Energy Level Spacing Research Articles

Construction of theoretical hybrid potential energy curves for LiH(XΣ+1)

Various all-electron and valence-electron potential energy curves for LiH(X (1)Sigma(+)) are compared and assessed. Hybrid potential energy curves are constructed from all-electron potentials at short range and a valence-electron calculation otherwise. This approach provides for the X state of LiH an overall potential curve, which is ionic at equilibrium, and presents an avoided crossing with the excited A state, leading to neutral dissociation products. The classical turning points predicted by these purely theoretical hybrid potentials are compared with those of the experimentally based inverted-perturbation approach (IPA) potentials for both (7)LiH and (7)LiD. Predicted vibrational energy-level spacings show reasonable (less than or approximately 1 cm(-1)) agreement with the corresponding IPA values. Rotation and vibration-rotation transition energies arising from the most accurate hybrid potential are shown to compare very favorably with recent high-resolution spectroscopic data on (7)LiH and (7)LiD.

Investigation of electron wave function hybridization in Ga0.25In0.75As/InP arrays

We present a measurement technique for quantifying coupling between semiconductor quantum dots in an array. This technique employs magnetoconductance fluctuations to probe the decrease in the average spacing of the quantum energy levels as the electron wave functions in the dots undergo hybridization. Focusing on Ga0.25In0.75As dots, we investigate hybridization as the coupling strength is varied and the number of dots in the array is increased. Our technique reveals a significant drop in the average energy level spacing for multiple dot arrays, which is strong evidence for wave function hybridization.

Importance of Coulomb interactions in bound-to-continuum THz quantum cascade lasers

We demonstrate the importance of Coulomb interactions in bound-to-continuum THz quantum cascade lasers, employing an ensemble Monte-Carlo analysis. In such structures, the electron-electron interactions between the closely spaced energy levels in the minibands tend to play a more important role than in resonant-phonon depopulation designs, where the energy levels are more energetically separated and LO phonon scattering prevails. Also a significant conduction band bending due to space charge effects is observed. Thus, especially for bound-to-continuum structures careful modelling of Coulomb interactions is crucial to obtain good agreement with experiment.

A back-to-front derivation: the equal spacing of quantum levels is a proof of simple harmonic oscillator physics

The dynamical behaviour of simple harmonic motion can be found in numerous natural phenomena. Within the quantum realm of atomic, molecular and optical systems, two main features are associated with harmonic oscillations: a finite ground-state energy and equally spaced quantum energy levels. Here it is shown that there is in fact a one-to-one mapping between the provision of equally spaced quantum energy levels and simple harmonic motion. The analysis establishes that the Hamiltonian of any system featuring quantized energy levels in an evenly spaced, infinite set must have a quadratic dependence on a pair of canonically conjugate variables. Moreover, specific physical inferences can be drawn. For example, exploiting this ‘back-to-front’ derivation, and based on the known existence of photons, it can be proved that an electromagnetic energy density is quadratic in both the electric and magnetic fields.

Charge dynamics of vortex cores in layered chiral triplet superconductors

In an accompanying paper (Sauls and Eschrig 2009 New J. Phys. 11 075008) we have studied the equilibrium properties of vortices in a chiral quasi-two-dimensional triplet superfluid/superconductor. Here we extend our studies to include the dynamical response of a vortex core in a chiral triplet superconductor to an external ac electromagnetic field. We consider in particular the response of a doubly quantized vortex with a homogeneous core in the time-reversed phase. The external frequencies are assumed to be comparable in magnitude to the superconducting gap frequency, such that the vortex motion is nonstationary but can be treated by linear response theory. We include broadening of the vortex-core bound states due to impurity scattering and consider the intermediate-clean regime, with a broadening comparable to or larger than the quantized energy level spacing. The response of the order parameter, impurity self-energy, induced fields and currents are obtained by a self-consistent calculation of the distribution functions and the excitation spectrum. Using these results we obtain the self-consistent dynamically induced charge distribution in the vicinity of the core. This charge density is related to the nonequilibrium response of the bound states and order parameter collective mode, and dominates the electromagnetic response of the vortex core.

Optical conductivity of rattling phonons in type-I clathrateBa8Ga16Ge30

A series of infrared-active optical phonons have been detected in type-I clathrate Ba$_8$Ga$_{16}$Ge$_{30}$ by terahertz time-domain spectroscopy. The conductivity spectra with the lowest-lying peaks at 1.15 and 1.80 THz are identified with so-called rattling phonons, i.e., optical modes of the guest ion Ba$^{2+}(2)$ with $T_{1u}$ symmetry in the oversized tetrakaidecahedral cage. The temperature dependence of the spectra from these modes are totally consistent with calculations based on a one-dimensional anharmonic potential model that, with decreasing temperature, the shape becomes asymmetrically sharp associated with a softening for the weight to shift to lower frequency. These temperature dependences are determined, without any interaction effects, by the Bose-factor for optical excitations of anharmonic phonons with the nonequally spaced energy levels.

Thermoelectric and Seebeck coefficients of granular metals

In this work we present a detailed study and derivation of the thermopower and thermoelectric coefficient of nanogranular metals at large tunneling conductance between the grains, ${g}_{T}⪢1$. An important criterion for the performance of a thermoelectric device is the thermodynamic figure of merit which is derived using the kinetic coefficients of granular metals. All results are valid at intermediate temperatures, ${E}_{c}⪢T/{g}_{T}>\ensuremath{\delta}$, where $\ensuremath{\delta}$ is the mean energy-level spacing for a single grain and ${E}_{c}$ is its charging energy. We show that the electron-electron interaction leads to an increase in the thermopower with decreasing grain size and discuss our results in light of future generation thermoelectric materials for low-temperature applications. The behavior of the figure of merit depending on system parameters such as grain size, tunneling conductance, and temperature is presented.

Conditional generation of highly excited Dicke states for N atoms via a single asymmetric atom–cavity interaction

We propose a scheme for the generation of highly excited Dicke states for multiple atoms. In the scheme N + 1 atoms dispersively interact with a vacuum cavity, with the last atom having a different coupling and driven by a classical field. As the first N atoms are symmetric, they remain in the symmetric Dicke subspace, with the spacing of energy levels being unequal due to the vacuum light shift. Employing the Stark shift induced by the classical field, one can tune the last atom in resonance with the transition between two specific Dicke states of the first N atoms. So the detection of the state of the last atom conditionally collapses the first N atoms to the desired Dicke state.

Modulated Conjugation as a Means of Improving the Intrinsic Hyperpolarizability

A new strategy for optimizing the first hyperpolarizability based on the concept of a modulated conjugated path in linear molecules is investigated. A series of seven novel chromophores with different types of conjugated paths were synthesized and characterized. Hyper-Rayleigh scattering experiments confirmed that modulated conjugation paths that include benzene, thiophene, and/or thiazole rings in combination with azo and/or ethenyl linkages between dihydroxyethylamino donor groups and various acceptor groups result in enhanced intrinsic hyperpolarizabilities that exceed the long-standing apparent limit for two of the chromophores. The experimental results are analyzed and interpreted in the context of quantum limits, which show that conjugation modulation of the bridge in donor/acceptor molecules simultaneously optimizes the transition moments and the energy-level spacing.

QUANTUM SIZE EFFECTS OF HYDROGEN IMPURITY AT OFF-CENTER DONOR ATOM IN SPHERICAL QUANTUM DOTS

We obtain the numerical solutions for a single electron energy level of a hydrogen impurity confined inside an off-center donor in a spherical quantum dot. The energy spectrum of single electron impurity for off-center donor atom subject to screened Coulomb's attractive potential barrier well is presented. Using B-spline finite element method, we generate a complete set of wavefunctions within the central unit cell, and through the perturbation theory, the whole energy spectrum of an impurity at low temperature can be explored. We find that the energy splitting and different related magnetic quantum number ml, i.e., (n, l, ml) varies with quantum dot size. This is due to the fact that quantum confinement effect lifts the degeneracy which comes from the Coulomb interaction. On the other hand, the phenomena of energy spectrum splitting under this model is an analogous case, considering that degeneracy of hydrogen energy levels in free space has been lifted by an external magnetic field (Zeeman effect).

Intermixing of InAs/GaAs quantum dots by proton implantation and rapid thermal annealing

Low-temperature photoluminescence (PL) experiments have been carried out to investigate In/Ga intermixing in InAs/GaAs self-assembled quantum dots (QDs) by studying the changes in the optical properties of the system by rapid thermal annealing (RTA) and by room temperature proton implantation at various doses followed by RTA. The study of the RTA effect on the investigated structure shows that thermal stability can be ensured for an annealing temperature below 675 °C. For higher annealing temperatures, the thermal intermixing is found to change both the optical transition energy and the inter-sublevel spacing of the QD energy levels. By using proton implantation at various doses and subsequent annealing at 675 °C for 30 s, a tunable energy shift up to 130 meV has been obtained. The band gap tuning limit for this system has been achieved for an implantation dose of 5×10 13 cm −2. Regardless of the intermixing technique employed, a pronounced PL peak broadening is found to occur at low annealing temperatures and/or proton implantation doses.

Carrier‐carrier and carrier‐phonon scattering in the low‐density and low‐temperature regime for resonantly pumped semiconductor quantum dots

AbstractWe study carrier relaxation due to Coulomb scattering and interaction with LO‐phonons in semiconductor quantum dots at low temperatures. Scattering for different relaxation process are evaluated for various carrier distributions that correspond to stages of the typical relaxation kinetics after optical excitation with a weak pulse, generating on average less than one electron per quantum dot.Even when the spacing of the quantum dot energy levels does not match the LO‐phonon energy, we.nd that carrier‐LO‐phonon scattering, in addition to electronelectron and electron‐hole interaction, provides efficient carrier relaxation. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Evolution of Quantum States in an Electro-Optic Phase Modulator

We present a quantum mechanical model to evaluate the time evolution of quantum states of light in an electro-optic phase modulator (EOPM). The phase modulator is analogous to a multilevel atomic system with equally spaced energy levels, each of which corresponds to the sidebands of the phase modulated optical field. Using perturbation theory, we solve the equations governing the interaction of the optical field with a modulating field both with and without phase velocity mismatch. A unitary operator that evolves any given quantum state allows us to evaluate the response of EOPM for single photon, coherent and squeezed states. We apply these results to the analysis of frequency coded quantum key distribution scheme.

Thermoelectric properties of granular metals

We investigate the thermopower and thermoelectric coefficient of nanogranular materials at large tunneling conductance between the grains, ${g}_{T}⪢1$. We show that at intermediate temperatures, $T>{g}_{T}\ensuremath{\delta}$, where $\ensuremath{\delta}$ is the mean energy-level spacing for a single grain, electron-electron interaction leads to an increase in the thermopower with decreasing grain size. We discuss our results in light of the next generation of thermoelectric materials and present the behavior of the figure of merit depending on the system parameters.

Using fundamental principles to understand and optimize nonlinear-optical materials

Our approach to the problem of understanding the nonlinear-optical response of a material focuses on fundamental concepts, which are exact and lead to broad results that encompass all material systems. For example, one can calculate precisely and without approximation the fundamental limit of the efficiency of any optical phenomenon. Such limits, in turn, when built into a scale-invariant figure of merit can be used to determine what makes a material optimal for maximizing a desired property, such as its nonlinear-optical response. However, since the results are broad and general (for example, fundamental guidelines for making better quantum systems may demand a particular shape of an electron cloud or energy level spacing), the difficulty arises in implementing such fine tuning using the approaches available to the synthetic chemist or nanotechnologist. Undoubtedly, an intimate interplay of empirical and fundamental approaches will need to be applied to the problem of optimizing molecules and materials. Much of the work in the field of nonlinear optics has by necessity focused on empirical approaches. Our work focuses on the fundamental approach, which is beginning to bear fruit in providing practical guidelines for making new materials. This paper reviews progress made over the last decade.

Structure and luminescence properties of antimony(III) complex compounds

The influence of the structure of the anion sublattice on the spectral luminescence properties is investigated using the example of antimony(III) complex compounds with outer-sphere organic cations. It is established that the factors favorable for the enhancement of the luminescence of antimony(III) ions are the island structure of the anion sublattice with minimum distortions of the antimony(III) coordination polyhedron and closely spaced energy levels of the outer-sphere organic cation and the antimony(III) ion.

Electronic Structure and Spectroscopy of Cadmium Telluride Quantum Wires

The size-dependent electronic structure of CdTe quantum wires is determined by density functional theory using the local density approximation with band-corrected pseudopotential method. The results of the calculations are then used to assign the size-dependent absorption spectrum of colloidal CdTe quantum wires synthesized by the solution-liquid-solid mechanism. Quantitative agreement between experiment and theory is achieved. The absorption features comprise transitions involving the highest 25-30 valence-band states and lowest 15 conduction-band states. Individual transitions are not resolved; rather, the absorption features consist of clusters of transitions that are determined by the conduction-band energy-level spacings. The sequence, character, and spacing of the conduction-band states are strikingly consistent with the predictions of the simple effective-mass-approximation, particle-in-a-cylinder model. The model is used to calculate the size dependence of the electron effective mass in CdTe quantum wires.

Theoretical Investigation of Excited States of Oligothiophene Anions

Electron-hole symmetry upon p- and n-doping of conducting organic polymers is rationalized with Hückel theory by the presence of symmetrically located intragap states. Since density functional theory (DFT) predicts very different geometries and energy level diagrams for conjugated pi-systems than semiempirical methods, it is an interesting question whether DFT confirms the existence of electron-hole symmetry predicted at the Hückel level. To answer this question, geometries of oligothiophene anions with 5-19 rings were optimized and their UV/vis spectra were calculated with time-dependent DFT. Although DFT does not produce symmetrically placed sub-band energy levels, spectra of cations and anions are almost identical. The similarity in transition energies and oscillator strengths of anions and cations can be explained by the fact that the single sub-band energy level of cations lies above the valence band by the same amount of energy as the single sub-band level of anions lies below the conduction band. This and the resemblance of the energy level spacings in valence bands of cations to those in conduction bands of anions give rise to peaks with equal energies and oscillator strengths.

Excited-state relaxation in PbSe quantum dots

In solids the phonon-assisted, nonradiative decay from high-energy electronic excited states to low-energy electronic excited states is picosecond fast. It was hoped that electron and hole relaxation could be slowed down in quantum dots, due to the unavailability of phonons energy matched to the large energy-level spacings ("phonon-bottleneck"). However, excited-state relaxation was observed to be rather fast (< or =1 ps) in InP, CdSe, and ZnO dots, and explained by an efficient Auger mechanism, whereby the excess energy of electrons is nonradiatively transferred to holes, which can then rapidly decay by phonon emission, by virtue of the densely spaced valence-band levels. The recent emergence of PbSe as a novel quantum-dot material has rekindled the hope for a slow down of excited-state relaxation because hole relaxation was deemed to be ineffective on account of the widely spaced hole levels. The assumption of sparse hole energy levels in PbSe was based on an effective-mass argument based on the light effective mass of the hole. Surprisingly, fast intraband relaxation times of 1-7 ps were observed in PbSe quantum dots and have been considered contradictory with the Auger cooling mechanism because of the assumed sparsity of the hole energy levels. Our pseudopotential calculations, however, do not support the scenario of sparse hole levels in PbSe: Because of the existence of three valence-band maxima in the bulk PbSe band structure, hole energy levels are densely spaced, in contradiction with simple effective-mass models. The remaining question is whether the Auger decay channel is sufficiently fast to account for the fast intraband relaxation. Using the atomistic pseudopotential wave functions of Pb(2046)Se(2117) and Pb(260)Se(249) quantum dots, we explicitly calculated the electron-hole Coulomb integrals and the P-->S electron Auger relaxation rate. We find that the Auger mechanism can explain the experimentally observed P-->S intraband decay time scale without the need to invoke any exotic relaxation mechanisms.

The quasi-quantum treatment of rotationally inelastic scattering from a hard shell potential: its derivation and practical use

The QQT is a quasi-quantum mechanical treatment of the collision between molecules. Instead of a partial wave expansion approach, it uses a kind of Feynman path integral method that exploits the path length differences originating from the different orientations of an anisotropic molecule. As a result, the QQT provides valuable physical insight while requiring very little computational effort. The current paper gives a systematic derivation of the QQT and explains its underlying principles. The expression for the scattering amplitude is shown to be self-consistent, without any normalisation factors, when the rotational energy level spacing is negligible. The constant curvature approximation that is presented makes the QQT conceptually even more simple, and its effect on the calculated differential cross-sections (DCSs) turns out to be small. As examples we present QQT calculations of the DCSs for Ne–CO(1Σ) and He–NO(2Π), at collision energies of, respectively, 511 cm−1 and 514 cm−1. The anisotropy of the hard shell potential energy surface for Ne–CO in terms of the incoming de Broglie wavelength is about twice as large as for He–NO. This leads to state-to-state DCSs that have up to three maxima of comparable amplitude, instead of only one large maximum as is found for He–NO. The QQT results for these two applications are compared with results from close coupling calculations.