Electron-Phonon Coupling and the Soft Phonon Mode in TiSe
We report high-resolution inelastic x-ray measurements of the soft phonon mode in the charge-density-wave compound TiSe. We observe a complete softening of a transverse optic phonon at the L point, i.e. , at . Renormalized phonon energies are observed over a large wavevector range . Detailed ab-initio calculations for the electronic and lattice dynamical properties of TiSe are in quantitative agreement with experimental frequencies for the phonon branch involving the soft mode. The observed broad range of renormalized phonon frequencies is directly related to a broad peak in the electronic susceptibility stabilizing the charge-density-wave ordered state. Our analysis demonstrates that a conventional electron-phonon coupling mechanism can explain a structural instability and the charge-density-wave order in TiSe although other mechanisms might further boost the transition temperature.
pacs:71.45.Lr, 63.20.kd, 63.20.dd, 63.20.dk
The origin of charge-density-wave (CDW) order, i.e., a periodic modulation of the electronic density, is a long-standing problem relevant to a number of important issues in condensed matter physics, such as the role of stripes in cuprates Kivelson03 () and charge fluctuations in the colossal magnetoresistive manganites Dagotto05 (). Chan and Heine derived the criterion for a stable CDW phase with a modulation wavevector as Chan73 ()
where is the electron-phonon coupling (EPC) matrix element associated with a mode at an unrenormalized energy of , is the dielectric response of the conduction electrons, and and are their Coulomb and exchange interactions. Static CDW order typically is taken as a result of a divergent electronic susceptibility due to nesting, i.e. parallel sheets of the Fermi surface (FS) separated by twice the Fermi wavevector . Electron-phonon coupling (EPC) is required to stabilize the structural distortion and, hence, an acoustic phonon mode at the CDW wavevector softens to zero energy at the transition temperature Chan73 (); Gruener88 (). However, when electronic probes reported only small and not well nested Fermi surfaces this scenario has been discarded for the prototypical CDW compound TiSe Wilson77 (); Hughes77 (); Rossnagel02 ().
Alternative scenarios such as indirect or band-type Jahn-Teller effects Hughes77 (); Rossnagel02 (); Whangbo92 () and, most prominently, exciton formation Wilson77 (); Cercellier07 (); Rohwer11 () are discussed in the theoretical as well as experimental literature. More recently, van Wezel et al. have invoked a model including exciton formation as well as EPC Wezel10a () and have shown that it can explain data from angle-resolved photoemission spectroscopy (ARPES) Wezel10b (), formerly taken as evidence of an excitonic insulating phase in TiSe Cercellier07 (). Determining the origin of CDW formation in TiSe is all the more important with respect to the nature of superconductivity, which emerges both as function of pressure Kusmartseva09 () and Cu intercalation Morosan06 (). In particular, pressure induced superconductivity is expected to be closely linked to the nature of the parent CDW state.
In this letter, we report a temperature dependent study of the CDW soft phonon mode in TiSe by high-resolution inelastic x-ray (IXS) scattering. We investigated both phonon energies as well as the phonon linewidths. We observe a complete softening of a transverse phonon branch at the L point close to the CDW transition temperature . Calculations of the lattice dynamical properties based on density-functional-perturbation theory (DFPT) describe our results very well and demonstrate that conventional EPC can stabilize CDW order contrary to the current discussion in the literature Rossnagel02 (); Cercellier07 (); Wezel10a ().
Our sample was a high-quality single crystal with dimensions ( that shows a sharp onset of a () CDW superstructure at as observed by synchrotron x-ray diffraction (XRD) Rosenkranz11 (). The high-resolution IXS experiments were carried out at the XOR 3-ID HERIX beamline of the Advanced Photon Source, Argonne National Laboratory. The incident energy was and the horizontally scattered beam was analyzed by a spherically curved silicon analyser (Reflection ). The full width at half maximum (FWHM) of the energy and wavevector space resolution was and Å, respectively. The components of the scattering vector are expressed in reciprocal lattice units (r.l.u.) ; with the lattice constants Å and Å of the hexagonal unit cell , ).
Measured energy spectra were fitted using a pseudo-Voigt function for the elastic line with a variable amplitude and fixed lineshape determined from scans through the CDW superlattice peak at base temperature. The phonon was fitted by a damped harmonic oscillator (DHO) function, where the energy of the damped phonon is given by Shirane02 (), where is the phonon energy renormalized only by the real part of the susceptibility, and denotes the phonon linewidth, which is closely related to the imaginary part of the susceptibility. The situation, where the phonon becomes overdamped and freezes into a periodic lattice distortion, is defined when the damping ratio reaches its critical value, .
Density-functional-theory calculations were performed in the framework of the mixed basis pseudopotential method Meyer (). The exchange-correlation functional was treated in the local-density approximation (LDA). Norm-conserving pseudo-potentials for Ti and Se were constructed with Ti and semicore states included in the valence space. Phonon frequencies and electronic contributions to the phonon linewidth were calculated using the linear response technique or density functional perturbation theory (DFPT) Baroni01 () in combination with the mixed-basis pseudopotential method Heid99 (). To resolve fine features related to the Fermi surface geometry, Brillouin-zone (BZ) integrations were performed with a dense hexagonanal -point mesh ( points in the irreducible BZ). The standard smearing technique was employed with a Gaussian broadening of . Displayed results were obtained for the undistorted hexagonal structure using experimental lattice constants and optimized internal parameters. The calculated electronic structure and lattice dynamical properties are in good agreement with previous results from LDA Jishi08 (); Calandra11 ().
Fig. 1 shows the temperature dependence of the soft phonon mode measured at the L-point, i.e. , from temperatures just above up to room temperature (Fig. 1a-c). At room temperature we find a phonon energy in good agreement with previous neutron scattering data Wakabayashi78 (). On lowering the temperature, softens to zero energy (within the experimental error) already at . At , the phonon is overdamped and can only be observed as a broad quasi-elastic peak underneath the sharp elastic line, which becomes gradually more dominant below (not shown). For low energy phonon scattering directly at the L point was undetectable due to the strong CDW superstructure peak. Overall, our results roughly agree with energies of the soft mode deduced from thermal diffuse scattering Holt01 ().
We studied in detail the dispersion of the transverse optic (TO) phonon starting at at A and ending in the CDW related soft mode at L, i.e. and 111Phonon scattering at was undetectable for due to strong elastic scattering.. We note that at there is an anti-crossing of this TO branch with an acoustic-like upward dispersing branch. The character of the soft mode is transferred to the energetically lower branch for . For simplicity, we speak of the soft-mode dispersion as represented by the dashed (blue) line in Fig. 3a, which involves parts of the TO branch () and the acoustic-like branch (). A set of energy scans taken at is shown in Fig. 2.
At (Fig. 2a), we observe two phonon peaks, where the one at lower energies corresponds to the character of the soft mode starting at at A. The second peak near represents the continuation of the upward dispersing acoustic-like branch in good agreement with structure factor calculations by DFPT. This mode does not show any significant temperature dependence. Results of its dispersion are not shown in Fig. 3 for the sake of clarity and will be discussed in detail elsewhere.
The energy of the phonon peak related to the soft mode already decreases for and, after the anti-crossing, shows a quasi-linear dispersion relation for (Fig. 2b-d), which extrapolates to a zero phonon energy at (Fig. 3a). At however, the strong CDW superlattice reflection made it impossible to directly measure the quasielastic scattering expected for the critically damped phonon. The calculated dispersions agree very well with these results. Corresponding to the strong renormalization of the phonon energy, we observe an increase of the damping ratio , which stays just below for . But just like the energy, the damping ratio also shows linear behavior for and extrapolates to an overdamped phonon at (Fig. 3b).
The range in q over which phonon energies are renormalized as function of temperature allows us to infer how sharp corresponding electronic signatures, e.g. in in a nesting scenario, might be Gruener88 (). We note that, in principle, the q dependence of the EPC matrix element can determine the periodicity of the low temperature superstructure as well Weber11 (). Because the soft mode at is still far from being unrenormalized (), we compare the dispersion at with results for in the CDW ordered state, as the latter give a good idea of the range of renormalized phonons. Fig. 3 clearly shows, that the phonon energies over the range are renormalized across the CDW transition. This large range in is in contrast to compounds, where a sharp FS nesting drives the CDW formation and renormalized phonon energies are observed only over a much smaller q range Renker73 (); Hoesch09 ().
Comparing the experimental results in more detail to the calculations, it is important to note that DFPT is a single particle response theory and, therefore, cannot include, e.g., exciton formation. However, EPC is calculated in great detail and, in particular, the electronic contribution to the phonon linewidth can be deduced (Fig. 4). of a specific phonon mode in turn can be approximately expressed as Heid02 ()
where the electronic joint density-of-states (JDOS) in turn is closely related to the imaginary part of the electronic susceptibility . We see that the q range over which shows a strong enhancement matches closely the range of observed renormalized phonon energies (Fig. 3a). Furthermore, also qualitatively explains the q dependence of the observed linewidth very well (see Fig. 4b). Quantitatively, the calculated linewidth is a factor of smaller than observed in experiment. Here, we note that similar observations have been made in the CDW compound NbSe Weber11 () and the conventional superconductor YNiBC Pintschovius08 (). In the latter, it is evident that only the lowest acoustic modes, which exhibit strong EPC, show this effect of too small calculated linewidths, whereas the linewidths of higher energy modes are predicted in much better agreement Weber11a (). Therefore, we think that soft modes as in TiSe are subject to an increased anharmonicity close to the structural phase transition reflected in an increased linewidth, which is not captured in our theory.
We now discuss the different contributions to , i.e. and the , according to equ. 2. The fact that for the soft phonon does not follow the increase of the total JDOS below is due to the presence of different bands near the Fermi energy. Whereas the JDOS at small is dominated by scattering processes within the Se bands around the point, at large scattering between Se and Ti bands dominate (Fig. 4a). Apparently, of the soft mode only samples the latter scattering processes. Using equation 2, and the partial JDOS of Se - Ti scattering processes, we can calculate . However, this is only possible for as scattering processes of the same type seem to contribute to the EPC of different phonon modes for and a clear determination of the relevant JDOS is not possible anymore. We calculate at and a decrease towards . For the same q range, the partial JDOS of Se - Ti scattering processes decreases by a factor of six. Hence, the dependence of the is the dominant factor in the large increase of the phonon linewidth. The corresponding phonon renormalization then leads to the structural instability at the point.
Our results explain the CDW phase transition in TiSe in the conventional picture of an increased electronic susceptibility at aided by a strong EPC matrix element. This is not in contradiction to ARPES data, on the basis of which FS nesting previously has been ruled out as a driving force of the CDW transition in TiSe Rossnagel02 (). Indeed, our results confirm that TiSe does not exhibit a divergent . However, the peak in the is connected with an enhancement of Calandra11 (), which can, according to equ. 1, tip the balance between undistorted and CDW ordered phases if stabilized by a sufficiently large . We note that DFPT is a zero temperature approach and, therefore, cannot calculate a transition temperature corresponding to, e.g., the calculated EPC strength. So, EPC might be only one ingredient for driving the CDW transition in TiSe, but our analysis shows that its importance has been greatly underestimated.
In conclusion we have reported a detailed analysis of the soft phonon behaviour in the CDW compound TiSe. The experimental observations are well reproduced by ab-initio calculations, which demonstrate that electron-phonon-coupling is strong enough to stabilize a structural phase transition in TiSe at finite temperatures. Our work emphasizes the essential part of electron-phonon coupling matrix elements and provides a new route to seemingly mysterious phase transitions as far as the electronic footprint is concerned.
Acknowledgements.We acknowledge valuable discussions with Jasper van Wezel and Mike Norman. Work at Argonne was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract No. DE-AC02-06CH11357.
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