Related content

LETTER

Charge density wave soliton liquid
To cite this article: T. Matsuura et al 2015 EPL 109 27005

View the article online for updates and enhancements.

- Single-valuedness problem on charge
density wave rings
T. Hosokawa, T. Matsuura, M. Tsubota etal.
- Field-induced dielectric response
saturation in o-TaS3
Yongchang Ma, Cuimin Lu, Xuewei Wanget al.
- High-frequency, 'quantum' andelectromechanical effects in quasi-onedimensional charge density wave
conductors
Vadim Ya Pokrovskii, Sergey G Zybtsev,
Maksim V Nikitin et al.

Recent citations- Hydrodynamic model of a Bose–Einsteincondensate with anisotropic short-range
interaction and bright solitons in a
repulsive Bose–Einstein condensate
Pavel A Andreev- From chiral anomaly to two-fluidhydrodynamics for electronic vortices
S. Brazovskii and N. Kirova- Possible origin of nonlinear conductivity
and large dielectric constant in thecommensurate charge-density-wave
phase of 1TTaS2
Yongchang Ma et al

This content was downloaded from IP address 133.87.184.25 on 29/03/2019 at 08:35

January 2015
EPL, 109 (2015) 27005
doi: 10.1209/0295-5075/109/27005

www.epljournal.org

Charge density wave soliton liquid
T. Matsuura1,2 , J. Hara1 , K. Inagaki3 , M. Tsubota4 , T. Hosokawa1 and S. Tanda1,2
1

Department of Applied Physics, Hokkaido University - Sapporo 060-8628, Japan
Center of Education and Research for Topological Science and Technology, Hokkaido University
Sapporo 060-8628, Japan3
Department of Physics, Asahikawa Medical University - Asahikawa 078-8510, Japan
4
Division of Quantum Science and Engineering, Hokkaido University - Sapporo 060-8628, Japan2

received 10 September 2014; accepted in ﬁnal form 13 January 2015
published online 2 February 201571.45.Lr – Charge-density-wave systems
72.15.Nj – Collective modes (e.g., in one-dimensional conductors)
PACS 05.45.Yv – Nonlinear dynamics and chaos: Solitons
PACS
PACS

Abstract – We investigate the charge density wave transport in a quasi–one-dimensional conductor, orthorhombic tantalum trisulﬁde (o-TaS3 ), by applying a radio-frequency ac voltage. We ﬁnd
a new ac-dc interference spectrum in the diﬀerential conductance, which appears on both sidesof the zero-bias peak. The frequency and amplitude dependences of the new spectrum do notcorrespond to those of any usual ac-dc interference spectrum (Shapiro steps). The results suggestthat CDW phase dynamics has a hidden degree of freedom. We propose a model in which 2πphase solitons behave as liquid. The origin of the new spectrum is that the solitons are depinned
from impurity potentials assisted by an ac ﬁeld when a small dc ﬁeld is applied. Our results
provide a new insight as regards our understanding of an elementary process in CDW dynamics.
c EPLA, 2015
Copyright Introduction. – Phase solitons are topological defects
in the phase ﬁeld of a charge density wave (CDW) [1]and are similar to the vortices in superconductors and elementary excitation in a vacuum. The degree of freedomof the phase soliton is the key to understanding the nature of a CDW. Phase solitons have been expected in the
one-dimensional CDW system of orthorhombic tantalum
trisulﬁde (o-TaS3 ) crystals to explain the CDW transport
properties [2–4], and the Aharonov-Bohm interference ino-TaS3 rings [5,6]. In addition, the coexistence of commensurate and incommensurate CDWs in o-TaS3 crystals wasobserved below the Peierls transition temperature TP =218 K by X-ray diﬀraction [7]. The satellite peaks of theincommensurate CDW of Qic ∼ 0.5a∗ + 0.125b∗ + 0.255c∗
develop ﬁrst just below TP , and those of the commensurate
CDW of Qc = 0.5a∗ + 0.125b∗ + 0.25c∗ develop with thedecrease of temperature. Furthermore, the incommensurate CDW was enhanced when the CDW was sliding. The
results suggest that there are unconventional degrees of
freedom associated with the solitons, and that they willaﬀect CDW dynamics.
In this letter, CDW dynamics in o-TaS3 crystals is investigated using a conventional ac-dc interference measurement in the temperature range where the commensurate
and incommensurate CDWs coexist. We ﬁnd a new typeof ac-dc interference spectrum, which does not originate
from the usual Shapiro interference mechanism. The results suggest that an o-TaS3 CDW system has 2π phasesolitons. The solitons are depinned from impurities andmove individually as a liquid when a large ac electric ﬁeld
is applied.
Experimental. – The ac-dc interference measurement
was performed as follows. The o-TaS3 crystals were synthesized using the chemical vapor transport method [8,9].
A needle-shaped o-TaS3 crystal was selected and placed ona sapphire substrate, and two silver wires were connectedto the crystal with gold evaporation ﬁlms and silver paste
for the two-probe measurement. The resistance of thesample present here was 7 Ω at room temperature (280 K).
The cross-section area of the sample was 16 × 7 μm2 , theminimum distance between two terminals was 110 μm, andthe room temperature resistivity was consistent with apreviously reported value [10]. The diﬀerential conductance of o-TaS3 crystals was measured as a function ofdc bias current while applying a radio-frequency (RF) ac
voltage. A diﬀerential conductance measurement with alock-in ampliﬁer (Stanford Research SR830DSP: a lowfrequency (LF) ac current ILF = 1 μA, and fLF = 13 Hz)
was performed. The sample temperature was controlled

27005-p1

T. Matsuura et al.

(a)

(a)

dI/dV (S)

0.01

0.005
T = 150 K, fRF = 1 MHz−0.5

0

T = 150 K, VRF = 200 mV

dI/dV [relative]

VRF =160 mV
100 mV
50 mV
0 mV

0.5

(b)

fRF =

50 mS

0.013

−0.4

−0.2

0.012

0.2

0.4

Idc (mA)−0.1

0(I CDW − I CDW)/2 (mA)

0.011

0.01

0

2.1 MHz
2.7 MHz
3.5 MHz
5.0 MHz

0.1

Fig. 1: (Colour on-line) (a) Diﬀerential conductance of a o-TaS3
crystal at 150 K when applying a RF ac voltage of fRF = 1 MHz
as a function of dc bias current Idc . Dc bias current swept
from negative to positive for all dV / dI vs. Idc curves. The−1= 5.0 × 10−3 S.
low-ﬁeld conductivity for VRF = 0 mV is RN(b) The magniﬁcation of a diﬀerential conductance curve for
VRF = 160 mV. New ac-dc interference peaks and the ﬁrst
Shapiro interference peaks are indicated by the red and blackarrows, respectively.

by using liquid nitrogen and an electrical heater with a±0.1 K accuracy.
Results. – Figure 1(a) shows the diﬀerential conductance of the o-TaS3 crystal as a function of the dc bias current at 150 K, when we applied an ac voltage of 1 MHz withvarious amplitudes. The threshold current and voltage
were Ith = 0.35 mA and Vth = 70 mV (Eth = 636 V/m)for VRF = 0, respectively, and the zero-bias peak became
sharper when VRF increased, then the unusual peaks appeared on both sides of the zero-bias peak for VRF =160 mV (see also the peaks indicated by red arrows inﬁg. 1(b)). We note that the vertical shifts of the dI/dV
curves in ﬁg. 1(a) are not artiﬁcial, but mean that there
are several CDW phase domains. A major domain gives
the Shapiro interference peaks indicated by black arrows inﬁg. 1, but other domains move incoherently. The threshold electric ﬁeld of the sample was relatively larger thanthose reported in the previous papers. One reason is that
it was a two-probe transport measurement. Generally thethreshold voltage observed by two-probe transport measurement is observed larger than that by four-probe transport measurement since the charge density wave phase
distortion and phase slip must be involved.
The new peak structures appearing under ac and dc
electric ﬁelds have never been reported explicitly before.
On the other hand, the step structures in the dI/dV curve
have been often observed in NbSe3 crystals [11,12], andother quasi–one-dimensional conductors. The step structures in the dI/dV curve are clearly diﬀerent from thesenew peak structures, since there is no decrease of dI/dV in

0.1

(b)

(c)

1st Shapiro peak
1st Shapiro
peak

(+)

(−)

Idc (mA)

0.2

new peaknew peak
0
0

100

200 0

VRF (mV)

2

4

fRF (MHz)

Fig. 2: (Colour on-line) (a) Frequency dependence of dI/ dV
vs. Idc curves at T = 150 K. The applied external ac voltage
is VRF = 200 mV. The curves are vertically shifted for clarity.
The red and black arrows indicate new ac-dc interference peaks
and the ﬁrst Shapiro step peaks, respectively. (b) Intervals(±)(±)
between positive and negative dc CDW current (ICDW = Idc −−1 (±)Vdc ) of the new peaks and the ﬁrst Shapiro peaks as aRNfunction of VRF for fRF = 1 MHz and (c) as a function of fRF
for VRF = 200 mV, both at 150 K. The solid lines are linear
ﬁtting curves.

the step structures. In addition, a Joule heat eﬀect cannot
explain the decrease in the diﬀerential conductivity because the conductivity of o-TaS3 crystals should increases
when the temperature increases [10] (see also ﬁg. 3 inset).
The RF amplitude and frequency dependences of thecurrent at the new peaks are not consistent with those of
the Shapiro peaks. The ﬁrst Shapiro peaks can also beseen in ﬁg. 1(b). The frequency dependence of the diﬀerential conductance is shown in ﬁg. 2(a). The peak positions and heights of the new peaks are symmetric, while
the dI/dV curves are asymmetric. The asymmetry suggests that there is a macroscopic polarization associatedwith metastable states of the phase ﬁeld in o-TaS3 crystals [13], however, metastability is not directly related to
the new peak structures.
From ﬁgs. 1(a) and 2(a), the RF amplitude and frequency dependences of the CDW current at the new peaks
and at the ﬁrst Shapiro peaks are obtained. Figure 2(b)shows the CDW current at the peaks as a function of theac amplitude for fRF = 1 MHz. The CDW current interval(+)(−)
of the ﬁrst Shapiro peaks is constant ((ICDW − ICDW )/2 =0.065 mA at fRF = 1 MHz). On the other hand, theCDW current at the new peaks increases as the RF amplitude increases. As shown in ﬁg. 2(c) the CDW current at
the ﬁrst Shapiro peaks is proportional to fRF , while the
CDW current at the new peaks decreases. Contrary to

27005-p2

Charge density wave soliton liquid
VRF = 600 mV, fRF = 1 MHz

and the peaks disappear at higher and lower temperatures.
This measurement was performed with both current sweep
directions. The new peak structures do not have hysteresisand dependence associated with current sweep direction.
We note that all samples did not show the new peak structures. We measured 10 o-TaS3 whiskers, and two samples
showed the unconventional ac-dc interference peaks. The
sample dependence has not been understood yet.

0.54

fRF = 1 MHz, VRF = 200 mV

0.5

T = 160 K

140 K
120 K

0.01

0.48

dI/dV (mS)dI/dV (S)

0.02

0.52
T = 150 K

0.18
0.16

T = 100 K

0.04−1

dI/dV (S)10

Idc = 0, VRF = 0

−2

10

0

−3

10

T = 80 K

110 K

0.03
0.009

TP = 218 K

0.008

−4

10
−0.4

0.005
−0.2

0.01 1/T (1/K)
0

Idc (mA)

0.2

0.4

0.007

T = 60 K
0.006
−0.02

0

0.02

Idc (mA)

Fig. 3: (Colour on-line) Left panel: temperature dependence of
dI/dV vs. Idc curves of o-TaS3 crystal for VRF = 200 mV andfRF = 1 MHz. The inset shows the temperature dependence ofthe diﬀerential conductance dI/dV for Vdc = 0 V, and VRF =0 V. The dashed line indicates the Arrhenius curve mentioned
in the text. Right panel: temperature dependence of dI/dV
vs. Idc curves of sample 2 for VRF = 600 mV and fRF = 1 MHz.

the Shapiro peaks following the conventional Shapiro interference mechanism, the new peaks have clearly diﬀerent
RF amplitude and frequency dependences. These results
suggest that the CDW dynamics has an unconventional
internal degree of freedom.
We also ﬁnd that the new peaks disappear at low temperature. Figure 3 (left panel) shows the temperaturedependence of the diﬀerential conductivity when an ac
voltage of VRF = 200 mV and fRF = 1 MHz is applied.
When the temperature decreases, the current at the new
peaks decreases, and the height of the zero-bias peak increases. Below 120 K, the new peaks are below the shoulder of the zero-bias peak. At the temperature range at
which the new peaks are observed, the commensurate
and incommensurate CDWs coexist simultaneously [7].
The synchrotron X-ray diﬀraction measurement could reveal high resolution of satellite diﬀraction structure of the
CDW in o-TaS3 . There were the peaks of commensurate
and incommensurate CDWs while the diﬀraction peaks ofthe mother lattice did not split. The results show that
the CDW in o-TaS3 has an internal degree of freedom
in the phase structures, which would be a key to understanding the origin of the new peak structures in the
dI/dV curve.
We note that the temperature dependence of the lowﬁeld conductivity (dI/dV for Idc = 0, and VRF = 0)in the inset of ﬁg. 3(a) is consistent with previous results [10]. The transition temperature is 218 K and the
conductivity above 110 K follows the Arrhenius curveσ(T ) = σ0 exp(−2Δ/T ) with a semiconducting energy gapof 2Δ = 828 K.
The new peak structures have been observed in another
o-TaS3 whisker (sample 2). Figure 3 (right panel) shows
the temperature dependence of the diﬀerential conductance under a RF electric ﬁeld. In this sample the new
peaks appear but no clear zero-bias peak are observed.
The new peak structures are observed at 80 K to 100 K,

Discussions. – The peak structures nearby the zerobias peak were clearly observed in sample 1, and 2. The
appearance of the peaks is a physical phenomenon, nota data error. The diﬀerential conductance without RF
bias has no special features. This means that there is nolarge crack in the crystals to make two or more threshold voltages. In the electrical measurement, there may
be a transversal current ﬂow. However, in our samples,the probe to probe lengths are more than 10 times larger
than the crystal width and thickness. The conductivity
anisotropy of the TaS3 is of the order of 100 at room temperature and increases at low temperature. Thus only
0.1% of the conductivity is aﬀected by the transverse current. Thus, we can neglect the eﬀect of transverse current.
The origin of the new peak structures is discussed. The
inhomogeneous CDW current explains the step structures
in dI/dV curves [11,12]. If the CDW current ﬂows inhomogeneously due to impurities or inhomogeneous current injection, the step structures might appear in dI/dV
curves. When there are parallel CDW chains with diﬀerent CDW currents ﬂow, total current vs. voltage curves
must have many threshold voltages. In dI/dV curves,
many step structures would be observed. Moreover, thefrequency and amplitude dependences of the new peaks
are completely diﬀerent from those of the Shapiro interference. Thus, the new peaks cannot be produced theShapiro interference of the inhomogeneous current.
Another possible internal degree of freedom causing thenew peak structures is the CDW phase solitons. Solitons
have been expected to exist in the commensurate CDWof an o-TaS3 system from electrical transport and opticalexcitation experiments [2,3]. The existence of a fractional
2π/M phase soliton (with a fractional charge 2e/M ) is
predicted in commensurate CDW systems with commensurability M , where M = 4 for o-TaS3 systems. Therefore,
we consider the soliton dynamics and its contribution into
IV characteristics with a RF ﬁeld. Solitons must play animportant role for the CDW dynamics, since a soliton is a
topologically stable and carries a 2e/M electrical charge.
Since the soliton has electrical charge, the solitons must be
pinned by impurities. Thus, the soliton pinning and depinning must be taken into account. When a RF electric ﬁeld
is applied, the solitons must be depinned and contributedc conduction at a moment, while the bulk CDW does notcontribute to dc conduction. The degree of freedom of thesolitons would modify CDW dynamics. The soliton ﬂow
would be associated with the origin of the unconventional
ac-dc interference peaks.

27005-p3

T. Matsuura et al.

(1)

where m∗ is the CDW eﬀective mass, ne = Qc /π is thebase one-dimensional electron density, gM is the commensurability coupling constant, τ is the damping
 time, E(t)is the external electric ﬁeld, and vph = m/m∗ vF is
the CDW phason velocity. Fimp (z, t) = gP i sin(Qc z +φ(z, t))δ(z − z i ) is the pinning potential force of impurities
at z = z i .
The phase solitons can exist without any excitation if
there is a mismatch between 2kF and 1/4c∗. With o-TaS3
systems, the true one-dimensional electron density mustbe Qic /2π, and the excess charges should make 2π/M
solitons. If there are many phase solitons in the system, ithe phase φ(z) is approximately written as φ(z, t) =i φ (z, t), where the i-th 2π phase soliton is expressed as



γ
4
tan−1 exp (z − z0i − vS t) .
φi (z, t) =(2)
MdHere vS is the soliton velocity, γ = [1 − (vS /vph )2 ]−1/2√
is the Lorentz factor, and d = vph /(M gM ) is the soliton width. Due to the experimental situation, the soliton
velocity vS = λc fNBN  vph , where λc = 2π/Qc is thecommensurate CDW wavelengths.
The X-ray diﬀraction [7] gave a hint about M [15]. In
the experiment, the commensurate and incommensurate
satellite diﬀraction spots were observed simultaneously atfrom 220 K to 50 K, while the Bragg diﬀraction spots ofthe mother lattice did not split. We found that 2π phasesolitons can only reproduce the commensurate and incommensurate satellite diﬀraction spots, as follows. By calculating the Fourier transformation of the electron density
ρ(z, t), we can expect the CDW satellite diﬀraction spectrum. The Fourier magnitudes have a spot at Qic , andhigh-order spots at Qic + nM ΔQ, where n is ±1, ±2, · · · ,and ΔQ = Qic − Qc , due to the Bessel function expansion.
To reproduce the commensurate peak at Qc [7], M shouldbe 1. The existence of the 2π solitons is reasonable since a
2π soliton can exist when a smallest phase dislocation loop
encircles a one-dimensional chain, while the chain-chain

(a)

(b)

6
c

20

ic

Edc = 0
Edc = 0.4 Eth
Edc = 0.8 Eth
Edc = Eth
Edc = 1.2 Eth
Edc = 1.6 Eth
Edc = 2 Eth

(c)

4

2

2

dφ/dz

4

0
0

0
200

z/λC

400

1

1.05

kz/QC

dφ/dz

0.4

(d)

d/λc

2
∂2φ
2 ∂ φ
−
v
+ gM sin(M φ) − Fimp
ph
∂t2
∂z 2
1 ∂φ πene
−
=−
E(t),
τ ∂t
m∗

6

FFT magnitude

Soliton liquid model. – Here we discuss the structureof o-TaS3 CDWs based on the sine-Gordon–type equation,comparing it with the structural information obtained by
the X-ray diﬀraction experiment [7]. The high resolved
X-ray diﬀraction experiment revealed that there are twocomponents corresponding to the commensurate (Qc =0.25c∗ ) and incommensurate (Qic = 0.255c∗ ) CDWs, andtheir intensities have a temperature dependence [14].
The electron density of the CDW on an o-TaS3 chain isexpressed as ρ(z, t) = |ρ0 | sin(Qc z + φ(z, t)), where |ρ0 | isthe amplitude of the CDW order parameter. The dynamical equation of the phase ﬁeld φ(z, t) of a commensurate
CDW with commensurability M is written as

10
0.2

0
−20

0

z/λc

20

0

1

2

0

Edc

Fig. 4: (Colour on-line) (a) dφ/dz of a one-dimensional CDW
for 2π phase soliton lattice (above the dashed line) and soliton liquid (below the dashed line) with several soliton width.
d increases from top to bottom. A 2π soliton exists every 50
commensurate CDW wavelengths (λc = 2π/Qc ). (b) Magnitude of the Fourier transform of electron density ρ(z) for soliton
lattice and soliton liquid states. (c) dφ/dt as a function of thedc electric ﬁeld calculated from eq. (1) without impurity potential. Eth is the threshold electric ﬁeld. (d) Soliton width d
as a function of the dc electric ﬁeld.

Coulomb interaction energy is minimized [5,6]. If a fractional soliton exists in a chain, the chain-chain Coulombinteraction energy between the neighboring chains must
be increased. In other words, the fractional solitons must
form a domain wall across the cross-section of the crystal
to match the CDW phase with those of the neighboring
chains.
If the soliton-soliton interaction is strong, the solitonsoliton distance will be constant. Then the 2π phase solitons form a one-dimensional lattice as the curve above thedashed line in ﬁg. 4(a) [16–18]. The Fourier magnitude ofthe electron density for the soliton lattice state shows not
only the commensurate and incommensurate spots, butalso high-order spots as the curve above the dashed line
in ﬁg. 4(b), contrary to the experimental results. The factthat only two spots at Qic and Qc = Qic − ΔQ were observed means that the soliton lattice has melted.
When the interaction becomes weaker than the impurity potential or thermal ﬂuctuation, the soliton lattice
will melt and the solitons will move individually [19,20].
This behavior is analogous to that in superconducting vortices [21]. The soliton positions are random in the liquidstate. The position of the i-th soliton z0i is assumed to be
λc Qc
distributed randomly in the MΔQ− d range. The curves
below the dashed line in ﬁg. 4(a) show dφ(z)/dz of thesoliton liquid state for several soliton widths.
The Fourier magnitudes for the soliton liquid state areshown below the dashed line in ﬁg. 4(b). For the soliton
liquid state of 2π phase solitons, both the commensurate
and incommensurate components remain in the Fourier
spectrum, and the randomness of the soliton positions
causes other high-order components to become diﬀused.
Moreover, increasing of the soliton width d reproduces
the crossover from the commensurate to the incommensurate CDWs. When the soliton width is small, thenthe commensurate peak is only observed. With the decrease of the soliton width, the CDW is approaching to
be incommensurate, then the incommensurate diﬀraction

27005-p4

Charge density wave soliton liquid

(c)

Eth

Eth

Idc

ES1

ES2

1/1

0.005

(e)

Sliding

2/1

Pinning
0.005(a)Soliton flow

0.01

Idc
0

0
0.01

(b)

(d)

2/1

(f)

2/1

0.02

3/2
1/1

1/1

0

0.005

1/2
Eth

ES2
ES1

Eth

New peakdI/dV

peak is enhanced. The results of the X-ray experiment [7]indicate that the soliton width d increases when temperature increases. The temperature dependence of d could
be interpreted as the development of the CDW order parameter, since the commensurate coupling constant gM isan increasing function of the CDW amplitude.
The existence of solitons qualitatively explains the enhancement of the incommensurate CDW at the sliding
state [7]. The experiment found that the intensity of theincommensurate CDW diﬀraction spots in the sliding stateis stronger than that in the pinning state. The numericalcalculation of eq. (1) in overdamped situation provides aconsistent result. Figure 1(d) shows dφ/dx as a functionof the applied dc electric ﬁeld. The peaks of dφ/dx correspond to the 2π-solitons. The soliton at zero external
electric ﬁeld have a maximum of the peak height. Thus,
the soliton width is smallest. The soliton width for 2π√solitons is expressed by d = vph / gM , while soliton-soliton
distance is larger than the soliton width. When the external electric ﬁeld exceeds the threshold ﬁeld, the bulk
CDW slides, and the soliton height becomes drasticallysmall, namely, the soliton width becomes larger, as shownin ﬁg. 1(d). The CDW moves when the external electricﬁeld exceeds the threshold ﬁeld, then the many parts of
the CDW do not stay the minimum of the commensurability energy potential. Then the constant gM eﬀectively
decreases for the moving CDW, since an incommensurate
CDW reduces the elastic energy instead of the commensurability energy in the sliding state. Hence, the soliton
width is increased at the CDW sliding state, and the CDW
phase ﬁeld becomes rather incommensurate, as shown inﬁg. 4(a) and (b). The one-dimensional soliton liquid model
naturally explains the enhancement of the incommensurate satellite spots at the sliding state.
We investigate CDW dynamics with the 2π soliton liquid and impurities in the presence of RF bias using numerical calculation of eq. (1). The dc (time-averaged) CDW
current dφ/dt is calculated as a function of dc electricﬁeld. All calculations are performed with overdamped situation, and the following parameters: vph = 3, gM = 0.5,
gP = 5, and τ = 0.01. If there is no soliton, and there is
no impurity potential, the calculation reproduces the wellknown nonlinear conduction of the CDW system, as shownin ﬁg. 5(a). When a RF ﬁeld is applied, the Shapiro interference occurs at dφ/dt = nfRF , where n is integer. The
steps in the IV curve correspond to the integer Shapiro
steps.
Secondly, if impurities are assumed to be induced randomly in the one-dimensional chain, the phase ﬁeld φ(z)is modiﬁed since the phase is locked locally by the impurities. Then the threshold electric ﬁeld is increased rather
than that without impurities, as shown in ﬁg. 5(b). In
the presence of the RF ﬁeld, the fractional Shapiro steps
appear. This must be caused by the interaction between
the impurities and the CDW phase ﬁeld.
Thirdly, if 2π phase solitons are introduced in the system, the current ﬂows when a low electric ﬁeld is applied,

1/2 Shapiro
1/1 Shapiro −0.02

Zero bias peak
0
0

0.5

1 0

Edc

0.5

1

0

Edc

0.5

Fig. 5: (Colour on-line) Current-voltage characteristics calculated numerically from eq. (1) taking eﬀects of soliton and impurity into consideration. (a) No soliton and no impurity inthe system. (b) No soliton and ﬁve impurities introduced at
random in the system of 100 crystal lattices. (c) A soliton is
introduced. (d) Both a soliton and ﬁve impurities are introduced. The black curves indicate dc current-voltage characteristics, where Idc is proportional to the time-average of ∂φ/∂t.
The red curves indicate the results when an alternative electricﬁeld of amplitude ERF = 2 and frequency fRF = 0.005/2π isapplied. (e) Magniﬁcation of current-voltage characteristics in(d) around Edc = 0 for ERF = 2. Both a soliton and ﬁve impurities are in the system. (f) Diﬀerential conductance obtainedby diﬀerentiation of curve for ERF = 2 in (e).

as shown in ﬁg. 5(c). Here the impurities were neglected.
The solitons move and carry charges, while the bulk CDW
is pinned by the commensurability energy potential. So,
the system shows an Ohmic conduction at zero voltage.
Finally, ﬁg. 5(d) shows the results when both the 2πphase solitons and the impurities are taken into accountfor the calculation. For ERF = 0, both the bulk CDW andthe solitons are pinned by impurities when Edc < Eth , anddepinned when Edc > Eth . The soliton depinning energy
is much larger than the CDW depinning energy from thecommensurability potential and impurity potential, then
only one threshold electric ﬁeld should be observed by theexperiment. We note that this is the same with our experimental results. If we assumed two or more separated CDW
domains, two or more threshold ﬁelds must be expected.
When a RF electric ﬁeld is applied, then the IV curveshows step-like structures around zero bias. Figure 5(e)shows magniﬁcation of the step-like structure. When
Edc < ES1 , solitons and CDW are moved by the RF
ﬁeld, however, the contribution to the dc conductivity iszero, as well as at the pinning state. For ERF  Eth ,the solitons depinned when Edc > ES1 with the help of
the RF ﬁeld, and contribute to dc conduction. The soliton conduction is much smaller than the bulk CDW conduction since the soliton charge is localized and sparse.
Therefore, a step-like structure appears in the currentvoltage characteristics. When Edc > ES2 , the bulk CDW
is depinned, and the sliding CDW causes the integer andfractional Shapiro steps. The diﬀerential conductivity of

27005-p5

T. Matsuura et al.
the calculation results indicates a peak structure besidethe zero-bias peak, as shown in ﬁg. 5(f), which is quantitatively consistent with our experimental results. Thismodel provides a possible explanation for both experimental results of the ac-dc interference measurement and the
X-ray diﬀraction measurement. While the strength of impurity potential has arbitrariness, this calculation suggeststhat the unconventional ac-dc interference eﬀect is associated with an interaction between solitons and impurities.
The sample dependence of strength of impurity potential
might be the reason why a few samples show the new ac-dc
interference peaks.
Summary. – In summary, we found new ac-dc interference peaks in the diﬀerential conductance of o-TaS3 crystals when we applied a radio-frequency (RF) ac electricﬁeld. We proposed the 2π soliton liquid model to explain
our results and the coexistence of commensurate and incommensurate CDWs. The existence of the soliton liquid
may provide a consistent explanation for the low-energy
excitations [2,3], and nonlocal voltages [4], and will play
an important role for understanding the quantum interference eﬀect in o-TaS3 systems [5,6].
∗∗∗
We thank K. Yamaya, K. Ichimura, J. Ishioka,
T. Kuroishi, T. Satoh, and K. Nakatsugawa for
fruitful discussions. This work is supported by the
JSPS KAKENHI Grant-in-Aid for Young Scientists (B)12837528 and Iketani Science and Technology Foundation
0241040-A.
REFERENCES[1] Monceau P., Adv. Phys., 61 (2012) 325.
[2] Bikiakovic K., Lasjaunias J. C. and Monceau P.,
Synth. Met., 29 (1989) F289.

[3] Zaitsev-Zotov S. V. and Minakova V. E., Phys. Rev.
Lett., 97 (2006) 266404.
[4] Inagaki K., Tsubota M. and Tanda S., Phys. Rev. B,
81 (2010) 113101.
[5] Tsubota M., Inagaki K. and Tanda S., Physica B, 404(2009) 416.
[6] Tsubota M., Inagaki K., Matsuura T. and Tanda S.,
EPL, 97 (2012) 57011.
[7] Inagaki K., Tsubota M., Higashiyama K., Ichimura
K., Tanda S., Yamamoto K., Hanasaki N., Ikeda N.,
Nogami Y., Ito T. and Toyokawa H., J. Phys. Soc.
Jpn., 77 (2008) 093708.
[8] Sambongi T., Tsutsumi K., Shiozaki Y., Yamamoto
M., Yamaya K. and Abe Y., Solid State Commun., 22(1977) 729.
[9] Tsutsumi K., Sambongi T., Kagoshima S. and
Ishiguro T., J. Phys. Soc. Jpn., 44 (1978) 1735.
[10] Thompson A. H., Zettl A. and Grüner G., Phys. Rev.
Lett., 47 (1981) 64.
[11] Latyshev Yu. I., Minakova V. E. and Rzhanov Yu.
A., JETP Lett., 46 (1987) 10.
[12] Maher M. P., Adelman T. L., McCarten J.,
DiCarlo D. A. and Thorne R. E., Phys. Rev. B, 43(1991) 9968.
[13] Tessema G. X., Alavi B. and Mihaly L., Phys. Rev.
B, 31 (1985) 6878.
[14] Roucau C., J. Phys. (Paris), Colloq., 44 (1983) C3-1725.
[15] Jacques V. L. R., Le Bolloc’h D., Ravy S., Dumas
J., Colin C. V. and Mazzoli C., Phys. Rev. B, 85(2012) 035113.
[16] Preobrazhensky V. B. and Taldenkov A. N., Synth.
Met., 29 (1989) F313.
[17] McMillan W. L., Phys. Rev. B, 14 (1976) 1496.
[18] Okwamoto Y., Takayama H. and Shiba H., J. Phys.
Soc. Jpn., 46 (1979) 1420.
[19] Miller J. H. jr., Wijesinghe A. I., Tang Z. and Guloy
A. M., Phys. Rev. B, 87 (2013) 115127.
[20] Miller J. H. jr., Wijesinghe A. I., Tang Z. and Guloy
A. M., Phys. Rev. Lett., 108 (2012) 036404.
[21] Koshelev A. E. and Vinokur V. M., Phys. Rev. Lett.,
73 (1994) 3580.

27005-p6