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Peierls transition in two-dimensional metallic monophosphate tungsten bronzes
Solid State Sciences 4 (2002) 387–396 www.elsevier.com/locate/ssscie
Peierls transition in two-dimensional metallic monophosphate tungsten bronzes P. Foury-Leylekian ∗ , J.-P. Pouget Laboratoire de Physique des Solides, CNRS-UMR 8502, Université de Paris Sud, 91405 Orsay cedex, France Dedicated to Martha Greenblatt
Abstract The two-dimensional metallic bronzes made of ReO3 -type layers of MoO6 or WO6 octahedra present quasi-one-dimensional (1D) electronic structures along three directions of preferential overlap of the t2g transition metal orbitals. They exhibit a Peierls instability towards the formation of charge density waves (CDW) at the 2kF critical wave vector allowing to nest simultaneously the Fermi surfaces associated to two quasi-1D band structures out of three. The Peierls transition is achieved through a periodic lattice distortion (PLD), that we analyse, in the present work, for the monophosphate tungsten bronzes. The Peierls critical temperature decreases in presence of disorder, which breaks the electron–hole pairs forming the CDW condensate, and in presence of misfit between the PLD wave vector and the 2kF nesting wave vector. The pair breaking effect accounts for the drop of the Peierls transition in the Nax P4 W12 O44 (x < 1) bronzes while the misfit effect, associated to a variation of the band filling, explains quantitatively the phase diagram of the Kx P4 W8 O32 (1 < x < 2) bronzes. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction The term bronze is attributed to a large variety of ternary transition metal (M) oxides of general formula Ax (MOn )y where A is a monovalent cation or an elemental group such as PO4 . The species A provide electrons to the d–t2g transition metal levels, in the crystalline field of the MO6 octahedra, which otherwise are unoccupied in the parent oxide MOn . In presence of this charge transfer these materials become conductors with a metallic luster recalling that of the bronze. The bronzes are known since the beginning of the 19th century, but it’s only since twenty years that these materials were recognized as a new class of low-dimensional metals [1,2]. Based on MO6 octahedral units, the structure of the bronzes can be one-, two- or three-dimensional (1D, 2D or 3D) with respect to the chemical bond network, but are 1D or 2D with respect to the bonding of a given set of t2g orbitals, because of the directional character of these orbitals. As this last dimensionality is lower than the structural dimensionality it appears that the bronzes possesses a ‘hidden’ low dimensionality from the electronic point of view. Although it had been noticed in the early * Correspondence and reprints.
E-mail address: [email protected] (P. Foury-Leylekian).
70’s that the Fermi surface (FS) of the metallic (3D cubic network) cation deficient perovskites, such as ReO3 and Nax WO3 , has a cylindrical shape and that the conduction bands of the conventional perovskites, such as BaTiO3 and SrTiO3 , possess a 2D anisotropy [3], this concept was more precisely expressed when extended Hückel semi-empirical band structure calculations performed in the layered (2D network) Mo and W bronzes [4,5] were able to show that their conduction band structure exhibits a subtle 1D anisotropy. These 1D features were recently confirmed by ab-initio band structure calculations performed in the 2D monophosphate tungsten bronze P4 W8 O32 [6]. With an electronic structure of reduced dimensionality, the FS of these metallic bronzes is expected to be enough anisotropic to possess good nesting properties. It is now well-known that such nesting properties drive density wave ground states (charge density wave (CDW) or spin density wave (SDW) if there is strong enough electron–phonon or electron–electron interactions, respectively) [2]. Such instabilities could eventually compete with the superconductivity, whose critical temperature exceeds 5 K in Ax WO3 and WO3−x . CDW ground states are effectively achieved in several 2D Mo and W bronzes [1,2]. This was first shown in the Mo purple bronze Ax Mo6 O17 (A = K, Rb, Tl; x ∼ 1) whose structure is made of ReO3 -type corner sharing MoO6 octahedra
1293-2558/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 1 ) 0 1 2 6 6 - 3
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slabs, separated by the monovalent atoms A and MoO4 tetrahedra. This is also the case of monophosphate W bronzes with pentagonal tunnels (MPTBp’s), (PO2 )4 (WO3 )2m (m 4), or hexagonal tunnels (MPTBh’s), Ax (PO2 )4 (WO3 )2m (A = K, Na), whose structure is also made of ReO3 -type slabs of WO6 octahedra separated by monophosphate PO4 tetrahedral groups [7]. The oxide Mo4 O11 , which structure resembles that of the m = 6 member of the MPTBp’s (γ phase) or of the MPTBh’s (η phase), with MoO4 tetrahedra replacing the PO4 groups, belongs also to these classes of 2D metals. All these 2D bronzes exhibit one or several successive CDW–Peierls transitions which have been associated to the nesting of several 1D parts of their FS [8]. The CDW ground state corresponds to a 2kF modulation of the electronic density (kF is the Fermi wave vector of the 1DFS) accompanied by a 2kF periodic lattice distortion (PLD). This latter modulation is detected in X-ray diffraction experiments [9]. In low-dimensional (generally quasi-1D) metals, the cost of elastic energy due to the PLD is overcome by the gain of electronic energy due to the opening of a gap at the Fermi level. However as the nesting mechanism of the 2D bronzes does not remove all the electronic states at the Fermi energy, the Peierls transition leads generally to a semimetallic state where cylindrical electron and hole pockets of small section remains. This 2D semi-metallic state presents interesting quantum phenomena such as those observed in η-Mo4 O11 . In this oxide a bulk quantum Hall effect and, in the quantum limit, the opening of a true Peierls gap, associated with a magnetic field induced electron- and hole-band inversion, are observed. 2. The CDW instability in 2D bronzes 2.1. The hidden 1D electronic structure Let us first discuss the cubic perovskite ‘prototype’ structure. For each type of (1 0 0) cubic plane there is a preferential overlap of one set of d–t2g orbitals of the transition metal; i.e., the dyz orbitals for the (1 0 0) plane [3]. As there is nearly no overlap between parallel planes, the band structure is 2D and its FS is a ‘cylinder’ perpendicular to the planes. The total band structure is thus the superimposition of the individual band structures of the 3 sets of (1 0 0) planes related by cubic symmetry and the global FS is the sum of 3 ‘cylinders’ running along the 3 cubic directions, as found in the calculation of the FS of ReO3 and Nax WO3 . The conducting layer of purple bronzes is obtained by slicing the ReO3 -like structure between two planes perpendicular to the 3-fold axis of the cubic perovskite structure. The cut leaves a trigonal layer in which run 3 sets of parallel infinite chains along the hexagonal-like intralayer directions. In each of these directions there is a preferential overlap of one set of the d–t2g Mo orbitals [4]. The band structure is thus 1D and the FS associated to each 1D band consists in a pair of parallel planes perpendicular to the chain direction and separated by 2kF . The conduction band
structure is the superimposition of three 1D band structures and the global FS is the sum of 3 couples of planar sheets related by trigonal symmetry, each having the same 2kF [4]. In the purple bronze each 1D band accommodates one electron given by the monovalent atom A and the MoO4 tetrahedra, leading to 2kF = 1/2 chain reciprocal unit (the factor 2 missing accounts for the spin degeneracy in the band filling). This FS topology has been recently confirmed by angle-resolved photo-emission spectroscopy (ARPES) measurements in the Na bronze [10]. Similarly the conducting layer of the MPTB’s results from the slicing of the ReO3 -like structure between two parallel (1¯ 1 2) planes of the cubic perovskite structure. The cut leaves a rectangular layer where the a and b directions are, respectively, perpendicular to the [1 1 0] and [1 1¯ 1] perovskite directions. Along each of the a, a + b and a − b directions run infinite parallel chains made of segments of m WO6 octahedra tilted with respect to the chain direction. There is a preferential overlap of one combination of d–t2g W orbitals for each kind of chains [5]. Similarly to the purple bronze, individual planar FS can be associated to the chains a and a ± b (see Fig. 1). However as the a and a ± b chains are not related by symmetry, the 2kF wave vector of their 1D band structure differs (below we shall note them 2kFa and 2kFd , respectively). The global FS is the sum of these three 1D-FS. In the MPTBp’s the total FS accommodates 2 electrons given by the PO4 groups to each segment of m WO6 octahedra. Thus, if 2kF is expressed in its corresponding chain reciprocal unit, one gets the simple relationship: 2kFa + 2 × 2kFd = 1.
(1)
In the m = 4 member, the ab-initio calculation [6] gives 2kFa ≈ 2kFd ≈ 1/3. ARPES studies of the Na purple bronze [11] and of MPTBp’s with different m values [12] show the occurrence of a well-defined Fermi edge separating the occupied electronic states from the unoccupied ones. This means that the concept of FS is validated and that the nesting mechanism, which will be introduced more quantitatively in the next section, can be used to describe the CDW instability of these bronzes. 2.2. The CDW FS-nesting instability The wave vectors q which achieve the best nesting of the global FS give rise to a maxima of the electron–hole (or Lindhard) response function χe (T , q). This response function has been calculated for the total FS in the MPTBp with m = 4 [6]. It exhibits, as schematically shown Fig. 2a in the (a ∗ , b∗ ) reciprocal plane, maxima of intensity along lines perpendicular to the a and a ± b chain direction. In the Brillouin zone these lines are approximately located at the 2kFa and 2kFd wave vectors at which there is nesting of the ideal 1D-FS’s associated to the chains a and a ± b, respectively.
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(a)
(a)
(b) (b) Fig. 1. Decomposition of the Fermi surface (FS) of the MPTBp’s into (a) a 1D-FS associated to chains running along the a directions and (b) two 1D-FS associated to chains running along the a + b and a − b directions.
For a 1D gas of non-interacting free electrons and a wave vector q close to the 2kF value of the optimal nesting, q = 2kF + δq, if one neglects matrix elements between Bloch functions in the electron–hole response function, one gets [13]: χe (T , q) = N(EF ) ln(Ec /kB T ) + Ψ (1/2) − 1/2 Ψ (1/2 + i h¯ vF δq/4πkB T ) + Ψ (1/2 − i h¯ vF δq/4πkBT ) . (2) In this expression, valid for a linear dispersion of the electron in the vicinity of the Fermi level, Ψ (x) is the digamma function which will be used later. N(EF ) is the density of states per spin at the Fermi level, Ec is a cutoff energy of the order of the Fermi energy (EF ) and vF is the Fermi velocity. For h¯ vF δq πkB T , the development of Ψ (x) gives: 2 χe (T , q) = N(EF ) ln(Ec /kB T ) − ξth (T )δq , (3)
Fig. 2. (a) Location in the (a∗ , b∗ ) section of the Brillouin zone of the 2kF CDW critical fluctuations associated to the nesting of the 1D-FS shown in Fig. 1. The crossing points qc at which the common nesting of two FS out of three occurs in the MPTBp’s is indicated: qc1 and qc4 for the chains a + b and a − b, qc2 for the chains a and a − b and qc3 for the chains a and a + b. The qc4 crossing point, when equal to b∗ /2 is relevant for the MPTBh’s, as shown in (b).
where:
√ ξth (T ) = hv ¯ F / 2 πkB T ,
(4)
is the thermal coherence length. With the electronic structure described in the last section the total electron–hole response function is the sum of the individual electron–hole response function of each 1D electron gas. The divergence of the total response function will be stronger at the intersection point qc (defined Fig. 2a) of the 2kF lines of extrema of intensity of χe (T , q). Here the electron–hole response functions of the two intersecting 2kF 1D-response functions (i = 1, 2) add: χei (T , qc ). χc (T , qc ) = (5) i
At qc the response function exhibits a
ln(Ec /kB T ) thermal divergence, which is, by its prefactor i Ni (EF ), stronger than the divergence on the isolated lines. As a consequence
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the electronic system will tend to stabilize a density wave at the qc wave vector, because the opening of a Peierls gap on two couples of 1D-FS, at the expense of a single structural distortion, will lead to a maximum gain of electronic energy. In the mean-field approximation the Peierls transition will occur at a Tc given by [13]: Vep (qc )χc (Tc , qc ) = 1,
(6)
where Vep (qc ) = 2g 2 (qc )/h¯ Ω(qc ). In this expression g(qc ) is the electron–phonon coupling constant (assumed here, for simplicity, to be the same for the two bands connected by qc ) and Ω(qc ) is the frequency of the bare phonon mode which will be critical by its coupling with the electronic degrees of freedom. The expression (6) leads to: kB Tc = Ec exp −1/2λep (qc ) , (7) where λep (qc ) = Vep (qc )N(EF ) is the reduced electron– phonon coupling (in this expression it is also assumed, for simplicity, that the two bands have the same density of states). Assuming that λep remains constant, Tc given by (7) is sizeably enhanced with respect to Tp of a standard Peierls transition achieving the nesting of a single pair of 1D-FS: kB Tp = Ec exp(−1/λep ).
(8)
For example, the formulae (7) leads to λep ≈ 0.16 from the upper CDW critical temperature Tc = 120 K of the m = 6 member of the MPTBp’s and the ab-initio calculated Fermi energy EF ≈ 0.2 eV for the a ± b chains [6]. With the same values of λep and EF the formulae (8) leads to a standard Peierls transition of only Tp = 5 K! The CDW ground state is achieved by the nesting of several sets of quasi-1D-FS. The nesting of one set of quasi1D-FS mixes the |kF + δq/2 and |−kF + δq/2 conduction electron wave functions. Their bonding (antibonding) combination enters in the wave functions of the electronic states below (above) the Peierls gap. With such wave functions there is an inversion of the phase between the CDW modulations built on the occupied and the unoccupied electronic states. This contrast inversion has been observed by scanning tunneling microscopy imaging of the CDW in the K purple bronze [14]. 2.3. Coupling to the lattice modes In the last section we have only considered the electronic couterpart of the Peierls transition. However in order to achieve the 2kF PLD which potential Vep (q) opens a gap at the Fermi level in the quasi-1D band structure, the electronic degrees of freedom must be coupled, at 1st order in the lattice displacements [13], to the structural degrees of freedom. The critical ones generally correspond to low frequency, Ω(q), phonon modes for which the reduced electron–phonon coupling constant, λep , is the strongest. This coupling requires that the associated lattice displacements should change at 1st order the intrachain
(a)
(b) Fig. 3. Schematic representation of the chains of m = 4 WO3 octahedra running in the a + b (a) and a (b) directions (the a − b chain is identical at the a + b chain shown in (a)). The x, y, z setting of the octahedron is indicated. In (a) a ‘ferroelectric’ distortion of the segment is indicated.
electronic quantities such as the transfer integral (t) or the site energy (ε). This coupling is achieved through the already introduced electron–phonon coupling constant g(q). Here we shall consider with more details the lattice degrees of freedom which could be involved in the structural–Peierls instability of the MPTBp’s. X-ray diffuse scattering investigations reveal that there are preexisting low frequency phonon modes in the MPTBp’s. They correspond to a displacement of the W atoms away from the center of the WO6 octahedra. This displacement, correlated inside the segment of m octahedra, but uncorrelated between the neighboring segments, is of the ferroelectric type. The associated low frequency phonon modes give rise to an enhanced X-ray diffuse scattering consisting of broad trails along the 2a∗ ± 3b∗ reciprocal directions perpendicular to the segment direction. Consistently the 2kFa and 2kFd pretransitional structural fluctuations of the Peierls–CDW transitions of the low m members of the MPTBp’s are located inside these broad trails [8]. Let us consider now how a ferroelectric distortion of a segment of m octahedra can be coupled at 1st order to the electronic states of a chain of octahedra. This requires that the displacement, u, of the W atoms changes, at 1st order in u, the overlap integral, t, between the d–t2g levels on which the 1D band structure of the chain is built. Let us, for example, consider the a + b chain of m = 4 octahedra shown Fig. 3a: the segments are tilted with respect to the chain direction and there is an inversion center in the middle of each segment. With this chain geometry,
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a displacement +u of all the W atoms of a segment leads to variation of the transfer integral t (u) of the d–t2g levels between neighboring segments which is different of that for the opposite displacement, −u, only if u is not perpendicular to the symmetry plane of the chain. For this polarisation of displacement, t is an odd function of u. The electron–phonon coupling constant previously introduced is proportional to its first order derivative: g ∝ dt/du [13b]. Note that such a displacement does not change at 1st order the molecular energy levels of the segment undergoing the distortion: because of the inversion symmetry of the segment the site energy, ε(u), is an even function of u. However this displacement changes at 1st order the site energy of the neighboring segment. This last process contributes also at the electron–phonon coupling constant. Now let us examine how a single ferroelectric displacement can modulate at 1st order the transfer integrals of two differently oriented chains in order to contribute at the electron–phonon coupling constants allowing the simultaneous development of a Peierls gap in the band structure of these two chains. Fig. 3b represents a chain a of the m = 4 member, and define the 3 elementary directions of displacement, x, y and z, for the W atom. The transfer integral of the chain a is only modulated at 1st order for W displacements in the xy plane of symmetry of the chain. The transfer integral of the chain a + b (a − b) is modulated at 1st order only for W displacements in the xz (yz) plane (Fig. 3a). A displacement along the x (y) direction modulates at 1st order simultaneously the transfer integrals of chains a and a + b (a − b). This is the polarisation of the W displacement wave found for the CDW modulated structure of the m = 10 member [15]. A displacement along the z direction modulates at 1st order the transfer integrals of both a ± b chains. More generally, displacements such as those along the x ± y directions, which possess components in all the planes of symmetry of the 3 kinds of chain, modulate simultaneously at 1st order the transfer integrals of all the chains. The structure factor analysis of the 1D-CDW pretransitional fluctuations of the m = 4, 6 and 7 members [8,16] as well as the structural refinement of the CDW modulated structure of the m = 4 member ([17], see also Fig. 43 in Ref. [7]), show that such a general displacement of the W atoms occurs in the low m members. 2.4. The specificity of the chains The finding in the low m members of a displacement wave of the W atoms with a polarization not directed along a symmetry direction of the segment does not mean that the 3 types of chain are simultaneously modulated at the same Peierls transition. Each Peierls transition selects a critical wave vector qc at which there is simultaneous nesting of two FS out of three and thus two chains modulated with this wave vector. The two chains involved in the upper CDW transition are those for which the reduced electron–phonon coupling constant λep is the strongest. Thus in order to interpret
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the sequence of CDW transitions of the low m members within the mean field scenario one should consider the qc dependence of the electron–phonon coupling constant and particularly its variation with the nature (i) of the chain since the chains a and a ±b are not identical. The electron–phonon coupling constant and the 1D density of states change with the parameters of the electron gas as [13b]: gi (qc ) ∝ (dti /du) sin(kFi d ),
(9)
and Ni (EF ) = |h¯ πviF |−1 ,
(10)
within the tight binding approximation: h¯ viF = −2d ti × sin(kFi d ). Since the frequency of the bare phonon mode is more likely the frequency of local ferroelectric displacements of W atoms inside the segment of m octahedra, Ω(qc ), it is not expected to vary strongly with the reciprocal wave vector 2kF . The reduced electron–phonon coupling of the chain i varies as: λep (2kFi ) ∝ (d ln ti /du)2 ti sin(kFi d ).
(11)
The reduced electron–phonon coupling is thus a function of the bandwidth (4ti ) of the band filling (proportional to kFi ) and of the direction of displacement of the W atoms with respect to the plane of symmetry of the chain i (this latter quantity fixing the value of the derivative d ln ti /du). For a given chain λep will change with m, especially if the direction of displacement of the W atoms varies with m. For a given member, λep will also change when the band filling increases upon doping. Both structural measurements and band structure calculations show that when m increases the band filling increases for the chains a ± b and decreases for the chain a [8]. This leads to an increase of the electron–phonon coupling on the chains a ± b and to a decrease of this coupling on the chain a. If one assumes that the others quantities do not vary, this variation provides a simple interpretation for the reversal in the sequence of CDW phase transitions between the m = 4 and m = 6 members [8]. Indeed in the m = 4 member the 80 K upper CDW transition occurs with the qc2 critical wave vector (defined Fig. 2a) which connects the FS associated to chains a and a + b, while in the m = 6 member the upper CDW transition occurs at 120 K with the qc1 critical wave vector which connects the chains a ± b. The decreasing role of the a chains when m increases is corroborated by the measurements of the anisotropy of the thermopower [18]. The thermoelectric measurements show that a gap opens in the electronic structure of the chains a at the 80 K upper CDW transition of the m = 4 member while it occurs only at 62 K for the second CDW transition of the m = 6 member. In addition the chains a of the m = 8 members do not apparently participate at the CDW transitions [19].
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3. The Ax (PO2 )4 (WO3 )2m bronzes The A = K, Na ternary phase diagram of the phosphate tungsten bronzes with m = 4 and m = 6 has been recently determined [20]. In the m = 6 member with A = Na and x 1 the MPTBp structure of the undoped bronze, with two WO3 layers per unit cell related by the 21 screw axis symmetry along c, is kept [20]. The Na atoms are located in the pentagonal tunnels, delimited by the monophosphate groups PO4 and the WO3 octahedra, which run along the a direction. In the more strongly K doped m = 4 member (0.75 x 2) a new structural series of bronzes, called MPTBh’s, is obtained. In their structure there is stacking of identical WO3 layers along c [21]. The monophosphate groups PO4 as well as the WO3 octahedra thus delimit hexagonal tunnels which are filled by the K atoms. The initial purpose of the study of these bronzes was to determine the evolution of the CDW transitions observed in the pure x = 0 bronzes when doping with alcaline atoms provides x/2 more electrons per segment of m octahedra. In addition as structural studies show that the alcaline atoms are generally disordered in the tunnels [20–22], these bronzes offer the opportunity to study the influence of a disorder external to the quasi-1D chains on the CDW transitions. In the m = 4 MPTBh’s and the m = 6 MPTBp’s the alcaline disorder appears to have a relatively weak effect on the electronic properties [20,22–24] since no weak localization effects have been observed at low temperature. This differs to the disorder due to stacking faults in the sequence of WO3 layers in the large m MPTBp’s [25]. Below, we shall consider two aspects of the (T , x) phase diagram of these bronzes which can be simply explained in the framework of the theory of the Peierls transition of quasi1D systems. 3.1. Pair-breaking effects in Nax P4 W12 O44 , for x < 1 The undoped m = 6 member undergoes 3 successives CDW transitions, which stabilizes at 120 K the qc1 = (0.385(5), 0, 0) modulation, then at 62 K the qc2 = (0.310(5), 0.295(5), 0) modulation and finally at about 30 K the qc3 = (0.29(2), 0.11(2), ?) modulation [8]. We have undergone a structural investigation of these CDW transitions in the case of weakly Na doped bronzes with x = 0.1 and x = 0.84 when the structure remains that of the MPTBp’s. The samples used are those whose resistivity measurements are presented Fig. 4 in Ref. [20]. Fig. 4 shows a X-ray pattern taken from the x = 0.1 Na bronze at 25 K. It exhibits only the qc1 satellite reflections of the pure bronze (with this doping the variation of the wave vector is small and its difference with that of the undoped bronze is not detectable with our experimental set up). Reminiscence of the two others CDW modulations is observed under the form of very broad diffuse spots centered around the qc2 and qc3 reciprocal positions. The qc1 satellite reflections have disappeared above 60 K. The two others
Fig. 4. X-ray diffraction pattern from Na0.1 P4 W12 O44 taken at 25 K. The white arrows point towards the qc1 satellites reflections and the black arrows towards the qc2 and qc3 diffuse spots.
CDW transitions exhibited by the m = 6 pure bronze are not detected in the temperature range of our measurement (above 25 K). This is in perfect agreement with the resistivity measurement [20] showing only a bump anomaly starting at 55 K. The X-ray investigation of the x = 0.84 Na bronze down to 25 K does not any more reveal the presence of the qc1 satellite reflections. This is also in agreement with the conductivity measurements which do not show any indication of a CDW transition down to 4 K. The drastic decrease of the upper CDW transition of the m = 6 member with Na doping can be explained if one considers the effect of disorder to break the electron– hole pairs forming the Peierls condensate. More precisely the electrons are scattered by the impurity potential. Their diffusion on localized charged impurity, such as the Na+ , can achieve momentum transfers as large as 2kF (‘backward’ scattering process) allowing to break the pairs formed by the |kF + δq/2 and |−kF + δq/2 coupled electronic states. Such a scattering process, giving a life time τ at the pairs, leads to a decrease of the Peierls critical temperature Tc . Its shift with respect to its value in absence of impurity, Tc0 , depends on τ by the expression [13,26]: Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2(1 + h¯ /2πkB Tc τ ) . (12) If one introduces the mean free path l = vF τ , the expression (12) thus becomes: √ Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2 1 + ξth (Tc )/ 2 l , (13) in which the thermal coherence length ξth (Tc ) is given by the expression (4). If there is a strong scattering by the Na+ potential, the mean free path will be of the order of the average distance between Na atoms. This distance is 2a/x in the tunnel direction a which is tilted by about 40◦ with respect to the a ± b chains √ which are involved in the qc1 transition. Thus with l ∼ ( 2 )a/x, the expression (13) becomes: Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2 1 + xξth (Tc )/2a ≈ −π 2 xξth (Tc )/8a.
(14)
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The right member of the expression (14) holds for xξth (Tc )/4a small. From it, one obtains ξth (Tc ) ≈ 34 Å, with a = 5.3 Å, Tc0 = 120 K (in x = 0) and Tc ≈ 55 K (in x = 0.1). This value agrees with ξth = 33 Å calculated at 55 K, using the 1/T dependence of the expression (4), from the RT ab-initio value of ξth for the a ± b chains of the m = 4 member [6]. ξth (Tc ) is also close to ξth ≈ 45 ± 10 Å estimated at 55 K, with the same 1/T dependence, from the experimental determination of ξth (300 K) = 9 ± 2 Å and ξth (200 K) = 11 ± 2 Å for the a ± b chains of the m = 6 member. With such ξth (T ), the expression (14) predicts a Tc less than 1 K for the x = 0.84 sample! Concerning the qc2 transition, which occurs at 62 K in the x = 0 bronze, the application of formulae (14) for chains a gives a Tc less than 10 K for the x = 0.1 bronze if one uses a ξth (T ) extrapolated from the experimental value of ξth (200 K) = 12 ± 2 Å in the m = 6 member. This critical temperature is below the temperature range experimentally investigated. Recently, similar pair breaking effects have been invoked to explain the decrease of the Peierls transition in alloys of the 1D organic conductor Per2 [M(mnt)2] [27].
393
(a)
3.2. Band filling effects in Kx P4 W8 O32 , for 1 < x < 2 The m = 4 series of K doped MPTBh’s exhibits around T0 = 150 K, for 1 < x < 2, an anomaly in its transport properties which recall those observed in the m = 4 undoped MPTBp’bronze [20,22,24]. Below this temperature a gap develops in the magnetic susceptibility, as expected for a Peierls transition [20]. However the critical temperature of this transition is two times higher than the critical temperature of the upper CDW transition of the undoped m = 4 bronze (150 K instead of 80 K). Thus in order to understand the origin of such a phase transition we have undertaken a structural study of the x = 1.05(5), 1.30(9) and 1.94(9) samples (the values of x have been obtained by microprobe analysis). The X-ray diffuse scattering investigation shows unexpected features compared to those previously reported in the low m MPTBp’s. Fig. 5a shows a X-ray pattern taken at room temperature in the x = 1.05 bronze. It reveals two types of diffuse scattering: • The first type of scattering is formed of diffuse lines perpendicular to the a ± b directions. Such lines, which have been already observed in the low m members of the MPTBp’s, are the fingerprint of the CDW pretransitional fluctuations located on the a ± b chains. One set of lines is roughly located at half the reciprocal distance between a∗ ± b∗ layers of main Bragg reflections and another set is located in these layers. Such diffuse lines are also observed in the x = 1.3 bronze, but not in the x = 1.94 bronze. • The second type of scattering is formed of diffuse segment perpendicular to the bh or ap direction (the
(b) Fig. 5. X-ray diffraction pattern from K1.05 P4 W8 O32 taken at room temperature (a) and 25 K (b). In (a) the black arrows point towards the diffuse lines perpendicular to the a ± b directions and the white arrows towards the diffuse segments perpendicular to the bh or ap directions. In (b) the dotted white arrows point towards the q0 satellite reflections.
index ‘h’ corresponds to the axis frame of the structure of the MPTBh’s as published in Ref. [21] and the index ‘p’ to the frame of the MPTBp’s used throughout this paper). This diffuse scattering already found in the Cs1−x P8 W8 O40 solid solution [28] corresponds to a local order between K atoms inside the tunnel where they are located, but not correlated between neighboring tunnels. From the position of the diffuse lines relatively to the neighboring Bragg layers one can deduce the K concentration (x). From the inverse of the half width at half maximum of the diffuse scattering (ξ ) one deduces the extent of the order in tunnel direction: L ∼ πξ . Such quantities are reported in Table 1 for the samples investigated. Note that the x value obtained from the X-ray diffuse scattering investigation agrees, within experimental errors, with the concentration determined by the microprobe analysis. This diffuse scattering is not observed in the x = 1.94 bronze, indicating that the K atoms are ordered. Fig. 5b shows an X-ray pattern of the x = 1.05 bronze taken at 25 K. The two types of diffuse scattering previously quoted are still visible, although the intensity of the
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Table 1 Summary of the structural results for three Kx P4 W8 O32 compounds studied. The tables gives also the critical temperatures Tc (resistivity measurements) and T0 (X-ray data) x (microprobe) x (X-ray) ξ (Å) at 300 K Tc (K) T0 (K) Isat /IBragg
1.05(5) 1.3(2) 12(3) 145 150(5) 5 × 10−3
1.30(9) 1.5(2) 15(3) 165 160(5) 10−3
1.94(9) No diffuse lines ? 80(10) 10−4
Fig. 6. Phase diagram T0 (x) of the Kx P4 W8 O32 determined by electrical (plain circles) and structural (empty circles) measurements. The continuous line is given by the expression (19) with ξth (T0 ) = 16 Å.
diffuse scattering perpendicular to the a ± b directions has decreased. This X-ray pattern however reveals extra Bragg reflections located at the commensurate wave vector q0 = 1/2a∗h = 1/2b∗p . These satellite reflections disappear upon heating at T0 , the same temperature at which the electronic properties exhibit anomalies. However, the resistivity of the x = 1.94 bronze does not exhibit an anomaly at the structural transition which occurs at 80 K (Table 1). In all the samples investigated the reflections are always observed at the same simple commensurate wave vector with an intensity which decreases when x increases (Table 1). However the temperature of transition is a non-monotonous function of x. Fig. 6 shows that T0 (x) exhibits a maximum at about 170 K for x ≈ 4/3. From the structural point of view the transition does not appear to be due to the usual CDW instability observed in the pure MPTBp’s because, firstly, the satellites are not located on the a∗ ± b∗ diffuse lines, which means that the structure factor of the lattice distortion is different to that of the CDW fluctuations on the a ± b chains. Secondly, the a∗ ± b∗ diffuse lines remains visible at low temperature while the q0 reflections are well-developed. Thirdly, it is expected for a CDW driven transition that the 2kF modulation wave vector should vary with the band filling which is a function of x; however q0 does not vary with x. Fourthly, the structural transition is observed in the x = 1.94 bronze which does not exhibit any anomaly at T0 in its electronic properties. However for the bronzes with lower values of x the structural
transition is accompanied by electronic anomalies recalling that observed for a Peierls transition. Let us now consider the band filling with more detail. The K atoms provide x/2 additional electrons to each segment of m octahedra. The relationship (1) thus becomes in the MPTBh’s: 2kFa + 2 × 2kFd = 1 + x/4.
(15)
The observation of 1D CDW-like fluctuations under the form of a broad diffuse line located at the 1/2(a∗ ± b∗ ) reduced wave vector means that 2kFd ∼ 1/2d ∗ for x = 1.05 and 1.30. This value can be obtained if starting from the band filling of the undoped m = 4 bronze (2kFa = 2kFd = 1/3 [6]) one assumes that the electrons provided by the K atoms fill only the bands associated to the chains a ± b. Thus with 2kFd = (1/3 + x/8)d ∗ , one gets 2kFd = 1/2d ∗ for x = 4/3 and 2kFd = 11/24d ∗ ≈ 0.46d ∗ for x = 1 (in that latter case the X-ray pattern shown Fig. 5 should present two diffuse lines at 0.46d ∗ and 0.54d ∗ , difficult to distinguish from a single broad diffuse line at 0.5d ∗ ). This situation is not that predicted by the tight-binding band calculation [5b]. To reconcile all these features we propose the existence of a coupling between a purely structural transition and the CDW instability in such a way that the MPTBh’s undergo a structural transition which provides an external periodicity q0 = 1/2a∗h = 1/2b∗p to the quasi-1D electron gas. Such a periodicity, close to 2kFd , will open a gap in the band structure of the chains a ± b, which induces a Peierls transition on the electronic subsystem. In our scenario the maximum value of T0 corresponds at a resonance between the band filling of the chains a ± b and the commensurate value of the modulation provided by the additional structural transition. In this case there is coincidence between the external periodicity and the qc4 crossing point, defined Fig. 3b, leading to the nesting of the FS associated to the chains a ± b. The x = 1.05 and x = 1.30 bronzes are close to the resonance condition, but not the x = 1.94 bronze (in addition its electronic properties do not exhibit any anomaly at T0 ). A recent structural refinement of the K doped MPTBh’s [29] shows that the q0 structural transition consists in an oscillation of the PO4 tetrahedron accompanied by a displacement of WO6 octahedra, possibly due to the shift of the K atoms. In our interpretation the mechanism of the coupled structural–electronic transition of the MPTBh’s is similar to that outlined in Section 2.2 for the Peierls transition with, in the expression (6), the potential provided by the structural transition Vst (q0 ) replacing the potential due to the electron–phonon coupling Vep (qc ). This transition however is not achieved by the divergence of 1D CDW fluctuations. However a gap is opened in the band structure because of the enhanced response of the Lindhard function at the periodicity of the external potential Vst (q0 ). As Vst (q0 ) provides the d ∗ /2 periodicity in chain direction, which can be different of 2kFd , the q dependent Lindhard function χe (T , q), given by the expression (2), must be used, with δq = q0 − 2kFd = (4/3 − x)d ∗ /8. The relationship (6)
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(a)
(b)
395
(c)
Fig. 7. Schematical representation of the location of the gap due to the external periodicity q0 in the 1D band structure of the a ± b chains of the K doped MPTBh’s for various band fillings corresponding to: (a) x < 4/3, (b) x = 4/3 and (c) x > 4/3.
gives the critical temperature T0 of the coupled structural– electronic transition of the MPTBh’s. This temperature T0 (x) can be expressed, with respect to its maximum value for x = 4/3, under the form: Ln T0 (x)/T0(4/3) = Ψ (1/2) − 1/2 Ψ 1/2 + i h¯ vF δq/4πkB T0 (x) + Ψ 1/2 − i h¯ vF δq/4πkB T0 (x) . (16) When δq increases the divergence of the Lindhard function decreases and T0 thus diminishes. If one introduces the thermal coherence length ξth (T0 ) previously defined, the expression (16) becomes: Ln T0 (x)/T0(4/3) √ = Ψ (1/2) − 1/2 Ψ 1/2 + iδqξth(T0 )/2 2 √ + Ψ 1/2 − iδqξth(T0 )/2 2 . (17) The shift of T0 can be also viewed as due to the difference of chemical potential (δµ = h¯ vF δq/2) between the Fermi energy and the electronic energy at the wave vector q0 at which the gap is opened. If δqξth (T0 ) is a small quantity, the expression (17) becomes: 2 Ln T0 (x)/T0(4/3) ≈ − δqξth (T0 ) . (18) This leads to a parabolic dependence in x: 2 Ln T0 (x)/T0(4/3) ≈ −(4/3 − x)2 ξth (T0 )d ∗ /8 ,
(19)
which is very well followed by the data of Fig. 6. Furthermore from this dependence one gets ξth (T0 ) ≈ 16 Å, which value is in perfect agreement with ξth estimated at 150 K from the experimental measurement of the CDW thermal coherence length in the m = 4 undoped member: (ξth (300 K) = 8 ± 2 Å for the chains a ± b [6]). At T0 a gap is opened in the 1D band structure. It kills the divergence of the Lindhard function and thus prevents the onset of the conventional CDW instability driven by the electron–phonon coupling, which generally occurs at
lower temperature (Tc = T0 /2) in the low m members of the MPTBp’s. This large value of T0 can be explained either by the replacement of the electron–phonon potential Vep (qc ) by a structural one Vst (q0 ) which can be stronger than Vep (qc ) or/and by the commensurability two of the q0 modulation. It is well-known [30] that for a two-fold commensurability the CDW ground state is particularly stable, and that an important shift δµ of chemical potential is required to destabilize it. In this scenario, Vst (q0 ) opens a gap near the Fermi level in the band structure of chains a ± b, and leaves unaffected the band structure of chains a.1 This differs of the undoped m = 4 MPTBp’s where after the successive CDW transitions all the FS’s are affected. As a consequence, large area of the Brillouin zone will remain unaffected by the structural transition in the K doped m = 4 MPTBh’s. The magnetoresistance effects will thus be much weaker at low temperature in them (14%) than in the pure m = 4 MPTBp’s (250%) [22]. Fig. 7 shows schematically the effect of a gap opening at q0 in the 1D band structure of the chains a ± b. In our simplified model the gap opens just at the Fermi level for x = 4/3 (Fig. 7b). For x < 4/3, it opens in the unoccupied part of the band, forming hole pockets (Fig. 7a), and for x > 4/3 it opens in the occupied part of the band, forming electron pockets (Fig. 7c). We believe that the formation of these hole (electron) pockets explains the positive (negative) jump of thermopower which starts at T0 in the bronzes with x values smaller (higher) than 4/3 [24]. In the m = 4 MPTBh’s the K remains disordered (except for the x ≈ 2 bronze). One should expect the occurrence 1 The short range order of the K atoms gives rise to a local potential whose main Fourier component in tunnel direction is located at x/2a∗p , within one reciprocal wave vector. For x = 4/3 its components are at 2/3a∗p and 1/3a∗p . In our scenario this latter wave vector corresponds to 2kFa . Thus the local potential of the K atoms can open a pseudo-gap at the Fermi level in the band structure of chains a, which effect diminishes when x deviates from 4/3.
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of a pair breaking phenomena which could, as observed in the Nax P4 W12 O44 bronze (see the previous section), destabilize the Peierls transition. This effects requires the presence of a 2kF backscattering process coming from the K+ potential. However the K atoms are enough ordered in tunnel direction to provide only Fourier components of the scattering potential which are spread on 2ξ−1 around a mean value of x/2b∗h = x/2a∗p . As this range of wave vectors does not overlap the critical wave vector of the structural transition, q0 = 1/2a∗h = 1/2b∗p , the K disorder will not affect the Peierls-like transition of the MPTBh’s.
[3] [4] [5] [6] [7] [8] [9] [10]
4. Conclusion This paper is written at the occasion of the 60th birthday of Professor Martha Greenblatt. She points out to us the interesting features exhibited by the phosphate tungsten bronzes just after her report of bump anomalies in the resistivity of the MPTBp’s [31]. This was the start of a collaboration of more than 10 years which has begun with the finding of CDW instabilities in these bronzes [32]. Since then, the scenario of a Peierls instability taking place, through a hidden nesting mechanism of quasi-1D bands associated to chains of WO3 octahedra running in the a and a ± b directions of the layers, has been confirmed for the 4 m 6 regular members of the series [8], as well as in a recent study of the CDW transition of the alternate 4/6 member [33]. We have reported here additional evidences of the relevance of the Peierls-like mechanism in these monophosphate tungsten bronzes through the quantitative analysis of the phase diagrams of ‘doped’ Nax P4 W12 O44 (x < 1) and Kx P4 W8 O32 (1 < x < 2) bronzes. We have also shown throughout this paper that a theoretical framework based on the physics of 1D conductors allows to simply understand the CDW properties exhibited by this family of 2D bronzes.
[11]
Acknowledgements
[24]
We are very grateful to our colleagues and collaborators for their contribution to the present work. In particular we would like to thank D. Groult for the crystal growth, S. Drouard, J. Dumas and C. Schlenker for the transport measurements and S. Ravy for helpful discussions. References [1] M. Greenblatt (Ed.), Oxide bronzes, Int. J. Mod. Phys. B 7 (1993) 3937–4164. [2] C. Schlenker, J. Dumas, M. Greenblatt, S. van Smaalen (Eds.), Physics and Chemistry of Low Dimensional Inorganic Conductors, NATO
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Peierls transition in two-dimensional metallic monophosphate tungsten bronzes P. Foury-Leylekian ∗ , J.-P. Pouget Laboratoire de Physique des Solides, CNRS-UMR 8502, Université de Paris Sud, 91405 Orsay cedex, France Dedicated to Martha Greenblatt
Abstract The two-dimensional metallic bronzes made of ReO3 -type layers of MoO6 or WO6 octahedra present quasi-one-dimensional (1D) electronic structures along three directions of preferential overlap of the t2g transition metal orbitals. They exhibit a Peierls instability towards the formation of charge density waves (CDW) at the 2kF critical wave vector allowing to nest simultaneously the Fermi surfaces associated to two quasi-1D band structures out of three. The Peierls transition is achieved through a periodic lattice distortion (PLD), that we analyse, in the present work, for the monophosphate tungsten bronzes. The Peierls critical temperature decreases in presence of disorder, which breaks the electron–hole pairs forming the CDW condensate, and in presence of misfit between the PLD wave vector and the 2kF nesting wave vector. The pair breaking effect accounts for the drop of the Peierls transition in the Nax P4 W12 O44 (x < 1) bronzes while the misfit effect, associated to a variation of the band filling, explains quantitatively the phase diagram of the Kx P4 W8 O32 (1 < x < 2) bronzes. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction The term bronze is attributed to a large variety of ternary transition metal (M) oxides of general formula Ax (MOn )y where A is a monovalent cation or an elemental group such as PO4 . The species A provide electrons to the d–t2g transition metal levels, in the crystalline field of the MO6 octahedra, which otherwise are unoccupied in the parent oxide MOn . In presence of this charge transfer these materials become conductors with a metallic luster recalling that of the bronze. The bronzes are known since the beginning of the 19th century, but it’s only since twenty years that these materials were recognized as a new class of low-dimensional metals [1,2]. Based on MO6 octahedral units, the structure of the bronzes can be one-, two- or three-dimensional (1D, 2D or 3D) with respect to the chemical bond network, but are 1D or 2D with respect to the bonding of a given set of t2g orbitals, because of the directional character of these orbitals. As this last dimensionality is lower than the structural dimensionality it appears that the bronzes possesses a ‘hidden’ low dimensionality from the electronic point of view. Although it had been noticed in the early * Correspondence and reprints.
E-mail address: [email protected] (P. Foury-Leylekian).
70’s that the Fermi surface (FS) of the metallic (3D cubic network) cation deficient perovskites, such as ReO3 and Nax WO3 , has a cylindrical shape and that the conduction bands of the conventional perovskites, such as BaTiO3 and SrTiO3 , possess a 2D anisotropy [3], this concept was more precisely expressed when extended Hückel semi-empirical band structure calculations performed in the layered (2D network) Mo and W bronzes [4,5] were able to show that their conduction band structure exhibits a subtle 1D anisotropy. These 1D features were recently confirmed by ab-initio band structure calculations performed in the 2D monophosphate tungsten bronze P4 W8 O32 [6]. With an electronic structure of reduced dimensionality, the FS of these metallic bronzes is expected to be enough anisotropic to possess good nesting properties. It is now well-known that such nesting properties drive density wave ground states (charge density wave (CDW) or spin density wave (SDW) if there is strong enough electron–phonon or electron–electron interactions, respectively) [2]. Such instabilities could eventually compete with the superconductivity, whose critical temperature exceeds 5 K in Ax WO3 and WO3−x . CDW ground states are effectively achieved in several 2D Mo and W bronzes [1,2]. This was first shown in the Mo purple bronze Ax Mo6 O17 (A = K, Rb, Tl; x ∼ 1) whose structure is made of ReO3 -type corner sharing MoO6 octahedra
1293-2558/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 1 ) 0 1 2 6 6 - 3
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slabs, separated by the monovalent atoms A and MoO4 tetrahedra. This is also the case of monophosphate W bronzes with pentagonal tunnels (MPTBp’s), (PO2 )4 (WO3 )2m (m 4), or hexagonal tunnels (MPTBh’s), Ax (PO2 )4 (WO3 )2m (A = K, Na), whose structure is also made of ReO3 -type slabs of WO6 octahedra separated by monophosphate PO4 tetrahedral groups [7]. The oxide Mo4 O11 , which structure resembles that of the m = 6 member of the MPTBp’s (γ phase) or of the MPTBh’s (η phase), with MoO4 tetrahedra replacing the PO4 groups, belongs also to these classes of 2D metals. All these 2D bronzes exhibit one or several successive CDW–Peierls transitions which have been associated to the nesting of several 1D parts of their FS [8]. The CDW ground state corresponds to a 2kF modulation of the electronic density (kF is the Fermi wave vector of the 1DFS) accompanied by a 2kF periodic lattice distortion (PLD). This latter modulation is detected in X-ray diffraction experiments [9]. In low-dimensional (generally quasi-1D) metals, the cost of elastic energy due to the PLD is overcome by the gain of electronic energy due to the opening of a gap at the Fermi level. However as the nesting mechanism of the 2D bronzes does not remove all the electronic states at the Fermi energy, the Peierls transition leads generally to a semimetallic state where cylindrical electron and hole pockets of small section remains. This 2D semi-metallic state presents interesting quantum phenomena such as those observed in η-Mo4 O11 . In this oxide a bulk quantum Hall effect and, in the quantum limit, the opening of a true Peierls gap, associated with a magnetic field induced electron- and hole-band inversion, are observed. 2. The CDW instability in 2D bronzes 2.1. The hidden 1D electronic structure Let us first discuss the cubic perovskite ‘prototype’ structure. For each type of (1 0 0) cubic plane there is a preferential overlap of one set of d–t2g orbitals of the transition metal; i.e., the dyz orbitals for the (1 0 0) plane [3]. As there is nearly no overlap between parallel planes, the band structure is 2D and its FS is a ‘cylinder’ perpendicular to the planes. The total band structure is thus the superimposition of the individual band structures of the 3 sets of (1 0 0) planes related by cubic symmetry and the global FS is the sum of 3 ‘cylinders’ running along the 3 cubic directions, as found in the calculation of the FS of ReO3 and Nax WO3 . The conducting layer of purple bronzes is obtained by slicing the ReO3 -like structure between two planes perpendicular to the 3-fold axis of the cubic perovskite structure. The cut leaves a trigonal layer in which run 3 sets of parallel infinite chains along the hexagonal-like intralayer directions. In each of these directions there is a preferential overlap of one set of the d–t2g Mo orbitals [4]. The band structure is thus 1D and the FS associated to each 1D band consists in a pair of parallel planes perpendicular to the chain direction and separated by 2kF . The conduction band
structure is the superimposition of three 1D band structures and the global FS is the sum of 3 couples of planar sheets related by trigonal symmetry, each having the same 2kF [4]. In the purple bronze each 1D band accommodates one electron given by the monovalent atom A and the MoO4 tetrahedra, leading to 2kF = 1/2 chain reciprocal unit (the factor 2 missing accounts for the spin degeneracy in the band filling). This FS topology has been recently confirmed by angle-resolved photo-emission spectroscopy (ARPES) measurements in the Na bronze [10]. Similarly the conducting layer of the MPTB’s results from the slicing of the ReO3 -like structure between two parallel (1¯ 1 2) planes of the cubic perovskite structure. The cut leaves a rectangular layer where the a and b directions are, respectively, perpendicular to the [1 1 0] and [1 1¯ 1] perovskite directions. Along each of the a, a + b and a − b directions run infinite parallel chains made of segments of m WO6 octahedra tilted with respect to the chain direction. There is a preferential overlap of one combination of d–t2g W orbitals for each kind of chains [5]. Similarly to the purple bronze, individual planar FS can be associated to the chains a and a ± b (see Fig. 1). However as the a and a ± b chains are not related by symmetry, the 2kF wave vector of their 1D band structure differs (below we shall note them 2kFa and 2kFd , respectively). The global FS is the sum of these three 1D-FS. In the MPTBp’s the total FS accommodates 2 electrons given by the PO4 groups to each segment of m WO6 octahedra. Thus, if 2kF is expressed in its corresponding chain reciprocal unit, one gets the simple relationship: 2kFa + 2 × 2kFd = 1.
(1)
In the m = 4 member, the ab-initio calculation [6] gives 2kFa ≈ 2kFd ≈ 1/3. ARPES studies of the Na purple bronze [11] and of MPTBp’s with different m values [12] show the occurrence of a well-defined Fermi edge separating the occupied electronic states from the unoccupied ones. This means that the concept of FS is validated and that the nesting mechanism, which will be introduced more quantitatively in the next section, can be used to describe the CDW instability of these bronzes. 2.2. The CDW FS-nesting instability The wave vectors q which achieve the best nesting of the global FS give rise to a maxima of the electron–hole (or Lindhard) response function χe (T , q). This response function has been calculated for the total FS in the MPTBp with m = 4 [6]. It exhibits, as schematically shown Fig. 2a in the (a ∗ , b∗ ) reciprocal plane, maxima of intensity along lines perpendicular to the a and a ± b chain direction. In the Brillouin zone these lines are approximately located at the 2kFa and 2kFd wave vectors at which there is nesting of the ideal 1D-FS’s associated to the chains a and a ± b, respectively.
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(a)
(a)
(b) (b) Fig. 1. Decomposition of the Fermi surface (FS) of the MPTBp’s into (a) a 1D-FS associated to chains running along the a directions and (b) two 1D-FS associated to chains running along the a + b and a − b directions.
For a 1D gas of non-interacting free electrons and a wave vector q close to the 2kF value of the optimal nesting, q = 2kF + δq, if one neglects matrix elements between Bloch functions in the electron–hole response function, one gets [13]: χe (T , q) = N(EF ) ln(Ec /kB T ) + Ψ (1/2) − 1/2 Ψ (1/2 + i h¯ vF δq/4πkB T ) + Ψ (1/2 − i h¯ vF δq/4πkBT ) . (2) In this expression, valid for a linear dispersion of the electron in the vicinity of the Fermi level, Ψ (x) is the digamma function which will be used later. N(EF ) is the density of states per spin at the Fermi level, Ec is a cutoff energy of the order of the Fermi energy (EF ) and vF is the Fermi velocity. For h¯ vF δq πkB T , the development of Ψ (x) gives: 2 χe (T , q) = N(EF ) ln(Ec /kB T ) − ξth (T )δq , (3)
Fig. 2. (a) Location in the (a∗ , b∗ ) section of the Brillouin zone of the 2kF CDW critical fluctuations associated to the nesting of the 1D-FS shown in Fig. 1. The crossing points qc at which the common nesting of two FS out of three occurs in the MPTBp’s is indicated: qc1 and qc4 for the chains a + b and a − b, qc2 for the chains a and a − b and qc3 for the chains a and a + b. The qc4 crossing point, when equal to b∗ /2 is relevant for the MPTBh’s, as shown in (b).
where:
√ ξth (T ) = hv ¯ F / 2 πkB T ,
(4)
is the thermal coherence length. With the electronic structure described in the last section the total electron–hole response function is the sum of the individual electron–hole response function of each 1D electron gas. The divergence of the total response function will be stronger at the intersection point qc (defined Fig. 2a) of the 2kF lines of extrema of intensity of χe (T , q). Here the electron–hole response functions of the two intersecting 2kF 1D-response functions (i = 1, 2) add: χei (T , qc ). χc (T , qc ) = (5) i
At qc the response function exhibits a
ln(Ec /kB T ) thermal divergence, which is, by its prefactor i Ni (EF ), stronger than the divergence on the isolated lines. As a consequence
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the electronic system will tend to stabilize a density wave at the qc wave vector, because the opening of a Peierls gap on two couples of 1D-FS, at the expense of a single structural distortion, will lead to a maximum gain of electronic energy. In the mean-field approximation the Peierls transition will occur at a Tc given by [13]: Vep (qc )χc (Tc , qc ) = 1,
(6)
where Vep (qc ) = 2g 2 (qc )/h¯ Ω(qc ). In this expression g(qc ) is the electron–phonon coupling constant (assumed here, for simplicity, to be the same for the two bands connected by qc ) and Ω(qc ) is the frequency of the bare phonon mode which will be critical by its coupling with the electronic degrees of freedom. The expression (6) leads to: kB Tc = Ec exp −1/2λep (qc ) , (7) where λep (qc ) = Vep (qc )N(EF ) is the reduced electron– phonon coupling (in this expression it is also assumed, for simplicity, that the two bands have the same density of states). Assuming that λep remains constant, Tc given by (7) is sizeably enhanced with respect to Tp of a standard Peierls transition achieving the nesting of a single pair of 1D-FS: kB Tp = Ec exp(−1/λep ).
(8)
For example, the formulae (7) leads to λep ≈ 0.16 from the upper CDW critical temperature Tc = 120 K of the m = 6 member of the MPTBp’s and the ab-initio calculated Fermi energy EF ≈ 0.2 eV for the a ± b chains [6]. With the same values of λep and EF the formulae (8) leads to a standard Peierls transition of only Tp = 5 K! The CDW ground state is achieved by the nesting of several sets of quasi-1D-FS. The nesting of one set of quasi1D-FS mixes the |kF + δq/2 and |−kF + δq/2 conduction electron wave functions. Their bonding (antibonding) combination enters in the wave functions of the electronic states below (above) the Peierls gap. With such wave functions there is an inversion of the phase between the CDW modulations built on the occupied and the unoccupied electronic states. This contrast inversion has been observed by scanning tunneling microscopy imaging of the CDW in the K purple bronze [14]. 2.3. Coupling to the lattice modes In the last section we have only considered the electronic couterpart of the Peierls transition. However in order to achieve the 2kF PLD which potential Vep (q) opens a gap at the Fermi level in the quasi-1D band structure, the electronic degrees of freedom must be coupled, at 1st order in the lattice displacements [13], to the structural degrees of freedom. The critical ones generally correspond to low frequency, Ω(q), phonon modes for which the reduced electron–phonon coupling constant, λep , is the strongest. This coupling requires that the associated lattice displacements should change at 1st order the intrachain
(a)
(b) Fig. 3. Schematic representation of the chains of m = 4 WO3 octahedra running in the a + b (a) and a (b) directions (the a − b chain is identical at the a + b chain shown in (a)). The x, y, z setting of the octahedron is indicated. In (a) a ‘ferroelectric’ distortion of the segment is indicated.
electronic quantities such as the transfer integral (t) or the site energy (ε). This coupling is achieved through the already introduced electron–phonon coupling constant g(q). Here we shall consider with more details the lattice degrees of freedom which could be involved in the structural–Peierls instability of the MPTBp’s. X-ray diffuse scattering investigations reveal that there are preexisting low frequency phonon modes in the MPTBp’s. They correspond to a displacement of the W atoms away from the center of the WO6 octahedra. This displacement, correlated inside the segment of m octahedra, but uncorrelated between the neighboring segments, is of the ferroelectric type. The associated low frequency phonon modes give rise to an enhanced X-ray diffuse scattering consisting of broad trails along the 2a∗ ± 3b∗ reciprocal directions perpendicular to the segment direction. Consistently the 2kFa and 2kFd pretransitional structural fluctuations of the Peierls–CDW transitions of the low m members of the MPTBp’s are located inside these broad trails [8]. Let us consider now how a ferroelectric distortion of a segment of m octahedra can be coupled at 1st order to the electronic states of a chain of octahedra. This requires that the displacement, u, of the W atoms changes, at 1st order in u, the overlap integral, t, between the d–t2g levels on which the 1D band structure of the chain is built. Let us, for example, consider the a + b chain of m = 4 octahedra shown Fig. 3a: the segments are tilted with respect to the chain direction and there is an inversion center in the middle of each segment. With this chain geometry,
P. Foury-Leylekian, J.-P. Pouget / Solid State Sciences 4 (2002) 387–396
a displacement +u of all the W atoms of a segment leads to variation of the transfer integral t (u) of the d–t2g levels between neighboring segments which is different of that for the opposite displacement, −u, only if u is not perpendicular to the symmetry plane of the chain. For this polarisation of displacement, t is an odd function of u. The electron–phonon coupling constant previously introduced is proportional to its first order derivative: g ∝ dt/du [13b]. Note that such a displacement does not change at 1st order the molecular energy levels of the segment undergoing the distortion: because of the inversion symmetry of the segment the site energy, ε(u), is an even function of u. However this displacement changes at 1st order the site energy of the neighboring segment. This last process contributes also at the electron–phonon coupling constant. Now let us examine how a single ferroelectric displacement can modulate at 1st order the transfer integrals of two differently oriented chains in order to contribute at the electron–phonon coupling constants allowing the simultaneous development of a Peierls gap in the band structure of these two chains. Fig. 3b represents a chain a of the m = 4 member, and define the 3 elementary directions of displacement, x, y and z, for the W atom. The transfer integral of the chain a is only modulated at 1st order for W displacements in the xy plane of symmetry of the chain. The transfer integral of the chain a + b (a − b) is modulated at 1st order only for W displacements in the xz (yz) plane (Fig. 3a). A displacement along the x (y) direction modulates at 1st order simultaneously the transfer integrals of chains a and a + b (a − b). This is the polarisation of the W displacement wave found for the CDW modulated structure of the m = 10 member [15]. A displacement along the z direction modulates at 1st order the transfer integrals of both a ± b chains. More generally, displacements such as those along the x ± y directions, which possess components in all the planes of symmetry of the 3 kinds of chain, modulate simultaneously at 1st order the transfer integrals of all the chains. The structure factor analysis of the 1D-CDW pretransitional fluctuations of the m = 4, 6 and 7 members [8,16] as well as the structural refinement of the CDW modulated structure of the m = 4 member ([17], see also Fig. 43 in Ref. [7]), show that such a general displacement of the W atoms occurs in the low m members. 2.4. The specificity of the chains The finding in the low m members of a displacement wave of the W atoms with a polarization not directed along a symmetry direction of the segment does not mean that the 3 types of chain are simultaneously modulated at the same Peierls transition. Each Peierls transition selects a critical wave vector qc at which there is simultaneous nesting of two FS out of three and thus two chains modulated with this wave vector. The two chains involved in the upper CDW transition are those for which the reduced electron–phonon coupling constant λep is the strongest. Thus in order to interpret
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the sequence of CDW transitions of the low m members within the mean field scenario one should consider the qc dependence of the electron–phonon coupling constant and particularly its variation with the nature (i) of the chain since the chains a and a ±b are not identical. The electron–phonon coupling constant and the 1D density of states change with the parameters of the electron gas as [13b]: gi (qc ) ∝ (dti /du) sin(kFi d ),
(9)
and Ni (EF ) = |h¯ πviF |−1 ,
(10)
within the tight binding approximation: h¯ viF = −2d ti × sin(kFi d ). Since the frequency of the bare phonon mode is more likely the frequency of local ferroelectric displacements of W atoms inside the segment of m octahedra, Ω(qc ), it is not expected to vary strongly with the reciprocal wave vector 2kF . The reduced electron–phonon coupling of the chain i varies as: λep (2kFi ) ∝ (d ln ti /du)2 ti sin(kFi d ).
(11)
The reduced electron–phonon coupling is thus a function of the bandwidth (4ti ) of the band filling (proportional to kFi ) and of the direction of displacement of the W atoms with respect to the plane of symmetry of the chain i (this latter quantity fixing the value of the derivative d ln ti /du). For a given chain λep will change with m, especially if the direction of displacement of the W atoms varies with m. For a given member, λep will also change when the band filling increases upon doping. Both structural measurements and band structure calculations show that when m increases the band filling increases for the chains a ± b and decreases for the chain a [8]. This leads to an increase of the electron–phonon coupling on the chains a ± b and to a decrease of this coupling on the chain a. If one assumes that the others quantities do not vary, this variation provides a simple interpretation for the reversal in the sequence of CDW phase transitions between the m = 4 and m = 6 members [8]. Indeed in the m = 4 member the 80 K upper CDW transition occurs with the qc2 critical wave vector (defined Fig. 2a) which connects the FS associated to chains a and a + b, while in the m = 6 member the upper CDW transition occurs at 120 K with the qc1 critical wave vector which connects the chains a ± b. The decreasing role of the a chains when m increases is corroborated by the measurements of the anisotropy of the thermopower [18]. The thermoelectric measurements show that a gap opens in the electronic structure of the chains a at the 80 K upper CDW transition of the m = 4 member while it occurs only at 62 K for the second CDW transition of the m = 6 member. In addition the chains a of the m = 8 members do not apparently participate at the CDW transitions [19].
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3. The Ax (PO2 )4 (WO3 )2m bronzes The A = K, Na ternary phase diagram of the phosphate tungsten bronzes with m = 4 and m = 6 has been recently determined [20]. In the m = 6 member with A = Na and x 1 the MPTBp structure of the undoped bronze, with two WO3 layers per unit cell related by the 21 screw axis symmetry along c, is kept [20]. The Na atoms are located in the pentagonal tunnels, delimited by the monophosphate groups PO4 and the WO3 octahedra, which run along the a direction. In the more strongly K doped m = 4 member (0.75 x 2) a new structural series of bronzes, called MPTBh’s, is obtained. In their structure there is stacking of identical WO3 layers along c [21]. The monophosphate groups PO4 as well as the WO3 octahedra thus delimit hexagonal tunnels which are filled by the K atoms. The initial purpose of the study of these bronzes was to determine the evolution of the CDW transitions observed in the pure x = 0 bronzes when doping with alcaline atoms provides x/2 more electrons per segment of m octahedra. In addition as structural studies show that the alcaline atoms are generally disordered in the tunnels [20–22], these bronzes offer the opportunity to study the influence of a disorder external to the quasi-1D chains on the CDW transitions. In the m = 4 MPTBh’s and the m = 6 MPTBp’s the alcaline disorder appears to have a relatively weak effect on the electronic properties [20,22–24] since no weak localization effects have been observed at low temperature. This differs to the disorder due to stacking faults in the sequence of WO3 layers in the large m MPTBp’s [25]. Below, we shall consider two aspects of the (T , x) phase diagram of these bronzes which can be simply explained in the framework of the theory of the Peierls transition of quasi1D systems. 3.1. Pair-breaking effects in Nax P4 W12 O44 , for x < 1 The undoped m = 6 member undergoes 3 successives CDW transitions, which stabilizes at 120 K the qc1 = (0.385(5), 0, 0) modulation, then at 62 K the qc2 = (0.310(5), 0.295(5), 0) modulation and finally at about 30 K the qc3 = (0.29(2), 0.11(2), ?) modulation [8]. We have undergone a structural investigation of these CDW transitions in the case of weakly Na doped bronzes with x = 0.1 and x = 0.84 when the structure remains that of the MPTBp’s. The samples used are those whose resistivity measurements are presented Fig. 4 in Ref. [20]. Fig. 4 shows a X-ray pattern taken from the x = 0.1 Na bronze at 25 K. It exhibits only the qc1 satellite reflections of the pure bronze (with this doping the variation of the wave vector is small and its difference with that of the undoped bronze is not detectable with our experimental set up). Reminiscence of the two others CDW modulations is observed under the form of very broad diffuse spots centered around the qc2 and qc3 reciprocal positions. The qc1 satellite reflections have disappeared above 60 K. The two others
Fig. 4. X-ray diffraction pattern from Na0.1 P4 W12 O44 taken at 25 K. The white arrows point towards the qc1 satellites reflections and the black arrows towards the qc2 and qc3 diffuse spots.
CDW transitions exhibited by the m = 6 pure bronze are not detected in the temperature range of our measurement (above 25 K). This is in perfect agreement with the resistivity measurement [20] showing only a bump anomaly starting at 55 K. The X-ray investigation of the x = 0.84 Na bronze down to 25 K does not any more reveal the presence of the qc1 satellite reflections. This is also in agreement with the conductivity measurements which do not show any indication of a CDW transition down to 4 K. The drastic decrease of the upper CDW transition of the m = 6 member with Na doping can be explained if one considers the effect of disorder to break the electron– hole pairs forming the Peierls condensate. More precisely the electrons are scattered by the impurity potential. Their diffusion on localized charged impurity, such as the Na+ , can achieve momentum transfers as large as 2kF (‘backward’ scattering process) allowing to break the pairs formed by the |kF + δq/2 and |−kF + δq/2 coupled electronic states. Such a scattering process, giving a life time τ at the pairs, leads to a decrease of the Peierls critical temperature Tc . Its shift with respect to its value in absence of impurity, Tc0 , depends on τ by the expression [13,26]: Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2(1 + h¯ /2πkB Tc τ ) . (12) If one introduces the mean free path l = vF τ , the expression (12) thus becomes: √ Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2 1 + ξth (Tc )/ 2 l , (13) in which the thermal coherence length ξth (Tc ) is given by the expression (4). If there is a strong scattering by the Na+ potential, the mean free path will be of the order of the average distance between Na atoms. This distance is 2a/x in the tunnel direction a which is tilted by about 40◦ with respect to the a ± b chains √ which are involved in the qc1 transition. Thus with l ∼ ( 2 )a/x, the expression (13) becomes: Ln Tc /Tc0 = Ψ (1/2) − Ψ 1/2 1 + xξth (Tc )/2a ≈ −π 2 xξth (Tc )/8a.
(14)
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The right member of the expression (14) holds for xξth (Tc )/4a small. From it, one obtains ξth (Tc ) ≈ 34 Å, with a = 5.3 Å, Tc0 = 120 K (in x = 0) and Tc ≈ 55 K (in x = 0.1). This value agrees with ξth = 33 Å calculated at 55 K, using the 1/T dependence of the expression (4), from the RT ab-initio value of ξth for the a ± b chains of the m = 4 member [6]. ξth (Tc ) is also close to ξth ≈ 45 ± 10 Å estimated at 55 K, with the same 1/T dependence, from the experimental determination of ξth (300 K) = 9 ± 2 Å and ξth (200 K) = 11 ± 2 Å for the a ± b chains of the m = 6 member. With such ξth (T ), the expression (14) predicts a Tc less than 1 K for the x = 0.84 sample! Concerning the qc2 transition, which occurs at 62 K in the x = 0 bronze, the application of formulae (14) for chains a gives a Tc less than 10 K for the x = 0.1 bronze if one uses a ξth (T ) extrapolated from the experimental value of ξth (200 K) = 12 ± 2 Å in the m = 6 member. This critical temperature is below the temperature range experimentally investigated. Recently, similar pair breaking effects have been invoked to explain the decrease of the Peierls transition in alloys of the 1D organic conductor Per2 [M(mnt)2] [27].
393
(a)
3.2. Band filling effects in Kx P4 W8 O32 , for 1 < x < 2 The m = 4 series of K doped MPTBh’s exhibits around T0 = 150 K, for 1 < x < 2, an anomaly in its transport properties which recall those observed in the m = 4 undoped MPTBp’bronze [20,22,24]. Below this temperature a gap develops in the magnetic susceptibility, as expected for a Peierls transition [20]. However the critical temperature of this transition is two times higher than the critical temperature of the upper CDW transition of the undoped m = 4 bronze (150 K instead of 80 K). Thus in order to understand the origin of such a phase transition we have undertaken a structural study of the x = 1.05(5), 1.30(9) and 1.94(9) samples (the values of x have been obtained by microprobe analysis). The X-ray diffuse scattering investigation shows unexpected features compared to those previously reported in the low m MPTBp’s. Fig. 5a shows a X-ray pattern taken at room temperature in the x = 1.05 bronze. It reveals two types of diffuse scattering: • The first type of scattering is formed of diffuse lines perpendicular to the a ± b directions. Such lines, which have been already observed in the low m members of the MPTBp’s, are the fingerprint of the CDW pretransitional fluctuations located on the a ± b chains. One set of lines is roughly located at half the reciprocal distance between a∗ ± b∗ layers of main Bragg reflections and another set is located in these layers. Such diffuse lines are also observed in the x = 1.3 bronze, but not in the x = 1.94 bronze. • The second type of scattering is formed of diffuse segment perpendicular to the bh or ap direction (the
(b) Fig. 5. X-ray diffraction pattern from K1.05 P4 W8 O32 taken at room temperature (a) and 25 K (b). In (a) the black arrows point towards the diffuse lines perpendicular to the a ± b directions and the white arrows towards the diffuse segments perpendicular to the bh or ap directions. In (b) the dotted white arrows point towards the q0 satellite reflections.
index ‘h’ corresponds to the axis frame of the structure of the MPTBh’s as published in Ref. [21] and the index ‘p’ to the frame of the MPTBp’s used throughout this paper). This diffuse scattering already found in the Cs1−x P8 W8 O40 solid solution [28] corresponds to a local order between K atoms inside the tunnel where they are located, but not correlated between neighboring tunnels. From the position of the diffuse lines relatively to the neighboring Bragg layers one can deduce the K concentration (x). From the inverse of the half width at half maximum of the diffuse scattering (ξ ) one deduces the extent of the order in tunnel direction: L ∼ πξ . Such quantities are reported in Table 1 for the samples investigated. Note that the x value obtained from the X-ray diffuse scattering investigation agrees, within experimental errors, with the concentration determined by the microprobe analysis. This diffuse scattering is not observed in the x = 1.94 bronze, indicating that the K atoms are ordered. Fig. 5b shows an X-ray pattern of the x = 1.05 bronze taken at 25 K. The two types of diffuse scattering previously quoted are still visible, although the intensity of the
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Table 1 Summary of the structural results for three Kx P4 W8 O32 compounds studied. The tables gives also the critical temperatures Tc (resistivity measurements) and T0 (X-ray data) x (microprobe) x (X-ray) ξ (Å) at 300 K Tc (K) T0 (K) Isat /IBragg
1.05(5) 1.3(2) 12(3) 145 150(5) 5 × 10−3
1.30(9) 1.5(2) 15(3) 165 160(5) 10−3
1.94(9) No diffuse lines ? 80(10) 10−4
Fig. 6. Phase diagram T0 (x) of the Kx P4 W8 O32 determined by electrical (plain circles) and structural (empty circles) measurements. The continuous line is given by the expression (19) with ξth (T0 ) = 16 Å.
diffuse scattering perpendicular to the a ± b directions has decreased. This X-ray pattern however reveals extra Bragg reflections located at the commensurate wave vector q0 = 1/2a∗h = 1/2b∗p . These satellite reflections disappear upon heating at T0 , the same temperature at which the electronic properties exhibit anomalies. However, the resistivity of the x = 1.94 bronze does not exhibit an anomaly at the structural transition which occurs at 80 K (Table 1). In all the samples investigated the reflections are always observed at the same simple commensurate wave vector with an intensity which decreases when x increases (Table 1). However the temperature of transition is a non-monotonous function of x. Fig. 6 shows that T0 (x) exhibits a maximum at about 170 K for x ≈ 4/3. From the structural point of view the transition does not appear to be due to the usual CDW instability observed in the pure MPTBp’s because, firstly, the satellites are not located on the a∗ ± b∗ diffuse lines, which means that the structure factor of the lattice distortion is different to that of the CDW fluctuations on the a ± b chains. Secondly, the a∗ ± b∗ diffuse lines remains visible at low temperature while the q0 reflections are well-developed. Thirdly, it is expected for a CDW driven transition that the 2kF modulation wave vector should vary with the band filling which is a function of x; however q0 does not vary with x. Fourthly, the structural transition is observed in the x = 1.94 bronze which does not exhibit any anomaly at T0 in its electronic properties. However for the bronzes with lower values of x the structural
transition is accompanied by electronic anomalies recalling that observed for a Peierls transition. Let us now consider the band filling with more detail. The K atoms provide x/2 additional electrons to each segment of m octahedra. The relationship (1) thus becomes in the MPTBh’s: 2kFa + 2 × 2kFd = 1 + x/4.
(15)
The observation of 1D CDW-like fluctuations under the form of a broad diffuse line located at the 1/2(a∗ ± b∗ ) reduced wave vector means that 2kFd ∼ 1/2d ∗ for x = 1.05 and 1.30. This value can be obtained if starting from the band filling of the undoped m = 4 bronze (2kFa = 2kFd = 1/3 [6]) one assumes that the electrons provided by the K atoms fill only the bands associated to the chains a ± b. Thus with 2kFd = (1/3 + x/8)d ∗ , one gets 2kFd = 1/2d ∗ for x = 4/3 and 2kFd = 11/24d ∗ ≈ 0.46d ∗ for x = 1 (in that latter case the X-ray pattern shown Fig. 5 should present two diffuse lines at 0.46d ∗ and 0.54d ∗ , difficult to distinguish from a single broad diffuse line at 0.5d ∗ ). This situation is not that predicted by the tight-binding band calculation [5b]. To reconcile all these features we propose the existence of a coupling between a purely structural transition and the CDW instability in such a way that the MPTBh’s undergo a structural transition which provides an external periodicity q0 = 1/2a∗h = 1/2b∗p to the quasi-1D electron gas. Such a periodicity, close to 2kFd , will open a gap in the band structure of the chains a ± b, which induces a Peierls transition on the electronic subsystem. In our scenario the maximum value of T0 corresponds at a resonance between the band filling of the chains a ± b and the commensurate value of the modulation provided by the additional structural transition. In this case there is coincidence between the external periodicity and the qc4 crossing point, defined Fig. 3b, leading to the nesting of the FS associated to the chains a ± b. The x = 1.05 and x = 1.30 bronzes are close to the resonance condition, but not the x = 1.94 bronze (in addition its electronic properties do not exhibit any anomaly at T0 ). A recent structural refinement of the K doped MPTBh’s [29] shows that the q0 structural transition consists in an oscillation of the PO4 tetrahedron accompanied by a displacement of WO6 octahedra, possibly due to the shift of the K atoms. In our interpretation the mechanism of the coupled structural–electronic transition of the MPTBh’s is similar to that outlined in Section 2.2 for the Peierls transition with, in the expression (6), the potential provided by the structural transition Vst (q0 ) replacing the potential due to the electron–phonon coupling Vep (qc ). This transition however is not achieved by the divergence of 1D CDW fluctuations. However a gap is opened in the band structure because of the enhanced response of the Lindhard function at the periodicity of the external potential Vst (q0 ). As Vst (q0 ) provides the d ∗ /2 periodicity in chain direction, which can be different of 2kFd , the q dependent Lindhard function χe (T , q), given by the expression (2), must be used, with δq = q0 − 2kFd = (4/3 − x)d ∗ /8. The relationship (6)
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(a)
(b)
395
(c)
Fig. 7. Schematical representation of the location of the gap due to the external periodicity q0 in the 1D band structure of the a ± b chains of the K doped MPTBh’s for various band fillings corresponding to: (a) x < 4/3, (b) x = 4/3 and (c) x > 4/3.
gives the critical temperature T0 of the coupled structural– electronic transition of the MPTBh’s. This temperature T0 (x) can be expressed, with respect to its maximum value for x = 4/3, under the form: Ln T0 (x)/T0(4/3) = Ψ (1/2) − 1/2 Ψ 1/2 + i h¯ vF δq/4πkB T0 (x) + Ψ 1/2 − i h¯ vF δq/4πkB T0 (x) . (16) When δq increases the divergence of the Lindhard function decreases and T0 thus diminishes. If one introduces the thermal coherence length ξth (T0 ) previously defined, the expression (16) becomes: Ln T0 (x)/T0(4/3) √ = Ψ (1/2) − 1/2 Ψ 1/2 + iδqξth(T0 )/2 2 √ + Ψ 1/2 − iδqξth(T0 )/2 2 . (17) The shift of T0 can be also viewed as due to the difference of chemical potential (δµ = h¯ vF δq/2) between the Fermi energy and the electronic energy at the wave vector q0 at which the gap is opened. If δqξth (T0 ) is a small quantity, the expression (17) becomes: 2 Ln T0 (x)/T0(4/3) ≈ − δqξth (T0 ) . (18) This leads to a parabolic dependence in x: 2 Ln T0 (x)/T0(4/3) ≈ −(4/3 − x)2 ξth (T0 )d ∗ /8 ,
(19)
which is very well followed by the data of Fig. 6. Furthermore from this dependence one gets ξth (T0 ) ≈ 16 Å, which value is in perfect agreement with ξth estimated at 150 K from the experimental measurement of the CDW thermal coherence length in the m = 4 undoped member: (ξth (300 K) = 8 ± 2 Å for the chains a ± b [6]). At T0 a gap is opened in the 1D band structure. It kills the divergence of the Lindhard function and thus prevents the onset of the conventional CDW instability driven by the electron–phonon coupling, which generally occurs at
lower temperature (Tc = T0 /2) in the low m members of the MPTBp’s. This large value of T0 can be explained either by the replacement of the electron–phonon potential Vep (qc ) by a structural one Vst (q0 ) which can be stronger than Vep (qc ) or/and by the commensurability two of the q0 modulation. It is well-known [30] that for a two-fold commensurability the CDW ground state is particularly stable, and that an important shift δµ of chemical potential is required to destabilize it. In this scenario, Vst (q0 ) opens a gap near the Fermi level in the band structure of chains a ± b, and leaves unaffected the band structure of chains a.1 This differs of the undoped m = 4 MPTBp’s where after the successive CDW transitions all the FS’s are affected. As a consequence, large area of the Brillouin zone will remain unaffected by the structural transition in the K doped m = 4 MPTBh’s. The magnetoresistance effects will thus be much weaker at low temperature in them (14%) than in the pure m = 4 MPTBp’s (250%) [22]. Fig. 7 shows schematically the effect of a gap opening at q0 in the 1D band structure of the chains a ± b. In our simplified model the gap opens just at the Fermi level for x = 4/3 (Fig. 7b). For x < 4/3, it opens in the unoccupied part of the band, forming hole pockets (Fig. 7a), and for x > 4/3 it opens in the occupied part of the band, forming electron pockets (Fig. 7c). We believe that the formation of these hole (electron) pockets explains the positive (negative) jump of thermopower which starts at T0 in the bronzes with x values smaller (higher) than 4/3 [24]. In the m = 4 MPTBh’s the K remains disordered (except for the x ≈ 2 bronze). One should expect the occurrence 1 The short range order of the K atoms gives rise to a local potential whose main Fourier component in tunnel direction is located at x/2a∗p , within one reciprocal wave vector. For x = 4/3 its components are at 2/3a∗p and 1/3a∗p . In our scenario this latter wave vector corresponds to 2kFa . Thus the local potential of the K atoms can open a pseudo-gap at the Fermi level in the band structure of chains a, which effect diminishes when x deviates from 4/3.
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of a pair breaking phenomena which could, as observed in the Nax P4 W12 O44 bronze (see the previous section), destabilize the Peierls transition. This effects requires the presence of a 2kF backscattering process coming from the K+ potential. However the K atoms are enough ordered in tunnel direction to provide only Fourier components of the scattering potential which are spread on 2ξ−1 around a mean value of x/2b∗h = x/2a∗p . As this range of wave vectors does not overlap the critical wave vector of the structural transition, q0 = 1/2a∗h = 1/2b∗p , the K disorder will not affect the Peierls-like transition of the MPTBh’s.
[3] [4] [5] [6] [7] [8] [9] [10]
4. Conclusion This paper is written at the occasion of the 60th birthday of Professor Martha Greenblatt. She points out to us the interesting features exhibited by the phosphate tungsten bronzes just after her report of bump anomalies in the resistivity of the MPTBp’s [31]. This was the start of a collaboration of more than 10 years which has begun with the finding of CDW instabilities in these bronzes [32]. Since then, the scenario of a Peierls instability taking place, through a hidden nesting mechanism of quasi-1D bands associated to chains of WO3 octahedra running in the a and a ± b directions of the layers, has been confirmed for the 4 m 6 regular members of the series [8], as well as in a recent study of the CDW transition of the alternate 4/6 member [33]. We have reported here additional evidences of the relevance of the Peierls-like mechanism in these monophosphate tungsten bronzes through the quantitative analysis of the phase diagrams of ‘doped’ Nax P4 W12 O44 (x < 1) and Kx P4 W8 O32 (1 < x < 2) bronzes. We have also shown throughout this paper that a theoretical framework based on the physics of 1D conductors allows to simply understand the CDW properties exhibited by this family of 2D bronzes.
[11]
Acknowledgements
[24]
We are very grateful to our colleagues and collaborators for their contribution to the present work. In particular we would like to thank D. Groult for the crystal growth, S. Drouard, J. Dumas and C. Schlenker for the transport measurements and S. Ravy for helpful discussions. References [1] M. Greenblatt (Ed.), Oxide bronzes, Int. J. Mod. Phys. B 7 (1993) 3937–4164. [2] C. Schlenker, J. Dumas, M. Greenblatt, S. van Smaalen (Eds.), Physics and Chemistry of Low Dimensional Inorganic Conductors, NATO
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