Ohlich Interaction Research Articles

Effect of spin-orbit coupling on the zero-point renormalization of the electronic band gap in cubic materials: First-principles calculations and generalized Fröhlich model

The electronic structure of semiconductors and insulators is affected by ionic motion through electron-phonon interaction, yielding temperature-dependent band gap energies and zero-point renormalization (ZPR) at absolute zero temperature. For polar materials, the most significant contribution to the band gap ZPR can be understood in terms of the Fr\"ohlich model, which focuses on the nonadiabatic interaction between an electron and the macroscopic electrical polarization created by a long-wavelength optical longitudinal phonon mode. On the other hand, spin-orbit interaction (SOC) modifies the bare electronic structure, which will, in turn, affect the electron-phonon interaction and the ZPR. We present a comparative investigation of the effect of SOC on the band gap ZPR of twenty semiconductors and insulators with cubic symmetry using first-principles calculations. We observe a SOC-induced decrease of the ZPR, up to 30%, driven by the valence band edge, which almost entirely originates from the modification of the bare electronic eigenenergies and the decrease of the hole effective masses near the $\mathrm{\ensuremath{\Gamma}}$ point. We also incorporate SOC into a generalized Fr\"ohlich model, addressing the Dresselhaus splitting which occurs in noncentrosymmetric materials, and confirm that the predominance of nonadiabatic effects on the band gap ZPR of polar materials is unchanged when including SOC. Our generalized Fr\"ohlich model with SOC provides a reliable estimate of the SOC-induced decrease of the polaron formation energy obtained from first principles and brings to light some fundamental subtleties in the numerical evaluation of the effective masses with SOC for noncentrosymmetric materials. We finally warn about a possible breakdown of the parabolic approximation, one of the most fundamental assumptions of the Fr\"ohlich model, within the physically relevant energy range of the Fr\"ohlich interaction for materials with high phonon frequencies treated with SOC.

Optical conductivity of an anharmonic large polaron gas at weak coupling

In a polar solid, electrons or other charge carriers can interact with the phonons of the ionic lattice, leading to the formation of polaron quasiparticles. The optical conductivity and optical absorption spectrum of a material are affected by this electron-phonon coupling, most notably leading to an absorption peak in the midinfrared region. Recently, a model Hamiltonian for anharmonic electron-phonon coupling was derived [M. Houtput and J. Tempere, Phys. Rev. B 103, 184306 (2021)] that includes both the conventional Fr\"ohlich interaction as well as an interaction in which an electron interacts with two phonons simultaneously. In this article, we calculate and investigate the optical conductivity of the anharmonic large polaron gas, and we show that an additional characteristic absorption peak appears due to this one-electron--two-phonon interaction. We calculate a semianalytical expression for the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ at finite temperatures and weak coupling using the Kubo formula. The electronic and phononic contributions can be split and treated separately, such that the many-body effects of the electron gas may be taken into account through the well-known dynamical structure factor $S(\mathbf{k},\ensuremath{\omega})$. From the resulting optical conductivity, we calculate the polaron effective mass, an estimate for the electron-phonon scattering times, and the optical absorption spectrum of the anharmonic polaron gas. It is shown that the effects are negligible for four common III-V semiconductors (BN, AlN, BP, AlP) in the zinc-blende structure, which justifies the commonly used harmonic approximation in these materials. We show that alongside the well-known polaron absorption peak at the phonon energy $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{LO}}$, the one-electron--two-phonon interaction leads to an additional absorption peak at $2\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{LO}}$. We propose this absorption peak as an experimentally measurable indicator for non-negligible one-electron--two-phonon interaction in a material, since the height of this peak is proportional to the strength of this anharmonic interaction.

Many-body Green's function approaches to the doped Fröhlich solid: Exact solutions and anomalous mass enhancement

In polar semiconductors and insulators, the Fr\"ohlich interaction between electrons and long-wavelength longitudinal optical phonons induces a many-body renormalization of the carrier effective masses and the appearence of characteristic phonon sidebands in the spectral function, commonly dubbed 'polaron satellites'. The simplest model that captures these effects is the Fr\"ohlich model, whereby electrons in a parabolic band interact with a dispersionless longitudinal optical phonon. The Fr\"ohlich model has been employed in a number of seminal papers, from early perturbation-theory approaches to modern diagrammatic Monte Carlo calculations. One limitation of this model is that it focuses on undoped systems, thus ignoring carrier screening and Pauli blocking effects that are present in real experiments on doped samples. To overcome this limitation, we here extend the Fr\"ohlich model to the case of doped systems, and we provide exact solutions for the electron spectral function, mass enhancement, and polaron satellites. We perform the analysis using two approaches, namely Dyson's equation with the Fan-Migdal self-energy, and the second-order cumulant expansion. We find that these two approaches provide qualitatively different results. In particular, the Dyson's approach yields better quasiparticle masses and worse satellites, while the cumulant approach provides better satellite structures, at the price of worse quasiparticle masses. Both approaches yield an anomalous enhancement of the electron effective mass at finite doping levels, which in turn leads to a breakdown of the quasiparticle picture in a significant portion of the phase diagram.

Resonant Raman scattering in UO2 revisited

Due to ${\mathrm{UO}}_{2}$ being a strong electron-phonon interaction system, resonant Raman scattering is particularly helpful in exploring the relationship between its optical properties and electronic structure. Fr\"ohlich interaction induced resonant 2LO(\ensuremath{\Gamma}) phonon Raman scattering exhibits excitation energy $({E}_{i}=1.58--3.06\phantom{\rule{0.16em}{0ex}}\mathrm{eV})$, temperature (77--873 K), and pressure (\ensuremath{\le}29 GPa) dependent weighted contributions of two resonances. The first (incoming) resonance occurs when the incident photon energy equals the energy of an optical transition positioned at \ensuremath{\delta} \ensuremath{\sim} 0.19 above the \ensuremath{\sim}2.1 eV band gap. The second (outgoing) resonance occurs when the scattered-photon energy is equal to that of the optical transition. The energy of this transition is proposedly related to the Brillouin zone \ensuremath{\Gamma} point electronic density of states, positioned to be slightly above the band gap, by recent density functional theory calculations. The interplay between the simultaneous temperature dependent effects of tuning the band gap and altering the lifetime broadening of the excited electronic state is elucidated and demonstrated to strongly depend on the excitation energy and its position with respect to the band gap. The core difference between pressure and temperature dependent Raman response is mostly attributed to the ``resonant-suppressing'' role played by excited-state lifetime shortening, which becomes increasingly dominant at increased temperatures.

Dynamics of a trapped ion in a quantum gas: Effects of particle statistics

We study the quantum dynamics of an ion confined in a radiofrequency trap in interaction with either a Bose or spin-polarized Fermi gas. To this end, we derive quantum optical master equations in the limit of weak coupling and the Lamb-Dicke approximations. For the bosonic bath, we also include the so-called "Lamb-shift" correction to the ion trap due to the coupling to the quantum gas as well as the extended Fr\"ohlich interaction within the Bogolyubov approximation that have been not considered in previous studies. We calculate the ion kinetic energy for various atom-ion scattering lengths as well as gas temperatures by considering the intrinsic micromotion and we analyse the damping of the ion motion in the gas as a function of the gas temperature. We find that the ion's dynamics depends on the quantum statistics of the gas and that a fermionic bath enables to attain lower ionic energies.

AbInitio Electron-Phonon Interactions in Correlated Electron Systems.

Electron-phonon (e-ph) interactions are pervasive in condensed matter, governing phenomena such as transport, superconductivity, charge-density waves, polarons, and metal-insulator transitions. First-principles approaches enable accurate calculations of e-ph interactions in a wide range of solids. However, they remain an open challenge in correlated electron systems (CES), where density functional theory often fails to describe the ground state. Therefore reliable e-ph calculations remain out of reach for many transition metal oxides, high-temperature superconductors, Mott insulators, planetary materials, and multiferroics. Here we show first-principles calculations of e-ph interactions in CES, using the framework of Hubbard-corrected density functional theory (DFT+U) and its linear response extension (DFPT+U), which can describe the electronic structure and lattice dynamics of many CES. We showcase the accuracy of this approach for a prototypical Mott system, CoO, carrying out a detailed investigation of its e-ph interactions and electron spectral functions. While standard DFPT gives unphysically divergent and short-ranged e-ph interactions, DFPT+U is shown to remove the divergences and properly account for the long-range Fröhlich interaction, allowing us to model polaron effects in a Mott insulator. Our work establishes a broadly applicable and affordable approach for quantitative studies of e-ph interactions in CES, a novel theoretical tool to interpret experiments in this broad class of materials.

Polaronic interaction in a single modulation-doped GaAs quantum well with the Feynman-Hellwarth-Iddings-Platzman approximation

Absolute far-infrared magnetotransmission experiments have been performed in magnetic fields up to 33.5 T on a series of single GaAs quantum wells doped with different electron concentrations. The transmission spectra have been simulated with a multilayer dielectric model. The imaginary part of the optical response function, which reveals singular features related to the electron-phonon interactions, has been extracted. In addition to the expected polaronic effects due to the longitudinal-optical phonon of GaAs, additional interactions with interface phonons are observed. The main interaction is analyzed quantitatively with the Feynman-Hellwarth-Iddings-Platzman model, which is shown to predict correctly the concentration of carriers beyond which the Fr\"ohlich interaction is completely screened.

Remote free-carrier screening to boost the mobility of Fröhlich-limited two-dimensional semiconductors

Van der Waals heterostructures provide a versatile tool to not only protect or control, but also enhance the properties of a 2D material. We use ab initio calculations and semi-analytical models to find strategies which boost the mobility of a current-carrying 2D semiconductor within an heterostructure. Free-carrier screening from a metallic "screener" layer remotely suppresses electron-phonon interactions in the current-carrying layer. This concept is most effective in 2D semiconductors whose scattering is dominated by screenable electron-phonon interactions, and in particular the Fr\"ohlich coupling to polar-optical phonons. Such materials are common and characterised by overall low mobilities in the small doping limit, and much higher ones when the 2D material is doped enough for electron-phonon interactions to be screened by its own free carriers. We use GaSe as a prototype and place it in a heterostructure with doped graphene as the "screener" layer and BN as a separator. We develop an approach to determine the electrostatic response of any heterostructure by combining the responses of the individual layers computed within density-functional perturbation theory. Remote screening from graphene can suppress the long-wavelength Fr\"ohlich interaction, leading to a consistently high mobility around $500$ to $600$ cm$^2$/Vs for carrier densities in GaSe from $10^{11}$ to $10^{13}$ cm$^{-2}$. Notably, the low-doping mobility is enhanced by a factor 2.5. This remote free-carrier screening is more efficient than more conventional manipulation of the dielectric environment, and it is most effective when the separator (BN) is thin.

Screening of hot electrons in the ferroelectric semiconductor In2Se3

In a conventional semiconductor, electrons $(e)$ scatter efficiently with longitudinal-optical phonons $(\mathrm{p}{\mathrm{h}}_{\mathrm{LO}})$ via the Fr\ohlich interaction, e.g., in polaron formation or hot-electron cooling. Recent studies on lead halide perovskites have suggested that the Fr\ohlich interaction may be sufficiently screened by the polar lattice. Here we show that hot-electron cooling in a layered ferroelectric semiconductor $\mathrm{I}{\mathrm{n}}_{2}\mathrm{S}{\mathrm{e}}_{3}$ slows down by 50% as temperature is increased from 115 to 387 K; this decrease is about four times more than that predicted by the Fr\ohlich model. We suggest that screening of Fr\ohlich interaction by polar domains or fluctuations in the ferroelectric semiconductor may be responsible for the reduced $\mathrm{e}\text{\ensuremath{-}}\mathrm{p}{\mathrm{h}}_{\mathrm{LO}}$ scattering. Such enhanced screening in polar materials featuring anharmonic lattice instability may also reduce carrier scattering with other charge carriers or with charged defects, leading to improved optoelectronic properties.

Contrasting temperature dependence of the band gap in CH3NH3PbX3 (X=I, Br, Cl) : Insight from lattice dilation and electron-phonon coupling

Although hybrid halide perovskites $(\text{MAPb}{X}_{3},$ $\mathrm{MA}={\mathrm{CH}}_{3}{\mathrm{NH}}_{3} \mathrm{and} X=\mathrm{I}, \mathrm{Br}, \mathrm{Cl})$ have been ubiquitously explored from the photovoltaic perspective, there are still a few unanswered questions which require a more fundamental understanding. One such unsettled issue is the puzzling behavior of the band gap. Unlike conventional semiconductors, $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$ is found to show a blueshift (increase) in the band gap $({E}_{g})$ with increasing temperature $(T)$, while ${\mathrm{MAPbCl}}_{3}$ shows an unusual redshift (decrease). In order to understand this, we performed a detailed $T$-dependent study of electronic, optical, and structural properties of $\text{MAPb}{X}_{3}$ combined with the state-of-the-art first-principles calculations. With increasing $T$, two dominant mechanisms which come into play are lattice dilation and electron-phonon coupling (EPC). The former (latter) is responsible for an increase (decrease) in ${E}_{g}$. We found that lattice dilation effect dominates in $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$, causing an enhancement in ${E}_{g}$. EPC involves various contributions, of which the interaction of charge carriers with the longitudinal optical phonon mode via Fr\ohlich interaction is the most dominant one at room temperature. We quantify this contribution using Fr\ohlich's theory of large polarons and show that the ${E}_{g}$ correction due to this effect in ${\mathrm{MAPbCl}}_{3}$ is almost double as compared to ${\mathrm{MAPbBr}}_{3}$ and thus explain the reduction in ${E}_{g}$ for the former.

Exciton-phonon coupling in InP quantum dots with ZnS and (Zn,Cd)Se shells

InP-based colloidal quantum dots are promising for optoelectronic devices such as light-emitting diodes and lasers. Understanding and optimizing their emission process is of scientific interest and essential for large-scale applications. Here we present a study of the exciton recombination dynamics in InP QDs with various shells: ZnS, ZnSe, and (Zn,Cd)Se with different amounts of Cd (5, 9, 12%). Phonon energies extracted from Raman spectroscopy measurements at cryogenic temperatures (4--5 K) are compared with exciton emission peaks observed in fluorescence line narrowing spectra. This allowed us to determine the position of both the bright $F=\ifmmode\pm\else\textpm\fi{}1$ state and the lowest dark $F=\ifmmode\pm\else\textpm\fi{}2$ state. We could identify the phonon modes involved in the radiative recombination of the dark state and found that acoustic and optical phonons of both the core and the shell are involved in this process. The Cd content in the shell increases electron wave-function delocalization, and thereby enhances the exciton-phonon coupling through the Fr\ohlich interaction.

Thermally enhanced Fröhlich coupling in SnSe

To gain insight into the peculiar temperature dependence of the thermoelectric material SnSe, we employ many-body perturbation theory and explore the influence of the electron-phonon interaction on its electronic and transport properties. We show that a lattice dynamics characterized by soft highly-polar phonons induces a large thermal enhancement of the Fr\"ohlich interaction. We account for these phenomena in ab-initio calculations of the photoemission spectrum and electrical conductivity at finite temperature, unraveling the mechanisms behind recent experimental data. Our results reveal a complex interplay between lattice thermal expansion and Fr\"ohlich coupling, providing a new rationale for the in-silico prediction of transport coefficients of high-performance thermoelectrics.

Adiabatic perturbation theory of nonequilibrium light-controlled superconductivity

Recent experiments, in which terahertz (THz) light has been used to induce nonequilibrium superconducting states, have raised a number of intriguing fundamental questions. Theoretically, these experiments are most often described within the Floquet formalism, which suffers a number of well-known limitations (e.g., Floquet heating). Alternative approaches rely on heavy numerical methods. In this paper, we develop an analytical theory of nonequilibrium superconductivity that combines path integrals on the Kostantinov-Perel' time contour with adiabatic perturbation theory [G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 (2008)]. We consider a general system of electrons and Raman phonons coupled by the Fr\"ohlich interaction, in the presence of a time-dependent external field which acts on the phonon subsystem. The latter is supposed to model the THz light-induced excitation of nonlinear interactions between infrared and Raman phonons. Assuming that the external field has a slow dependence on time, we derive equations for the dynamical superconducting gap, calculating the leading adiabatic term and the first nonadiabatic correction. Our nonequilibrium formulas can be solved numerically with a minimal increase of computational complexity with respect to that needed to calculate the superconducting gap at equilibrium.

Accelerated Carrier Relaxation through Reduced Coulomb Screening in Two-Dimensional Halide Perovskite Nanoplatelets

For high-speed optoelectronic applications relying on fast relaxation or energy-transfer mechanisms, understanding of carrier relaxation and recombination dynamics is critical. Here, we compare the differences in photoexcited carrier dynamics in two-dimensional (2D) and quasi-three-dimensional (quasi-3D) colloidal methylammonium lead iodide perovskite nanoplatelets via differential transmission spectroscopy. We find that the cooling of excited electron–hole pairs by phonon emission progresses much faster and is intensity-independent in the 2D case. This is due to the low dielectric surrounding of the thin perovskite layers, for which the Fröhlich interaction is screened less efficiently leading to higher and less density-dependent carrier-phonon scattering rates. In addition, rapid dissipation of heat into the surrounding occurs due to the high surface-to-volume ratio. Furthermore, we observe a subpicosecond dissociation of resonantly excited 1s excitons in the quasi-3D case, an effect which is suppressed in the 2D nanoplatelets due to their large exciton binding energies. The results highlight the importance of the surrounding environment of the inorganic nanoplatelets on their relaxation dynamics. Moreover, this 2D material with relaxation times in the subpicosecond regime shows great potential for realizing devices such as photodetectors or all-optical switches operating at THz frequencies.

Control of two-dimensional electronic states at anatase TiO2(001) surface by K adsorption

The nature of the intriguing metallic electronic structures appearing at the surface of anatase titanium dioxide (a-$\mathrm{Ti}{\mathrm{O}}_{2}$) remains to be elucidated, mainly owing to the difficulty of controlling the depth distribution of the oxygen vacancies generated by photoirradiation. In this study, K atoms were adsorbed onto the (001) surface of a-$\mathrm{Ti}{\mathrm{O}}_{2}$ to dope electrons into the a-$\mathrm{Ti}{\mathrm{O}}_{2}$ and to confine the electrons in the surface region. The success of the electron doping and its controllability were confirmed by performing in situ angle-resolved photoemission spectroscopy as well as core-level measurements. Clear subband structures were observed in the surface metallic states, indicating the creation of quasi-two-dimensional electron liquid (q2DEL) states in a controllable fashion. With increasing electron doping (K adsorption), the q2DEL states exhibited crossover from polaronic liquid states with multiple phonon-loss structures originating from the long-range Fr\ohlich interaction to ``weakly correlated metallic'' states. In the q2DEL states in the weakly correlated metallic region, a kink due to short-range electron--phonon coupling was clearly observed at about $80\ifmmode\pm\else\textpm\fi{}10\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$. The characteristic energy is smaller than that previously observed for the metallic states of a-$\mathrm{Ti}{\mathrm{O}}_{2}$ with three-dimensional nature $(\ensuremath{\sim}110\phantom{\rule{0.16em}{0ex}}\mathrm{meV})$. These results suggest that the dominant electron--phonon coupling is modulated by anisotropic carrier screening in the q2DEL states.

Magneto-optical transport properties of monolayer MoS2 on polar substrates

We theoretically study the magneto-optical transport properties of monolayer molybdenum disulfide (${\mathrm{MoS}}_{2}$) on polar substrates in the presence of a perpendicular magnetic field. The magneto-optical absorption coefficient (MOAC) is investigated as a function of the incident photon energy when carriers are scattered by three different types of phonons: the intrinsic ${\mathrm{MoS}}_{2}$ acoustic, optical phonons, and the surface optical (SO) phonons induced by polar substrates. Among the substrates considered, the largest magnitude of MOAC and full-width at half maximum (FWHM) are observed for a ${\mathrm{SiO}}_{2}$ substrate over the entire temperature and magnetic field range considered due to its strongest electron-SO phonon scattering, while an h-BN substrate displays the lowest one. The piezoelectric (PE) coupling to the transverse (TA) phonon is shown to dominate the MOAC and FWHM due to intrinsic acoustic phonon scattering. Meanwhile, these properties for intrinsic optical phonons are dominated by zero-order deformation potential (DP) couplings and the Fr\"ohlich interaction. The dependence of the MOAC and FWHM on temperature, magnetic field, and the effective ${\mathrm{MoS}}_{2}$-substrate distance is also examined. The present results for monolayer ${\mathrm{MoS}}_{2}$ are compared with those in conventional two-dimensional systems as well as in graphene. Our results show that SO phonons play a crucial role at high temperature depending on the substrates and have a non-negligible effect on the magneto-optical transport properties of monolayer ${\mathrm{MoS}}_{2}$, which could be further experimentally and theoretically investigated in the future.

Two-dimensional Fröhlich interaction in transition-metal dichalcogenide monolayers: Theoretical modeling and first-principles calculations

We perform ab initio calculations of the coupling between electrons and small-momentum polar-optical phonons in monolayer transition metal dichalcogenides of the 2H type: MoS2, MoSe2, MoTe2, WS2, and WSe2. The so-called Fr\"ohlich interaction is fundamentally affected by the dimensionality of the system. In a plane-wave framework with periodic boundary conditions, this coupling is affected by the spurious interaction between the 2D material and its periodic images. To overcome this, we perform density functional perturbation theory calculations with a truncated Coulomb interaction in the out-of-plane direction. We show that the 2D Fr\"ohlich interaction is much stronger than assumed in previous ab initio studies. We provide analytical models depending on the effective charges and dielectric properties of the materials to interpret our ab initio calculations. Screening is shown to play a fundamental role in the phonon-momentum dependency of the polar-optical coupling, with a crossover between two regimes depending on the dielectric properties of the material relative to its environment. The Fr\"ohlich interaction is screened by the dielectric environment in the limit of small phonon momenta and sharply decreases due to stronger screening by the monolayer at finite momenta. The small-momentum regime of the ab initio Fr\"ohlich interaction is reproduced by a simple analytical model, for which we provide the necessary parameters. At larger momenta, however, direct ab initio calculations of electron-phonon interactions are necessary to capture band-specific effects. We compute and compare the carrier relaxation times associated to the scattering by both LO and A1 phonon modes. While both modes are capable of relaxing carriers on timescales under the picosecond at room temperature, their absolute and relative importance vary strongly depending on the material, the band, and the substrate.

Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs

We generalize the Wannier interpolation of the electron-phonon matrix elements to the case of polar-optical coupling in polar semiconductors. We verify our methodological developments against experiments, by calculating the widths of the electronic bands due to electron-phonon scattering in GaAs, the prototype polar semiconductor. The calculated widths are then used to estimate the broadenings of excitons at critical points in GaAs and the electron-phonon relaxation times of hot electrons. Our findings are in good agreement with available experimental data. Finally, we demonstrate that while the Fr\"ohlich interaction is the dominant scattering process for electrons/holes close to the valley minima, in agreement with low-field transport results, at higher energies, the intervalley scattering dominates the relaxation dynamics of hot electrons or holes. The capability of interpolating the polar-optical coupling opens new perspectives in the calculation of optical absorption and transport properties in semiconductors and thermoelectrics.

Hot-electron cooling by acoustic and optical phonons in monolayers ofMoS2and other transition-metal dichalcogenides

We study hot-electron cooling by acoustic and optical phonons in monolayer ${\mathrm{MoS}}_{2}$. The cooling power $P$ (${P}_{e}=P/n$) is investigated as a function of electron temperature ${T}_{e}$ (0--500 K) and carrier density $n\phantom{\rule{0.28em}{0ex}}({10}^{10}--{10}^{13} {\mathrm{cm}}^{\ensuremath{-}2}$) taking into account all relevant electron-phonon (el-ph) couplings. We find that the crossover from acoustic phonon dominated cooling at low ${T}_{e}$ to optical phonon dominated cooling at higher ${T}_{e}$ takes place at ${T}_{e}\ensuremath{\sim}50--75$ K. The unscreened deformation potential (DP) coupling to the TA phonon is shown to dominate $P$ due to acoustic phonon scattering over the entire temperature and density range considered. The cooling power due to screened DP coupling to the LA phonon and screened piezoelectric (PE) coupling to the TA and LA phonons is orders of magnitude lower. In the Bloch-Gr\"uneisen (BG) regime, $P\ensuremath{\sim}{T}_{e}^{4}\phantom{\rule{4pt}{0ex}}({T}_{e}^{6})$ is predicted for unscreened (screened) el-ph interaction and $P\ensuremath{\sim}{n}^{\ensuremath{-}1/2}\phantom{\rule{4pt}{0ex}}({P}_{e}\ensuremath{\sim}{n}^{\ensuremath{-}3/2})$ for both unscreened and screened el-ph interaction. The cooling power due to optical phonons is dominated by zero-order DP couplings and the Fr\"ohlich interaction, and is found to be significantly reduced by the hot-phonon effect when the phonon relaxation time due to phonon-phonon scattering is large compared to the relaxation time due to el-ph scattering. The ${T}_{e}$ and $n$ dependence of the hot-phonon distribution function is also studied. Our results for monolayer ${\mathrm{MoS}}_{2}$ are compared with those in conventional two-dimensional electron gases (2DEGs) as well as monolayer and bilayer graphene.

Spin-Orbit Splitting in Single-LayerMoS2Revealed by Triply Resonant Raman Scattering

Although new spintronic devices based on the giant spin-orbit splitting of single-layer MoS(2) have been proposed, such splitting has not been studied effectively in experiments. This Letter reports the valence band spin-orbit splitting in single-layer MoS(2) for the first time, probed by the triply resonant Raman scattering process. We found that upon 325 nm laser irradiation, the second order overtone and combination Raman modes of single-layer MoS(2) are dramatically enhanced. Such resonant Raman enhancement arises from the electron-two-phonon triple resonance via the deformation potential and Fröhlich interaction. As a sensitive and precise probe for the spin-orbit splitting, the triply resonant Raman scattering will provide a new and independent route to study the spin characteristics of MoS(2).