Showing papers on "Brillouin zone published in 1977"
PARC1
TL;DR: In this paper, the efficiency of two different methods for obtaining "special" points useful for Brillouin-zone integrations of periodic functions is compared, and it is shown that the method suggested by Monkhorst and Pack leads to different and sometimes less efficient point sets than those previously obtained by Chadi and Cohen.
Abstract: The efficiency of two different methods for obtaining "special" points useful for Brillouin-zone integrations of periodic functions is compared. We find that for some Bravais lattices (such as body-centered cubic and hexagonal), the method suggested by Monkhorst and Pack leads to different and sometimes less efficient point sets than those previously obtained by Chadi and Cohen. For a two-dimensional oblique lattice, special points twice as efficient as those suggested by Cunningham are given.
2,617 citations
TL;DR: For all cubic lattices the method of Monkhorst and Pack generates not only point sets identical to those obtained by Chadi and Cohen, but also intermediate sets with the same properties as mentioned in this paper.
Abstract: It is again emphasized that for all cubic lattices the method of Monkhorst and Pack generates not only point sets identical to those obtained by Chadi and Cohen, but also intermediate sets with the same properties. In addition, a comparison of these two methods for the hexagonal lattice reveals our method to be more flexible and efficient.
2,310 citations
TL;DR: In this article, the static ground state energy of a two-dimensional Wigner crystal has been obtained for each of the five 2D Bravais lattices, and the dispersion curves for wave vectors along the symmetry directions in the first Brillouin zone for the hexagonal lattice are calculated.
Abstract: The static ground-state energy of a two-dimensional Wigner crystal has been obtained for each of the five two-dimensional Bravais lattices. At constant electron number density the hexagonal lattice has the lowest energy. Phonon dispersion curves have been calculated for wave vectors along the symmetry directions in the first Brillouin zone for the hexagonal lattice. In the long-wavelength limit one of the two branches of the dispersion relation vanishes with vanishing two-dimensional wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ as $q$, the second as ${q}^{\frac{1}{2}}$. The coefficient of $q$ in the former branch is pure imaginary for certain directions of propagation in the square lattice, implying a dynamical instability of this lattice; the hexagonal lattice is stable. The vibrational zero-point energy and low-temperature thermodynamic functions have been obtained for the hexagonal lattice. The dielectric susceptibility tensor of a two-dimensional Wigner crystal ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ has been determined in the long-wavelength limit, in the presence of a static magnetic field perpendicular to the crystal, and the result has been used to obtain the dispersion relation for plasma oscillations in the electron crystal.
386 citations
TL;DR: In this article, a fully-consistent numerical-basis-set linear-combination-of-atomic-orbitals calculation of the electronic structure of Ti${\mathrm{S}}_{2}$ is reported using the method described previously.
Abstract: A fully-self-consistent numerical-basis-set linear-combination-of-atomic-orbitals calculation of the electronic structure of Ti${\mathrm{S}}_{2}$ is reported using the method described previously. The calculated band structure differs considerably from those previously obtained by non-self-consistent muffin-tin models. Comparison with experiment shows that the calculated optical properties for energies below 16 eV and the various characteristics of the valence and conduction bands agree very well with optical-absorption and electron-energy-loss data as well as with photoemission, x-ray absorption, and appearance-potential spectra. A small indirect gap (0.2-0.3 eV) occurs at the points $M$ and $L$ in the Brillouin zone with a larger direct gap (0.8 eV) at $\ensuremath{\Gamma}$. We suggest that the characteristic semi-metallic large $g$ value observed experimentally originates from a near coincidence of the band gap with the enhanced spin-orbit splitting which is consistent with the soft-x-ray data and our band model. The bonding mechanism in Ti${\mathrm{S}}_{2}$ is discussed in detail; it is shown by a direct calculation of the self-consistent charge density and the transverse effective charge that the system is predominantly covalent with small static ionic character and large dynamic ionicity. In contrast with muffin-tin $X\ensuremath{\alpha}$ models, the bonding is found to be largely due to Ti $4s4p$ to S $3p$ bonds and a much weaker Ti $3d$ to S $3p$ bond. The effects of muffin-tin approximation and self-consistency are discussed in detail. Extrapolation of these results to the case of Ti${\mathrm{Se}}_{2}$ is made and the possible origin of its charge-density wave is discussed.
130 citations
TL;DR: The first experimental observation of resonant Brillouin scattering of excitonic polaritons in a semiconductor was reported in this paper, where the incident light energy was scanned through the $n=1$ exciton resonance and allowed a direct measurement of the polariton dispersion curve.
Abstract: We report the first experimental observation of resonant Brillouin scattering of excitonic polaritons in a semiconductor The usual symmetric Brillouin doublet becomes an asymmetric multiplet when the incident light energy is scanned through the $n=1$ exciton resonance and allows a direct measurement of the polariton dispersion curve In GaAs we determine a longitudinal-transverse splitting of 008 \ifmmode\pm\else\textpm\fi{} 002 meV and a translational mass ${M}_{\mathrm{ex}}=(06\ifmmode\pm\else\textpm\fi{}01){m}_{0}$ for the [100] heavy exciton The excitions couple preferentially to LA phonons
121 citations
TL;DR: In this article, band structures and Fermi surfaces for three quasi-octahedral layer compounds are presented, and the dependence of both properties on the component kz of the wavevector normal to the layers is shown to be important particularly in its effect on the formation of charge density waves.
Abstract: Band structures and Fermi surfaces are presented for three quasi-octahedral layer compounds. The dependence of both properties on the component kz of the wavevector normal to the layers is shown to be important particularly in its effect on the formation of charge density waves. Whereas the nesting of the Fermi surfaces is rather weak in the two-dimensional approximation it becomes very strong if the critical wavevector q is a allowed to have a qz component and equal to one third of the height of the Brillouin zone. Two additional (related) features are discussed namely the three-fold (as opposed to six-fold) symmetry and the small trigonal splitting of the t2g triplet at the centre of the zone.
94 citations
TL;DR: In this paper, the de Haas-van Alphen (dHvA) effect in LaB 6 single crystal has been studied in detail using the field modulation technique and the electron density calculated from the Fermi surface is 1.39×10 22 cm -3 which corresponds to just one electron per unit cell.
Abstract: The de Haas-van Alphen (dHvA) effect in LaB 6 single crystal has been studied in detail using the field modulation technique. Observed dHvA frequencies in the (010) and (1\bar10) planes range from the order of 10 6 to 10 8 gauss. The angular dependence of the dHvA frequencies and the disappearance of the oscillation in some angular regions indicate that the Fermi surface of LaB 6 consists of a multiply connected surface, large pieces of which are centered at the point X (or M) in the simple cubic Brillouin zone. The electron density calculated from the Fermi surface is 1.39×10 22 cm -3 which corresponds to just one electron per unit cell.
90 citations
TL;DR: In this article, the authors present a unified account of the lattice dynamics of unrelaxed (001) faces of seven crystals having the rocksalt structure, three of which have an absolute gap between the acoustical and optical bulk bands (RbF, RbCl, and NaI) and four of which overlap between these bulk bands(NaF, NaCl, LiF, and MgO).
Abstract: We present a comprehensive, unified account of the lattice dynamics of unrelaxed (001) faces of seven crystals having the rocksalt structure, three of which have an absolute gap between the acoustical and optical bulk bands (RbF, RbCl, and NaI) and four of which have overlap between these bulk bands (NaF, NaCl, LiF, and MgO). The results are obtained from calculations of the normal modes of a thin crystal film in which the ionic interactions are described by the rigid-shell model. An outline of the formulation of the shell model for the film is given, and the rapid convergence of surface-phonon frequencies with increasing film thickness is demonstrated. Surface- and pseudosurface-phonon dispersion curves and bulk bands are displayed in the form of the dispersion curves of a 15-layer film over the two-dimensional surface Brillouin zone. The surface-excess phonon density of states ${f}^{s}(\ensuremath{\omega})$ is given, and its peaks and valleys are correlated with features in the film dispersion curves. The domains of existence and the vibrational character of the more prominent surface-phonon bands are discussed. We establish the relation between the surface modes of the lattice models and the macroscopic surface waves of the dielectric and elastic continuum theories (Fuchs-Kliewer and Rayleigh waves, respectively). Comparison with experiments is made; the agreement with low-energy atom scattering from single-crystal surfaces of LiF and with inelastic neutron scattering from microcrystallite samples of MgO is particularly notable, although the latter comparison also displays some significant discrepancies.
75 citations
TL;DR: In this paper, the spin wave dispersion relations along three principal directions were systematically explored across the whole magnetic Brillouin zone at 4 K, and the results were analyzed using the Heisenberg Hamiltonian with long range magnetic interactions.
Abstract: The spin waves in a Heusler alloy Cu 2 MnAl were investigated at several temperatures by neutron inelastic scattering. The spin wave dispersion relations along three principal directions were systematically explored across the whole magnetic Brillouin zone at 4 K. The results were analysed using the Heisenberg Hamiltonian with long range magnetic interactions. At T =493 K ( T / T c =0.78), the renormalization of the spin wave energy at large momentum transfer regions was found to be very small. This result is interpreted in terms of the temperature dependence of the s - d interaction at large distances.
66 citations
TL;DR: In this paper, a tight-binding band structure of TiO2 is calculated by using estimated values for the LCOAO parameters, and the major interband transitions giving rise to the pronounced optical structure around 4 eV occur between energy band regions near the Sigma symmetry line of the Brillouin zone.
Abstract: For pt.I see ibid., vol.10, no.19, p.3893 (1977). The interband optical properties of TiO2 are analysed on the basis of a tight-binding band structure which is calculated by using estimated values for the LCOAO parameters. The major interband transitions giving rise to the pronounced optical structure around 4 eV occur between energy band regions near the Sigma symmetry line of the Brillouin zone. The broad structure of the electroreflectance (ER) is mainly caused by the energy shift of these bands due to the dielectric polarisation. Values for four elements of the polarisation-potential tensor have been deduced from the experimental data which agree with the results of a quantitative model developed for the electro-optic effect. Some features of the ER fine structure are unambiguously identified as Franz-Keldysh oscillations and are designated to specific critical points.
61 citations
TL;DR: In this article, the results of non-vertical interband transitions of two-dimensional band structures were discussed in terms of electron energy loss measurements, and the results were shown to be similar to those in this paper.
Abstract: Wave-vector (q)-dependent energy loss functions, Im [−ϵ⟂(ω, q⟂)]−1, and dielectric functions, ϵ2⟂(ω, q⟂), of hexagonal boron nitride and graphite up to q⟂-values of almost one third of the Brillouin zone and up to energies of 40 eV were derived from electron energy loss measurements. The results are discussed in terms of non-vertical interband transitions of two-dimensional band structures.
Aus Elektron-Energieverlustmessungen wurden die vom Wellenvektor abhangigen Energieverlustfunktionen Im [−ϵ⟂(ω, q⟂)]−1 und dielektrischen Funktionen ϵ2⟂(ω, q⟂) von hexagonalem Bornitrid und Graphit bis hin zu q⟂ -Werten von ungefahr einem Drittel der Brillouinzone und bis hin zu Energieverlusten von 40 eV bestimmt. Die Ergebnisse werden in der Form nichtsenkrechter Interbandubergange fur zweidimensionale Bandstrukturen diskutiert.
TL;DR: In this paper, a narrow-band cw dye laser through the A-exciton transition at helium temperatures was tuned for resonance enhancement, and the authors observed a pronounced asymmetry of the Stokes and anti-Stokes shift and a dramatic increase of the Brillouin shifts as a function of laser frequency.
TL;DR: In this article, the absorption of intense laser light is reduced when targets are irradiated by 1.06-m light with long pulse widths (150-400 ps) and large focal spots (100-250 m).
Abstract: The absorption of intense laser light is found to be reduced when targets are irradiated by 1.06-\ensuremath{\mu} m light with long pulse widths (150-400 ps) and large focal spots (100-250 \ensuremath{\mu}m). Estimates of Brillouin scatter which account for the finite heat capacity of the underdense plasma predict this reduction. Spectra of the back-reflected light show red shifts indicative of Brillouin scattering.
TL;DR: In this article, the generalized Brillouin Theorem multiconfiguration method (GBT-MC) is extended and applied to the calculation of excited states, and quadratic convergence can be guaranteed.
Abstract: The Generalized Brillouin Theorem Multiconfiguration Method (GBT-MC) of Grein and Chang is extended and applied to the calculation of excited states. Orthogonality constraints to lower states as well as second-order interaction effects of states lying close together have been taken into account. In this way quadratic convergence can be guaranteed. Difficulties with coupling coefficients and Lagrangian multipliers of SCF methods can be circumvented. Test calculations have been performed on valence electron excited states of C, H2O, and CH2O, and on core excited states of Li.
TL;DR: In this article, the authors studied the Brillouin zone edge phonons of mixed CdS 1−x Se x and showed that these phonons have a two-mode behavior.
TL;DR: In this article, angle-resolved photoemission (ARP) spectra from polycrystalline copper single crystals were used to demonstrate the importance of the Debye-Waller factor in photo-emission, leading to a breakdown of the direct transition model.
Abstract: With the advent of synchrotron radiation in the 32-280 eV range at the Stanford Synchrotron Radiation Project, it has become possible to elucidate the transition from ultraviolet to X-ray-induced photoemission. This has been accomplished by studies of noble metals. Polycrystalline copper shows a valence-band (VB) profile that approaches the X-ray induced shape at hv ~ 100 eV. In polycrystalline silver, the 4d cross section follows the atomic curve, with a reversal of VB peak intensities near hv = 110 eV. Strongly anisotropic behavior is observed in copper single crystals, using angle-resolved photoemission (ARP). Normal ARP spectra from Cu(100), (110), and (111) crystals follow the band dispersion through the Brillouin Zone, including a dramatic resonance between the Fermi level (EF) and 2 eV binding energy for hv = 43-52 eV. High temperature and high photon energy studies demonstrate the importance of the Debye-Waller factor in photoemission leading to a breakdown of the direct transition model. In adsorption studies of CO on Ni and Pt, CO is shown to stand up with oxygen out. For Pt, electrons are found to flow from t2g orbitals near EF to CO, and the CO 1π and 5σ binding energies are reversed relative to the gas phase. At higher photon energies, hv = 150 eV, the CO orbitals are very prominent on a Pt substrate. An inversion of the angular distribution of these orbitals and energy-dependent resonances in their intensities provide evidence for final state scattering effects at photon energies above 40 eV.
TL;DR: The band structure of CdI2 has been calculated using a modified semi-empirical tight-binding method and the results obtained have been compared with both angularly averaged and angularly resolved photoemission spectra as mentioned in this paper.
Abstract: The band structure of CdI2 has been calculated using a modified semi-empirical tight-binding method and the results obtained have been compared with both angularly averaged and angularly resolved photoemission spectra. The theoretically computed density of states distribution is in excellent agreement with angularly averaged results and all the main features observed experimentally are reproduced in the theory. Angularly resolved spectra have been used to draw up energy band dispersion curves directly and agreement with calculated bands in both the Gamma M and Gamma K directions of the Brillouin zone is good.
TL;DR: In this paper, a nonlocal self-consistent pseudopotential scheme is employed to calculate the electronic structure of the ideal (001) surface of Nb and the two-dimensional band structure for the surface is presented and discussed.
Abstract: A nonlocal self-consistent pseudopotential scheme is employed to calculate the electronic structure of the ideal (001) surface of Nb. Charge densities, electronic local density of states, and the two-dimensional band structure for the surface are presented and discussed. Prominent surface bands and surface resonances are identified and analyzed throughout the two-dimensional Brillouin zone. The projected Nb bulk band structure on the (001) surface is also obtained. Surface states of different angular momentum character are found to exist over a wide range of energies and over different portions of the two-dimensional Brillouin zone. Our calculations predict strong surface features in the density of states in the range 0-2 eV above the Fermi energy.
TL;DR: In this paper, a variety of specular intensity patterns associated with bound state resonances near the Brillouin zone boundaries are exhibited, and the (10) and (11) resonances give rise to two split specular minima with the splitting depending essentially on v 10 for a fixed beam energy; however, the detailed features are dependent on other periodic components.
TL;DR: In this paper, a very detailed experimental band structure of the layered compound GaSe has been obtained in all the symmetry directions of the Brillouin zone for photon energy varying between 21 and 45 eV.
TL;DR: These calculations are valid for well aligned etalons when the phonon lineshape is Lorentzian and cover a wide range of instrumental and phonon spectral widths.
Abstract: The sharper instrumental profile of the multipassed Fabry-Perot etalon introduces distortions to Brillouin spectra which may differ considerably from those introduced by the single-pass etalon. The half-width at half-height of the observed profile is calculated for five-pass and three-pass spectrometers. Phonon spectral widths may be extracted from the graphs and appropriate relationships are presented. These calculations are valid for well aligned etalons when the phonon lineshape is Lorentzian and cover a wide range of instrumental and phonon spectral widths.
TL;DR: In this article, the De Haas-van Alphen effect frequencies for a single crystal of Gd of resistance ratio 500 were reported for the Fermi surfaces obtained by exchange splitting the nonmagnetic RAPW bandstructure.
Abstract: De Haas-van Alphen effect frequencies are reported for a single crystal of Gd of resistance ratio 500. Ten frequencies were found for H within 60 degrees for the c axis and four frequencies for H in or near the basal plane. The six highest frequencies, in the range 1.5-8*107 G, were identified with orbits on the Fermi surfaces obtained by exchange splitting the nonmagnetic RAPW bandstructure. There appears to be three surfaces; 3rd and 4th zone majority surfaces and a 3rd zone minority surface. Orbits were identified on all three surfaces, but to fit the results it was necessary to allow the exchange splitting to vary between 0.060 Ryd and 0.070 Ryd over the Brillouin zone. The RAPW bandstructure predicts very few small orbits and could not account for the eight lowest frequencies, in the range 1.6*107 to 2*105 G.
TL;DR: In this paper, the Raman frequencies, the infrared absorption and reflexion spectra are given and interpreted on the basis of a mechanical and a rigid-ion lattice-dynamical model.
Abstract: The 'pure' ferroelastic behaviour of P-rich Pb3(PxV1 - xO4)2 is of the improper type. The ferroelastic phase transition is antiferrodistortive and accompanied by a second one at a lower temperature. The temperature range of the intermediate phase is approximately 20°C. At the transition points singularities of phonon spectra, dielectric and optical behaviour, and lattice constants are found. The Raman frequencies, the infrared absorption and reflexion spectra are given and interpreted on the basis of a mechanical and a rigid-ion lattice-dynamical model. From the results a critical mode is predicted at the boundary of the Brillouin zone (L in R{\bar 3}m). Further, the Γ phonon becomes weak at the lower transition point. The normal coordinates of both modes give parallel and antiparallel deviations of Pb positions from the [111] axis of the rhombohedral R{\bar 3}m high-temperature phase.
TL;DR: In this paper, the effect of the enhanced acoustic mode softening at T c on the wave velocity and Young's modulus surfaces is discussed, as well as the effects of the acoustic phonon branch with wave vector (110), particle displacement (110) near the Brillouin zone center.
Abstract: Indium rich, In-Cd alloys undergo a f.c.c. to f.c.t. structural phase transition which the presence of a cubic invariant in the Hamiltonian renders first order. Ultrasonic wave velocity measurements have been made in single crystals of both the f.c.c. and f.c.t. phases, emphasis being placed on behaviour near the phase transition. The shear modulus ½(C 11-C 12) tends to very small values in both phases as the transition temperature T c is approached: the onset of instability is manifested as softening of the acoustic phonon branch with wave vector (110), particle displacement (110) near the Brillouin zone centre. However, in contrast to its behaviour in the In-Tl alloys, ½(C 11-C 12) does not go to zero in the In-Cd alloys. This finding corresponds to the more recognizable first-order character of the transition in the In-Cd compared to that in In-Tl alloys. The effect of the enhanced acoustic mode softening at T c on the wave velocity and Young's modulus surfaces is discussed. In addition to the...
TL;DR: In this article, the phonon frequencies have been determined as a function of temperature and the integrated intensities of the shear modes in KCN and NaCN were also measured as the temperature increases.
Abstract: Brillouin spectra of acoustic phonons in the orientationally disordered phase in KCN and NaCN have been studied. The phonon frequencies have been determined as a function of temperature. The frequency of the transverse acoustic phonon propagating along (110) and polarized along (001) corresponds well with mean field theory for the phase transition for the two cyanide crystals. The elastic constants C11, C12,, and C44 have been determined as a function of temperature in the cubic disordered phase. The Cauchy relation for cubic crystals is found to be invalid in KCN and NaCN. The two types of phase transitions in KCN and NaCN are discussed in terms of the vanishing shear modulii C44 and (C11−C12)/2, each of which induces lattice instability when it becomes small. The integrated intensities of the shear modes in KCN and NaCN were also measured as a function of temperature. The intensity enhancement upon approaching the phase transition temperature has been found to be due to the decrease of C44. The relative...
TL;DR: In this article, the phonon energies at the edge of the Brillouin zone (M point) are calculated by means of the force constants model including the atoms of a single layer.
Abstract: Reflection and absorption of light by InSe single crystals are investigated in the range of wave numbers 500 to 100 cm-1. The measurements are carried out at room temperature in E ⟂ C orientation of the electric vector. The linear chain model is used for calculating the phonon frequencies at k = 0. The phonon energies at the edge of the Brillouin zone (M point) are calculated by means of the force constants model including the atoms of a single layer.
TL;DR: In this paper, Raman scattering and luminescence have been observed in coal and the similarity of the results to those of disordered graphite is striking, and the disorder-induced line in graphite was shown to be due to the high phonon density of states at 0 and K points of the Brillouin zone for graphite.
TL;DR: The Raman spectra of an oriented single crystal of β-LiNH4SO4 show that the first order phase transition involves a doubling of the primitive unit cell and is characterized by an unstable pair of degenerate phonons at the Brillouin zone boundary as mentioned in this paper.
TL;DR: In this article, the successive phase transitions in KCdF 3 have been studied by using X-ray diffraction and differential type thermal analysis; three transitions are observed at 485, 471 and 243 K; these are of 1st-, 2nd-, and 2nd-order respectively.