Journal of the Physical Society of Japan
Vol. 75, No. 2, February, 2006, 024706#2006 The Physical Society of Japan

Supercluster of Electrons in Ultrathin TaSe2 Nanocrystals
Takeshi TOSHIMA, Katsuhiko I NAGAKI, Noriyuki H ATAKENAKA1 and Satoshi T ANDA
1

Department of Applied Physics, Hokkaido University, Sapporo 060-8628
Faculty of Intergrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8521(Received August 24, 2005; accepted November 28, 2005; published February 10, 2006)

We discover a supercluster of electrons in the ultrathin TaSe2 crystals synthesized by our developed
de-chalcogenide
61 electrons denoted
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ method. The supercluster
pﬃﬃﬃ containing
pﬃﬃﬃ
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃhas an unexpected structureby 61  61, in addition to a 49  49 supercluster based on 7  7 clusters expected in triangle
lattices. This new hierarchal structure and successive clustering cannot be explained by a conventional
electron clustering mechanism assisted by background lattice distortions, i.e., charge-density-wave
formation mechanism based on Fermi surface nesting. We propose an alternative possible formation
mechanism based on saddle-point nesting, taking into account a trigonal symmetry as well as lattice
distortion energy.
KEYWORDS: CDW, crystal growth, two-dimensional system, TaSe2
DOI: 10.1143/JPSJ.75.024706

1.

Introduction

Successive clustering leads to supercluster with hierarchy,1) i.e., a cluster of clusters, found in various systems
ranging from supermolecule dendrimers,2) icosahedral packing of B12 3,4) to supergalaxy.5) Clustering as a building blockof supercluster needs an attractive force like gravitation insupergalaxy, while a repulsive force does not promoteclustering. In fact, electrons governed by Coulomb repulsion
only form a homogeneous trigonal lattice known as Wigner
crystal.6,7) In crystals, an attractive force between electrons
is introduced by the electron–phonon interaction. At lower(one or two) dimensions, electric instability occurs due to anelectron–phonon interaction related to Fermi surface nesting
characterized by the Fermi wave vector kF. This leads to a
new type of condensed phase called a charge density wave
(CDW)8–10) where the conduction electron density is
modulated like a wave, and the lattices are simultaneously
modulated from their original positions, resulting in a kindof electron’s cluster. 1T–TaSe2 is a typical two-dimensional
CDW material with a width and length of 0.1–10 mm and a
thickness of 10 –100 mm, and which is usually synthesized
11)
by the chemical
pﬃﬃﬃﬃﬃvapour
pﬃﬃﬃﬃﬃtransport method. The resulting
structure is a 13  13 cluster containing 13 electrons12)
as shown in Fig. 1. However, it appears to be diﬃcult for a
thick system to form a cluster of clusters, i.e., an electrons
supergalaxy in solids because of crystal rigidity. Thinner
systems with ﬂexibility are required for the creation of asupercluster.
2.

Experimental

We develop a new method to enhance two dimensionalityof the system by making a thinner ﬁlm using the ‘‘dechalcogenide method’’. The starting material is TaSe3 ﬁbre,
which is used as a template. The TaSe3 crystals aresynthesized by the conventional chemical vapour transport
method. The size of the TaSe3 crystals is typically 10 nm–
1 mm (thickness and width) and 100 nm–1 cm (length). We
convert the TaSe3 crystals with nanoscale to TaSe2 ultrathin
ﬁlms. The TaSe3 crystals are set on a quartz plate in asample chamber (Thermo Riko IVF298) evacuated to about
1  105 Torr. The sample chamber at approximately 1 103 Torr is ﬁlled with hydrogen gas in order to remove theoxygen gas. The TaSe3 crystals are heated rapidly fromroom temperature to 600 K and then reacted with hydrogengas for about 30 min. The chemical reaction is expressed as
TaSe3 þ H2 (gas) ! TaSe2 þ H2 Se(gas). One selenium
atom is removed from the TaSe3 template per unit cell
(TaSe3 ! TaSe2 þ Se), resulting in the growth of a TaSe2
thin ﬁlm on the surface of TaSe3 ﬁbre as shown in Fig. 2(f).
The surface TaSe3 crystal is converted into a TaSe2
monolayer near the surface during the de-chalcogenide
processes [from Figs. 2(a) to 2(b)]. These obtained crystals
are suspended in isopropyl alcohol by sonic dispersion of
30 min to strip ultrathin TaSe2 ﬁlm from TaSe3 surface. The
size of the TaSe2 crystals is typically 10 nm–10 mm (widthand length) and 10 nm (thickness). Submicron TaSe2 crystals
are deposited on a substrate. We analyze these crystals by
their transmission electron diﬀraction (TED) pattern.
3.

Experimental Result and Discussion

Figure 3(a) shows the TED pattern of the TaSe2 thin ﬁlm.
From the Bragg refraction and satellite spots, the crystal is ahexagonal structure with lattice constants a ¼ b ¼ 3:49 A
(0:03 A) modulated by a CDW characterized by the CDW
wave vectors, Q1 ¼ ð0:286  0:002Þa þ ð0:143  0:002Þb ,Q2 ¼ ð0:142  0:002Þa þ ð0:425  0:002Þb , and jQj ¼
 1 ). Figure 3(b), which is a Fourier trans0:378(0:005 A
formation of Fig. 3(a), shows real-space images of thecrystal. Figure
pﬃﬃﬃ3(b) clearly shows diﬀerent translational
vectors with 7 times the lattice constant in contrast
theﬃﬃﬃ
pﬃﬃﬃ to p
trigonal vectors. This crystal is then identiﬁed as a 7  7
CDW structure involving seven atoms, a centre atom (boldblack spot) surrounded by six atoms (faint spots), in the unit
cell as shown in Fig. 4(b). The same structure has beenreported in an intercalated system.13)
Figure 3(c) shows TED patterns of the crystal shown in
Fig. 3(a) obtained from an adjacent location. This reveals
the existence of an unprecedented CDW structure with (0:005 A).

the same lattice constants a ¼ b ¼ 3:42 A
Figure 3(d), which is a Fourier transformation of Fig. 3(c),characterizes the CDW structure by using conventional
CDW wave vectors, Q1 ¼ ð0:064  0:002Þa þ ð0:082 
0:002Þb , Q2 ¼ ð0:146  0:002Þa þ ð0:065  0:002Þb ,024706-1

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Fig. 1. A model of clustering in a two-dimensional trigonal system. The blue spheres are atoms or electrons. In a bulk system, an attractive force compels
the system to form a cluster only once due to Fermi instability whenforce caused by Fermi surface nesting is stronger than the force caused
pﬃﬃﬃﬃﬃ thepattractive
ﬃﬃﬃﬃﬃ
by the Van Hove singularity. For example, the system shows a 13  13 CDW phase. In thinner systems, the attractive force caused by the Van Hove
singularity could be stronger than the force caused by Fermi surface nesting. The ﬁrst generation shows no atom modulation. In the second generation, acentre atom attracts six neighbouring atoms. The clusters form a trigonal lattice again. In an ideal third generation, the centre cluster attracts sixneighbouring clusters. With the real third generation, however, a centre cluster attracts six neighbouring clusters where 12 atoms lie on the vertex of ahexagonal supercluster and act like a cushion.

Fig. 2. A schematic diagram of de-chalcogenide processes. The TaSe2 crystal is synthesized from TaSe3 crystal in the sequence (a)–(c)–(d)–(e)–(b). In (a),the TaSe3 crystal starting material is shown by the b-axis. The red, green, and pink spheres represent selenium atoms and the blue and yellow spheres aretantalum atoms. (c), (d), and (e) are enlarged views of the square yellow area in (a). Selenium atoms belonging to the TaSe3 surface (pink spheres) areremoved [(c) to (d)], and then new dangling bonds form in the removed area. The dangling bonds immediately vanish owing to their highly reactive
properties by combining with the neighbouring TaSe3 [(d) to (e)]. Finally the surface TaSe3 crystal becomes TaSe2 crystal [(e) to (b)]. (f) is atransmission electron microscope (TEM) image of TaSe2 crystals growing on the TaSe3 crystal. Inset is an enlarged image of the blue-squared area. The
needle-like structure is TaSe3 , which is the mother material, and the hexagonal crystal attached to the needle crystal is TaSe2 thin crystal.

 1
and
pﬃﬃﬃﬃﬃ jQj ¼ 0:12 (0:01 A ) or, R ¼ 5a þ 4b and jRj is
61 times the lattice
Thus the new CDW crystal ispﬃﬃﬃﬃﬃ constant.
pﬃﬃﬃﬃﬃ
identiﬁed
as
a
61

61
CDW
structure. p
Inﬃﬃﬃ addition,pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ
pﬃﬃﬃ thenew 61  61 CDW is constructed from 7  7 CDW
clustersp
ﬃﬃﬃﬃﬃ pasﬃﬃﬃﬃﬃshown in the inset of Fig. 3(d). Therefore this61  61 CDW is a supercluster.
Let us consider the supercluster formationpﬃﬃﬃﬃﬃ mechanism.
pﬃﬃﬃﬃﬃ
1T–TaSe2 crystal ordinarily forms a 13  13 CDW
structure (13 electron cluster) aspaﬃﬃﬃﬃﬃresultof Fermi instapﬃﬃﬃﬃﬃ
bility.12) In contrast, the observed 61  61 superstructure

is not explained by conventional CDW nesting theory
because there is no possibility of a phase transition. The
attractive force required for clustering acts only once when
the system has undergone a CDW transition. Therefore, wehave to ﬁnd an alternative superclustering mechanism.
We employ the CDW theory proposed by Rice andScott14) to account for the hierarchal structure we discovered. According to their theory, electrons may form a CDW
structure without reference to Fermi surface nesting 2kF if
an electron density of states has saddle points in its band

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Fig. 3. (a) and (c) are TED images of a TaSe2 thin ﬁlm. (b) and (d) are fast Fourier transform (FFT) images of (a) and (c), respectively. The bold black spot
in (a), are caused
thepﬃﬃﬃ by p
ﬃﬃﬃ Bragg reﬂection of TaSe2 , a and b are reciprocal lattice vectors. The faint spots around the bold black spot are satellites
resulting from 7  7 CDW modulation. Q1 and Q2 are CDW wave vectors. The inset FFT image (b) is an enlarged image of the area outlined with ared square. The spots show the tantalum atom positions. The p
distance
between neighbouring
spots
ﬃﬃﬃ
pﬃﬃﬃ pﬃﬃﬃ corresponds to the lattice constant indicated in theinset
pﬃﬃﬃﬃﬃ aspaﬃﬃﬃﬃﬃand b. The distance between bold black spots is 7  jaj, reﬂecting the 7  7 CDW structure. (c) is TED image of TaSe2 with a
61  61 CDW state. The inset is an enlarged image of the area outlined with a blue square. We observed CDW wave vectors Q1 and Q2 as well as
the reciprocal lattice vectors a and b . The inset in FFT image (d) is an enlarged image of the centre.
pﬃﬃﬃﬃﬃ A new translational vector described by the vector
R ¼ 5a þ 4b is observed. Moreover, an internal structure is also discovered in the unit of the 61  jaj structure.

Fig. 4. A model of the hierarchal structure in the two-dimensional trigonal
TaSe2 system. The coloured spheres are tantalum atoms. The black lines
show the basic trigonal lattice. Vectors a and b are translational vectors of
TaSe2 . (a) is the ﬁrst generation without the CDW state. The basic unit
has one atom and occupies the green hexagonal area. The centre atoms
areﬃﬃﬃ represented
by the vector R1 ¼ a. (b) is the second generation with appﬃﬃﬃ
7  7 CDW state. The cluster has seven atoms (one red and six blue
atoms) and occupies the green hexagonal area. The centre atoms are
expressed
pﬃﬃﬃﬃﬃ bypthe
ﬃﬃﬃﬃﬃvector R2 ¼ 2a þ b. (c) is the ideal third generation withthe49  49 CDW state. The supercluster has seven secondgeneration clusters and occupies the green non-smooth area. The centre
atoms are combined for the vector r ¼ 5a þ 3b. If we draw a smoothhexagonal boundary such as the pink line, the areas outlined with whitecirclesp
ﬃﬃﬃﬃﬃ act
pﬃﬃﬃﬃﬃas load positions. (d) is an observed third generationpﬃﬃﬃﬃﬃwithpthe
ﬃﬃﬃﬃﬃ
61  61 CDW state. The supercluster includes a49  49
structure and twelve atoms (gold spheres), and occupies thepﬃﬃﬃﬃﬃsmoothhexagonal green area. The distance between red spheres is 61  jaj.
The centre atoms are combined for the vector R3 ¼ 5a þ 4b.

structure. At the saddle points, a large number of electrons
resulting from the Van Hove singularity15) contribute to thescattering processes, resulting in the same electric instability. We consider that the ﬁrst generation hierarchal
structure is a non-CDW structure, where electrons form a

trigonal structure on a two-dimensional system, as shown in
Fig. 4(a). Studies of Wigner crystals have shown that twodimensional trigonal electron systems have saddle points.16)
Note that 2H–TaSe2 crystals also have saddle points.17) Suchthe saddle points can also cause CDW formation.14,17) We

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Table I. Clustering number and coordinate. n mean number of atoms belonging to each generation of cluster. An and Bn mean coordinate of an atom
position of neighbor cluster. Supercluster vector is a vector between centre atom of a cluster and centre atom of neighbor cluster. (Show in Fig. 4)
Generations

nðAn ; Bn ÞFirst generation

1(1, 0)

R1 ¼ a [Fig. 4(a)]Second generation
Third generation

7
61(2, 1)(5, 4)

R2 ¼ 2a þ b [Fig. 4(b)]R3 ¼ 5a þ 4b [Fig. 4(c)]Fourth generation

547

Nth generation(14, 13)
 n1
3
þ 1 3n1  1;
2
2

R4 ¼ 14a þ 13b

32n1 þ 1
4

pﬃﬃﬃ
assumethat saddle points promote the formation of a 7 pﬃﬃﬃ
7 CDW structure, in which an atom (electron) attracts six
atoms (electrons) aroundit,ﬃﬃﬃ as shown in Fig. 4(b). This ispﬃﬃﬃ preminiscent of the 7  7 CDW structure in a quantum
Hall liquid system with a weak external magnetic ﬁeld.18)
However, nesting vector between the saddle points onpﬃﬃﬃ–J
16,17)
line
does not coincide with the wave vectors of 7 p
ﬃﬃﬃ
7 CDW we discovered. The position of saddle points
probably does not lie on  –J line due to diﬀerence of crystal
structure between 1T–TaSe2 and 2H–TaSe2 , the size of
crystals and possibility of expansion of distance between
layers.13)
There is an advantage in the assumption of the saddle
points.
pﬃﬃﬃOnepfree
ﬃﬃﬃ electron remains in the resulting cluster ofthe 7  7 CDW
as described in a previous
pﬃﬃﬃﬃﬃ structure,pﬃﬃﬃﬃﬃ
19)
report on the 13  13 CDW structure.
pﬃﬃﬃ The excess
electrons again form a trigonal lattice with 7 times the ﬁrst
generation. If such excess electrons form new saddle points
as with the ﬁrst generation, the resulting cluster will attract
six neighbouring clusters. The supercluster again has sevenclusters and one free electron regardless of the crystal
structure, as shown in Fig. 4(c). A cluster of superclusters
would be produced by the same rule. This model canaccount for the observed hierarchal structure.
pﬃﬃﬃﬃﬃThepnext
ﬃﬃﬃﬃﬃ stablecluster is therefore expected to be a 49  49 supercluster, where a cluster (electron) attracts six clusters(electrons)
shows
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃaround it. However, ourpresult
ﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ that the61  61 p
supercluster
includesa4949 superclusﬃﬃﬃ pﬃﬃﬃ
ter based on 7  7 clusters.
In
otherwords,
twelve extra
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ
Ta atoms are attached to the 49  49 supercluster in thethird generation as shown in Fig. 4(d). This is due to theenergy relaxationpﬃﬃﬃﬃﬃ ofpthe
ﬃﬃﬃﬃﬃ lattice displacement at the boundaries of the 49  49 supercluster structure asshown
inpﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
Fig. 4(c). Note that the boundary of the
49  49
supercluster is rough where the strain energies are accumulated (white circled area). Twelve extra atoms are necessarytosmooththe boundary. The actual third generation is then apﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
61  61 CDW structure, as shown in Fig. 4(d).
Our supercluster on a two-dimensional trigonal system issummarized by introducing a clustering number n andcoordinate ðAn ; Bn Þ where n stands for generation in ahierarchy (Table I). The clustering number n is the numberof atoms or electrons that belong to each unit cell expressedasn ¼ 7  n1 þ ð6=3Þ  ðn1  1Þ ¼ 9  n1  2;

ð1Þ

where 1 ¼ 1. The formula consists of a ‘‘hierarchal’’ term
and a ‘‘relaxation’’ term. The prior unit attracts the sixneighbouring units and forms a new unit. With trigonal

Supercluster vector

Rn ¼ An a þ Bn b

symmetry, there are seven atoms in the hierarchal term. If
there is only the ﬁrst term, n ¼ 7n1 is written with theusual simple fractal formula.20) The second term is due to thebackground, and the lattice deformation energy acts as thecrack. The factor 6/3 means that each cluster shares six load
points that lie on their vertexes with three neighbouring
clusters in Fig. 4(d). Formula (1) is reduced to a simple
formulan ¼

32n1 þ 1:
4

ð2Þ

The clustering coordinate speciﬁes the centre position ofeach cluster using a translation vector, Rn ¼ An  a þ Bn b, of the trigonal lattice system a and b as shown in
Fig. 4(a). The clustering coordinate is then written as
 n1

3
þ 1 3n1  1
ðAn ; Bn Þ ¼
;
:
ð3Þ
2
2
According to p
this
ﬃﬃﬃﬃﬃﬃﬃﬃ formula,pﬃﬃﬃﬃﬃﬃﬃﬃ the fourth generation is
expected to be a 547  547 structure with R4 ¼ 14a þ
13b. The hierarchal structures led by the formula are
diﬀerent from a simple fractal structure such as a Kochcurve or a Sierpinski gasket, but resemble a natural network
such as the distribution of sunﬂower seeds, the texture of asnail shell or a honeycomb. The shape of the boundary
between the clusters distinguishes these hierarchal structures. The concept of relaxation, namely smoothing, leads us
to consider another hierarchal structure.
4.

Conclusion

We have synthesized the nanoscale TaSe2 crystals by ourdeveloped de-chalcogenide method using the template of
TaSe3 nanoﬁbre. We also observed the crystals by transmission electron microscope image and diﬀraction
pﬃﬃﬃ analysis.
pﬃﬃﬃ
Theelectrons
on
such
the
thin
crystalsform
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 7  7 and61  61 CDW structure. The 61  61 structure has
another periodicity inner; this result shows hierarchal supercluster of electrons clusters is realized. Conventional CDW
mechanism of Fermi surface nesting dose not promote suchthe hierarchal structure, we apply the saddle-point nestingmechanism to explain
pﬃﬃﬃ thepﬃﬃﬃ origin of hierarchy. Once trigonal
system forms
7  7 structure due to saddle-point
nesting,
new system also becomes the trigonal system withp
ﬃﬃﬃ
7 times scale and new saddle points also born. From theseresults, we introduce a clustering number and coordinate oftrigonal system. Further studies both experimental andtheoretical of other superclustering systems will unveil theprinciple of the clustering mechanism.

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Acknowledgment
We wish to acknowledgement T. Sambongi, K. Yamaya,
M. Nishida and D. Meacock for stimulating discussions.
This work has been partially supported by Grant-in-Aid forthe 21st Century COE program on ‘‘Topological Science andTechnology’’ from the Ministry of Education, Culture, Sport,
Science and Technology of Japan.
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