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§9, problem Statement. Used models


Last years

Polar phonon modes widely obsuzh
Were gave in the literature as the important channel of a dissipation of energy both in volume, and in various sorts CP and structural structures: ([Z і], [39-4 і], [52j [53], [86-96]-theory, [97-101], [ill] - experiment).bolshaja the part of such examinations was spent within the limits of the theory of the dielectric continuum, well applicable for the description of the long-wave lattice oscillations [102]. Along with existence of volume polar optical phonons existence of the interface (superficial) phonons and their important role in an electron-PHONON vzaimo-activity in the difficult systems [102-104] is well an established fact.
At the same time, in a number of the operations devoted to dispersion of the free carriers of charges on polar optical phonons in structures with KJA and infrared uptake of light with the assistance of polar optical phonons in similar systems, the renormalization of an electronic spectrum leading to occurrence of razmerno-quantized electronic subbands whereas for phonons it was used, in essence, volume-like aproksimatsija was considered only.
So, in operations ([52, 5з] - a dispersion problem, [40,41,9б] — a problem of uptake of light) for the description of an electronic spectrum is-used model KJA with infinite barriers that allows to consider the basic features related with kvazidvumerizatsiej of a motion of electrons in such systems, but by viewing of the lattice oscillations and an interaction electron-hole following models are applied:
Model I. It is supposed, that the phonon spectrum is, on-sou - shchestvu, a volume spectrum. EFV also it is featured by a Hamiltonian of type Pekara-Freliha for an unlimited crystal, and the behaviour of electrons is that, that the law of conservation of momentum for an electron in a hole, so-called approach of maintenance of an impulse is carried out. In these frameworks viewing of process of dispersion in Ridley's operation [52] is spent.
This model also has been used in [95]. For the description of uptake of light with participation polar optical, and also deformation (ultrasonic and optical) phonons. Within the limits of the specified approaches it is possible to gain analytical effects, however in the field of small thickness use in model I PSI gives
To incorrect effect.
Model 2. Same Hamiltonian EFV is used, but matrix devices EFV are calculated precisely, without application PSI. With reference to a problem of dispersion it has been made in operation Riddocha and Ridley [bz]. We will score also, that „the incorrectness of an ultraquantum limit (problem restriction nizhajshej razmerno - a quantized subband) even for films with small enough thickness (-10.0) in the same place has been illustrated. The analysis of this model with reference to process of uptake of light ^ і] вщіво, one ^ to, importance of the account of exact view EFV in field a quantum - of a phonon resonance.
Let's score, that with reference to a dispersion problem were, attempts to consider and a renormalization of a spectrum of optical phonons. In particular, in Rudin's operation and Rejnike [z і] comparison of effects for a scattering factor of electrons on optical phonons is spent in
Frameworks of several macroscopical models an electron-phonon of the interaction, differing different boundary conditions with the effects corresponding to microscopic model.
The Hamiltonian used Rudinom and Rejnike for the description of the interface (superficial) modes, is a Hamiltonian of the dielectric medium and thus undertakes as additional to the hectares-miltonianu, based on microscopic model? Ю9]. This approach leans on pure quality standards vzaimoodnoznachnogo conformity kontinualnoj and dispersion models? і02] and consequently cannot apply for the quantitative description of a problem. Besides, a Hamiltonian used in [z і] for the description of the interface modes (a Hamiltonian of operation C^oJ) п0 ~ luchen without the exact solution of a problem on partitioning of phonon modes (see the head І). All it leads to essential distinctions in dependences of the superficial contributions on a thickness of a hole (d) in comparison with corresponding effects of strict viewing on Hamiltonian EFV Cl7-I9] which are gained in the present operation. Besides it, it appears impossible to ooze unequivocally influence of correct boundary conditions on behaviour of contributions depending on

For konfajnment phonons. We will score takzhegchto at the solution of a problem on dispersion of carriers in a polar stratum authors
Were restricted to viewing of transitions with the account only two nizhajshih subbands (type 1 І, 2 І). It does not allow to gain the correct limit (gained in operations С52, 5з]) for a scattering factor at the energies considerably exceeding a threshold of issue.
We suggest to use for the analysis of the specified problems model for EFV and phonons which has already been applied by us to the analysis of not kinetic processes in two previous heads. We will call it further Model 3 to have possibility
More operative comparison to the effects which are available in the literature (Model I and 2).
Model 3. Hamiltonian EFV for plates, receptions by the strict solution of a problem on a vibrational spectrum of systems of viewed type [l7-I ^.matrichnye ale-cops EFV is used are calculated precisely outside the limits of PSI.
For the description of an electronic spectrum in §10 and §12 we will be, as well as in the previous heads, to use model of a rectangular hole with the perpetual walls. We will score, that with reference to the given problem it allows to verify in addition the validity is-polzuemoj models EFV. Really, dispersion on phonons, no less than light uptake in process with participation of phonons should disappear in process of disappearance of a radiant of polarisation fields - a polar material; that is valid for any type of phonons (both superficial, and volume). By viewing of structural semiconductor system (not polar-polar-not polar semi-conductors) with electron localisation on the average a stratum (§10 and §12) we, thus, at any model of an electronic subsystem of dolzh th to observe disappearance of the processes related to a relaxation avr - gii on phonons at

If model EFV is correct.
However, at use for an electron folnovoj the function supposing possibility of a delocalization of an electron from centre, the incorrectness of description EFV can appear disguised.
We also guess valid approach of effective masses for carriers in a hole, the elementary model for zonnoj structures (the dimensional quantization on an axis

The parabolic law of a variance for a motion in a plane) and performance of statistics Bolshchana for electrons.


Let's feature now short Hamiltonians used in a problem.
The problem geometry coincides with viewed in the head
The Hamiltonian of a problem without EFV and interactions with light looks like:



Where all labels coincide with entered in chapter 2. The electron from a plate conduction band interreacts with two atop-nostnymi

And dilatational konfajnment

opticheskimi'sh - damiv Hamiltonian EFV looks like:
Let's score, that the used Hamiltonians EFV learnt. In Sb] on the basis of offered in С8} partitioning plans kole bate l - nyh modes, except obvious partitioning on konfajnment and superficial (even Si and odd S ^) components, have a number of properties important for the further analysis:
First, in difference from a Hamiltonian neperenormirovanno - go a volume spectrum, in them there is the correct dimensional dependence which is easily giving in physical intertsretatsii;
Secondly, constants EFV and frequencies of superficial (in difference from volume) modes contain the material parametres of mediums gra-nichashchih with a stratum of localisation of electrons that provides possibility of guidance with dispersion and uptake sveta.putem izme-nenija inductivities of the next mediums;
Thirdly, apparently from formulas (2.3в) takes place thickness - naja and dispersion dependence of frequencies of the superficial oscillations that can serve as additional difference of processes EFV with participation of the specified type of phonons.
- For radiation, including and spontaneous radiation.


Let's score, that further we guess a vector of polarisation of a light wave

Perpendicular axes
§10о Dispersion of electrons in structures with the quantum
Holes
10.I. A scattering factor in structures with KJA. T.o. The electron wave function in a quantum hole looks like, already given by us before (formulas 0.25, 0.30).
Scattering factor of an electron with a wave vector

On optical phonons in the first order of a perturbation theory it is spotted under the formula
At last, we will write down Hamiltonians of interaction of an electron with light



Here

Wave vector of an absent-minded electron (on which integration is spent).

- Energy optical - a phonon

The upper sign in
Expression (JUL) corresponds to issue, inferior - absorptions fo-nona. The matrix device in (JUL) can be presented in a view:
\, Т7П r, V I OF 77 % Л* 1,77'I 4*1, AND
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A source: Kalinovsky Vladislav Vjacheslavovich. DISPLAY the INTERACTION ELECTRON-PHONON In ELECTRONIC and eksitonnyh STATES, UPTAKE of LIGHT And DISPERSION of the FREE CARRIERS In POLAR STRUCTURAL STRUCTURES. 1992

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