PHYSICAL REVIEW B 73, 153403 共2006兲

Quenching of phase coherence in quasi-one-dimensional ring crystals
K. Shimatake,1 Y. Toda,1,2 and S. Tanda1
1Department of Applied Physics, Hokkaido University, Kita 13 Nishi 8 Kita-ku, Sapporo 060-8628, Japan
2PRESTO Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama, Japan

共Received 14 November 2005; revised manuscript received 17 January 2006; published 10 April 2006兲
A comparison of the single-particle 共SP兲 dynamics of whisker and ring NbSe3 crystals provides new insight
into the phase transition properties of quasi-one-dimensional charge density wave 共CDW兲 systems. In theincommensurate CDW phase, SP relaxation triggered by an ultrafast laser pulse reflects the formation ofcollective states, and reveals the divergence of the relaxation time when approaching a transition temperature.
The degree of divergence is less pronounced in rings than in whiskers, suggesting a loss of phase coherence inring crystals characterized by a closed-loop topology.
DOI: 10.1103/PhysRevB.73.153403

PACS number共s兲: 78.47.⫹p, 71.45.Lr

Over the last few decades, there has been intense interestin understanding the properties of charge density waves
共CDW兲 in quasi-one-dimensional 共1D兲 metals, where one of
the most widely discussed issues is the influence of phasefluctuations on CDW transitions.1 In an uncoupled-1D system, large fluctuations greatly reduce the CDW transition
temperature 共Tc兲 below the mean-field transition temperature. CDWs undergo a phase transition only when long-range
three-dimensional 共3D兲 order develops.1 Indeed, a pronounced finite size effect has been observed in NbSe3, inwhich a reduction in the number of parallel chains makes thetransitions less pronounced and reduces Tc.2 In this sense, itis also expected that crystal topology will make a substantial
contribution to the phase transition since the topology imposes additional constraints on interchain correlations. However, the effects of crystal topology on CDW properties arestill poorly understood.
Recently, several 1D transition metal chalcogenides of thetype MX3 were found to have various types of topologicalcrystals, where the whisker crystals naturally form ring,
Möbius, and figure-of-eight geometries.3,5 The small sizes
and damage-free formations of these structures make it possible for CDWs to maintain their coherence within individual
chains with a closed loop, thus making them good candidates
for studying the effects of topology on the long-range ordering. In this work, we have investigated topological crystalsof NbSe3 by measuring time-resolved optical reflectivity
changes. A comparison of the single-particle 共SP兲 decay ofwhisker and ring structures reveals a significant difference,suggesting that the crystal topology influences their phasetransitions.
Figures 1共a兲 and 1共b兲 show scanning electron micrographsof a NbSe3 whisker and ring, respectively. Both crystals wereprepared by the chemical vapor transport method under virtually the same conditions. The whisker has a standard bulk
with a length of a few mm 共along the conducting axis b兲 anda width of 50 ␮m. The ring has a somewhat different structure with an outer diameter of 50 ␮m and an internal diameter of several ␮m 共conducting axis b in the tangential direction兲. The circumference of the center hole is around 10 ␮mand this is comparable to the correlation length of CDWs
共␰储b ⬎ 2.5 ␮m兲4 in this compound. The details of the growthmechanism for the ring and its structural analysis based on
x-ray diffraction have been described in detail elsewhere.5
To evaluate the CDW properties, we employed an opticalmeasurement technique that is ideal for observing topological structures since the photon can act as a noncontact probe
and therefore preserve the crystal topology. A time-resolved
measurement was achieved by using a conventional pumpprobe technique with a micro-optical setup. For the excitation source, we used a mode-locked Ti:Sapphire laser with apulse width of ⬃130 fs centered at an energy of 1.56 eVwith a repetition rate of 76 MHz. We detected the reflectivity
change 共⌬R兲 of the probe pulse as a function of the delay

1098-0121/2006/73共15兲/153403共4兲/$23.00

FIG. 1. 共Color online兲. Electron micrographs of the topologicalsamples: 共a兲 Whisker and 共b兲 ring. The corresponding transient reflectivity changes, ⌬R, at various temperatures below and above Tc2
in 共c兲 and 共d兲. The solid lines indicate the results of least-squares
fitting.

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from the pump pulse. The pump and probe pulses were orthogonally polarized and focused through an objective lens
onto a single-crystal region. The probe polarization was set
along the chain direction of the whisker, where an efficienttransient signal was observed. On the other hand, for the ringcrytal, we set the probe polarization along any arbitrary axis
in the b-c plane. The pulse overlap had a diameter of 10 ␮mon the sample surface. This overlap was monitored using acharge coupled device camera, and was kept at a fixed position during the measurement of each sample. For the ringsample, the position was fixed so that it was near the innerhole in order to emphasize its topological character. The
pump and probe fluences were ⬃40 ␮J / cm2 and⬃10 ␮J / cm2, respectively. The steady-state heating causedby the laser was accounted for by measuring the excitationpower dependence of ⌬R.
NbSe3 consists of three pairs of metallic chains parallel tothe conducting axis, and exhibits two incommensurate
CDWs with transition temperatures at Tc1 = 145 K and Tc2= 59 K. Note that similar Tc’s 共Tc1 = 140.8 K and Tc2= 57.4 K兲 have been found in ring samples by resistivity
measurement.5 Figures 1共c兲 and 1共d兲 show several transient⌬R’s of a whisker and a ring, respectively, for various temperatures below and above Tc2. As observed in K0.3MoO3and several other CDW compounds,6,7 the signal at the lowest temperature is dominated by two features: A combinationof exponential responses and damped sinusoidal oscillations.
We assign the exponential part to the transient responses ofthe SPs, while the oscillation part reflects coherent
motions—including phonon and collective CDW modes induced by instantaneously photoexcited SPs.
We will now focus our attention on the SP dynamics. Theinterpretation of the transient ⌬R in CDW systems has been
established by analogy to that in widely used materials, suchas metals and semiconductors.6,8,9 Since a pump pulse withnear-infrared energy can excite SPs into continuum states far
above the CDW gap, relaxation to states near the band edge
results in an abrupt increase in ⌬R. The CDW gap thencauses carriers to accumulate at its upper edge. A subsequent
decay thus reflects the transient density change of these accumulated carriers. Since the CDW gap depends on thesample temperature, this fast decay exhibits a temperaturedependence associated with the formation of the gap.6,8 A
long-lived decay can be attributed to relaxation from phasonstates pinned just above the ground state and/or impurityrelated trapped states within the gap.9
When the temperature is increased to Tc2, the decay timein both types of crystal increases—followed by an abrupt
decrease above Tc2. The relaxation process across the gap ismanifested as phonon emissions and absorptions. Since theenergy gap of CDWs is strongly temperature dependent anddecreases as it approaches Tc2 from below, the increase inphonon density—which can contribute to the reabsorption
process—results in an increase in the relaxation time. The
divergence of the decay time just below Tc2 thus reflects theformation of a collective gap.
It is important to note that no significant changes occur in⌬R at around Tc1 under the present experimental conditions
with a probe energy of 1.56 eV. On the other hand, ⌬R witha probe energy of 1.1 eV shows characteristically diverse

FIG. 2. 共Color online兲 Temperature dependences of ␶s in thewhisker 共a兲 and ring 共b兲 around Tc2. The solid line in 共a兲 shows atheoretical fit 共Ref. 8兲 to the data, and this is also shown in 共b兲 forcomparison, where a large reduction in ␶s and disagreement withthe theoretical curve are clearly seen. Insets: Sketches of CDWsmodulated on neighboring chains. Below the mean-field temperature, Coulomb repulsion tends to align the CDWs out of phase witheach other.

behavior at each transition temperature. In the latter case,however, the SP response from another chain can contribute
more to the signal below Tc2 than in the former case. Inaddition, Tc2 is suitable for analyzing characteristic SP dynamics because the transition at a lower temperature reduces
the thermal fluctuation and makes the collective gap formation clearer than that at a higher temperature.
Figures 2共a兲 and 2共b兲 show plots of the fast decay time
共␶s兲 as a function of temperature in the whisker and ring,
respectively. In accordance with the SP relaxation processes
described above, we evaluated ␶s by a least-squares fitting
procedure with the sum of two exponential functions. Below⬃40 K, ␶s in both samples is nearly constant 共⬍1 ps兲. Incontrast, a clear difference is seen above T = 50 K, where ␶sdiverges due to the reduction in the gap energy. However, thedegrees of divergence of ␶s are very different in these twosamples. For a better comparison, Fig. 3 shows simultaneousplots of transient signals associated with the longest decaytime. The signal in the whisker exhibits an extremely longexponential decay whose evaluated ␶s reaches almost 10 ps.
In contrast, ␶s in the ring is almost 2 ps.
For another qualitative analysis of ␶s, we fit the data interms of the Bardeen–Cooper–Schrieffer 共BCS兲-type temperature dependence of the gap 关2⌬g共T兲兴. The solid curve in
Fig. 2共a兲 represents the theoretical fit given by Kabanov etal.8 to the data with ⌬g共0兲 ⬇ 50 meV. Although 2⌬g共0兲 inNbSe3 obtained in previous experiments varies
significantly,10 ⌬g共0兲 ⬇ 50 meV obtained in our optical measurement is in good agreement with the maximum gap valueof 45± 10 meV obtained with angle-resolved photoemission
spectroscopy.11 On the other hand, we cannot accurately fix
the parameter using the ring data. This is due to the quenching behavior of ␶s at around Tc2. Instead, the curve optimized
for the whisker is represented by the dashed line in Fig. 2共b兲.
A marked deviation from the theoretical fit can be seen in thevicinity of Tc2. This discrepancy suggests that the transient⌬R reflects the critical Ginzuburg–Landau fluctuations,where ⌬g共T兲 cannot be reproduced by conventionally used

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FIG. 4. Fourier transform of the oscillation signal obtained bysubtracting the exponential contributions from the data: 共a兲 Spectrum for the whisker at 28 K and 共b兲 spectrum for the ring at 27 K.
The peak at the lower frequency is that of an amplitude mode
共AM兲.
FIG. 3. 共Color online兲 Semilogarithmic plots of the reflectivitychanges for the whisker 共T = 56 K兲 and the ring 共T = 52 K兲, wherethe longest ␶s was obtained in each crystal. The plots are shiftedvertically for clarity. The inset shows another plot for the ring. Thesolid line indicates the slope of exp共−t / ␶兲0.3, where ␶ is 1.61.

theoretical analysis based on the mean-field approximation.
We now consider the topological effect on the phase transition properties on the basis of the different ␶s temperaturedependences of the crystals. The difference in the longest ␶sat Tc2 is attributed to the difference between the coherencesof the collective states of the samples. In this case, the suppressed divergence of the decay time observed in the ringindicates quenching of the phase coherence. The insets inFigs. 2共a兲 and 2共b兲 are schematic illustrations of the candidate spatial distributions of charge density along the individual chains for the whisker and the ring, respectively. Let
us recall the phase transition process in CDW materials. In a
quasi-1D system, the Coulomb correlation between adjacent
chains should be taken into account in the phase transition.1
In a whisker, 3D ordering can be easily realized by adjusting
the neighboring chains with a phase difference of ␲. In contrast, for a ring crystal, CDWs with the same oscillation period in the closed loop reduce the Coulomb repulsion solely
by undergoing a lattice expansion, resulting in a loss ofphase coherence between adjacent chains. This topologicallyinduced phase fluctuation is also consistent with deviations
from the BCS-type temperature dependence in the vicinity of
Tc2. The enhanced phase fluctuation prevents the divergenceof the relaxation time represented by the mean-field approximation. As a result, the transient ⌬R in the ring renders thecritical phase fluctuations clear at around Tc2. So far, only thesize effect has been focused on in discussions of the interchain interaction in quasi-1D systems.2 However, based on
the mechanism discussed above, crystal topology can alsoenhance the phase fluctuations between neighboring chains.
Note that it is also possible to consider that the difference
between the ⌬R values of the two crystals arises from thenonconservative polarization axis characterized by the ringshape. However, as far as the decay constant of ⌬R is concerned, almost no polarization dependence was observed
even in the whisker, while the magnitude of ⌬R reveals astrong anisotropy with respect to the conducting axis. Therefore, we conclude that the polarization effect can be excluded

from our discussions based on the differences in the decay
properties.
We now briefly comment on the relaxation function of thetransient ⌬R in the vicinity of Tc2. Recently, Nogawa et al.12
proposed a phase field model for CDWs in ring crystals remarking on the frustration between intra- and interchain couplings. By using Monte Carlo simulations, they found thatthe relaxation function in the low-temperature-ordered phaseexhibits a power-law decay instead of the usual exponential
decay. On the other hand, we should note that the transient⌬R for the ring is well fitted by a function of exp共−t / ␶兲0.3 incomparison with the simple exponential decay for the whisker, as highlighted in Fig. 3.13 These facts suggest that thetype of phase transition for ring-shaped CDWs is essentiallydifferent from that for whiskers. In the slow relaxation expressed by a power or stretched exponential function, theenhancement of phase fluctuations originating from the ring
topology must play an important role.
Nevertheless, it is also possible to explain the different
transient dynamics in terms of the differences between the
crystal inhomogeneities of the samples. Since the phase fluctuation is sensitive to impurities, dislocations, and other disorders, we cannot completely exclude the contribution of thecrystal inhomogeneities to the present results. As a simple
check, we evaluated a coherent oscillation of CDWs induced
by instantaneous optical excitation. Since the dephasing timeof collective motion is determined by the CDW coherence
length, we can qualitatively compare the crystal inhomogeneities.
Figures 4共a兲 and 4共b兲 show the fast Fourier transform
spectra of the whisker and the ring, respectively, at almost
the same temperature. In each spectrum, two distinct peaks
are clearly visible. The lower mode at around 1.1 THz showsa softening behavior with increasing temperature and completely disappears above Tc2, indicating the collective excitation of the CDW 共amplitude mode: AM兲 associated with
the incommensurate phase of Tc2, while the higher mode isidentified as a coherent phonon because of its less pronounced temperature dependence. Note that this is the firstobservation of the AM mode in NbSe3. However, its frequency and temperature dependences are qualitatively similar to the AM oscillation observed in several other CDW
compounds.6,7 The linewidths of the AM obtained from the153403-3

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Lorentzian fits to the spectra are ⬃75 GHz for the whisker
and ⬃48 GHz for the ring, which are comparable values. We
thus believe that the contributions made by crystal inhomogeneities to the phase fluctuation are almost the same.
In summary, we have investigated the ultrafast SP dynamics in ring and whisker NbSe3 crystals, in terms of the crystal
topology. In the phase transition around Tc2, the temperaturedependence of the SP recombination time observed in thering is quantitatively and qualitatively different from that in

the whisker: The divergence of ␶s is quenched and there is adivergence from BCS-type temperature dependence. Theseresults can be explained in terms of the enhanced phase fluctuation in the closed-loop topology.

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The authors sincerely thank K. Inagaki, T. Nogawa and K.
Nemoto for fruitful discussions. This work was supported by
a Grant-in-Aid for the 21st Century COE program “Topological Science and Technology.”

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