and identification oftemperature scales below which tan θH becomes large and Hindependent.

The antimonates also exhibit diamagnetism, indicating thatthese materials are also spin–orbit coupled, but their magneticsusceptibility is not well fitted by an expression of the type ofEq. 2. Instead, the susceptibility is mostly temperature indepen-dent, closer to the expectation for Landau diamagnetism forquadratic bands (36). These materials also exhibit a MR that is∼H 2. These facts, as well as the smallness of tan θH in the anti-monates, are all well explained if we postulate that these systemsare compensated semimetals described by a two-band model(SI Appendix, section 1). In compensated semimetals, tan θH issmall, as long as mobilities of carriers remain in a similar range,and the MR is ∼H 2, consistent with observation. The MR isnonsaturating for perfect compensation, but will eventually sat-urate at a value ∼1/δn2, where δn is the difference betweenelectron and hole densities. We ascribe the lack of saturation ofMR up to 31 T to the systems being close enough to compen-sation that we do not hit the saturation value at experimentallyaccessible fields. Here we note that, despite high mobilities ofboth carriers in antimonates (SI Appendix, Table S1) that sat-isfy the condition of νB 1, tan θH is found to be much lessthan unity. This implies that the system effectively remains in thelimit of ωcτ 1 and the MRs of the antimonates should not besaturated within the experimentally accessible field range of thiswork.

The field dependence of ρxy is also informative. With smalldeviation from perfect compensation, one expects (SI Appendix,section 4) that ρxy ∼H for systems with linear dispersion, butfor quadratic dispersion one expects ρxy ∼H at low fields, with acrossover to ρxy ∼H 3 at higher fields. The data in Fig. 5 are moreconsistent with the latter behavior, suggesting that the NbSb2should be thought of as compensated semimetals with effec-tively quadratic dispersion (i.e., appreciable band curvature onthe scale of the doping level). This conclusion is also consistentwith the magnetic susceptibility data, which are reminiscent ofthe Landau diamagnetism of quadratic bands.

SummaryWe investigated magnetotransport and magnetic susceptibilityof four different semimetals. We find that the combination ofsusceptibility and magnetotransport measurements allows us tocleanly characterize the nonsaturating behavior of MR. Thetwo phosphide materials that we study (NbP and TaP) are welldescribed by a model of uncompensated semimetals with lin-ear dispersion, wherein the MR is well described by guidingcenter diffusion with tan θH 1 and field independent. Thecombination of measurements that we have made also allowsus to extract the disorder strength and disorder correlationlength in these materials, as well as the doping level. Mean-while, the antimonates (NbSb2 and TaSb2) are well describedas compensated semimetals governed by a two-band modelwith effectively quadratic bands and tan θH 1. The criteriareported here highlight a distinct set of traits for nonsatu-rating MR and will serve as a primary touchstone to classifyMR phenomena in materials, which in turn will provide designprinciples for material platforms and devices for technologicalapplication.

ACKNOWLEDGMENTS. We thank David Graf for technical assistance withthe high-field measurement. This work was supported by the University ofColorado Boulder Office of Research Innovation. High–magnetic-field datawere obtained at the National High Magnetic Field Laboratory, which issupported by National Science Foundation Cooperative Agreement DMR-1157490 and the State of Florida. This research was sponsored in part bythe Army Research Office and was accomplished under Grant W911NF-17-1-0482 (to Y.-P.L. and R.M.N.). J.C.W.S. acknowledges the support of theSingapore National Research Foundation (NRF) under NRF Fellowship AwardNRF-NRFF2016-05. The views and conclusions contained in this documentare those of the authors and should not be interpreted as representing theofficial policies, either expressed or implied, of the Army Research Officeor the US Government. The US Government is authorized to reproduce anddistribute reprints for Government purposes notwithstanding any copyrightnotation herein.

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