Optical properties of wurtzite GaN/AlN quantum dots grown on non-polar planes: the effect of stacking faults in the reduction of the internal electric field

The optical emission of non-polar GaN/AlN quantum dots has been investigated. The presence of stacking faults inside these quantum dots is evidenced in the dependence of the photoluminescence with temperature and excitation power. A theoretical model for the electronic structure and optical properties of non-polar quantum dots, taking into account their realistic shapes, is presented which predicts a substantial reduction of the internal electric field but a persisting quantum confined Stark effect, comparable to that of polar GaN/AlN quantum dots. Modeling the effect of a 3 monolayer stacking fault inside the quantum dot, which acts as zinc-blende inclusion into the wurtzite matrix, result…

Inhomogeneous free-electron distribution in InN nanowires: Photoluminescence excitation experiments

Photoluminescence excitation (PLE) spectra have been measured for a set of self-assembled InN nanowires (NWs) and a high-crystalline quality InN layer grown by molecular-beam epitaxy. The PLE experimental lineshapes have been reproduced by a self-consistent calculation of the absorption in a cylindrical InN NW. The differences in the PLE spectra can be accounted for the inhomogeneous electron distribution within the NWs caused by a bulk donor concentration $({N}_{D}^{+})$ and a two-dimensional density of ionized surface states $({N}_{ss}^{+})$. For NW radii larger than 30 nm, ${N}_{D}^{+}$ and ${N}_{ss}^{+}$ modify the absorption edge and the lineshape, respectively, and can be determined f…

Deformation profile in GaN quantum dots: Medium-energy ion scattering experiments and theoretical calculations

Medium energy ion scattering (MEIS) has been used to measure at the scale of the monolayer the deformation profile of self-organized GaN quantum dots grown on AlN by molecular-beam epitaxy. The effect of capping the GaN dots by a thin layer of AlN has also been studied. It is shown that GaN dots are partially relaxed in every situation. Capping them with AlN has little effect on the basal plane, as expected, but strongly modifies the strain of the upper part of dots. The experimental results are compared with theoretical calculations, allowing one to conclude that GaN quantum dots experience a nonbiaxial strain, which drastically decreases when going from the basal plane up to the apex of t…

LDA+Uand tight-binding electronic structure of InN nanowires

In this paper we employ a combined ab initio and tight-binding approach to obtain the electronic and optical properties of hydrogenated Indium nitride InN nanowires. We first discuss InN band structure for the wurtzite structure calculated at the LDA+U level and use this information to extract the parameters needed for an empirical tight-binging implementation. These parameters are then employed to calculate the electronic and optical properties of InN nanowires in a diameter range that would not be affordable by ab initio techniques. The reliability of the large nanowires results is assessed by explicitly comparing the electronic structure of a small diameter wire studied both at LDA+U and…

Anisotropic optical response of GaN and AlN nanowires.

We present a theoretical study of the electronic structure and optical properties of free-standing GaN and AlN nanowires. We have implemented the empirical tight-binding method, with an orbital basis sp(3), that includes the spin-orbit interaction. The passivation of the dangling bonds at the free surfaces is also studied, together with the effects on the electronic structure of the nanowire. For both GaN and AlN nanowires, we have found a remarkable anisotropy of the optical absorption when the light-polarization changes, showing in the case of GaN a dependence on the nanowire size.

Resonant hyper-Raman scattering in spherical quantum dots

A theoretical model of resonant hyper-Raman scattering by an ensemble of spherical semiconductor quantum dots has been developed. The electronic intermediate states are described as Wannier-Mott excitons in the framework of the envelope function approximation. The optical polar vibrational modes of the nanocrystallites (vibrons) and their interaction with the electronic system are analized with the help of a continuum model satisfying both the mechanical and electrostatic matching conditions at the interface. An explicit expression for the hyper-Raman scattering efficiency is derived, which is valid for incident two-photon energy close to the exciton resonances. The dipole selection rules f…

Finite Size Effects on the Optical Transitions in Quantum Rings under a Magnetic Field

We present a theoretical study of the energy spectrum of single electron and hole states in quantum dots of annular geometry under a high magnetic field along the ring axis in the frame of uncorrelated electron-hole theory. We predict the periodic disappearance of the optical emission of the electron-hole pair as the magnetic field increases, as a consequence of the finite height of the barriers. The model has been applied to semiconductor rings of various internal and external radii, giving as limiting cases the disk and antidot.

Dynamic carrier distribution in quantum wells modulated by surface acoustic waves

We have investigated the dynamics of photogenerated carriers under surface acoustic wave (SAW) fields in GaAs quantum wells using spatially and time-resolved photoluminescence (PL). The frequency and phase of the PL oscillations under a SAW yield information about the carrier distribution and the band-gap modulation induced by the SAW. We directly prove that the transport properties of the carriers ultimately control their distribution, storage and, subsequent recombination in the modulated potential.

Energy levels of a quantum ring in a lateral electric field

Abstract The electronic states of a semiconductor quantum ring (QR) under an applied lateral electric field are theoretically investigated and compared with those of a quantum disk of the same size. The eigenstates and eigenvalues of the Hamiltonian are obtained from a direct matrix diagonalization scheme. Numerical calculations are performed for a hard-wall confinement potential and the electronic states are obtained as a function of the electric field and the ratio r2/r1, where r2 (r1) is the outer (inner) radius of the ring. The effects of decreasing symmetry and mixing on the energy levels and wave functions due to the applied electric field are also studied. The direct optical absorpti…

Inhomogeneous electron distribution in InN nanowires: Influence on the optical properties

In this work, we study theoretically and experimentally the influence of the surface electron accumulation on the optical properties of InN nanowires. For this purpose, the photoluminescence and photoluminescence excitation spectra have been measured for a set of self-assembled InN NWs grown under different conditions. The photoluminescence excitation experimental lineshapes have been reproduced by a self-consistent calculation of the absorption in a cylindrical InN nanowires. With the self-consistent model we can explore how the optical absorption depends on nanowires radius and doping concentration. Our model solves the Schrodinger equation for a cylindrical nanowire of infinite length, a…

Spontaneous core–shell elemental distribution in In-rich In(x)Ga1-xN nanowires grown by molecular beam epitaxy.

International audience; The elemental distribution of self-organized In-rich InxGa1-xN nanowires grown by plasma-assisted molecular beam epitaxy has been investigated using three different techniques with spatial resolution on the nanoscale. Two-dimensional images and elemental profiles of single nanowires obtained by x-ray fluorescence and energy-dispersive x-ray spectroscopy, respectively, have revealed a radial gradient in the alloy composition of each individual nanowire. The spectral selectivity of resonant Raman scattering has been used to enhance the signal from very small volumes with different elemental composition within single nanowires. The combination of the three techniques ha…

Resonant Raman scattering in self-assembledGaN∕AlNquantum dots

Self-assembled $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ quantum dots have been investigated by means of Raman scattering. A resonant enhancement of the Raman peaks has been observed when the excitation is tuned above the GaN band-gap energy. The polar mode nature, either quasiconfined or interfacial, has been assigned after comparing with the polar optical modes of spheroidal dots calculated within the framework of the anisotropic dielectric continuum model. The built-in strain of the GaN dots induced a substantial blueshift of the nonpolar ${E}_{2H}$ Raman mode frequency. A theoretical model that analyzes the three-dimensional strain distribution in the quantum dots has been employed …

Influence of the InAs coverage on the phonon-assisted recombination in InAs/GaAs quantum dots

6 páginas, 3 figuras.

Influence of strain in the reduction of the internal electric field in GaN/AlN quantum dots grown ona-plane 6H-SiC

The strain state of stacks of GaN/AIN quantum dots (QDs) grown on (0001) and (1120) 6H-SiC has been investigated by means of Raman spectroscopy. Depending on the orientation of the wurtzite axis with respect to the growth direction it is found that the piezoelectric contribution to the electrostatic potential may either reinforce that arising from the spontaneous polarization or oppose it. The experimental results are compared with a theoretical model for the strain and polarization field in QDs of both orientations that allows the calculation of the electrostatic potential in the QDs. Both the experimental results and the theoretical model indicate that the internal electric field and elec…

Reversed polarized emission in highly strained a-plane GaN/AlN multiple quantum wells

The polarization of the emission from a set of highly strained $a$-plane GaN/AlN multiple quantum wells of varying well widths has been studied. A single photoluminescence peak is observed that shifts to higher energies as the quantum well thickness decreases due to quantum confinement. The emitted light is linearly polarized. For the thinnest samples the preferential polarization direction is perpendicular to the wurtzite $c$ axis with a degree of polarization that decreases with increasing well width. However, for the thickest well the preferred polarization direction is parallel to the $c$ axis. Raman scattering, x-ray diffraction, and transmission electron microscopy studies have been p…

Tight-Binding Simulations of Nanowires

Carrier Transport in GaAs Nanowires Using Surface Acoustic Waves

ABSTRACTThe oscillating piezoelectric field of a surface acoustic wave (SAW) is employed to transport photoexcited electrons and holes in GaAs nanowires (NWs) transferred to a SAW beam line on a LiNbO3 crystal. We show that carriers generated in the NW by a focused light spot can be acoustically transported to a second location, where they recombine emitting short light pulses. The results presented here demonstrate the high-frequency manipulation of carriers in NWs without the use of electrical contacts, which opens new perspectives for applications in opto-electronic devices operating at GHz frequencies.

Resonant Raman scattering of core-shell GaN/AlN nanowires.

Abstract We have analyzed the electron–phonon coupling in GaN/AlN core–shell nanowires by means of Raman scattering excited at various wavelengths in the ultraviolet spectral range (335, 325 and 300 nm) and as a function of the AlN shell thickness. The detailed analysis of the multi-phonon spectra evidences important differences with excitation energy. Under 325 and 300 nm excitation the Raman process is mediated by the allowed A 1(LO) phonon mode, where the atoms vibrate along the NW axis. Considering its selection rules, this mode is easily accessible in backscattering along the wurtzite c axis. Interestingly, for 335 nm excitation the scattering process is instead mediated by the E 1(LO)…

Elemental distribution and structural characterization of GaN/InGaN core-shell single nanowires by Hard X-ray synchrotron nanoprobes

Improvements in the spatial resolution of synchrotron-based X-ray probes have reached the nano-scale and they, nowadays, constitute a powerful platform for the study of semiconductor nanostructures and nanodevices that provides high sensitivity without destroying the material. Three complementary hard X-ray synchrotron techniques at the nanoscale have been applied to the study of individual nanowires (NWs) containing non-polar GaN/InGaN multi-quantum-wells. The trace elemental sensitivity of X-ray fluorescence allows one to determine the In concentration of the quantum wells and their inhomogeneities along the NW. It is also possible to rule out any contamination from the gold nanoparticle …

Excitonic model for second-order resonant Raman scattering.

A theoretical model for second-order resonant Raman scattering is presented. The effect of Coulomb interaction between electrons and holes is fully taken into account in the framework of the effective-mass approximation. By introducing discrete and continuous excitonic intermediate states in the Raman process, an explicit expression for the Raman scattering efficiency is given for long-range Fr\"ohlich electron-phonon interaction. The model developed can be used to evaluate Raman profiles around the resonant region. A closed-form expression for all matrix elements of the exciton-phonon interaction is obtained once the Coulomb problem for the relative electron-hole motion is separated in sph…

The generalized plane piezoelectric problem: Theoretical formulation and application to heterostructure nanowires

We present a systematic methodology for the reformulation of a broad class of three-dimensional (3D) piezoelectric problems into a two-dimensional (2D) mathematical form. The sole underlying hypothesis is that the system geometry and material properties as well as the applied loads (forces and charges) and boundary conditions are translationally invariant along some direction. This class of problems is commonly denoted here as the generalized plane piezoelectric (GPP) problem. The first advantage of the generalized plane problems is that they are more manageable from both analytical and computational points of view. Moreover, they are flexible enough to accommodate any geometric cross secti…

Raman study and theoretical calculations of strain in GaN quantum dot multilayers

Changes in strain and phonon mode energy in stacks of self-assembled GaN quantum dots embedded in AlN have been studied by means of Raman spectroscopy as a function of the number of periods. The ${E}_{2H}$ phonon modes related to the quantum dots and AlN spacers are clearly resolved, and their energies allow monitoring the state of strain of the dots and AlN spacers simultaneously. The evolution of the measured phonon frequencies and the associated strains are discussed in comparison with theoretical calculations of the inhomogeneous strain distribution in a system of coherent misfitting inclusions.

Electronic and optical properties of InN nanowires

Abstract We have employed a multiband envelope function method to study wurtzite [0001] InN nanowires of cylindrical cross section. The electronic subband structure and optical absorption spectrum are calculated as a function of the nanowire radius. The energies of the Γ -point conduction band states show a monotonous increase with decreasing radius. On the other hand, the size dispersion of the valence band states is more involved, showing various crossing and anticrossing effects due to the interplay between the confinement and band mixing effects. Thus, for small ( R 2 nm ) nanowires the highest valence band state has dominant p z -symmetry, but for R > 2 nm the highest state has dominan…

Structural properties of GaN quantum dots

The strain state and the deformation profile of GaN quantum dots embedded in AlN have been measured by high resolution electron microscopy, medium energy ion scattering and grazing incidence X-ray diffraction. The results are compared with theoretical calculations, allowing one to conclude that GaN quantum dots experience a non biaxial strain which drastically decreases when going from the basal plane up to the apex of the dots. We also demonstrate that AlN is distorted in the surroundings of the dots, which provides the driving force for vertical correlation of GaN dots when the AlN spacer between successive planes is thin enough.

Electric-field-induced Raman scattering in GaAs: Franz-Keldysh oscillations

We have studied the influence of strong electric fields on the Raman scattering intensity from LO phonons in GaAs (100) at room temperature using laser excitation energies above the fundamental ${\mathit{E}}_{0}$ gap. Striking oscillations are found in the scattering intensity for configurations where either the deformation potential or Fr\"ohlich electron-phonon interaction contribute. The oscillations in the deformation-potential-mediated scattering intensity can be related to Franz-Keldysh oscillations derived from the ${\mathit{E}}_{0}$ gap, whereas a more complicated mechanism has to be invoked for processes where Fr\"ohlich interaction is responsible.

Topology driven g-factor tuning in type-II quantum dots

We investigate how the voltage control of the exciton lateral dipole moment induces a transition from singly to doubly connected topology in type-II InAs/GaAsxSb1−x quantum dots. The latter causes visible Aharonov-Bohm oscillations and a change of the exciton g factor, which are modulated by the applied bias. The results are explained in the frame of realistic →k⋅→p and effective Hamiltonian models and could open a venue for new spin quantum memories beyond the InAs/GaAs realm.

Microscopic carrier dynamics in quantum wells modulated by high-frequency lateral fields

Abstract We have investigated the dynamics of photogenerated carriers in GaAs quantum wells under the influence of high-frequency fields produced by metal gratings and by surface acoustic waves (SAW's) using spatially and time-resolved photoluminescence (PL). The frequency and phase of the PL oscillations induced by the high-frequency field yield information about the spatial distribution of the carriers and, in the case of SAW's, about the band-gap modulation induced by the SAW strain.

Raman study of strain in GaN/AlN quantum dot multilayered structures

Raman spectroscopy has been used to investigate self-assembled stacks of GaN/AlN quantum dots with increasing number of periods. The E2H phonon modes associated to GaN and AlN are clearly resolved with visible excitation, and their energies allow the simultaneous monitoring of the dot and barrier strain states. The compression of the quantum dots is evidenced by a shift of the E2H phonon mode of circa 29 cm–1 to higher energies with respect to its relaxed value. The strain of the AlN spacer is found to be correlated to that of the dot, with an increase in its tensile component for the samples with fewer periods and a partial relaxation for samples over 50 periods. Additionally, resonant eff…

Tight-binding study of the optical properties of GaN/AlN polar and nonpolar quantum wells

The electronic structure of wurtzite semiconductor superlattices (SLs) and quantum wells (QWs) is calculated by using the empirical tight-binding method. The basis used consists of four orbitals per atom (sp3 model), and the calculations include the spin-orbit coupling as well as the strain and electric polarization effects. We focus our study on GaN/AlN QWs wells grown both in polar (C) and nonpolar (A) directions. The band structure, wave functions and optical absorption spectrum are obtained and compared for both cases.

Resonant hyper-Raman scattering in semiconductors

A theoretical model for resonant hyper-Raman scattering by LO phonons is developed, taking into account excitonic effects. The model is valid for energies below and above an allowed absorption edge. The matrix elements corresponding to the exciton-photon and exciton-phonon interactions are derived analytically, and their contributions to the total scattering efficiency are analyzed. The two main electron-phonon interaction mechanisms present in polar semiconductors, deformation potential, and Fr\"ohlich interaction, are considered. It is shown that the one-phonon resonance hyper-Raman scattering mediated by the deformation potential interaction is dipole forbidden, whereas it is allowed whe…

Semiempirical pseudopotential approach for nitride-based nanostructures and {\it ab initio} based passivation of free surfaces

We present a semiempirical pseudopotential method based on screened atomic pseudopotentials and derived from \textit{ab initio} calculations. This approach is motivated by the demand for pseudopotentials able to address nanostructures, where \textit{ab initio} methods are both too costly and insufficiently accurate at the level of the local-density approximation, while mesoscopic effective-mass approaches are inapplicable due to the small size of the structures along, at least, one dimension. In this work we improve the traditional pseudopotential method by a two-step process: First, we invert a set of self-consistently determined screened {\it ab initio} potentials in wurtzite GaN for a ra…

Excitonic effects in two-LO-phonon resonant Raman scattering.

Abstract The role of electron-hole correlation in resonant Raman scattering by two LO-phonons is analyzed. A theoretical model including excitons belonging to the discrete and continuous spectra as virtual intermediate states in the Raman process and valid for incident energies below and above the gap is developed. For the exciton-phonon coupling the Frohlich Hamiltonian is considered. The most important contribution to the Raman scattering efficiency corresponds to the continuous→discrete→discrete exciton transition, followed by the discrete→discrete→discrete and the continuous→continuous→discrete ones. The model is tested for GaP, where the scattering efficiency data are available around …

Spatiotemporal carrier dynamics in quantum wells under surface acoustic waves

We present a theoretical study of transport and recombination of electrons and holes in quantum wells under the piezoelectric field induced by a surface acoustic wave (SAW). Our model calculations, which include free carriers and excitons in the framework of the drift-diffusion equations, describe the spatial and time dependences of the photoluminescence intensity on excitation density and SAW amplitude, and show overall agreement with recent microphotoluminescence experiments performed on GaAs/(Al,Ga)As quantum wells and quantum wires.

Carrier recombination in InAs/GaAs self-assembled quantum dots under resonant excitation conditions

5 páginas, 4 figuras.-- PACS: 73.21.La;73.63.Kv;78.55.Cr;78.67.Hc;S7.12.-- Trabajo presentado en la 7th International Conference on Optics and Excitons in Confined Systems (OECS7).

Optical transitions and excitonic recombination in InAs/InP self-assembled quantum wires

InAs self-assembled quantum wire structures have been grown on InP substrates and studied by means of photoluminescence and polarized-light absorption measurements. According to our calculations, the observed optical transitions in each sample are consistent with wires of different heights, namely from 6 to 13 monolayers. The nonradiative mechanism limiting the emission intensity at room temperature is related to thermal escape of carriers out of the wires.

Resonant hyper-Raman scattering in semiconductors: Excitonic effects

Abstract A theoretical model of resonant hyper-Raman scattering involving two incident photons of frequency ωL is developed. The model is valid for energies 2ℏωL around the absorption edge of the semiconductor, and takes into account Wannier excitons as intermediate states in the scattering process. Both deformation potential and Frohlich interaction are included in the model: It is found that Frohlich-mediated scattering is a dipole-allowed process, in contrast to one-phonon Raman scattering, where the Frohlich mechanism is dipole-forbidden. We have performed numerical calculations of the resonance profile (hyper-Raman cross-section versus 2ℏωL) and applied our model to materials with dipo…

Surface-acoustic-wave-induced space-charge waves in electron–hole systems

Abstract Space-charge waves in an electron–hole system are studied, which are excited by a moving grating provided by a surface acoustic wave (SAW). The SAW induces a constant current that may change its sign, when a constant electric field is applied opposite to the wave propagation direction. Current resonances are predicted to appear, when the SAW wavelength and frequency match the ones of the space-charge wave.

Anisotropic polarization of non‐polar GaN quantum dot emission

We report on experimental and theoretical studies of the polarization selection rules of the emission of non-polar GaN/AlN self-assembled quantum dots. Time-integrated and time-resolved photoluminescence measurements have been performed to determine the degree of polarization. It is found that the emission of some samples can be predominantly polarized parallel to the wurtzite c axis, in striking difference with the previously reported results for bulk GaN and its heterostructures, in which the emission was preferentially polarized perpendicular to the c axis. Theoretical calculations based on an 8-band k·p model are used to analyze the relative importance of strain, confinement and quantum…

Thermally Tunable Surface Acoustic Wave Cavities

We experimentally demonstrate the dynamical tuning of the acoustic field in a surface acoustic wave (SAW) cavity defined by a periodic arrangement of metal stripes on LiNbO3 substrate. Applying a dc voltage to the ends of the metal grid results in a temperature rise due to resistive heating that changes the frequency response of the device up to 0.3%, which can be used to control the acoustic transmission through the structure. The timescale of the switching is demonstrated to be of about 200 ms. In addition, we have also performed finite-element simulations of the transmission spectrum of a model system, which exhibits a temperature dependence consistent with the experimental data. The adv…

Fermi energy dependence of the optical emission in core/shell InAs nanowire homostructures.

InAs nanowires grown by vapor–liquid–solid (VLS) method are investigated by photoluminescence. We observe that the Fermi energy of all samples is reduced by ~20 meV when the size of the Au nanoparticle used for catalysis is increased from 5 to 20 nm. Additional capping with a thin InP shell enhances the optical emission and does not affect the Fermi energy. The unexpected behavior of the Fermi energy is attributed to the differences in the residual donor (likely carbon) incorporation in the axial (low) and lateral (high incorporation) growth in the VLS and vapor–solid (VS) methods, respectively. The different impurity incorporation rate in these two regions leads to a core/shell InAs homost…

Optical and Vibrational Properties of Self-assembled GaN Quantum Dots

Publisher Summary This chapter describes quantum dots (QDs) based on group III nitrides (III-N). They are expected to be the active medium of new optoelectronic devices operating at high powers and high temperatures. Besides the well-known advantages of their bulk and quantum well (QW) counterparts, III-N QDs provide strong confinement of carriers in nearly perfect zero-dimensional boxes. Quantum effects provide new degrees of freedom for the design of advanced devices. The chapter reviews the systems of dots that appear spontaneously during epitaxial growth without the need of artificial post-processing and designate them as self-assembled or self-organized QDs regardless of the mechanism …

Acoustically driven photon antibunching in nanowires.

The oscillating piezoelectric field of a surface acoustic wave (SAW) is employed to transport photoexcited carriers, as well as to spatially control exciton recombination in GaAs-based nanowires (NWs) on a subns time scale. The experiments are carried out in core-shell NWs transferred to a SAW delay line on a LiNbO(3) crystal. Carriers generated in the NW by a focused laser spot are acoustically transferred to a second location, leading to the remote emission of subns light pulses synchronized with the SAW phase. The dynamics of the carrier transport, investigated using spatially and time-resolved photoluminescence, is well-reproduced by computer simulations. The high-frequency contactless …

Vibrational modes and strain in GaN/AlN quantum dot stacks: dependence on spacer thickness

We have investigated the influence of spacer thickness on the vibrational and strain characteristics of GaN/AlN quantum dot multilayers (QD). The Raman shift corresponding to the E2h vibrational mode related to the QDs has been analyzed for AlN thicknesses ranging from 4.4 nm to 13 nm, while the amount of GaN deposited in each layer remained constant from sample to sample. It is shown that there is a rapid blue shift of the GaN vibrational mode with spacer thickness when its value is smaller than 7 nm while it remains almost constant for thicker spacers. A rapid increase of the Raman line-width in the thicker samples is also observed. The experimental behavior is discussed in comparison wit…

Electronic structure of a quantum ring in a lateral electric field

The electronic states of novel semiconductor quantum rings (QR's) under applied lateral electric fields are theoretically investigated for different values of the ratio ${r}_{2}{/r}_{1},$ where ${r}_{2}$ ${(r}_{1})$ is the outer (inner) radius of the ring. The eigenstates and eigenvalues of the Hamiltonian are obtained from a direct matrix diagonalization scheme. Numerical calculations are performed for a hard-wall confinement potential and the electronic states are obtained as a function of the electric field and the ratio ${r}_{2}{/r}_{1}.$ An anomalous behavior in the energy vs. electric-field fan plot due to the break of symmetry is predicted. Analytical expressions for the energy level…