Electron scattering in tantalum monoarsenide

TL;DR

This study investigates electron scattering in TaAs, a Weyl semimetal, revealing how transport properties change near Weyl nodes and supporting the chirality-protected shift theory.

Contribution

It provides detailed analysis of electron scattering and transport parameters in TaAs, highlighting the enhancement of the transport to quantum lifetime ratio near Weyl nodes.

Findings

Transport lifetime to quantum lifetime ratio increases near Weyl nodes

Transport parameters are obtained through two-band model fitting and Shubnikov de Haas analysis

Results support the chirality-protected shift interpretation in Weyl semimetals

Abstract

We report comprehensive studies of the single crystal growth and electrical transport properties for various samples of TaAs, the first experimentally confirmed inversion symmetry-breaking Weyl semimetal. The transport parameters for different samples are obtained through the fitting of the two band model and the analysis of Shubnikov de Haas oscillations. We find that the ratio factor of transport lifetime to quantum lifetime is intensively enhanced when the Fermi level approaches the Weyl node. This result is consistent with the side-jump interpretation derived from a chirality-protected shift in the scattering process for a Weyl semimetal.

Tables

Table 1. Table 1: Summarized growth conditions, residual resistance ratio (RRR) and magnetoresistance (MR) at 2 K and 9 T for representative samples of TaAs. All the growth started from 250 mg of TaAs powders and one of the agents listed below.

$Sample\#$	$Agent$(mass)	$T$	$l_{tube}$	$RRR$	$MR$(9T)
	mg	o$C$	cm	$\frac{{\rho }_{xx}\left(300K\right)}{{\rho }_{xx}\left(2K\right)}$	$\Delta \rho /{\rho }_0$
D1	SnI4(60)	800$\to$1000	23	25	1329
D2	TeI4(60)	1000(1 zone)	23	6	81
D3	TeI4(70)	1000(1 zone)	23	5	86
D5	NbI5(50)	700$\to$900	23	2	3
D6	SnI4(60)	800$\to$1000	23	11	840
D10	SnI4(60)	930$\to$1000	23	22	2473
D11	SnI4(60)	850$\to$930	23	38	5143
B4	TaBr5(150)	970$\to$830	23	7	293
B5	TaCl5(120)	970$\to$830	23	8	261
B6	TaCl5(170)	970$\to$830	23	7	226
B7	TaBr5(220)	970$\to$830	23	41	3180
B8	TaCl5(120)	970$\to$830	23	11	308
B9	TaCl5(170)	970$\to$830	23	13	423
S1	SnI4(60)	1000(1 zone)	30	49	5449
S3	SnI4(80)	800$\to$1000	24	12	873
S4	SnI4(80)	800$\to$1000	24	17	1476
S5	I2(50)	1000(1 zone)	30	5	4
S9	SnI4(100)	800$\to$1000	24	21	1593
S14	SnI4(100)	800$\to$1000	24	N/A	1244

Table 2. Table 2: Fitting results of the Hall data (2 K and 300 K) for representative samples S1, S2, S5, S9 and S14. n 𝑛 n and μ 𝜇 \mu are obtained from the two-band model.

	$n_e$(2K)	$n_h$(2K)	${\mu }_e$(2K)	${\mu }_h$(2K)	$n_h$(300K)
unit	${10}^{17}{\mathrm{cm}}^{-3}$	${10}^{17}{\mathrm{cm}}^{-3}$	${\mathrm{cm}}^2{\left(\mathrm{Vs}\right)}^{-1}$	${\mathrm{cm}}^2{\left(\mathrm{Vs}\right)}^{-1}$	${10}^{18}{\mathrm{cm}}^{-3}$
S1	$7.1$	$4.6$	$4.8\times {10}^5$	$5\times {10}^4$	$13$
S2	$41$	$10$	$2.5\times {10}^3$	$–$	$87$
S5	$–$	$9.1$	$–$	$4.5\times {10}^4$	$15$
S9	$6.4$	$1.0$	$1.5\times {10}^4$	$–$	$33$
S14	$130$	$90$	$2.1\times {10}^4$	$–$	$–$

Table 3. Table 3: Summarized transport parameters for various samples of TaAs.

$TaAs$	EF	$\mu$(W1)	kF	mc	TD	${\tau }_Q$	${\tau }_{tr}$	${\tau }_{tr}$/${\tau }_Q$
	meV	m/(vs)	${\AA }^{-1}$	me	K	s	s
Sample#						10-12	10-12
S1	11.5	31	0.015	0.15	3.6	0.34	26	79
S3	16.1	11	0.015	0.11	5.4	0.23	6.7	30
S5	-45.4	0.77	0.021	0.07	9.8	0.12	0.27	2
S9	16.8	0.56	0.015	0.1	3.8	0.32	17	54
S14	24.1	2.1	0.021	0.14	$–$	$–$	$–$	$–$
S16	13.6	1.5	0.017	0.16	1.1	1.1	0.52	0.5
S17	18.2	29	0.017	0.11	1.9	0.66	0.98	1.5

Equations

$${\sigma }_{xx}=e{n}_h{\mu }_h\frac{1}{1+({\mu }_{h}B{)}^2}+e{n}_e{\mu }_e\frac{1}{1+({\mu }_{e}B{)}^2},$$

$${\sigma }_{xy}=[n_h{\mu }_h^2\frac{1}{1+({\mu }_{h}B{)}^2}-n_e{\mu }_e^2\frac{1}{1+({\mu }_{e}B{)}^2}]eB,$$

$${\rho }_{xx}={\rho }_0[1+A(B,T)cos2\pi (F/B+\gamma )],$$

$$A(B,T)\propto exp(-2{\pi }^2{k}_B{T}_D/\hslash {\omega }_c)\frac{\lambda (T)}{sinh(\lambda (T))},$$

$$\mathrm{MR}\equiv \frac{\Delta \rho }{{\rho }_0}=\frac{{\sigma }_e{\sigma }_h({\mu }_e-{\mu }_h{)}^2{B}^2}{({\sigma }_e+{\sigma }_h{)}^2+B^2({\mu }_e{\sigma }_h+{\mu }_h{\sigma }_e{)}^2},$$

$$\frac{\Delta {\rho }_{xx}(T,H)}{{\rho }_{xx}(T,0)}=\mathrm{F}(\frac{H}{{\rho }_{xx}(T,0)}),$$

Full text

I i. INTRODUCTION

A semimetal has a small overlap between the bottom of the conduction band and the top of the valence band. The small occupation of the Fermi surface (FS) in a semimetal can cause some unique electrical transport features unlike those in typical metals. For instance, the compensation of the electrons and holes at the Fermi level (FL) in a semimetal leads to a resonance in the charge transport Ali et al. (2014). Large magnetoresistance ($\mathrm{MR}\equiv\Delta\rho/\rho_{H=0}$) has been observed in several compensated semimetals Ali et al. (2014); Kopelevich et al. (2003); Alers and Webber (1953), including bismuth, which is an archetypical semimetal: high-purity bismuth has a carrier concentration as low as \sim$$10^{17} $\mathrm{cm^{-3}}$ and its FS occupies a mere $10^{-5}$ of the Brillouin zone. Bismuth can serve as a model system for studying the dilute Dirac electron gas Behnia et al. (2007); Li et al. (2008). Discovery of a new topological semimetal with different band structures and electron characteristics has been one of the central task for the condensed matter physics society. The newly discovered topological semimetals include Dirac semimetals Cd3As2 Wang et al. (2013) , Na3Bi Wang et al. (2012) and the Weyl semimetal TaAs isostructural family Xu et al. (2015a); Lv et al. (2015); Yang et al. (2015); Xu et al. (2015b, 2016, c); Huang et al. (2015a); Weng et al. (2015). The TaAs family exhibit many interesting transport properties that are attributed to their exotic low-energy excitations. These properties include negative longitudinal MR induced by the chiral anomaly Zhang et al. (2016); Huang et al. (2015b); Yang et al. (2017); Hu et al. (2016); Du et al. (2016), extremely large linear MR and ultrahigh carrier mobilities Zhang et al. (2015); Shekhar et al. (2016); Ghimire et al. (2015); Shekhar et al. (2015). However when we compare the results of the transport experiments on each compound of the TaAs family, we found that the properties of different samples are indeed very different. Understanding the mechanism underneath the sample difference should be very important for addressing the Weyl quasiparticles. Unfortunately, researchers usually only present the data from their ‘best’ samples in one report.

We here present our results in an extensive manner after a systematic exploration on the methods of sample growth, electrical measurements on different samples and comprehensive analyses on transport parameters. We show that the growth procedures significantly affect the sample’s quality and transport properties for TaAs. The position of the FL is one of the most important factors to affect the scattering procedures. The large lifetime ratio factor at low temperatures indicates that the large mobility comes from the strong backscattering protection in TaAs. These exotic properties promise TaAs as a platform for investigating the topological electrons.

II ii. materials and methods

Previous study reported that the single crystals of TaAs can be grown via a standard chemical vapor transfer (CVT) method Murray et al. (1976). However we found that the yield and quality of the single crystals are not stable in different batches when we followed the procedure in Ref. Murray et al. (1976). Therefore we optimized the CVT method by attempting different temperature gradients and agents. Below we describe the modified growth procedures in detail.

Polycrystalline samples were prepared from stoichiometric mixtures of Ta (99.98%) and As (99.999%) powders in an evacuated quartz ampoule at 1473 K. Then we grew the single crystals via the CVT method with a positive temperature gradient similar as the procedure in Ref. Murray et al. (1976). The powder of TaAs (300 mg) and the transport agent (I2, 10 $\mathrm{mg/cm^{3}}$ or SnI4, 30-50 mg) were sealed in a 30 cm-long evacuated quartz ampoules. The end of the sealed ampoule with charges was placed horizontally at the center of a single-zone tube furnace. The center was slowly heated up to 1273 K and kept the temperature for 5 days. Small crystals about 0.5 mm in scale were obtained at the cold end which was about 973 K during the growth. The yield is not stable from batch to batch in this procedure.

We found that the CVT growth with reversed temperature gradients and different transport agents can produce large crystals in a more reproducible manner. Polycrystalline TaAs (300 mg) and one of the transport agents (SnI4, NbI5, TeI4, TaBr5, TaCl5 or BiBr3; 20-40 mg) were sealed in 23 cm-long evacuated quartz ampoules. The ampoules were placed in a three-zone furnace with the end containing the charges at the center of the central zone while another end at the center of the side zone. The central and side zones were slowly heated up to 1073 K and 1273 K, respectively, and then kept these temperatures for 8 days.

The procedure described above yields different shapes of single crystals like pyramids or flat blocks (Fig. 2(a)). All the samples can be seen as part of truncated octahedron indeed. X-ray diffraction measurements confirmed that the square surfaces on the crystals are the crystallographic $\bf{c}$ planes with four-fold rotational axes. The size of the crystals is about 0.5-2 mm in general, but bigger crystals can be obtained by extending the growing period to one month.

All the measurements were preformed on the polished $\bf{c}$ planes with the electric current passing along the $\bf{a}$ direction in this paper. The crystal growth conditions, residual resistance ratio (RRR) and the MR at 2 K and 9 T of the samples are summarized in Table I. We empirically found that the samples with golden metallic lustre have the highest RRR and mobility among all the samples. We also found that some large crystals ($>$ 1 mm) have many stripes on their side faces, which are likely due to the stacking disorders along c-axis Besara et al. (2016). These samples in general have much smaller RRR values. It has been reported that the pnictide deficiency induces stacking disorders in the isostructural compounds Boller and Parth (1963). According to our observation, the stacking disorders affect the quality of large crystals seriously, which is an obstacle for obtaining large high-quality crystals.

The values of MR and RRR for various samples of TaAs were shown in a double-logarithmic plot in Fig. 1. The MR at 2 K and 9 T changes from 104 to less than 10 for the samples grown with different agents. It is noteworthy that the MR at 2 K seems to follow the power-law of MR $\propto$ RRR2.4. Such power law was reported in WTe2 as well Ali et al. (2015).

All the temperature-dependent and field-dependent resistivity measurements were performed in a Quantum Design physical property measurement system (PPMS-9). A standard four-probe method for resistivity measurements was adopted with employing silver paste contacts on the samples. All the magneto-transport measurements were carried out from field -$H$ to $H$, and then $\rho_{xx}$ and $\rho_{yx}$ were calculated using the formulas $\rho_{xx}(H)$ = ($\rho_{xx}(H)$ $+$ $\rho_{xx}(-H)$)/2 and $\rho_{yx}(H)$ = ($\rho_{yx}(H)$ $-$ $\rho_{yx}(-H)$)/2, which eliminates the nonsymmetrical effect of the contacts.

III iii. experiment

We show the electrical transport properties for sample S1 in detail which has the largest RRR value. The data for other representative samples with smaller RRR are shown for comparison as well. Figure 3(a) and (b) show the temperature dependent resistivity in different magnetic fields and the $\mathrm{MR}$ at different temperatures, respectively. When a low magnetic field (0.3 T) is applied, the temperature dependent resistivity changes to an insulating profile and emerges with a plateau at low temperatures (Fig. 3(a)). The metal-to-insulator-like ‘transition’ in magnetic field is commonly observed in semimetals such as bismuth and graphite Du et al. (2005). We noticed that this behavior is not like that of Cd3As2 Liang et al. (2014), in which the ‘turn-on’ temperatures (defined as the temperatures when d$\rho$/d$T$ changes the sign) in the same magnetic fields are much lower Liang et al. (2014). The resistivity plateaus at low temperatures in magnetic fields are commonly observed in semimetals as well Ali et al. (2014).

Large $\mathrm{MR}$ has been observed in various semimetals including the TaAs family Liang et al. (2014); Mun et al. (2012); Wang et al. (2014); Takatsu et al. (2013); Ali et al. (2014), but the power law varies from linear to parabolic for different compounds, and the mechanism of the large MR underneath is still under debate. Figure 3(c) is a logarithmic plot for the field dependence of the $\mathrm{MR}$ at different temperatures for sample S1. The $\mathrm{MR}$ at low temperatures changes from a parabolic to a linear dependence with a crossover manner. The crossover field changes from 0.5 T at 2 K to 6 T at 75 K.

Figure 3(d) shows the MR for several representative samples at 2 K. For the samples S2 and S5 with RRR about 5, their MR at 2 K is two orders of magnitude less than those of S9 and S14. It is noteworthy that the MR for S2 and S5 do not show linear-field dependence but intends to be saturated in a moderate field. We observed clear SdH oscillations at 2 K even for the sample with RRR less than 5. The dependence of the MR and RRR is discussed in the following part.

Figure 4 shows that the Hall resistivity ($\rho_{yx}$) above 150 K for sample S1 is positive, linearly dependent on the magnetic field up to 9 T. When the temperature is below 100 K, a large, negative field-dependent signal occurs in high magnetic field. The Hall signal is dominated by strong SdH oscillations at 2 K, while the non-oscillatory part shows a large and negative linear-field dependence. Figure 4(b) and (c) show $\rho_{yx}$ for different samples at 300 K and 2 K, respectively. All the samples show similar magnitude of positive Hall resistivity at high temperatures while $\rho_{yx}$ for different samples at low temperature are distinct. For the samples with large RRR and MR (such as S9 and S1), their $\rho_{yx}$ is large negative with clear SdH oscillations at 2 K. Sample S5 with RRR = 5 shows positive $\rho_{yx}$ at 2 K while the SdH oscillations are very weak.

IV iv. Data analysis

Band structure calculations, angle-resolved photoemission spectroscopy (ARPES) and electrical transport experiments have revealed that TaAs is a semimetal possessing multi-conductive channels Zhang et al. (2016). We adopt a two-band model based on the Boltzmann equation to analyze the electrical transport data. In this model, the longitudinal conductivity reads as M.Hurd (1972)

[TABLE]

and the Hall conductivity tensor reads as M.Hurd (1972)

[TABLE]

where $n_{e}$ ($n_{h}$) and $\mu_{e}$ ($\mu_{h}$) denote the carrier concentrations and mobilities for the electrons (holes), respectively.

The analysis of the Hall data is based on the two-band model we outline above. We applied two constraints for the four free parameters in the formula in the fitting process Ando (2013). They are the zero-field resistivity and the Hall resistivity in the large B-field limit. In high field, the Hall resistivity reads $\rho_{yx}=\frac{B}{e}\cdot\frac{1}{n_{e}-n_{h}}$ and the value of $n_{e}-n_{h}$ can be estimated by a linear fitting of $\rho_{yx}$. The procedure of the data analysis is standard in previous works Ando (2013). The fittings for sample S1 at 2 K and 100 K are shown in Fig. 5. The $n_{e}$ and $n_{h}$ are well-compensated below 75 K Zhang et al. (2016). The parameters of the fittings for different samples are summarized in Table II.

In order to understand the difference of the transport properties for various samples of TaAs, we analyzed the SdH oscillations in their field-dependent resistivity at different temperatures. Figure 6(a) shows the oscillatory components $\Delta\rho_{xx}$ of several representative samples at 2 K. We adopted the Lifshitz-Kosevich (L-K) formula of $\rho_{xx}$ for a 3D system Murakawa et al. (2013):

[TABLE]

where A(B,T) is expressed as

[TABLE]

where $\rho_{0}$ is the non-oscillatory part of the resistivity, $A(B,T)$ is the amplitude of SdH oscillations, $B$ is the magnetic field, $\gamma$ is the Onsager phase, $k_{B}$ is Boltzmann’s constant, and $T_{D}$ is the Dingle temperature. The cyclotron frequency is $\omega_{c}$ = e$B$/$m_{cyc}$ and $\lambda(T)$ = $2\pi^{2}k_{B}T/\hbar\omega_{c}$. The frequency of the oscillations is $\mathrm{F=\frac{\hbar}{2\pi{e}}A_{F}}$, where $\mathrm{A_{F}}$ is the extremal cross-sectional area of the FS associated with the LL index n, $e$ is the electron charge, and $h$ is the Planck’s constant($\hbar=h/{2\pi}$). The Landau fan diagram applied here show the Berry phase for the electrons of pocket close to $\pi$ Zhang et al. (2016). The cyclotron mass was obtained by fitting to temperature-dependent SdH amplitude damping by L-K formula (Fig. 6(b)). In order to obtain the quantum lifetime $\tau_{Q}$, the Dingle temperature ($T_{D}$) is fitted by using Eq. (4) for representative samples (Fig. 6(c)). The values of $\tau_{Q}$ is calculated through $\tau_{Q}$ = $\frac{\hbar}{2\pi{k_{B}}T_{D}}$. All the parameters are summarized in Table III.

V v. Discussion

In the framework of the two-band model. The MR can be written as Ziman (1972)

[TABLE]

where $\sigma_{e}$ ($\mu_{e}$) and $\sigma_{h}$ ($\mu_{h}$) are the conductivities (mobilities) associated with electrons and holes, respectively. The Drude conductivity writes as $\sigma_{i}$ = $n_{i}e^{2}\tau_{i}/m_{i}^{*}$ (i = e or h), and the corresponding mobility appears as $\mu_{i}$ = $e\tau_{i}/m_{i}^{*}$. This MR follows a quadratic $B$ dependence in low field, and is saturated in high field. If the electrons and holes share the same life time $\tau$ ($\tau_{e}$ = $\tau_{h}$), the MR will scale as a function of $\tau{B}$ and then follows a Kohler’s law which states as

[TABLE]

where the replacement of $\tau$ by $\rho_{xx}(0)^{-1}$ is due to that $\tau$ is inversely proportional to $\rho_{xx}(0)$. However the Kohler’s law does not constrain itself on the two-band model, its physics only underlies in the single lifetime $\tau$ assumption Ziman (1960).

Figure 7(a) shows the Kohler’s plot for sample S1 at different temperatures. The Kohler’s plot is such assemblies of MR data in one sample obtained at different temperatures, or MR data of different specimens of the same compound at identical temperature versus $\frac{B}{\rho_{xx}(T,0)}$. The data all fall on a same curve in Kohler’s framework. Here the $\mathrm{MR}$ for S1 violates Kohler’s rule over the whole fields. Figure 7(b) and (c) show the Kohler’s plots for different samples at 2 K and 300 K, respectively. The MR for different samples severely violates the Kohler’s rule at 2 K, but the MR at 300 K approximately fall on a same curve. We point out that the Hall resistivity for different samples is similar and positive at 300 K but distinct at 2 K.

We realize that the MR at different temperatures doesn’t follow the rule in Kohler’s framework. The breakdown of Kohler’s law can be triggered by multiple reasons Pippard (2009), while here we emphasize three plausible reasons under the background of semimetals. The first reason for breakdown of Kohler’s law is the violation of the assumption of single $\tau$, saying that there are more than one $\tau$ with different values in a system. The second reason for breakdown of Kohler’s law is the temperature variation of the constants of Eq. (5). In semimetals, the carrier concentration ($n$) of each pocket changes at different temperatures due to the thermal population effect. This effect can be treated as breaking of Kohler’s law, but the single lifetime $\tau$ assumption may not be violated Ding et al. (2014). The third reason comes from the phonon scattering which is dominant at high temperature. This phonon dominated scattering will alter the scattering pattern, and then the single $\tau$ assumption is violated. The breakdown of Kohler’s law in TaAs over the whole temperature regime (Fig. 7(a)) is pertinent to all the three reasons by considering its semimetal-like electronic structures. The violation of Kohler’s law for different samples at 2 K seems to be obvious because the carrier’s type and concentration are different.

The information of the FL and band structure of TaAs have been obtained from the SdH oscillations, ARPES experiments and first-principle calculations Zhang et al. (2016). The main frequency of the SdH oscillations for TaAs comes from the electron pockets near the W1 nodes Zhang et al. (2016). We plot the mobility of the carriers near W1 ($\mu$) at 2 K which are obtained from the fitting by the two-band model and the main frequencies ($F$) for different samples in Fig. 8. The mobilities show a dip-like feature with respect to $F$ when 10 T $<$ $F$ $<$ 12 T. When $F$ is less than 10 T, the mobilities show exponential-like enhancement with the decreasing of $F$. When the FL is close to W1, the mobility is significantly enhanced. The external cross sectional area about 10 to 12 T corresponds to the Lifshitz transition point Lee et al. (2015) which has a flat top of the inverse band. This flat band may induce significant scattering of the electrons and suppress the mobility.

The large enhancement of the mobility can be understood by analyzing the scattering process where the FL is close to the W1 node. By taking sample S1 for example, We calculated quantum life time $\tau_{Q}$ = $\frac{\hbar}{{2\pi}k_{B}T_{D}}$ = $3.4\times 10^{-13}$ $\mathrm{s}$ by fitting the Dingle temperature $T_{D}$ ($T_{D}$ = 3.6 K). The transport life time $\tau_{tr}$ is estimated from the expression $\tau_{tr}$ = $\frac{\mu_{e}{\hbar}k_{F}}{ev_{F}}$ = $2.6\times 10^{-11}$ $\mathrm{s}$. It is well-known that $\tau_{tr}$ measures backscattering processes that relax the current while $\tau_{Q}$ is sensitive to all processes that broaden the LLs. The large ratio $R_{\tau}\equiv\tau_{tr}/\tau_{Q}$ = 79 for sample S1 indicates that the small-angle scatterings are dominant while the backscatterings are strongly protected at low temperature. The same transport parameters are also obtained in other samples, and summarized in Table III. We select the values of the samples from a same batch, and plot them in a double-logarithmic plot in Fig. 9. We can see an evident enhancement of $\tau_{tr}$/$\tau_{Q}$ when the FL approaches the W1 node. A recent theoretical work unveiled the underlying mechanism of the protection against the backscattering Jiang et al. (2016). In a Weyl/Dirac system, the electrons will gain an extra side momentum ($\Delta_{ch}$) to side jump because of the definite chirality. This backscattering protection effect is enhanced when the FL approaches the Weyl/Dirac point Jiang et al. (2016).

Our analyses of the electrical transport properties for various samples of Weyl semimetal TaAs reveal that the mobility and MR are indeed strongly dependent on the samples’ qualities. Particularly the positions of the FL strongly affect the scattering process and mobilities. Fine tuning the FL of the Weyl semimetal TaAs family is crucial for exploring the unique properties of the Weyl quasiparticles.

Acknowledgements.

We thank Yuan Li and Ji Feng for using their instruments. S. Jia is supported by National Basic Research Program of China (Grant Nos. 2013CB921901 and 2014CB239302). C.L. Zhang treats his research work as precious homework from Shui-Fen Fan.