PHYSICAL REVIEW B 84, 245125 (2011)

Charge-parity symmetry observed through Friedel oscillations in chiral charge-density waves
J. Ishioka,1 T. Fujii,2 K. Katono,1 K. Ichimura,1,3 T. Kurosawa,4 M. Oda,3,4 and S. Tanda1,3,*
1Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Department of Physics, Asahikawa Medical College, Asahikawa 078-8510, Japan
3
Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan
4
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan(Received 21 July 2011; revised manuscript received 21 November 2011; published 27 December 2011)2

We discovered a chiral behavior of Friedel oscillations (FOs), which have both triangular symmetry andinverse chirality from chiral charge-density waves (CDWs) in 1T-TiSe2 by using scanning tunneling microscopy(STM). Surprisingly, the existence of FOs with the opposite chirality to the underlying CDWs suggests that theexcess electron induced by impurity reduces the local helicity of CDWs without CDW breaking. This FOs canbe explained with a two-branch model with different chiralities, which permits excitation from one branch toanother, that is, Dirac-type dispersion. Moreover, we found two states that correspond to electron addition andelectron removal. In terms of the chirality of electrons, these two states are the same. This result provides anevidence for the charge-parity (CP) symmetry conservation in CDW systems.
DOI: 10.1103/PhysRevB.84.245125

PACS number(s): 71.45.Lr, 68.37.Ef, 72.80.Ga, 73.22.Gk

I. INTRODUCTION

In nature, there are parity violations with several different
scales. For example, in particle physics, parity symmetry is
broken in the β decay of 60 Co.1 DNA mainly has a right-handed
double-helix structure.2 In liquid crystal, the molecules arearranged with helical stacking in the cholesteric phase.3 Recently, it has been reported that a chiral p-wave superconductor
breaks parity symmetry.4 Thus, the macroscopic quantum
order in materials can also break mirror symmetry. Using amacroscopic quantum system, we have now reached the stageof being able to approach the issue of the asymmetric propertyof the universe.
We have discovered chiral charge-density waves (CDWs) in
1T-TiSe2 ,5 in which the macroscopic quantum state exhibits
helicity in a static structure. If a charged impurity is added
to an electron system, the Friedel oscillation for screening is
induced.6–8 Does the FO in a chiral CDW have helicity? If
so what is the relationship between the FO and underlying
CDWs? In systems composed of atoms or molecules, it isdifficult for the antihelix of the system to emerge spontaneously once the helicity of the entire system has been
fixed. Moreover, a change in the number of components will
not affect the overall helicity. However, this argument does
not necessarily hold for a macroscopic quantum system. A
CDW system is the most promising system for realizing thedirect and local observability of the interaction between amacroscopic quantum state and the electronic perturbation
caused by impurities.
In this study, we discovered a “triangular” FO attributed
to the chiral CDWs of TiSe2 . By using an in situ cleavage
method in an ultra-high vacuum STM chamber, we foundthat triangular FOs have chirality based on the intensityof the tunnel current. Unexpectedly, the observed Friedel
oscillations have inverse chirality from underlying CDWs.
To develop locally inverted helicity, electron systems needinternal right-handed and left-handed states. That is, ourresults strongly suggest that the CDWs in 1T-TiSe2 are Diractype CDWs composed of right- and left-handed components.
Essentially, the electron system conserves parity symmetry,1098-0121/2011/84(24)/245125(5)however CDW excitation breaks the parity symmetry. Our
discovery of a Dirac-type CDW provides another approach
for investigating Dirac fermions. The charge-parity (CP)symmetry conservation in CDW systems is revealed by FO.
1T-TiSe2 9–19 crystallizes in a 1T-CdI2 type structure. The
crystal is constructed of Se-Ti-Se layers. These Se-Ti-Se layers
consist of a plane occupied by Ti atoms that is sandwiched
between similar planes of Se atoms on either side. All the
Se-Ti-Se layers 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 = 3.54 Å,c0 = 6.00 Å are lattice constants) at Tc = 200 K.9,10
II. EXPERIMENTAL

We synthesized a single crystal of 1T-TiSe2 with Ti selfintercalation at 800 ◦ C, and the STM images were obtained asin Ref. 5. By in situ cleavage method in the specimen chamber
at 77 K and 10−8 Pa, we obtained an atomically defect-free Se
surface.
III. RESULTS AND DISCUSSION

Figure 1(a) shows a tunneling current image of the Se
surface of TiSe2 . We observed chiral CDWs as reported in
our previous paper.5 We found triangular FOs as denoted by
the white and black solid circles [Fig. 1(a)]. The brightness
corresponds to the intensity of the tunneling current. The brightpoint corresponds to Se atoms, which form triangular lattices
with a lattice constant a0 . We confirmed that a 2a0 × 2a0
charge modulation develops in the dark region enclosed witha rectangle [Fig. 1(a)]. We found clearly triangular charge
localizations with the same orientation. On the assumption that
this charge localization is formed by the three planar waves,each component of the planar waves oscillates spatially witha wavelength 2a0 and maximum decay length of 1 nm. This
wavelength is equal to the planar components of the CDW
q vectors in TiSe2 , and the decay length is comparable to theThomus-Fermi screening length. Therefore this is consistent
with a FO in terms of these characteristic lengths.20,21

245125-1©2011 American Physical Society

J. ISHIOKA et al.

PHYSICAL REVIEW B 84, 245125 (2011)

(a)

(b)

(d)

q1

q1

q2

3

q3

1

2

Counter clockwise FO

q2

(c)

q3

(e)

2

3

1
3

7 nm

2

Clockwise CDWs

1

Clockwise FO

FIG. 1. (Color online) (a) 20 nm × 20 nm STM image measured at Vs = 180 mV and It = 0.2 nA at T = 84 K, (b) Fourier transformationof dark areas in (a), (c) interpretation of underlying CDW distribution in real space, and (d),(e) schematic images of bright FO and dark FO,
respectively.

We analyzed the underlying CDWs using a Fourier transformation (FT). Figure 1(b) shows a FT image of the darkarea shown by the area outlined with a rectangle in Fig. 1(a).
The background chiral CDW shows the clockwise phase.5
The interpretation of the FT image in real space is shown in
Fig. 1(c). The charge concentration is indicated by the deeplycolored part. Each color corresponds to the CDW components.
The difference between the three CDW amplitudes causeschirality. Fourier analysis corresponds to real space analysis.
As indicated by the black and white circles in Fig. 1(a), we
found two types of triangular FOs, that were reproducible.
And we found an intensity difference for each atom as shown
schematically in Figs. 1(d) and 1(e). To investigate the chirality
in FOs, we analyzed the intensity of the charge on each atom.
Figures 2(a) and 2(b) show bright FO and dark FOs
indicated in yellow circles. We integrated the tunneling currentof the area on each atom outlined with the yellow circles in
Figs. 2(a) and 2(b). Then, we averaged the intensity ratiosof the three sites for the observed FOs. In Fig. 2(a), theintensity ratio of the triad peaks is 1 ± 0.22 : 0.53 ± 0.08 :0.40 ± 0.07. We defined the chirality of the bright FOs basedon their height from the background level. The chirality
is anticlockwise. We defined the chirality of the dark FOs
based on their depth from the background level. The averageintensity on clockwise dark FOs and anticlockwise dark FOs
are 1 ± 0.12 : 0.75 ± 0.10 : 0.49 ± 0.16 and 1 ± 0.11 :0.87 ± 0.08 : 0.63 ± 0.08, respectively. From those results,
a number-weighted mean ratio in dark FOs is 1 ± 0.12 :0.80 ± 0.09 : 0.55 ± 0.13. The chirality is clockwise. Insummary, bright FOs are anticlockwise and dark FOs areclockwise, as shown in Fig. 2. It must be emphasized that
if the bright FOs and dark FOs are superimposed, a constantcharge density cannot occur because of the opposite chiralityof those FOs.
We performed the same analysis on anticlockwise CDWs
[Figs. 2(c) and 2(d)]. The results are listed in Table I with theresults of Fig. 1. With anticlockwise CDWs, we confirmed thetendency for bright FOs to be clockwise and for dark FOs to

be anticlockwise [Figs. 2(c) and 2(d)]. Therefore, there is arelationship whereby the chirality of bright FOs is opposite tothat of the underlying CDWs and that of dark FOs is the sameas that of the CDWs.
To understand the origin of the triangular FOs, we comparedthem with results for FOs in CDW material, TaS2 and TaSe2 .8
FOs were observed with hexagonal or spherical symmetry
attributed to the two-dimensional CDWs. On the other hand,(a)underlying CDW

(c)

1 nm
(b)

underlying CDW

1 nm

underlying CDW

1 nm
(d)

underlying CDW

1 nm

FIG. 2. (Color online) Friedel oscillations in clockwise CDW[(a),(b)] and anticlockwise CDW [(c),(d)]. STM images of (c)Vs =180 mV and It = 0.35 nA at T = 84 K and (d) Vs = 800 mV and
It = 0.40 nA at T = 6.3 K. Three-dimensional plots are inset in each
image, and each peak and gap is indicated by yellow arrows. The
red and blue arrows indicate clockwise and anticlockwise CDWs andFOs, respectively.

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PHYSICAL REVIEW B 84, 245125 (2011)

TABLE I. Statistical analysis for chirality of FOs in clockwise and anticlockwise CDWs. We calculated the “averaged” intensity after
sorting them in descending order of the value. With a dark FO, we averaged the intensity with weight ratio based on the number of clockwise
and anticlockwise FOs.

Underlying CDWs
Type of FO
Sample Number(clockwise:anti-clockwise)
Average intensity of FOs

7(0 : 7)
1 ± 0.22 :0.53 ± 0.08 :0.40 ± 0.07

11
(7 : 4)
1 ± 0.12 :0.80 ± 0.09 :0.55 ± 0.13

1
(1 : 0)1:0.73 :
0.38

3(0 : 3)
1 ±0.34 :0.67 ±0.30 :0.45 ± 0.27

Chirality of FOs

our results show a triangular symmetry structure. What is thedifference between TaS2 and TiSe2 ? We focused on the phasepinning of CDWs on Friedel oscillation. As regards screening
the impurities in CDW systems, the static energy is smallerwhen the fixing CDW phase and the phase of the FOs are
fixed. When phase pinning occurs with the three components ina two-dimensional CDW, hexagonal charge localization froman impurity center will be induced. This picture corresponds to
the FOs observed in TaS2 . On the other hand, in TiSe2 , the c∗
vectors (∗ indicates a reciprocal vector) means that FOs include
c-axis components.12 Basically, a chiral CDW is a state withthree CDW components with different wave function phases.
When the three CDW components have different phases, thewavefront is not fixed at the point. Therefore the FOs do notform a hexagon, and it would be more proper to think triangularcharge localizations are contributed from collective electronic
states of 1T-TiSe2 .
Recently, an explanation for triangular charge localizations
has been attempted from the viewpoint of crystallographic
structure. STM measurements on 1T-TiSe2 at room temperature was performed by A. S. Razinkin et al., and theauthors also found triangular charge localizations similar to
our results.22 A. S. Razinkin et al. had discussed that atriangular charge localization is attributed to some defects
which change the local crystallographic structure from 1T(octahedral coordination for Ti) to 2H (prismatic coordination
for Ti). However, this model does not seem to cover the casesof either 1T-TaS2 or 1T-TaSe2 ,8 systems which have the same
coordination for metal atoms. Moreover, we found that inmagnetic field at low temperature, those FOs are enhanced.
That effect cannot be explained within crystallographic structure change, and the magnetic field effect will be reported
elsewhere. It is more appropriate to think that triangularcharge localizations are mainly contributed from collective
electronic states of 1T-TiSe2 rather than to think that result
from some crystallographic bonding reconstruction related to
Frenkel defects.23
For further understanding, we discuss why the chirality of
the FOs is the inverse of that of the underlying CDWs. Chiralinversion needs a mirror plane, but once chiral charge density
waves are excited, all of the mirror planes disappear. One

possible reason is that the local mirror inversion is induced by
Ti atoms. We focused on the crystallographic structure with
1T-TiSe2 with the self-intercalation of Ti in the van der Waals
gap. Figure 3 shows the arrangement of each atom in the
Se-Ti-Se lattice. 1T-TiSe2 is composed of a triangular latticeof Se and Ti. The unit sheets of Se-Ti-Se repeat along the c axis
in an ordinary sequence. For excess atoms, a site immediately
above the Ti atoms is fitted to intercalate in the van der Waals
gap because of the neighboring Se lattices. An intercalated Ti
atom induces a set of mirror planes for space inversion along
the z axis (Fig. 3). If a chiral field is attached in the entire
space, the added atom plays a role in the chiral inversion. On
the other hand, the removal of an atom from the original layer
does not make a mirror plane. The inversion of chirality at an(a) Top view
Se

Se
Se

Ti

Ti
Se
Se

Se
Se

Ti
Se

Ti

Se

Se
Se

(b) Side view

Se

Se

Se

Ti

Ti

Ti

Se

Se

Se

unit sheet
Se

Se

Se

Se

Se
Se

Se

Se

Se

Se

Mirror Plane A

Intercalated Atom

Se

Se

Se
Ti
Se

Se
Se

Se

Se

Se

van der Waals gap
Se

Mirror Plane B

FIG. 3. (Color online) (a) Top view and (b) side view ofcrystallographic structure of 1T-TiSe2 with self-intercalated Ti atom.

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J. ISHIOKA et al.

PHYSICAL REVIEW B 84, 245125 (2011)z = 2c0(a)z = 2c0

(b)

(a)

(b)

q3

q2

left-handed

q1

right-handedE(-h < 0)z = 4c0/3(h > 0)z = 4c0/3

HCDW < 0
HFO > 0q´i = -qiφ´i = φi

HCDW < 0
HFO < 0q´i = qiφ´i = φileft-handed FOs (electron)p(c)z = 2c0/3

z = 2c0/3

right-handed Chiral CDWs

counter clockwise

clockwiseyyz = 2c0/3

z = 4c0/3
z = 2c0/3 z = 0

z = 4c0/3

right-handed FOs (hole)z=0
z = 2c0z = 2c0

(d)

(e)

x

x

Removal of Ti or Se

Adding Ti or Se

FIG. 4. (Color online) Schematic representation of chirality atFriedel oscillation around (a) vacancy of Se and (b) intercalation of
Ti atom.

FO can be understood by the inversion of each wave vector at
an impurity.
To illustrate qualitatively that a local mirror plane induces
inversed chirality around an FO, we performed a numericalcalculation of the electronic distribution by the simplest
model considering planer waves of either CDWs or screening
electron. Figure 4 shows a numerical calculation of thesummation of three CDW components. This image showselectronic distribution as a function of x,y,
ρCDW (r) =cos(qi · r + φi ),(1)i=1,2,3∗

where q1 = 1/2a + 1/2c∗ , q2 = 1/2b∗ + 1/2c∗ , and q3 =−1/2(a∗ + b∗ ) + 1/2c∗ , respectively. Each CDW component
has a different initial phase, that is, φ1 = 0, φ2 = 2π/3, and
φ3 = 4π/3. As HCDW ≡ q1 · (q2 × q3 ) < 0, this distribution isdefined as left-handed chiral CDWs [Fig. 4(a)]. In addition, we
suppose that there is a charged impurity at (x,y,z) = (0,0,0).
And then we assumed that the charge distribution
for screening
is simply induced by ρscreen (r) = A/|r| i=1,2,3 cos(qi · r +φi ), where A/|r| expresses a decay factor.20 In Fig. 4,
the bright red areas are charge density peaks formed by
the intersection of wave planes in each xy plane. To focus
on the position change of the charge density peak near the
impurity, we show the density peak as a black solid circle andthe impurity as a red cross. As the z position goes deeper,
the peak position moves clockwise around the impurity in
Fig. 4(a). When qi = qi , φi = φi , the peak position moves inthe same way as the CDWs. In this case, the local chirality
HFO ≡ q1 · (q2 × q3 ) < 0 so the FO is locally left-handed.
This state describes the FOs without an inversion center causedby an atomic vacancy. Holes screen a negative charge, which isinduced by a lack of Ti atoms as shown in Figs. 2(b) and 2(d).
This is seen as a dark triangle. In contrast, when qi = −qi ,φi =
φi , The peak position moves anticlockwise [Fig. 4(b)].

left-handed

right-handed(-h < 0)(h > 0)E

right-handed FOs (electron)p(f)left-handed Chiral CDWs

left-handed FOs (hole)

FIG. 5. (Color online) Schematic illustrations of a model of Diracsea in CDWs. Each picture corresponds to (a) right-handed chiral
CDWs, (b) left-handed bright FOs, (c) right-handed dark FOs, (d) lefthanded chiral CDWs, (e) right-handed bright FOs, and (f) left-handed
dark FOs. The filled red and blue circles are right-handed electron(h > 0) and left-handed electron (−h < 0), respectively. The opened
red and blue circles are right-handed hole (−h < 0) and left-handed
hole (h > 0), respectively.

HFO ≡ q1 · (q2 × q3 ) > 0. The FO is locally right-handed.
This state describes the FOs with a mirror inversion causedby an intercalated atom. The electrons screen the positive
charge induced by Ti intercalation into the van der Waals gap.
This is seen as a bright triangle. The experimental results are
consistent with the emergence of inverted chirality around FO
from the underlying CDWs.
The relation between the FO and CDW can be understoodfrom the particle picture (Fig. 5). Figure 5 shows the energy
dispersion in momentum space. In Fig. 5, a branch denoted bya red line corresponds to a right-handed branch with positive
helicity (h > 0), and that denoted by a blue line corresponds
to a left-handed branch with negative helicity (−h < 0).
Figure 5(a) shows the initial state of the right-handed chiral
CDW. In this state, the difference of the number of the electronin the right-handed branch and the left-handed branch makes
the right-handed chirality of CDWs. Then, adding an electron
to the left-handed branch corresponds to Fig. 5(b). The removalof an electron from the right-handed branch corresponds toFig. 5(c). Because removing an electron can be interpreted

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PHYSICAL REVIEW B 84, 245125 (2011)

to adding a hole to the system, these pictures represent thebright FOs and dark FOs, respectively. It reveals that adding
and removing electrons decreases the helicity of the electron
system in the same way. In terms of helicity, a bright FO anda dark FO are the same. With left-handed chiral CDWs, thechirality relation is the inverse of that in right-handed chiral
CDWs [Figs. 5(d)–5(f)]. If there is only a single branch witha single helicity, the number of electrons simply correspondto number of helicities in the system. Thus, a single branchcannot explain why an increase in electrons causes a reductionin helicity. Therefore it is possible to understand our results bysupposing that the CDW system has a Dirac-type dispersion
curve with a set of right-handed and left-handed branches.
This picture corresponds to the experimental results shown in
Table I (insets in Fig. 5). The mechanism of the semiclassicalpicture of the quantum anomaly seems to work as if it were in
the Dirac electron system.
For understanding chiral CDWs, another theoretical approach is proposed by introducing a concept of orbital

ordering.24 The chiral-inverse FO may be understood by meansof this concept, although it is not clear whether the addition ofexcess orbit of electron reduces a local helicity.
IV. CONCLUSION

In conclusion, we have discovered a chiral FO in chiral
CDWs. We showed the relationship between chiral CDWs
and chiral Friedel oscillations. From our analysis of chirality
in an FO, we discovered that the CDW system consists ofDirac-type CDWs. Based on the fact that it can convert anelectronic structure into a pattern in real space, an FO can beconsidered a “window” through which to see the ground stateof an electronic system from real space to reciprocal space.
ACKNOWLEDGMENTS

We thank Y. H. Liu, K. Yamaya, T. Toshima, K. Inagaki,
T. Matsuura, and H. Nobukane.

*

14

1

15

tanda@eng.hokudai.ac.jp
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We have performed the same experiments on 1T-TiSe2 at 300 K
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