Lifting Of Degeneracy Research Articles

Symmetry and structure of N–O shallow donor complexes in silicon

Shallow donors in silicon related to nitrogen–oxygen complexes have been investigated by piezospectroscopy of their hydrogenic transitions in the far infrared. Complete stress dependences up to 0.25GPa were obtained for the 1s→2p0 and 1s→2p± transitions of the most prominent members of the (N, O)-family, N–O-3 and N–O-5. Very unusual for shallow donors in silicon, the symmetry of the ground state wave function is T2-like. The lifting of orientational degeneracy for stress in the 〈100〉, 〈111〉, and 〈110〉 directions is compatible with a C2v defect symmetry. Data from the other species of the (N, O)-family are indicative for the same symmetry. The microscopic structure of these centers, in part contradictory to present theoretical models, is discussed.

Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex–vortex interactions are shown to lead to the lifting of topological degeneracy and the energy is found to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low-energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find the necessary conditions for the model to approximate the spectrum of the honeycomb lattice model, and show that bi-partite interactions are responsible for the lifting of degeneracy also in many-vortex systems.

Subband Structure Engineering in Silicon-On-Insulator FinFETs Using Confinement

Splitting between equivalent valleys larger than the spin splitting energy is observed in confined electron systems, e.g. Si films grown either on SiGe substrate or Si dioxide and Si/SiGe quantum dots. Understanding the contribution of different factors in the valley degeneracy lifting is of key importance for the development of spin-based devices in Si. We demonstrate that the splitting between equivalent valleys strongly depends on the confinement direction and that it is orientation dependent. To explain the effect we use a simple but accurate two-band k·p model for the conduction band in silicon. Our data is in good agreement with recent results obtained by first-principle calculations.

Modeling of complex oxide materials from the first principles: systematic applications to vanadates RVO3 with distorted perovskite structure

"Realistic modeling" is a new direction of electronic structure calculations, where the main emphasis is made on the construction of some effective low-energy model entirely within a first-principle framework. Ideally, it is a model in form, but with all the parameters derived rigorously, on the basis of first-principles electronic structure calculations. The method is especially suit for transition-metal oxides and other strongly correlated systems, whose electronic and magnetic properties are predetermined by the behavior of some limited number of states located near the Fermi level. After reviewing general ideas of realistic modeling, we will illustrate abilities of this approach on the wide series of vanadates RVO3 (R= La, Ce, Pr, Nd, Sm, Gd, Tb, Yb, and Y) with distorted perovskite structure. Particular attention will be paid to computational tools, which can be used for microscopic analysis of different spin and orbital states in the partially filled t2g-band. We will explicitly show how the lifting of the orbital degeneracy by the monoclinic distortion stabilizes C-type antiferromagnetic (AFM) state, which can be further transformed to the G-type AFM state by changing the crystal distortion from monoclinic to orthorhombic one. Two microscopic mechanisms of such a stabilization, associated with the one-electron crystal field and electron correlation interactions, are discussed. The flexibility of the orbital degrees of freedom is analyzed in terms of the magnetic-state dependence of interatomic magnetic interactions.

Theoretical studies on vibronic Jahn-Teller systems using group theory

A general form of Hamiltonian for vibronic Jahn-Teller systems is derived on the basis of adiabatic approximation and group theory. The electronic operators and active Jahn-Teller modes appearing in a vibronic system are also discussed. Further calculations of excited states in minima are carried out using unitary transformation method and energy minimization procedure. The results of energy splitting for an electronic triplet Jahn-Teller system are analyzed and compared with particular reference to tetrahedral and its related crystal systems. It is shown that the lift of electronic degeneracy can be quantitatively described by the decomposition of irreducible representations of related group and subgroups.

Lattice dynamics of a rotor-stator molecular crystal: Fullerene-cubaneC60⋅C8H8

The dynamics of fullerene-cubane $({\text{C}}_{60}\ensuremath{\cdot}{\text{C}}_{8}{\text{H}}_{8})$ cocrystal is studied combining experimental [x-ray diffuse scattering, quasielastic and inelastic neutron scattering (INS)] and simulation (molecular dynamics) investigations. Neutron scattering gives direct evidence of the free rotation of fullerenes and of the libration of cubanes in the high-temperature phase, validating the ``rotor-stator'' description of this molecular system. X-ray diffuse scattering shows that orientational disorder survives the order/disorder transition in the low-temperature phase, although the loss of fullerene isotropic rotational diffusion is featured by the appearance of a 2.2 meV mode in the INS spectra. The coupling between INS and simulations allows identifying a degeneracy lift of the cubane librations in the low temperature phase, which is used as a tool for probing the environment of cubane in this phase and for getting further insights into the phase transition mechanism.

Quantum anomalous Hall state in bilayer graphene

We present a symmetry-based analysis of competition between different gapped states that have been proposed in bilayer graphene (BLG), which are all degenerate on a mean field level. We classify the states in terms of a hidden SU(4) symmetry, and distinguish symmetry protected degeneracies from accidental degeneracies. One of the states, which spontaneously breaks discrete time reversal symmetry but no continuous symmetry, is identified as a Quantum Anomalous Hall (QAH) state, which exhibits quantum Hall effect at zero magnetic field. We investigate the lifting of the accidental degeneracies by thermal and zero point fluctuations, taking account of the modes softened under RG. Working in a 'saddle point plus quadratic fluctuations' approximation, we identify two types of RG- soft modes which have competing effects. Zero point fluctuations, dominated by 'transverse' modes which are unique to BLG, favor the QAH state. Thermal fluctuations, dominated by 'longitudinal' modes, favor a SU(4) symmetry breaking multiplet of states. We discuss the phenomenology and experimental signatures of the QAH state in BLG, and also propose a way to induce the QAH state using weak external magnetic fields.

Piezospectroscopy of nitrogen-oxygen shallow donor complexes in silicon

Electronic transitions of N-O related shallow donors in silicon doped with nitrogen and oxygen have been studied under uniaxial stress by Fourier transform spectroscopy in the far infrared. In first approximation, the stress dependence of the hydrogeniclike transitions is described by ground and excited state wave functions belonging to the ${T}_{2}$ representation of the tetrahedral group. Small splittings of the lines originate from the lifting of orientational degeneracy of low-symmetry centers. Nonlinear stress responses of some of the transitions are attributed to interactions with higher-lying $1s$ states. Full analysis of the two most prominent shallow donors N-O-3 and N-O-5 in terms of selection rules and line intensities under polarized light prove that the electronic states are perturbed by a defect potential of apparent ${C}_{2\text{v}}$ symmetry, related to centers with an alignment along the $⟨110⟩$ and a twofold symmetry axis along the $⟨001⟩$ lattice direction. The donor wave function itself can be constructed from Bloch states belonging to a single pair of conduction band minima. It is concluded that present theoretical models for the microstructure of these complexes must be reconsidered, taking into account that any deviations from ${C}_{2\text{v}}$ symmetry must be small and have no measurable effect on the stress pattern.

Controlling mode degeneracy in a photonic crystal nanocavity with infiltrated liquid crystal

We demonstrate the control of the mode degeneracy when a liquid crystal (LC) is infiltrated into an InGaAsP membrane photonic crystal nanocavity with embedded InAs quantum dots. Mode splitting exists in the anisotropic nematic LC state, and not in the unfilled or isotropic LC state. The degeneracy lifting of the quadrupole mode is attributed to the different interactions of the two orthogonal basis modes of the degenerate mode with the two components of the refractive index of the LC. The interpretation is supported by the quantitative agreement between the experimental results and the three-dimensional finite-difference time-domain computations.

High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes

We study theoretically the energy and spatially resolved local density of states (LDoS) in graphene at high perpendicular magnetic field. For this purpose, we extend from the Schr\"odinger to the Dirac case a semicoherent-state Green's-function formalism, devised to obtain in a quantitative way the lifting of the Landau-level degeneracy in the presence of smooth confinement and smooth disordered potentials. Our general technique, which rigorously describes quantum-mechanical motion in a magnetic field beyond the semiclassical guiding center picture of vanishing magnetic length (both for the ordinary two-dimensional electron gas and graphene), is connected to the deformation (Weyl) quantization theory in phase space developed in mathematical physics. For generic quadratic potentials of either scalar (i.e., electrostatic) or mass (i.e., associated with coupling to the substrate) types, we exactly solve the regime of large magnetic field (yet at finite magnetic length, formally, this amounts to considering an infinite Fermi velocity) where Landau-level mixing becomes negligible. Hence, we obtain a closed-form expression for the graphene Green's function in this regime, providing analytically the discrete energy spectra for both cases of scalar and mass parabolic confinement. Furthermore, the coherent-state representation is shown to display a hierarchy of local energy scales ordered by powers of the magnetic length and successive spatial derivatives of the local potential, which allows one to devise controlled approximation schemes at finite temperature for arbitrary and possibly disordered potential landscapes. As an application, we derive general analytical nonperturbative expressions for the LDoS, which may serve as a good starting point for interpreting experimental studies. For instance, we are able to account for many puzzling features of the LDoS recently observed by high magnetic field scanning tunneling spectroscopy experiments on graphene, such as a roughly $\sqrt{m}$ increase in the $m\text{th}$ Landau-level linewidth in the LDoS peaks at low temperatures, together with a flattening of the spatial variations in the Landau-level effective energies at increasing $m$.

Anomalous hyperfine structure of NSF3in the degenerate vibrational statev5=1: Lifting of the parity degeneracy by the fluorine spin-rotation interaction

Author Institution: Institut fur Physikalische Chemie, Universitat Kiel, 24098; Kiel, Germany; Dipartimento di Chimica Fisica e Inorganica, Universita di; Bologna, 40136 Bologna, Italy; Department of Physics and Astronomy, University of British; Columbia, BC V6T 1Z1, Canada

Thermodynamics of polar anisotropy in morphotropic ferroelectric solid solutions

A classical 2-4-6 Landau polynomial is used to describe the generic pseudo-binary composition–temperature (c–T) diagram of ferroelectric solid solutions that display a morphotropic boundary (MB). This polynomial is separated into an isotropic part that depends on the absolute value of the polarization and an anisotropic part that depends also on the direction of polarization. Composition and temperature are taken as external thermodynamic variables and the direction of the polarization vector is treated as an internal relaxing variable of the system. Conditions leading to the observation of stable rhombohedral (FR), tetragonal (FT) and orthorhombic (FO) phases in the c–T plane are determined. In the lowest-order approximation, the isothermal composition change producing the FR to FT transition at the MB is dictated by the condition that the polarization anisotropy energy vanishes. A small lifting of the orientational degeneracy of the polarization leads to differing phase diagram topologies wherein a stable FO phase may interleave the FR and FT phase fields along the MB line. The theory simultaneously accounts for disparate physical phenomena observed in morphotropic ferroelectric solid solutions, which include a sharp reduction in the ferroelectric domain size and the appearance of a dipole glass state connected with intermediate ferroelectric phases, and large extrinsic contributions to the electromechanical properties near the MB. The curving of the MB line, the nearly continuous transitions between adjacent ferroelectric phases and the change in the order of the paraelectric to ferroelectric transitions with composition also arise naturally in the theory.

Exchange interactions in TbMnO3

The most general form of the exchange interaction for Mn atoms in TbMnO3is derived, including nearest neighbour and next-nearest neighbour interactions. It is found that for each exchange path there are six independent exchange terms, leading to a rather complicated Hamiltonian. Experimentally determined constraints on the Hamiltonian are discussed. Degeneracy lifting in the spin-wave excitation spectrum is also discussed.

Valley Symmetry Breaking in Bilayer Graphene: A Test of the Minimal Model

Physical properties reflecting the valley asymmetry of Landau levels in a biased bilayer graphene under a magnetic field are discussed. Within the 4-band continuum model with a Hartree-corrected self-consistent gap and finite damping factor we predict the appearance of anomalous steps in quantized Hall conductivity due to the degeneracy lifting of Landau levels. Moreover, the valley symmetry breaking effect appears as a nonsemiclassical de Haas-van Alphen effect where the reduction of the oscillation period to half cannot be accounted for through quasiclassical quantization of the orbits in reciprocal space.

Chirality and magnetism: Free electron on an infinite helix, NCD, MCD, and magnetochiral dichroism

The free electron on a helix with periodic boundary conditions is undoubtedly one of the simplest quantum chemical models exhibiting both chirality and orbital angular momentum. Subjected to a light beam traveling parallel to the helix axis, the model exhibits natural circular dichroism and, assuming a lifting of the magnetic degeneracy by an external static magnetic field, also magnetic circular dichroism and magnetochiral dichroism. We believe that the model illustrates well the interplay of chirality and magnetism in a pure magnetic state, and that it could also be of interest in the study of the recently discovered chiral ferromagnets.

Analysis of enantiotopic discrimination in the NMR spectra of prochiral solutes in chiral liquid crystals by symmetry factorization of the Saupe ordering matrix

The enantiotopic discrimination in the NMR spectrum of prochiral molecules dissolved in chiral liquid crystals (CLCs) is governed by the ordering of the solute molecules. The lifting of the spectral degeneracy of enantiotopic sites in such solvents stems from the nonequivalence in their effective ordering. This, in turn, is brought about by a change in the directions (or the equivalence) of the principal axes of their ordering tensor, relative to those in achiral liquid crystals. This discrimination mechanism can only occur for solute molecules belonging to a limited number of (“allowed”) improper point groups, viz., D2d, C2v, Cs, and S4 [D. Merlet, J. W. Emsley, P. Lesot, and J. Courtieu, J. Chem. Phys. 111, 6890 (1999)]. In this work it is shown that the ordering tensor of such prochiral solutes in CLC, SCLC, and likewise, the tensors, Tk, describing the anisotropic magnetic resonance interactions of enantiotopic pairs, can be partitioned into symmetric and antisymmetric (and irrelevant) parts. The NMR results in such solvents can be cast into separate sets of equations depending on either the symmetric or the antisymmetric parts of SCLC and Tk. The discrimination observed in such solvents depends only on the latter set of equation, while the former applies to the average splitting of enantiotopic pairs as well as to diasterotopic or homotopic sites in the prochiral molecules. The factorization procedure greatly facilitates the analysis of the ordering properties of prochiral solutes in CLC and provides new insight on the discrimination mechanism. In particular, it allows correlation between independent enantiotopic partners and the identification of NMR signals belonging to common prosterogenic faces in the molecule. Expressions relating NMR observables with the symmetric and antisymmetric parts of SCLC are derived for each of the four allowed groups. Model examples are presented and discussed.

Charge2eSkyrmions in Bilayer Graphene

Quantum Hall states that result from interaction induced lifting of the eightfold degeneracy of the zeroth Landau level in bilayer graphene are considered. We show that at even filling factors electric charge is injected into the system in the form of charge 2e Skyrmions. This is a rare example of binding of charges in a system with purely repulsive interactions. We calculate the Skyrmion energy and size as a function of the effective Zeeman interaction and discuss the signatures of the charge 2e Skyrmions in the scanning probe experiments.

Crystal electric field excitations in ferromagnetic Ce TX compounds

A ferromagnetic ground state was identified for the compounds CeCuGe (T c = 10 K), CeCuSi (T c = 15 K) [F. Yang, et al., J. Appl. Phys. 69 (1991) 4705] and CeAuGe (T c = 10K) [R. Pottgen, J. Magn. Magn. Mater. 152 (1996) 196]. The observed saturation magnetic moment values at low temperatures for all three compounds are considerably less than the theoretically expected value g J J = 2.14μ B for the free Ce 3+ ion involving the entire six-fold J = 5/2 multiplet, and thus provide a first indication of partial lifting of the f-electron level degeneracy in these compounds. Specific heat data yield crystal electric field (CEF) excitation energies (Δ Sch ) equivalent to 140 K for CeCuGe, 110 K for CeCuSi and 280 K for CeAuGe. To confirm the presence of CEF excitations directly, we have carried out inelastic neutron scattering (INS) measurements on all three compounds, using the HET spectrometer at ISIS Facility. Here, we present a detailed analysis of the INS spectra of CeCuSi on the basis of a CEF model and the detailed analysis of the INS of the other two compounds will be reported elsewhere.

Low-energy Landau levels of Bernal zigzag graphene ribbons

Low-energy Landau levels of Bernal zigzag graphene ribbons in the presence of a uniform perpendicular magnetic field (B) are investigated by the Peierls coupling tight-binding model. State energies and associated wave functions are dominated by the B-field strength and the kz-dependent inter-ribbon interactions. The occupied valence bands are asymmetric to the unoccupied conduction bands about the Fermi level. Many doubly degenerate Landau levels and singlet curving magnetobands exist along the kx and kz directions, respectively. The kz-dependent inter-ribbon interactions dramatically modify the magnetobands, such as the lift of double degeneracy, the change in state energies, and the production of two groups of curving magnetobands. They also change the characteristics of the wave functions and cause the redistribution of the charge-carrier density. The kz-dependent wave functions are further used to predict the selection rule of the optical transition.

HIGH FIELD MAGNETO-OPTICAL SPECTROSCOPY OF HIGHLY ALIGNED INDIVIDUAL AND ENSEMBLE SINGLE-WALLED CARBON NANOTUBES

The tubular nature of single-walled carbon nanotube (SWCNT) crystals allows them to exhibit non-intuitive quantum phenomena when threaded by a magnetic flux, which breaks the time reversal symmetry and adds an Aharonov-Bohm phase to the circumferential boundary conditions on the electronic wave function. We demonstrate that such a symmetry-breaking magnetic field can dramatically "brighten" an optically-inactive, or dark, exciton state at low temperature. This phenomenon, magnetic brightening, can be understood as a consequence of interplay between the strong intervalley Coulomb mixing and field-induced lifting of valley degeneracy. Most recently, we made the direct observation of the dark excitonic state in individual SWCNTs using low-temperature micro-photoluminescence (PL) and and verified the importance of a parallel, tube-threading magentic field with ensemble spectroscopy. For micro-PL, a magnetic field up to 5 T, applied along the nanotube axis, brightened the dark state, leading to the emergence of a new emission peak. The peak rapidly grew in intensity with increasing field at the expense of the originally-dominant bright exciton peak and finally became dominant at fields > 3 T. The directly measured dark-bright splitting values were 1-4 meV for tube diameters 1.0-1.3 nm. For ensemble PL, we used fields up to 55 T in two collection geometries to demonstrate the importance of the tube-threading component. These experiments have provided one of the most critical tests for recently-proposed theories of 1-D excitons taking into account the strong 1-D Coulomb interactions and unique band structure on an equal footing.