Lattice vibrations in CaV₂O₅ and their manifestations: a theoretical study based on density functional theory

J Spitaler^1,2,5, C Ambrosch-Draxl^1,2 and E Ya Sherman^3,4

¹ Chair of Atomistic Modelling and Design of Materials, University of Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
² Institut für Physik, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
³ Departamento de Química Física, Universidad del País Vasco—Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain
⁴ IKERBASQUE Basque Foundation for Science, Alameda Urquijo 36-5, 48011, Bilbao, Bizkaia, Spain

⁵ Author to whom any correspondence should be addressed.

E-mail: juergen.spitaler@unileoben.ac.at

Received 30 July 2009
Published 5 November 2009

Contents

• 1. Introduction
• 2. Computational methods
• 3. Phonon modes
  • 3.1. Eigenvectors
  • 3.2. Frequencies
• 4. Comparison with experiment
  • 4.1. Raman spectra
  • 4.2. Infrared active modes
• 5. Model parameters
• 6. Conclusions
• Acknowledgments
• References

1. Introduction

Fascinating physical properties of ladder compounds have been stimulating very active studies in this field over the last 15 years [1], (and references there in [2]). The common feature of these crystals are the pairs of metal ions linked by an oxygen, which form the rungs of the ladder. These rungs are arranged in one dimension in the direction of the ladders' legs. The main electron–electron interactions and electron motion occur within one ladder, while the interactions between neighboring ladders in the same plane are considerably weaker, and ladders in different planes are hardly coupled. In this situation, the coupling of the charge, spin and lattice degrees of freedom causes a rich variety of physical phenomena.

The most intensively studied examples of ladder compounds are formed by copper ions with spin 1/2 per magnetically active site and have half-filled upper bands, such as SrCu₂O₃ or (Sr, Ca)₁₄Cu₂₄O₄₁. However, there exist also two half-filled V-based ladder compounds, namely CaV₂O₅ and MgV₂O₅. In spite of the fact that Ca and Mg have the same valence configuration and both compounds are very similar in their V subsystem, they form crystals with different space groups: Cmnm and Pmmm for MgV₂O₅ and CaV₂O₅, respectively. The structural differences have a strong impact on the magnetic properties [3, 4]. While CaV₂O₅ is best described as a collection of well-isolated single two-leg spin ladders [5, 6], MgV₂O₅ exhibits an additional strongly frustrated interladder coupling [7, 8]. This leads to a very large gap of 500 K for the triplet excitations in the former compound [5], while in the latter a small gap of only 15 K is observed experimentally [7].

In addition to half-filled systems, V-based ladders show the unique realization of quarter-filling in NaV₂O₅ [10], which emphasizes the crucial importance of the chemical composition. Since a spin-Peierls transition in this compound was discovered NaV₂O₅ has become the most intensively investigated vanadium ladder system. It is especially interesting in the context of a comparison with CaV₂O₅ since both compounds crystallize in the same Pmmn structure with two formula units per unit cell (figure 1) at room temperature.

Since the properties of the ladder compounds are expected to be strongly influenced by the interplay of the electron motion, the lattice and the magnetic degrees of freedom, quantitative information about spin–lattice and charge–lattice coupling is strongly desirable, however, so far still lacking. To fill this gap and to gain an understanding of the physics of these compounds, we perform, to the best of our knowledge, the first ab initio investigation of the lattice dynamics of all non fully symmetric phonons in a half-filled ladder system using the example of CaV₂O₅. We report the theoretical phonon frequencies and eigenvectors for the B_1g, B_2g and B_3g modes, which are visible in Raman measurements, as well as all infrared-active vibrations, seen in light absorption and reflection experiments. A detailed comparison of the theoretical results, including the calculated Raman spectra, to experiment is presented and allows for a better understanding of the inelastic light scattering in this compound. In addition, we study electron– and spin–phonon coupling by mapping the ab initio results on a model band structure and identify the modes showing strong coupling to the electron subsystem. We relate the properties of CaV₂O₅ to those of NaV₂O₅ to understand the role of the electron occupancy at the vanadium site, which determines the filling of the upper electron band.

2. Computational methods

All total-energy and band-structure calculations are performed within density functional theory (DFT) using the full-potential augmented plane waves + local orbitals (FP-APW + lo) [11] formalism implemented in the WIEN2k code [12]. Exchange and correlation terms are described within the generalized gradient approximation (GGA) [13]. The atomic sphere radii are chosen as 1.6 au for V, 1.8 au for Ca and 1.4 au for the O atoms. A K_max value of 4.64 is used in all calculations, corresponding to R_MT×K_max values of 6.5 for O, 7.43 for V and 8.36 for Ca.

The number of k points in the full Brillouin zone (BZ) for the self-consistent field cycles is typically about 600. For the calculation of the dielectric function, which serves as input for the Raman spectra, a number of 3800 k points in the full BZ is used. The G_max used for charge density and potential is 14. The internal geometry is fully relaxed, taking the experimental atomic positions reported in [14] as starting point. Details about the relaxation can be found in [15].

The phonon eigenfrequencies and eigenvectors are determined in the frozen-phonon approximation, where for each degree of freedom corresponding to the symmetry coordinates four displacements, two in positive and two in negative direction, are considered. The symmetry coordinates for each irreducible representation can be inferred from the degrees of freedom appearing in the corresponding table and figure, respectively.

3. Phonon modes

A factor group analysis yields, after subtraction of three acoustic (B_1u + B_2u + B_3u) and three silent A_u modes,

where the letters in parenthesis indicate the polarization of the incoming and outgoing light, respectively. Hence, there are 24 Raman-active A_g and B_ng modes and 18 infrared-active B_nu modes (n{1, 2, 3}). The eigenvectors and frequencies of the Raman-active B_1g, B_2g and B_3g, and all infrared-active phonon modes are presented in table 1, while the fully symmetric Γ point phonons have already been published in [15]. The eigenvectors e_αζ (where α denotes the atom and ζ indicates the phonon mode) are normalized with respect to the whole unit cell including two formula units of CaV₂O₅. For each set of equivalent atoms only the components corresponding to one atom are reported, where this reference atom is highlighted in the respective figures using bold italic characters.

3.1. Eigenvectors

Due to the structural similarity to NaV₂O₅ most of the phonon modes of both compounds are close in frequency [16], and the eigenvectors differ only very little. Figure 2 shows a selection of phonon modes which most strongly differ from their NaV₂O₅ counterparts.

As in the case of NaV₂O₅ the eigenvectors of the B_1g modes represent a V–O2 stretching, an O3–V–O2 bending, and a vibration where the two legs of one ladder are shifted anti-parallelly along , respectively. The same situation is found for the B_2g modes, where the features of the atomic displacements are the same as in the Na compound.

Within the B_3g vibrations, two CaV₂O₅ modes exhibit a particular behavior. The O1–V–O2 B_3g mode with 392 cm^–1 has some contribution from the Ca atom moving in the -direction, in contrast to Na which hardly participates in the corresponding NaV₂O₅ mode. For the 195 cm^–1 vibration with the same symmetry, where Ca moves out of phase with respect to the ladder and parallel to the -axis, the Ca eigenvector component is twice as large as the corresponding Na component in NaV₂O₅. Moreover, in CaV₂O₅ also the in-rung oxygen O1 takes part. Also two of the B_1u phonon modes differ from their counterparts in NaV₂O₅: in CaV₂O₅, the V–O2–V bending mode (559 cm^–1) does not involve a shift of the in-rung oxygen O1, which is present in NaV₂O₅, and the apex oxygen O3 has a considerable z-component. In the 483 cm^–1B_1u mode the O3 atom has a larger x-component, while the x-component of the in-leg oxygen O2 is quite small compared to the NaV₂O₅ mode. Among the phonons with B_2u symmetry, the 401 cm^–1 vibration shows a noticeable Ca contribution, which makes it differ from the O1–V–O3 bending mode of NaV₂O₅. The B_3u vibration exhibiting the clearest deviation from its NaV₂O₅ counterpart is the one with 479 cm^–1. This mode represents a pure O2–V–O2 bending in the x–y plane in CaV₂O₅, while in NaV₂O₅ the in-rung oxygen O1 participates considerably.

3.2. Frequencies

Given the close relation between the two compounds in terms of their eigenvectors, it is of special interest to compare the frequencies in CaV₂O₅ and NaV₂O₅ (presented in [16]) since they can give a direct information on the bonding and the interaction between the atoms in the two materials. It turns out that the phonon modes can be divided into several groups showing different tendencies in terms of the phonon frequency.

The first group is formed by the V–O3 stretching modes and exhibits a strong frequency decrease compared to NaV₂O₅. In detail, the changes are –14% for the mode with B_2g symmetry (842 cm^–1), –7% for the one with B_1u symmetry (903 cm^–1) and –10% for the B_3u V–O3 stretching mode (845 cm^–1). A very similar tendency is found for the V–O2 stretching modes, where the frequencies are lowered by 10 and 15% in case of the B_1g (593 cm^–1) and the B_3g vibration (605 cm^–1), respectively. In this context the V–O2 stretching mode of the B_2u species (550 cm^–1) represents an exception, since its frequency is only slightly decreased with respect to NaV₂O₅. The main reason for the lower energy of the V–O stretching in CaV₂O₅ is found in the chemical difference between Na and Ca. While Na contributes only one s electron to the system, thus yielding V^4.5 +  ions, the extra valence electron in Ca reduces the formal V charge to 4. Correspondingly, a Coulomb picture would imply a frequency decrease by a factor of , i.e. approximately 6%.

A trend to lower frequencies is also found for the group of V–O2–V bending modes with components perpendicular to the x–y plane. These vibrations also change the length of the V–O3 bond, such that the same argument as for the stretching modes applies. In case of the B_2g vibration with 259 cm^–1 and the B_3u vibration with 367 cm^–1 the decrease is about 10%, while it is more moderate for the 342 cm^–1B_2g mode with 5%. The B_1u483 cm^–1 and the B_1u348 cm^–1 vibrations have very similar frequencies in both CaV₂O₅ and the NaV₂O₅.

The situation is different for the V–O2–V bending modes parallel to the x–y plane. Here the vibrations in the Ca compound rather exhibit a frequency increase, which is only 1% for the B_1u mode with 559 cm^–1, but 6% for the 499 cm^–1B_2g mode and even 8% for the B_3u mode with 479 cm^–1. This tendency can be related to the stronger ladder–ladder interaction, which shows up in the slightly smaller inter-ladder distance in CaV₂O₅ (1.98 Å, compared to 2.01 Å in NaV₂O₅) and in a much larger inter-ladder hopping parameter t_i [15]. The latter can be explained by a stronger overlap of orbitals of the V and O2 atoms in neighboring ladders, a consequence of the fact that the z coordinates of these two atoms differ by 0.06 Å less in CaV₂O₅ (Δz_V,O2 = 0.42 Å) compared to NaV₂O₅ (Δz_V,O2 = 0.48 Å).

The last group is represented by low-energy modes involving Ca. These modes show much higher frequencies compared with their counterparts in NaV₂O₅. In detail, the increase is 21% in the B_2g mode with 190 cm^–1, 4% in the B_3g mode with 195 cm^–1, 24% in the 235 cm^–1B_1u mode, 27% in the B_1u vibration with 195 cm^–1, 13% for the B_2u mode with 196 cm^–1 and 20% in the 174 cm^–1B_3u vibration. A straightforward explanation of this tendency is delivered by the larger ionicity of Ca^2 +  than Na ⁺  ions. This effect alone would imply even larger frequency increases, but it is partly compensated by the larger mass of Ca (40.08 au) compared with Na (22.99 au).

4. Comparison with experiment

4.1. Raman spectra

In order to compute the Raman intensities the atoms are displaced according to the calculated eigenvectors, and the dielectric function for the distorted crystal is calculated. The changes of the dielectric function with respect to such displacements along an eigenvector of mode ζ allow the determination of the corresponding Raman intensity I_ij. To this end we first define the operator corresponding to the phonon coordinate via the following relation with the actual displacement coordinate u_ζ^α:

Here, M_α is the mass of ion α and ω_ζ defines the corresponding frequency. Then, I_ij is given by

where ω_R is the Raman shift, the Cartesian indices ij correspond to the polarizations of incident and scattered light, respectively, |1 and |0 denote the one-phonon and the phonon-less states, respectively, n_B(ω_ζ) = 1/[exp(ω_ζ/T)–1] is the phonon Bose distribution function, ω_I is the frequency of the incoming light, and L(ω_R, ω_ζ, Γ) is the Lorentzian shape of the phonon line with a broadening Γ, which is chosen to be 15 cm^–1 for all modes. Further information on the applied method can be found in [17].

The results for the B_1g, B_2g and B_3g modes are presented in figure 3. The highest peaks are found at 593 cm^–1 for xy polarization, at 135, 259 and 652 cm^–1 for xz polarization, and at 254 cm^–1 for yz polarization. A specially subtle feature is found for the 219 cm^–1B_2g peak, which is surrounded by much higher neighbors. The signal of the B_3g mode at 195 cm^–1 is completely covered by the strongly dominating 254 cm^–1 peak.

Figure 4 shows the change of the dielectric function for atomic displacements along the respective eigenvector, i.e. |∂ε_ij/∂Q|², for selected B_1g, B_2g and B_3g modes, where modes with maximum changes smaller than 0.01 have been omitted. Large values of |∂ε_ij/∂Q|² at a certain photon energy imply resonances observed with the corresponding laser frequency. Figure 4 shows that some of the modes exhibit a strong dependence on the applied photon energy, among them the B_2g mode with 253 cm^–1, the B_2g mode with 652 cm^–1 and the B_3g mode with 254 cm^–1. Moreover, it is found that the experimentally used 514.5 nm (2.41 eV) line of an Ar ⁺  ion laser [18]–[20] leads to a very small response of the system for all but the 254 cm^–1B_3g mode. Instead, all modes exhibit resonances for photon energies below 2.0 eV, and many of the modes at photon energies between 2.5 and 3.0 eV.

In literature one polarized Raman measurement using a polycrystalline sample [18] and two unpolarized ones [19, 20] are reported, but there are no spectra obtained from CaV₂O₅ single crystals. For this reason no unambiguous experimental evidence for the symmetry of the peaks is found. Table 2 shows a comparison of the measured results with theoretical frequencies.

	Experiment
	[18]	[18]*	[20]*	[19]	[19]*	Theory	Assignment
B1g	638	638	636	636	636	593	V–O2 stretching
	292	–	–	292	–	253	O3–V–O2 bending
	212	137	–	182	138.6	136	antiparallel leg-shift
B2g	–	–	–	–	–	842	V–O3 stretching
	–	–	–	–		652	V–O1 stretching
	–	–	–	–	–	499	O2–V–O1 bending
	–	–	310	334	311	342	O2–V–O3 bending
	–	292	293	311	292	259	O2–V–O3 bending
	–	212	211	213	213	219	Ca-ladder shear
	–	–	–	–	182	190	Ca
	–	137	–	–	138.6	135	chain rotation
B3g	–	–	–	636	–	605	V–O2 stretching
	–	369	–	366	366	392	O1–V–O3 bending
	–	212	212	280	213	254	rung versus O3 shear
	–	–	–	–	–	195	(ladder versus Ca)
	–	112	–	–	112	113	antiparallel ladder shift

Theory reveals that the experimental feature at 638 cm^–1 goes back to the highest-frequency B_1g mode, which shows up as a very pronounced peak in the theoretical Raman spectra. A B_2g mode at almost the same energy is probably covered by the B_1g signal in experiment. A measured feature at 138 cm^–1, which was interpreted as A_g mode in [19], and not assigned any symmetry in [18], can neatly be traced back to a superposition of the B_1g and the B_2g peak lowest in energy. A peak measured at about 370 cm^–1 is explicitly addressed in [19] and also clearly visible in the crossed-polarized spectra of [18]. This peak is in reasonable agreement with the low theoretical B_3g peak at 392 cm^–1. The situation is comparable for the modest B_2g feature at 342 cm^–1, which is resolved in the unpolarized measurements [19, 20] at a slightly lower energy of 311 cm^–1. Very good agreement is found for the theoretical B_2g peak at 190 cm^–1 and the experimental peak at 182 cm^–1. A B_3g feature at virtually the same energy has very low intensity and is therefore either not visible in experiment or integrated in the stronger B_2g peak. Another B_3g feature, which is not explicitly addressed in the experiment, but still distinguishable, is found at 113 cm^–1, where the position of the experimental and the theoretical peak agree perfectly.

The comparison of ab initio results and measurements is more intriguing for the frequency range between 200 and 300 cm^–1: while experiments show two pronounced peaks at 212 and 292 cm^–1, theory yields high-intensity peaks at 259 cm^–1 in the B_2g spectra and at 254 cm^–1 in the B_3g spectra. Since the peak belonging to the latter is by far the highest, it most probably can be related to the measured feature at 212 cm^–1, while the intensity of the former mode is in good qualitative agreement with the experimental peak at 292 cm^–1. A third phonon mode with almost the same theoretical frequency is found among the B_1g vibrations. However, the intensity of this mode is much smaller, which implies that in the experimental spectra it is probably masked by the A_g peak at 238 cm^–1. The tiny calculated B_2g peak appearing at 212 cm^–1 is most probably integrated in the large experimental feature at 219 cm^–1.

For the rest of the Raman peaks no experimental counterpart is found, in perfect accordance with theory yielding very low scattering intensities.

4.2. Infrared active modes

In table 3, the theoretical frequencies for the infrared-active phonon modes are compared to experiment. To the best of our knowledge up to now only one, unpolarized measurement is found in literature [19], where a symmetry assignment was only done by comparison with polarized results for NaV₂O₅. Thus theory can substantially contribute to a proper symmetry assignment of the peaks. The root mean square deviation between measured and calculated frequencies is less than 5%, with the largest difference (–9 cm^–1 = –7.3%) found for the B_3u chain-rotation mode with a theoretical frequency of 123 cm^–1. As in [16] we have estimated the theoretical intensities of the infrared spectra. To do so, we have calculated the dipole moment P induced by a phonon distortion using a simple model with fixed ion charges (V^4.5 + , Ca^2 + , O^2–), but taking into account the real displacements according to the eigenvectors.

	Experiment
	[19]	[19]*	Theory	Assignment
B1u	957	957	903	V–O3 stretching
	–	579	559	V–O2 stretching
	–	–	483	O1–V–O2 bending
	–	–	348	O1–V–O2 bending  +  O3x
	264	264	282	rung stretching
	–	244	235	O1–V–O3 bending  +  Ca
	198	198	195	Ca
B2u	579	515	550	V–O2 stretching
	369	408	401	O1–V–O3 bending
	–	–	229	O3 versus ladder shear
	123	198	196	Ca
B3u	892	892	845	V–O3 stretching
	515	–	658	O1
	408	–	479	O2–V–O1 bending
	–	369	367	V–O2 bending
	244	–	229	V–O3 bending
	–	–	174	Ca
	–	123	114	chain rotation

In the experimental infrared spectra, we identify five features as stemming from B_1u phonons: in addition to the ones at 957, 282 and 198 cm^–1, already identified in [19], the theoretical results show that also the 579 cm^–1 and the peak at 198 cm^–1 are related to a B_1u vibration. In case of the former not only the theoretical frequency but also the rather small induced dipole moment imply that this small experimental peak is associated with the calculated mode at 559 cm^–1. The latter turns out to represent a superposition with a B_2u mode at the same frequency.

Other B_2u modes that can be distinguished in experiment are found at 515 cm^–1, where the large phonon-induced dipole moment obtained by theory agrees nicely with experiment, and the feature at 408 cm^–1, which just perfectly corresponds to the calculated phonon energy. This is in contrast to the assignment obtained by comparison with NaV₂O₅, which related these two peaks to B_3u modes [19].

The peak at 892 cm^–1, which is the second highest in frequency in the infrared spectra, is readily identified as belonging to the B_3u species. While the two phonon modes next lower in frequency are not found in the measurement, the peak at 369 cm^–1 fits perfectly to the theoretical result of 367 cm^–1. The feature with the lowest energy in the measurement, i.e. the one at 123 cm^–1, clearly belongs to the B_3u species, which means that the assignment based on the comparison with NaV₂O₅ fails in this case.

5. Model parameters

In order to extract effective model parameters from the Kohn–Sham band structure six bands where mapped onto a tight-binding model, namely the bonding and anti-bonding pair of the V 3 d_xy bands at the Fermi level and the two highest lying O 2p bands. We restrict ourselves to the direction and take into account the hopping along the ladder t_||, the in-rung hopping t_⊥, and the inter-ladder transfer t_i. Since the overall features of the electronic structure, except for the Fermi energy, are very similar to the ones in NaV₂O₅, the same mapping procedure as described [16] is applicable.

The values of the hopping parameters in the undistorted structure, t_|| = 0.143, t_⊥ = 0.321 and t_i = 0.244 eV, as well as the vanadium Hubbard parameter U = 2.45 eV have been presented in [15]. With the knowledge of these parameters, one can calculate the spin–spin exchange. Below we consider exchange along the ladder J_||, using the expression J_|| = t_||²/E_g, where E_g is the charge transfer gap between V and O [21]. In order to compare the electron– and spin–phonon coupling parameters for the different phonon modes with each other, the displacements are normalized with respect to the dimensionless phonon coordinate Q defined in equation (1). Since the contribution to E_g which is linear in Q vanishes for all but the fully symmetric phonons, we consider below only the effect of the lattice vibrations on the hopping and spin–spin exchange.

The largest parameters of electron–phonon and spin–phonon coupling for the Raman active B_1g, B_2g and B_3g vibrations and all infrared active phonon modes of CaV₂O₅ (δt_||/δQ≥0.008 eV), are presented in table 4. Comparing the coupling of t_|| to lattice distortions corresponding to the Γ point vibrations, similarly to NaV₂O₅ , the strongest effect is related to the V–O2 stretching modes. Again, the largest value (δt_||/δQ = 0.040 eV) is found for the B_1g mode highest in frequency. This vibration is followed by the B_3g V–O2 stretching with 605 cm^–1 (figure 5), with a δt_||/δQ of 0.016 eV, where the corresponding parameter is much less significant in the respective mode of NaV₂O₅. The phonon mode with the next smaller change of t_|| coincides with its counterpart in NaV₂O₅ again, i.e. it is the B_1u mode second highest in energy.

Symmetry	ωζ	Assignment
B1g	593	V–O2 stretching	0.040	0.601
B3g	605	V–O2 stretching	0.016	0.210
B3g	254	rung versus O3 shear	0.009	0.139
B3g	113	antipar. ladder shift	0.008	0.134
B1u	903	V–O3 stretch.	0.008	0.104
B1u	559	V–O2 stretch.	0.013	0.155
B1u	483	O1–V–O2 bend.	0.008	0.106

The t_⊥ show very weak coupling to phonon-distortions for all phonon modes considered here, with all values of δt_⊥/δQ less than 0.007 eV, and are therefore not presented in the table. This means that the A_g vibrations [15] have by far the strongest effect on the hopping matrix element along the rung.

The spin–phonon coupling parameters expressed in terms of relative changes of J_|| as induced by phonon distortions are presented in the last column of table 4. In CaV₂O₅, the highest values for δJ_||/(J_||δQ) are found for the V–O2 stretching modes with B_1g symmetry (0.601), B_3g symmetry (0.210), A_g symmetry (0.186) and B_1u symmetry (0.155). In fact, the mentioned vibrations are also the ones dominating the electron–phonon coupling of hoppings along the ladder, which seems reasonable due to the interrelation of both quantities. Besides, also in NaV₂O₅ the V–O2 stretching modes turn out to give the largest respective parameters. All other phonon modes lead to changes of δJ_||/J_|| per δQ smaller than 0.15. Since t_⊥ changes very little, all phonon-induced changes of J_⊥ for the considered phonon modes are very small, too.

6. Conclusions

We have determined the frequencies and eigenvectors of all Raman- and infrared-active low-symmetry Γ point phonons of CaV₂O₅, a representative example of a half-filled compound with a ladder-like crystal structure. Our results provide detailed information on the lattice dynamics, which is desirable for both, the understanding of experimental data and the development of better models to describe such compounds. Our results provide the first direct symmetry assignment of the phonon modes, thus complementing former assignments based on a mere frequency-based comparison with polarized Raman experiments for NaV₂O₅ single crystals [19].

We have compared the phonon eigenvectors to the ones of the isostructural NaV₂O₅, and we have found that, in contrast to the latter, not only the low-frequency vibrations but also the O2–O1–O2 bending modes with B_3g symmetry (392 cm^–1) and B_2u symmetry (401 cm^–1) mode exhibit a significant Ca contribution. This shows that the interaction between the in-leg oxygen O1 and the interstitial ion is significantly larger than in NaV₂O₅, which is also reflected in the smaller distance between them (2.31 Å in CaV₂O₅, compared with 2.43 Å in NaV₂O₅).

In terms of frequencies we have distinguished four different groups of vibrations: the first two of them, i.e. the V–O3 stretching modes and the V–O2–V bending modes out of the x–y plane, show significantly lower frequencies compared to their NaV₂O₅ counterparts, which is due to the different chemical situation where Ca provides one more valence electron to the two V sites of one rung, thus increasing its own ionicity and lowering the positive charge on the V sites. In contrast, V–O2–V bending modes within the x–y plane show rather higher frequencies compared to NaV₂O₅, which is due to the stronger overlap of orbitals of the V and O2 atoms in neighboring ladders, a consequence of the fact that the z-coordinates of these atoms differ less in CaV₂O₅ compared with NaV₂O₅. The group of low-frequency vibrations involving Ca^2 + , finally, exhibits strongly increased phonon frequencies due to the much stronger ionicity compared to Na ⁺  in the sodium ladder compound.

We have calculated the spectra of all Raman active Γ point phonons from first principles, thus providing a sound basis for the analysis of the experimental spectra. This way we can make a direct assignment of the experimental peaks in terms of their symmetry. We have shown that, for example, the experimental peak at 139 cm^–1 goes back to a superposition of a B_1g and a B_2g mode, and that a small feature at 113 cm^–1, which is clearly visible but not explicitly addressed in the experimental works, is related to the anti-parallel ladder shift with B_3g symmetry. In addition to the Raman spectra, we have determined the dependence of the Raman intensities on the energy of the exciting photons. It turns out that for the photon energy applied in the available experiments the response of the system is rather weak, while strong resonances for all modes are found at energies below 2 eV, and for many phonon modes at the energies between 2.5 and 3.0 eV. Hence, we propose to perform Raman experiments with corresponding laser frequencies, which would allow us to see phonon modes not observed before.

The theoretical eigenmodes of the infrared active phonons, together with an estimate of their induced dipole moment, elucidate experiments as well. We have improved the interpretation of the measured spectra, and shown, e.g. that the experimental feature at 579 cm^–1 stems from a B_1u mode, while the peak at 198 cm^–1 represents a superposition of a B_1u and a B_3u peak. The measured peak at 123 cm^–1, in turn, can be assigned to a B_3u vibration.

Finally, we have determined the electron–phonon and spin–phonon coupling parameters by mapping the ab initio eigenvalues for phonon-distorted structures on a 6-band Hamiltonian. We have found that t_⊥, and consequently also J_⊥, are little affected by the non fully symmetric Γ point phonons. Regarding the hopping along the legs t_||, and the corresponding exchange coupling J_||, the V–O2 stretching modes show the largest changes, where the B_1g593 cm^–1 mode reveals the strongest electron- and spin–phonon coupling of all Γ point phonons. In summary, a quantitative description of the interactions that can strongly influence the electron subsystem has been achieved, yielding a profound understanding of this representative half-filled ladder compound.

Acknowledgments

The work was financed by the Austrian Science Fund (FWF), project P15520. We also appreciate support by FWF projects P15834 and P16227 and the EU RTN network EXCITING (contract HCPR-CT-2002-00317).

References

[1]

Dagotto E and Rice T M 1996 Surprises on the way from one- to two-dimensional quantum magnets: the ladder materials Science 271 618 
CrossRef

[2]

Lemmens P, Güntherodt G and Gros C 2003 Magnetic light scattering in low-dimensional quantum spin systems Phys. Rep. 375 1–103 
CrossRef

[3]

Korotin M A, Elfimov I S, Anisimov V I, Troyer M and Khomskii D I 1999 Exchange interactions and magnetic properties of the layered vanadates CaV₂O₅, MgV₂O₅, CaV3O7 and CaV4O9 Phys. Rev. Lett. 83 1387 
CrossRef

[4]

Korotin M A, Anisimov V I, Saha-Dasgupta T and Dasgupta I 2000 Electronic structure and exchange interactions of the ladder vanadates CaV₂O₅ and MgV₂O₅ J. Phys.: Condens. Matter 12 113 
IOPscience

[5]

Iwase H, Isobe M, Ueda Y and Yasuoka H 1996 Observation of Spin Gap in CaV₂O₅ by NMR J. Phys. Soc. Japan 65 2397 
CrossRef

[6]

Isobe M and Ueda Y 1998 Doping effects in AV₂O₅ (A = Li, Na and Ca) J. Magn. Magn. Mater. 177–81 671–2 
CrossRef

[7]

Isobe M, Ueda Y, Takizawa K and Goto T 1998 Observation of a Spin Gap in MgV2O5 from high field magnetization measurements J. Phys. Soc. Japan 67 755 
CrossRef

[8]

Millet P, Satto C, Bonvoisin J, Normand B, Penc K, Albrecht M and Mila F 1998 Magnetic properties of the coupled ladder system MgV2O5 Phys. Rev. B 57 5005 
CrossRef

[9]

Kokalj A 2003 Comput. Mater. Sci. 28 155 (http://www.xcrysden.org/) 
CrossRef

[10]

Smolinski H, Gros C, Weber W, Peuchert U, Roth G, Weiden M and Geibel C 1998 NaV₂O₅ as a quarter filled ladder compound Phys. Rev. Lett. 80 5164 
CrossRef

[11]

Sjöstedt E, Nordström L and Singh D J 2000 An alternative way of linearizing the augmented plane-wave method Solid State Commun. 114 15 
CrossRef

[12]

Blaha P, Schwarz K and Luitz J 1997 WIEN2k (Release 97.8) (Improved and updated Unix version of the original copyright WIEN code, which was published in Blaha P, Schwarz K, Sorantin P and Trickey S B 1990 Comput. Phys. Commun.59399

[13]

Perdew J P, Burke K and Ernzerhof M 1996 Generalized gradient approximation made simple Phys. Rev. Lett. 77 3865–68 
CrossRefPubMed

[14]

Onoda M and Nishiguchi N 1996 Crystal structure and Spin Gap state of CaV₂O₅ J. Solid State Chem. 127 359 
CrossRef

[15]

Spitaler J, Sherman E Ya, Evertz H G and Ambrosch-Draxl C 2004 Optical properties, electron–phonon coupling, and Raman scattering of vanadium ladder compounds Phys. Rev. B 70 125107 
CrossRef

[16]

Spitaler J, Sherman E Ya and Ambrosch-Draxl C 2007 Zone-center phonons in NaV2O5: a comprehensive ab initio study including Raman spectra and electron–phonon interaction Phys. Rev. B 75 014302 
CrossRef

[17]

Ambrosch-Draxl C, Auer H, Kouba R, Sherman E Ya, Knoll P and Mayer M 2002 Raman scattering in YBa₂Cu₃O₇: a comprehensive theoretical study in comparison with experiments Phys. Rev. B 65 064501 
CrossRef

[18]

Konstantinović M J, Popovic Z V, Isobe M and Ueda Y 2000 Raman scattering from magnetic excitations in the spin–ladder compounds CaV₂O₅ and MgV₂O₅ Phys. Rev. B 61 15185 
CrossRef

[19]

Popović Z V, Konstantinović M J, Gajić R, Popov V N, Isobe M, Ueda Y and Moshchalkov V V 2002 Phonon dynamics in AV₂O₅ (A Na, Ca, Mg, Cs) oxides Phys. Rev. B 65 184303 
CrossRef

[20]

Popović Z V, Stergiou V, Raptis Y S, Konstantinović M J, Isobe M, Ueda Y and Moshchalkov V V 2002 High pressure Raman study of CaV₂O₅ J. Phys.: Condens. Matter 14 L583 
IOPscience

[21]

Sherman E Ya, Fischer M, Lemmens P, van Loosdrecht P H M and Güntherodt G 1999 Electron–phonon and spin–phonon coupling in NaV₂O₅: charge fluctuation effects Europhys. Lett. 48 648 
IOPscience

Notes

Note1  Crystal structures are produced with XCrySDen [9].