They consider a onedimensional lattice, described by a periodic potential of rectangular barrier see fig. Kronig penney model supports the existence of these bands of energies allowed bands and forbidden bands. The energy spectrum of imperfect kronigpenney model. Real imperfect crystals are even more interesting, but thats a discussion that can lead to a career. This model is called kronig penney model of potentials. The sum of s is over all atoms in a unit length, which means over 1a atoms. Blochs theorem, kronig penney model exercise 2 kronig penney model one of the simplest models of a periodic potential where the band structure can be computed analytically is the kronig penney model in one dimension. The development requires no quantum mechanics beyond that taught in most physical chemistry courses and can be taught in a single onehour lecture. The resultant spectrum of solutions will then have gaps in the admissible energies. Under such circumstances, surface states can appear and the aun of this letter is to demonstrate this possibility. Detailed description of the effective mass concept print file in pdf format 2.
The solution is defined only in the shaded region, i. Penney developed a useful onedimensional model of the electron lattice interaction in. To simplify the analysis, we only consider a onedimensional system where atoms are aligned and equally spaced. The kspace diagram is the relation of particle energy e versus the momentum wave number k. B, condensed matter 8411 september 2011 with 6 reads. Think of this like the hydrogen atom in elementary. Dec 05, 2018 this topic kronig penney model is from band theory of solids from solid state physics.
Considering the kronig penney kp model we propose a new type of. Energy band analysis of mqw structure based on kronig. Here we show only the result of the kronig penny model. Derivation of the kronigpenney model the solution to schrodingers equation for the kronigpenney potential previously shown in figure 2. Recitation kroning penney massachusetts institute of. Ive got an exam tomorrow, and this is one of the topics i cant get my head around. The possible states that the electron can occupy are determined by the schrodinger equation, in the case of the kroningpenney model, the potential vx is a periodic square wave. Grossman department of materials science and engineering, massachusetts institute of technology, 77 massachusetts avenue, cambridge, massachusetts 029 received 3 august 2011. An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. The kronigpenney model 1 is a simplified model for an electron in a onedimensional periodic potential.
Blochs theorem and kronig penney model free download as powerpoint presentation. This implies that there are only certain allowed energy bands in crystal solids. Recitation kronig penney the kronig penney model helps understand the nature of bands and band gaps starting. A lecture note on blochs theorem and kronig penney model. The periodic potential has the form ux 2 2m 0 x1 n1 x na 6 with aas lattice constant and. The possible states that the electron can occupy are determined by the schrodinger equation, in the case of the kroning penney model, the potential vx is a periodic square wave. January 25, 2012 the kronig penney model describes electron motion in a period array of rectangular barriers fig. Spectrum and transmission periodic in the strength of the barriers m. Kronig penney free download as powerpoint presentation. The kronigpenney model wolfram demonstrations project. It is often used as a pedagogical model to illustrate how periodic potentials can lead to the formation of electronic energy bands and band gaps.
A lecture note on blochs theorem and kronigpenney model. Lecture xx i energy bands central equation and kronig. The kronig penney model is a simple, idealized quantummechanical system that consists of an infinite periodic array of square potential barriers. We assume that the potential energy of an electron has the form of a periodic array of square wells. Think of this like the hydrogen atom in elementary quantum mechanics, wherein the energy was allowed to only take on certain values due to quantization. A comparison of the exact with the approximate solutions as a function of potential. Energy gap where there are no waves that can travel in crystal energy gap k. Kronigpenney model of scalar and vector potentials in graphene.
Here i go over the results of the kronig penney model. Quantitative methods for bandstructure calculations. The tightbinding formulation of the kronigpenney model. In this section, we present the free electron model and the kronigpenney model. Origin of energy bands and nearly free electron model. For the imperfect kronigpenny model we observe some energy states in the band gap area. The repulsivepotential kronigpenney kp model for a onedimensional band structure is well known. Blochs theorem and kronigpenney model electronic band. This relation has to satisfy the kronigpenney mathematic model. The model is closely related to the more conceptual empty lattice approximation. Kronig penney electron hole electronic band structure.
Kronig penney model 17 is a potential field model with periodic array of rectangular potential wells. Shockley states in a semiinfinite kronigpenney model. The spectrum of a periodic succession of function barriers kronig penney model in bilayer graphene is periodic in p with period 2. In a conductor, the highest energy band containing electrons is only partially. Next step for understanding electrons in crystals simplest extension of the electron gas model nearly free electron gas very small potential variation with the periodicity of the crystal. In this case, we no longer have a discrete set of energies, but instead bands of energies that form admissible eigenfunctions, with. Periodic potentials kronig penney model electrons in a lattice see a periodic potential due to the presence of the atoms, which is of the form shown in figure 1. For the solution of kronig penney model mentioned above, if the value of p is large i. It is a generalization of the free electron model, which assumes zero potential inside. General numerical method for the kronig penney model of energy bands in solids.
Physics 4750 physics of modern materials chapter 5. A simple model of a crystalline solid that leads to an electronic band structure is presented. Consider a semiinfinite kronig penney model with the potential plot as shown in fig. For the solution of kronigpenney model mentioned above, if the value of p is large i. In section 4 we consider a model in which the magnetic and. As will be shown shortly, this periodic potential will. Periodic potential in the kronigpenney model we now consider a schrodinger equation, 2 2 2 2 x v x x e x dx d m. A single lecture illustrating the band structure of solids donald a. This forms the preliminary basis of classifying solids into conductors and insulators. Energy gap where there are no waves that can travel in crystal. Kronig penney model in reciprocal space by central equation we use the kronig penney model of a periodic delta function potential. The purpose of the solution is to illustrate much of bandstructure physics and also to develop a bag of. Still, the kronigpenny model does a remarkable job of isolating the essential e. Energy bands in crystals this chapter will apply quantum mechanics to a one dimensional, periodic lattice of potential wells which serves as an analogy to electrons interacting with the atoms of a crystal.
The model enables understanding and calculating the electronic band structure of especially metals. The kronig penney model it is worthwhile to reproduce the most essential fragment of the calculations of the original paper of kronig and penney 1. In other words, for certain energies, there will be no proper solutions to the kronig penney model. Quantitative methods for bandstructure calculations 15. Derivation of the kronigpenney model the solution to schrodingers equation for the kronig penney potential previously shown in figure 2. January 25, 2012 the kronigpenney model describes electron motion in a period array of rectangular barriers fig.
This relation has to satisfy the kronig penney mathematic model. Feb 14, 2010 in kronigpenney model we seek solutions for time independent scrodinger equation inside and outside of the periodic potential. What is the significance of the kronigpenney model. The situation, however, changes when a strong positive electric field such as used in field ion microscopes is applied to a crystal surface. Introduction metals generally have a crystalline structure, that is, the ions are arranged in a way that exhibits a spatial periodicity. Blochs theorem and kronigpenney model free download as powerpoint presentation. Puszkarski kronig penney model with energy dependent potential,vx. For details of the calculation, see for example kittel introduction to solid state physics. Kronigpenney model 17 is a potential field model with. In order to satisfy the kronig penney model, the ke diagram is not continuous, which determines the allowed. Mar 28, 2018 here i go over the results of the kronig penney model. Energy bands and band gaps the kronig penney model michael a. Scribd is the worlds largest social reading and publishing site.
In solidstate physics, the nearly free electron model or nfe model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. It illustrates how a periodic potential say for a one dimensional lattice can give rise to a band gap. Origin of energy bands and nearly free electron model intuitive picture braggreflection for nearly free electrons gap opening for nearly free electrons solution of the schrodinger equation kronig penney model empty lattice approximation number of states in a band and filing of the bands. Kronigpenney model overview and the ek diagram youtube. Penney developed a useful onedimensional model of the electron lattice interaction in 1931. Superpose the curves for a large gap, the whole of.
Energy gap of kronigpenneytype hydrogenated graphene. Kronigpenney model in reciprocal space by central equation we use the kronigpenney model of a periodic delta function potential. Recitation kronig penney the kronig penney model helps understand the nature of bands and band gaps starting form and very easy and simplified atomistic model. In quantum mechanics, the particle in a onedimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice.
Kronigpenney model of scalar and vector potentials in. For p smaller than a critical value p c, the spectrum exhibits two dirac points while for p larger than p c an energy gap. In order to satisfy the kronigpenney model, the ke diagram is not continuous, which determines the allowed. The kronig penney model demonstrates that a simple onedimensional periodic potential yields energy bands as well as energy band gaps. In the perfect crystal, no electron of forbidden energy can exist in its bulk. General numerical method for the kronig penney model of. Pdf the kronigpenney model extended to arbitrary potentials via. The schematic kronig penney model of the crystal is. Mathematica using the mathematica, one can derive the fundamentak equation for the energy diagram in the. The wave functions and characteristic energies of the kronig penney model, essentially an electron in a chain of rectangular well potentials, are obtained starting from bloch theorem and integrating numerically. As will be shown shortly, this periodic potential will open gaps in the dispersion relation.
Energy gap of kronig penney type hydrogenated graphene superlattices joohyoung lee and jeffrey c. This results in the decrease of allowed energy and. Homework statement homework equations the attempt at a solution ive calculated that alpha4. Kronig penney model of scalar and vector potentials in graphene m ramezani masir 1, p vasilopoulos2 and f m peeters. The onedimensional kronig penney model 10 consists of an electron moving in a periodic potential as depicted in fig. The first assumption is to consider a row of evenly spaced atoms with average distance a. This model is called kronigpenney model of potentials. Of course, this is for electrons of the correct energy, falling in one of the energy bands. Energy bands find, read and cite all the research you need on researchgate. We will show that as the number of wells becomes large, the allowed energy levels for the electron form nearly continuous energy. This means that the allowed bands are narrower and forbidden bands arc wider. This diagram determines which states the electrons are allowed to occupy, and is the fundamental result we. Energy bands from the kronigpenney model physics forums. While it is an oversimplification of the threedimensional potential and bandstructure in an actual semiconductor crystal, it is an instructive tool to demonstrate how the band structure can be calculated for a periodic potential, and how allowed and forbidden.
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