Doublewell and tightbinding models pdf lecture notes. The important point is that the electron in a periodic potential is accelerated relative to the lattice in an applied electric or magnetic field as if its mass is equal to an effective mass. Application of berrys phase to the effective mass of bloch. Effective mass of electron depends on band structure nihar mahapatra. Effective mass of electron depends on band structure youtube. Im just curious to learn how experimentally, for a given material gaas, ge. A simpleminded derivation of the equation of motion is solids. A table is not feasible, because, unlike the ordinary mass, the effective mass of an electron or hole in a material is not a constant. Next we introduce and explore a new term called the effective mass. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it works pretty well, so that it is able to describe many important properties of metals. Nature the graph between the energy and the propagation constant k is in the parabolic form. Consider a solid rubber ball which has some weight acting downwards. The effective mass is a tensor and may be obtained experimentally or numerically. We assess the accuracy of the approximations in calculating selected.
Therefore, the density of states effective mass is expressed as 3 1 2 d l m t 11. Introduction of the concepts of hole and effective mass. Consider an electron in the solid subject to an external force f. Problems forsolid state physics 3rdyearcourse6 hilary term2011. Application of berrys phase to the effective mass of bloch electrons to cite this article. Electron wavefunction the position of an electron is described by a wavefunction \ zx y. Derivation of effectivemass expressions for electrons and. An electron with wavevector k0can scatter into a state with wavevector k k0 q mod g by absorption of a phonon of wavevector q or emission of a phonon of wavevector q. Nov 11, 2016 i have seen that the energy of the band structure is defined in the following form. In fact, in real metals, the measured mass of the electron is generally not equal to the well known value me 9.
For electrons or electron holes in a solid, the effective mass is usually stated in units of the rest mass of an electron, m e 9. Problems forsolid state physics 3rdyearcourse6 hilary. Free electron model a free electron model is the simplest way to represent the electronic structure of metals. Consider an electron trapped in an energy well with infinite potential barriers. It is usual to obtain the effective mass by analyzing results of tm in the limit m 0. In physics, the reduced mass is the effective inertial mass appearing in the twobody problem of newtonian mechanics. The ek diagram for direct band gap semiconductors will be studied and the hole. Electron wavepacket coulomb potential due to nuclei.
Charge and mass of the electron michigan state university. Derivation of density of states concept first a needed tool. Additionally, how can you obtain the effect mass of the electron. The interaction of the conduction electrons with the periodic potential of the rigid crystal lattice band effective mass. I have only considered the maximum valence band and the minimum. The effective mass has to be used in the derivation of the kinetic energy, particle density, average electron energy and so forth.
Equation 44 recognises this fact by considering that the electrons effective mass in the medium will be different to its free mass at rest. The effective mass from solid state physics is indeed infinite. Measuring the effective mass physics stack exchange. Limited accuracy of conduction band effective mass equations. The effective mass of a semiconductor is obtained by fitting the actual ek diagram around the conduction band minimum or the valence band maximum by a paraboloid. Electron charge and mass quantity symbol value units mass me 9. Thisalteredmassiscalledasan effectiveeffectivemass mass.
February 15, 2011 this note gives a simple classical interpretation of the semiclassical equations of electron motion. Therefore, the densityofstates effective mass is expressed as 3 1 2 d l m t 11. Do you just manipulate algebraically the equation of the energy of the band structure and solve it. In solid state physics, a particles effective mass often denoted m. Box 3000, fi90014 university of oulu, finland dated. Effective mass of electrons electrons in solids can be localized or delocalized. Accordingly we evaluate the electrons effective mass by separately writing down the expressions for the force on the electron, and its acceleration, and then solving for the electrons effective mass. The dynamics of the electron in free space and in the lattice will be compared. Why effective mass of hole is greater than electron.
Invent a new particle to describe the current carried by the whole band with a vacancy, which has a positive charge and positive mass which. Effective mass approximation an overview sciencedirect. I am trying to calculate the effective mass of electrons and holes manually from the band structure of a 2d system i am working on. Finally, we get an expression for electrical conductivity of both the conduction and valence bands. Charge to mass ratio of the electron vanderbilt university. The important point is that the electron in a periodic potential is accelerated relative to the lattice in an applied electric or magnetic field as if. The effective mass tensor ma is of the form 4 3 2 4 1 a 0 0 0 0 m m m m m m, 6 where 1, 2, and 3 refer to the binary x, the bisectrix y, and the trigonal z axes, respectively.
Effective masses of electrons and heavy holes in gaas, inas. In silicon the longitudinal electron mass is m e,l 0. Collective modes zero sound and spin waves in fermi liquids. Introduction of the concepts of hole and effective mass using. For example, without an understanding of the concept of ef fective mass, positive hall voltages cannot be explained in terms of electrons 2, which can lead. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it works pretty well, so that it is able. Later on we will discuss electron phonon scattering, which is inelastic. We begin by deriving an expression for the force on an electron within an intrinsic semiconductor in an electric field. Now in the situation of an electron moving inside a crystal i. Wave packet formed by superposition of plain waves velocity of wave packet 1 v k k k k v v h r v v.
The effective mass may be anisotropic, and it may even be negative. Fermi surface in the free electron sommerfeld theory of metals. In the free electron gas model an electron had the mass me always written straight and not in italics because it is not a variable but a constant of nature. Accordingly we evaluate the electron s effective mass by separately writing down the expressions for the force on the electron, and its acceleration, and then solving for the electron s effective mass.
When you put it inside water or any liquid, there is a buoyant force that acts upward. Notable among these properties is anisotropy of the effective masses of both electrons and holes. In this approximation, an exciton is considered to be confined to a spherical volume of the crystallite and the mass of electron and hole is replaced with effective masses me and mh to define the wave function 10. External force on a crystal electron dt dv m dt dk f v r h v this yields equation of motion. Band structure of solids a physicist point of view martin wolf fritzhaberinstitut of the maxplancksociety, berlin, germany 1. Theoretical studies on the effect of confinement on quantum.
The top of the valence band tends to be flatter due to the asymmetry of the situation. The effective mass m is the second order of derivative of energy with respect to wavevector, which is representative of the local curvature of the dispersion relation in three dimensional space. Density of states derivation the density of states gives the number of allowed electron or hole states per volume at a given energy. Similarly, an electron of wavevector k can scatter into the state k0by emission of a phonon of. The derivation allows us to classify various forms of the effective mass equations in terms of a hierarchy of approximations. The equivalent three dimensional formula in the derivation of one of the problems is. Simple understanding of the e ective mass erkki thuneberg department of physical sciences, p. Limited accuracy of conduction band effective mass. Electron responds to external force as if it had an effective mass. Effective mass renormalization in 2d electron gases experiment.
We know that an electron has a well defined mass, and it obey newtonian mechanics when accelerated by an electric field e. It is a quantity which allows the twobody problem to be solved as if it were a onebody problem. From this follows that the hole effective mass is often larger than the electron effective mass. The derivation of the electronphonon hamiltonian hintand its quantization is relatively straightforward, and i will sketch it here. Where can i find a table with effective masses in metals. The derivation allows us to classify various forms of the effective mass equations in terms of a. When we ac celerate an electron in a uniform electric field e maintained over a distance d, the potential difference over this distance is ed and the electron will gain a kinetic energy ke given by ke mu22 ev.
Related content semiclassical quantization of electrons in magnetic fields. Definition, description and derivation of effective mass of electron,effective mass is that mass which an electron seems to have when it is present in the periodic potential in a lattice crystal. The crystal lattice is not explicitly taken into account in the free electron model, but a quantummechanical justification was given a year later 1928 by blochs theorem. Thompson discovered the first elementary particle, the electron, by measuring the ratio of its charge to mass in a manner similar to the experiment that we will.
It has a negative effective mass same as an electron in the vacancy. Thisalteredmassiscalledasan effectiveeffective mass mass. Here we present a systematic derivation of a conductionband effective mass equation for a selfassembled semiconductor quantum dot in a magnetic. The interaction of the conduction electrons with the periodic potential of. In most semiconductors the effective mass is a tensor quantity. This is a result of bandstructurein metals, which we will explain later in the course. This uncompensated electron has the negative velocity of an electron if it were at the vacancy position. Perturbation theory and nearlyfree electron model pdf. Ravindran, phy02e semiconductor physics, 30 january 20. For electrons or electron holes in a solid, the effective mass is usually stated in units of the.
Nov 03, 2017 definition, description and derivation of effective mass of electron,effective mass is that mass which an electron seems to have when it is present in the periodic potential in a lattice crystal. Later on we will discuss electronphonon scattering, which is inelastic. Mar 12, 2016 effective mass of electron depends on band structure nihar mahapatra. Note, however, that the mass determining the gravitational force is not reduced.
I have seen that the energy of the band structure is defined in the following form. Effective masses are usually tabulated in terms of free electron masses, e. Chapter 11 density of states, fermi energy and energy bands. How to calculate effective mass of an electron and effective. Equation 44 recognises this fact by considering that the electron s effective mass in the medium will be different to its free mass at rest. Related content the electron effective mass in heavily doped gaas a raymond, j l robert and c bernardcyclotron resonance with epitaxial films of. Discovery of the electron in the 1890s, based on several independent experiments a negative charge e, and mass m were assigned to the electron, recognition and verification of the increase of effective mass with velocity in 1905 and 1906, assignment of the compton wavelength. The other two electron ellipsoids b and c pockets are obtained by rotations. On the cyclotron effective mass of gaas to cite this article. Analogous relation for the effective mass m of the electron of the fermi energy accounting for both the band nonparabolicity and the above increase in effective masses due to the dopant concentration reads in turn as follows 29. Recall that the reflection coefficient for infinite potential was 1 so the electron can not penetrate the barrier. The charge to mass ratio em for the electron introduction in 1897 j. The power dissipated by the force per electron must be equal to the change of the electron energy p vf dek dt. It can be shown that the electrons placed in these bands behave as free electrons except with a different mass.
Effective mass at k0, the e,k relationship near the minimum is usually parabolic. In summary, we have learnt that, for small particle spacing, the electrons can be regarded collectively as a gaseous system where the kinetic energy is the only relevant energy. Introduction the effective mass of a semiconductor is obtained by fitting the actual ek diagram around the conduction band minimum or the valence band maximum by a paraboloid. So the electrons inside the crystal will have a different mass than that of the electron in vacuum. If one talks about zero effective mass dirac fermions in graphene, this comes from the massless dirac equation which has the same dispersion relation.
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