Main

A Weyl semimetal (WSM) is a three-dimensional analogue of graphene, in which the conduction and valence bands cross near the Fermi energy. The band-crossing point, the so-called Weyl point, acts as a magnetic monopole (a singular point of Berry curvature) in momentum space and always comes in a pairs. Unusual transport properties and surface states such as Fermi arcs are predicted, stimulating strong interest in realizing the WSM state in real materials2,12,13. If the time-reversal and inversion symmetries are respected, a pair of Weyl points can become degenerate in energy as a result of the crystal symmetry, forming another topological phase called a Dirac semimetal14,15. WSMs and Dirac semimetals usually exhibit very high mobilities, possibly attributed to the high Fermi velocity of massless Dirac states, as observed in transport experiments (such as Cd3As2; refs 16, 17). Generally, semimetals are new platforms to realize a huge magnetoresistance (MR; refs 18, 19), an effect that has been pursued intensively in emerging materials in recent years, because of its significant application in state-of-the-art information technologies20. Electrical transport in a semimetal usually consists of two types of carriers (electrons and holes), leading to a large MR when a magnetic field is applied at an electron–hole resonance1,21. In a simple Hall effect set-up, the transverse current carried by a particular type of carrier may be non-zero, although no net transverse current flows when the currents carried by the electrons and holes compensate for each other. These non-zero transverse currents will experience a Lorentz force caused by the magnetic field in the inverse-longitudinal direction. Such a back flow of carriers eventually increases the apparent longitudinal resistance, resulting in an extremely high MR that is much stronger than that in normal metals and semiconductors. Thus, it is crucial to obtain high-purity samples to realize a balance between electrons and holes and a high carrier mobility (μ) as well, both of which will enhance the MR effect.

Elemental Bi (refs 22, 23, 24, 25) and WTe2 (ref. 1) exhibit a high MR as typical examples of semimetals, in which electron and hole pockets coexist on the Fermi surface (Fig. 1a). There is a special type of semimetal whose conduction-band bottom and valence-band top touch the Fermi surface at the same point in momentum (k) space (Fig. 1b, c). Many such semimetals exhibit a high carrier mobility and relatively large MR, with a linear dependence on the magnetic field, such as zero-gap topological-insulator silver chalcogenides26,27 and Heusler compounds28,29,30, the Dirac semimetal Cd3As2 (μ = 9 × 106 cm2 V−1 s−1 at 5 K, MR = 1,500% at 1.5 K and 14.5 T; refs 16, 31, 32, 33), and the WSM TaAs (μ = 5 × 105 cm2 V−1 s−1 at 2 K, MR = 5.4 × 105% at 10 K and 9 T; ref. 34). The high mobility may originate from the linear or nearly linear energy dispersion. The unsaturated linear MR is interpreted as a classical effect due to the strong inhomogeneity in the carrier density35 or as a quantum effect due to the linear energy dispersion at the band touching point36. The semimetal NbP combines the main features of the WTe2-type (showing extremely large MR) and Cd3As2-type (showing ultrahigh mobility) semimetals in the band structure (Fig. 1d), exhibiting hole pockets from normal quadratic bands and electron pockets from linear Weyl bands. As we will see, NbP exhibits an ultrahigh carrier mobility, comparable to that of Cd3As2, and an extremely large MR, surpassing that of WTe2.

Figure 1: Band structure for different semimetals.
figure 1

Schematic illustration of different types of semimetals and representative materials. a, Normal semimetal with the coexistence of electron and hole pockets. b, Semimetal with quadratic conduction and valence bands touching at the same momentum point. c, Weyl semimetal and Dirac semimetal. d, Semimetal with one hole pocket from the normal quadratic band and one electron pocket from the linear Weyl semimetal band for NbP. The valence and conduction bands are indicated by blue and green shading, respectively. The Fermi energy is marked in each case by a horizontal line.

Full size image

The single crystal of NbP used for the present study and the respective X-ray diffraction patterns are shown in Fig. 2a. The crystal structure of NbP is non-centrosymmetric space group I41md (Fig. 2b). No indication of twinning was found in the diffraction experiments. Both atom types have the same coordination number of six, and the same coordination environment in the form of a trigonal prism. A detailed overview of the structural characterization is presented in the Supplementary Information. A measurement of the temperature dependence of the resistivity, ρxx(T), is a simple way to identify the electronic states of a material. On the basis of our high-quality single crystals of NbP grown via chemical vapour transport reactions, ρxx(T) is measured under various transverse magnetic fields ranging from 0 to 9 T, as shown in Fig. 2c. At zero field, we observe metallic behaviour with ρxx(300 K) = 73 μΩ cm and a residual resistivity ρxx(2 K) = 0.63 μΩ cm. This results in a residual resistivity ratio [ρxx(300 K)/ρxx(2 K)] = 115, which is directly related to the metallicity and quality of the crystal. Compared to other similar materials at low temperature (2 K), NbP exhibits a resistivity that is about 30 times lower than that of WTe2 (ref. 1) but 30 times higher than that of Cd3As2 (ref. 16). NbP does not become superconducting for temperatures above 0.10 K. After applying a magnetic field, we observe a remarkable change in the resistivity. ρxx(T) changes from a positive slope (metallic) to a negative slope (semiconducting) at a very small field of 0.1 T, and becomes completely semiconducting at a field of 2 T. This may be due to the opening of a gap at the Weyl point. In general, a conventional semimetal does not exhibit such behaviour, whereas some small-gap or gapless semimetals (for example, WTe2 (ref. 1) and Cd3As2 (ref. 16)) exhibit a similar trend, usually at very high fields and low temperature. Another important fact observed in the present material is that ρxx(T) also increases markedly owing to the application of magnetic fields at room temperature (300 K).

Figure 2: Crystal structure, magnetoresistance and mobility.
figure 2

a, Orientation of the measured single-crystal NbP with the respective X-ray diffraction axial oscillation patterns. b, Crystal structure of NbP in a body-centred-tetragonal lattice. c, Temperature dependence of the resistivity, ρxx, measured at different transverse magnetic fields, as labelled next to the corresponding curve. The inset of c shows the temperature dependence of resistance measured in zero field. d, Transverse magnetoresistance measured at different temperatures in magnetic fields up to 9 T. The inset shows the magnetoresistance at higher temperatures. e, Temperature dependence of the mobility (left ordinate) and the carrier density (right ordinate). The inset shows the evolution of the Hall coefficient with temperature. The temperature regimes where electrons and holes act as the main charge carriers are marked with blue and red shading, respectively. f, Magnetoresistance measured at different angles, θ, between the current (I) and the magnetic field (B) as shown schematically in the inset.

Full size image

We now focus on the MR measurement in NbP. The MR is commonly calculated as the ratio of the change in resistivity due to the applied magnetic field (H), [(ρ(H) − ρ(0))/ρ(0)] × 100%. Figure 2d shows the MR measured in transverse magnetic fields up to 9 T at different temperatures. At low temperatures, we find that NbP exhibits an extremely large MR = 8.5 × 105% at 1.85 K in a field of 9 T. This MR is five times as large as that measured for the same field in WTe2 (ref. 1) and nearly twice as large as that of TaAs (ref. 34), another WSM predicted in the same family as NbP (refs 10, 11). On increasing the temperature, the MR of NbP remains almost unchanged up to 20 K and then starts to decrease at higher temperatures. We note that the MR is still as high as 250% in a field of 9 T at room temperature (inset of Fig. 2d). We have also measured the dependence of the MR on the direction of the magnetic field. Figure 2f shows the MR observed at different angles (θ) between the field direction and the current direction. The MR decreases slightly from 8.5 × 105% at θ = 90° (transverse) to 2.5 × 105% at θ = 0° (longitudinal). Thus, the MR varies by a factor of only 3.4, implying a relatively isotropic nature of the material compared with the layered semimetal WTe2.

A large MR is usually associated with a high mobility. The carrier mobility and concentration are two important parameters of a material that can be derived from the Hall coefficient. We have performed Hall effect measurements in both temperature-sweep and field-sweep modes to improve the accuracy of our data. The field dependence of the Hall resistivity ρxy(H) exhibits a linear characteristic at high fields (see Supplementary Information). However, the nonlinear behaviour in low fields indicates the involvement of more than one type of charge carrier in the transport properties. As seen from the inset of Fig. 2e, NbP exhibits a negative Hall coefficient, RH(T), up to 125 K, which changes sign for temperatures above 125 K. For the sake of simplicity, we use the single-carrier Drude band model, ne, h(T) = 1/[eRH(T)], to calculate the carrier density and μe, h(T) = RH(T)/ρxx(T) to estimate the mobility, where ne(nh) and μe(μh) are the charge density and mobility of the electron (hole), respectively. We use the slope of ρxy(H) at high fields to calculate the Hall coefficient (Fig. 2e). The electron carrier concentration, ne, is found to be 1.5 × 1018 cm−3 at 1.85 K, and increases slowly with temperature, exhibiting a semimetal-like or very small gap-like behaviour. The mobility plays a major role in the charge transport in a material and consequently determines the efficiency of various devices. Here, NbP exhibits an ultrahigh mobility of 5 × 106 cm2 V−1 s−1 at 1.85 K. This value is close to that of Cd3As2 (9 × 106 cm2 V−1 s−1; ref. 16), one order magnitude higher than that of TaAs (ref. 34) at 2 K. We note that the current mobility is extracted from a simple one-band model that neglects anisotropy. This averaged mobility is below the record mobility measured in Bi (108 cm2 V−1 s−1; ref. 37). Furthermore, it has been shown that the mobility in Cd3As2 scales with the residual resistivity. Hence, it can be proposed that materials with a low residual resistivity exhibit high mobilities, with the present case being a good example.

The low-field measurements (Fig. 2d, f) show no saturation of the MR up to 9 T in the entire temperature range. To pursue this finding to even higher fields, we have performed transverse MR measurements up to 30 T in d.c. magnetic fields, as shown in Fig. 3a. The MR increases to 3.6 × 106% for a field of 30 T at 1.3 K, and still shows no tendency of saturation. We have further corroborated this trend of a non-saturating magnetoresistance by performing experiments in pulsed magnetic fields up to 62 T at 1.5 K (inset of Fig. 3a). The MR continues to increase with magnetic field up to 62 T, the maximum field reached in the pulsed magnetic field experiments, when we find a MR of 8.1 × 104 (or 8.1 × 106%). In a field as high as 60 T, we note that WTe2 (ref. 1) and Bi (ref. 25) are reported to exhibit a MR of 1.3 × 105 and 1 × 106, respectively. As already indicated in the 9 T MR data depicted in Fig. 2d, f, SdH quantum oscillations appear for T ≤ 30 K. At the lowest temperature (1.85 K) the oscillations start for fields as low as 1 T. Because SdH oscillations appear only when the energy spacing between two Landau levels is larger than their broadening due to disorder, we can estimate a lower limit to the quantum mobility of the carriers involved, μq > 105 cm2 V−1 s−1, in agreement with the large electron (transport) mobility extracted from the Hall effect.

Figure 3: High-field magnetoresistance and SdH oscillation.
figure 3

a, Transverse MR measured in static magnetic fields of up to 30 T at different temperatures. The inset of a shows the MR measured in a pulsed magnetic field up to 62 T. b, SdH oscillations after subtracting the background from the 9 T ρxx measurements. The inset of b shows the temperature dependence of the relative amplitude of Δρxx for the SdH oscillation at 8.2 T. The solid line is a fit to the Lifshitz–Kosevich formula. c, SdH oscillations after subtracting the background from the 30 T ρxx measurements.

Full size image

The physical parameters of the charge carriers are derived from the Fermi surface that is measured by the oscillations observed in the transport properties. Both ρxx and ρxy measured up to 9 T exhibit very clear SdH oscillations starting from 1 T. This indicates a very low effective mass, resulting in a high mobility. To obtain the amplitude of the SdH oscillations, Δρxx, from ρxx, we subtracted a smooth background. The results are plotted as a function of 1/(μ0H) at various temperatures in Fig. 3b. As expected, the oscillations are periodic in 1/(μ0H). They occur owing to the quantization of energy levels, which directly gives the effective mass of the charge carriers. We observe three different oscillation frequencies of F = 7 T, 13 T and 32 T from the data shown in Fig. 3b, which is consistent with the existence of tiny carrier pockets near the Weyl points in the calculated Fermi surface (Fig. 4). We note that these frequencies depend sensitively on the field direction. For example, the frequency of F = 32 T, which is equivalent to the periodicity 1/(μ0ΔH) = 0.03127 T−1, corresponds to a cross-sectional area of the Fermi surface AF = 0.003 Å−2 from the Onsager relationship F = (Φ0/2π2)AF, where Φ0 is the magnetic flux quantum. This is a tiny area, only 0.08% of the cross-sectional area of the first Brillouin zone. Supposing a circular cross-section, a very small Fermi momentum kF = 0.0312 Å−1 is obtained. The cyclotron effective mass of the carriers is determined by fitting the temperature dependence of amplitude of the oscillations to the Lifshitz–Kosevich formula,

where kB is Boltzmann’s constant, β = ehB/2πm∗ and TD = h/4π2τkB are the fitting parameters, which directly results in the effective mass m∗ and quantum lifetime τ of the charge carriers. The value of m∗ calculated from the temperature-dependent SdH oscillations in a field of 8.2 T is 0.076 m0, where m0 is the bare mass of the electron. This value is comparable to that reported for Cd3As2 (ref. 16). We also find a very large Fermi velocity vF = ℏkF/m∗ = 4.8 × 105 m s−1. The large Fermi velocity and low effective mass are responsible for the observed ultrahigh mobility in NbP.

Figure 4: Bulk band structures of NbP.
figure 4

a, Ab initio band structure. The dispersions along ky connecting a pair of Weyl points with opposite chirality are plotted for the W1- and W2-types of Weyl points in the right panels. Red and green circles represent positive and negative chiral Weyl points, respectively. The Fermi energy is shifted to zero. The valence and conduction bands are indicated by the red and blue lines, respectively. b, The first Brillouin zone in momentum space for the primitive unit cell. All twelve pairs of Weyl points are illustrated, in which W1 and W2-types of Weyl points are indicated. c, Bulk Fermi surfaces. The blue and red surfaces represent electron (W1 and W2) and hole (H) pockets, respectively. The Fermi energy is chosen as the slightly electron-doped case to qualitatively match the experiment.

Full size image

The SdH oscillations at different temperatures obtained from the 30 T dc magnetic field measurements are plotted in Fig. 3c. At the lowest temperature, marked by arrows, the SdH maxima at 8 T (0.125 T−1) and 11 T(0.091 T−1) start to split into two distinct peaks, and the maximum at 16 T (0.0625 T−1) develops into four peaks. This can be assigned to the lifting of the spin degeneracy and the degeneracy of the dual Weyl point.

To further understand the transport properties and verify the Weyl physics of NbP, we have performed ab initio band-structure calculations. NbP crystallizes in a body-centred-tetragonal lattice with the non-symmorphic space group I41md. The lack of inversion symmetry of the lattice leads to the lifting of spin degeneracy in the band structure. Near the Fermi energy, twelve pairs of Weyl points lie aside the central planes in the Brillouin zone, consistent with recent calculations10,11. Twelve pairs of Weyl points can be classified into two groups, labelled as W1 and W2 (Fig. 4b). The W1 pair that lies in the kz = 0 plane is lower in energy that the W2 pair off the kz = 0 plane. An important feature in the band structure is that the Fermi energy crosses the quadratic-type valence bands and also the linear Weyl-type conduction bands, leading to very small hole and electron pockets on the Fermi surface, respectively. Four equivalent hole pockets appear around the Z point (labelled as H in Fig. 4c). Four larger and eight smaller electron pockets exist near the Σ and N points, respectively, corresponding to W1 and W2 Weyl bands. This is consistent with multiple tiny Fermi surface areas with low effective masses extracted from SdH oscillations. We note that the specific shape of the Fermi surfaces relies sensitively on the position of the Fermi energy, which is determined by the dopant concentration of the sample. In Fig. 4c we set the Fermi energy as the ideal electron–hole compensation case. Furthermore, from the bulk to the surface, as a manifestation of topology, Fermi arcs exist between the projection of Weyl points with opposite chirality. Angle-resolved photoemission spectroscopy (ARPES) experiments are called for to verify NbP as a WSM by investigating the surface states.

As we have demonstrated here, NbP is an exotic semimetal with interesting transport properties. As seen in the band structure, it combines the electronic structures of a normal semimetal and a WSM together. Similar to normal semimetals such as WTe2 and Bi, NbP exhibits both electron and hole pockets at different positions in the Brillouin zone; however, unlike WTe2 and Bi, its electron pockets are relevant to the linear Weyl bands, from which the high mobility of the high-quality samples may originate. In contrast to Weyl semimetals (for example, TaAs) and Dirac semimetals (for example, Cd3As2), in which the Fermi energy may cross only one type of band (electron or hole), NbP naturally hosts both types of carriers and consequently exhibits a huge MR in an electron–hole resonance situation.

Methods

High-quality single crystals of NbP were grown via a chemical vapour transport reaction using iodine as a transport agent. Initially, a polycrystalline powder of NbP was synthesized by a direct reaction of niobium (Chempur 99.9%) and red phosphorus (Heraeus 99.999%) kept in an evacuated fused silica tube for 48 h at 800 °C. Starting from this microcrystalline powder, the single crystals of NbP were synthesized by chemical vapour transport in a temperature gradient starting from 850 °C (source) to 950 °C (sink) and a transport agent with a concentration of 13.5 mg cm−3 iodine (Alfa Aesar 99.998%; ref. 38). The orientation and crystal structure of the present single crystal were investigated using the diffraction data sets collected on a Rigaku AFC7 diffractometer equipped with a Saturn 724+ charge-coupled device detector (monochromatic Mo Kα radiation, λ = 0.71073 Å). Structure refinement was performed by full-matrix least-squares on F using the program package WinCSD.

The transport measurements were performed in various physical property measurement systems (PPMS, Quantum Design, ACT option, home build adiabatic demagnetization stage). The 30 T static magnetic field measurements were performed at the High Field Magnet Laboratory HFML-RU/FOM in Nijmegen, and the pulsed magnetic field experiments were carried out at the Dresden High Magnetic Field Laboratory HLD-HZDR; both laboratories are members of the European Magnetic Field Laboratory (EMFL).

The ab initio calculations were performed within the framework of density functional theory (DFT), implemented in the Vienna ab initio simulation package39. The core electrons were represented by the projector-augmented-wave potential and generalized gradient approximation (GGA) are employed for the exchange correlation functional. We interpolated the bulk Fermi surface using maximally localized Wannier functions (MLWFs; ref. 40).