quantum system

To explain many properties of microparticles (photons, electrons, etc.), special laws and approaches of quantum mechanics are required. The quantum properties of the microcosm are manifested through the properties of macrosystems. Micro-objects make up a certain physical system, which is called quantum. Examples of quantum systems are: photon gas, electrons in metals. Under terms quantum system, quantum particle one should understand a material object, which is described using a special apparatus of quantum mechanics.

Quantum mechanics explores the properties and phenomena of the world of microparticles that cannot be interpreted by classical mechanics. Such features, for example, are: wave-particle duality, discreteness, the existence of spins. The methods of classical mechanics cannot describe the behavior of the particles of the microworld. The simultaneously wave and corpuscular properties of a microparticle make it impossible to determine the state of the particle from the classical point of view.

This fact is reflected in the Heisenberg uncertainty relation ($1925$):

where $\triangle x$ is the inaccuracy in determining the coordinate, $\triangle p$ is the error in determining the momentum of the microparticle. This ratio can be written as:

where $\triangle E$ is the energy uncertainty, $\triangle t$ is the time uncertainty. Relations (1) and (2) indicate that if one of the quantities in these relations is determined with high accuracy, then the other parameter has a large error in the determination. In these ratios $\hbar =1.05\cdot (10)^(-34)J\cdot s$. Thus, the state of a microparticle in quantum mechanics cannot be described using coordinates and momentum at the same time, which is possible in classical mechanics. A similar situation applies to energy at a given time. States with a specific energy value can only be obtained in stationary cases (that is, in cases that do not have an exact definition in time).

Having corpuscular and at the same time wave properties, a microparticle does not have an exact coordinate, but is "smeared" in a certain region of space. If there are two or more particles in a certain region of space, it is not possible to distinguish them from each other, since it is impossible to track the movement of each. From the foregoing follows the identity of particles in quantum mechanics.

Some parameters related to microparticles take discrete values, which cannot be explained by classical mechanics. In accordance with the provisions and laws of quantum mechanics, in addition to the energy of the system, the angular momentum of the system can be discrete:

where $l=0,1,2,\dots $

spin can take the following values:

where $s=0,\ \frac(1)(2),\ 1,\ \frac(3)(2),\dots $

Projection magnetic moment on the direction of the external field takes the following values:

where $m_z$ is a magnetic quantum number that takes the values: $2s+1: s, s-1,...0,...,-(s-1), -s.$

$(\mu )_B$ is the Bohr magneton.

For the purpose of a mathematical description of the quantum features of physical quantities, each quantity is assigned an operator. So, in quantum mechanics, physical quantities are represented by operators, while their values ​​are determined by the averages over the eigenvalues ​​of the operators.

The state of the quantum system

Any state in a quantum system is described by a wave function. However given function predicts the parameters of the future state of the system with a certain degree of probability, and not reliably, this is a fundamental difference from classical mechanics. Thus, for the parameters of the system, the wave function determines the probabilistic values. Such uncertainty, inaccuracy of predictions most of all caused controversy among scientists.

Measured parameters of a quantum system

The most global differences between classical and quantum mechanics lie in the role of measuring the parameters of the quantum system under study. The problem of measurements in quantum mechanics is that when trying to measure the parameters of a microsystem, the researcher acts on the system with a macro device, which changes the state of the quantum system itself. So, when trying to accurately measure the parameter of a micro-object (coordinate, momentum, energy), we are faced with the fact that the measurement process itself changes the parameters that we are trying to measure, and significantly. It is impossible to make precise measurements in the microcosm. There will always be errors in accordance with the uncertainty principle.

In quantum mechanics, dynamic variables represent operators, so it makes no sense to talk about numerical values, since the operator determines the action on the state vector. The result is represented, also by a Hilbert space vector, not by a number.

Remark 1

Only if the state vector is an eigenvector of a dynamic variable operator, then its action on the vector can be reduced to multiplication by a number without changing the state. In such a case, a dynamic variable operator can be mapped to a single number that is equal to the operator's eigenvalue. In this case, we can assume that the dynamic variable has a certain numerical value. Then the dynamic variable has a quantitative value independent of the measurement.

In the event that the state vector is not an eigenvector of the operator of a dynamic variable, then the result of the measurement does not become unambiguous and one speaks only of the probability of one or another value obtained in the measurement.

The results of the theory, which are empirically verifiable, are the probabilities of obtaining a dynamic variable in a dimension with a large number of dimensions for the same state vector.

The main characteristic of a quantum system is the wave function, which was introduced by M. Born. physical meaning most often, it is determined not for the wave function itself, but for the square of its modulus, which determines the probability that the quantum system is at a given point in space at a given point in time. The basis of the microworld is probability. In addition to knowing the wave function, describing a quantum system requires information about other parameters, for example, about the parameters of the field with which the system interacts.

The processes that take place in the microcosm lie beyond the limits of human sensory perception. Consequently, the concepts and phenomena that quantum mechanics uses are devoid of visualization.

Example 1

Exercise: What is the minimum error with which one can determine the speed of an electron and a proton if the coordinates of the particles are known with an uncertainty of $1$ µm.

Solution:

As a basis for solving the problem, we use the Heisenberg uncertainty relation in the form:

\[\triangle p_x\triangle x\ge \hbar \left(1.1\right),\]

where $\triangle x$ is the uncertainty of the coordinate, $\triangle p_x$ is the uncertainty of the projection of the particle's momentum onto the X axis. The magnitude of the momentum uncertainty can be expressed as:

\[\triangle p_x=m\triangle v_x\left(1.2\right).\]

Substitute right side expression (1.2) instead of the uncertainty of the momentum projection in expression (1.1), we have:

From formula (1.3) we express the required velocity uncertainty:

\[\triangle v_x\ge \frac(\hbar )(m\triangle x)\left(1.4\right).\]

It follows from inequality (1.4) that the minimum error in determining the particle velocity is:

\[\triangle v_x=\frac(\hbar )(m\triangle x).\]

Knowing the mass of the electron $m_e=9,1\cdot (10)^(-31)kg,$ we will carry out the calculations:

\[\triangle v_(ex)=\frac(1,05\cdot (10)^(-34))(9,1\cdot (10)^(-31)\cdot (10)^(-6) )=1,1\cdot (10)^2(\frac(m)(c)).\]

the proton mass is equal to $m_p=1.67\cdot (10)^(-27)kg$, we calculate the error in measuring the proton velocity under given conditions:

\[\triangle v_(px)=\frac(1.05\cdot (10)^(-34))(1.67\cdot (10)^(-27)\cdot (10)^(-6) )=0.628\cdot (10)^(-1)(\frac(m)(s)).\]

Answer:$\triangle v_(ex)=1.1\cdot (10)^2\frac(m)(s),$ $\triangle v_(px)=0.628\cdot (10)^(-1)\frac( m)(s).$

Example 2

Exercise: What is the minimum error in measuring the kinetic energy of an electron if it is in a region whose size is l.

Solution:

As a basis for solving the problem, we use the Heisenberg uncertainty relation in the form:

\[\triangle p_xl\ge \hbar \to \triangle p_x\ge \frac(\hbar )(l)\left(2.1\right).\]

From inequality (2.1) it follows that the minimum momentum error is equal to:

\[\triangle p_x=\frac(\hbar )(l)\left(2.2\right).\]

The kinetic energy error can be expressed as:

\[\triangle E_k=\frac((\left(\triangle p_x\right))^2)(2m)=\frac((\left(\hbar \right))^2)((\left(l\ right))^22\cdot m_e).\]

Answer:$\triangle E_k=\frac((\left(\hbar \right))^2)((\left(l\right))^22\cdot m_e).$

Kabardin O.F. Nuclear spectra // Kvant. - 1987. - No. 3. - S. 42-43.

By special agreement with the editorial board and the editors of the journal "Kvant"

As you know, atomic nuclei consist of nucleons - protons and neutrons, between which nuclear forces of attraction and Coulomb repulsive forces act. What can happen to the nucleus when it collides with another nucleus, particle or gamma-ray? The experiments of E. Rutherford, performed in 1919, showed, for example, that under the influence of an alpha particle a proton can be knocked out of the nucleus. In experiments conducted by D. Chadwick in 1932, it was found that alpha particles can also knock out neutrons from atomic nuclei (“Physics 10”, § 106). But does the collision process always end like this? Can't an atomic nucleus absorb the energy received in a collision and redistribute it between its constituent nucleons, thereby changing its internal energy? What will happen to such a core next?

Answers to these questions were given by direct experiments on the study of the interaction of protons with atomic nuclei. Their results are very similar to the results of the experiments of Frank and Hertz on the study of collisions of electrons with atoms ("Physics 10", § 96). It turns out that with a gradual increase in the energy of protons, only elastic collisions with atomic nuclei are observed at first, kinetic energy is not converted into other types of energy, but only redistributed between the proton and the atomic nucleus as one particle. However, starting from a certain value of the proton energy, inelastic collisions can also occur, in which the proton is absorbed by the nucleus and completely transfers its energy to it. The nucleus of each isotope is characterized by a strictly defined set of “portions” of energy that it can accept.

Transformation of the nitrogen nucleus with the capture of an alpha particle and the emission of a proton.

These experiments prove that nuclei have discrete spectra of possible energy states. Thus, the quantization of energy and a number of other parameters is a property not only of atoms, but also of atomic nuclei. State atomic nucleus with a minimum amount of energy is called the ground, or normal, states with excess energy (compared to the ground state) are called excited.

Atoms are usually in excited states for about 10 -8 seconds, and excited atomic nuclei get rid of excess energy in a much shorter time - about 10 -15 - 10 -16 seconds. Like atoms, excited nuclei are released from excess energy by emitting quanta of electromagnetic radiation. These quanta are called gamma quanta (or gamma rays). A discrete set of energy states of the atomic nucleus corresponds to a discrete spectrum of frequencies emitted by them gamma rays. Gamma rays are transverse electromagnetic waves, the same as radio waves, visible light or x-rays. They are the shortest-wavelength type of electromagnetic radiation known, and their corresponding wavelengths range from approximately 10 -11 m to 10 -13 m.

The energy states of atomic nuclei and the transitions of nuclei from one state to another with the absorption or emission of energy are usually described using energy diagrams similar to the energy diagrams of atoms (“Physics 10”, § 94). The figure shows the energy diagram of the nucleus of the iron isotope - \(~^(58)_(26)Fe\), obtained on the basis of proton bombardment experiments. Note that while the energy diagrams of atoms and nuclei are qualitatively similar, there are significant quantitative differences between them. If an energy of several electron volts is required to transfer an atom from the ground state to an excited state, then energy of the order of hundreds of thousands or millions of electron volts is required to excite the atomic nucleus. This difference is due to the fact that the nuclear forces acting between the nucleons in the nucleus largely exceed the forces of the Coulomb interaction of electrons with the nucleus.

Energy level diagram of an iron isotope nucleus.

The ability of atomic nuclei to spontaneously transition from states with a large supply of energy to a state with less energy explains the origin of not only gamma radiation, but also the radioactive decay of nuclei.

Many patterns in nuclear spectra can be explained using the so-called shell model of the structure of the atomic nucleus. According to this model, nucleons in the nucleus are not mixed in disorder, but, like electrons in an atom, they are arranged in bound groups, filling the allowed nuclear shells. In this case, the proton and neutron shells are filled independently of each other. The maximum numbers of neutrons: 2, 8, 20, 28, 40, 50, 82, 126 and protons: 2, 8, 20, 28, 50, 82 in filled shells are called magic. Nuclei with magic numbers of protons and neutrons have many remarkable properties: an increased value of the specific binding energy, a lower probability of entering into a nuclear interaction, resistance to radioactive decay etc.

The transition of the nucleus from the ground state to the excited state and its return to the ground state, from the point of view of the shell model, is explained by the transition of the nucleon from one shell to another and back.

With a large number of advantages, the shell model of the nucleus is not able to explain the properties of all nuclei in various types interactions. In many cases, the concept of the nucleus as a drop of nuclear liquid, in which nucleons are bound by nuclear forces, Coulomb forces, and surface tension forces, turns out to be more fruitful. There are other models, but none of the proposed ones can still be considered universal.

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 the general case, 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 infinite a large number discrete states, between to-rymi energetic. 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. relies 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 moment, 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 more moments. 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 specular 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.

Quantum systems and their properties.

Probability distribution over energies in space.

Boson statistics. Fermi-Einstein distribution.

fermion statistics. Fermi-Dirac distribution.

Quantum systems and their properties

In classical statistics, it is assumed that the particles that make up the system obey the laws of classical mechanics. But for many phenomena, when describing micro-objects, it is necessary to use quantum mechanics. If a system consists of particles that obey quantum mechanics, then we will call it a quantum system.

The fundamental differences between a classical system and a quantum one include:

1) Corpuscular-wave dualism of microparticles.

2) Discreteness of physical quantities describing micro-objects.

3) Spin properties of microparticles.

The first implies the impossibility of accurately determining all the parameters of the system that determine its state from the classical point of view. This fact is reflected in the Heisandberg uncertainty relation:

In order to mathematically describe these features of microobjects in quantum physics, the quantity is assigned a linear Hermitian operator that acts on the wave function .

Eigenvalues operator determine the possible numerical values ​​of this physical quantity, the average over which coincides with the value of the quantity itself.

Since the momenta and coefficients of the microparticles of the system cannot be measured simultaneously, the wave function is presented either as a function of coordinates:

Or, as a function of impulses:

The square of the modulus of the wave function determines the probability of detecting a microparticle per unit volume:

Wave function describing specific system, is found as an eigenfunction of the Hamelton operator:

Stationary Schrödinger equation.

Non-stationary Schrödinger equation.

The principle of indistinguishability of microparticles operates in the microworld.

If the wave function satisfies the Schrödinger equation, then the function also satisfies this equation. The state of the system will not change when 2 particles are swapped.

Let the first particle be in state a and the second particle be in state b.

The system state is described by:

If the particles are interchanged, then: since the movement of the particle should not affect the behavior of the system.

This equation has 2 solutions:

It turned out that the first function is realized for particles with integer spin, and the second for half-integer.

In the first case, 2 particles can be in the same state:

In the second case:

Particles of the first type are called spin integer bosons, particles of the second type are called femions (the Pauli principle is valid for them.)

Fermions: electrons, protons, neutrons...

Bosons: photons, deuterons...

Fermions and bosons obey non-classical statistics. To see the differences, let's count the number of possible states of a system consisting of two particles with the same energy over two cells in the phase space.

1) Classical particles are different. It is possible to trace each particle separately.

classical particles.

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 by several orders of magnitude thermal energy carriers. Typical work function values ​​in most solids have a value of W = 4 -5 e. 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 Quantization principle 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. common feature 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 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 increases high levels. 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 carrier energy 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 (kxky) plane 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 The distribution of quantum states in low-dimensional structures The density of states in a two-dimensional film can also be represented as - whole 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 come to 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 E 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 Electron 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 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 lower subband filling 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 degeneracy, and simple formulas can only be written in extreme cases. In a nondegenerate 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