DOCSLIB.ORG

The Plasmon Density of States of a Layered Electron Gas H

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Plasmon Density of States of a Layered Electron Gas H Z. Phys. B 90, 277-281 (1993) Condensed Zeitsehrift far Physik B Matter Springer-Verlag 1993 The plasmon density of states of a layered electron gas H. Morawitz ~, I. Bozovic 2, V.Z. Kresin 3, G. Rietveld 4, D. van der Marel 4 IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120-6099, USA 2 Varian Research Center, 611 Hansen Way, Palo Alto, CA 94303, USA 3 Materials and Chemical Physics Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA 4 Delft University of Technology, Faculty of Applied Physics, Lorentzweg 1, 2628 CJ Delft, The Netherlands Received: 24 September 1992 Abstract. The plasmon density of states (DOS)p(co) of a mon side-band measured with such a local probe. The layered electron gas (LEG) is studied theoretically. It is plasmon-pole approximation replaces the response func- shown that p (co) is a linear function of frequency ~o as co tion with the density of states of the layer plasmons, goes to 0, and then increases more rapidly as co tends to which is defined by the screened, three dimensional plasma frequency cop. We also study the partial densities of state for constant K and p(~o)=(2n) -3 ~ O(~-Op(tc, qz)). (1) q~, where K and qz are the magnitude of the transverse K~ qz and perpendicular momentum transfers. A possible ex- The nature of the collective excitations in layered systems perimental probe for the plasmon DOS is discussed. has been studied extensively [5, 6] theoretically, because of continuing interest in superlattices and naturally layered materials such as graphite and the transition metal I. Introduction chalcogenides (TaS2, TeS2, etc.). The outline of this paper is 'as follows: In Sect. 2 we The new superconducting cuprates [1-4] with high su- rederive the layer plasmon dispersion relations [6] using perconducting transition temperatures T~ are all charac- the random phase approximation (RPA). In Sect. 3 we terized by a layered structure of conducting (CuO2) calculate the partial density of states, keeping either of planes. We propose to use the layered electron gas (LEG) the variables K (the absolute value of the LEG plasmon model [5] to describe their collective charge excitation wavevector parallel to the conducting planes) or qz (the (layer plasmons) and to calculate their density of states LEG plasmon wavevector perpendicular to the planes) (DOS) as a function of frequency in this paper. The struc- constant. In Sect. 4 we derive a simple form of the total ture of the plasmon spectrum in layered, conducting sys- plasmon DOS, based on an approximate form of the LEG tems differs in a striking way from those of the isotropic dispersion relations. In Sect. 5 we calculate both ana- (3 D) conductors. In the latter case, the plasmon branch lytically and numerically the exact layer plasmon density is described by a dispersion relation co (q)=oJ 0 +aq 2, of states and discuss the various frequency regimes whereas a layered conductor is characterized by a highly ~o/cop< l, = 1 and > 1. Finally - in Sect. 6 - discuss the anisotropic plasmon band D 0c, qz) without a gap at tc = 0 implications of these results for the shake-up spectra of except for the single branch for which qz = 0 (to is a two- core levels in a layered compound. dimensional wave-vector in the plane of the sheets and qz has been chosen perpendicular to the layers). That 2. Plasmon dispersion relations produces noticeable changes in the spectrum of collective We assume that the carriers are localized in N(N~) modes of these materials. In experimental probes, such independent layers, spaced a distance D apart. The dis- as absorption of light, the dielectric response is subject persion relation for two-dimensional carriers is assumed to strong momentum selection rules. As a result only the to be given by the effective mass approximation: ek = long-wavelength dielectric response is probed in such ex- k 2/2 m*. Electrons on different sheets are coupled by the periments. In local experimental probes, such as the sud- Coulomb interaction. We can write an electronic Ham- den creation of a core hole by X-ray photo-electron spec- iltonian H el. troscopy, one can, in principle, couple to the full spectrum of plasmons. This shows up as a side-band of the core- H ~1-= ~, eKIA~,+ A,~l hole excitation spectrum due to plasmon shake-up. It is Ir conceivable that the plasmon-pole approximation can be + Z Z V~~ + + ,,"z A,,+ " - ,," z" A,,I" A,~,z used for the calculation of the spectral shape of the plas- K,,,,,,~" l,t" (2) 278 where t~, te' and K" are two-dimensional momentum var- the background dielectric constant of the medium due to iables, and l, l' label different layers, At,. l, A k,z+ are carrier all other high frequency excitations. (A typical value is annihilation and creation operators and the interaction v~ 0.05). With these definitions, and solving for the roots matrix element V~ (l, l') is the two-dimensional Fourier of e (te, qz; co) = 0, we obtain with (8) transform of the Coulomb interaction between electrons on layers l and l'. For an infinite system V(I,l')= F(y,z) [ cop ] Vq-P). 0=1 v~y [COZp_(VFy)2]I/2 1 . (10) 2 7re 2 V~~ ') =-- e -~Dll-r I (3) From this equation we solve the layer plasmon dispersion ~Mte relation F2(Y'Z) (11) If we make a Fourier transformation of the variable c~ v~ Y2-~ v~+ 2F(y,z)/y" r = l- l' we obtain Alternatively we can express this as a dispersion relation q~0(te;q~)= ~. V(O)(r)exp(iq~Dr) for O as a function of te and q~. When written in this r = --ao form the dispersion relation reads 2 7~e2 --- F(teD, q~D), (4) Op (n, qz) CM te 1/2 where we use the dimensionless function F (the 'layer 2 1 .~-VFte 1+ [ 2F(xD'q~D)] form factor'), introduced by Fetter [5b] in the hydro- 4F(xD, q~D) dynamic treatment of the layer plasmon dispersion rB x A r B K sinh teD F (te D, q~D ) - cosh teD - cos q~D " (5) For later use it is convenient to express F as a function of COp and y using (10), and also to express z in terms of The study of the screening properties and collective modes F and y, by inverting (5). The resulting expressions are requires the knowledge of the fully screened interaction found to be V(1, l'). Within the random phase approximation, the v~yV~2p-(vFy) 2 latter obeys the equation F(mp,y) - co, - Vco - (v y)2 " V(I,I')= V~ ") (12) sinh ( y ) ~'~. +H~176 Z V~ V(l",l'), (6) z(Ogp, y)=arccos (cosh (y) ~/ l" where the single layer polarizability H ~ is given by It should be noted here that (11) goes beyond the earlier RPA results as it used the exact form of the single sheet 2 f(8~+q)--f(ek) (7) polarizability )~0 (9) and not its expansion valid for H~ =A - ~ gk+q--ek -ico co ~ VFX only. On the other hand, we have not included the effects of the non-local exchange and correlation part Here A is the area of one layer and f (e,~) is the Fermi of the electron-electron interactions. Some exploration of function. The layer plasmons are defined as the zeroes of the inclusion of such effects by a local field correction the layer dielectric function for the in-plane electron-electron interaction has been recently made in [8]. We note that the general conse- e (te, q~; co) = 1 - ~0 (te, q~)Re {H(~ (8) quence of such treatments as the GRPA (generalized ran- dom phase approximation) is to reduce the strength of In the frequency regime co > vete, the real part of the the electron-electron interactions. polarizability Z0 of a single sheet is [5, 7] Re{H~ (te; co)} =)-~o,[ [CO~_(Vvte)2],/2--1.] (9) 3. Plasmon partial density of states Let us first address the question of the LEG plasmon In order to keep the expressions simple we introduce frequency dependence for co ~ 1. In this region (see Fig. 1) dimensionless variables for the in-plane and out-of-plane y~ 1 holds also, and the hyperbolic functions may be wave-vector components te and qz and the LEG plas- expanded in a Taylor series for small y. The LEG plasmon mon frequency co_, by defining y-=teD, z=-qzD, dispersion relations then reduce to the much simpler form cop (y, z) ~ Op (y, z)f0 o with ~o2 = 2 vZ/(DrB) and VF = (y, z) = 2 OF/(D(20) (rB/2D) 1/2. Here h2e~ is the effec- y2 rB = ~177"e (13) tive Bohr radius, m* the effective carrier mass and c g is [ + 1 -- cos z] [2 + VFI.gy2 .'1 2 + 1 -cosz)] 279 1.5 i r i // / / 1.0 t.O i ',/,t ~- I //Z p(qD,w) 0.5 0 i I i I r I i 0 1 2 3 KD 0.30 I i o 0.5 1 .o y=KD -- 0.2 0.25 --.-- 0.4 --- 0.6 ...... 0.8 0.20 .... 1,0 1.2 8 0.15 l o.10 - I rl i ! dlnp 0.05 i, !!, l\iAjl dlnw 4 i .4.~_//~ ! ~ 0 0 0.5 1.0 c a,,/w 0 Fig. la-d. a Layer plasmon dispersion for v 2= 0.05 and (from top to bottom) z/~ =0, 0.2, 0.4, 0.6, 0.8, 1.
Load more

Recommended publications
  • Lecture 3: Fermi-Liquid Theory 1 General Considerations Concerning Condensed Matter
    Phys 769 Selected Topics in Condensed Matter Physics Summer 2010 Lecture 3: Fermi-liquid theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 General considerations concerning condensed matter (NB: Ultracold atomic gasses need separate discussion) Assume for simplicity a single atomic species. Then we have a collection of N (typically 1023) nuclei (denoted α,β,...) and (usually) ZN electrons (denoted i,j,...) interacting ∼ via a Hamiltonian Hˆ . To a first approximation, Hˆ is the nonrelativistic limit of the full Dirac Hamiltonian, namely1 ~2 ~2 1 e2 1 Hˆ = 2 2 + NR −2m ∇i − 2M ∇α 2 4πǫ r r α 0 i j Xi X Xij | − | 1 (Ze)2 1 1 Ze2 1 + . (1) 2 4πǫ0 Rα Rβ − 2 4πǫ0 ri Rα Xαβ | − | Xiα | − | For an isolated atom, the relevant energy scale is the Rydberg (R) – Z2R. In addition, there are some relativistic effects which may need to be considered. Most important is the spin-orbit interaction: µ Hˆ = B σ (v V (r )) (2) SO − c2 i · i × ∇ i Xi (µB is the Bohr magneton, vi is the velocity, and V (ri) is the electrostatic potential at 2 3 2 ri as obtained from HˆNR). In an isolated atom this term is o(α R) for H and o(Z α R) for a heavy atom (inner-shell electrons) (produces fine structure). The (electron-electron) magnetic dipole interaction is of the same order as HˆSO. The (electron-nucleus) hyperfine interaction is down relative to Hˆ by a factor µ /µ 10−3, and the nuclear dipole-dipole SO n B ∼ interaction by a factor (µ /µ )2 10−6.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties Lattice Vibrations
    Solid State Physics Lecture 5 Last week: Phys 446: (Ch. 3) • Phonons Solid State Physics / Optical Properties • Today: Einstein and Debye models for thermal capacity Lattice vibrations: Thermal conductivity Thermal, acoustic, and optical properties HW2 discussion Fall 2007 Lecture 5 Andrei Sirenko, NJIT 1 2 Material to be included in the test •Factors affecting the diffraction amplitude: Oct. 12th 2007 Atomic scattering factor (form factor): f = n(r)ei∆k⋅rl d 3r reflects distribution of electronic cloud. a ∫ r • Crystalline structures. 0 sin()∆k ⋅r In case of spherical distribution f = 4πr 2n(r) dr 7 crystal systems and 14 Bravais lattices a ∫ n 0 ∆k ⋅r • Crystallographic directions dhkl = 2 2 2 1 2 ⎛ h k l ⎞ 2πi(hu j +kv j +lw j ) and Miller indices ⎜ + + ⎟ •Structure factor F = f e ⎜ a2 b2 c2 ⎟ ∑ aj ⎝ ⎠ j • Definition of reciprocal lattice vectors: •Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law): σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • What is Brillouin zone kl kl 2 2 C •Elastic wave equation: ∂ u C ∂ u eff • Bragg formula: 2d·sinθ = mλ ; ∆k = G = eff x sound velocity v = ∂t 2 ρ ∂x2 ρ 3 4 • Lattice vibrations: acoustic and optical branches Summary of the Last Lecture In three-dimensional lattice with s atoms per unit cell there are Elastic properties – crystal is considered as continuous anisotropic 3s phonon branches: 3 acoustic, 3s - 3 optical medium • Phonon - the quantum of lattice vibration. Elastic stiffness and compliance tensors relate the strain and the Energy ħω; momentum ħq stress in a linear region (small displacements, harmonic potential) • Concept of the phonon density of states Hooke's law: σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • Einstein and Debye models for lattice heat capacity.
    [Show full text]
  • Chapter 3 Bose-Einstein Condensation of an Ideal
    Chapter 3 Bose-Einstein Condensation of An Ideal Gas An ideal gas consisting of non-interacting Bose particles is a ¯ctitious system since every realistic Bose gas shows some level of particle-particle interaction. Nevertheless, such a mathematical model provides the simplest example for the realization of Bose-Einstein condensation. This simple model, ¯rst studied by A. Einstein [1], correctly describes important basic properties of actual non-ideal (interacting) Bose gas. In particular, such basic concepts as BEC critical temperature Tc (or critical particle density nc), condensate fraction N0=N and the dimensionality issue will be obtained. 3.1 The ideal Bose gas in the canonical and grand canonical ensemble Suppose an ideal gas of non-interacting particles with ¯xed particle number N is trapped in a box with a volume V and at equilibrium temperature T . We assume a particle system somehow establishes an equilibrium temperature in spite of the absence of interaction. Such a system can be characterized by the thermodynamic partition function of canonical ensemble X Z = e¡¯ER ; (3.1) R where R stands for a macroscopic state of the gas and is uniquely speci¯ed by the occupa- tion number ni of each single particle state i: fn0; n1; ¢ ¢ ¢ ¢ ¢ ¢g. ¯ = 1=kBT is a temperature parameter. Then, the total energy of a macroscopic state R is given by only the kinetic energy: X ER = "ini; (3.2) i where "i is the eigen-energy of the single particle state i and the occupation number ni satis¯es the normalization condition X N = ni: (3.3) i 1 The probability
    [Show full text]
  • High Pressure Band Structure, Density of States, Structural Phase Transition and Metallization in Cds
    Chemical and Materials Engineering 5(1): 8-13, 2017 http://www.hrpub.org DOI: 10.13189/cme.2017.050102 High Pressure Band Structure, Density of States, Structural Phase Transition and Metallization in CdS J. Jesse Pius1, A. Lekshmi2, C. Nirmala Louis2,* 1Rohini College of Engineering, Nagercoil, Kanyakumari District, India 2Research Center in Physics, Holy Cross College, India Copyright©2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The electronic band structure, density of high pressure studies due to the development of different states, metallization and structural phase transition of cubic designs of diamond anvil cell (DAC). With a modern DAC, zinc blende type cadmium sulphide (CdS) is investigated it is possible to reach pressures of 2 Mbar (200 GPa) using the full potential linear muffin-tin orbital (FP-LMTO) routinely and pressures of 5 Mbar (500 GPa) or higher is method. The ground state properties and band gap values achievable [3]. At such pressures, materials are reduced to are compared with the experimental results. The fractions of their original volumes. With this reduction in equilibrium lattice constant, bulk modulus and its pressure inter atomic distances; significant changes in bonding and derivative and the phase transition pressure at which the structure as well as other properties take place. The increase compounds undergo structural phase transition from ZnS to of pressure means the significant decrease in volume, which NaCl are predicted from the total energy calculations. The results in the change of electronic states and crystal structure.
    [Show full text]
  • Lecture 24. Degenerate Fermi Gas (Ch
    Lecture 24. Degenerate Fermi Gas (Ch. 7) We will consider the gas of fermions in the degenerate regime, where the density n exceeds by far the quantum density nQ, or, in terms of energies, where the Fermi energy exceeds by far the temperature. We have seen that for such a gas μ is positive, and we’ll confine our attention to the limit in which μ is close to its T=0 value, the Fermi energy EF. ~ kBT μ/EF 1 1 kBT/EF occupancy T=0 (with respect to E ) F The most important degenerate Fermi gas is 1 the electron gas in metals and in white dwarf nε()(),, T= f ε T = stars. Another case is the neutron star, whose ε⎛ − μ⎞ exp⎜ ⎟ +1 density is so high that the neutron gas is ⎝kB T⎠ degenerate. Degenerate Fermi Gas in Metals empty states ε We consider the mobile electrons in the conduction EF conduction band which can participate in the charge transport. The band energy is measured from the bottom of the conduction 0 band. When the metal atoms are brought together, valence their outer electrons break away and can move freely band through the solid. In good metals with the concentration ~ 1 electron/ion, the density of electrons in the electron states electron states conduction band n ~ 1 electron per (0.2 nm)3 ~ 1029 in an isolated in metal electrons/m3 . atom The electrons are prevented from escaping from the metal by the net Coulomb attraction to the positive ions; the energy required for an electron to escape (the work function) is typically a few eV.
    [Show full text]
  • Lecture Notes for Quantum Matter
    Lecture Notes for Quantum Matter MMathPhys c Professor Steven H. Simon Oxford University July 24, 2019 Contents 1 What we will study 1 1.1 Bose Superfluids (BECs, Superfluid He, Superconductors) . .1 1.2 Theory of Fermi Liquids . .2 1.3 BCS theory of superconductivity . .2 1.4 Special topics . .2 2 Introduction to Superfluids 3 2.1 Some History and Basics of Superfluid Phenomena . .3 2.2 Landau and the Two Fluid Model . .6 2.2.1 More History and a bit of Physics . .6 2.2.2 Landau's Two Fluid Model . .7 2.2.3 More Physical Effects and Their Two Fluid Pictures . .9 2.2.4 Second Sound . 12 2.2.5 Big Questions Remaining . 13 2.3 Curl Free Constraint: Introducing the Superfluid Order Parameter . 14 2.3.1 Vorticity Quantization . 15 2.4 Landau Criterion for Superflow . 17 2.5 Superfluid Density . 20 2.5.1 The Andronikoshvili Experiment . 20 2.5.2 Landau's Calculation of Superfluid Density . 22 3 Charged Superfluid ≈ Superconductor 25 3.1 London Theory . 25 3.1.1 Meissner-Ochsenfeld Effect . 27 3 3.1.2 Quantum Input and Superfluid Order Parameter . 29 3.1.3 Superconducting Vortices . 30 3.1.4 Type I and Type II superconductors . 32 3.1.5 How big is Hc ............................... 33 4 Microscopic Theory of Bosons 37 4.1 Mathematical Preliminaries . 37 4.1.1 Second quantization . 37 4.1.2 Coherent States . 38 4.1.3 Multiple orbitals . 40 4.2 BECs and the Gross-Pitaevskii Equation . 41 4.2.1 Noninteracting BECs as Coherent States .
    [Show full text]
  • Chapter 13 Ideal Fermi
    Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz.
    [Show full text]
  • Density of States
    Density of states A Material is known to have a high density of states at the Fermi energy. (a) What does this tell you about the electrical, thermal and optical properties of this material? (b) Which of the following quasiparticles would you expect to observe in this material? (phonons, bipolarons, excitons, polaritons, suface plasmons) Why? (a): A high density of states at the fermi energy means that this material is a good electrical conductor. The specific heat can be calculated via the internal energy. Z 1 Z 1 E · D(E) u(E; T ) = E · D(E) · f(E)dE = dE (1) −∞ −∞ 1 + exp E−µ kB ·T du We know that cv = dT . So we get the following expression: E−µ 1 Z E · D(E) · (E − µ) · exp k T c = B dE (2) v 2 −∞ 2 E−µ kBT 1 + exp kbT Hence we deal with a metal, we will have a good thermal conductor because of the phonon and electron contribution. Light will get reflected out below the plasma frequency !P . (b): Due to fact that phonons describe lattice vibrations, it is possible to observe them in this material. They can be measured with an EELS experiment or with Raman Spectroscopy. A polaron describes a local polarisation of a crystal due to moving electrons. They are observable in materials with a low electron density and describe a charge-phonon coupling. There are two different kinds of polarons, the Fr¨ohlich Polaron and the Holstein Polaron. The Fr¨ohlich polarons describe large polarons, that means the distortion is much larger than the lattice constant of the material so that a lot of atoms are involved.
    [Show full text]
  • Density of States Explanation
    www.Vidyarthiplus.com Engineering Physics-II Conducting materials- - Density of energy states and carrier concentration Learning Objectives On completion of this topic you will be able to understand: 1. Density if energy states and carrier concentration Density of states In statistical and condensed matter physics , the density of states (DOS) of a system describes the number of states at each energy level that are available to be occupied. A high DOS at a specific energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level. Explanation Waves, or wave-like particles, can only exist within quantum mechanical (QM) systems if the properties of the system allow the wave to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of www.Vidyarthiplus.com Material prepared by: Physics faculty Topic No: 5 Page 1 of 6 www.Vidyarthiplus.com Engineering Physics-II Conducting materials- - Density of energy states and carrier concentration certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels. For example, the density of states for electrons in a semiconductor is shown in red in Fig.
    [Show full text]
  • Optical Properties and Electronic Density of States* 1 ( Manuel Cardona2 > Brown University, Providence, Rhode Island and DESY, Homburg, Germany
    t JOU RN AL O F RESEAR CH of th e National Bureau of Sta ndards -A. Ph ysics and Chemistry j Vol. 74 A, No.2, March- April 1970 7 Optical Properties and Electronic Density of States* 1 ( Manuel Cardona2 > Brown University, Providence, Rhode Island and DESY, Homburg, Germany (October 10, 1969) T he fundame nta l absorption spectrum of a so lid yie lds information about critical points in the opti· cal density of states. This informati on can be used to adjust parameters of the band structure. Once the adju sted band structure is known , the optical prope rties and the de nsity of states can be generated by nume ri cal integration. We revi ew in this paper the para metrization techniques used for obtaining band structures suitable for de nsity of s tates calculations. The calculated optical constants are compared with experim enta l results. The e ne rgy derivative of these opti cal constants is di scussed in connection with results of modulated re fl ecta nce measure ments. It is also shown that information about de ns ity of e mpty s tates can be obtained fro m optical experime nt s in vo lving excit ati on from dee p core le ve ls to the conduction band. A deta il ed comparison of the calc ul ate d one·e lectron opti cal line s hapes with experime nt reveals deviations whi ch can be interpre ted as exciton effects. The acc umulating ex pe rime nt al evide nce poin t· in g in this direction is reviewed togethe r with the ex isting theo ry of these effects.
    [Show full text]
  • The Single Particle Density of States
    Physics 112 Single Particle Density of States Peter Young (Dated: January 26, 2012) In class, we went through the problem of counting states (of a single particle) in a box. We noted that a state is specified by a wavevector ~k, and the fact that the particle is confined within a box of size V = L × L × L implies that the allowed values of ~k form a regular grid of points in k-space, We showed that the number of values of ~k whose magnitude lie between k and k + dk is ρ˜(k) dk where V 2 ρ˜(k) dk = 4πk dk. (1) (2π)3 This result is general, and does not depend on the dispersion relation, i.e. the relation between the 2 energy, ǫ, and the wavevector k (for example ǫ = ¯hck for photons and ǫ = ¯h k2/2m for electrons). However, since Boltzmann factors in statistical mechanics depend on energy rather than wavevector we need to convert Eq. (1) to give the density of states as a function of energy rather than k, i.e. we need the number of states per unit interval of energy, ρ(ǫ). This does require a knowledge of the dispersion relation. We now consider several cases. 1. Photons Consider first electromagnetic radiation in a cavity (black body radiation) for which the quanta are called photons. Here, the dispersion relation is linear: ǫ = ¯hω = ¯hck , (2) where c is the speed of light. Changing variables in Eq. (1) from k to ǫ gives the number of k-values whose energy is between ǫ and ǫ + dǫ to be V 4π 2 ǫ dǫ.
    [Show full text]
  • Density of States in the Bilayer Graphene with the Excitonic Pairing Interaction
    Eur. Phys. J. B (2017) 90: 130 DOI: 10.1140/epjb/e2017-80130-8 THE EUROPEAN PHYSICAL JOURNAL B Regular Article Density of states in the bilayer graphene with the excitonic pairing interaction Vardan Apinyana and Tadeusz K. Kope´c Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wroclaw 2, Poland Received 3 March 2017 / Received in final form 16 May 2017 Published online 5 July 2017 c The Author(s) 2017. This article is published with open access at Springerlink.com Abstract. In the present paper, we consider the excitonic effects on the single particle normal density of states (DOS) in the bilayer graphene (BLG). The local interlayer Coulomb interaction is considered between the particles on the non-equivalent sublattice sites in different layers of the BLG. We show the presence of the excitonic shift of the neutrality point, even for the noninteracting layers. Furthermore, for the interacting layers, a very large asymmetry in the DOS structure is shown between the particle and hole channels. At the large values of the interlayer hopping amplitude, a large number of DOS at the Dirac’s point indicates the existence of the strong excitonic coherence effects between the layers in the BLG and the enhancement of the excitonic condensation. We have found different competing orders in the interacting BLG. Particularly, a phase transition from the hybridized excitonic insulator phase to the coherent condensate state is shown at the small values of the local interlayer Coulomb interaction. 1 Introduction ences [10–12]. A very large excitonic gap, found in a suffi- ciently broad interval of the repulsive interlayer Coulomb The problem of excitonic pair formation in the graphene interaction parameter, given in reference [12], suggests and bilayer graphene (BLG) systems is one of the long- that the level of intralayer density and chemical poten- standing and controversial problems in the modern solid tial fluctuations in the BLG system, discussed in refer- state physics [1–12].
    [Show full text]