In
solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, a particle's effective mass (often denoted
) is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
that it ''seems'' to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a
thermal distribution. One of the results from the
band theory of solids is that the movement of particles in a periodic potential, over long distances larger than the lattice spacing, can be very different from their motion in a vacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of a free particle with that mass. For some purposes and some materials, the effective mass can be considered to be a simple constant of a material. In general, however, the value of effective mass depends on the purpose for which it is used, and can vary depending on a number of factors.
For
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s or
electron hole
In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is a quasiparticle which is the lack of an electron at a position where one could exist in an atom or atomic lattice. Since in a normal atom or ...
s in a solid, the effective mass is usually stated as a factor multiplying the
rest mass of an electron, ''m''
e (9.11 × 10
−31 kg). This factor is usually in the range 0.01 to 10, but can be lower or higher—for example, reaching 1,000 in exotic
heavy fermion material
In solid-state physics, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, t ...
s, or anywhere from zero to infinity (depending on definition) in
graphene
Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure. . As it simplifies the more general band theory, the electronic effective mass can be seen as an important basic parameter that influences measurable properties of a solid, including everything from the efficiency of a solar cell to the speed of an integrated circuit.
Simple case: parabolic, isotropic dispersion relation
At the highest energies of the valence band in many semiconductors (Ge, Si, GaAs, ...), and the lowest energies of the conduction band in some semiconductors (GaAs, ...), the band structure can be locally approximated as
:
where is the energy of an electron at
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
in that band, is a constant giving the edge of energy of that band, and is a constant (the effective mass).
It can be shown that the electrons placed in these bands behave as free electrons except with a different mass, as long as their energy stays within the range of validity of the approximation above. As a result, the electron mass in models such as the
Drude model must be replaced with the effective mass.
One remarkable property is that the effective mass can become ''negative'', when the band curves downwards away from a maximum. As a result of the
negative mass
In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as the ...
, the electrons respond to electric and magnetic forces by gaining velocity in the opposite direction compared to normal; even though these electrons have negative charge, they move in trajectories as if they had positive charge (and positive mass). This explains the existence of
valence-band holes, the positive-charge, positive-mass
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
s that can be found in semiconductors.
In any case, if the band structure has the simple parabolic form described above, then the value of effective mass is unambiguous. Unfortunately, this parabolic form is not valid for describing most materials. In such complex materials there is no single definition of "effective mass" but instead multiple definitions, each suited to a particular purpose. The rest of the article describes these effective masses in detail.
Intermediate case: parabolic, anisotropic dispersion relation
In some important semiconductors (notably, silicon) the lowest energies of the conduction band are not symmetrical, as the constant-energy surfaces are now
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
s, rather than the spheres in the isotropic case. Each conduction band minimum can be approximated only by
:
where , , and axes are aligned to the principal axes of the ellipsoids, and , and are the inertial effective masses along these different axes. The offsets , , and reflect that the conduction band minimum is no longer centered at zero wavevector. (These effective masses correspond to the principal components of the inertial effective mass tensor, described later.
)
In this case, the electron motion is no longer directly comparable to a free electron; the speed of an electron will depend on its direction, and it will accelerate to a different degree depending on the direction of the force. Still, in crystals such as silicon the overall properties such as conductivity appear to be isotropic. This is because there are multiple
valley
A valley is an elongated low area often running between Hill, hills or Mountain, mountains, which will typically contain a river or stream running from one end to the other. Most valleys are formed by erosion of the land surface by rivers ...
s (conduction-band minima), each with effective masses rearranged along different axes. The valleys collectively act together to give an isotropic conductivity. It is possible to average the different axes' effective masses together in some way, to regain the free electron picture. However, the averaging method turns out to depend on the purpose:
General case
In general the dispersion relation cannot be approximated as parabolic, and in such cases the effective mass should be precisely defined if it is to be used at all.
Here a commonly stated definition of effective mass is the ''inertial'' effective mass tensor defined below; however, in general it is a matrix-valued function of the wavevector, and even more complex than the band structure.
Other effective masses are more relevant to directly measurable phenomena.
Inertial effective mass tensor
A classical particle under the influence of a force accelerates according to
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
, , or alternatively, the momentum changes according to . This intuitive principle appears identically in semiclassical approximations derived from band structure.
The force gives a rate of change in
crystal momentum :
:
where is the ''
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
''.
Acceleration for a wave-like particle becomes the rate of change in
group velocity:
:
where is the
del operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
in
reciprocal space
In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fu ...
. The last step follows from using
the chain rule for a total derivative for a quantity with indirect dependencies, because the direct result of the force is the change in given above, which indirectly results in a change in .
Combining these two equations yields
:
using the
dot product rule with a uniform force ().
is the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of in reciprocal space. We see that the equivalent of the Newtonian reciprocal
inertial mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
for a free particle defined by has become a tensor quantity
:
whose elements are
:
This tensor allows the acceleration and force to be in different directions, and for the magnitude of the acceleration to depend on the direction of the force.
* For parabolic bands, the off-diagonal elements of are zero, and the diagonal elements are constants
* For isotropic bands the diagonal elements must all be equal and the off-diagonal elements must all be equal.
* For parabolic isotropic bands, , where is a scalar effective mass and is the identity.
* In general, the elements of are functions of .
* The inverse, , is known as the effective mass tensor. Note that it is not always possible to invert
For bands with linear dispersion
such as with photons or electrons in
graphene
Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure. , the group velocity is fixed, i.e. electrons travelling with parallel with to the force direction cannot be accelerated and the diagonal elements of are obviously zero. However, electrons travelling with a component perpendicular to the force can be accelerated in the direction of the force, and the off-diagonal elements of are non-zero. In fact the off-diagonal elements scale inversely with , i.e. they diverge (become infinite) for small . This is why the electrons in graphene are sometimes said to have infinite mass (due to the zeros on the diagonal of ) and sometimes said to be massless (due to the divergence on the off-diagonals).
Cyclotron effective mass
Classically, a charged particle in a magnetic field moves in a helix along the magnetic field axis. The period ''T'' of its motion depends on its mass ''m'' and charge ''e'',
:
where ''B'' is the
magnetic flux density
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
.
For particles in asymmetrical band structures, the particle no longer moves exactly in a helix, however its motion transverse to the magnetic field still moves in a closed loop (not necessarily a circle). Moreover, the time to complete one of these loops still varies inversely with magnetic field, and so it is possible to define a ''cyclotron effective mass'' from the measured period, using the above equation.
The semiclassical motion of the particle can be described by a closed loop in k-space. Throughout this loop, the particle maintains a constant energy, as well as a constant momentum along the magnetic field axis. By defining to be the area enclosed by this loop (this area depends on the energy , the direction of the magnetic field, and the on-axis wavevector ), then it can be shown that the cyclotron effective mass depends on the band structure via the derivative of this area in energy:
:
Typically, experiments that measure cyclotron motion (
cyclotron resonance
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
,
De Haas–Van Alphen effect, etc.) are restricted to only probe motion for energies near the
Fermi level.
In
two-dimensional electron gas
A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion ...
es, the cyclotron effective mass is defined only for one magnetic field direction (perpendicular) and the out-of-plane wavevector drops out. The cyclotron effective mass therefore is only a function of energy, and it turns out to be exactly related to the density of states at that energy via the relation
, where is the valley degeneracy. Such a simple relationship does not apply in three-dimensional materials.
Density of states effective masses (lightly doped semiconductors)
In semiconductors with low levels of doping, the electron concentration in the conduction band is in general given by
:
where is the
Fermi level, is the minimum energy of the conduction band, and is a concentration coefficient that depends on temperature. The above relationship for can be shown to apply for any conduction band shape (including non-parabolic, asymmetric bands), provided the doping is weak (); this is a consequence of
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
limiting towards
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
.
The concept of effective mass is useful to model the temperature dependence of , thereby allowing the above relationship to be used over a range of temperatures. In an idealized three-dimensional material with a parabolic band, the concentration coefficient is given by
:
In semiconductors with non-simple band structures, this relationship is used to define an effective mass, known as the density of states effective mass of electrons. The name "density of states effective mass" is used since the above expression for is derived via the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
for a parabolic band.
In practice, the effective mass extracted in this way is not quite constant in temperature ( does not exactly vary as ). In silicon, for example, this effective mass varies by a few percent between absolute zero and room temperature because the band structure itself slightly changes in shape. These band structure distortions are a result of changes in electron–phonon interaction energies, with the lattice's thermal expansion playing a minor role.
Similarly, the number of holes in the valence band, and the density of states effective mass of holes are defined by:
:
where is the maximum energy of the valence band. Practically, this effective mass tends to vary greatly between absolute zero and room temperature in many materials (e.g., a factor of two in silicon), as there are multiple valence bands with distinct and significantly non-parabolic character, all peaking near the same energy.
Determination
Experimental
Traditionally effective masses were measured using
cyclotron resonance
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
, 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 structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...
s using techniques such as angle-resolved
photo emission (
ARPES
Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoelec ...
) 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
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
at constant volume
. 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
In solid-state physics, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, t ...
materials. Since carrier
mobility
Mobility may refer to:
Social sciences and humanities
* Economic mobility, ability of individuals or families to improve their economic status
* Geographic mobility, the measure of how populations and goods move over time
* Mobilities, a conte ...
depends on the ratio of carrier collision lifetime
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. The
optical Hall effect
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviol ...
is an emerging technique for measuring the free charge carrier density, effective mass and mobility parameters in semiconductors. The optical Hall effect measures the analogue of the quasi-static electric-field-induced electrical Hall effect at optical frequencies in conductive and complex layered materials. The optical Hall effect also permits characterization of the anisotropy (tensor character) of the effective mass and mobility parameters.
Theoretical
A variety of theoretical methods including
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
,
k·p perturbation theory
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.
It is pronounced "k dot p", and is ...
, and
others are used to supplement and support the various experimental measurements described in the previous section, including interpreting, fitting, and extrapolating these measurements. Some of these theoretical methods can also be used for predictions of effective mass in the absence of any experimental data, for example to study materials that have not yet been created in the laboratory.
Significance
The effective mass is used in transport calculations, such as transport of electrons under the influence of fields or carrier gradients, but it also is used to calculate the carrier density and
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
in semiconductors. These masses are related but, as explained in the previous sections, are not the same because the weightings of various directions and wavevectors are different. These differences are important, for example in
thermoelectric materials
Thermoelectric materials show the thermoelectric effect in a strong or convenient form.
The ''thermoelectric effect'' refers to phenomena by which either a temperature difference creates an electric potential or an electric current creates a t ...
, where high conductivity, generally associated with light mass, is desired at the same time as high
Seebeck coefficient
The Seebeck coefficient (also known as thermopower, thermoelectric power, and thermoelectric sensitivity) of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material ...
, generally associated with heavy mass. Methods for assessing the electronic structures of different materials in this context have been developed.
Certain group
III–
V compounds such as
gallium arsenide
Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a Zincblende (crystal structure), zinc blende crystal structure.
Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monoli ...
(GaAs) and
indium antimonide
Indium antimonide (InSb) is a crystalline compound made from the elements indium (In) and antimony (Sb). It is a narrow- gap semiconductor material from the III- V group used in infrared detectors, including thermal imaging cameras, FLIR systems ...
(InSb) have far smaller effective masses than
tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
group IV
The carbon group is a periodic table group consisting of carbon (C), silicon (Si), germanium (Ge), tin (Sn), lead (Pb), and flerovium (Fl). It lies within the p-block.
In modern IUPAC notation, it is called group 14. In the field of semicon ...
materials like
silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic tab ...
and
germanium
Germanium is a chemical element with the symbol Ge and atomic number 32. It is lustrous, hard-brittle, grayish-white and similar in appearance to silicon. It is a metalloid in the carbon group that is chemically similar to its group neighbors s ...
. In the simplest
Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass:
, where
with
being the
electronic charge. The ultimate speed of
integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
s 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
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
applications like
cellular telephony
Mobile telephony is the provision of telephone services to phones which may move around freely rather than stay fixed in one location. Telephony is supposed to specifically point to a voice-only service or connection, though sometimes the l ...
.
In April 2017, researchers at Washington State University claimed to have created a fluid with negative effective mass inside a
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
, by engineering the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
.
See also
Models of solids and crystals:
*
Tight-binding model
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon Quantum superposition, superposition of wave functions for isolated ...
*
Free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
*
Nearly free electron model
In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model i ...
Footnotes
References
*{{cite book , author= Pastori Parravicini, G. , title=Electronic States and Optical Transitions in Solids , publisher=
Pergamon Press
Pergamon Press was an Oxford-based publishing house, founded by Paul Rosbaud and Robert Maxwell, that published scientific and medical books and journals. Originally called Butterworth-Springer, it is now an imprint of Elsevier.
History
The cor ...
, year=1975 , isbn=978-0-08-016846-3 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).
External links
NSM archive
Condensed matter physics
Mass