Journal of the Physical Society of Japan
Vol. 74, No. 6, June, 2005, pp. 1787–1791#2005 The Physical Society of Japan

Violation of Kohler’s Rule in the Magnetoresistancenear the Lower Charge-Density-Wave Instability in NbSe3
Syuma Y ASUZUKA, Kazuhiko Y AMAYA, Yoshitoshi O KAJIMAy, Satoshi TANDA,
Nao T AKESHITA1 z, Hiroyuki M ITAMURA1 , Takeshi N AKANISHI2 x and Nobuo M ÔRI1 }Department of Applied Physics, Hokkaido University, Sapporo 060-8628
Institute for Solid State Physics (ISSP), University of Tokyo, Chiba 277-8581
2
Department of Physics, College of Humanities and Science, Nihon University, Tokyo 156-8550
1(Received February 13, 2005; accepted March 17, 2005)

This paper reports the measurements of resistance (R) and magnetoresistance of NbSe3 near the criticalpressure (Pc ¼ 7:5 kbar), where the lower charge-density-wave (CDW) phase is on the verge ofdestruction. For P ¼ 7:6 kbar, the temperature dependence of dR=dT exhibits a weak anomaly at T  ¼
15 K and the superconducting transition is observed at Tc ¼ 2:8 K. From the observation of a large
magnetoresistance due to an imperfect nesting of Fermi surface, we show that the weak anomaly isclosely related to the lower CDW phase. Moreover, an excess conductance and violation of Kohler’s rule
are found below T  . The origins of the excess conductance and the violation of Kohler’s rule are
discussed in terms of charge ﬂuctuation associated with the lower CDW state.
KEYWORDS: NbSe3 , charge-density-wave, superconductivity, critical pressure, magnetoreistance, Kohler’s rule
DOI: 10.1143/JPSJ.74.1787

1.

ln TIntroduction

A physical system that crosses the boundary between
crystalline ordered and itinerant phases changes its properties in a fundamental way. It may, for example, melt orfreeze. This macroscopic change is driven by microscopic
ﬂuctuation. When the temperature of the system approaches
zero, all thermal ﬂuctuations die out. This prohibits phasetransitions in classical systems at zero temperature. In aquantum system, however, ﬂuctuations are present even atzero temperature, so-called quantum ﬂuctuations. These
quantum ﬂuctuations may be strong enough to drive atransition from one phase to another, bringing about amacroscopic change.
In this paper, we focus on the pressure–temperature (P–T)
phase diagram in a typical quasi-one dimensional conductor
NbSe3 ,1–3) in which there is remarkable competition of thesuperconducting (SC) and the lower charge-density-wave(T2 -CDW) states at a critical pressure of Pc ¼ 7:5 kbar (seeFig. 1).4,5) This competition has been understood in terms of
the Fermi surface (FS) nesting4–6) and the quantum ﬂuctuations are ignored. The quantum ﬂuctuations, however, may
play an important role for the competition near Pc . Because
the SC state is the itinerant oﬀ-diagonal long-range order(ODLRO) by Bose condensation of the Cooper pairs, while
the CDW is the diagonal long-range order (DRLO) withstructural ordering with electrons and the competition occurs
Present address: Department of Molecular Materials Science, Graduate
School of Science, Osaka City University, Sumiyoshi-ku, Osaka 5588585. E-mail: yasuzuka@sci.osaka-cu.ac.jp
yPresent address: Asahikawa National College of Technology, 2-2-1-6
Shunkodai, Asahikawa, Hokkaido 071-8142.
z
Present address: Correlated Electron Research Center (CERC), National
Institute of Advanced Industrial Science and Technology (AIST),
Tsukuba, Ibaraki 305-8562.
x
Present address: High Pressure Science Section, Max-Planck-Institute for
the Chemical Physics of Solids, Noethnitzer Str. 40, D-01187 Dresden,
Germany.
}
Present address: Department of Physics, Faculty of Science, Saitama
University, 255 Shimo-Ohkubo, Saitama 338-8570.

T1

Metal58 K

T2

3.5 K

Lower
CDW
phase

UpperCDW
phase

142 KTc
Superconducting
phase

Pc
(7.5 kbar)

P (kbar)Fig. 1. Schematic pressure–temperature phase diagram of NbSe3 . NearPc ¼ 7:5 kbar, the lower CDW phase competes with the superconducting
phase at low temperatures.

at low temperatures down to 3 K. We are also interested in
the lower CDW phase near Pc where the exotic phasebehavior due to the quantum ﬂuctuations of the lower CDW
phase is expected at low temperatures. Actually, Snow etal.7) recently observed the quantum melting of a CDW state
tuned by a pressure at low temperatures. In this paper, we
report the experimental results of the resistance (R) andmagnetoresistance (MR) of NbSe3 near Pc .
2.

ExperimentalSingle crystals of NbSe3 were grown by chemical vapor
transport. The crystals gave residual resistance ratio
R(300 K)/R(4.2 K) in the range 50–60. The R and MR were
measured by a usual four probe dc method, where current
and magnetic ﬁeld are parallel to the b- and c-axes,respectively. The pressure was generated by use of a WC
piston and a copper–beryllium cylinder with the internal
diameter of 8 mm. Transmitting liquid was 1 : 1 mixture ofFluorinert FC70 and FC77. The pressure was applied at room
temperature and kept constant during the measurements incooling and heating processes. Pressures were estimatedfrom NH4 F I–II transition and the pressure dependence of T11787

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S. YASUZUKA et al.

and T2 in NbSe3 based on a previous result.5)
It is known that the lower CDW phase is accompaniedwith a large magnetoresistance (LMR) just below T2 whenmagnetic ﬁeld is applied parallel to the c-axis.8–11) The
origin of the LMR has been explained by considering thenormal carriers on the small pockets created by the imperfectnesting of the FS.10–12) As a unique method to detect thelower CDW phase near Pc , we took advantage of theoccurrence of the LMR. The LMR appears as long as thelower CDW phase exists, while the LMR disappears as soon
as the lower CDW phase is suppressed totally, showing thatthe observation of the LMR is useful for probing the lower
CDW phase near Pc .10,11)6(a) P = 7.2 kbar
5320T

Results

30.2

P = 7.6 kbar
B=0T0.15

2
1.5

0.1

T*1dR/dT (Ohm / K)

R (Ohm)2.50.050.50051015202530035

T (K)Fig. 2. Temperature dependence of R (left-hand scale, dots) and dR=dT(right-hand scale, squares) for P ¼ 7:6 kbar.

0510152025302025

T (K)2(b) P = 7.6 kbar
1.517 T

T*R (Ohm)

Figure 2 shows the temperature (T) dependence of the R(left-hand scale, dot) and dR=dT (right hand scale, squares)for P ¼ 7:6 kbar. With decreasing temperature, R decreases
monotonically and the superconducting transition takes
place at Tc ¼ 2:8 K. There is no anomaly associated withthe lower CDW transition. These results agree well withprevious results.4,5) Besides these conventional phenomena,
we ﬁnd that the T-dependence of dR=dT exhibits a weakanomaly at T  ¼ 15 K. The slope of dR=dT below T  isgentler than that above T  . The origin of the weak anomaly
is unclear. Near Pc , one expects the presence of the lower
CDW phase. In order to know whether the weak anomaly isrelated to the lower CDW transition or not, we measured theresistance under magnetic ﬁelds for P ¼ 7:6 kbar.
Figure 3(a) shows the T-dependence of the resistance inmagnetic ﬁelds ranging from 0 to 17 T under a pressure of
7.2 kbar. The lower CDW transition is observed at T2 ¼ 22 K
under zero magnetic ﬁeld. The magnetic ﬁeld enhances theresistance anomaly due to the lower CDW formation.
However, the value of T2 is almost independent of themagnetic ﬁelds. The origin of the ﬁeld-induced enhancementof the lower CDW resistance anomaly has been explained bynormal carriers on the small pockets created by the imperfectnesting of the FS, leading to the occurrence of theLMR.10–12) On the other hand, a small increment ofresistance is observed above T2 . Here the magnitude ofT2

10

3.

17 T

4R (Ohm)

1788

1

0.5

0T
0

0

5

10

15T (K)
Fig. 3. Temperature dependence of R in magnetic ﬁelds ranging from 0 to
17 T. The magnetic ﬁelds are 0, 0.5, 1, 1.5, 2.5, 5, 7, 12, 14.5, and 17 T inorder of increasing resistance for (a) P ¼ 7:2 kbar and (b) P ¼ 7:6 kbar.

MR is negligibly small at ambient pressure because of thenearly perfect nesting of FS for the upper CDW phase.13)
However, applying pressure causes a pressure-induced
imperfect nesting of FS, leading to the appearance of MR
which is called the pressure-induced MR (PIMR).10,11) Theseobservations agree well with our previous results.10,11)
In addition, we ﬁnd the superconducting phase transition
at Tc ¼ 2:7 K. This result does not agree with previous
results by Ido et al.5) It is known that the superconductivityof NbSe3 is very sensitive to the pressure inhomogeneity
below Pc . However, the observed resistive transition is very
sharp and very reproducible. In addition, a recent highpressure study14) by cubic anvil device shows that thesuperconductivity coexists with the upper CDW phase near
critical pressure for the upper CDW phase. These observations may suggest that the coexistence of superconductivity
and the lower CDW phase is intrinsic.
Figure 3(b) shows the T-dependence of the resistance inmagnetic ﬁelds ranging from 0 to 17 T under a pressure of
7.6 kbar. We observe metallic behavior (dR=dT > 0) below
2.5 T while upturn around below T  above 2.5 T. Thestriking feature below T  is reminiscent of the ﬁeld-induced
enhancement of the lower CDW resistance anomaly as seenin Fig. 3(a).

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S. YASUZUKA et al.

In Fig. 3, the SC transitions vanish above 0.5 T. Theseobservations are consistent with a previous result.15) The
small upper critical magnetic ﬁeld, Hc2 , has been explainedby the eﬀective mass model for Hc2 .16)
Figure 4(a) shows the T-dependence of MR, =0 ,under the pressure of 7.2 kbar. The LMR is clearly observed
around below T2 , while relatively small PIMR above T2 . The
occurrences of the LMR and PIMR agree well with previous

B // c6∆ρ / ρ

00.5 T
1T
1.5 T
2.5 T
5T

427T
12 T
14.5 T
17 T

T

205

1015

2025

30

35

T (K)
5(b) P = 7.6 kbar∆ρ / ρ04B // c
0.5 T
1T
1.5 T
2.5 T
5T

327T
12 T
14.5 T17 TT*
10010

515202530Discussion

The LMR can be explained by semiclassical transport
theory.10,11) According to the semiclassical transport theory,
Kohler’s rule holds if there is a single species of charge
carrier and the scattering time is the same at all points on theFS.17) Kohler’s rule is given by

B:
ð1Þ
¼ Fð!c Þ ¼ f
0
Rð0; TÞ(a) P = 7.2 kbar

0results.10,11) Figure 4(b) shows the T-dependence of MR,
=0 , under the pressure of 7.6 kbar. The T-dependence ofMR for 7.6 kbar is very similar to that for 7.2 kbar. MR
gradually increases with decreasing temperature and then it
rises sharply around T  . For P ¼ 7:6 kbar, the value of MR
at 3 K are about 400% and much larger than that of thePIMR appearing above 15 K.10,11) These results suggest thatthe weak anomaly at T  in dR=dT is closely related to thelower CDW phase transition.
4.

835T (K)Fig. 4. Temperature dependence of magnetoresistance under pressure of(a) P ¼ 7:2 kbar and (b) P ¼ 7:6 kbar.

The corresponding plots are known as Kohler plots. Here !cis the cyclotron frequency,  is the scattering time, andRð0; TÞ is the zero-ﬁeld resistance. If Kohler’s rule holds, allof MR curves would collapse onto a single curve. Let us
check whether the MR for 7.2 and 7.6 kbar obey Kohler’srule or not.
Figure 5(a) shows the MR data for P ¼ 7:2 kbar in theform of Kohler plots. In this case, we see that MR curves
collapse onto an upper and a lower curves for T < T2 =2 andT > T2 , respectively. This implies Kohler’s rule holds for
T < T2 =2 and T > T2 . Note that here the T-dependence ofthe lower CDW gap is very little. On the other hand, theviolation of Kohler’s rule is clearly observed for T2 =2 <T < T2 where the magnitude of the lower CDW gap varies
signiﬁcantly. These observations are consistent with the Tdependence of the lower CDW gap.
Next, we consider the MR data in the form of Kohler plotsfor P ¼ 7:6 kbar. Figure 5(b) shows the most surprising andstriking results of the present work. We ﬁnd Kohler’s ruleholds for temperatures T > T  and does not below T < T  .
Comparing with the result observed at P ¼ 7:2 kbar [see125(a) P = 7.2 kbar

T < T 2 /24(b) P = 7.6 kbar

90.5 T

0.5 T0∆ρ / ρ∆ρ / ρ01T
1.5 T

6

2.5 T
5T

31T
1.5 T
2.5 T

25T
7T

7T

12 T

12 T

31

14.5 T

14.5 T
17 T

17 T

T>T*T > T200.1

01

10100

B / R (0 , T ) [T / Ohm]1789

0.1

1

10100

B / R (0 , T ) [T / Ohm]

Fig. 5. Magnetoresistance data for (a) P ¼ 7:2 kbar and (b) P ¼ 7:6 kbar in the form of Kohler plots.

1790

J. Phys. Soc. Jpn., Vol. 74, No. 6, June, 2005

S. YASUZUKA et al.

Fig. 5(a)], it might be natural to assume that T  corresponds
to the lower CDW transition temperature at P ¼ 7:6 kbar.
However, MR does not follow Kolher’s rule even at 3 K
despite the T-dependence of the lower CDW gap isconsidered to be very little for temperatures T < T  =2 
7:5 K, indicating that T  does not necessarily correspond tothe lower CDW transition temperature where the 3D longrange order is formed.
To consider the origin of the violation of Kohler’s rule, wecarefully analyze the T-dependence of R and dR=dT. Let us
consider the T-dependence of R in the absence of the weakanomaly at T  in dR=dT. If there is no anomaly at T  indR=dT, the T-dependence of dR=dT is expected to be astraight line which is obtained by extrapolating the dR=dTabove T  (see Fig. 2). Integrating the straight line withrespect to T, we can calculate the T-dependence of R in theabsence of the weak anomaly at T  as seen in Fig. 6. Fromthe analysis, we ﬁnd that the weak anomaly of dR=dT at T yields an excess conductance below T  . This behavior iscontrary to the case of a usual CDW transition.1–3) The
excess conductance around below T  cannot be readily
explained in the case of a pinned static CDW state.
One may think that the excess conductance is caused bythe sliding motion of the lower CDW state. But we canobserve the excess conductance in the ohmic regime. So it isunlikely that the excess conductance is attributed to thesliding CDW state. Then we consider that a charge
ﬂuctuation associated with the lower CDW state as thepossible origin of the excess conductance. Because the CDW
domains may be very mobile in the so-called ﬂuctuation
regime,18) leading to the lack of the depinning electric
threshold ﬁeld. It is, therefore, not surprising that the RðTÞbehaves as if there were no CDW ﬂuctuation. Our observed
excess conductance is reminiscent of a metallic phase in
TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane)with a strongly temperature-dependent conductivity due tothe ﬂuctuating CDW.19,20) In the ﬂuctuation regime of
NbSe3 , a pseudo gap will open.18) Then the electronic
structure changes and the charge ﬂuctuation along the chain
direction will bear enhancement of anisotropy of scattering

time. Moreover, the pseudo gap will develop with decreasing temperature as similar to the CDW gap.21) These willlead to the violation of the Kohler’s rule. Therefore, weconclude that the excess conductance and the violation of theKohler’s rule are due to the charge ﬂuctuation associatedwith the lower CDW state.
The violation of Kohler’s rule at low temperatures showsthat the lower CDW transition temperature T2 is below 3 K
at least, or will go to absolute zero at Pc , where the thermal
ﬂuctuations frozen out, while the quantum ﬂuctuations
prevail. This observation suggests importance of a quantum
mechanical nature in the lower CDW phase near Pc , which is
not considered by the mean-ﬁeld theory by Yamaji.6)
Recently, Snow et al.7) revealed in a CDW system 1TTiSe2 that the CDW softens and exhibits enhanced ﬂuctuations when the collapse of the CDW state occurs near acritical pressure at low temperatures. Destruction of thelower CDW phase in NbSe3 has been explained in terms ofcollapse of the conventional FS nesting.4–6) The lower CDW
state near Pc , however, may be very similar to the quantum
melting of the CDW state in 1T-TiSe2 .7)
5.

Conclusions

We have performed the measurements of R and MR nearPc (¼ 7:5 kbar) where the lower CDW phase is on the vergeof destruction. For P ¼ 7:6 kbar, we ﬁnd the weak anomaly
at T  ¼ 15 K in the T-dependence of dR=dT and thesuperconducting transition takes place at Tc ¼ 2:8 K. Fromthe measurements of the LMR, we show that the weakanomaly is closely related to the lower CDW phase.
Moreover, the excess conductance and the violation of
Kohler’s rule are found below T  . Based on the aboveﬁndings, we claim the presence of the charge ﬂuctuation
related to the lower CDW phase below T  near Pc . The
experimental results reported here demonstrate the exotic
phase behavior due to the quantum ﬂuctuations of the lower
CDW phase near Pc , which is not predicted by the meanﬁeld theory by Yamaji.6)
Acknowledgment
The authors thank Dr. M. Ohashi for experimental
assistance. The authors thank Professor Y. Okayama for
useful discussion and suggestions. A part of the experimental work was carried out at the Institute for Solid State
Physics, the University of Tokyo.

0.7

P = 7.6 kbar
B=0TR (Ohm)

0.6

No anomaly at T = T*in dR / dT
0.5

T*
0.4

Actual T-dependence
0.3

910111213141516T (K)Fig. 6. Temperature dependence of resistance in the absence (calculation,open circles) and presence (experiment, solid circles) of the weakanomaly at T  .

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