SCIENCE CHINA
Physics, Mechanics & Astronomy
. Article .
Special Topic: New Unconventional Superconductors and Weyl Semimetal
May 2016 Vol. 59 No. 5: 657406
doi: 10.1007/s11433-016-5798-4
Large unsaturated positive and negative magnetoresistance
in Weyl semimetal TaP
JianHua Du1 , HangDong Wang2 , Qin Chen1 , QianHui Mao1 , Rajwali Khan1 , BinJie Xu1 ,
YuXing Zhou1 , YanNan Zhang1 , JinHu Yang2 , Bin Chen2 , ChunMu Feng1 , and MingHu Fang1,3*
1 Department
of Physics, Zhejiang University, Hangzhou 310027, China;
of Physics, Hangzhou Normal University, Hangzhou 310036, China;
3 Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
2 Department
Received January 25, 2016; accepted February 17, 2016; published online February 24, 2016
After successfully growing single-crystal TaP, we measured its longitudinal resistivity (ρxx ) and Hall resistivity (ρyx ) at magnetic
fields up to 9 T in the temperature range of 2-300 K. At 8 T, the magnetoresistance (MR) reached 3.28 × 105 % at 2 K, 176%
at 300 K. Neither value appeared saturated. We confirmed that TaP is a hole-electron compensated semimetal with a low carrier
concentration and high hole mobility of µh =3.71 × 105 cm2 /V s, and found that a magnetic-field-induced metal-insulator transition
occurs at room temperature. Remarkably, because a magnetic field (H) was applied in parallel to the electric field (E), a negative
MR due to a chiral anomaly was observed and reached −3000% at 9 T without any sign of saturation, either, which is in contrast
to other Weyl semimetals (WSMs). The analysis of the Shubnikov-de Haas (SdH) oscillations superimposed on the MR revealed
that a nontrivial Berry’s phase with a strong offset of 0.3958, which is the characteristic feature of charge carriers enclosing a Weyl
node. These results indicate that TaP is a promising candidate not only for revealing fundamental physics of the WSM state but
also for some novel applications.
Weyl semimetal, positive and negative magnetoresistance, Weyl fermions
PACS number(s): 71.30.+h, 73.43.Qt, 72.20.Jv
Citation:
J. H. Du, H. D. Wang, Q. Chen, Q. H. Mao, R. Khan, B. J. Xu, Y. X. Zhou, Y. N. Zhang, J. H. Yang, B. Chen, C. M. Feng, and M. H. Fang,
Large unsaturated positive and negative magnetoresistance in Weyl semimetal TaP, Sci. China-Phys. Mech. Astron. 59, 657406 (2016), doi:
10.1007/s11433-016-5798-4
1 Introduction
In Weyl semimetal (WSM) phase, the bulk electronic bands
disperse linearly along the momentum direction through a
node, called the Weyl point, in a three-dimensional (3D) analog of graphere. It can be viewed as an intermediate phase
between a trivial insulator and topological insulator [1-5]. A
number of candidates for a WSM have previously been proposed, such as Y2 Ir2 O7 [6] and HgCr2 Se4 [7] compounds in
which the magnetic order breaks the time-reversal symmetry,
and the LaBi1−x Sb x Te3 [8] compound in which fine-tuning
the chemical composition is necessary to break the inversion
symmetry. However, none of these compounds have been
used to realize a WSM experimentally. This is because the
magnetic domain is not large or it is very difficult to tune
the chemical composition within 5%. Very recently, the theoretical proposal [9, 10] for a WSM in a class of stoichiometric materials including TaAs, TaP, NbAs, and NbP that
break crystalline inversion symmetry was confirmed in the
experiments [11-15], except for TaP. This was due to the difficulty of growing large crystals of TaP. The exotic transport properties exhibited by these materials has ignited extensive interest in both the condensed matter physics and ma-
*Corresponding author (email: mhfang@zju.edu.cn)
c Science China Press and Springer-Verlag Berlin Heidelberg 2016
phys.scichina.com
link.springer.com
J. H. Du, et al.
Sci. China-Phys. Mech. Astron.
Intensity (103 a.u.)
terial science communities, especially because of their extremely large magnetoresistance (MR) and ultrahigh mobility
of charge carriers.
Materials with a large MR are used as magnetic sensors
[16], in magnetic memory [17], and in hard drives [18] at
room temperature. A large MR is an uncommon property,
and is mostly found in magnetic compounds. Examples include the giant magnetoresistance (GMR) [19] of Fe/Cr thin
films, and colossal magnetoresistance (CMR) of manganese
based perovskites [20, 21]. In contrast, ordinary MR, a relatively weak effect, is commonly found in non-magnetic compounds and elements [22]. Magnetic materials typically have
a negative MR. Positive MR is seen in metals, usually at the
level of a few percent, and in some semiconductors, such as
200% at room temperature in Ag2+δ (Te,Se) [23], comparable to materials showing CMR [24], and in some semimetals,
such as high-purity bismuth [22], graphite [25], and WTe2
[26], in which it can reach 4.5 × 104 %. In semimetals, the
very large MR is attributed to a balanced hole-electron “resonance” condition, as described in [26]. WSM provides another possibility to realize extremely large MR, as observed
in TaAs [13, 14], NaAs [27] and NbP [12]. Moreover, the
helical Weyl fermions with unprecedented mobility are well
protected from defect scattering by real spin conservation associated with the chiral Weyl nodes. Thus, searching for
more WSM compounds and characterizing their MR behavior is very interesting not only for fundamental physics but
also for practical applications.
TaP crystallizes in a body-centered tetragonal lattice with
the nonsymmorphic space group I41 md (No. 109), which
lacks inversion symmetry, as shown in Figure 1(a). It is a
member of the group predicted to have as a WSM in refs.
[9,10]. In this study, we successfully grew single-crystal TaP,
and then measured its longitudinal resistivity (ρ xx ) and Hall
resistivity (ρyx ). At 8 T its MR reached 3.28 × 105 % at 2
K and 176% at 300 K. Neither value appeared saturated. A
(O)
Figure 1 (Color online) (a) Crystal structure of TaP with a body-centered
tetragonal structure; (b) photo of TaP crystal; (c) XRD pattern of powder
obtained by grinding TaP crystals, the line shows its Rietveld refinement; (d)
energy dispersive X-ray spectrometer (EDXS) for the TaP crystal.
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magnetic-field-induced metal-insulator transition occurred at
room temperature. The Hall resistivity measurements confirmed that TaP is a hole-electron compensated semimetal
with a low carrier concentration and high mobility of µh =3.71
× 105 cm2 /V s. Remarkably, when a magnetic field (H) was
applied in parallel to the electric field (E), a negative MR due
to a chiral anomaly was observed that reached near −3000%
at 9 T without any sign of saturation. This is distinct from
other WSMs and may originate from the larger distance between Weyl nodes in the momentum space. We also confirmed that a nontrivial Berry’s phase with strong offset of
0.3958 realizes in TaP, which is the characteristic feature of
charge carriers enclosing a Weyl node.
2 Experimental
Single crystals of TaP were grown by using a chemical vapor
transport method. Previously prepared polycrystalline TaP
was used to fill in a quartz tube with 10 mg/cm3 iodine as
a transporting agent. After being evacuated and sealed, the
quartz tube was heated for 3 weeks in a tube furnace with
a temperature field of ∆T =(950-850)◦C. Large polyhedral
crystals with dimensions of up to 1.0 mm were obtained, as
shown in Figure 1(b). Energy dispersive X-ray spectrometry (EDXS) was used to determine the crystal composition,
and stoichiometric TaP was confirmed. The X-ray diffraction
(XRD) pattern [see Figure 1(c)] at room temperature of TaP
powder fabricated by grinding pieces of crystals confirmed
its tetragonal structure, and its Rietveld refinement (reliability
factor: Rwp = 7.18%, χ2 = 2.973) gave the lattice parameters a
= 3.3184(±0.0001) Å and c = 11.3388(±0.0001) Å. Electrical
resistivity and Hall resistivity measurements were carried out
by using a Quantum Design physical property measurement
system (PPMS).
3 Results and discussion
Figure 2 summarizes the resistivity (ρ xx ) results for TaP crystal measured at various temperatures (T) and in different magnetic fields (H), which were applied along the c axis and normal to the current. As shown in Figure 2(a), the temperature dependence of resistivity, ρ xx (T ), at H = 0 T displayed
a metallic behavior with ρ(300 K) = 77.98 µΩ cm, and ρ(2
K) = 1.933 µΩ cm. Thus, the residual resistivity ratio [RRR]
was estimated to be ρ(300 K)/ρ(2 K) ∼ 40. A crossover from
“metallic” to “insulating” behavior clearly occurred with the
application of magnetic field that appeared to be a magneticfield-induced metal-insulator transition. The resistivity increased with decreasing temperature, as it does in an insulator, but then saturated towards a field-dependent constant
value at the lowest temperature. This is more clearly seen
in the log-log plot in the main panel of Figure 2(a). Similar
behavior has also been observed in conventional semimetals,
such as high-purity Bi [22], graphite [28, 29], WTe2 [26] and
β-Ag2 Te [23, 30], and in the Dirac semimetal Cd3 As2 [31]
J. H. Du, et al.
Sci. China-Phys. Mech. Astron.
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(c)
(b)
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Figure 2 (Color online) (a) Temperature dependence of the longitudinal resistivity ρ xx (T ) measured at different magnetic fields for single-crystal TaP; (b)
MR scaling law with magnetic field. Magnetic field dependence of MR below 100 K (c) and above 100 K (d).
and other Weyl semimetals TaAs [10,14], NbAs [32] and NbP
[12, 15]. These similarities invite the interpretation that this
interesting behavior can be ascribed to the properties shared
by these compounds, a low carrier density and equal number
of electrons and holes (compensation). In contrast, metallic
behavior remains at higher temperatures, even with high field
in the WTe2 [26] and β-Ag2 Te [23, 30], the metal-insulator
transition occurs at room temperature above the 2 T field in
the semimetal TaP, as shown in the inset in Figure 1(a), as
well as in TaAs [14].
Figures 2(c) and (d) plot the relative change of the MR at
various temperatures as a function of the magnetic field based
∆ρ
on the typical definition of MR [33]: ρ(0)
= [ ρ(H)−ρ(0)
ρ(0) ]×100%,
where ρ(H) is the resistivity measured at H for each given
isotherm. At 2 K, the MR reached 3.28 × 105 % at 8 T, which
is two times larger than that in the recently reported compound WTe2 at 2 K in 8 T and did not appear saturated in the
highest field in our measurements, although the RRR of the
former was one to two orders of magnitude larger than that
of the latter. Remarkably, the MR at room temperature (300
K) reached 176% at 8 T, and increased almost linearly with
the field without any saturation. These indicate that TaP can
be used in the magnetic devices in the future. There was an
explicit transition from quadratic MR at low magnetic fields
to linear MR at high magnetic fields at a given temperature,
as expected theoretically for the WSMs [34]. The relative
MR of many metal and semimetals can be represented by the
form commonly referred to as Kohler’s rule [34], ∆ρ/ρ(0) =
F[H/ρ(0)], where F(H/ρ(0)) usually follows a power law. As
shown in Figure 2(b), although all of the MR data below 100
K collapsed onto the same curve, the MR did not follow a
simple power law. In addition, the MR data above 100 K did
not collapse this curve, which indicates that phonon scattering plays a role at higher temperatures. The following Hall
resistivity and SdH oscillation analysis help with understanding this complexity. Shubnikov-de Haas (SdH) oscillations
were superimposed on the MR. The SdH data were analyzed
in detail, as given below.
Here, we discuss the charge carrier information for a TaP
compound. Figure 3(a) shows the Hall resistivity as a function of the magnetic field, ρyx (H), measured at various temperatures. At higher temperatures (above 140 K), the positive slope of ρyx (H) indicates that the holes dominated the
main transport processes. At lower temperatures (below 140
K), even at the lowest temperature (2 K), the slope of ρyx (H)
changed from a positive to a negative value as the field increased, as shown in the inset of Figure 3(a). This indicates
the coexistence of both electron and hole carrier charges.
SdH oscillations superimposed on the ρyx (H) at higher fields
were also clearly observed below 10 K. Similar to ref. [14]
for the TaAs compound, we fitted the Hall conductivity tensor with σ xy =ρyx /(ρ2xx + ρ2yx ) by adopting a two-carrier model
[35], σ xy = [nh µ2h 1+(µ1h H)2 − ne µ2e 1+(µ1e H)2 ], where ne and nh denotes the carrier concentrations, and µe and µh denote the mobilities of the electrons and holes, respectively. Figure 3(b)
presents the fitting for σ xy data measured at five representative temperatures. The Hall conductivity above 180 K was
mainly from the hole band due to the thermal depopulation
of hole-like pockets just above the Fermi level. Thus, the nh
and µh values above 180 K were estimated by using a single
band fitting. The inset of Figure 3(b) shows the ne , nh , µe
and µh values obtained by fittings over the whole temperature
range. It is remarkable that the ne and nh values were almost
the same at lower temperatures (below 100 K). For example,
J. H. Du, et al.
Sci. China-Phys. Mech. Astron.
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(c)
(b)
(d)
Figure 3 (Color online) (a) Hall resistivity versus magnetic field from 2 to 300 K. A strong SdH oscillation was observed below 10 K. Inset: Hall resistivity
for the higher and lower temperatures in lower fields; (b) Hall conductivity σyx at five representative temperatures as a function of the field. The solid lines are
the fitting curves from the two-carrier model. Inset: Temperature dependence of the mobilities and carrier concentrations of the electrons and holes; (c) MR
versus the magnetic field measured at 2 K for various H alignments with respect to I; (d) MR versus the magnetic field measured at 2 K for H when near to
parallel to I.
such as at 2 K, ne = 2.58 × 1018 cm−3 and nh = 2.90 × 1018
cm−3 . This indicates that TaP is indeed a hole-electron compensated semimetal with a low carrier concentration, which is
consistent with the above discussions about extremely large
MR. Note that nh showed an obvious increase at 140 K with
increasing temperature. At lower temperatures, the hole mobility, µh , was one order of magnitude larger than µe . For
example, at 20 K, µh = 3.71 × 105 cm2 /V s, and µe = 3.04
× 104 cm2 /V s. This is in contrast to that in other WSMs,
such as TaAs [10, 14], NbAs [33] and NbP [12], for which
it is usually µe > µh . With increasing temperature, both µh
and µe showed an obvious decrease around 140 K due to the
enhanced phonon thermal scattering. The mobility of carriers
in all recently verified WSMs, such as TaAs, NbAs and NbP,
as well as TaP reported here are comparable, but much higher
than that in conventional semimetals such as WTe2 [26], high
purity Bi [22], and graphite [28, 29]. This is related to the
existence of Weyl fermions in these compounds.
In WSMs the presence of chiral Weyl node pairs would
lead to an unconventional negative MR induced by the AdlerBell-Jackiw anomaly [36, 37] (also named as the chiral
anomaly) when H is applied in parallel to the electric field E,
i.e., the current direction I. Thus, we measured the MR at 2 K
for different angles θ between H and I, as shown in the inset
of Figure 3(c). As shown in Figure 3(d), when θ = 90◦ , i.e.,
H k I, a negative MR was indeed observed. It reached near
−3000% at 9 T, and exhibited no sign of saturation even at the
highest field in our measurements. This behavior is different
from that of other WSMs such as TaAs [13] and NbP [15],
in which the MR changes soon from a negative to positive
value as H increases. This may be due to TaP having a larger
distance between Weyl nodes in the momentum space [38].
Note that the origin of the negative MR over 100% for H k
I is deputed, although we confirmed this behavior in many
measurements and eliminated the influence of the Hall resistivity by using different samples. The MR curve showed a
small positive peak below 2 T, which has also been observed
in Bi1−x Sb x [39], TaAs [13, 14] and NbP [15]. It is attributed
to weak anti-localization. The SdH oscillation was clearly superimposed onto the MR. When θ was decreased to 88◦ , the
unsaturated negative MR behavior remained, but its value decreased. When θ 6 85◦ , the MR became positive and reached
its maximum value at θ = 0◦ , as shown in Figure 3(c).
In order to obtain information on the Fermi surface of TaP,
we also measured the magnetoresistivity ρ xx (H) (not shown
here) for different angles β between H and the c axis, represented in the inset of Figure 4, while keeping H ⊥ I. Figure
4(c) shows the angle dependence of resistance measured at 2
K and 3.0 T. In a magnetic field, the resistance exhibited its
maximum at β = 0 and reached its minimum value at β = 45◦ .
This indicates the anisotropy of the Fermi surface topology.
In fact, for all the angles, strong Shubnikov-de Haas (SdH)
oscillations were superimposed on ρ xx (H).
Here, we discuss another interesting consequence due to
the presence of chiral Weyl node pairs: a nontrivial Berry’s
phase (ΦB ) was observed in TaP. This is the characteristic
feature of charge carriers within k-space cyclotron orbits enclosing Dirac points [40-43]. We used the expression [44]
ρ xx = ρ0 + ∆ρ xx = ρ0 [1 + A(B,T)cos2π(F/B + γ)] to analyze
the SdH oscillations in ρ xx (H) for H k c axis, i.e., β = 0◦ as
shown in Figure 4. Here, ρ0 is the non-oscillatory part of the
resistivity, A(B,T) is the amplitude of SdH oscillations, B =
J. H. Du, et al.
Sci. China-Phys. Mech. Astron.
µ0 H is the magnetic field, and γ is the Onsager phase, F =
~
2eπ AF is the frequency of the oscillations, where AF is the
extremal cross-sectional area of the Fermi surface (FS) associated with the Landau level index n, e is the elementary
h
charge, and ~ = 2π
, h is the Plank’s constant. Figure 4(b)
shows ∆ρ xx as a function of 1/µ0 H at 2 K, 5 K, and 10 K after
ρ0 is subtracted. Two sets of oscillations were clearly observed corresponding to the two frequencies of F1 = 25.6 T
and F1 = 49.6 T. This was confirmed by the Fourier transformation (β = 0), as shown in Figure 4(a). According to the
Onsager relation, the extremal cross-sectional areas of the
FS were estimated as AF = 2.439 × 10−3 Å2 , 4.725 × 10−3
Å2 corresponding to F1 and F2 , respectively, which are very
small compared with the whole cross-sectional area of the
first Brilloun zone (BZ) [13]. We then considered the SdH
oscillation with F1 . To determine the Berry’s phase of the
carriers, we plotted the Landau fan diagram, as shown in Figure 4(d). Linear fitting gave an Onsager phase of γ =–0.3958.
The existence of a nontrivial Berry’s phase with a strong offset of 0.3958 confirmed the Weyl fermion behavior. It has
been predicted [10] by ab initio calculations that there are 12
pairs of Weyl nodes in the BZ. The eight pairs are located at
kz ∼ ± 0.6π/c′ (named W1) where c′ = c/2 and the other four
pairs sit on the kz = 0 plane (W2). Because H is applied along
the c axis, possible WSM electron pockets and hole pockets
are quantized equivalently because of the four-fold rotation
symmetry of the BZ [9]. Thus, we expect that the two major
oscillation frequencies at β=0◦ can be assigned to these two
types of Weyl nodes.
We then focused on the angle (β) dependence of the SdH
oscillation frequency. As shown in Figure 4(a), with increasing β, the F1 peak at β = 0◦ shifted to a slightly higher frequency and then turned back to exhibit nearly isotropic behavior, which is expected for W1 cones [9]. For F2 at β = 0,
however, which corresponded to the anisotropic W2 nodes,
In summary, the longitudinal resistivity (ρ xx ) and Hall resistivity (ρyx ) of single-crystal TaP were measured. At 8 T, the
MR reached 3.28 × 105 % at 2 K, and 176% at 300 K. The
Hall resistivity results confirmed that TaP is indeed a holeelectron compensated semimetal with a low carrier concentration and high mobility. Remarkably, when a magnetic field
(H) was applied in parallel to the electric field (E), a negative
MR due to a chiral anomaly was observed. It reached to near
−3000% at 9 T, and exhibited no sign of saturation, in contrast to other WSMs. Based on the analysis of Shubnikov-de
Haas (SdH) oscillations superimposed on the MR, a nontrivial Berry’s phase with a strong offset of 0.3958 was observed
in this compound, which confirms its Weyl fermion behavior.
This work was supported by the National Basic Research Program of China
(Grant Nos. 2015CB921004, 2012CB821404 and 2011CBA00103), the
National Natural Science Foundation of China (Grant Nos.
11374261
and 11204059), Zhejiang Provincial Natural Science Foundation of China
(Grant No. LQ12A04007) and the Fundamental Research Funds for the
Central Universities of China.
1
2
3
4
8
9
10
(d)
11
12
Figure 4 (Color online) (a) Fourier transformation for the MR after the
non-oscillation part is subtracted when measured at 2 K for various H alignments with respect to the c axis and H ⊥ I; (b) ∆ρ xx as a function of 1/µ0 H
at 2 K, 5 K, and 10 K, where ∆ρ xx is the oscillation part in the MR ρ xx after ρ0 is subtracted for β = 0; (c) angle dependence of the resistance R xx (β)
measured at 2 K and 3 T; (d) linear fitting of the Landau fan diagram for F1
= 25.6 T showing a non-trivial Berry’s phase with a strong offset of 0.3958.
657406-5
4 Summary
7
(c)
Vol. 59 No. 5
it was difficult to observe a clear trend with a changing angle
β. Interestingly, there were another two peaks at F ′ = 112
T and 146 T for β= 45◦ and 50◦ curves, except for the major peak at 32.8 T corresponding to W1 nodes. These high
frequency peaks may have originated from the other massive
bands near FS, which seems to explain the minimum MR observed at β =45 ◦ .
5
6
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