Effective mass - Citizendia

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. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating 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. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J An electron hole is the conceptual and mathematical Opposite of an Electron, useful in the study of Physics and Chemistry. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges This mass is usually stated in units of the ordinary mass of an electron m_e (9. 11×10^-31 kg).

1 Definition
2 Effective mass for some common semiconductors (for density of states calculations)
3 Experimental determination
4 Significance
5 See also
6 External links

Definition

When an electron is moving inside a solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of electron in Newton's law. The effective mass can be negative or different due to circumstances. Generally, in the absence of electric and magnetic field, the concept of effective mass does not apply.

Effective mass is defined by analogy with Newton's second law $\vec{F}=m\vec{a}$ . Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Using quantum mechanics it can be shown that for an electron in an external electric field E:

$a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE$

where $a$ is acceleration, $\hbar = h/2\pi$ is reduced Planck's constant, $k$ is the wave number (often loosely called momentum since $k$ = $p/\hbar$ for free electrons), $ε(k)$ is the energy as a function of $k$ , or the dispersion relation as it is often called. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. From the external electric field alone, the electron would experience a force of $\vec{F}=q\vec{E}$ , where q is the charge. Hence under the model that only the external electric field acts, effective mass $m *$ becomes:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 \varepsilon} \over {d k^2}} \right]^{-1}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! 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.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Effective mass, being generally dependent on direction (with respect to the crystal axes), is a tensor. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Reduced mass is the "effective" Inertial mass appearing in the Two-body problem of Newtonian mechanics. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements Effective mass can only be understood with quantum mechanics.

==Derivati $v = {{d \omega} \over {d k}}$ = ${{1} \over {\hbar}} \cdot {{d \varepsilon} \over {d k}}$

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

$d \varepsilon = {{d \varepsilon} \over {d k}} {d k} = -eE{d x} = -eEv{d t} = {-eE \over {\hbar}} {{d \varepsilon} \over {d k}}{dt}$

Now we can say:

$\hbar \cdot {{d k} \over {d t}} = {{d p} \over {d t}} = m \cdot {{d v} \over {d t}}$

where p is the electron's momentum. Substitute the expression for the group velocity into this last equation and we get:

${{\hbar} \over {m}} \cdot {{d k} \over {d t}} = {{1} \over {\hbar}} \cdot {{d} \over {d t}}{{d \varepsilon} \over {d k}} ={{1} \over {\hbar}} \cdot {{d^{2} \varepsilon} \over {dk^{2}}}{{d k}\over {d t}}$

From this follows the definition of effective mass:

${{1} \over {m}} = {{1} \over {\hbar^{2}}} \cdot {{d^{2} \varepsilon} \over {d k^2}}$

An alternative derivation can be given by considering the Hamiltonian of a free particle and using the de Broglie relation:

${H={{p^{2}} \over {2m}} = {{\hbar^2 k^2} \over {2m}}}$

The same result is obtained, identifying the Hamiltonian with the total energy:

${{1} \over {m}} = {{1} \over {\hbar^{2}}} \cdot {{d^{2} \varepsilon} \over {d k^2}}$

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

Material	Electron effective mass	Hole effective mass
Group IV
Si (4. Silicon (ˈsɪlɪkən or /ˈsɪlɪkɒn/ silicium is the Chemical element that has the symbol Si and Atomic number 14 2K)	1. 08 m_e	0. 56 m_e
Ge	0. Germanium (dʒɚˈmeɪniəm is a Chemical element with the symbol Ge and Atomic number 32 55 m_e	0. 37 m_e
III-V
GaAs	0. Gallium arsenide ( GaAs) is a compound of two elements Gallium and Arsenic. 067 m_e	0. 45 m_e
InSb	0. Indium antimonide ( is a Narrow gap Semiconductor material from the III - V group used in Infrared detectors including Thermal 013 m_e	0. 6 m_e
II-VI
ZnO	0. Zinc oxide is a Chemical compound with the formula ZnO It is nearly insoluble in water but soluble in Acids and Bases It occurs 19 m_e	1. 21 m_e
ZnSe	0. Zinc selenide ( Zn[[Selenium Se]] is a light yellow binary solid compound 17m_e	1. 44 m_e

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.

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 $\omega_c = \frac{eB}{m^* c}$ . Electron cyclotron resonance is a phenomenon observed both in Plasma physics and Condensed matter physics. 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. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron The de Haas–van Alphen effect, often abbreviated to dHvA, was discovered in 1930 by Wander Johannes de Haas and PM van Alphen Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic specific heat at constant volume $c v$ . Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity 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. In Physics, electron mobility (or simply mobility) is a quantity relating the Drift velocity of Electrons to the applied Electric field

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: $\vec{v} = \begin{Vmatrix}\mu\end{Vmatrix} \cdot \vec{E}$ where $\begin{Vmatrix}\mu\end{Vmatrix} = \frac{e \tau}{\begin{Vmatrix}m^*\end{Vmatrix}}$ with $e$ being the electronic charge. The Drude model of Electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of Electrons in materials (especially metals The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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. Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside

External links

NSM archive
Pastori Parravicini, G. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron (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.
S. Pekar, The method of effective electron mass in crystals, Zh. Eksp. Teor. Fiz. 16, 933 (1946).

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Contents

Definition

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

Experimental determination

Significance

See also

External links