ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS The principle of size quantization The whole complex of phenomena usually understood by the words "electronic properties of low-dimensional electronic systems" is based on a fundamental physical fact: a change in the energy spectrum of electrons and holes in structures with very small sizes. Let us demonstrate the basic idea of ​​size quantization using the example of electrons in a very thin metal or semiconductor film of thickness a.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Quantization principle The electrons in a film are in a potential well with a depth equal to the work function. The depth of the potential well can be considered infinitely large, since the work function exceeds the thermal energy of the carriers by several orders of magnitude. Typical values ​​of the work function in most solids are W = 4 -5 Oe. B, several orders of magnitude higher than the characteristic thermal energy of the carriers, which is of the order of magnitude k. T, equal at room temperature to 0.026 e. C. According to the laws of quantum mechanics, the energy of electrons in such a well is quantized, i.e., it can take only some discrete values ​​En, where n can take integer values ​​1, 2, 3, …. These discrete energy values ​​are called size quantization levels.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization For a free particle with an effective mass m*, whose motion in the crystal in the direction of the z axis is limited by impenetrable barriers (i.e., barriers with infinite potential energy), the energy of the ground state increases compared to the state without limitation This increase in energy is called the size quantization energy of the particle. Quantization energy is a consequence of the uncertainty principle in quantum mechanics. If the particle is limited in space along the z-axis within the distance a, the uncertainty of the z-component of its momentum increases by an amount of the order of ħ/a. Correspondingly, the kinetic energy of the particle increases by the value E 1. Therefore, the considered effect is often called the quantum size effect.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization The conclusion about the quantization of the energy of electronic motion refers only to motion across the potential well (along the z axis). The well potential does not affect the motion in the xy plane (parallel to the film boundaries). In this plane, the carriers move as free and are characterized, as in a bulk sample, by a continuous energy spectrum quadratic in momentum with an effective mass. The total energy of carriers in a quantum-well film has a mixed discretely continuous spectrum

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization In addition to increasing the minimum energy of a particle, the quantum-size effect also leads to quantization of the energies of its excited states. Energy spectrum of a quantum-dimensional film - the momentum of charge carriers in the plane of the film

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization Let the electrons in the system have energies less than E 2 and therefore belong to the lower level of size quantization. Then no elastic process (for example, scattering by impurities or acoustic phonons), as well as scattering of electrons by each other, can change the quantum number n by transferring the electron to a higher level, since this would require additional energy costs. This means that during elastic scattering electrons can only change their momentum in the plane of the film, i.e., they behave like purely two-dimensional particles. Therefore, quantum-dimensional structures in which only one quantum level is filled are often called two-dimensional electronic structures.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization There are other possible quantum structures where the movement of carriers is limited not in one, but in two directions, as in a microscopic wire or filament (quantum filaments or wires). In this case, the carriers can move freely only in one direction, along the thread (let's call it the x-axis). In the cross section (the yz plane), the energy is quantized and takes on discrete values ​​Emn (like any two-dimensional motion, it is described by two quantum numbers, m and n). The full spectrum is also discrete-continuous, but with only one continuous degree of freedom:

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Quantization principle It is also possible to create quantum structures resembling artificial atoms, where the movement of carriers is limited in all three directions (quantum dots). In quantum dots, the energy spectrum no longer contains a continuous component, i.e., it does not consist of subbands, but is purely discrete. As in the atom, it is described by three discrete quantum numbers (not counting the spin) and can be written as E = Elmn , and, as in the atom, the energy levels can be degenerate and depend on only one or two numbers. A common feature of low-dimensional structures is the fact that if the motion of carriers along at least one direction is limited to a very small region comparable in size to the de Broglie wavelength of the carriers, their energy spectrum changes noticeably and becomes partially or completely discrete.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Definitions Quantum dots - quantum dots - structures whose dimensions in all three directions are several interatomic distances (zero-dimensional structures). Quantum wires (threads) - quantum wires - structures, in which the dimensions in two directions are equal to several interatomic distances, and in the third - to a macroscopic value (one-dimensional structures). Quantum wells - quantum wells - structures whose size in one direction is several interatomic distances (two-dimensional structures).

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Minimum and maximum sizes The lower limit of size quantization is determined by the critical size Dmin, at which at least one electronic level exists in a quantum-size structure. Dmin depends on the conduction band break DEc in the corresponding heterojunction used to obtain quantum size structures. In a quantum well, at least one electronic level exists if DEc exceeds the value h - Planck's constant, me* - the effective mass of an electron, DE 1 QW - the first level in a rectangular quantum well with infinite walls.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Minimum and maximum dimensions If the distance between energy levels becomes comparable to thermal energy k. BT , then the population of high levels increases. For a quantum dot, the condition under which the population of higher levels can be neglected is written as E 1 QD, E 2 QD are the energies of the first and second size quantization levels, respectively. This means that the benefits of size quantization can be fully realized if This condition sets upper limits for size quantization. For Ga. As-Alx. Ga 1-x. As this value is 12 nm.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Along with its energy spectrum, an important characteristic of any electronic system is the density of states g(E) (the number of states per unit energy interval E). For three-dimensional crystals, the density of states is determined using the Born-Karman cyclic boundary conditions, from which it follows that the components of the electron wave vector do not change continuously, but take a number of discrete values, here ni = 0, ± 1, ± 2, ± 3, and are the dimensions crystal (in the form of a cube with side L). The volume of k-space per one quantum state is equal to (2)3/V, where V = L 3 is the volume of the crystal.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Thus, the number of electronic states per volume element dk = dkxdkydkz, calculated per unit volume, will be equal here, the factor 2 takes into account two possible spin orientations. The number of states per unit volume in the reciprocal space, i.e., the density of states) does not depend on the wave vector In other words, in the reciprocal space the allowed states are distributed with a constant density.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures It is practically impossible to calculate the function of the density of states with respect to energy in the general case, since isoenergetic surfaces can have a rather complex shape. In the simplest case of an isotropic parabolic dispersion law, which is valid for the edges of energy bands, one can find the number of quantum states per volume of a spherical layer enclosed between two close isoenergetic surfaces corresponding to energies E and E+d. E.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures The volume of a spherical layer in k-space. dk is the layer thickness. This volume will account for d. N states Taking into account the relationship between E and k according to the parabolic law, we obtain From here the density of states in energy will be equal to m * - the effective mass of the electron

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Thus, in three-dimensional crystals with a parabolic energy spectrum, as the energy increases, the density of allowed energy levels (density of states) will increase in proportion to the density of levels in the conduction band and in the valence band. The area of ​​the shaded regions is proportional to the number of levels in the energy interval d. E

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us calculate the density of states for a two-dimensional system. The total energy of carriers for an isotropic parabolic dispersion law in a quantum-well film, as shown above, has a mixed discretely continuous spectrum. In a two-dimensional system, the states of a conduction electron are determined by three numbers (n, kx, ky). The energy spectrum is divided into separate two-dimensional En subbands corresponding to fixed values ​​of n.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Curves of constant energy represent circles in reciprocal space. Each discrete quantum number n corresponds to the absolute value of the z-component of the wave vector. Therefore, the volume in the reciprocal space, bounded by a closed surface of a given energy E in the case of a two-dimensional system, is divided into a number of sections.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us determine the energy dependence of the density of states for a two-dimensional system. To do this, for a given n, we find the area S of the ring bounded by two isoenergetic surfaces corresponding to the energies E and E+d. E: Here The value of the two-dimensional wave vector corresponding to the given n and E; dkr is the width of the ring. Since one state in the plane (kxky) corresponds to the area where L 2 is the area of ​​a two-dimensional film of thickness a, the number of electronic states in the ring, calculated per unit volume of the crystal, will be equal, taking into account the electron spin

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Since here is the energy corresponding to the bottom of the n-th subband. Thus, the density of states in a two-dimensional film is where Q(Y) is the unit Heaviside function, Q(Y) =1 for Y≥ 0, and Q(Y) =0 for Y

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures The density of states in a two-dimensional film can also be represented as an integer part equal to the number of subbands whose bottom is below the energy E. Thus, for two-dimensional films with a parabolic dispersion law, the density of states in any subband is constant and does not depend on energy. Each subband makes the same contribution to the total density of states. For a fixed film thickness, the density of states changes abruptly when it does not change by unity.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures. Dependence of the density of states of a two-dimensional film on energy (a) and thickness a (b).

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures In the case of an arbitrary dispersion law or with another type of potential well, the dependences of the density of state on energy and film thickness may differ from those given above, but the main feature, a nonmonotonic course, will remain.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us calculate the density of states for a one-dimensional structure - a quantum wire. The isotropic parabolic dispersion law in this case can be written as x is directed along the quantum filament, d is the thickness of the quantum filament along the y and z axes, kx is a one-dimensional wave vector. m, n are positive integers characterizing where the axis is quantum subbands. The energy spectrum of a quantum wire is thus divided into separate overlapping one-dimensional subbands (parabolas). The motion of electrons along the x axis turns out to be free (but with an effective mass), while the motion along the other two axes is limited.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Energy spectrum of electrons for a quantum wire

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire versus energy Number of quantum states per interval dkx , calculated per unit volume where is the energy corresponding to the bottom of the subband with given n and m.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire as a function of energy Thus Hence In deriving this formula, the spin degeneracy of states and the fact that one interval d. E corresponds to two intervals ±dkx of each subband, for which (E-En, m) > 0. The energy E is counted from the bottom of the conduction band of the bulk sample.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire on energy Dependence of the density of states of a quantum wire on energy. The numbers next to the curves show the quantum numbers n and m. The degeneracy factors of the subband levels are given in parentheses.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire as a function of energy Within a single subband, the density of states decreases with increasing energy. The total density of states is a superposition of identical decreasing functions (corresponding to individual subbands) shifted along the energy axis. For E = Em, n, the density of states is equal to infinity. The subbands with quantum numbers n m turn out to be doubly degenerate (only for Ly = Lz d).

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot as a function of energy With a three-dimensional limitation of particle motion, we arrive at the problem of finding allowed states in a quantum dot or zero-dimensional system. Using the effective mass approximation and the parabolic dispersion law, for the edge of an isotropic energy band, the spectrum of allowed states of a quantum dot with the same dimensions d along all three coordinate axes will have the form n, m, l = 1, 2, 3 ... - positive numbers numbering the subbands. The energy spectrum of a quantum dot is a set of discrete allowed states corresponding to fixed n, m, l.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot as a function of energy The degeneracy of the levels is primarily determined by the symmetry of the problem. g is the level degeneracy factor

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot versus energy Degeneracy of levels is primarily determined by the symmetry of the problem. For example, for the considered case of a quantum dot with the same dimensions in all three dimensions, the levels will be three times degenerate if two quantum numbers are equal to each other and not equal to the third, and six times degenerate if all quantum numbers are not equal to each other. A specific type of potential can also lead to an additional, so-called random degeneracy. For example, for the considered quantum dot, to a threefold degeneracy of the levels E(5, 1, 1); E(1, 5, 1); E(1, 1, 5), associated with the symmetry of the problem, a random degeneration E(3, 3, 3) is added (n 2+m 2+l 2=27 in both the first and second cases), associated with the form limiting potential (infinite rectangular potential well).

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot versus energy Distribution of the number of allowed states N in the conduction band for a quantum dot with the same dimensions in all three dimensions. The numbers represent quantum numbers; the level degeneracy factors are given in parentheses.

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures Three-dimensional electron systems The properties of equilibrium electrons in semiconductors depend on the Fermi distribution function, which determines the probability that an electron will be in a quantum state with energy E EF is the Fermi level or electrochemical potential, T is the absolute temperature , k is the Boltzmann constant. The calculation of various statistical quantities is greatly simplified if the Fermi level lies in the energy band gap and is far from the bottom of the conduction band Ec (Ec – EF) > k. T. Then, in the Fermi-Dirac distribution, the unit in the denominator can be neglected and it passes into the Maxwell-Boltzmann distribution of classical statistics. This is the case of a non-degenerate semiconductor

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures Three-dimensional electron systems The distribution function of the density of states in the conduction band g(E), the Fermi-Dirac function for three temperatures, and the Maxwell-Boltzmann function for a three-dimensional electron gas. At T = 0, the Fermi-Dirac function has the form of a discontinuous function. For Е EF the function is equal to zero and the corresponding quantum states are completely free. For T > 0, the Fermi function. The Dirac smears in the vicinity of the Fermi energy, where it rapidly changes from 1 to 0 and this smearing is proportional to k. T, i.e., the more, the higher the temperature. (Fig. 1. 4. Edges)

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of Carriers in Low-Dimensional Structures Three-Dimensional Electronic Systems The electron density in the conduction band is found by summing over all states Note that we should take the energy of the upper edge of the conduction band as the upper limit in this integral. But since the Fermi-Dirac function for energies E >EF decreases exponentially rapidly with increasing energy, replacing the upper limit with infinity does not change the value of the integral. Substituting the values ​​of the functions into the integral, we obtain the -effective density of states in the conduction band

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Two-dimensional electron systems Let us determine the charge carrier concentration in a two-dimensional electron gas. Since the density of states of a two-dimensional electron gas We obtain Here also the upper limit of integration is taken equal to infinity, taking into account the sharp dependence of the Fermi-Dirac distribution function on energy. Integrating where

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures Two-dimensional electron systems For a nondegenerate electron gas, when In the case of ultrathin films, when only the filling of the lower subband can be taken into account For a strong degeneracy of the electron gas, when where n 0 is an integer part

ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures It should be noted that in quantum-well systems, due to the lower density of states, the condition of complete degeneracy does not require extremely high concentrations or low temperatures and is quite often implemented in experiments. For example, in n-Ga. As at N 2 D = 1012 cm-2, degeneracy will already take place at room temperature. In quantum wires, the integral for calculation, in contrast to the two-dimensional and three-dimensional cases, is not calculated analytically by arbitrary degeneration, and simple formulas can be written only in limiting cases. In a non-degenerate one-dimensional electron gas in the case of hyperthin filaments, when only the occupation of the lowest level with energy E 11 can be taken into account, the electron concentration is where the one-dimensional effective density of states is

Energy levels (atomic, molecular, nuclear)

1. Characteristics of the state of a quantum system
2. Energy levels of atoms
3. Energy levels of molecules
4. Energy levels of nuclei

Characteristics of the state of a quantum system

At the heart of the explanation of St. in atoms, molecules and atomic nuclei, i.e. phenomena occurring in volume elements with linear scales of 10 -6 -10 -13 cm lies quantum mechanics. According to quantum mechanics, any quantum system (ie, a system of microparticles, which obeys quantum laws) is characterized by a certain set of states. In general, this set of states can be either discrete (discrete spectrum of states) or continuous (continuous spectrum of states). Characteristics of the state of an isolated system yavl. the internal energy of the system (everywhere below, just energy), the total angular momentum (MKD) and parity.

System energy.
A quantum system, being in different states, generally speaking, has different energies. The energy of the bound system can take any value. This set of possible energy values ​​is called. discrete energy spectrum, and energy is said to be quantized. An example would be energy. spectrum of an atom (see below). An unbound system of interacting particles has a continuous energy spectrum, and the energy can take arbitrary values. An example of such a system is free electron (E) in the Coulomb field of the atomic nucleus. The continuous energy spectrum can be represented as a set of an infinitely large number of discrete states, between which the energy. gaps are infinitely small.

The state, to-rum corresponds to the lowest energy possible for a given system, called. basic: all other states are called. excited. It is often convenient to use a conditional scale of energy, in which the energy is basic. state is considered the starting point, i.e. is assumed to be zero (in this conditional scale, everywhere below the energy is denoted by the letter E). If the system is in the state n(and the index n=1 is assigned to main. state), has energy E n, then the system is said to be at the energy level E n. Number n, numbering U.e., called. quantum number. In the general case, each U.e. can be characterized not by one quantum number, but by their combination; then the index n means the totality of these quantum numbers.

If the states n 1, n 2, n 3,..., nk corresponds to the same energy, i.e. one U.e., then this level is called degenerate, and the number k- multiplicity of degeneration.

During any transformations of a closed system (as well as a system in a constant external field), its total energy, energy, remains unchanged. Therefore, energy refers to the so-called. conserved values. The law of conservation of energy follows from the homogeneity of time.


Total angular momentum.
This value is yavl. vector and is obtained by adding the MCD of all particles in the system. Each particle has both its own MCD - spin, and orbital momentum, due to the motion of the particle relative to the common center of mass of the system. The quantization of the MCD leads to the fact that its abs. magnitude J takes strictly defined values: , where j- quantum number, which can take on non-negative integer and half-integer values ​​(the quantum number of an orbital MCD is always an integer). The projection of the MKD on the c.-l. axis name magn. quantum number and can take 2j+1 values: m j =j, j-1,...,-j. If k.-l. moment J yavl. the sum of two other moments , then, according to the rules for adding moments in quantum mechanics, the quantum number j can take the following values: j=|j 1 -j 2 |, |j 1 -j 2 -1|, ...., |j 1 +j 2 -1|, j 1 +j 2 , a . Similarly, the summation of a larger number of moments is performed. It is customary for brevity to talk about the MCD system j, implying the moment, abs. the value of which is ; about magn. The quantum number is simply spoken of as the projection of the momentum.

During various transformations of a system in a centrally symmetric field, the total MCD is conserved, i.e., like energy, it is a conserved quantity. The MKD conservation law follows from the isotropy of space. In an axially symmetric field, only the projection of the full MCD onto the axis of symmetry is preserved.

State parity.
In quantum mechanics, the states of a system are described by the so-called. wave functions. Parity characterizes the change in the wave function of the system during the operation of spatial inversion, i.e. change of signs of the coordinates of all particles. In such an operation, the energy does not change, while the wave function can either remain unchanged (even state) or change its sign to the opposite (odd state). Parity P takes two values, respectively. If nuclear or el.-magnets operate in the system. forces, parity is preserved in atomic, molecular and nuclear transformations, i.e. this quantity also applies to conserved quantities. Parity conservation law yavl. a consequence of the symmetry of space with respect to mirror reflections and is violated in those processes in which weak interactions are involved.

Quantum transitions
- transitions of the system from one quantum state to another. Such transitions can lead both to a change in energy. the state of the system, and to its qualities. changes. These are bound-bound, freely-bound, free-free transitions (see Interaction of radiation with matter), for example, excitation, deactivation, ionization, dissociation, recombination. It is also a chem. and nuclear reactions. Transitions can occur under the influence of radiation - radiative (or radiative) transitions, or when a given system collides with a c.-l. other system or particle - non-radiative transitions. An important characteristic of the quantum transition yavl. its probability in units. time, indicating how often this transition will occur. This value is measured in s -1 . Radiation probabilities. transitions between levels m And n (m>n) with the emission or absorption of a photon, the energy of which is equal to, are determined by the coefficient. Einstein A mn , B mn And B nm. Level transition m to the level n may occur spontaneously. Probability of emitting a photon Bmn in this case equals Amn. Type transitions under the action of radiation (induced transitions) are characterized by the probabilities of photon emission and photon absorption , where is the energy density of radiation with frequency .

The possibility of implementing a quantum transition from a given R.e. on k.-l. another w.e. means that the characteristic cf. time , during which the system can be at this UE, of course. It is defined as the reciprocal of the total decay probability of a given level, i.e. the sum of the probabilities of all possible transitions from the considered level to all others. For the radiation transitions, the total probability is , and . The finiteness of time , according to the uncertainty relation , means that the level energy cannot be determined absolutely exactly, i.e. U.e. has a certain width. Therefore, the emission or absorption of photons during a quantum transition does not occur at a strictly defined frequency , but within a certain frequency interval lying in the vicinity of the value . The intensity distribution within this interval is given by the spectral line profile , which determines the probability that the frequency of a photon emitted or absorbed in a given transition is equal to:
(1)
where is the half-width of the line profile. If the broadening of W.e. and spectral lines is caused only by spontaneous transitions, then such a broadening is called. natural. If collisions of the system with other particles play a certain role in the broadening, then the broadening has a combined character and the quantity must be replaced by the sum , where is calculated similarly to , but the radiat. transition probabilities should be replaced by collision probabilities.

Transitions in quantum systems obey certain selection rules, i.e. rules that establish how the quantum numbers characterizing the state of the system (MKD, parity, etc.) can change during the transition. The most simple selection rules are formulated for radiats. transitions. In this case, they are determined by the properties of the initial and final states, as well as the quantum characteristics of the emitted or absorbed photon, in particular its MCD and parity. The so-called. electric dipole transitions. These transitions are carried out between levels of opposite parity, the complete MCD to-rykh differ by an amount (the transition is impossible). In the framework of the current terminology, these transitions are called. permitted. All other types of transitions (magnetic dipole, electric quadrupole, etc.) are called. prohibited. The meaning of this term is only that their probabilities turn out to be much less than the probabilities of electric dipole transitions. However, they are not yavl. absolutely prohibited.

In the first and second parts of the textbook, it was assumed that the particles that make up macroscopic systems obey the laws of classical mechanics. However, it turned out that in order to explain many properties of micro-objects, instead of classical mechanics, we must use quantum mechanics. The properties of particles (electrons, photons, etc.) in quantum mechanics are qualitatively different from the usual classical properties of particles. The quantum properties of micro-objects that make up a certain physical system are also manifested in the properties of a macroscopic system.

As such quantum systems, we will consider electrons in a metal, a photon gas, etc. In what follows, by the word quantum system or particle we will understand a certain material object described by the apparatus of quantum mechanics.

Quantum mechanics describes the properties and features inherent in the parts of the microworld, which we often cannot explain on the basis of classical concepts. Such features include, for example, the wave-particle duality of micro-objects in quantum mechanics, discovered and confirmed by numerous experimental facts, the discreteness of various physical parameters, "spin" properties, etc.

The special properties of micro-objects do not allow one to describe their behavior by the usual methods of classical mechanics. For example, the presence of a microparticle that manifests itself at the same time and wave and corpuscular properties

does not allow simultaneously accurately measuring all the parameters that determine the state of the particle from the classical point of view.

This fact is reflected in the so-called uncertainty relation, discovered in 1925 by Heisenberg, which consists in the fact that inaccuracies in determining the position and momentum of a microparticle are related by the relation:

A consequence of this relationship is a number of other relationships between different parameters and in particular:

where is the uncertainty in the value of the energy of the system and the uncertainty in time.

Both of the above relations show that if one of the quantities is determined with high accuracy, then the second value is determined with low accuracy. The inaccuracies here are determined through Planck's constant, which practically does not limit the accuracy of measurements of various quantities for macroscopic objects. But for microparticles with low energies, small sizes and momenta, the accuracy of the simultaneous measurement of the noted parameters is no longer sufficient.

Thus, the state of a microparticle in quantum mechanics cannot be simultaneously described using coordinates and momenta, as is done in classical mechanics (Hamilton's canonical equations). In the same way, it is impossible to speak about the value of the energy of a particle at a given moment. States with a certain energy can only be obtained in stationary cases, i.e., they are not determined exactly in time.

Possessing corpuscular-wave properties, any microparticle does not have an absolutely precisely defined coordinate, but turns out to be, as it were, “smeared” over space. In the presence of a certain region of space of two or more particles, we cannot distinguish them from each other, since we cannot trace the movement of each of them. This implies the fundamental indistinguishability or identity of particles in quantum mechanics.

Further, it turns out that the quantities characterizing some parameters of microparticles can only change in certain portions, quanta, from which the name of quantum mechanics comes from. This discreteness of many parameters that determine the states of microparticles cannot be described in classical physics either.

According to quantum mechanics, in addition to the energy of the system, discrete values ​​can take the angular momentum of the system or spin, magnetic moment, and their projections to any preferred direction. So, the square of the angular momentum can only take the following values:

Spin can only take values

where can be

The projection of the magnetic moment onto the direction of the external field can take the values

where the Bohr magneton and the magnetic quantum number, which takes the value:

In order to describe these features of physical quantities mathematically, each physical quantity had to be associated with a certain operator. In quantum mechanics, therefore, physical quantities are represented by operators, and their values ​​are defined as averages over the eigenvalues ​​of the operators.

When describing the properties of micro-objects, it was necessary, in addition to the properties and parameters encountered in the classical description of microparticles, to introduce new, purely quantum parameters and properties. These include the “spin” of a particle, which characterizes the intrinsic moment of momentum, the “exchange interaction”, the Pauli principle, etc.

These features of microparticles do not allow describing them using classical mechanics. As a result, micro-objects are described by quantum mechanics, which takes into account the noted features and properties of microparticles.

A.G. Akmanov, B.G. Shakirov

Fundamentals of quantum and optoelectronic devices

UDC 621.378.1+621.383.4

Reviewers

Department of "Telecommunication Systems" USATU

Malikov R.F., Doctor of Physical and Mathematical Sciences,

BSPU professor

Minutes No. 24 dated 24.06.2003 plenum of the UMO Council for Education in

the field of telecommunications.

Akmanov A.G., Shakirov B.G.

A40 Fundamentals of quantum and optoelectronic devices. Tutorial.

Ufa: RIO BashGU, 2003. - 129 p.

This work is a textbook for the disciplines "Optoelectronic and quantum devices and devices", "Quantum radiophysics" in the specialties "Physics and technology of optical communication" and "Radiophysics and electronics".

The physical foundations, principle of operation, and characteristics of solid-state, gas, and semiconductor lasers, and questions of controlling their parameters are considered. The physical bases and characteristics of elements of optoelectronic devices are stated.

UDC 621.378.1 + 621.383.4

Lakmanov A.G., Shakirov B.G., 2003

ã BashSU, 2003

INTRODUCTION

Quantum electronics as a field of science and technology is understood as a science that studies the theory and method of generation and amplification of electromagnetic waves by induced radiation in thermodynamically non-equilibrium quantum systems (atoms, molecules, ions), the properties of generators and amplifiers obtained in this way and their applications.

The basis of quantum electronics is formed by the physical provisions formulated back in 1916 by A. Einstein, who theoretically predicted the existence of induced radiation and pointed out its special property - coherence to driving radiation.

The possibility of creating quantum devices was substantiated in the early 1950s. In 1954, microwave molecular quantum generators (or masers1) were developed at the Physical Institute of the Academy of Sciences of the USSR (A. M. Prokhorov, N. G. Basov) and at Columbia University (Ch. Towns). The next step, natural for the development of quantum electronics, was taken in the direction of creating quantum devices in the optical range. The theoretical substantiation of this possibility (Ch. Townes, A. Shavlov, 1958), the proposal of an open resonator as an oscillatory system in the optical range (AM Prokhorov, 1958) stimulated experimental research. In 1960, a ruby ​​laser 1 was created (Meiman T., USA), in 1961 - a laser based on a mixture of helium with neon (Javan A., USA), and in 1962 - the first semiconductor lasers (USA, USSR ).

Optoelectronics (OE) is a field of science and technology related to the development and application of electro-optical devices and systems for transmitting, receiving, processing, storing and displaying information.

Depending on the nature of the optical signal, coherent and incoherent optoelectronics are distinguished. Coherent OE is based on the use of laser radiation sources. Incoherent OE include discrete and matrix incoherent emitters and indicator devices built on their basis, as well as photodetectors, optocouplers, optocoupler integrated circuits, etc.

Laser radiation has the following properties:

1. Temporal and spatial coherence. The coherence time can be up to 10 -3 s, which corresponds to a coherence length of the order of 10 5 m (l coh =c coh), i.e. seven orders of magnitude higher than for conventional light sources.

2. Strict monochromaticity (<10 -11 м).

3. High energy flux density.

4. Very small angular discrepancy in the medium.

The efficiency of lasers varies widely - from 0.01% (for a helium-neon laser) to 75% (for a semiconductor laser), although for most lasers the efficiency is 0.1-1%.

The unusual properties of laser radiation are now widely used. The use of lasers for processing, cutting and microwelding of hard materials is more economically advantageous. Lasers are used for high-speed and accurate detection of defects in products, for the most delicate operations (for example, a CO 2 laser beam as a bloodless surgical knife), for studying the mechanism of chemical reactions and influencing their course, for obtaining ultrapure substances. One of the important applications of lasers is the production and study of high-temperature plasma. This area of ​​their application is associated with the development of a new direction - laser controlled thermonuclear fusion. Lasers are widely used in measuring technology. Laser interferometers are used for ultra-precise remote measurements of linear displacements, refractive indices of a medium, pressure, and temperature.

Laser radiation sources are widely used in communication technology.

PHYSICAL FOUNDATIONS OF LASERS

Amplification of a light wave in lasers is based on the phenomenon of induced emission of a photon by an excited particle of a substance (atom, molecule). In order for stimulated emission to play the main role, it is necessary to transfer the working substance (amplifying medium) from an equilibrium state to a nonequilibrium state, in which an inversion of the populations of the energy levels is created.

The so-called open resonator, which is a system of two highly reflective mirrors, is used as an oscillatory system in lasers. When a working substance is placed between them, a condition is created for repeated passage of the amplified radiation through the active medium, and thus a positive feedback is realized.

The process of excitation of an active medium in order to create a population inversion in it is called pumping, and the physical system that provides this process is called a pumping system.

Thus, in the structural scheme of any type of laser, three main elements can be distinguished: the active medium, the pumping system, and the open resonator.

In accordance with this, Chapter I sets out the fundamentals of the theory of quantum amplification and generation in the interaction of light radiation with matter, pumping methods, and the theory of an open resonator.

optical radiation

Optical radiation or light is called electromagnetic waves, the wavelengths of which are in the range from a few nanometers to hundreds of micrometers. In addition to the visible radiation perceived by the human eye ( l\u003d 0.38-0.76 microns), distinguish ultraviolet ( l=0.01-0.38 µm) and infrared ( l=0.78-100 µm) radiation.

Let us recall some provisions and formulas of wave and quantum optics. Wave optics is based on the equations of classical electrodynamics, which is based on Maxwell's equations:

[ E]=rot E=

[ H]=rot H= (1.1) where E, D, H, B are the intensity and induction vectors of the electric and magnetic fields, respectively (system (1.1) is written for the case of the absence of currents and charges in the medium). In a homogeneous isotropic medium D And B associated with fields E And H ratios (in the SI system):

D=ε 0 e E, B=μ 0 m h,(1.2) where e is the relative dielectric, m- relative magnetic permeability of the medium, e 0– electric, m0 are the magnetic constants. System (1.1) reduces to the wave equation for (or ): (1.3) Equation (1.3) has a solution , (1.4) which describes a plane wave propagating in the direction determined by the wave vector with phase velocity:

(1.5)

Where c= is the speed of light in vacuum. For non-magnetic environment m=1, n= and for the wave speed we get: (1.5а)

The volumetric energy density carried by an electromagnetic wave is given by: r=(1/2)ε 0 eE2+ (1/2)μ 0 mH2= ε 0 eE2. (1.6)

Spectral volume energy density rn is determined by the ratio: (1.7)

Module of the Umov-Poynting vector (1.8)

determines the flux density of light energy, .

Light intensity is understood as the time-averaged energy flux (1.9)

The processes of absorption and emission of light can only be explained within the framework of quantum optics, which considers optical radiation in the form of a stream of elementary particles - photons that do not have a rest mass and electric charge, and have energy Ef =hn, momentum p= h k and moving at the speed of light.

Photon Flux Density F=I/(hn)=ru/(hn)(1.10)

Where [ hn]=J, [ F]=1/(m 2 s).

Energy states of a quantum system. Populations of quantum levels

The most important property of quantum systems (an ensemble of atoms, molecules) is that their internal energy can only take on discrete values E 1 ,E 2 ,..E n determined by solutions of the corresponding Schrödinger equations. The set of energy levels possible for a given quantum system is called the energy spectrum. In an energy level diagram, energy is expressed in joules, reciprocal centimeters, or electron volts. The state with the lowest energy, which is the most stable, is called the ground state. All other states, which correspond to a large energy, are called excited.

In general, one can imagine that several different excited states are characterized by the same value of internal energy. In this case, the states are said to be degenerate, and the degree of degeneracy (or the statistical weight of the level gi.) is equal to the number of states.

Consider a macrosystem consisting of N0 identical weakly interacting microsystems (atoms) with a certain spectrum of energy levels. Such a macrosystem is the laser active medium.

The number of atoms per unit volume that are at a given energy level i, is called the population of this level N i . The distribution of populations over levels under conditions of thermodynamic equilibrium obeys the Boltzmann statistics:

(1.11)

Where T is the absolute temperature, k is the Boltzmann constant, gi is the multiplicity of level degeneracy, , Where E i - energy i-th quantum level. From (1.11) it follows that , i.e. the sum of the populations of all energy levels is equal to the number of particles N0 in the ensemble under consideration.

In accordance with (1.11), in the ground state with energy E 1 at thermodynamic equilibrium, there is the largest number of atoms, and the populations of the upper levels decrease with increasing level energy (Fig. 1.1). The ratio of the populations of two levels in the equilibrium state is given by the formula: (1.12)

For simple non-degenerate levels g 1 \u003d g 2 \u003d 1 and formula (1.12) takes the form: (1.12a)

Instantaneous, level jumping E i to the level E j called a quantum transition. At E i >E j the quantum system gives off energy equal to ( E i -E j), and at E i <E j- absorbs it. A quantum transition with the emission or absorption of a photon is called optical. The energy of an emitted (absorbed) photon is determined by the Bohr relation:

hn ij = E i -E j (1.13)

1.3 Elementary interaction processes
optical radiation with matter

Let us consider in more detail the quantum transitions that can occur between two arbitrarily chosen energy levels, for example, 1 and 2 (Fig. 1.2), which correspond to the energy E 1 And E 2 and population N 1 And N 2.

N 2
a B C)
N 2
N 2
E 2
E 2
E 2

Rice. 1.2 . Quantum transitions in a two-level system.

There are three types of optical transitions: spontaneous,forced with takeover And forced with radiation.

Let us introduce quantitative characteristics for these probabilistic processes, as was first done by A. Einstein.

Spontaneous transitions

If an atom (or molecule) is in state 2 at time t=0, then there is a finite probability that it will go to state 1, while emitting a quantum of light (photon) with energy hn 21 \u003d (E 2 -E 1)(Fig. 1.2a). This process, which occurs without interaction with the radiation field, is called spontaneous transition, and the corresponding radiation is spontaneous emission. The probability of spontaneous transitions is proportional to time, i.e. (dw 21) cn \u003d A 21 dt, (1.14)

Where A 21 -Einstein coefficient for spontaneous emission and determines the transition probability per unit time, =1/c.

Let's assume that at the time t the population of level 2 is N 2. The rate of transition of these atoms to the lower level due to spontaneous emission is proportional to the transition probability A 21 and the population of the level from which the transition occurs, i.e.

(dN 2 /dt) cn \u003d -A 21 N 2.(1.15)

It follows from quantum mechanics that spontaneous transitions occur from a given state only to states that are lower in energy, i.e. there are no spontaneous transitions from state 1 to state 2.

Forced transitions

Let us consider the interaction of a group of identical atoms with a radiation field whose energy density is uniformly distributed over frequencies near the transition frequency. When an atom is exposed to electromagnetic radiation of resonant frequency ( n \u003d ν 21 \u003d (E 2 -E 1) / h) there is a finite probability that the atom will pass from state 1 to the upper level 2, absorbing an electromagnetic field quantum (photon) with energy hn(Fig. 1.2b).

Energy difference (E 2 -E 1) necessary for the atom to make such a transition is taken from the energy of the incident wave. This is the process takeovers, which can be described using the rate equation (dN 1 /dt) n \u003d W 12 N 1 \u003d r n B 12 N 1,(1.16)

Where N 1 is the level 1 population, W 12 \u003d r v B 12 is the absorption probability per unit time, r v - spectral volume energy density of the incident radiation, AT 12 is the Einstein coefficient for absorption.

Another expression for the probability is also used W 12 as:

W 12 \u003d s 12 F,(1.17)

Where F is the incident photon flux density, s 12- a quantity called absorption cross section, = m 2.

Let us now assume that the atom is initially at the upper level 2 and a wave with a frequency n=n 21. Then there is a finite probability that this wave initiates the transition of an atom from level 2 to level 1. In this case, the energy difference (E 2 -E 1) will be released in the form of an electromagnetic wave, which will be added to the energy of the incident wave. This is the phenomenon stimulated (induced) radiation.

The process of stimulated emission can be described using the rate equation: (dN 2 /dt) vyn \u003d W 21 N 2 \u003d r n B 21 N 2,(1.18)

Where N 2 is the level 2 population, W 21 \u003d r v B 21 is the probability of a forced transition per unit of time, B21-Einstein coefficient for forced transition. And in this case, the following relation holds for the transition probability: W 21 \u003d s 21 F,(1.19)

Where s 21 is the stimulated emission cross section for the 2→1 transition.

There is a fundamental difference between the processes of spontaneous and stimulated emission. The probabilities of induced transitions are proportional to the spectral volume density of the electromagnetic field, while spontaneous ones do not depend on the external field. In the case of spontaneous emission, an atom emits an electromagnetic wave, the phase of which has no definite relation to the phase of the wave emitted by another atom. Moreover, the emitted wave can have any direction of propagation.

In the case of stimulated emission, since the process is initiated by an incident wave, the radiation of any atom is added to this wave in the same phase. The incident wave also determines the polarization and propagation direction of the emitted wave. Thus, as the number of forced transitions increases, the intensity of the wave increases, while its frequency, phase, polarization, and direction of propagation remain unchanged. In other words, in the process of forced transitions from the state E 2 into a state E 1 going on coherent amplification of electromagnetic radiation at frequency n 21 \u003d (E 2 -E 1) / h. Of course, in this case, reverse transitions also occur. E 1 ®E 2 with absorption of electromagnetic radiation.

Spontaneous emission

Integrating expression (1.15) over time with the initial condition N 2 (t=0)=N 20 we get: N 2 (t) \u003d N 20 exp (-A 21 t).(1.20)

The spontaneous emission power is found by multiplying the photon energy hv 21 on the number of spontaneous transitions per unit of time:

P cn \u003d hν 21 A 21 N 2 (t) V \u003d P cn 0 exp (-A 21 t)(1.21)

Where P cn 0 \u003d hn 21 A 21 N 20 V, V - the volume of the active medium.

We introduce the concept about the average lifetime of atoms in an excited state relative to spontaneous transitions. In the two-level system under consideration, atoms that leave the excited state 2 in the time from t before t+Dt, obviously, were in this state for a period of time t. The number of such atoms is N 2 A 21 Dt. Then their average life span in an excited state is determined by the relation:

Let's represent the formula (1.22) in the form:

(1.21 a)

the value t cn can be found experimentally, since it appears as a parameter in the decay law of spontaneous luminescence, defined by formula (1.21 a).


Similar information.


The atomic nucleus, like other objects of the microworld, is a quantum system. This means that the theoretical description of its characteristics requires the involvement of quantum theory. In quantum theory, the description of the states of physical systems is based on wave functions, or probability amplitudesψ(α,t). The square of the modulus of this function determines the probability density of detecting the system under study in a state with characteristic α – ρ (α,t) = |ψ(α,t)| 2. The argument of the wave function can be, for example, the coordinates of the particle.
The total probability is usually normalized to one:

Each physical quantity is associated with a linear Hermitian operator acting in the Hilbert space of wave functions ψ . The spectrum of values ​​that a physical quantity can take is determined by the spectrum of eigenvalues ​​of its operator.
The average value of the physical quantity in the state ψ is

() * = <ψ ||ψ > * = <ψ | + |ψ > = <ψ ||ψ > = .

The states of the nucleus as a quantum system, i.e. functions ψ(t) , obey the Schrödinger equation ("u. Sh.")

(2.4)

The operator is the Hermitian Hamilton operator ( Hamiltonian) systems. Together with the initial condition on ψ(t), equation (2.4) determines the state of the system at any time. If it does not depend on time, then the total energy of the system is the integral of motion. The states in which the total energy of the system has a certain value are called stationary. Stationary states are described by the eigenfunctions of the operator (Hamiltonian):

ψ(α,t) = Eψ(α,t);

ψ (α ) = Eψ( α ). 		(2.5)

The last of the equations - stationary Schrödinger equation, which determines, in particular, the set (spectrum) of energies of the stationary system.
In the stationary states of a quantum system, in addition to energy, other physical quantities can also be conserved. The condition for the conservation of the physical quantity F is the equality 0 of the commutator of its operator with the Hamilton operator:

[,] ≡ – = 0. (2.6)

1. Spectra of atomic nuclei

The quantum nature of atomic nuclei is manifested in the patterns of their excitation spectra (see, for example, Fig. 2.1). Spectrum in the region of excitation energies of the 12 C nucleus below (approximately) 16 MeV It has discrete character. Above this energy the spectrum is continuous. The discrete nature of the excitation spectrum does not mean that the level widths in this spectrum are equal to 0. Since each of the excited levels of the spectrum has a finite average lifetime τ, the level width Г is also finite and is related to the average lifetime by a relation that is a consequence of the uncertainty relation for energy and time ∆t ∆E ≥ ћ :

The diagrams of the spectra of nuclei indicate the energies of the levels of the nucleus in MeV or keV, as well as the spin and parity of the states. The diagrams also indicate, if possible, the isospin of the state (since the diagrams of the spectra give level excitation energy, the energy of the ground state is taken as the origin). In the region of excitation energies E< E отд - т.е. при энергиях, меньших, чем энергия отделения нуклона, спектры ядер - discrete. It means that the width of the spectral levels is less than the distance between the levels G< Δ E.