In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron m_e (9.11×10^-31 kg). Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Quartz crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules... e- redirects here. ... For the following two reasons the electron hole was introduced into calculations: If an electron is excited into higher state it leaves a hole in its old state. ... Quartz crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules... It has been suggested that optical field be merged into this article or section. ... For other senses of this term, see magnetic field (disambiguation). ... The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. It was assigned to the United States in 1889 and is periodically recertified and traceable to the primary international standard, The Kilogram, held at the Bureau International des Poids et...

Definition

Effective mass is defined by analogy with Newton's second law . Using quantum mechanics it can be shown that for an electron in an external electric field E: Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Fig. ...

where a is acceleration, is reduced Planck's constant, , k is the wave number (often loosely called momentum since k = for free electrons), ε(k) is the energy as a function of k, or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of , where q is the charge. Hence under the model that only the external electric field acts, effective mass m ^* becomes: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... In classical mechanics, momentum (pl. ... The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept. f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where are real numbers and . ...

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes), is a tensor. However, for most calculations the various directions can be averaged out. This article or section is not written in the formal tone expected of an encyclopedia article. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics. Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ...

Quantum Calculation

In the free electron model, the electronic wave function can be in the form of . For a wave packet the group velocity is given by:

=

In presence of an electric field E, the energy change is:

Now we can say:

where p is the electron's momentum. Just put previous results in this last equation and we get:

From this follows the definition of effective mass:

Effective mass for some common semiconductors (for density of states calculations)

Sources:
S.Z. Sze, Physics of Semiconductor Devices, ISBN 0-471-05661-8.
W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN 0-486-66021-4.
This site gives the effective masses of Silicon at different temperatures. It has been suggested that Silicons ranking be merged into this article or section. ... General Name, Symbol, Number germanium, Ge, 32 Chemical series metalloids Group, Period, Block 14, 4, p Appearance grayish white Atomic mass 72. ... This article is about the chemical compound. ... Indium antimonide (InSb) is a narrow gap semiconductor group the (III-V group) used in infrared detectors, including thermal imaging cameras. ... Zinc oxide is a chemical compound with formula ZnO. It is nearly insoluble in water but soluble in acids or alkalis. ... Zinc selenide (ZnSe), is a light yellow binary solid compound. ...

Experimental determination

Traditionally effective masses were measured using cyclotron resonance, a method in which microwave absorption of a semiconductor immersed in a magnetic field goes through a sharp peak when the microwave frequency equals the cyclotron frequency . In recent years effective masses have more commonly been determined through measurement of band structures using techniques such as angle-resolved photoemission (ARPES) or, most directly, the de Haas-van Alphen effect. Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic specific heat at constant volume c_v. The specific heat depends on the effective mass through the density of states at the Fermi level and as such is a measure of degeneracy as well as band curvature. Very large estimates of carrier mass from specific heat measurements have given rise to the concept of heavy fermion materials. Since carrier mobility depends on the ratio of carrier collision lifetime τ to effective mass, masses can in principle be determined from transport measurements, but this method is not practical since carrier collision probabilities are typically not known a priori. Electron cyclotron resonance is a phenomenon observed both in plasma physics and condensed matter physics. ... In solid state physics, the electronic band structure, or simply band structure, refers to the dispersion relation (the relation between energy versus momentum) of electrons in a crystal. ... The photoelectric effect is the emission of electrons from a surface (usually metallic) upon exposure to, and absorption of, electromagnetic radiation (such as visible light and ultraviolet radiation) that is above the threshold frequency particular to each type of surface. ... The de Haas-van Alfven effect {dHvA} was discovered in 1930 by Wander Johannes de Haas and PM van Alfven. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... This does not adequately cite its references or sources. ...

Significance

As the table shows, III-V compounds based on GaAs and InSb have far smaller effective masses than tetrahedral group IV materials like Si and Ge. In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass: where with e being the electronic charge. The ultimate speed of integrated circuits depends on the carrier velocity, so the low effective mass is the fundamental reason that GaAs and its derivatives are used instead of Si in high-bandwidth applications like cellular telephony. The Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials (especially metals). ... e- redirects here. ... Integrated circuit showing memory blocks, logic and input/output pads around the periphery Microchips with a transparent window showing the integrated circuit inside. ... This article or section does not cite its references or sources. ... Cellular redirects here. ...

External link

• NSM archive
• Pastori Parravicini, G. (1975). Electronic States and Optical Transitions in Solids. Pergamon Press. ISBN 0-08-016846-9.  This book contains an exhaustive but accessible discussion of the topic with extensive comparison between calculations and experiment.