PHYSICAL REVIEW B 93, 075423 (2016)

Anisotropic magnetoresistance of charge-density wave in o-TaS3
Katsuhiko Inagaki,1 Toru Matsuura,2,3 Masakatsu Tsubota,4 Shinya Uji,5 Tatsuya Honma,1 and Satoshi Tanda2,3
1

Department of Physics, Asahikawa Medical University, 2-Jo 1-Chome, Midorigaoka-Higashi, Asahikawa 078-8510, Japan
2
Division of Applied Physics, Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo 060-8628, Japan
3
Center of Education and Research for Topological Science and Technology, Hokkaido University, Kita 13 Nishi 8,Kita-ku, Sapporo 060-8628, Japan
4
Division of Quantum Science and Engineering, Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo 060-8628, Japan
5
National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japan(Received 25 July 2014; revised manuscript received 29 June 2015; published 16 February 2016)
We report the magnetoresistance of a charge-density wave (CDW) in o-TaS3 whiskers at 4.2 K under a magnetic
field up to 5.2 T. An anisotropic negative magnetoresistance was found in the nonlinear regime of current-voltage
characteristics. The angle dependence of the magnetoresistance, studied by rotating the magnetic field upon thec axis, exhibited a twofold symmetry. The magnetoresistance amplitude exhibited maxima when the field wasparallel to the a axis, whereas it vanished to the b axis. The observed anisotropy may come from a difference
in interchain coupling of adjacent CDWs along the a and b axes. Comparison of the anisotropy to the scanningtunneling microscope image of CDWs allows us to provide a simple picture to explain the magnetoresistance interms of delocalization of quantum interference of CDWs extending over the b-c plane.
DOI: 10.1103/PhysRevB.93.075423

Dynamics of charge density waves (CDWs) has beenof interest for decades [1]. In particular, the effect of amagnetic field to CDW dynamics remains unsolved. Phaseof a CDW determines the initial position of the density wave,
and collective motion of the CDW is described in terms ofvariation of the phase. Naively, no magnetic responses would
be expected since CDWs consist of hole and electron pairs,
whose charge is neutral. However, Aharonov-Bohm (AB)oscillations of CDWs have been reported in two systems,
first in the ion-beam-radiated NbSe3 [2], then in the ring
crystals of TaS3 [3]. The observed oscillation period was h/2e
for both cases, suggesting that quantum interference mightoccur for the phase of a CDW with the charge of 2e, which
coupled with a vector potential. Theoretical interpretation ofthe AB oscillations in CDWs has not been established. Direct
coupling between the CDW phase and a vector potential
was first proposed [4], then CDW solitons were studied as
carriers affected by a magnetic field [3,5]. Therefore, it isimportant to focus on magnetoresistance of CDWs in a trivial
topology such as a simple whisker in order to reveal themechanism of such a quantum interference. To study themagnetoresistance of CDWs, systems with imperfect nestingshould be avoided because it might hinder possible effects
to CDW dynamics. For example, NbSe3 , which is known as
an imperfect nesting CDW system, exhibits a large positive
magnetoresistance at low temperatures [6], resulting fromuncondensed electrons remained on the Fermi surface evenbelow the Peierls temperature.
In this article, we report the magnetoresistance of a CDW
in o-TaS3 whiskers. The whole Fermi surface of o-TaS3 disappears below its Peierls temperature; hence at low temperatureselectric conduction is only due to the CDW. We measured theresistance in the nonlinear regime of o-TaS3 under magnetic
field up to 5.2 T at 4.2 K, which is low enough to prevent
thermally exciting quasiparticles. The magnetoresistance wasnegative in sign. The angle dependence of the magnetoresistance was studied by rotating the magnetic field upon the c2469-9950/2016/93(7)/075423(4)axis, namely, the chain axis of the crystal. A twofold symmetry
was found in the magnetoresistance. The magnetoresistance
amplitude exhibited maxima when the field was parallel tothe a axis, whereas it vanished to the b axis. The observed
anisotropy may come from a difference in interchain couplingof adjacent CDWs along the a and b axes. Comparison of
the anisotropy to the scanning tunneling microscope (STM)
image of CDWs allows us to provide a simple picture to
explain the magnetoresistance in terms of delocalization ofquantum interference of CDWs extending over the b-c plane.
Our observation is an unexpected phenomenon, which will
provide an important key to understand the AB oscillations ofthe CDWs.
Single crystals of o-TaS3 were grown using a standard
chemical vapor transportation method. A pure tantalum sheet
and sulfur powder were placed in a quartz tube. The quartz
tube was evacuated to 1 × 10−6 Torr and heated in a furnace
at 530 ◦ C for two weeks. The grown crystals were ribbonlike
whiskers. The chain direction of o-TaS3 is along the c axis,
and the flat surface of the ribbon is reported to be a b-c plane,
perpendicular to the a axis [7–9]. The crystal orientation ofeach sample was determined and confirmed to be consistent
with the previous reports by the electron backscattering
diffraction (EBSD) technique (OIM TSL). The electrodes weremade using 50-μm-diameter silver wires glued with silver
paint. Gold thin film was deposited on the crystal before thesilver wires were attached to reduce the contact resistance to1  at room temperature.
The resistance of the sample was measured with a standard
four-probe technique. As described in a previous study [10],a high-impedance digital voltmeter (Keithley 6512; Zin >200 T) was employed. All measurements were performed
with constant currents generated by a current source (Keithley220). A magnetic field was applied with a couple of superconducting coils. The sample holder was rotated along an axis
perpendicular to the magnetic line. In this experiment the axisof rotation was aligned with the chain axis of the sample.

075423-1©2016 American Physical Society

INAGAKI, MATSUURA, TSUBOTA, UJI, HONMA, AND TANDAPHYSICAL REVIEW B 93, 075423 (2016)

810R (Ω)

Resistance (Ω)

6101610410V (V)8102100420

400−1101000/T (K )−1

T=4.2 K

210

100
200
Temperature (K)−505I (nA)

FIG. 1. Temperature dependence of o-TaS3 . The discontinuity iscaused by a change in the measurement current. Inset shows three
Arrhenius fits for high, intermediate, and low temperature ranges,
denoted by broken, sold, and dotted lines, respectively. The lowtemperature fit is also shown in the main panel by the dotted line,
by which the extrapolation of the ohmic resistance at 4.2 K appears
to exceed 109 .

The sample was glued to the sample holder with the ribbon
surface facing the holder. Since Joule heat induced by eddy
current might have caused the temperature to increase when
the holder was rotated, each measurement was performed after
the temperature was stabilized in less than 4 mK rise.
Figure 1 shows the typical temperature dependence ofresistance (R-T ). The sample cross section is 15 × 0.5 μm2 .
The room temperature resistivity of the sample is 2.8 ×
10−6  m, which is consistent with previous reports (∼3 ×10−6  m) [11,12]. By lowering the temperature, the system
undergoes a Peierls transition at 220 K, below which theelectrons at the Fermi surface condense into a charge density
wave state. In the 100 to 200 K temperature range, theresistance obeys an Arrhenius law with an activation energyof 860 K (broken line in the inset of Fig. 1). Discrepancy fromthe Arrhenius law is found below 100 K. In the 40 to 100 K
temperature range, a smaller activation energy (∼200 K; solidline) is applicable, and it becomes higher (∼400 K; dotted line)
at temperatures below 30 K. Such behavior is reproducible andalso consistent with previous reports [11,13].
Figure 2 shows the current-voltage (I -V ) characteristicsof the sample at 4.2 K with flowing current I = 2 nA. At
this temperature, the ohmic resistance exceeds 109 , asdeduced from an extrapolation of the R-T curve. However,the slope of the I -V curve corresponds to 1.3 × 109 ,
which comes from a tiny current accompanied by relaxation.
No nonlinear conduction threshold was observed in the I -V
characteristics at this temperature where there were almost
no thermally activated quasiparticles. A slight hysteresis wasalso observed in the I -V curve in the neighborhood of I = 0.
This phenomenon can be interpreted as a rearrangement of theCDW dislocations, which hold electric charges, as reported forblue bronze [14,15] and TaS3 [16]. In a higher current range

FIG. 2. Current-voltage (I -V ) characteristics of the o-TaS3
sample observed at 4.2 K. The I -V curve is significantly nonlinear.
A slight hysteresis is found near I = 0 as shown by the arrows.

|I | > 1 × 10−9 A, the hysteresis became insignificant. The
following experiments were performed in this current range.
Figure 3 shows the magnetoresistance of the sampleobserved at 4.2 K. The magnetic field was applied perpendicular to the current flow (c axis). θ = 0◦ means thatthe field is directed along the b axis, and at θ = 90◦ thefield is along the a axis, as shown in the inset. This result
revealed negative magnetoresistance with anisotropy, and wasreproduced in several samples. We also confirmed that theoθ=0
ΔR(B)/R(0)00afield

−0.01

b

θ=90

current
02o4

B (T)FIG. 3. Magnetoresistance of the o-TaS3 sample observed at
4.2 K. θ = 0 (open circles) means the magnetic field is parallel to thecurrent flow, and at θ = 90◦ (solid circles) the field is perpendicular
to the ribbon face, as shown in the inset. The current is 2 nA, whichstays in the nonlinear regime of the I -V characteristics. The error bar
represents ±2σ , where σ is the standard deviation of the data. Solidlines are the guide to the eyes.

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observed magnetoresistance is independent of the current
direction. Moreover, the ratio V (B)/V (0) was constant overthe nonlinear regime of the I -V characteristics in the fieldof B = 5.2 T. As shown by the error bars in Fig. 3, whichcorrespond to ±2σ where σ is a standard deviation of raw
data, the data seem to be noisy, probably because of influenceof the CDW’s collective motion, e.g., narrow band noise.
Magnetoresistance of quasi-one-dimensional conductors
has been intensively studied [17]. A magnetic field changes
electron motion on the Fermi surface. This provides anincrease of resistivity, namely, positive magnetoresistance withanisotropic angle dependence according to the shape andtopology of the Fermi surface. On the contrary, the sign of themagnetoresistance of o-TaS3 was negative (Fig. 3). Moreover,no normal carrier in o-TaS3 is left at the Fermi surface in
the CDW state, and thermally activated quasiparticles arenegligible at 4.2 K. Hence our observation should not be
understood in terms of the conventional magnetoresistance.
A comparison with the magnetoresistance of the NbSe3 case[6] is also noteworthy. NbSe3 has two CDW transitions at T1 =145 K and T2 = 59 K. Even below T2 there remain normal carriers on the Fermi surface. A large positive magnetoresistancewas observed below T2 , accompanied by an increase of thenumber of CDW carriers. Magnetic response to the dynamics
of a CDW was hindered in the previous experiment. Our result
also excludes the possibility of a spin-related phenomenon
being a major contributor to the observed magnetoresistance,
as with the negative magnetoresistance of TaS2 [18], which isessentially isotropic.
Figure 4 shows the angle dependence of the magnetoresistance, which reveals the twofold symmetry of themagnetoresistance. As the first approximation for twofold
symmetry, we tried to fit the angle dependence of the observedmagnetoresistance with the formula: R = −A sin2 (θ − θ0 ),where A is the amplitude of magnetoresistance and θ0 is theoffset angle. The parameters were determined by nonlinear
least square fitting to be A = 5.78 × 106  and θ0 = −7◦ .
The solid line in Fig. 4 shows the result of the fit. The residual
error of the fit was roughly the same as the distribution of theobserved data, which is shown with the error bars of 2σ inFig. 4. The magnetoresistance amplitude exhibited maxima
when the field was applied along the a axis, whereas it
vanished when the field was along the b axis. The offset angle
θ0 = −7◦ represents a slight misalignment of the crystal axes
to the magnetic field direction. This angle coincides with thedirection of the b axis of the sample determined with the EBSD
technique.
It is necessary to look carefully at the difference between
the a and b axes. The STM image [9] demonstrates thatmaxima of CDW intensity on the b-c plane are canted
and form an angle of 86◦ with the chain axis. This angle
corresponds to arctan(8b0 /4c0 ) ∼ 84◦ , where b0 and c0 are thelattice constants of o-TaS3 . This implies the wave functionof CDW extends over the b-c plane, and it is not firmly
bound to a particular chain. The STM image is consistent
with the CDW vector of o-TaS3 , qCDW = (0.5,0.125,0.25),
formerly determined with diffraction studies [7,19]. CDWs ofadjacent chains are out of phases along the a axis. This resultsfrom energetically favorable configuration among independent
chains coupled with the Coulomb interaction [1]. Hence theΔR(θ)/R(0)ANISOTROPIC MAGNETORESISTANCE OF CHARGE- . . .

B=5.2 T

0−0.01bc−0.02
L0180360θ (degrees)
FIG. 4. Angle dependence of the magnetoresistance in the field
B = 5.2 T observed at 4.2 K. The error bars represent ±2σ as in Fig. 3.
A twofold symmetry is clearly exhibited. The solid line shows a resultof a least-square fit to the formula R = −A sin2 (θ − θ0 ), where theoffset angle θ0 = −7◦ . This angle coincides with the direction of baxis of the sample determined with the EBSD technique. The inset
shows a schematic image of a CDW loop formed in domain structure.
The thin lines represent the wave front of CDWs, and the bold line
shows a loop, surrounding an area of L2 sin α, where L is the lengthof the loop, and α is the angle between CDW and the b axis. The
angle α is exaggerated in the inset.

response to the magnetic field may not be symmetric for therotation upon the c axis.
The canting of CDW wave vector provides a simple picture
as follows. There are two possibilities for the CDW to settleon the pristine lattice by canting left or right. Once either
left or right is chosen, a certain area of CDWs are aligned toform a domain. If CDWs consist of such domains, a closed
loop can exist as shown in the inset of Fig. 4. The areasurrounded by the loop is represented as S = L2 sin α, where
L is the coherent length along the c axis, and α is the angle
between the CDW wave front and the b axis. In fact, thesize of domains is distributed and such loops may be formed
randomly in the sample. Self-interference of a wave function
in a random media, known as the Anderson localization [20],can explain the magnetic response. A magnetic field destroys
the interference of waves which go around the closed loop.
This phenomenon is known as delocalization and provides
a negative magnetoresistance [21,22]. A relative
 shift of theelectron phase ψ is represented as ψ = he A dl, where
h is Planck’s constant, e is the charge quantum of the carrier,and A is the vector potential. In a two-dimensional system,
electrons can only move on a conduction plane. With a closedpath, which surrounds an area S, the phase shift is proportional
to the flux  = BS, where B = rotA is an applied field.
Since Anderson localization results from the superposition ofpossible closed paths of electrons, this leads to the anisotropyof the magnetoresistance when the applied field is inclined byan angle θ to the conducting plane, and the effective area
becomes S sin θ . Without spontaneous magnetization [23],075423-3

INAGAKI, MATSUURA, TSUBOTA, UJI, HONMA, AND TANDAPHYSICAL REVIEW B 93, 075423 (2016)

magnetoresistance is represented as an even function of themagnetic flux, e.g., R ∝ −2 ; hence magnetoresistancewould have a component of sin2 θ . Therefore, the observed
twofold symmetry in magnetoresistance (Fig. 4) can beinterpreted as a natural consequence of the delocalization
picture.
The coherent length L was estimated both by the STM
image and by x-ray diffraction. The STM image (Fig. 3 inRef. [9]) shows CDWs in 22.5 × 22.5 nm2 area, where twodomains of CDW can be distinguished. This image suggeststhat the size of each domain is much larger than 2 × 10−8
m. A synchrotron x-ray study [24] exhibits the coexistenceof incommensurate and commensurate CDWs in o-TaS3 . Twosatellite spots are separated by two pixels apart at the detector,
and each spot is as narrow as one-pixel width (Fig. 1 inRef [24]). This gives an estimation of the correlation length
longer than 3 × 10−7 m in c∗ direction. If L ∼ 3 × 10−7 m isassumed, the corresponding area of the CDW loop becomes1 × 10−14 m2 , which gives the field of 0.2 T for a flux quantum
h/2e. This field can be interpreted as the minimum field forthe negative magnetoresistance to occur. The observed magnetoresistance ranges above the field of 0.2 T, as shown in Fig. 3;
hence it is consistent quantitatively with the delocalization
picture.
What kinds of carrier are consistent with the delocalization
picture? Quantum interference can occur for any kind of carrier

as far as their phase is not lost. We have already ruled out
magnetoresistance of thermally activated quasiparticles as apossible candidate for our observation. If the collective motionof CDW plays a major role for the observed negative magnetoresistance, such a quantum interference would increase thepinning rate of the CDW. In the case of soliton transport, soliton
travels along such a closed path may be inactive for a carrier.
The negative magnetoresistance with the twofold symmetry is
consistent for both cases.
Finally, our observation and interpretation will ease conditions to observe the AB oscillations in CDWs. The previous
studies were performed with the ion-beam-radiated NbSe3 [2]and the ring crystal of TaS3 [3]. Tsubota et al. proposed thata CDW soliton might be confined and move along a single
chain whose ends coalesced [3]. This proposal was based on
the growth mechanism of the ring crystals [25]. However,
the magnetoresistance is interpreted as quantum interferenceof the CDWs extending over the b-c plane, and CDWs canmaintain their quantum phase across the chains. Therefore, ourobservation is not only consistent with the previous reports,
but also suggesting the possibility of the AB oscillations in aclosed CDW loop realized by other methods.

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We are grateful to K. Yamaya, S. Takayanagi, and K.
Ichimura for fruitful discussions, and to T. Ikeda and T. Kannofor experimental support.

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