A new quantum-theoretical derivation of the elastic and inelastic scattering probability of He atoms from a metal surface, where the energy and momentum exchange with the phonon gas can only occur through the mediation of the surface free-electron density, shows that the Debye-Waller exponent is directly proportional to the electron-phonon mass coupling constant $λ$. The comparison between the values of $λ$ extracted from existing data on the Debye-Waller factor for various metal surfaces and the $λ$ values known from literature indicates a substantial agreement, which opens the possibility of directly extracting the electron-phonon coupling strength in quasi-2D conducting systems from the temperature or incident energy dependence of the elastic Helium atom scattering intensities.
We show that the method of calculations of the Debye-Waller factors for IC structures by J.D. Axe, Phys. Rev. (1971), B 21, 4181, is not applicable for such calculations. The set of variables, which should be used for such calculations, is the phase $ϕ$ and the amplitude of the IC modulation, but not the real and imaginary parts of the complex IC modulation function.
An ab initio equation of motion method is introduced to calculate the temperature-dependent mean square vibrational amplitudes which appear in the Debye-Waller factors in x-ray absorption, x-ray scattering, and related spectra. The approach avoids explicit calculations of phonon-modes, and is based instead on calculations of the displacement-displacement time correlation function from ab initio density functional theory molecular dynamics simulations. The method also yields the vibrational density of states and thermal quantities such as the lattice free energy. Illustrations of the method are…
The Debye-Waller factor has been calculated for stabilized delta-phase plutonium with 5% aluminum. A quasi-harmonic Born-von Karman force model with temperature dependent phonon frequencies was used to calculate the mean-square thermal atomic displacement from absolute zero to 800 K. Implementation of the observed anomalous softening of the long wavelength phonons with increasing temperature cannot account for the softening of the measured thermal parameter at high temperatures nor for its rather high value at low temperatures. The implications for diffraction measurements on delta-phase stabi…
The lattice vibrations are determined in the quasi-harmonic approximation for elemental Au and Cu to twice their normal density by first-principles electronic band-structure calculations. It is found for these materials that the important moments of the phonon density of states can be obtained to high accuracy from short-ranged force constant models. We discuss the implications for the Grueneisen parameters on the basis of calculated phonon moments and their approximations by using bulk moduli and Debye-Waller factors.
By means of first principles calculations, we computed the effective electron-phonon coupling constant $G_0$ governing the electron cooling in photoexcited bismuth. $G_0$ strongly increases as a function of electron temperature, which can be traced back to the semi-metallic nature of bismuth. We also used a thermodynamical model to compute the time evolution of both electron and lattice temperatures following laser excitation. Thereby, we simulated the time evolution of (1 -1 0), (-2 1 1) and (2 -2 0) Bragg peak intensities measured by Sciaini et al [Nature 458, 56 (2009)] in femtosecond elect…
An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption spectra. These factors are represented in terms of the cumulant expansion up to third order. They account respectively for the net thermal expansion $σ^{(1)}(T)$, the mean-square relative displacements $σ^2(T)$, and the asymmetry of the pair distribution function $σ^{(3)}(T)$. Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes $u^2(T)$. Our method is based on density functional theory calculations of the dynamical matri…
The bandgap of insulating materials is renormalized in various ways by the electron-phonon interaction owing to the dynamical and quantum fluctuations of nuclei. These fluctuation effects are considered in the perturbative Allen-Heine-Cardona theory using the formulae for the Fan-Migdal and Debye-Waller terms. However, the material dependence is not clear in the formulae. Thus, in this study, we focus on the analytical form of the Debye-Waller term and find that the term can be reformulated using the momentum matrix. In addition, the optical selection rule is found to play a role. For diamond-…
We present some theoretical results on the lattice vibrations that are necessary for a concise derivation of the Debye-Waller factor in the harmonic approximation. First we obtain an expression for displacement of an atom in a crystal lattice from its equilibrium position. Then we show that an atomic displacement has the Gaussian distribution. Finally, we obtain the computational formula for the Debye-Waller factor in the Debye model.
The enhancement of mobility at the surface of an amorphous alloy is studied using a combination of molecular dynamic simulations and normal mode analysis of the non-uniform distribution of Debye-Waller factors. The increased mobility at the surface is found to be associated with the appearance of Arrhenius temperature dependence. We show that the transverse Debye-Waller factor exhibits a peak at the surface. Over the accessible temperature range, we find that the bulk and surface diffusion coefficients obey the same empirical relationship with the respective Debye-Waller factors. Extrapolating…
We use ultrafast electron diffraction to detect the temporal evolution of phonon populations in femtosecond laser-excited ultrathin single-crystalline gold films. From the time-dependence of the Debye-Waller factor we extract a 4.7 ps time-constant for the increase in mean-square atomic displacements. We show from the increase of the diffuse scattering intensity that the population of phonon modes near the X and K points in the Au fcc Brillouin zone grows with timescales of 2.3 and 2.9 ps, respectively, faster than the Debye-Waller average. We find that thermalization continues within the init…
We have studied the topological insulator Bi$_2$Te$_3$(111) by means of helium atom scattering. The average electron-phonon coupling $λ$ of Bi$_2$Te$_3$(111) is determined by adapting a recently developed quantum-theoretical derivation of the helium scattering probabilities to the case of degenerate semiconductors. Based on the Debye-Waller attenuation of the elastic diffraction peaks of Bi$_2$Te$_3$(111), measured at surface temperatures between $110~\mbox{K}$ and $355~\mbox{K}$, we find $λ$ to be in the range of $0.04-0.11$. This method allows to extract a correctly averaged $λ$ and to addre…
Combining a precise ab initio electron band structure calculation of the TiO2 rutile structure with the temperature evolution of the Ti mean-square displacements, we reproduce a puzzling temperature increase of the electric field gradient at Ti sites in TiO2, observed experimentally. Our method employs a procedure of averaging two quadrupole electron density components (L = 2) inside a sphere vibrating with the Ti nucleus at its center, where the key factor introducing the temperature dependence is the square root of the Debye-Waller factor. Although the Debye-Waller factor always reduces the …
Probing Temperature at Nanoscale through Thermal Vibration Characterization using Scanning Precession Electron Diffraction
State Key Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Future Material Innovation Center, Zhangjiang Institute for Advanced Study, Shanghai Jiao Tong University（上海交通大学）；School of Physics and Electronics, Hunan Key Laboratory of Super Microstructure and Ultrafast Process, State Key Laboratory of Powder Metallurgy, Central South University（中南大学）；Research Center for Scientific Data Hub, Zhejiang Lab（浙江实验室）；School of Physics and Astronomy, Shanghai Jiao Tong University, Key Laboratory for Particle Astrophysics and Cosmology (MOE), SKLPPC（上海交通大学）；School of Mechanical Engineering, Shanghai Jiao Tong University（上海交通大学）
The effects of thermal diffuse scattering on the transmission and eventual diffraction of highly accelerated electrons are investigated with a method that incorporates the frozen phonon approximation to the exact numerical solution of the time-dependent Schrödinger equation. Unlike other methods in the related literature, in this approach the attenuation of diffraction features arises in a natural way by averaging over a number of wave-packet realizations, thus avoiding any additional experimentally obtained Debye-Waller factors or artificial modulations. Without loss of generality, the method…
He atom scattering has been demonstrated to be a sensitive probe of the electron-phonon interaction parameter $λ$ at metal and metal-overlayer surfaces. Here it is shown that the theory linking $λ$ to the thermal attenuation of atom scattering spectra (the Debye-Waller factor), can be applied to topological semimetal surfaces, like the quasi-one dimensional charge-density-wave system Bi(114) and the layered pnictogen chalcogenides.
High-resolution neutron backscattering techniques are exploited to study the elastic and quasi-elastic response of the high-density amorphous (HDA), the low-density amorphous (LDA) and the crystalline ice Ic upon temperature changes. Within the temperature ranges of their structural stability (HDA at T > 80 K, LDA at T > 135 K, ice Ic at T < 200 K) the Debye-Waller factors and mean-square displacements characterise all states as harmonic solids. During the transformations HDA->LDA (T ~ 100 K), LDA->Ic (T ~ 150K) and the supposed glass transition with Tg ~ 135 K no relaxation processes can be d…
We report on ultrafast electron diffraction on high quality single crystal silicon. The ultrafast dynamics of the Bragg peaks exhibits a giant photo-induced response which can only be explained in the framework of dynamical diffraction theory, taking into account multiple scattering of the probing electrons in the sample. In particular, we show that lattice heating following photo-excitation can cause an unexpected increase of the Bragg peak intensities, in contradiction with the well-known Debye-Waller effect. We anticipate that multiple scattering should be systematically considered in ultra…
The relation between ab initio molecular dynamics formalism and the electron-phonon interaction formalism [P.B. Allen and V. Heine, J. Phys. C 9, 2305 (1976)] is explored. The fundamental quantity obtained in the AIMD formalism - total energy for any configuration - is also obtained from the formalism (ES-DWF) that incorporates the role of Debye-Waller Factor in electronic structure calculations. The two formalisms are exactly equivalent and represent the direct and perturbation theory approaches to determine total energy. This equivalence allows either formalism to be used depending on the re…
Atom scattering is becoming recognized as a sensitive probe of the electron-phonon interaction parameter $λ$ at metal and metal-overlayer surfaces. Here, the theory is developed linking $λ$ to the thermal attenuation of atom scattering spectra (in particular, the Debye-Waller factor), to conducting materials of different dimensions, from quasi-one dimensional systems such as W(110):H(1$\times$1) and Bi(114), to quasi-two dimensional layered chalcogenides and high-dimensional surfaces such as quasicrystalline 2ML-Ba(0001)/Cu(001) and d-AlNiCo(00001). Values of $λ$ obtained using He atoms compar…
Recent theory has demonstrated that the value of the electron-phonon coupling strength $λ$ can be extracted directly from the thermal attenuation (Debye-Waller factor) of Helium atom scattering reflectivity. This theory is here extended to multivalley semimetal systems and applied to the case of graphene on different metal substrates and graphite. It is shown that $λ$ rapidly increases for decreasing graphene-substrate binding strength. Two different calculational models are considered which produce qualitatively similar results for the dependence of $λ$ on binding strength. These models predi…
The classical motion of gliding dislocation lines in slip planes of crystalline solid helium leads to plastic deformation even at temperatures far below the Debye temperature and can affect elastic properties. In this work we argue that the gliding of dislocations and plasticity may be the origin of many observed elastic anomalies in solid He-4, which have been argued to be connected to supersolidity. We present a dislocation motion model that describes the stress-strain $τ$-$ε$ curves and work hardening rate $dτ/dε$ of a shear experiment performed at constant strain rate $\dotε$ in solid heli…
We report a fully {\it ab-initio} calculation of the temperature dependence of the electronic band structure of PbTe. We address two main features relevant for the thermoelectric figure of merit: the temperature variations of the direct gap and the difference in energies of the two topmost valence band maxima located at L and $Σ$. We account for the energy shift of the electronic states due to thermal expansion, as well as electron-phonon interaction computed using the non-adiabatic Allen-Heine-Cardona formalism within density functional perturbation theory and the local density approximation.…
A first principles analysis of order-disorder transition in alloys shows that ordering energy is a function of temperature due to thermal vibrations. The inter-nuclear potential energy term converges if zero point vibrations are incorporated and this method can replace the Ewald sum method. Core energy contributions to the ordering energy are stored exclusively in superlattice lines. The effect of electron-phonon interactions on ordering energy is of the same order of magnitude as ordering energy near transition temperatures and cannot be ignored. Ising model and variants are incorrect in expl…
The elastic moduli, elastic anisotropy coefficients, sound velocities and Poisson's ratio of hcp solid helium have been calculated using density functional theory in generalized gradient approximation (up to $30$ TPa), and pair+triple semi-empirical potentials (up to 100 GPa). Zero-point vibrations have been treated in the Debye approximation assuming $^4$He isotope (we exclude the quantum-crystal region at very low pressures from consideration). Both methods give a reasonable agreement with the available experimental data. Our calculations predict significant elastic anisotropy of helium ($\t…
We present a combined experimental and theoretical study of the surface vibrational modes of the topological insulator (TI) Bi$_2$Se$_3$ with particular emphasis on the low-energy region below 10 meV that has been difficult to resolve experimentally. By applying inelastic helium atom scattering (HAS), the entire phonon dispersion was determined and compared with density functional perturbation theory (DFPT) calculations. The intensity of the phonon modes is dominated by a strong Rayleigh mode, in contrast to previous experimental works. Moreover, also at variance with recent reports, no Kohn-a…
German Aerospace Center (DLR), Institute of Quantum Technologies（德国航空航天中心量子技术研究所）；Institut des Sciences Moléculaires d’Orsay (ISMO), Centre national de la recherche scientifique (CNRS), University Paris-Sud, Université Paris-Saclay（奥赛分子科学研究所，法国国家科学研究中心，巴黎第十一大学，巴黎-萨克雷大学）；University of Vienna, Faculty of Physics（维也纳大学物理学院）
In ultrafast electron diffraction (UED) experiments, accurate retrieval of time-resolved structural parameters, such as atomic coordinates and thermal displacement parameters, requires an accurate scattering model. Unfortunately, kinematical models are often inaccurate even for relativistic electron probes, especially for dense, oriented single crystals where strong channeling and multiple scattering effects are present. This article introduces and demonstrates dynamical scattering models tailored for quantitative analysis of UED experiments performed on single-crystal films. As a case study, …
At low temperature T, the experimental gap of silicon decreases as E_g(T)=E_g(0)-AT^4. The main reason is electron-phonon renormalization. The physics behind the T^4-power law is more complex than has been realized. Renormalization by intraband scattering requires a careful non-adiabatic treatment in order to correctly include acoustic phonons and avoid divergences from piezoacoustic phonon interactions. The result is an unexpected low T term E_g(0)+A' T^p with positive coefficient A', and power p=4 for non-piezoelectric materials, and power p=2 for piezoelectric materials. The acoustic phonon…
