Phonon spectrum in the spin-Peierls phase of CuGeO$_3$

Abstract

CuGeO$_3$ has long been studied as a prototypical example of the spin-Peierls transition in a $S = 1/2$ Heisenberg chain. Despite intensive investigation of this quasi-one-dimensional material, systematic measurements and calculations of the phonon excitations in the dimerized phase have not to date been possible, leaving certain aspects of the spin-Peierls phenomenon unresolved. We perform state-of-the-art density functional theory (DFT) calculations to compute the electronic structure and phonon dynamics in the low-temperature dimerized phase. We also perform high-resolution neutron spectroscopy to measure the full phonon spectrum over multiple Brillouin zones. We find excellent agreement between our numerical and experimental results that extend to all measurement temperatures. Notable features of our phonon spectra include a number of steeply dispersive modes, nonmonotonic dispersion features, and specific phonon anticrossings, which we relate to the mode eigenvectors. By calculating the magnetic interactions within DFT and studying the effects of different phonon modes on the superexchange paths, we discuss the possibility of observing spin-phonon hybridization effects in experiments performed both in and out of equilibrium.

Figures (11)

• Figure 1: (a) Atomic structure of CuGeO$_3$ represented using three planes of CuO$_2$ chains. The upper gray box shows one unit cell (which contains two formula units) of the orthorhombic undimerized structure and the lower gray box one primitive monoclinic unit cell (four formula units) of the dimerized structure. (b) Schematic representation of the uniform (undimerized) spin chain. (c) Representation of the dimerized spin chain, with the two different Cu atoms numbered. The structural dimerization causes the magnetic interaction parameter of the undimerized chain, $J$, to be changed to alternating stronger ($J_1$) and weaker ($J_1^\prime$) interactions. (d) Schematic representation of the magnetic excitation spectrum of the uniform $S = 1/2$ Heisenberg chain, which is a characteristic continuum of two-spinon processes. (e) Excitation spectrum of the alternating chain, which contains a low-energy triplon and at higher energies shows remnants of the two-spinon continuum.
• Figure 2: Phonon spectra of CuGeO$_3$ at $T = 5$ K measured with $E_{\rm i} = 54.4$ meV (a,b) and $E_{\rm i} = 35.0$ meV (c,d). Upper panels show INS measurements and lower panels DFT calculations. (a) $L$ direction at $H = 0$ with integration ranges $-1 < K < 1$ and $-0.3 < H < 0.3$; the wide integration range in $K$ is justified by the invariance of the dominant phonons in this direction and is chosen to optimize the detector coverage of these branches. (b) $L$ direction at $K = -5$ and $H = 0$ with integration ranges $-5.1 < K < -4.9$ and $-0.2 < H < 0.2$. (c) $K$ direction at $L = 2$ and $H = 0$ with integration ranges $1.85 < L < 2.15$ and $-0.15 < H < 0.15$. (d) $H$ direction at $L = 1$ and $K = -6$ with integration ranges $-6.15 < K < -5.85$ and $0.85 < L < 1.15$. (e-h) Analogous spectra computed by DFT and displayed with the same integration ranges.
• Figure 3: Eigenvectors of selected phonons in CuGeO$_3$, representing the atomic displacements in six phonon modes of the dimerized phase. Green arrows represent the relative phases of the atomic motions in each panel and their relative magnitudes across all panels. (a) B$_{3u}$ mode at 2.81 THz. (b) B$_{1g}$ mode at 3.03 THz. (c) A$_{g}$ mode at 6.05 THz. (d) B$_{3u}$ mode at 8.96 THz. (e) B$_{3g}$ mode at 16.36 THz. (f) A$_{g}$ mode at 21.83 THz.
• Figure 4: Avoided crossings in the phonon spectrum of CuGeO$_3$. INS spectra are shown in the first column and DFT spectra in the second, where orange circles mark prominent phonon anticrossings. The third and fourth columns show the eigenvectors of the two phonon modes involved in the anticrossings marked by the dark orange circles. (a-d) $L$ direction at $H = 0$ with integration ranges $-0.3 < H < 0.3$ and $-1 < K < 1$. (e-h) $L$ direction at $K = -5$ with integration ranges $-0.2 < H < 0.2$ and $-5.1 < K < -4.9$. (i-l) $K$ direction at $L = 2$ and $H = 0$ with integration ranges $-0.15 < H < 0.15$ and $1.85 < L < 2.15$. The incident energies are $E_{\rm i} = 54.4$ meV in panels (a) and (e) and $E_{\rm i} = 35.0$ meV in panel (i).
• Figure 5: Phonon spectra of CuGeO$_3$ measured at 5 K (a,e,i,m), 20 K (b,f,j,n), and 110 K (c,g,k,o) for four selected $(H~K~L)$ paths in the Brillouin zone at incident energies $E_{\rm i} = 54.4$ meV (a-c), $E_{\rm i} = 35.0$ meV (e-g,i-k), and $E_{\rm i} = 24.4$ meV (m-o). Panels (d,h,l,p) show the corresponding DFT calculations. In panels (a-c), (e-g), and (i-k), only minimal thermal effects are discernible, primarily due to the disappearance of the magnetic response in panel (a), whereas in panels (m-o) some more significant changes are evident.