Electron-phonon Coupling Matrix Elements Research Articles

Electron–phonon physics from first principles using the EPW code

EPW is an open-source software for ab initio calculations of electron–phonon interactions and related materials properties. The code combines density functional perturbation theory and maximally localized Wannier functions to efficiently compute electron–phonon coupling matrix elements, and to perform predictive calculations of temperature-dependent properties and phonon-assisted quantum processes in bulk solids and low-dimensional materials. Here, we report on significant developments in the code since 2016, namely: a transport module for the calculation of charge carrier mobility under electric and magnetic fields using the Boltzmann transport equation; a superconductivity module for calculations of phonon-mediated superconductors using the anisotropic multi-band Eliashberg theory; an optics module for calculations of phonon-assisted indirect transitions; a module for the calculation of small and large polarons without supercells; and a module for calculating band structure renormalization and temperature-dependent optical spectra using the special displacement method. For each capability, we outline the methodology and implementation and provide example calculations.

Intervalley scattering induced significant reduction in lattice thermal conductivities for phosphorene.

The thermal transport properties of buckled phosphorene (β-P) and antimonene (β-Sb) are investigated using first-principles methods. The large acoustic-optical phonon gaps of 3.8 THz and 2.2 THz enable the four-phonon interaction to play an important role in phonon scattering for both β-P and β-Sb. Considering the electron-phonon coupling, the lattice thermal conductivity can further undergo 84% decrease to 4.9 W mK-1 for p-type β-P at n = 5 × 1013 cm-2. By quantitatively describing the scattering probability of electrons in different paths combined with electron-phonon coupling matrix element analysis, it is found that multi-valley features of electronic band structure and strong electron-phonon coupling strength make electrons have strong intervalley scattering behavior in β-P. The former plays an important role in the energy conservation condition of the scattering process, and the latter determines the selection rule. Our work elucidates the contribution of higher-order phonon interactions as well as electron-phonon coupling effects to lattice thermal conductivity, and provides a new idea for finding materials with low lattice thermal conductivity induced by intervalley scattering.

Nonadiabatic electron-phonon coupling and its effects on superconductivity

As one of the most striking phenomena in many-body systems, the breakdown of the adiabatic Born-Oppenheimer (ABO) approximation is essential for understanding and predicting the dynamics of quantum materials. Here, we demonstrate that strong nonadiabatic (NA) effects can emerge in electron-phonon coupling (EPC) and resultant superconductivity, yielding a surprisingly large contribution (up to 20%) to the total EPC strength \ensuremath{\lambda} of doped graphene/graphane and modulating the superconductivity transition temperature ${T}_{c}$ by \ensuremath{\sim}20%. The NA EPC, with a detectable isotopic effect, results from the renormalized EPC matrix elements (up to 40%) combined with the Fermi surface modulation, due to electron distributions deviating from the equilibrium. Our results imply that nonadiabaticity plays a wider and more important role than previously conceived, offering new understandings and strategies for dynamic engineering of quantum materials.

Doping dependence of the electron-phonon coupling in two families of bilayer superconducting cuprates

While electron-phonon coupling (EPC) is crucial for Cooper pairing in conventional superconductors, its role in high-$T_c$ superconducting cuprates is debated. Using resonant inelastic x-ray scattering at the oxygen $K$-edge, we studied the EPC in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) and Nd$_{1+x}$Ba$_{2-x}$Cu$_3$O$_{7-\delta}$ (NBCO) at different doping levels ranging from heavily underdoped ($p =0.07$) to overdoped ($p=0.21$). We analyze the data with a localized Lang-Firsov model that allows for the coherent excitations of two phonon modes. While electronic band dispersion effects are non-negligible, we are able to perform a study of the relative values of EPC matrix elements in these cuprate families. In the case of NBCO, the choice of the excitation energy allows us to disentangle modes related to the CuO$_3$ chains and the CuO$_2$ planes. Combining the results from the two families, we find the EPC strength decreases with doping at $\mathbf{q_\parallel}=(-0.25, 0)$ r.l.u., but has a non-monotonic trend as a function of doping at smaller momenta. This behavior is attributed to the screening effect of charge carriers. We also find that the phonon intensity is enhanced in the vicinity of the charge-density-wave (CDW) excitations while the extracted EPC strength appears to be less sensitive to their proximity. By performing a comparative study of two cuprate families, we are able to identify general trends in the EPC for the cuprates and provide experimental input to theories invoking a synergistic role for this interaction in $d$-wave pairing.

Stability and phonon-limited mobility for CsSnI3 and CsPbI3

Phase stability of CsSnI3 and CsPbI3 is revealed by analyzing molecular volume, free energy and lattice vibration. Furthermore, the carrier effective mass is calculated by constant energy surface. Surprisingly, the values for electron/hole are as low as 0.06/0.03 m0 for B-α CsSnI3, and 0.08/0.06 m0 for B-α CsPbI3. They gradually increase in the order of B-α, B-β and B-γ. The effective mass for CsSnI3 is lower than that for CsPbI3, implying a high mobility in CsSnI3. At 300 K, the mobility is predicted by empirical models and electron-phonon coupling (EPC) matrix element based on Boltzmann transport equation (BTE). The value from longitudinal acoustic phonon (LAP) model is much higher than that from longitudinal optical phonon (LOP) model. So the mobility is mainly determined by the scattering from polar optical phonon. These perovskites exhibit small high frequency dielectric constant (ϵ∞) and large dielectric constant difference (Δϵ = ϵ0 − ϵ∞), which limit the mobility. Finally, phonon-limited mobility is predicted by the EPC matrix element. The values of CsSnI3/CsPbI3 are μe = 311/118 and μh = 450/70 cm2V−1s−1 in B-β, μe = 99/70 and μh = 218/46 cm2V−1s−1in B-γ.

Ab initio study of photoelectric properties in ZnO transparent conductive oxide

Zinc oxide is regarded as a third generation transparent conductive film which can replace indium doped tin oxide. It can be widely used in the electrodes of solar cells, the display of electronic instruments and other devices. In this paper, the photoelectric properties of ZnO thin films and their different growth directions were studied, and the best growth direction was obtained. The ZnO system was calculated by density functional theory. The band structure, density of states and other electronic properties of the stable structure are calculated by using Quantum Espresso program. It is found that the Valence Band Maximum and Conduction Band Minimum of ZnO (002) crystal plane both pass through the Fermi level, and the system exhibits semi-metallic properties, which has an important effect on the electrical properties. The band structure is interpolated by wannier90 program, and the precise maximum localized Wannier functions are obtained. Phonons properties are studied by density functional perturbation theory, and the accurate phonon dynamic matrix and the electron phonon coupling matrix elements are obtained. Based on that, the Perturbo program is used to solve the Boltzmann Transport Equation by using relaxation time approximation and electron phonon coupling theory. The scattering rate and relaxation time under different temperature are obtained and the reason why the conductivity and mobility decrease with the increase of temperature is analyzed. In the optical properties of the calculation, the time-dependent density functional theory and epsilon.x program are used to solve the Kohn–Sham equation, and then find out the refractive index, absorption coefficient, visible light transmittance and other important optical properties by obtaining the dielectric function. Combined with the analysis of electrical and optical properties, the (002) crystal plane is determined to be the best growth direction, which is well consistent with the preferred orientation obtained from the experiment. Therefore, it provides a theoretical basis and support for the preparation of better ZnO thin films. The work in this paper fills the blank of theoretical research on ZnO.

Behaviour of induced states of substitutional and adatom impurity doping on electronic transport properties of single-layer black phosphorus

Substitutional and adatom impurity doping can be considered quite effective or even better for modulating the electronic transport coefficients. Herein, we have studied the impact of substitutional doping by Al and Cl multi atoms and adatom adsorption by O, Cu, K, and Ag atoms on the electronic transport properties of single-layer black phosphorus using the first-principles method based density functional theory framework in conjunction with Boltzmann transport equation at room temperature. We found that these impurity atoms introduce new donor and acceptor induced eigenstates and flat bands within the fundamental bandgap. We have calculated the scattering rates and relaxation time by solving the Boltzmann transport equation with the ab-initio calculated electron-phonon interactions. The electron-phonon coupling matrix elements are computed through maximally localized Wannier functions and density functional perturbation theory using Wannier-Fourier interpolation on a fine sampling of the Brillouin zone . The binding energies illustrated the structural stability of all doped systems and found that phosphorene showed a stronger adsorption capability as compared with the graphene and MoS 2 sheet. Moreover, binding energy for the substitutionally doped system is found to decrease with the increasing atomic number. Furthermore, the achieved electron mobility, electrical conductivity ( σ ), and Seebeck coefficient (S) of pristine single-layer black phosphorus are ~279 cm 2 V −1 s −1 , 6.1 × 10 6 Ω −1 m −1 and 1.4 mV/K respectively. The calculated values of carrier mobility , electrical conductivity and Seebeck coefficient have found a rich diversity in each impurity doped layer, and the negative differential behaviour is observed. These results provide comprehensive information about chemical functionalization and bandgap tunability for modelling of phosphorene based devices. • Substitutional and adatom impurity doping are effective for modulating the electronic transport coefficients of phosphorene. • We have studied the impact of substitutional doping by Al and Cl multi atoms and adatom adsorption by O, Cu, K, and Ag atoms. • We have calculated the scattering rates and relaxation time by solving the Boltzmann transport equation. • The binding energies illustrated the structural stability and found that phosphorene showed a stronger adsorption capability. • The calculated values of carrier mobility, electrical conductivity and Seebeck coefficient have found a rich diversity.

AICON2: A program for calculating transport properties quickly and accurately

Calculating the transport properties, such as electrical conductivity, has been a great challenge in materials modeling fields because of its complexity. We have implemented an algorithm to calculate the electronic transport properties using the generalized Kane band model and perturbation theory in the framework of the relaxation time approximation. Three scattering mechanisms affect the total relaxation time: acoustic phonon scattering, polar optical phonon scattering, and ionized impurity scattering. All the necessary parameters can be calculated from first principles. The capability of the program was tested on a group of semiconductors, and the obtained results show reasonable agreement with experiment. The program works fast, and is robust and especially appropriate for high-throughput screening of thermoelectric materials. Program summaryProgram title: AICON2CPC Library link to program files:https://doi.org/10.17632/s9b8y8t92c.2Code Ocean capsule:https://codeocean.com/capsule/7509547Licensing provisions: GPLv3Programming language: Python3External routines/libraries: Numpy, Scipy, spglib, pymatgen, atomate, emcNature of problem: Calculation of electrical and thermal conductivity from first principles requires a large number of computing resources in order to construct electron-phonon coupling matrix elements, integrate over the Brillouin zone and construct high-order force constants matrix.Solution method: Combining the perturbation theory with the deformation potential theory to calculate the electronic transport properties, using the modified Debye-Callaway model to calculate the phonon transport properties.

Impact of spin-orbit coupling on the physical properties and superconductivity of Ir-rich superconductor Mg2Ir3Si: A first-principles investigation

In this study, we elucidate the impact of spin-orbit coupling (SOC) on the physical properties and superconductivity of ternary Laves compound Mg2Ir3Si by performing full relativistic and scalar relativistic ab initio pseudopotential calculations. The consideration of SOC shifts the scalar relativistic bands in energy and changes degeneracy of the bands, but the elastic and mechanical properties are marginally afftected. SOC decreases the electron-phonon coupling strength λ by 11% (from 1.05 to 0.93) A critical assessment of phonon spectrum and Eliashberg spectral function reveals that the origin of the SOC-induced diminishment of λ lies both in slight hardening of some low-frequency phonon modes and in a decrease in the electron-phonon coupling matrix elements. As a consequence, the inclusion of SOC reduces the value of superconducting transition temperature from 7.64 K to 7.05 K, resulting in improved agreement with its recent experimental value of 7 K.

A coupled cluster framework for electrons and phonons.

We describe a coupled cluster framework for coupled systems of electrons and harmonic phonons. Neutral and charged excitations are accessed via the equation-of-motion version of the theory. Benchmarks on the Hubbard-Holstein model allow us to assess the strengths and weaknesses of different coupled cluster approximations, which generally perform well for weak to moderate coupling. Finally, we report progress toward an implementation for ab initio calculations on solids and present some preliminary results on finite-size models of diamond with a linear electron-phonon coupling. We also report the implementation of electron-phonon coupling matrix elements from crystalline Gaussian type orbitals within the PySCF program package.

Enhanced Electron-Phonon Coupling for Charge-Density-Wave Formation in La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}.

Charge density wave (CDW) correlations are prevalent in all copper-oxide superconductors. While CDWs in conventional metals are driven by coupling between lattice vibrations and electrons, the role of the electron-phonon coupling (EPC) in cuprate CDWs is strongly debated. Using Cu L_{3} edge resonant inelastic x-ray scattering, we study the CDW and Cu-O bond-stretching phonons in the stripe-ordered cuprate La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}. We investigate the interplay between charge order and EPC as a function of doping and temperature and find that the EPC is enhanced in a narrow momentum region around the CDW ordering vector. By detuning the incident photon energy from the absorption resonance, we extract an EPC matrix element at the CDW ordering vector of M≃0.36 eV, which decreases to M≃0.30 eV at high temperature in the absence of the CDW. Our results suggest a feedback mechanism in which the CDW enhances the EPC which, in turn, further stabilizes the CDW.

Towards theoretical framework for probing the accuracy limit of electronic transport properties of SnSe2 using many-body calculations

A theoretical framework to investigate the accuracy limit of phonon-limited carrier mobility in intrinsic n-type SnSe2 has been developed by solving the Boltzmann transport equation based on ab initio calculated electron-phonon interactions. The electron-phonon coupling matrix elements have been computed using maximally localized Wannier functions and density functional perturbation theory. The intrinsic electron mobility of ∼ 8.75 cm2V-1s-1 has been achieved at 300 K, incorporating spin-orbit coupling, many-body quasiparticle corrections, and iterative solution of the Boltzmann transport equation using dense sampling of the Brillouin zone. The calculated intrinsic mobility is in close agreement with the experimental values. Furthermore, the electron mean-free path, electrical conductivity, and Seebeck coefficient have been calculated for SnSe2 under varying temperatures. The maximum mean-free path of 20 nm has been achieved for electrons at 300 K. This contribution provides a comprehensive method to investigate the transport properties and presents a framework towards the accuracy limit of prototypical SnSe2.

Determining the electron-phonon coupling in superconducting cuprates by resonant inelastic x-ray scattering: Methods and results on Nd1+xBa2−xCu3O7−δ

The coupling between lattice vibration quanta and valence electrons can induce charge density modulations and decisively influence the transport properties of materials, e.g. leading to conventional superconductivity. In high critical temperature superconductors, where electronic correlation is the main actor, the actual role of electron-phonon coupling (EPC) is being intensely debated theoretically and investigated experimentally. We present an in-depth study of how the EPC strength can be obtained directly from resonant inelastic x-ray scattering (RIXS) data through the theoretical approach derived by Ament et al. [EPL 95, 27008 (2011)]. The role of the model parameters (e.g. phonon energy $\omega_0$, intermediate state lifetime $1/\Gamma$, EPC matrix element $M$, and detuning energy $\Omega$) is thoroughly analyzed, providing general relations among them that can be used to make quantitative estimates of the dimensionless EPC $g = (M/\omega_0)^2$ without detailed microscopic modeling. We then apply these methods to very high resolution Cu $L_3$ edge RIXS spectra of three Nd$_{1+x}$Ba$_{2-x}$Cu$_3$O$_{7-\delta}$ films. For the insulating antiferromagnetic parent compound the value of $M$ as a function of the in-plane momentum transfer is obtained for Cu-O bond-stretching (breathing) and bond-bending (buckling) phonon branches. For the underdoped and the nearly optimally doped samples, the effects of Coulomb screening and of charge-density-wave correlations on $M$ are assessed. In light of the anticipated further improvements of the RIXS experimental resolution, this work provides a solid framework for an exhaustive investigation of the EPC in cuprates and other quantum materials.

First-principles study of the phonon replicas in the photoluminescence spectrum of 4H -SiC

The free-exciton photoluminescence (PL) spectrum of $4H$-SiC exhibits a detailed fine structure due to the different phonons involved in the indirect gap transition. Here a first-principles calculation of these phonons, their symmetry labeling and their contribution to the photoluminescence spectrum is presented. The calculation uses phonons and electron-phonon coupling matrix elements computed via density functional perturbation theory and energy bands and optical phonon matrix elements calculated in density functional theory. The results are in excellent agreement with the experiment for the phonon energies and the polarization dependence of the spectrum. The relative intensities are also in fair agreement if we allow for some phonons within a few meV to be interchanged. There is however a remarkable discrepancy that the experimental spectrum shows a distinct behavior for phonons with energy below $\ensuremath{\sim}55$ meV and above that energy, which is absent in the theory. The experimental PL lines corresponding to phonon energies below 55 meV are about a factor 5--10 smaller in intensity. This is not found in our calculations. The calculations show a similar peak distribution as the experiment in this range but with intensities comparable to those above 55 meV. This indicates that another mechanism outside the scope of the electron-phonon mediated transitions is operative for photon energies of the PL lines closer to the indirect exciton gap than this 55 meV cutoff, which reduces the overall intensity of these lines. We propose that this may result from competition between the phonon-assisted PL and trapping of the electron in the available unoccupied hexagonal site N-donor shallow level at 53-meV binding energy. As part of this study, we also present the phonon dispersions and density of states in $4H$-SiC and the electronic band structure including quasiparticle corrections.

Sensitivity of Thermoelectric Properties from the EPA Method and Its Variants on Variations in Phonon Frequencies

The electron–phonon coupling phenomenon is an important scattering source of charge carriers in thermoelectric materials. As the result, consideration of the scattering mechanism due to electron–phonon coupling is an essential part of the first-principles prediction methods for thermoelectric properties of materials, i.e., the Seebeck coefficient and electrical conductivity. However, a direct, high-resolution treatment of electron–phonon coupling in a realistic situation is too expensive for practical purposes, and approximation methods have been developed for the phenomena at a reasonable level of computational cost. The electron–phonon averaged (EPA) method and its variants have been recently introduced as such approximation methods. These methods comprise two major approximations. First, the electron–phonon coupling matrix elements, which are explicitly dependent on the momentum of charge carriers and that of scattering phonons, are approximated as a function of two energies only, i.e., the energy of the incoming charge carrier before scattering and that of the outgoing charge carrier after scattering. Second, the phonon frequencies in the formula of scattering rate are replaced with their averages in the Brillouin zone. Although these methods have achieved remarkable successes in predicting thermoelectric properties of materials, uncertainty introduced by these approximations and sensitivity of the results to variations of input parameters in these approximations have not been assessed completely. In this study, the uncertainty and sensitivity of thermoelectric properties, predicted with the EPA method and its variants, on variations in phonon frequencies are investigated. A thermoelectric p-type half-Heusler compound, HfCoSb, is used as an example. An empirical bootstrap method is employed to assess the impact of sampled phonon frequencies during the averaging process, whose results show that the impact of variations in phonon frequencies induces a minor but detectable change in predicted thermoelectric properties.

Theoretical prediction of intrinsic electron mobility of monolayer InSe: first-principles calculation

Recently, a novel two-dimensional (2D) semiconductor, InSe, has attracted great attention due to its potential applications in optoelectronic devices and field effect transistors. In this study, phonon-limited mobility is investigated by the first-principles calculation. At 300 K, the intrinsic electron mobilities calculated from the electron–phonon coupling (EPC) matrix element are as high as (zigzag direction) and (Armchair direction), respectively. The deformation potential theory (DPT) based on longitudinal acoustic and optical phonon scattering is also employed to investigate electron mobility. The mobility from optical phonon scattering is much higher than that from longitudinal acoustic phonon scattering. If the polarization characteristics of InSe are not considered, the electron mobility calculated from EPC matrix element is closed to that from the longitudinal acoustic phonon DPT. In this study, we have also investigated the effect of polarization properties in 2D InSe on electron mobility. At 300 K, the electron mobility for including Fröhlich interaction is reduced to and . Therefore, the electron mobility for InSe is controlled by the scattering from polar phonons. The mobility can be increased to and under 4% biaxial strain. This result is compared with the experiment, and some disagreements are explained.

Electron-Phonon Coupling as the Source of 1/f Noise in Carbon Soot

Two 1/f noise peaks were found in a carbon soot resistor at voltages characteristic of Kohn anomalies in graphite. The ratio of the electron-phonon coupling matrix elements at the anomalies calculated from the noise peak intensities is the same as the one obtained from the Raman frequencies. This demonstrates that the electron-phonon coupling is the microscopic source of 1/f noise in carbon soot. A new, very general formula was deduced for the frequency exponent, wherein nonlinearity and dispersion are the only ingredients. The interplay between nonlinearity and dispersion in this formula describes the sublinear-supralinear transitions experimentally observed at both anomalies in the voltage dependence of the frequency exponent. A quadratic dependence of the 1/f noise parameter on the matrix element is proposed and applied to explain the M-shape of the 1/f noise in graphene. We found that the frequency exponent mimics the dependence of the noise intensity in the whole voltage range, while both are the image of the graphite phonon spectrum. This implies that the source of nonlinearity is in the electron-phonon coupling which modulates the slope of the spectrum. It requires the presence of 1/f noise in the thermal noise background of the resistor till phonon frequencies.

Phonon-limited carrier mobility and temperature-dependent scattering mechanism of 3C -SiC from first principles

Electron-phonon coupling is at the core of various regimes of material-based science and technology. Taking $3C$-silicon carbide ($3C$-SiC) as an example, despite its very wide application in high-temperature and high-power devices, the transport properties of $3C$-SiC are not yet fully understood at the microscopic level because of inadequate knowledge in electron-phonon coupling. In this paper, with electron-phonon coupling matrix elements calculated from first principles, the phonon limited carrier mobility of $3C$-SiC is quantified by solving the Boltzmann transport equation. The calculated mobilities for both holes and electrons are in reasonable agreement with the experimental data. Unlike other polar semiconductors such as GaAs, where the polar-longitudinal-optical (LO)-phonon interactions are the dominant scattering mechanism, the mobilities of electrons and holes are dominated by the intravalley longitudinal acoustic phonon scattering at 300 K due to the low occupation number of high-frequency polar LO phonons in $3C$-SiC. The dominant scattering mechanism in $3C$-SiC varies with temperature. At high temperature (800 K), LO phonons govern the scattering instead. The maximum mean-free paths of electrons and holes at room temperature are found to be 40 nm and 15 nm, respectively.

Vibrational and electron-phonon coupling properties of β-Ga2O3 from first-principles calculations: Impact on the mobility and breakdown field

The wide band gap semiconductor β-Ga2O3 shows promise for applications in high-power and high-temperature electronics. The phonons of β-Ga2O3 play a crucial role in determining its important material characteristics for these applications such as its thermal transport, carrier mobility, and breakdown voltage. In this work, we apply predictive calculations based on density functional theory and density functional perturbation theory to understand the vibrational properties, phonon-phonon interactions, and electron-phonon coupling of β-Ga2O3. We calculate the directionally dependent phonon dispersion, including the effects of LO-TO splitting and isotope substitution, and quantify the frequencies of the infrared and Raman-active modes, the sound velocities, and the heat capacity of the material. Our calculated optical-mode Grüneisen parameters reflect the anharmonicity of the monoclinic crystal structure of β-Ga2O3 and help explain its low thermal conductivity. We also evaluate the electron-phonon coupling matrix elements for the lowest conduction band to determine the phonon mode that limits the mobility at room temperature, which we identified as a polar-optical mode with a phonon energy of 29 meV. We further apply these matrix elements to estimate the breakdown field of β-Ga2O3. Our theoretical characterization of the vibrational properties of β-Ga2O3 highlights its viability for high-power electronic applications and provides a path for experimental development of materials for improved performance in devices.

Role of spin-orbit coupling in the physical properties of LaX3 ( X=In , P, Bi) superconductors

We report a comprehensive and complementary study on structural, elastic, mechanical, electronic, phonon, and electron-phonon interaction properties of $\mathrm{La}{X}_{3}$ ($X$ = In, Pb, and Bi) using first-principles density functional calculations within the local density approximation with and without the spin-orbit coupling (SOC). The calculated lattice parameters for these intermetallic compounds with and without SOC are found to differ by less than $2%$ from their experimental values. The effect of SOC on the elastic, mechanical, electronic, phonon, and electron-phonon interaction properties is more profound for ${\mathrm{LaPb}}_{3}$ and ${\mathrm{LaBi}}_{3}$ containing heavier $X$ elements rather than ${\mathrm{LaIn}}_{3}$ containing lighter $X$ element. The inclusion of SOC considerably removes the degeneracies of some bands near the Fermi level and makes some phonon branches in ${\mathrm{LaPb}}_{3}$ and ${\mathrm{LaBi}}_{3}$ softer and increases the strength of dominant peaks in their Eliashberg spectral functions. Thus the SOC related enhancement of their electron-phonon coupling parameter values can be related to both a softening of their phonon dispersion curves and an increase in their electron-phonon coupling matrix elements. The superconducting transition temperature with SOC is computed to be 0.69 K for ${\mathrm{LaIn}}_{3}$, 4.23 K for ${\mathrm{LaPb}}_{3}$, and 6.87 K for ${\mathrm{LaBi}}_{3}$, which agree very well with the respective measured values of 0.70, 4.18, and 7.30 K.