PHYSICAL REVIEW B 97, 115432 (2018)

Lock-in transition of charge density waves in quasi-one-dimensional conductors:
Reinterpretation of McMillan’s theory
Katsuhiko Inagaki
Department of Physics, Asahikawa Medical University, Midorigaoka Higashi 2-1, Asahikawa, 078-8510, Japan

Satoshi Tanda
Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo 060-8628, Japan(Received 9 August 2017; revised manuscript received 3 February 2018; published 20 March 2018)
We investigated the lock-in transition of charge density waves (CDWs) in quasi-one-dimensional conductors,based on McMillan’s free energy. The higher-order umklapp terms play an essential role in this study. McMillan’s
theory was extended by Nakanishi and Shiba to treat multiple CDW vectors. Although their theories were aimed
at understanding CDWs in quasi-two-dimensional conductors, we applied them to the quasi-one-dimensional
conductors, including K0.3 MoO3 , NbSe3 , and m-TaS3 , and confirmed its validity for these cases. Then we discussed
our previous experimental result of o-TaS3 , which revealed the coexistence of commensurate and incommensurate
states. We found that the coexistence of multiple CDW vectors is essential for the lock-in transition to occur ino-TaS3 . The even- and odd-order terms in the free energy play roles for amplitude development and phasemodulation, respectively. Moreover, consideration of the condition of being commensurate CDWs allowed us torelate it with that of the weak localization in random media.
DOI: 10.1103/PhysRevB.97.115432

I. INTRODUCTION

II. PREVIOUS EXPERIMENTS

Lock-in transition between incommensurate and commensurate charge density waves (CDWs) has been studied since
the mid-1970s [1–14]. It is induced by the coupling of alattice periodicity with a charge density wave. The transition
is often accompanied with the formation of a discommensuration lattice between commensurate and incommensurate
phases. Occurrence of discommensuration was predicted bytheory [2] and found in quasi-two-dimensional conductors,e.g., in 2H-TaSe2 [5,7] and 1T-TaS2 [9,11,14], both of which
are typical quasi-two-dimensional conductors with Peierls
transition. In contrast, the lock-in transition of CDWs in
quasi-one-dimensional conductors remains unsubstantiated,
although several experiments were reported [6,8,10,12]. We
previously performed a synchrotron x-ray study in o-TaS3and suggested that the discommensuration lattice is formed
when commensurate and incommensurate phases coexist [13].
However, this preliminary report lacked theoretical interpretation. In this paper, we review the theoretical treatments of thelock-in transition and apply them to the quasi-one-dimensional
conductors, including K0.3 MoO3 , NbSe3 , and m-TaS3 . The
validity of the theory is confirmed for these cases. We then go
on to discuss the synchrotron data of o-TaS3 . The coexistenceof the commensurate and incommensurate phases is found tobe essential for the lock-in transition. The even- and odd-orderterms in free energy play roles for amplitude development andphase modulation, respectively. Moreover, consideration of thecondition of being commensurate CDWs allows us to relate
it with that of the weak localization in random media [15].
This explains why quantum interference phenomena have been
observed in CDWs [16,17].

Let us take an overview of the lock-in transition in o-TaS3 .
This quasi-one-dimensional conductor undergoes a Peierls
transition at 220 K. At the transition, all the electrons on
the Fermi surfaces contribute to form the Peierls gap and thesystem becomes an insulator, contrary to similar materials,
such as NbSe3 and m-TaS3 , in which remained electrons
contribute to metallic conduction even after Peierls transitions
occur. Hence, by absence of remained normal electrons, o-TaS3
is one of the most appropriate materials for CDW studies. The
first x-ray study was made by Tsusumi et al. who determined
the CDW vector of o-TaS3 [18]. This work was followedby Roucau, who found that the CDW vector shifts frombeing 0.255c∗ (incommensurate) to 0.250c∗ (commensurate)
at low temperatures [8]. The details of the lock-in transition
were revealed by use of synchrotron diffraction [13]. Bylowering the temperature, the CDW vector shifts from being
incommensurate closer to commensurate; however, it stops
at 0.252c∗ . The commensurate CDW independently appears
at 130 K. In addition, coexistence of the commensurate andincommensurate CDWs was found in the temperature range
down to 50 K, then the complete lock-in was observed at thelowest temperature, as shown in Fig. 1 [19]. The observed
diffraction pattern of coexistence of two CDWs is clearly
distinguished from those in the current-induced discommensuration lattice, which induces symmetric subpeaks at both sidesof the main satellite [20,21].
The observed CDW characteristics in o-TaS3 differ fromthose in other quasi-one-dimensional conductors. Blue bronze
K0.30 MoO3 undergoes a Peierls transition at 180 K withk1 = 0.263b∗ [12]. By lowering the temperature to 100 K,

2469-9950/2018/97(11)/115432(4)115432-1©2018 American Physical Society

KATSUHIKO INAGAKI AND SATOSHI TANDAPHYSICAL REVIEW B 97, 115432 (2018)
III. MODEL AND RESULTS

Theoretically, the lock-in transition in quasi-twodimensional conductors has been discussed, initially by
McMillan [2] and followed by Nakanishi and Shiba [4]. Theirtreatment is based on free energy with higher-order umklapp
terms. McMillan’s free energy has the following form:
1F1 = F 0 d 2 s − |φ|2 − βY Re(φ 3 ) + |φ|4222(1)+ β|∇φ + iφ| + γ | − q1 × ∇φ| ,FIG. 1. Phase diagram of o-TaS3 deduced from this study. Solidcircles represent the intensity of a commensurate CDW, Ic , normalized
by the total intensity of both the commensurate and incommensurate
satellites, Iic + Ic . The commensurate CDW begins to develop at
around 180 K (solid line). The two q’s of the CDWs coexist until
the system undergoes the lock-in transition at a temperature between
30 K and 50 K (broken line). The insets are diffraction profiles ofsatellite peaks at 30, 130, and 180 K.

its CDW wave vector shifts to be nearly commensurate witha slight residual incommensurability (∼ 0.250b∗ ). Lock-in
transition to the commensurate state does not occur in thisconductor (incomplete lock-in). In NbSe3 [10], as well as
m-TaS3 [6], there are three conducting chains, two of which
contribute to form CDWs. Neither NbSe3 nor m-TaS3 exhibits
lock-in transition. The first CDW wave vector in m-TaS3 is
independent of temperature with q1 = 0.245b∗ , whereas q1of that in NbSe3 shifts from 0.245b∗ at 150 K to 0.241b∗
at low temperatures [22]. In these conductors, the second
CDW (q2 = 0.247b∗ for m-TaS3 , q2 = 0.260b∗ for NbSe3 )appears at lower temperatures. As summarized in Table I,
it seems difficult to treat such various behaviors of CDWs
in quasi-one-dimensional conductors with a simple theory,in particular, for such vector shift and complete/incomplete
lock-in phenomena.
TABLE I. Characteristics of quasi-one-dimensional CDW systems. The wave vectors of each material and its ground state areshown (commensurate, incommensurate, or nearly commensurate).
Material

CDW 1

o-TaS3
K0.3 MoO3
m-TaS3
NbSe3

∗

CDW 2
∗

0.255c → 0.252c
0.263b∗ → ∼ 0.250b∗
0.253b∗0.245b∗ → 0.241b∗∗0.250c−0.247b∗0.260b∗Ground StateCNCIC
IC

where φ is a phase of CDW defined as ψ = ψ0 eiK·r/3 φ(r),and ψ is a complex order parameter. Here the coefficients F 0 ,β, Y , and γ werethe same as those defined in the originalliterature [23]. Equation (1) was derived for commensurability
index, namely, the ratio of CDW and lattice periodicities,
M = 3. The commensurability energy originates from thethird-order umklapp term, proportional to the coefficient β.
Though McMillan’s discussion aimed to understand the behavior of quasi-two-dimensional conductors, e.g., 2H-TaSe2 ,it also includes quasi-one-dimensional cases. To apply their
theories to our case, M = 4, we should know what happens
in McMillan’s free energy. By substituting ψ = ψ0 eiK·r/4 φ(r)for the order parameter, a simple calculation gives a result
similar to Eq. (1); however, it lacks the βY Re(φ 3 ) term,
because the umklapp term becomes fourth-order in this case,namely, proportional to |φ|4 . In contrary to the M = 3 case, thiscalculation provides an unfamiliar result. The umklapp term
gives no energy gain if a phase modulation φ = e−iθ(x) alone isconsidered as in McMillan. His calculation for the case M = 3
provided the free energy as F 0 (− 21 + β(1 − Y )). A first-orderlock-in transition takes place at the point Y = 1. On the otherhand, from our calculation for the case M = 4, the free energyof the commensurate state is a constant value F 0 (− 21 + β),which is always larger than that of incommensurate stateF 0 (−1/2). This explains the absence of the lock-in transitionin the charge density wave of blue bronze, whose CDW vector
becomes nearly commensurate at low temperatures. On thecontrary, the origin of the CDW vector shift from 0.263b∗ to∼ 0.250b∗ remains unsolved. We will discuss this issue later.
Nakanishi and Shiba’s extension of McMillan’s theory
covers the systems with multiple CDW vectors [4]. They
treated the lock-in transition of a two-dimensional conductor
1T-TaS2 , whose nesting vectors ki (i = 1,2,3) satisfy a relation
3ki − kj = Gi , where Gi are reciprocal vectors, leading to thecommensurability energy through the fourth term of umklapp
processes. Also, after a simple calculation, this fourth-orderterm is found to give the energy gain only when coefficients of
the nesting vectors in such a relation are odd numbers (1 or 3)
for combining them to the reciprocal vector. This explains theabsence of the lock-in transition in NbSe3 and m-TaS3 , bothof which have the nesting vectors satisfying 2k1 + 2k2 = G.
As shown above, even-order processes in the free energy
develop the amplitude, while odd-order processes induce
phase-related phenomena. Figure 2 shows whether the fourthorder umklapp terms couple to the lattice periodicity or not.
The (2,2) case, namely, 2k1 + 2k2 = G, which is satisfied inNbSe3 and m-TaS3 , provides the same potential modulation as115432-2

LOCK-IN TRANSITION OF CHARGE DENSITY WAVES IN …

PHYSICAL REVIEW B 97, 115432 (2018)

FIG. 3. Umklapp process in M = 4 CDW, responsible to commensurability (left). Schematic diagram of Bergmann’s condition(right) [15].
FIG. 2. Visualization of the fourth-order umklapp terms: ψ 4 ,ψ ψ 2 , and ψψ 3 ; charge density wave of commensurability index
M = 4, and lattice potential (from top to bottom).
2that in blue bronze. Therefore, the absence of lock-in transitionin these conductors is found to be of the same origin. On theother hand, the (1,3) case provides sufficient contribution to thelock-in transition also in quasi-one-dimensional conductors.
This case was discussed to explain the lock-in transition of anorganic conductor, TTF-TCNQ [3].
Now we will apply these theoretical considerations to ourexperimental results. CDWs in o-TaS3 were not assumed as
those in multiple chains, such as NbSe3 and m-TaS3 . However,the coexistence of commensurate and incommensurate CDWs
in o-TaS3 , as shown in Fig. 1, suggests this possibility.
By lowering the temperature, CDWs split into two kinds:
commensurate and incommensurate ones. The commensurate
CDW vector appears at kc = 0.250c∗ from even-order terms
in the free energy, whereas the incommensurate CDW vector
remains at kic = 0.252c∗ . The fourth-order umklapp term,
which satisfies kc + 3kic  G, couples to the lattice periodicity and obtains commensurability energy. At a temperature
between 50 K and 30 K, a transition may occur, allowing
the system to be complete lock-in. This scenario perfectly
explains our synchrotron data [13]. The incommensurate phasein the coexistence regime may have discommensurations,as discussed in the previous report, because a discommensuration state is energetically preferable to incommensurate
CDW, according to McMillan [2]. In addition, the transition
temperature coincides with that of occurrence of glasslike
behavior [24]. This behavior can be understood as a resultof the lock-in transition, which freezes global motion of theCDWs.
IV. DISCUSSIONS

Our discussion does not rule out the possibility for the generation of an individual discommensuration, namely soliton,in commensurate CDWs. According to Bak and Emery [3], asinusoidal potential in CDW leads to the sine-Gordon equation,
whose solution includes a phase soliton with the charge e/M.
Moreover, such a sinusoidal modulation of potential can bederived only by commensurability [25]. This agrees with previous experimental results, including the discrepancy between
longitudinal and transverse conductivity at low temperatures[26], the existence of unexpected carriers [27], and the nonlocal
transportation [28].
According to the microscopic theory [25], the sinusoidal potential in commensurate CDWs is rooted in thecondition
k+MQ = k ,(2)where k is the energy of momentum k, and Q = 2kF is a CDWvector. Equation (2) means that the sum of each vector equals
the reciprocal vector, i.e., MQ = G, and the energy of anelectron-hole pair conserves after it is interacted M times by theCDW momentum of 2kF . This leads to the phase dependenceof the gap energy as 2||M (cos Mφ − 1) [29].
If a system is purely one-dimensional, Eq. (2) merely
provides M’th order of umklapp process, whereas in twodimensional systems, another interpretation becomes possible
as follows: it is similar to that of Anderson localization, inparticular, in the weak localization regime [15]. Anderson
localization results from self-interference of a wave function
by multiple elastic scattering in random media. Bergmann’scondition for the localization to occur has a form i gi = 0,
where gi denotes scattering vectors by impurities. It should
be noted that a moment in the lattice can stay in any arbitrary
Brillouin zone. Since all the scattering processes are elastic,
the energy of the wave function conserves. Therefore, byconsidering Bergmann’s condition in substitution of Q for gi ,one may obtain Eq. (2), as shown in Fig. 3.
This interpretation agrees with previous experimental results in o-TaS3 . At low temperatures, the system undergoes
complete lock-in, as shown in Fig. 1, where quantum interference phenomena were discovered in o-TaS3 [16,17]. In
particular, the localization phenomenon in the commensurate
state suggests that CDWs have a two-dimensional correlation
over the b-c plane, and the closed path of CDW trajectory plays
a crucial role [17].
Finaly, here we will mention a limitation to our discussion.
The lock-in energy has been found to relate with the oddorder terms in McMillan’s free energy. By applying thisto quasi-one-dimensional conductors with M = 4, most ofcharacteristics summarized in Tabel I are explained within
this framework, except for the vector shift observed in blue
bronze. One plausible explanation is the excitation of soliton
and antisoliton pairs [30]. Each excitation of the soliton andantisoliton pair has been observed as a discrete step [31]. Since
similar steps have also been observed in o-TaS3 [32], further115432-3

KATSUHIKO INAGAKI AND SATOSHI TANDAPHYSICAL REVIEW B 97, 115432 (2018)

investigation must be necessary to clarify the lock-in transitionof CDWs.

should be more focused, since it must contain far richer physics
than previously thought.

V. CONCLUSIONACKNOWLEDGMENTS

In summary, we provide a unified view for the lock-intransition both in quasi-one- and two-dimensional conductors,
based on the difference of roles between even- and odd-orderterms in the free energy. The study of commensurate CDWs

The authors thank K. Ichimura, K. Yamaya, T. Honma, M.
Hayashi, and K. Nakatsugawa for fruitful discussions, andM. Tsubota, T. Matsuura, S. Uji, N. Ikeda, and Y. Nogami
for experimental support.

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