PRL 105, 176401 (2010)

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PHYSICAL REVIEW LETTERS

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Chiral Charge-Density Waves
J. Ishioka,1 Y. H. Liu,2 K. Shimatake,1 T. Kurosawa,2 K. Ichimura,1,3 Y. Toda,1,3 M. Oda,2,3 and S. Tanda1,3,*
1Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan2
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
3
Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan(Received 10 January 2010; published 18 October 2010)
We discovered the chirality of charge-density waves (CDW) in 1T-TiSe2 by using STM and timedomain optical polarimetry. We found that the CDW intensity becomes Ia1 :Ia2 :Ia3 ¼ 1:0:7  0:1:0:5 0:1, where Iai (i ¼ 1; 2; 3) is the amplitude of the tunneling current contributed by the CDWs. There were
two states, in which the three intensity peaks of the CDW decrease clockwise and anticlockwise. Thechirality in CDW results in the threefold symmetry breaking. Macroscopically, twofold symmetry wasindeed observed in optical measurement. We propose the new generalized CDW chirality HCDW q1  ðq2  q3 Þ, where qi are the CDW q vectors, which is independent of the symmetry of components.
The nonzero HCDW —the triple-q vectors do not exist in an identical plane in the reciprocal space—shouldinduce a real-space chirality in CDW system.
DOI: 10.1103/PhysRevLett.105.176401

PACS numbers: 71.45.Lr, 68.37.Ef, 72.80.Ga, 73.22.Gk

Chirality is an important concept in particle physics andthe condensed matter field in relation to baryogenesis [1] orspontaneous symmetry breaking. In condensed matterphysics, many researchers have attempted to study thechiral properties of 3 He-A [2] and chiral p-wave superconductors [3,4]. Is it possible to break the chiral symmetryin electron systems that also have macroscopic quantum
order such as charge-density waves (CDWs)?
Chirality is often characterized by using the parameterof helicity h ¼   p=jpj (, spin; p, momentum) in aphoton, or h ¼ v  ! (v, velocity; !, vorticity) in fluids.
If the helicity is nonzero in a system, the system haschirality. Chirality describes the character of an object orstructure that cannot be superimposed on its mirror image.
So far, there is no evidence of the chirality in onedimensional (1D) CDWs and two-dimensional CDWs, asusual, since a CDW has the form  ¼ cosðkx þ Þ (,charge-density; k, wave number; x, position; , phase),
which is invariant by space inversion, or parity transformation [k ! ðkÞ and x ! ðxÞ].
In this Letter we report the chirality of a CDW in
1T-TiSe2 [5–12] measured directly by using STM andtime-domain optical polarimetry. STM is used to investigate the microscopic electron distribution and the phase of
the CDW order parameter in real space [5,6,13]. In themeasurement, the CDW intensity becomes Ia1 :Ia2 :Ia3 ¼
1:0:7  0:1:0:5  0:1, where Iai (i ¼ 1; 2; 3) is the amplitude of the tunneling current contributed by the CDWs. We
defined a rotation of the phase in terms of the observed
CDW intensity differences: clockwise and anticlockwise.
Two chiral phases were observed. Moreover, the two-color
pump-probe pulse method was performed to determine themacroscopic property by changing the linearly polarized
angle of the probe pulse. A twofold property in transientreflectivity was detected. To explain these results, we
propose a new CDWs configuration based on the 3D
0031-9007=10=105(17)=176401(4)CDW q vectors of 1T-TiSe2 in reciprocal space. In real
space, the relative phase shifts between three CDW qvectors induce the helical structure in CDW unit cells.
We provide the first evidence of chirality in a CDW.
Moreover, we propose generalized CDW chirality intriple-q systems as HCDW  q1  ðq2  q3 Þ.
1T-TiSe2 crystallizes in a 1T-CdI2 -type structure. The
crystal is constructed of Se-Ti-Se layers. All the Se-Ti-Selayers are weakly accumulated by the van der Waals attraction along the c axis. 1T-TiSe2 undergoes a CDW
transition into a 2a0  2a0  2c0 superlattice (a0 ¼ are lattice constants) at Tc ¼ 200 K c0 ¼ 6:00 A
3:54 A,[10].
We synthesized a single crystal of 1T-TiSe2 using theself-vapor transportation method [7]. By controlling thegrowth temperature, we obtained sample A at 700  C andsample B at 800  C. The resistivity behavior of each sample was consistent with that described in the previous
report [8,14].
In the STM measurements, the samples were cleaved
in situ just before the approach of the STM tip toward thesurface in an ultrahigh vacuum at 77 K. We were able toobserve atomically resolved STM images in the constantheight mode with a constant sample voltage Vs applied between the tip and the sample. The STM images of samples A and B were obtained at 84 and 6.3 K, respectively.
The two-color pump-probe pulse method [15–17] wasemployed with the cleaved TiSe2 surface using a microoptical setup as mentioned in Ref. [18]. The pump andprobe pulse polarization was varied to investigate thedominant direction of the deviation of reflectivity RðtÞ.
The sample was measured at 3 K.
Figures 1(a) and 1(b) show STM images and line profiles
along unit vectors, respectively. Figure 1 shows the low
symmetry structure of the electron distribution in TiSe2
with atomic resolution in sample A. In Fig. 1(a), the bright

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Ó 2010 The American Physical Society

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PHYSICAL REVIEW LETTERS

area is the point at which many electrons are tunneling. We
observed a triangular lattice of Se atoms with a lattice
constant a0 . In addition, we saw a 2a0 wavelength CDW
along the unit vectors. To compare the amplitude of theCDW intensity for each CDW q direction, we show the line
profiles of the tunneling currents along the unit vectors a1 ,a2 , and a3 [Fig. 1(b)]. In Fig. 1(b), there is obvious
difference in CDW amplitude among each direction of
the unit vector. The CDW amplitude Iai was specified foreach magnitude and labeled in order starting with thelargest amplitude. In sample A, Ia1 :Ia2 :Ia3 ¼ 1:0:7:0:5.
This STM image does not have the threefold symmetry
owing to a triangular lattice structure. In general, the line
profiles consist of the sum of two of the three density waves[Fig. 1(a)]. It is necessary to separate the contribution ofeach CDW.
Figures 1(c) and 1(d) show a Fourier transformation(FT) image of sample A and line profiles along the CDW
q vectors, respectively. In Fig. 1(c), the bright spots correspond to the intensity peak of each wave vector. The outerintensity peaks are the Bragg peaks of the selenium lattice
whose wavelength is a0 . The inner peaks correspond to a
CDW satellite whose wavelength is 2a0 . In Fig. 1(d), there
is no difference in the intensity of Bragg peaks whichcorrespond to each a vector, where a is a basis ofreciprocal lattice. In contrast, we can see an obvious difference between the three sets of CDW intensity peaks.
Then, we denote the wave vectors q1 , q2 , and q3 in order
starting with the highest intensity, too [Fig. 1(d)]. In
sample A, Iq1 :Iq2 :Iq3 ¼ 1:0:9:0:5. In terms of the rotationin which the indexed number increases [inset in Fig. 1(c)],FIG. 1 (color). (a) 3:5 nm  3:5 nm STM images and (b) lineprofiles along unit vector a1 , a2 , and a3 . Iai is the amplitude ofthe line profiles along ai . Sample A was cleaved in 108 Torr at
77 K and measured in situ at a sample bias voltage of Vs ¼
150 mV and an initial tunneling current of It ¼ 0:3 nA at T ¼
84 K. (c) FT image of sample A and (d) line profiles along q1 , q2 ,q3 wave vector. Iqi is the intensity of the line profiles along qi .
FT was performed over the entire field of view (25 nm  25 nm)of STM image.

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we name the phase the anticlockwise phase. In further
STM experiment, we also found a clockwise phase in thesame sample.
Figure 2(a) shows 12 nm  12 nm STM images insample B, and Figs. 2(b) and 2(c) show the FT of two areas
each enclosed with a square. In Fig. 2(a), the clockwise andanticlockwise phases were observed simultaneously.
Figure 2(b) is clockwise and Fig. 2(c) is anticlockwise
phase; that is, Fig. 2(b) is a mirror image of Fig. 2(c).
Therefore, Fig. 2(b) cannot be superimposed on Fig. 2(c)solely with rotational transformation. Thus, the results
show the parity symmetry breaking. Although Slough
et al. [5] and Coleman et al. [6] observed anisotropy inthe conductive plane in a previous STM measurement, themirror image relation was not demonstrated within theiranalysis by contour plotting. We discovered two phaseswith different chirality. The blue line in Fig. 2(a) shows theboundary between the clockwise and anticlockwise phases.
Because we found no steps of atomic layer and no changesof the lattice direction, this is not a crystallographic defectin the observed layer. Thus, the boundary between the twophases is a chiral domain wall.
Figures 3(a) and 3(b) show the polarization dependenceof the transient reflectivity variation of a 700  C grown
1T-TiSe2 measured at 3 K and 1T-TaS2 measured at 12 K,respectively. Figure 3 shows the clear twofold symmetry in
TiSe2 . The transient variation in optical reflectance RðtÞwas measured by changing the polarization angle Epol of
the incident probe laser pulse. In general, the peak intensityof transient signals corresponds to the number of electrons
excited over the CDW gap [19,20]. But in Fig. 3(a), thepeak intensity behaved anisotropically. Surprisingly, when
we plot the peak intensity as a function of polarization
angle Epol [Fig. 3(c)], we observed twofold symmetry
although 1T-TiSe2 has a trigonal lattice structure. In con-FIG. 2 (color). (a) 12 nm  12 nm STM image and (b),(c) FTof two areas each enclosed with a square in sample B. Sample B
was measured at Vs ¼ 350 mV and It ¼ 0:4 nA at T ¼ 6:3 K.
The blue line is the domain wall between the clockwise phaseand the anticlockwise phase.

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PHYSICAL REVIEW LETTERS

trast, this clear twofold property is not seen with 1T-TaS2 ,which is the most common 2D CDW material with thesame lattice structure [Figs. 3(b) and 3(d)]. In general, thetwofold symmetry does not emerge solely in most twodimensional CDW systems because three CDWs generatea triangular lattice that has threefold and twofold symmetrysimultaneously. Thus, at least within the light penetration
length, a spatial twofold structure was achieved in theCDW system. The macroscopic twofold property is consistent with the microscopic twofold symmetry observedwith STM as shown in Fig. 1(b).
To understand these results, let us consider the static
charge distribution in real space. We propose a simple
model of the CDW stacking. The CDW stacking is notconventional layer stacking with 2D CDW sheets [21–23]
but CDW component stacking that makes internal structure
in a CDW unit cell. Figure 4 shows the configuration of
three CDW components. Charge concentration is indicatedby the deeply colored part. Each color corresponds to theCDW q vector illustrated in the inset of Fig. 4. When weconsider the charge-density corresponding to the wavevector qi (i ¼ 1; 2; 3), qi ¼ A cosðqi  r þ i Þ, where A
is the amplitude and i is the initial phase. In our model,
we considered relative phase shift between each CDW. In
the TiSe2 system, the wave vector qi have c component
which corresponds to the 2c0 periodicity. We assumed
1 ¼ 0, 2 ¼ 2=3, and 3 ¼ 4=3. This assumption
leads to the CDW density peak shift along the c direction
at the interval of 2c0 =3. The shifted density peaks exist in
virtual layers which are indicated as the dotted lines in

FIG. 3 (color online). Polarization dependence of transientreflectivity variation in 1T-TiSe2 measured at 3 K (a) and
1T-TaS2 measured at 12 K (b). The height of each peak isplotted in the (c) and (d) as function of polarization angle
Epol . The pump pulse wavelength is 1160 nm and the power is
150 W. The probe pulse is 800 nm, 80 W.

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Figs. 4(a) and 4(b). Chirality is achieved by twisted stacking such as with cholesteric liquid crystals [24]. Thus,
CDW component stacking results in helical stacking andhelical axis parallel to the c axis. Note that by putting threedensity peaks in 2c0 , there are two ways of CDW stacking;that is, when we permutate the sequence of two coloreddensity waves (blue and green), the helix is inverted[Fig. 4(a)]. A charge helix along the c axis was achievedin the CDWs, and the helicity difference results from thestacking sequence.
This configuration can explain the low symmetry charge
distribution in the STM measurement and the twofold
symmetry in the optical measurement. In the STM measurement, the tunneling current can obtain information
about one or two atomic layers. In one atomic layer, theintensity of three CDWs differs due to the distance between
the observed surface and each CDW peak. A CDW whose
peak was closer to the surface was more clearly observed(Iq1 ) than a CDW that was further away (Iq3 ). This difference breaks the threefold symmetry. Let us consider theCDW layers depicted as red and blue layer in our model.
The CDW distribution forms twofold symmetry in the redlayer and the CDW also forms twofold symmetry in theblue layer. Consequently, the whole layers form twofold
symmetry of the system since the optical measurement
reflects the superposition of all of the layers. In terms ofthe intensity difference between Iq1 and Iq3 , the FT image
obtained with STM (Fig. 2) reveals the direction of thehelical stacking. Clockwise is the ‘‘left-handed’’ and anticlockwise is the ‘‘right-handed’’ state.
To consider the origin of the helical stacking of CDW,
we take account of the each q vector with a c component
in 1T-TiSe2 [9] unlike those in 1T-TaS2 . In typical 2D
CDW materials, such as 1T-TaS2 , at first the in-plane
triple-q vectors form a hexagonal superlattice structure ina S-Ta-S layer. Then, by the Coulomb interaction, theS-Ta-S layers stack with the next CDW layer so as notwo density peaks have the same position [Fig. 4(c)]. We
call this conventional layer stacking. The phase shift in theneighboring layer results from interlayer coupling. On theother hand, in 1T-TiSe2 , the 3D q vectors, that originallyhave a c-axis component, induce a charge modulation
directly along the q vectors. Accordingly, in 1T-TiSe2
there is no such conventional layer stacking. Moreover,relative phase shifting along the c axis between q1 , q2 ,and q3 can reduce the Coulomb interaction. As a result,the configuration achieves a helical structure (Fig. 4).
Moreover, since there are three q vectors, CDWs canform two types of helical stacking based on the stacking
sequence: clockwise and anticlockwise. Therefore, threedimensional triple q vectors make chirality in CDW
systems.
From the above discussion, we propose generalized
CDW chirality in triple q in reciprocal space:

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HCDW  q1  ðq2  q3 Þ;(1)

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FIG. 4 (color). Schematic representation of (a) left-handedchiral CDWs and (b) right-handed chiral CDWs in a TiSe2
CDW unit cell in real space and (c) typical 2D CDW. Charge
concentration is indicated by the deeply colored part. The colors
correspond to the colored CDW q vector in the inset. In a CDW
unit cell, the density peaks of three CDW are shifted at intervalsof 2c0 =3. If we look at one layer, the different intensities of threeCDWs form a low symmetry structure. There are two phases
caused by difference in stacking direction (red-blue-green or redgreen-blue). As with cholesteric liquid crystals, the stacking
results in the helicity of CDW systems. There is a mirror-image
relation between the two configurations in (a) and (b). As well as
the stacking of CDWs layers in real space, CDW helicity definedin reciprocal space, HCDW  q1  ðq2  q3 Þ, reverses by space
inversion [qi ! ðqi Þ]. HCDW is positive in a right-handed chiral
CDW, and HCDW is negative in a left-handed chiral CDW.

where q1 , q2 , and q3 are the CDW q vectors. If the HCDW iszero (three CDW q vectors in one atomic plane), a triple-q
CDW exists simultaneously in one layer in a CDW unit
cell. But if the HCDW is nonzero (inset in Fig. 4), there is adegree of freedom owing to charge helicity; that is,HCDW < 0 represents a left-handed chiral CDW and
HCDW > 0 represents a right-handed chiral CDW. The
space inversion of three q axes inverts the sign of HCDW
(right-handed to left-handed) (Fig. 4). Therefore, we predict that there will be CDW component stacking in a CDW
system where HCDW Þ 0 and therefore the CDWs themselves will have chirality.
Finally, we briefly discuss the phase transition in TiSe2 .
Originally, TiSe2 is a mystery because the Fermi surface
nesting scenario cannot explain all of the phase transition[11,12]. Based on our discovery, we can suggest a new
scenario about the unconventional phase transition considering the chiral phases. That is, the anomalously broad
peak with a width of 100 K in the resistivity in the phasetransition [8] may be a consequence of the frustration ofleft-handed and right-handed phases separated by chiraldomain walls. A domain wall, which is a boundary between the superlattices, will behave as a conductive path.
Therefore, chiral CDW will not have sharp resistivity
peaks. Moreover, it is interesting that the CDW system inwhich the wave function is generally based on the s-wave
symmetry has the chirality such as cholesteric liquid crystals despite no isotropic component. Our discovery sheds
new light on the old and rather controversial problem of
CDW instability and opens the possibility to explore di-week ending
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rectly the interplay between chirality and transport inelectron systems. Our findings could also pave the way
towards the investigation of charge helicity and externalfields.
We thank K. Yamaya and T. Toshima for providing
advice on sample preparation. We also thank K. Inagaki,T. Matsuura, and H. Nobukane for measuring the electrical property and for fruitful discussions. And we thankN. Hatakenaka for helpful comments. This work was supported by the 21COE program on ‘‘Topological Scienceand Technology’’ from Ministry of Education, Culture,
Science and Technology of Japan.

*Corresponding author.
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