Pseudo-angular Momentum Research Articles

Momentum-Dark Intervalley Exciton in Monolayer Tungsten Diselenide Brightened via Chiral Phonon.

Inversion symmetry breaking and 3-fold rotation symmetry grant the valley degree of freedom to the robust exciton in monolayer transition-metal dichalcogenides, which can be exploited for valleytronics applications. However, the short lifetime of the exciton significantly constrains the possible applications. In contrast, the dark exciton could be long-lived but does not necessarily possess the valley degree of freedom. In this work, we report the identification of the momentum-dark, intervalley exciton in monolayer WSe2 through low-temperature magneto-photoluminescence spectra. Interestingly, the intervalley exciton is brightened through the emission of a chiral phonon at the corners of the Brillouin zone (K point), and the pseudoangular momentum of the phonon is transferred to the emitted photon to preserve the valley information. The chiral phonon energy is determined to be ∼23 meV, based on the experimentally extracted exchange interaction (∼7 meV), in excellent agreement with the theoretical expectation of 24.6 meV. The long-lived intervalley exciton with valley degree of freedom adds an exciting quasiparticle for valleytronics, and the coupling between the chiral phonon and intervalley exciton furnishes a venue for valley spin manipulation.

Lattice Dynamics, Phonon Chirality, and Spin–Phonon Coupling in 2D Itinerant Ferromagnet Fe3GeTe2

AbstractFe3GeTe2 has emerged as one of the most fascinating van der Waals crystals due to its 2D itinerant ferromagnetism, topological nodal lines, and Kondo lattice behavior. However, lattice dynamics, chirality of phonons, and spin–phonon coupling in this material, which set the foundation for these exotic phenomena, have remained unexplored. Here, the first experimental investigation of the phonons and mutual interactions between spin and lattice degrees of freedom in few‐layer Fe3GeTe2 is reported. The results elucidate three prominent Raman modes at room temperature: two A1g(Γ) and one E2g(Γ) phonons. The doubly degenerate E2g(Γ) mode reverses the helicity of incident photons, indicating the pseudoangular momentum and chirality. Through analysis of temperature‐dependent phonon energies and lifetimes, which strongly diverge from the anharmonic model below Curie temperature, the spin–phonon coupling in Fe3GeTe2 is determined. Such interaction between lattice oscillations and spin significantly enhances the Raman susceptibility, allowing to observe two additional Raman modes at the cryogenic temperature range. In addition, laser radiation‐induced degradation of Fe3GeTe2 in ambient conditions and the corresponding Raman fingerprint is revealed. The results provide the first experimental analysis of phonons in this novel 2D itinerant ferromagnet and their applicability for further fundamental studies and application development.

Chiral phonons in kagome lattices

Chiral phonons have been theoretically proposed [Zhang and Niu, Phys. Rev. Lett. 115, 115502 (2015)] and experimentally verified [Zhu et al., Science 359, 579 (2018)] in materials with hexagonal lattice. Here, we theoretically predict and investigate the chiral phonons in the kagome lattice. We show that with broken inversion symmetry, chiral phonons with large circular polarization can emerge at the $K$ and ${K}^{\ensuremath{'}}$ high symmetry points. Different from the hexagonal lattice, these chiral phonons have all three sublattices vibrating with the same chirality, and the vibration orbit can take an elliptical shape. They possess valley-contrasting Berry curvatures sharply peaked at the $K$ and ${K}^{\ensuremath{'}}$ valleys, which can be used to separate the chiral phonons according to their valley label via the valley phonon Hall effect. The pseudoangular momentum for these chiral phonons has also been studied, which determines the selection rules of electron-phonon interaction in the intervalley scattering process. Furthermore, we show that by tuning the relative bond strength, there can be a change in phonon chirality associated with a phonon branch switching. Our work enriches the studies on chiral phonons and offers a probe for many fascinating quantum materials which possess the kagome type lattice.

Nondegenerate Chiral Phonons in Graphene/Hexagonal Boron Nitride Heterostructure from First-Principles Calculations.

Triggered by the recent successful observation of previously predicted phonon chirality in the monolayer tungsten diselenide [ Science 2018 , 359 , 579 ], we systematically study the chiral phonons in the classical heterostructure of graphene/hexagonal boron nitride (G/ h-BN) by first-principles calculations. It is found that the broken inversion symmetry and the interlayer interaction of G/ h-BN not only open the phononic gaps but also lift the degeneracy of left-handed and right-handed chiral phonons at the first-Brillouin-zone corners (valleys). At valleys, the hybridization makes chiral phonon modes solely contributed from one individual layer. Moreover, we demonstrate that the vertical stress is effective to tune the degenerated phononic gap while keeping the valley-phonon chirality of G/ h-BN heterostructure, which is favorable for the Raman or ultrafast infrared spectroscopy measurement. We also analyze the pseudoangular momentum of valley-phonon modes, which provide important references for the excitation and measurement of the chiral phonons in the process of electronic intervalley scattering. Collectively, our results on the chiral phonons in the G/ h-BN heterostructure system could stimulate more experimental and theoretical studies and promote the future applications on the phonon-chirality-based phononics.

Conservation law of angular momentum in helicity-dependent Raman and Rayleigh scattering

In first-order Raman scattering, helicity of circularly polarized incident light is either conserved or changed depending on the Raman modes. When the helicity of incident light changes in the scattered light, the angular momentum of a photon is transferred to the material. Here, we present the conservation law of pseudoangular momentum in the helicity-dependent Raman scattering for a $N$-fold ($N=1--4,6$) rotational symmetry of a crystal. Furthermore, the conservation law of electron-phonon interaction is discussed by considering the vibration direction of a phonon that has the same or lower symmetry than the symmetry of the crystal, which is essential to allow the helicity change in Raman scattering in a highly symmetric material, such as graphene. We also discuss the conservation law of pseudoangular momentum in Rayleigh scattering and show that the helicity change is allowed only in the crystal with one- or twofold rotational symmetry.

Phonon angular momentum and chiral phonons

In traditional physics, phonon is widely regarded as being linearly polarized, which means that phonon carries zero angular momentum. Thus the angular momentum of lattice related to mechanical rotation only reflects the lattice rigid-body motion. Recently, in a magnetic system with time reversal symmetry broken by spin-phonon interaction, one found that the phonon angular momentum is nonzero and an odd function of magnetization. At zero temperature, phonon was reported to have a zero-point angular momentum and zero-point energy. Thus the gyromagnetic ratio obtained through the Einstein-de Haas effect needs correcting by considering the nonzero phonon angular momentum. As is well known, if phonon has nonzero angular momentum, which means that phonon can have rotation, it can be right-handed or left-handed, that is, the phonon is chiral. Actually, we can define the polarization of phonon to represent the phonon chirality, which comes from the circular vibration of sublattices. When the phonon polarization is larger (less) than zero, the phonon is right (left)-handed. In non-magnetic honeycomb AB lattices, with inversion symmetrybrocken, the chiral phonons are found to be of valley contrasting circular polarization and concentrated in Brillouin-zone corners. At valleys, there is a three-fold rotational symmetry endowing phonons with quantized pseudo angular momentum. Then conversation of pseudo angular momentum, which determines the selection rules in phonon-involved intervalley scattering of electrons, must be satisfied. Chiral valley phonons can be measured by polarized infrared absorption or emission. In addition, since the phonon Berry curvature is reported to be nonzero at valley, it can distort phonon transport under a strain gradient, which can act as an effective magnetic field. Thus, a valley phonon Hall effect is theoretically predicted, which is probably a method of measuring chiral valley phonons. In consideration of phonons angular momentum and chiral phonons, photon helicity changed by phonons at Gamma point will be explained reasonably. In conclusion, chiral phonons are present in systems that break time reversal or spatial inversion symmetries. In a magnetic system, where time reversal symmetry is broken, phonons generally carry a nonzero angular momentum, which can influence the classic Einstein-de Haas effect. In a nonequilibrium system, the phonon Hall effect can be observed due to the chiral phonons. In a non-magnetic crystal, with inversion symmetry brocken, phonons in the Brillouin-zone center and corners are chiral and have a quantized pseudo angular momentum, providing an alternative to valleytronics in insulators. We believe that the findings of the phonon angular momentum and the chiral phonons together with phonon pseudoangular momentum, selection rules, and valley phonon Hall effect will lead to the relevant exploration and new development of phonon related subject in condensed matter physics.

The “Forgotten” Pseudomomenta and Gauge Changes in Generalized Landau Level Problems: Spatially Nonuniform Magnetic and Temporally Varying Electric Fields

By perceiving gauge invariance as an analytical tool in order to get insight into the states of the "generalized Landau problem" (a charged quantum particle moving inside a magnetic, and possibly electric field), and motivated by an early article that correctly warns against a naive use of gauge transformation procedures in the usual Landau problem (i.e. with the magnetic field being static and uniform), we first show how to bypass the complications pointed out in that article by solving the problem in full generality through gauge transformation techniques in a more appropriate manner. Our solution provides in simple and closed analytical forms all Landau Level-wavefunctions without the need to specify a particular vector potential. This we do by proper handling of the so-called pseudomomentum K (or of a quantity that we term pseudoangular momentum Lz), a method that is crucially different from the old warning argument, but also from standard treatments in textbooks and in research literature (where the usual Landau wavefunctions are employed - labeled with canonical momenta quantum numbers). Most importantly, we go further by showing that a similar procedure can be followed in the more difficult case of spatially-nonuniform magnetic fields: in such case we define K and Lz as plausible generalizations of the previous ordinary case, namely as appropriate line integrals of the inhomogeneous magnetic field - our method providing closed analytical expressions for all stationary state wavefunctions.. (see paper) ... also provide explicit Berry's phase calculations and their connection to probability currents and ...issues in elementary Quantum Mechanics and Condensed Matter Physics. As an added feature, we also show how the possible presence of an additional electric field can be treated through a further generalization of pseudomomenta and their proper handling.

Generators of dynamical symmetries and the correct gauge transformation in the Landau level problem: use of pseudomomentum and pseudo-angular momentum

Due to the importance of gauge symmetry in all fields of physics, and motivated by an article written almost three decades ago that warns against a naive handling of gauge transformations in the Landau level problem (a quantum electron moving in a spatially uniform magnetic field), we point out a proper use of the generators of dynamical symmetries combined with gauge transformation methods to easily obtain exact analytical solutions for all Landau level-wavefunctions in arbitrary gauge. Our method is different from the old argument and provides solutions in an easier manner and in a broader set of geometries and gauges; in so doing, it eliminates the need for extra procedures (i.e. a change of basis) pointed out as a necessary step in the old literature, and gives back the standard simple result, provided that an appropriate use is made of the dynamical symmetries of the system and their generators. In this way the present work will at least be useful for university-level education, i.e. in advanced classes in quantum mechanics and condensed matter physics. In addition, it clarifies the actual role of the gauge in the Landau level problem, which often appears confusing in the usual derivations provided in textbooks. Finally, we go further by showing that a similar methodology can be made to apply to the more difficult case of a spatially non-uniform magnetic field (where closed analytical results are rare), in which case the various generators (pseudomomentum and pseudo-angular momentum) appear as line integrals of the inhomogeneous magnetic field; we give closed analytical solutions for all cases, and show how the old and rather forgotten Bawin–Burnel gauge shows up naturally as a ‘reference gauge’ in all solutions.

Chiral phonons at high-symmetry points in monolayer hexagonal lattices.

In monolayer hexagonal lattices, the intravalley and intervalley scattering of electrons can involve chiral phonons at Brillouin-zone center and corners, respectively. At these high-symmetry points, there is a threefold rotational symmetry endowing phonon eigenmodes with a quantized pseudoangular momentum, which includes orbital and spin parts. Conservation of pseudoangular momentum yields selection rules for intravalley and intervalley scattering of electrons by phonons. Concrete predictions of helicity-resolved optical phenomena are made on monolayer molybdenum disulfide. The chiral phonons at Brillouin-zone corners excited by polarized photons can be detected by a valley phonon Hall effect. The chiral phonons, together with phonon circular polarization, phonon pseudoangular momentum, selection rules, and valley phonon Hall effect will extend the basis for valley-based electronics and phononics applications in the future.

Conservation laws, radiative decay rates, and excited state localization in organometallic complexes with strong spin-orbit coupling.

There is longstanding fundamental interest in 6-fold coordinated d6 () transition metal complexes such as [Ru(bpy)3]2+ and Ir(ppy)3, particularly their phosphorescence. This interest has increased with the growing realisation that many of these complexes have potential uses in applications including photovoltaics, imaging, sensing, and light-emitting diodes. In order to design new complexes with properties tailored for specific applications a detailed understanding of the low-energy excited states, particularly the lowest energy triplet state, T1, is required. Here we describe a model of pseudo-octahedral complexes based on a pseudo-angular momentum representation and show that the predictions of this model are in excellent agreement with experiment - even when the deviations from octahedral symmetry are large. This model gives a natural explanation of zero-field splitting of T1 and of the relative radiative rates of the three sublevels in terms of the conservation of time-reversal parity and total angular momentum modulo two. We show that the broad parameter regime consistent with the experimental data implies significant localization of the excited state.

Spectroscopic fingerprints of toroidal nuclear quantum delocalization via ab initio path integral simulations

We investigate the quantum-mechanical delocalization of hydrogen in rotational symmetric molecular systems. To this purpose, we perform ab initio path integral molecular dynamics simulations of a methanol molecule to characterize the quantum properties of hydrogen atoms in a representative system by means of their real-space and momentum-space densities. In particular, we compute the spherically averaged momentum distribution n(k) and the pseudoangular momentum distribution n(kθ). We interpret our results by comparing them to path integral samplings of a bare proton in an ideal torus potential. We find that the hydroxyl hydrogen exhibits a toroidal delocalization, which leads to characteristic fingerprints in the line shapes of the momentum distributions. We can describe these specific spectroscopic patterns quantitatively and compute their onset as a function of temperature and potential energy landscape. The delocalization patterns in the projected momentum distribution provide a promising computational tool to address the intriguing phenomenon of quantum delocalization in condensed matter and its spectroscopic characterization. As the momentum distribution n(k) is also accessible through Nuclear Compton Scattering experiments, our results will help to interpret and understand future measurements more thoroughly.

Direct and two-phonon Orbach-Aminov type spin-lattice relaxation in molecular magnet V15

In this article we propose a model of spin-phonon relaxation in K6[VIV 15As6O42(H2O)]-8H2O, the so called V15 cluster exhibiting the unique layered magnetic structure. The work is motivated by the recent observation of the Rabi oscillation [1] in this system and aimed to elucidate the role of spin-phonon interaction as a source of decoherence. The spin-phonon coupling is assumed to appear as a result of the modulation of the isotropic and antisymmetric (Dzyaloshinsky-Moriya) exchange interactions in the central triangular layer of vanadium ions by the acoustic lattice vibrations. The relaxation rates are estimated within the Debye model for the lattice vibrations. Within the pseudo-angular momentum representation the selection rules for the direct (one-phonon) transitions between Zeeman levels are derived and a special role of the antisymmetric exchange is underlined. The probabilities of the two-phonon Orbach-Aminov type processes are evaluated as well, while the Raman type relaxation is shown to have a negligible importance at low temperatures at which the Rabi oscillations have been detected.

Magnetic relaxation in V 15 cluster: Direct spin-phonon transitions

In this article we propose a model of spin-vibronic relaxation in K 6[V IV 15As 6O 42(H 2O)]·8H 2O, the so called V 15 cluster exhibiting the unique layered structure. The work is motivated by the recent observation of the Rabi oscillation [1] in this system and aimed to elucidate the nature of the relaxation processes. The model assumes that the spin-phonon coupling arises as a result of modulation of the isotropic and antisymmetric (Dzyaloshinsky–Moriya) exchange interactions in the central triangular layer of vanadium ions by the acoustic lattice vibrations. Within the pseudo-angular momentum representation the selection rules for the direct (one-phonon) transitions between Zeeman levels are derived and a special role of the antisymmetric exchange is underlined. The relaxation times related to one-phonon transitions in different ranges of the field are estimated within the Debye model for the lattice vibrations.

Quasiatomic orbitals forab initiotight-binding analysis

Wave functions obtained from plane-wave density-functional theory (DFT) calculations using norm-conserving pseudopotential, ultrasoft pseudopotential, or projector augmented-wave method are efficiently and robustly transformed into a set of spatially localized nonorthogonal quasiatomic orbitals (QOs) with pseudoangular momentum quantum numbers. We demonstrate that these minimal-basis orbitals can exactly reproduce all the electronic structure information below an energy threshold represented in the form of environment-dependent tight-binding Hamiltonian and overlap matrices. Band structure, density of states, and the Fermi surface are calculated from this real-space tight-binding representation for various extended systems (Si, SiC, Fe, and Mo) and compared with plane-wave DFT results. The Mulliken charge and bond order analyses are performed under QO basis set, which satisfy sum rules. The present work validates the general applicability of Slater and Koster's scheme of linear combinations of atomic orbitals and points to future ab initio tight-binding parametrizations and linear-scaling DFT development.

Low temperature EPR spectra of the mesoscopic cluster V15: The role of antisymmetric exchange

The low temperature EPR spectra of the unique nanometer-scale molecular magnet V(15) are analyzed within the three-spin model preserving trigonal symmetry. The Hamiltonian includes isotropic and antisymmetric (AS) exchange interactions introduced by Dzyaloshinsky (Zh. Eksp. Teor. Fiz. 32, 1547 (1957) [Sov. Phys. JETP 5, 1259 (1957)]) and [Moria Phys. Rev. 120, 91 (1960)]. With the aid of pseudoangular momentum representation the exact selection rules for the EPR transitions are deduced. AS exchange in the frustrated triangular system gives rise to a first order zero-field splitting of two low lying spin doublets and to a second order splitting of the excited quadruplet. This leads to a peculiar series of strong intramultiplet transitions and weak intermultiplet transitions. We show that the intramultiplet transitions are allowed only when the vector of the AS exchange is normal to the plane of vanadium triangle, but the series of weak intermultiplet transitions are a consequence of the in-plane part of the AS exchange. We have revealed a special temperature dependence of the EPR pattern with a peculiar shape of the full spectrum. Experimental data on high-frequency EPR of V15 at ultralow temperature are discussed.

States for phase estimation in quantum interferometry

Ramsey interferometry allows the estimation of the phase $\phi$ of rotation of the pseudospin vector of an ensemble of two-state quantum systems. For $\phi$ small, the noise-to-signal ratio scales as the spin-squeezing parameter $\xi$, with $\xi = (|N,0>+|0,N>)/\sqrt{2}$) using several different properties, including $\xi$, $\zeta$, the coefficients in the pseudo angular momentum basis (in the three primary directions) and the angular Wigner function $W(\theta,\phi)$. Finally we discuss the experimental options for creating phase squeezed states and doing single-shot phase estimation.

Solution of radial equation of Kepler’s problem by pseudo-angular-momentum method and normalization of eigenstate and coherent state

By using the pseudoangularmomentum method, the radial equation for bound state of Kepler's problem is solved and the analytic expression for eigenstate is derived. The result shows that the normalization of eigenfunction that should be done carefully is quite peculiar. The corresponding coherent state is also discussed.

Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices.

We study acoustical vortices (AVs) and their connection with optical vortices (OVs). We show that AVs and OVs have the same properties if the concept of pseudomomentum is used. Radiation stress and conservation of topological charge are obtained with the pseudomomentum rule. In a weak nonlinear regime, the conservation of pseudo angular momentum imposes a linear increase of the topological charge with the harmonic order. This last result is experimentally verified.

Magnetic Exchange between Orbitally Degenerate Metal Ions: The Problem of Magnetic Anisotropy

In this paper we show that a strong magnetic anisotropy appears in exchange mixed–valence clusters containing orbitally degenerate metal ions. Combining an effective Hamiltonian approach with the technique of the irreducible tensor operators (ITO) and pseudoangular momentum representation we have solved the problem of magnetic exchange in localized and delocalized (mixed–valence) systems with different overall symmetries (D2h, D3h, D4h). The energy pattern as well as the character of the magnetic anisotropy is closely related to the ground term of the ions, electron transfer pathways, and overall symmetry of the system being affected also by the local crystal fields, spin–orbital interactions, and vibronic interactions. Special attention is paid to the origin of the magnetic anisotropy in the face–shared (D3h) binuclear unit [Ti2Cl9]−3. For this case a very good agreement between calculated and measured χ (T) is obtained.

Magnetic exchange interaction in a pair of orbitally degenerate ions: Magnetic anisotropy of [Ti2Cl9]−3

The theory of the kinetic exchange in a pair of orbitally degenerate ions developed by the authors [J. Phys. Chem. A 102, 200 (1998)] is applied to the case of face-shared bioctahedral dimer (overall D3h-symmetry). The effective kinetic exchange Hamiltonian is found for a T22–T22 system taking into account all relevant transfer pathways and charge-transfer crystal field states. The influence of different transfer integrals involved in the kinetic exchange on the energy pattern and magnetic properties of the system is examined. The role of other related interactions (trigonal crystal field, spin–orbit coupling) is also discussed in detail. Using the pseudoangular momentum representation and the technique of the irreducible tensor operators of R3-group we give a general outlook on the nontrivial symmetry properties of the effective Hamiltonian for the D3h-pair, and on the magnetic anisotropy arising from the orbital interactions specific for the case of orbital degeneracy. The magnetic properties of the binuclear unit [Ti2Cl9]−3 in Cs3Ti2Cl9 are discussed with a special emphasis on the magnetic anisotropy experimentally observed in this system. The existing exchange models for [Ti2Cl9]−3 and the concept of the effective Hamiltonian are discussed in the context of the present study.