Download Charge density waves in low-dimensional metals

Univerza v Ljubljani
Fakulteta za matematiko in fiziko
Oddelek za fiziko
Seminar 2.2
Charge density waves in low-dimensional metals
Andrej Kocan
Date: 9. 10. 2007
Mentor: prof. dr. Albert Prodan
Abstract
In quasi one- and two-dimensional metals the electrons can organize themselves into
regular patterns known as charge density waves (CDW). This phenomenon was
theoretically predicted by Peierls in 1950s and first observed in 1970s. Since then CDWs
were found in numerous inorganic low-dimensional structures, like the transition-metal
chalcogenides TaS2, NbSe3 and TaS3. All these structures undergo a phase transition
towards a low-temperature modulated ground state, which can be detected by numerous
techniques, i.a. X-ray and electron diffraction and scanning tunneling microscopy. Some
of these techniques and the corresponding information about the CDW states will be
discussed. A short overview of materials exhibiting CDWs and a theoretical description
of a simple 1-D metal will be presented. It will be shown that the transition into the CDW
state is connected with a periodic structural distortion and a non-linear electrical
conductivity, caused by a partial opening of the Fermi surface.
Table of Contents
Introduction......................................................................................................................... 2
Theory ................................................................................................................................. 2
Free electron Fermi gas................................................................................................... 2
Nearly free electron model.............................................................................................. 3
Origin of the energy gap ................................................................................................. 4
Charge density wave ........................................................................................................... 5
CDW sliding ................................................................................................................... 7
Low dimensional crystals ................................................................................................... 9
Properties of CDWs .......................................................................................................... 10
CDW peculiarities......................................................................................................... 12
CDW FET ..................................................................................................................... 14
Conclusions....................................................................................................................... 15
Literature:.......................................................................................................................... 16
1
Introduction
The nearly free electron model in which electrons are described as propagating through a
lattice without interacting with the static ion cores is sufficient in describing many
electrical transport properties of solids. When metals are cooled, they often undergo a
phase transition into a state exhibiting a new type of order. Metals such as iron and nickel
become ferromagnetic below temperatures of several hundred degrees Celsius. Other
materials such as lead and aluminum, become superconductors at cryogenic temperatures.
The interaction between electrons and the lattice vibrations, or electron-phonon
interaction, is essential for a complete description of such transport phenomena. Since the
mid-1970s, a wide range of quasi-one-dimensional metals have been discovered that
undergo a different type of phase transition, both above and below room temperature:
they become charge-density-wave (CDW) conductors. These materials show striking
nonlinear and anisotropic electrical properties, gigantic dielectric constants, unusual
elastic properties and rich dynamic behavior. Several groups of both organic and
inorganic materials are known today and some have been investigated in detail by a wide
array of experimental techniques.[1]
Because density waves arise in their simplest form in highly anisotropic materials some
fundamental aspects of the one-dimensional electron gas will be discussed first. Next
chapter focuses on materials, i.e. the various groups of so-called linear chain compounds.
Finally, a discussion on the basic experimental observations is given.
Theory
Free electron Fermi gas
We can understand many physical properties of metals in terms of the free electron model
– the valence electrons of the constituent atoms become conduction electrons and move
freely through the volume of the metal. They form a free electron Fermi gas, which is
subject to the Pauli principle.
Consider a free electron gas in one dimension (1D), taking account of quantum theory
and of the Pauli principle. The wavefunction ψ ( x) of the electron is a solution of the
Schroedinger equation Hψ = Eψ ; with the neglect of potential energy we have
H = p 2 / 2m where p is the momentum. In quantum theory p may be represented by the
operator − i d / dx , so that
Hψ = −
d 2ψ
= Eψ ,
2m dx 2
2
where E is the energy of the electron. If the electrons are confined to a linear chain of
atoms of length L, the boundary condition is ψ (0) = ψ ( L) = 0 . This condition is satisfied
2
by the sine-like wavefunction of the standing wave ψ n ( x) = A sin(kx) = A sin(π n x / L) ,
where n is a positive integer. It is convenient to introduce wavefunctions that satisfy
periodic boundary conditions – we now require the wavefunctions to be periodic in x with
period L, thus ψ ( x + L) = ψ ( x) .[2] Wavefunctions satisfying the free-particle
Schroedinger equation and the periodicity condition are of the form of a traveling plane
wave ψ k ( x) = exp(ikx) , provided that the values of the wavevector k are of the form
2 n π / L , where n is a positive or negative integer, the energy of the orbital with value k
being
Ek =
2
2m
k2 .
To calculate the ground state of N electrons confined to the length L, assuming the
electrons don’t interact with each other (independent electron approximation), means first
finding the energy levels of a single electron and then filling these levels up in a manner
consistent with the Pauli exclusion principle, which permits at most one electron to
occupy any single electron level. Electron orbitals with discrete values of k can
accommodate two electrons with opposite spins.
Nearly free electron model
The free electron model of metals gives a good insight into the heat capacity, thermal and
electrical conductivity, magnetic susceptibility, and electrodynamics of metals. But the
model fails in case of other important questions. One of them is of importance for this
discussion: the distinction between metals, semiconductors and insulators. To understand
the difference between insulators and conductors, we must extend the free electron model
and take into account the periodic lattice of the solid with the weak periodic potential of
the ion cores. We know that a Bragg reflection is a characteristic feature of the wave
propagation through crystals. At Bragg reflections wavelike solutions of the Schroedinger
equation do not exist, as shown in fig 1.
Figure 1: (a) Plot of energy E versus wavevector k for a free electron. (b) Plot of energy
versus wavevector for an electron in a monatomic linear lattice of lattice constant a. The
3
energy gap E g shown is associated with the first Bragg reflection at k = ±π / a ; other
gaps are found at higher energies at ± nπ / a , for integral values of n. [3]
The only allowed scattering vectors of an X-ray, a neutron or an electron are the
reciprocal lattice vectors {G}.[4] The Bragg condition (k+G)2 = k2 for diffraction of a
wave of wavevector k, is given in one dimension
k = ± 12 G = ± n
π
a
,
where G = 2π n / a is a 1D reciprocal lattice vector and n is an integer. The first
reflections and the first energy gap occur at k = ±π / a : At these special values of k the
time-independent state is represented by standing waves. The region in k-space between
−π / a and π / a is called the first Brillouin zone of this lattice. From the two travelling
waves exp(±iπ x / a ) we can form two different standing waves:
ψ (+) = exp(iπ x / a ) + exp(−iπ x / a ) = 2 cos(π x / a)
ψ (−) = exp(iπ x / a) − exp(−iπ x / a) = 2i sin(π x / a)
The standing waves are labeled (+) or (-) according to whether or not they change sign
when − x is substituted for x . Both standing waves are composed of equal parts of rightand left-directed travelling waves.[2]
Origin of the energy gap
The two standing waves pile up electrons in different regions, and therefore the two
waves have different values of the potential energy in the vicinity of the ions of the
lattice. This is the origin of the energy gap. For a pure travelling wave the charge density
is constant, while it is not constant for linear combinations of plane waves. For the
standing waves we have
2
ρ (+) = ψ (+ ) ∝ cos 2 π x / a
2
ρ (−) = ψ (−) ∝ sin 2 π x / a
The first function piles up electrons on the positive ions centered at x = 0, a, 2a,… , where
the potential energy is lowest. The other standing wave concentrates electrons away from
the ion cores, as is shown in figure 2. Potential energy of ρ (+) is lower than that of the
travelling wave, whereas the potential energy of ρ (−) is higher than the travelling wave.
We get an energy gap of width Eg .
4
Figure 2: Distribution of probability density in the periodic potential for standing wave 1
- ρ (−) and 2 - ρ (+ ) . The standing wave piles up charges in the region between the ion
cores while standing wave 2 piles up charges around the core points. [3]
Charge density wave
In 1955, Sir Rudolph Peierls first predicted that at absolute zero temperature a structural
modulation ( q = 2k F ) corresponding to a distortion of the crystal lattice and conduction
electron density results in splitting of the energy spectrum at the Fermi level and in
formation of an additional energy gap (Fig. 3).[5]
Figure 3: A quasi 1D metal, as represented in a, can reduce its energy by developing a
CDW, as shown in b. A CDW consists of coupled modulations of the conduction electron
density and the atomic positions. The modulations have wavelength λC = π / k F and
produce an energy gap at the Fermi surface k = ± k F . The modulations are usually quite
small; atomic displacements are only about 1% of the interatomic spacing, and the
conduction electron density varies by several percent.[1]
5
As a result, the CDW state is the ground state of a 1D metal at T = 0 K. At temperatures
above absolute zero, some electrons become thermally excited across the gap. This
lowers the reduction in energy associated with the CDW formation and eventually
quenches the CDW at a critical temperature TCDW . Above that temperature the metallic
state is stable.
The charge density of the conduction band exhibits a periodic variation with the same
wavevector q as the periodic lattice distortion (PLD). For a crystal with a half filled band,
the CDW state corresponds to a periodic structure with a lattice constant that is twice as
large as the periodicity of the basic structure. In general however, the fillling of the
conduction band is not related to the lattice periodicity of the crystal, and k F can be any
fraction of the basis vector of the reciprocal lattice. If the new lattice constant a′ is an
integer multiple of the unperturbed lattice constant a, the CDW is referred to as
commensurate. Contrary, the CDW is incommensurate if the ratio a′/a is an irrational
fraction. If a modulated structure is incommensurate, it lacks translational symmetry and
can no longer be considered truly periodic.[5]
Since an energy gap forms within the former conduction band, a 1D metal would be
expected to become insulating below TCDW. The concept of a 1D metal is of course an
idealization and real materials exhibiting CDWs are quasi-1D with coupled adjacent
metal atom chains and a 3-D character. Thus the ideal planar Fermi surfaces are
deformed and a single CDW wavevector q may fail to remove the entire Fermi surface.
As a result, the CDW is generally associated with an anomaly in the electrical resistance
which remains however metallic, as shown in fig 4.
Figure 4: 2D sections of the Fermi surfaces of quasi-one-dimensional crystals. (a) Noninteracting metallic chains parallel to [001] with (k1 , k 2 , 0) describing k vectors
perpendicular to the chains. (b) Weakly interacting metallic chains result in a warping of
the Fermi surface. Perfect nesting is obtained for q = (0,1 / 2, q3 ) , where q3 fulfills the
nesting condition. Notice that the second half of the Fermi surface is nested by
q ′ = q − b * .[5]
Interchain interactions thus explain the metallic character of the CDW state (as well as
non-zero perpendicular components of the modulation wavevectors). These features are
6
illustrated by NbSe3. NbSe3 exhibits two (independent?) CDW phase transitions, each of
which removes part of the Fermi surface as expressed by the anomalous increase of the
electrical resistivity on cooling down through the phase transitions (fig 5) (Monceau,
1985).[5]
Figure 5: Temperature dependence of the electrical resistance (R) of NbSe3, normalized
to the value at 300 K. Data obtained on cooling (squares) and heating (circles) coincide.
[5]
While commensurate modulations do occur in CDW materials, such as pure Nb3Te4, in
general a Peierls transition leads to an incommensurate CDW. For the incommensurate
case, the phase of the CDW is independent of the lattice. However, the CDW phases can
be “pinned” by defects or impurities in the crystal. In some materials, depinning can also
occur as a result of an electric field applied to the crystal. If this field is sufficient to
overcome the pinning energy, the incommensurate CDW can effectively carry a
current.[1]
CDW sliding
The simplest model that describes the behavior of density waves is called the classical
particle model. The CDW is represented by a single massive particle positioned at its
center of mass. The behavior of this particle reflects that of the entire array. When there
are no external electric fields, in the commensurate case the particle sits on a ribbed
surface, like a marble in a cup of an egg tray. The marble is free to move around the
bottom of the eggcup and can therefore readjust its position sensitively in response to
applied electric fields. The marble usually adjusts its position so as to reduce the electric
field acting on it. Thus, materials with CDWs have a large dielectric constant, so large
that they could be called superdielectrics. Measurments on CDWs give values for the
dielectric constant more than one million times larger than that of ordinary
semiconductors.[6]
What happens if a DC voltage is applied to an incommensurate CDW, where the state of
the crystal is independent of the phase of the modulation? The CDW becomes mobile in
the present of the electric field, but the phase of the modulation is pinned by impurities,
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lattice defects and the surface. Pinning of the CDW can be overcome by sufficiently large
electrical fields. For fields larger than a threshold field ( ET ), the sliding CDW provides a
second conduction path next to a single-particle electron conduction. Macroscopically
this leads to non-linear electrical conductivity for large fields. Deformations of a sliding
CDW have been observed by shifts of the positions of satellite reflections (i.e. changes of
the magnitudes of q ) dependent on the distance of the area illuminated by the X-ray
beam from the electrodes.
Figure 6: Charged-particle model illustrating how the current flow in CDW differs from
that in normal metals. In a metal (left ) the particle rests on a flat (electrical potential)
surface. If we apply a voltage, the surface tilts, and the particle starts to move: there is a
current. For a charge-density wave (right ), the surface is ribbed. If the applied voltage is
low - that is, the tilt is small - the particle changes position only slightly, and there is no
current. If the tilt is large enough for the particle to cross the barrier, the particle runs
down the ribbed surface. The resulting current oscillates as the particle climbs over each
barrier. Non-linear conductivity due to this sliding CDW is known as Fröhlich
conductivity [6,5].
Figure 7: Current versus voltage is plotted schematically for metals and charge-density
waves. In a metal (blue) the current increases linearly with voltage. For a charge-density
wave material, there is no current until the voltage increases to a critical value; only then
does the current start to flow (red). If in addition to a direct voltage an alternating voltage
is applied, the curve shows plateaus (purple). The plateaus correspond to a “mode
locking” when the flow of the CDW matches the alternating frequency.[6]
8
Low dimensional crystals
Ideal 1D crystals do not exist. Materials occupy a finite volume in space, and thus are 3D
by definition. However, 1D electron bands can reside in 3D crystals if the atomic orbitals
comprising the conduction band show an appreciable overlap in one direction while they
have much smaller overlaps in the perpendicular directions. The crystal then is composed
of parallel metallic chains, with weak interchain interactions only. The metallic chains
can be embedded in a matrix of non-metallic atoms, such that the crystal structure
contains chemical bonds in all three directions of space. Consequently, quasi-1D
electronic crystals are 3D as far as the phonons are concerned.[5]
Figure 8: (a) Despite widely varying crystal morphologies, CDW materials share a
common architecture consisting of weakly coupled molecular chains. Single-crystal
whiskers of NbSe3, grown by chemical vapor transport.[1]
A large number of organic and inorganic solids have crystal structures in which the
fundamental structural units form linear chains. While most of these materials are
insulators or semiconductors, several groups have partially filled electron bands, and
consequently display metallic behavior at high temperatures[7]; among them, three have
been explored in detail:
•
•
•
Mixed valence platinum chain compounds are composed of a columnar array of
units which incorporate a chain of Pt atoms with strongly overlapping d orbitals;
most of the experiments have been performed on the material K2Pt(CN)4Br0.3
3.2H2O, known simply as KCP or Krogmann’s salt. It consists of columnar
stacked array of Pt(CN)4 units.
Transition metal chalcogenides, MX3 and (MX4)nY: Group IV or V transition
metals, Nb or Ta, when combined with chalcogen atoms, S, Se and Te, form a
variety of linear chain compounds.
Transition metal bronzes. The term bronze is applied to a variety of crystalline
phases of the transition metal oxides. Examples are the ternary molybdenum
oxides of formula A0.3MoO3, where the alkali metal A can be K, Rb or Tl.
9
Figure 9: The crystal structure of KCP (left) , blue bronze (middle) and NbSe3 (right)[1]
Two-dimensional CDWs have been found in several classes of compounds; most
extensively studied are the transition-metal dichalcogenids MX2 (M=Nb, Ta; X=S, Se),
which are layered compounds known for their polytypism. M atoms are at the nodes of a
planar hexagonal lattice, and they are sandwiched between planes of X atoms, with either
trigonal prismatic coordination of M (T layers) or octahedral coordination (H layers). [8]
Figure 10: Example of stacking NbSe2. In the 2H polytype the stacking is ABAB.[9]
Properties of CDWs
The CDW of the conduction band and the PLD describing displacements of atoms are
different aspects of a single phenomenon. Depending on the property that is studied,
either the CDW or the PLD is probed. For example, scanning tunneling microscopy
(STM) depends on the density of states (DOS) at the Fermi level, and thus probes the
CDW directly.[10] The electrical resistance is determined by electron-phonon coupling,
and thus depends on both the CDW and the PLD. X-ray diffraction almost exclusively
probes the positions of atoms, because it does not depend on the details of the valenceelectron distribution.
10
The periodicity q of the PLD is responsible for the occurrence of additional Bragg
reflections in the X-ray diffraction (satellite reflections) at positions ±mq around each
Bragg reflection of the underlying basic structure. Satellite reflections may have equal
widths as main reflections, indicating true long range order of the CDW. Alternatively,
satellites broader than main reflections indicate a limited order of the CDW. Usually,
CDW satellite reflections are weak ( 10 −2 −10 −5 × I main ), and they can be observed for
values of m equal to 1 or 2 only.[5]
Electron diffraction experiments confirmed the existence of CDW in NbSe3 in 1977.[11]
The electron diffraction of NbSe3 was performed with the acceleration voltage of the
electron beam of 100kV. Figure 11 left shows the b*-c* plane of the reciprocal space
above 140K. The anomaly in diffuse scattering can be clearly seen to be one-dimensional
(the chain axis is the b axis). In figure 11 right, which shows the a*-b* plane below
140K, the satellite spots are incommensurate with the underlying lattice. The width of the
diffuse scattering, which is oriented perpendicular to the 1D chains, usually increases
with increasing temperature, indicating a temperature dependent finite correlation length
of the fluctuations along the chains.[11]
Figure 11: Electron diffraction pattern of NbSe3. Left: above 140 K showing onedimensional anomaly in diffuse scattering (marked by arrows). Right: below 140 K
showing the satellite spots (marked by arrows).. [11]
More direct observations of the CDWs can be achieved by STM imaging, as shown in
figure 12 for the case of NbSe2 at liquid He temperature, i.e. well below the critical
temperature ( TCDW ≈ 35 K ).[9]
11
Figure 12: Image on the left shows a 13 nm2 topografic scan. Inset on the right: Fourier
transphorm (FT) showing two sets of reflections. The outer peaks correspond to the
hexagonal lattice of the atoms and the inner ones to those of the CDW. The CDW is
commensurate with the atomic lattice with the wavelength 3× the atomic lattice. [9]
CDW peculiarities
Although theory predicts that the CDW state is a ground state for a 1-D metal structure,
not all linear chain compounds show the modulation. One example are isostrucural
compounds Nb3(S4, Se4, Te4). Of these only Nb3Te4 with the largest lattice constants
shows CDWs, which run along the zigzag octahedral columns shown in figure 13.[12]
Figure 13: The crystal structure of Nb3S4, Nb3Se4, and Nb3Te4.[12]
Furthermore, not all CDW materials exhibit nonlinear conductivity behavior ascribed to
CDW sliding. A considerable attention was paid in the past to isostructural compunds,
especially NbSe3 and m-TaS3, which show nonlinear transport properties. The peculiarity
about these two compounds is that they belong to both, the quasi one- as well as quasi
two-dimensional families of structures. The 1D character is provided by the trigonal
prismatic columns along their b-direction and 2D character by layers in the b-c plane,
held together by very weak van der Waals (vdW) forces as shown in figure 14. As was
shown in figure 5, the material shows two peaks in the resistance versus temperature
curve. This indicates an onset of two independent CDW states running along two of the
12
three slightly different pairs of trigonal-prismatic chains (named type I, II and III, which
form the enlarged unit cell). [13]
Figure 14: Crystal structure of NbSe3 viewed perpendicular to the b axis of linear chains.
[13]
NMR and high resolution X-ray diffraction measurements of NbSe3 show that the HTCDW (Tc = 145K) primarily affects type III chains, while the LT-CDW (Tc = 59K)
primarily affects type I chains [14, 15]. However, early STM results at liquid helium
temperature (1991) suggest that all three types of chains present a strong CDW
modulation [16]. It is still unclear whether all chains are CDW modulated and some
efforts were put into theoretical studies of the phenomenon. It is believed that highresolution LT STM experiments can clarify the experimental observations. LT-STM
pictures of an in-situ cleaved sample of NbSe3 and the corresponding diffraction patterns
are shown in figure 15, 16 and 17. Both the underlying molecular lattice and two CDW
superlattices were observed simultaneously at 5K with molecular resolution. The CDW
wavevectors extracted from 2D fourier transform are in good agreement with bulk
results.[17]
Figure 15: 50*50 nm2 STM image of
(b,c) plane of NbSe3 at T~62K. Vbias=300mV, It=60pA. Both Q1 and Q2
CDWs are seen above T2 the bulk
critical temperature for the Q2 CDW, as
confirmed in the FT of this image in Fig
16.[18]
Figure 16: 2D FT of Fig.1 showing Q1
and Q2 superlattice spots with C* Bragg
spot. The intensity of the LT-CDW(Q2)
is stronger than that of HTCDW(Q1).[18]
13
Figure 17: Constant current mode topographical image of the (100) plane of NbSe3 at 5K.
Arrows 1, 2 and 3 indicate the observed type I, II and III chains. The image was recorded
with negative bias voltage (-100mV, 100 pA); the surface vectors 4b and 2c are indicated
by two arrows.[17]
CDW FET
Recently (1995), a novel electronic device, a CDW field-effect-transistor (CDW FET),
has been produced.[19] The structure of this device is similar to that of the metal-oxidesemiconductor field-effect transistor (MOSFET), but the channel layer is made of a CDW
material. The CDW FET has the potential to become a useful functional device, because
its characteristics are different from those of conventional FETs, therefore a short
description of such a device follows. The channel layer of this particular CDW FET,
fabricated by Ryo Kurita [20], is made of NbSe3 single crystals grown by the direct
chemical-gas-transport reaction. A micrograph and a schematic illustration of the CDW
FET are shown in Figs. 18 and 19, respectively. During the fabrication process, a cleaved
NbSe3 crystal of 0.2 μm thickness was placed on a glass substrate. The source and drain
electrodes made of a of gold paste were formed at both ends of the NbSe3 crystal, and a
SiO2 insulator of 100nm thickness was deposited by rf sputtering. Finally, the gate
electrode of indium was formed on the SiO2 insulator. In these samples, the gate
electrode was formed between the source and drain electrodes.
Figure 19: Schematic illustration of
CDW FET.[20]
Figure 18: Top view of CDW FET. The
arrow indicates a thin NbSe3 crystal.[20]
14
Figure 20 shows the I DS − VDS characteristics for several gate voltages at 35K for one of
the samples, where I DS and VDS are, respectively, the current and the voltage between the
drain and source electrodes. The conduction is ohmic below the threshold voltage VT , but
I DS increases nonlinearly above VT and is strongly modulated by the gate bias VG . The
nonlinear current is reduced by the positive gate bias and is enhanced by the negative
one.
Figure 20: I DS − VDS characteristics at 35K for several gate voltages VG for a CDW FET
sample. The straight line indicates the ohmic current component. The open arrows
indicate the steplike structures.[20]
Conclusions
In this seminar a simple theoretical description of the origin of the CDW ground state
was given. Transition towards a modulated structure requires a strong electron-phonon
interaction and the nesting condition of the modulation wavevector q fulfilled for a large
part of Fermi surface. From the discovery of low dimensional metals exhibiting CDWs in
1970 on, some of these compounds have been thoroughly studied, but adequate theoretic
explanation of CDW properties and their dynamic behavior still remains an open
problem. Possible future practical (commercial) CDW abusing devices have been
proposed. The opportunities these developments provide should ensure that CDW
conductors remain interesting enough for many years to come.
15
Literature:
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[2] Charles Kittel: Introduction to solid state physics, 8th edition, John Wiley & Sons,
Inc., 2005
[3] Energy band formation, last updated: 4.19.2002 [cited 9. 10. 2007]:
http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm
[4] A. W. Overhauser: Charge-density waves and isotropic metals. Adv. in Phys., 27 (3),
1978
[5] Sander van Smaalen: The Peierls transition in low-dimensional electronic crystals.
Acta Cryst. (2005). A61, 51-61
[6] Stuart Brown and George Grüner: Charge and spin density waves. Sci. Am. April (5056), 1994
[7] George Grüner: Density waves in solids, Addison-Wesley, 1994
[8] J. A. Wilson, F. J. Di Salvo and S. Mahajan: Charge-density waves and superlattices
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NanoSciences de Paris:
http://www.insp.upmc.fr/axe1/Dispositifs%20quantiques/AxeI2_more/CDW/cdw_more.
HTM
[10] G. Gammie et al: Scanning tunneling microscopy of NbSe3 and orthorombic TaS3.
Phys. Rev. B, 40 (14), 1989
[11] K. Tsutsumi, T. Takagaki: Direct electron diffraction evidence of charge densitywave formation in NbSe3. Phys. Rev. Lett. 39 (1977)
[12] F. W. Boswell, J. C. Bennett, A. Prodan: Charge density wave transitions in Nb3Se4
and Nb3S4 by indium intercalation. J. Sol. St. Chem., 144, 454-460 (1999)
[13] A. Prodan, N. Jug, H. J. P. van Midden, H. Böhm, F. W. Boswell and J. C. Bennet:
An alternative model for the structural modulation in NbSe3 and TaS3. Phys. Rev. B 64,
115423 (2001)
[14] F. Devreux, J. Phys. (Paris) 43, 1489 (1982)
[15] Van. Smaalen , J. L. de Boer,A. Meetsma, H. Graafsma, H.-S. Sheu, A. Darovskikh
and P. Coppens, Phys. Rev. B 45, 3103 (1992)
[16] Z. Dai, C.G. Slough and R.V. Coleman, Phys. Rev. Lett. 66, 1318, (1991)
[17] C. Brun, Z.Z. Wang and P. Monceau: Charge density waves modulation in different
chains of NbSe3. J. Phys. IV France 131 (2005) 225-226
[18] Z.Z. Wang: Surface Charge Density wave in low dimensional system measured by
LT-UHV-STM. Workshop „Recent developments in low dimensional charge density
wave conductors“ Skradin, Croatia June 29.-July 3. 2006.
[19] T. L. Adelman, S. V. Zaitsev-Zotov and R. E. Thorne: Phys. Rev. Lett. 74 (1995)
5264.
[20] Ryo Kurita: Current modulation of charge-density-wave field-effect transistors with
NbSe3 channel. Journal of the Physical Society of Japan Vol 69, No. 8, August, 2000, pp.
2604-2608
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