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LETTER

Single-valuedness problem on charge density
wave rings
To cite this article: T. Hosokawa et al 2015 EPL 112 27005- Charge density wave soliton liquid
T. Matsuura, J. Hara, K. Inagaki et al.
- Aharonov-Bohm effect in charge-density
wave loops with inherent temporal current
switching
M. Tsubota, K. Inagaki, T. Matsuura et al.
- Persistent currents in a Peierls insulator
M. I. Visscher, B. Rejaei and G. E. W.
Bauer

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October 2015
EPL, 112 (2015) 27005
doi: 10.1209/0295-5075/112/27005

www.epljournal.org

Single-valuedness problem on charge density wave rings
T. Hosokawa1 , T. Matsuura1,2 , M. Tsubota3 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, Japan
3
Division of Quantum Science and Engineering, Hokkaido University - Sapporo 060-8628, Japan2

received 20 June 2015; accepted in ﬁnal form 20 October 2015
published online 5 November 201571.45.Lr – Charge-density-wave systems
72.15.Nj – Collective modes (e.g., in one-dimensional conductors)
PACS 03.65.Vf – Phases: geometric; dynamic or topological
PACS
PACS

Abstract – We investigate the bimodal ﬂuctuation of charge density wave (CDW) current throughring-shaped crystals (typical diameter is a few ten μm) of orthorhombic tantalum trisulﬁde(o-TaS3 ). We ﬁnd that the CDW current changes between two types of the bimodal ﬂuctuation,and exhibits hysteretic behavior as a function of the dc voltage. The hysteresis of the ﬂuctuation modes suggests that the CDW rings involve several metastable sliding states, which might
be a manifestation of single-valuedness of the CDW phase ﬁeld with a strong periodic boundary
condition of the ring geometry.
c EPLA, 2015
Copyright Introduction. – Single-valuedness of wave functions
is a central role in physics on multi-connected spaces [1].
When a particle propagates around a closed ring, thewave function Ψ(θ) of the particle will interfere with itself,since Ψ(θ) is a single-valued function of position coordinate angle θ; Ψ(θ) = Ψ(θ + 2π). A signiﬁcant example
is the Aharonov-Bohm eﬀect for an electron propagatingaround a magnetic ﬂux [2–4]. Macroscopic wave functionsof condensed-matter phases, such as superﬂuid heliumand superconductivity, also possess single-valuedness, andshow quantized circulation of superﬂuid-helium persistentcurrent, and quantized magnetic ﬂux in superconducting
rings, respectively.
Analogously, charge and spin density waves (CDWand SDW), degenerated macroscopic wave functions
developed in low-dimensional electron systems [5],are expressed as a function of position x, such as
ρCDW/SDW (x) = ρ0 cos(QCDW/SDW x + φ(x)). However,the single-valuedness problem of CDW and SDW has
never been studied explicitly.
Recent progress on synthesizing seamless ring-shaped
single crystals of one-dimensional CDW materials witha diameter of a few ten micrometers opened a way forexperimental investigation of CDWs in multi-connected
space [6–10]. Tsubota et al., revealed quantum behaviorof CDW phase ﬁeld dynamics using an Aharonov-Bohm
quantum interference experiment with orthorhombic

tantalum trisulﬁde (o-TaS3 ) rings [11,12]. The authors
found that a magnetoresistance oscillation with φ0 =h/2e, which might be the quantum interference of a CDW
2π phase soliton. Furthermore, they found that the CDW
sliding current ﬂuctuates temporally between two discretevalues simultaneously with the magnetoresistance oscillation. The current switching behavior was clearly observed
when a dc voltage 1.6 times larger than the threshold voltage of the CDW sliding was applied, while thepreviously observed current switching in NbSe3 [13–17],
K0.30 MoO3 [18], and o-TaS3 [19] whiskers had been observed only in the vicinity of the threshold voltage. Theseresults suggested that a discrete CDW dynamics would beenhanced by the ring geometry.
In this manuscript, we report the eﬀect on the singlevalued function of CDWs revealed by the observation ofthe voltage hysteresis of CDW current ﬂuctuation throughring-shaped o-TaS3 crystals. At the CDW sliding regime,
the CDW current shows bimodal ﬂuctuation. Furthermore, the bimodal ﬂuctuation has two kinds of switchingmodes [20]. An eﬀect of the single-valuedness of CDW
rings predicts four metastable states, which would correspond to each current state.
Experimental. – The ring-shaped o-TaS3 crystals
were synthesized by the chemical vapor transport
method [11,12]. Tantalum wires and sulfur powder were27005-p1

T. Hosokawa et al.
0.61(b)no crystal dislocation is introduced for stress relaxation
when the radial thickness of the ring-shaped crystals is
less than ∼ 1 μm [21–24]. Therefore, thin ring-shaped
crystals have as good a crystal quality as whisker crystals,
and no CDW phase disorder is induced along the radial direction [9]. The nesting vector along the one-dimensional
axis c is QCDW = 1/4c∗ at T < 100 K [25,26].
The room temperature resistivity is estimated to be
6 × 10−6 Ωm, where the ring is treated as a parallel circuit.
The CDW transition is observed at 220 K. No hysteresis of
the low-ﬁeld resistance is visible between the cooling andheating processes. The room temperature resistivity andthe CDW transition temperature are consistent with theprevious reported values for o-TaS3 whisker crystals [27].

104I (μA)

30.52617

ICDW 0.2dI/dV (μS)

0.459008Inormal
0.5Vth(c)11.5

0Vdc (V)130 nA370

4I (nA)

762

7ΔI

370

920

1010300500

Time (sec)

1000p (a.u.)

Fig. 1: (Color online) (a) Scanning electron microscopy image of a 45 μm diameter ring-shaped o-TaS3 crystal, which isﬁxed on the substrate with overexposed PMMA ﬁlms, and isattached to two gold electrodes using the electron beam lithography technique. (b) Current-voltage characteristic of the ringshaped o-TaS3 crystal at 80 K. The threshold voltage Vth is
0.2 V. The red broken line is the regression line of the Ohmic
regime, which corresponds to the normal current of thermally
excited electrons. The numbers on the curve indicate the orderof the current ﬂuctuation measurement. (c) Left panels: current ﬂuctuation as a function of time for measurement 1, 4, 7, 9,and 10. A change of a switching mode is observed in measurement 4 and 10 at the moment indicated by the vertical red
broken lines. The range of the vertical axis is 30 nA for allpanels. Right panels: normalized probability distribution p of
the measured current values. The colors of the distribution
curves correspond to those of the time ranges indicated in theleft panels.

heated for 3 hours in an evacuated quartz tube witha temperature gradient of 500 to 600 ◦C. Several ringshaped crystals were found in ﬂuﬀ of o-TaS3 whisker crystals. We picked each ring up and put it on an insulating
SiO2 substrate. Two gold electrical contacts were fabricated on the ring using an electron beam lithography
technique.
The main data reported in this manuscript were obtained with the ring-shaped crystal shown in ﬁg. 1(a).
The crystal had a diameter of 45 μm, a radial thicknessof 0.05 μm, and a width of 2.9 μm. Previous studies of
the residual stress in ring-shaped crystals reported thatResults. – Figure 1(b) shows the dc voltage dependence of the dc current through the ring-shaped crystal at80 K. The nonlinear conduction caused by the CDW sliding is observed above the threshold voltage (Vth = 0.2 V).
At the same temperature, we measure the current ﬂuctuation of the CDW current at several constant dc voltages
indicated by the ﬁlled circles. The numbers show the orderof the temporal ﬂuctuation measurement. The measurement starts at Vdc = 1 V, which is in the sliding regime.
Then the dc voltage is changed to 0.8, 1.2, 1.4 V · · · . The
current value is measured with a picoammeter (Keithley)
every second, and a total of 50000 measurement points areobtained for each constant dc voltage.
The current ﬂuctuations for several dc voltages over aperiod of 1000 seconds selected from all the measurement
points are shown in the left panels of ﬁg. 1(c). The rightpanels show the normalized probability distributions of
the current values. In measurement 1 (Vdc = 1 V), we
observe that the current temporally switches between thelower and the higher values. The bimodal ﬂuctuation isconsistent with that reported in a previous work [12]. This
discretization is not caused by the minimum digit of the picoammeter or accuracy of the measurement system. Notethat we have observed no clear magnetic-ﬁeld dependence
in the switching phenomena in this ring-shaped crystal.
The reason for this remains unclear. Here we focus on thedc voltage dependence of the temporal current switching.
In measurement 1–4, when the dc voltage is increased,current diﬀerence between the lower and higher current(ΔI) and one-state ﬂuctuation are increased. There isa tendency for the probability of the higher current toincrease when the applied dc voltage increases.
In measurement 4 (Vdc = 1.4 V), after several hundred seconds from the measurement started, the switchingmode changes at the moment indicated by the red broken
line in the second panel on the left in ﬁg. 1(c). The changeof the switching mode is clearly seen also in the probability
distributions shown in the right panel. In the black probability distribution is that for before the mode change, andthe lower current state is predominant. After the modechange, the probability of the higher current state changes
that drawn in red. Hereafter, we call the ﬁrst and second

27005-p2

Single-valuedness problem on charge density wave rings(a)Phigh (%)

100710698 5021 341

ICDW (μA)

ΔI (nA)(b)
10Vth000.5

11.50Vdc (V)Fig. 2: (Color online) (a) Hysteretic behavior of probability
at the higher current state as a function of the CDW current.
The numbers on the circles correspond to the order numbers inﬁg. 1(b). The vertical arrows show the change from the blacktrace (switching mode A) to the red trace (switching mode B).
(b) Voltage dependence of the CDW current (blue ﬁlled circles)
and the diﬀerence between the current values of the higher andlower current states (green ﬁlled circles). The broken curves are
guides for the eye.

switching modes as “mode A” and “mode B”, respectively,
for convenience.
We ﬁnd that the mode change is hysteretic. In measurements 5–7, the mode remains B even the dc voltage is reduced below 1.4 V. The result of measurements 7 is clearly
diﬀerent from that of measurements 1. When ICDW = 0(in measurement 9), no switching behavior was observed(ΔI = 0). After that time, we observed the change of theswitching mode from A to B again at Vdc = 1.6 V in measurement 10. The mode change is reproducible, and canbe reset by reducing the dc voltage to below the thresholdvoltage. The hysteresis of the current switching mode wasobserved in NbS3 rings at 100 K.
The probability at the higher current state Phigh ≡Nhigh /(Nhigh + Nlow ) is displayed in ﬁg. 2(a) as a functionof the dc voltage, where Nhigh , and Nlow are the number of data points recognized to higher and lower current
states, respectively. In mode A, Phigh follows the blacktrace. It appears that when the dc voltage exceeds 1.4 V,
the switching mode can change easily from A to B. After the mode change, Phigh follows the red trace. WhenICDW = 0, the switching mode is reset to A, and thenPhigh again follows the black trace.
It seems that amplitude of ﬂuctuation ΔI might havea physical meaning since it is conserved before and after
the switching mode change. We plot ΔI as a function of
the dc voltage in ﬁg. 2(b). ΔI is a linear function of Vdc ,while the CDW current ICDW is not inear with respect
to the dc voltage. The results implies that the current
ﬂuctuation is caused by extra charge carrier, such as CDW
phase solitons, but not ﬂuctuation of conventional CDW
sliding current.

Discussion. – The discrete probability distribution
suggests that the CDW current ﬂuctuates between manymetastable states. Furthermore, there is several switchingmodes. For the bimodal ﬂuctuation at 80 K, ΔI is conserved before and after the switching mode change. We
plot ΔI as a function of the dc voltage in ﬁg. 2(b). ΔI isa linear function of Vdc , while the CDW current ICDW is
not linear with respect to the dc voltage. The results imply that the current ﬂuctuation is caused by extra charge
carriers, such as CDW phase solitons, but not ﬂuctuationof conventional CDW sliding current.
The metastable states were considered to be causedby the CDW phase ﬁeld modulation. As regards
NbSe3 [13–17], K0.30 MoO3 [18], and o-TaS3 [19] whiskers,impurities or electrical contacts act as strong pinning
centers. Depending on the impurity positions, several
metastable states of the CDW phase ﬁeld will emerge.
The current (or voltage) induced switching between two
metastable states could be considered the origin of theelectrical current switching. Such quantized behavior wasalso observed in Ohmic regime for small one-dimensional
CDW materials of o-TaS3 , NbSe3 and K0.30 MoO3 [28–30].
The number of one CDW wavelength between the electrical contacts was quantized. We note that the X-ray
observation of QCDW distribution in the sliding regime
showed that observed phase slipping region modulated byapplied electric ﬁeld is over mm [31]. Thus our CDW rings
are too small to exhibit the phase slipping phenomenon.
In the case of the ring-shaped crystals, the ring geometry would dominate the phase ﬁeld modulation because
the boundary condition restricts CDWs. A single-valued
function of the CDW is imposed to ensure that ρCDW (x) =ρCDW (x + 2πR), where R is radius of the ring. The phasefactor of the CDW is required the following condition:∂x (QCDW x + φ(x)) · dx = 2πnCDW ,(1)C

where x is coordinate on the ring of the one-dimensional
chain C. nCDW is an integer number. Equation (1)is a strong boundary condition for φ(x). In the real
CDW rings, the extra phase φ(x) must be ﬁnite, except
QCDW R = an integer number. Therefore, the ring-shaped
crystals at a CDW state become electrically charged, since∂φ/∂x is proportional to extra electron desnity. An extra
charge in a CDW ring is analogous with a magnetic ﬂux
penetrating a superconducting ring.
There are two possible ways for ﬁnite φ(x). The ﬁrst
one is homogeneous phase modulation, and the second
one is generation of CDW phase solitons, as shown inﬁg. 3. A homogeneous phase modulation changes the CDW
wavelength. On the other hand, the CDW rings can makea fractional/irrational phase soliton. The electrical charge
and phase diﬀerence of a soliton must be less than 2eand 2π. Such solitons should be localized across the wholecross-section and become a soliton wall, due to the interchain coupling energy. The fractional/irrational soliton

27005-p3

T. Hosokawa et al.
Lower current statex

Higher current state

the soliton states are current switching. The hystereticchange between the negatively charged states and the positively charged states could correspond to the mode change
between A and B. The free energies of the four metastablestates will diﬀer depending on the value of QCDW R foreach ring-shaped crystal. Then the probability distribution of modes A and B will be diﬀerent.

Mode ACCurrentswitching

Positive soliton
Mode changes
at a high voltage

Mode changes
at V<VthLower current state

Higher current state

Mode BCurrentswitching

Negative soliton

Fig. 3: (Color online) Schematics of one-dimensional CDWs ofQCDW R = 17.5 at lower and higher current states, respectively.
Black dots indicate crystal lattice. The upper panel shows current switching between the CDW ring of QCDW R = 18 and thatof QCDW R = 17.5 with a positive half (π) phase soliton. Thelower panel shows current switching between the CDW ring ofQCDW R = 17 and that of QCDW R = 17.5 with a negative half(−π) phase soliton.

Summary. – We demonstrated the dc voltage hysteresis of the CDW current switching probability ino-TaS3 ring-shaped crystals. We discussed that the hysteresis is caused by the eﬀect of single-valuedness for theCDW rings. The previous experiments of the AharonovBohm magnetoresistance oscillation suggest the singlevaluedness of the quantum wave function of phase solitons,
but the coupling mechanism between the current switching is unclear yet [12]. To solve this problem, a CDW
might be treated as a quantum wave function within theframework of the Ginzburg-Landau theory [36].
∗∗∗We thank T. Tsuneta, N. Matsunaga, K. Yamaya,K. Nakatsugawa, K. Yamaguchi, K. Katono andK. Ichimura, for fruitful discussions and experimenis diﬀerent from the 2π solitons [32–34] or the amplitude tal support. This work is supported by Iketani Science and Technology Foundation 0241040-A, Hitachi
solitons [35] that can exist individually.
The fractional/irrational phase soliton can explain the Metals · Materials Science Foundation, and the JSPS
experimental results that ΔI/ICDW ∼ 10−3 . The soliton KAKENHI Grant Nos. 12837528, and 26287069.
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