Spatial Distortion of Vibration Modes via Magnetic Correlation of Impurities

Long wavelength vibrational modes in the ferromagnetic semiconductor ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{As}$ are investigated using time resolved x-ray diffraction. At room temperature, we measure oscillations in the x-ray diffraction intensity corresponding to coherent vibrational modes with well-defined wavelengths. When the correlation of magnetic impurities sets in, we observe the transition of the lattice into a disordered state that does not support coherent modes at large wavelengths. Our measurements point toward a magnetically induced broadening of long wavelength vibrational modes in momentum space and their quasilocalization in the real space. More specifically, long wavelength vibrational modes cannot be assigned to a single wavelength but rather should be represented as a superposition of plane waves with different wavelengths. Our findings have strong implications for the phonon-related processes, especially carrier-phonon and phonon-phonon scattering, which govern the electrical conductivity and thermal management of semiconductor-based devices.

Figure 1

Measurements (filled circles) and simulations (solid lines) of time-dependent x-ray diffracted intensity from ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{As}$. Panels (a),(b),(c) show data measured at room temperature, whereas panels (d),(e),(f), at 60 K. The data have been compared with predictions of dynamical theory of x-ray diffractions that assume acoustic modes with well-defined spatial periodicity $q$ (red solid lines). The black solid line in panel (d) represents a damped elastic wave with $q=0.018$ $\pi /a$, ${\omega }_q=2\pi /(19\mathrm{ps})$ and damping time constant $\tau =12\mathrm{ps}$.

Figure 2

The temperature dependence of the measured time-dependent diffracted intensities probing the acoustic mode $q=0.015\pi /a$. The data for different temperatures have been shifted along the intensity axis for better visibility. Inset: the measured magnetization curve (left) and, the inverse of the damping time (diamonds, right) and calculated susceptibility (solid line, right) [37].

Figure 3

Eigenvectors of a linear chain of atoms with randomly distributed impurities. Panels (a) and (b) show a schematic drawing of a mode eigenvector without and with magnetic correlation of spins, respectively. In (a), mode eigenvectors can be described by a sinusoidal wave with a well-defined wavelength ${\lambda }_q$. In (b), ferromagnetic coupling of randomly distributed magnetic impurities leads to a destruction of a sinusoidal pattern of a vibrational mode. The average distance between two phase interruptions is represented by the coherence length ${\mathrm{\Lambda }}_q$. Panel (c) shows a simulated mode $q=0.018$ $\pi /a$ eigenvector without spin-phonon interaction ($\mathrm{\Delta }{k}_{\mathrm{mag}}=0$), whereas panel (d) displays modification of the mode eigenvector when the magnetic contribution to the spring constants around impurities has been set to $\mathrm{\Delta }{k}_{\mathrm{mag}}=-24.7\mathrm{meV}/{\AA }^2$. Panel (e) shows the Fourier transformation power spectrum of selected acoustic mode eigenvectors, and panel (f) compares the dispersion relation with that of the unperturbed crystal.

Figure 4

Simulated coherence length as a function of wave vector $q$. The critical coherence length ${\mathrm{\Lambda }}_q^{\mathrm{crit}}$ marks the region below which the coherence length of the mode $q$, ${\mathrm{\Lambda }}_q$, is smaller than the wavelength ${\lambda }_q=2\pi /q$. In this region phonons cannot sample the periodicity of the lattice and the wave vectors cannot rationalize them anymore. Blue triangles represent the oscillation amplitudes of the x-ray diffraction signal measured at 60 K normalized to those at room temperature. The data corresponding to the acoustic modes $0.018\pi /a$ and $0.015\pi /a$ follow the functional dependence of ${\mathrm{\Lambda }}_q(q)$ multiplied by a constant factor.