Quasiparticles

one of the fundamental concepts of the theory of the condensed state of matter and particularly of solid-state theory. The theoretical description and explanation of the properties of condensed media (solids and liquids) based on the properties of the particles (atoms and molecules) constituting them present great difficulties, first because the number of particles is very great (about 10²² particles per cu cm) and second because they interact strongly with one another. Because of the interaction of the particles, the total energy of such a system— which determines many of its properties—is not the sum of the energies of the individual particles, as is the case in an ideal gas. The particles of a condensed medium obey the laws of quantum mechanics; therefore, the properties of an aggregate of particles constituting a solid or liquid can be understood only on the basis of quantum concepts. The development of the quantum theory of condensed media led to the creation of special concepts in physics, particularly the concept of quasiparticles—elementary excitations of the aggregate of interacting particles. The concept of quasiparticles has provided particularly fruitful results in the theory of crystals and liquid helium.

Properties. It was found that the energy ε of a crystal or liquid helium may be approximately regarded as consisting of two parts: the energy of the ground (unexcited) state ε0 (the lowest energy, corresponding to the state of the system at absolute zero), and the total energy ε_λ of elementary (irreducible) motions or excitations:

The index λ characterizes the type of elementary excitation, and n_λ is an integer that indicates the number of elementary excitations of type λ.

Thus, it has proved possible to write the energy of the excited state of a crystal (or of helium) in the same manner as the energy of an ideal gas, in the form of an energy sum. However, in the case of a gas the energies of its particles (atoms and molecules) add, whereas in the case of a crystal the energies of the elementary excitations of an entire set of atoms add (hence the term “quasiparticles”). In the case of a gas consisting of free particles, the index λ denotes the momentum p of the particle, ε_λ denotes its energy (ε_λ = p²/2m, where m is the mass of the particle), and n denotes the number of particles that have momentum p. The velocity is v = p/m.

An elementary excitation in a crystal is also characterized by the vector p, whose properties are similar to momentum; it is called quasimomentum. The energy ε_λ of an elementary excitation depends on the quasimomentum, but the relationship ε_λ (p) is not as simple as in the case of a free particle. The velocity of propagation of an elementary excitation also depends on the quasimomentum and the form of the function ε_λ (p). In the case of quasiparticles the index λ includes a designation of the type of elementary excitation, since elementary excitations that differ in nature are possible in a condensed medium (an analogue is a gas containing particles of different types).

The introduction of the term “quasiparticles” for elementary excitations resulted not only from the external similarity in the description of the energy of the excited state of a crystal or liquid helium and an ideal gas but also from the deep-seated analogy between the properties of a free (quantum-mechanical) particle and the elementary excitation of a set of interacting particles, which is based on the particle-wave duality. The state of a free particle in quantum mechanics is described by a monochromatic wave of frequency ѡ = ε/ℏ and wavelength λ = 2πℏ/p (where ε and p are the energy and momentum of the free particle, and ℏ is Planck’s constant). In a crystal the excitation of one particle (such as absorption of a photon by one of the atoms), which leads to excitation of neighboring particles because of the interaction (bond) between the atoms, does not remain localized but is passed on to its neighbors and propagates a wave of excitation. This corresponds to a quasiparticle with a quasimomentum p = ℏk and an energy ε = ℏω(k), where k is the wave vector and the wavelength is λ = 2π/k.

The dependence of frequency on the wave vector k makes it possible to determine the dependence of the energy of quasiparticles on quasimomentum. This relation, ε_λ = ε_(p) called the dispersion law, is the main dynamic characteristic of a quasiparticle and, in particular, defines its velocity (v = ∂ε/∂p). A knowledge of the dispersion law of quasiparticles makes it possible to study the motion of quasiparticles in external fields. A quasiparticle, in contrast to an ordinary particle, is not characterized by a specific mass. However, in emphasizing the similarity between a quasiparticle and a particle it is sometimes convenient to introduce a quantity that has the dimension of mass. It is called the effective mass, m* (as a rule the effective mass depends on the quasimomentum and on the form of the dispersion law).

All of the above makes it possible to regard an excited condensed medium as a gas consisting of quasiparticles. The similarity between a gas consisting of particles and a gas consisting of quasiparticles is also manifested in the fact that since the concepts and methods of the kinetic theory of gases may be used to describe the properties of a gas of quasiparticles. In particular, one may speak of collisions of quasiparticles (in which the specific laws of conservation of energy and quasimomentum are valid), the mean free time, and the mean free path. The Boltzmann kinetic equation may be used to describe a gas of quasiparticles.

One of the important distinguishing properties of a gas of quasiparticles (in comparison with a gas consisting of ordinary particles) is that quasiparticles may appear and disappear, that is, their number is not conserved. The number of quasiparticles depends on the temperature. For T = 0°K no quasiparticles are present. The energy spectrum (the set of energy levels) may be determined for a gas of quasiparticles as for a quantum system, and it may be considered as the energy spectrum of a crystal or liquid helium. The diversity of the types of quasiparticles is great, since their nature depends on the atomic structure of the medium and the interaction among particles. Several types of quasiparticles may exist in a single medium.

Like ordinary particles, quasiparticles may have an intrinsic mechanical moment, or spin. Quasiparticles may be divided into bosons and fermions, depending on the magnitude of the spin (expressed as an integer or half-integer times ℏ). Bosons are produced and disappear one by one, but fermions are produced and disappear in pairs.

For fermion quasiparticles the distribution over the energy levels is determined by the Fermi distribution function; for boson quasiparticles, by the Bose distribution function. Fermi and Bose “branches” may be distinguished in the energy spectrum of a crystal (or liquid helium), Which is the set of energy spectra of all possible types of quasiparticles. In some cases a quasiparticle gas may behave like a gas, obeying Boltzmann statistics (for example, a gas of conduction electrons and holes in a nondegen-erate semiconductor; see below).

The theoretical explanation of the observed macroscopic properties of crystals or liquid helium, which is based on the concept of quasiparticles, requires knowledge of the dispersion law of the quasiparticles and also of the probability of collisions of quasiparticles with each other and with crystal defects. The numerical values of these characteristics can be obtained only by means of computers. In addition, the semi-empirical approach has developed significantly: the quantitative characteristics of quasiparticles are determined by comparing theory with experiment and are then used to calculate the characteristics of the crystals (or liquid helium).

To determine the characteristics of quasiparticles, the scattering of neutrons, the scattering and absorption of light, ferromagnetic and antiferromagnetic resonance, and ferroacoustic resonance are used, and the properties of metals and semiconductors in strong magnetic fields, particularly cyclotron resonance and galvanomagnetic phenomena, are studied.

The concept of quasiparticles is applicable only at comparatively low temperatures (near the ground state), when the properties of a quasiparticle gas are close to those of an ideal gas. As the number of quasiparticles increases, the probability of collisions grows, the mean free time of quasiparticles decreases, and, in accordance with the uncertainty relation, the uncertainty of the energy of the quasiparticles increases. The very concept of quasiparticles loses its meaning. Therefore it is clear that not all motions of atomic particles in condensed media can be described in terms of quasiparticles. For example, quasiparticles are unsuitable for describing self-diffusion (the random wandering of atoms through a crystal).

However, even at low temperatures not all possible motions in a condensed medium can be described by means of quasiparticles. Although all atoms of a body generally take part in an elementary excitation, the excitation is microscopic: the energy and momentum of each quasiparticle are of atomic scale, and every quasiparticle moves independently of the others. The atoms and electrons in a condensed medium may take part in a motion of a totally different nature, a motion that is essentially macroscopic (hydrodynamic motion), but the medium does not lose its quantum properties. Examples of such motions are su-perfluid motion in helium II and the electric current in superconductors. The distinguishing feature here is the strict coordination (coherence) of the motion of the individual particles.

The concept of quasiparticles has found application not only in solid-state theory and in the theory of liquid helium but also in other fields of physics, such as the theory of the atomic nucleus, plasma theory, and astrophysics.

Phonons. In a crystal, the atoms perform small oscillations that propagate through the crystal in wave form. At low temperatures, longwave acoustic oscillations—ordinary sound waves, which have the lowest energy—play the primary role. The quasiparticles corresponding to the waves of atomic displacements are called phonons. Phonons are bosons; the number of phonons at low temperatures T increases in proportion to T³. This fact, which is connected with the linear dependence of the energy of a phonon ε_ph, on its quasimomentum p when the quasimomen-tum is sufficiently small (ε_ph = sp, where s is the speed of sound), explains why the specific heat of nonmetallic crystals is proportional to T₃ at low temperatures.

Phonons in superfluid helium. The ground state of helium resembles an extremely degenerate Bose gas. As in any fluid, sound waves (waves of density fluctuation) may propagate in helium. Sound waves are the only type of microscopic motion possible in helium near the ground state. Since in a sound wave the frequency a) is proportional to the wave vector k, ω = sk, the corresponding phonons have the dispersion law ε = sp. As the momentum increases, the curve ε = ε(p) deviates from linear behavior. Helium phonons also obey Bose statistics. The concept of the energy spectrum of helium as a phonon spectrum not only describes its thermodynamic properties, such as the dependence of the specific heat of helium on temperature, but also explains the phenomenon of superconductivity.

Magnons. In ferromagnetics and antiferromagnetics the spins of the atoms are strictly ordered at T = 0°K. The state of excitation of a magnetic system is associated with the deviation of the spin from the “correct” position. This deviation is not localized on a specific atom but rather is transmitted from atom to atom. The elementary excitation of a magnetic system assumes the form of a wave of spin-axis rotations (a spin wave), and the quasiparticle corresponding to it is called a magnon. Magnons are bosons. The energy of a magnon has a square-law dependence on the quasimomentum (in the case of low quasimo-menta). This is reflected in the thermal and magnetic properties of ferromagnets and antiferromagnets (for example, at low temperatures the deviation of the magnetic moment of a ferromagnet from saturation varies as T^3/2). The high-frequency properties of ferromagnets and antiferromagnets are described in terms of magnon “creation.”

Frenkel exciton. A Frenkel exciton is an elementary excitation of the electron system of an individual atom or molecule that propagates through the crystal in the form of a wave. Excitons usually have high energy (on an atomic scale), of the order of several electron volts. Therefore, the contribution of excitons to the thermal properties of solids is small. Excitons manifest themselves in the optical properties of crystals. The average number of excitons is usually very low; therefore, they may be described by classical Boltzmann statistics.

Conduction electrons and holes. In solid dielectrics and semiconductors elementary excitations caused by processes analogous to the ionization of an atom coexist with excitons. As a result of such “ionization” two independently propagating quasiparticles arise: a conduction electron and a hole (a deficiency of an electron in an atom). A hole behaves as a positively charged particle, although its motion is an electron charge exchange wave rather than the motion of a positive ion. Conduction electrons and holes are fermions. They are the carriers of electric current in a solid. Semiconductors whose “ionization” energy is low always have an appreciable number of conduction electrons and holes. The conductivity of semiconductors decreases with temperature, since the number of electrons and holes decreases as the temperature drops

An electron and a hole, since they are attracted to each other, may form a Mott exciton (quasiatom), which is manifested in the optical spectra of crystals as hydrogen-like absorption lines.

Polarons. The interaction of an electron with lattice vibrations leads to polarization of the lattice near the electron. The interaction of the electron and the crystal lattice is sometimes so strong that the motion of the electrons through the crystal is accompanied by a polarization wave. The corresponding quasiparticle is called a polaron.

Conduction electrons in metals. The conduction electrons of a metal, which interact with each other and with the ionic field of a crystal lattice, are equivalent to a quasiparticle gas with a complicated dispersion law. The charge of each quasiparticle is equal to the charge of a free electron, and the spin is equal to 1/2. Their dynamic properties, which are due to the dispersion law, differ significantly from the properties of ordinary free electrons. Conduction electrons are fermions. In quasimomentum space at T = 0°K they fill a region bounded by a Fermi surface. The excitation of conduction electrons signifies the appearance of a pair: an electron “above” the Fermi surface and a vacant site (hole) “below” the surface. An electron gas is strongly degenerate not only at low temperatures but also at room temperature. This determines the temperature dependence of most of the characteristics of a metal (in particular, the linear dependence of specific heat on temperature when T —→ 0°).

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