Shear Acoustic Phonons in Multilayer Arsenide Semiconductor Nanostructures

Using the elastic continuum model, exact analytical solutions for the equations of motion for the elastic medium of a multilayer resonant tunneling nanosystem describing the shear modes of acoustic phonons are obtained. The expressions describing the components of the stress tensor arising in the studied nanostructure and boundary conditions for the components of the elastic displacement vector and the components of the stress tensor are obtained. Using the obtained equations of motion for the elastic medium and boundary conditions, the theory of the spectrum and phonon modes for shear acoustic phonons is developed in the proposed work for a plane arsenide semiconductor nanostructure. It is shown that the spectrum of the displaced acoustic phonons of the studied nanosystem is obtained from the dispersion equation following from the boundary conditions using transfer-matrix method. Using the orthonormality condition, the normalized modes of shear acoustic phonons are obtained. For the parameters of the three-barrier nanostructure – the active zone of a quantum cascade detector – the calculation of the spectrum of acoustic phonons and its dependencies on the wave vector and the geometric parameters of the nanostructure has been performed. It is shown that the calculated dependences of the spectrum of acoustic phonons on the wave vector form three groups with boundary values equal to the corresponding energies of acoustic phonons in massive crystals. Also it is obtained that an increase in the thickness of the internal barrier at constant other geometrical parameters of the nanosystem leads to a steady decrease in the values of the phonon energy levels energies. The proposed theory can be used to study the scattering of electron fluxes on acoustic phonons in multilayer resonant-tunneling structures.


INTRODUCTION
The rapid development of nanoscience as a part of solid state physics determines the current relevance and wide range of nanotechnology applications [1][2][3]. In particular, in the physics of semiconductor nanosystems, the creation of quantum cascade lasers (QCL) [4,5] and detectors (QCD) [6,7] operating in the terahertz range of electromagnetic waves is significant. To expand the capabilities of the above-mentioned nanodevices, it is important to take into account the effects of various kinds arising in multilayer semiconductor resonant tunneling structures (RTS), which are their functional elements.
Despite the fact that the effects of the interaction of electrons tunneled through a RTS with optical phonons, constant electric and high-frequency electromagnetic fields, have been sufficiently clarified and analyzed [8][9][10][11], studies of acoustic phonons and their influence on electron tunneling transport are practically absent. Attention should be paid only to a few papers, calculations of the acoustic phonons spectrum in which were performed for single quantum wells [12] and nanosystems of cylindrical and spherical symmetry [13,14]. It should also be noted that the classification of acoustic phonons arising in a quantum well placed in a massive semiconductor medium and calculations of their spectra was investigated in [15]. However, for RTS, used in QCL and QCD due to the mutual consistency of their cascades, this theory is not applicable, primarily as it is necessary to use other boundary conditions for the displacement vector and the components of the stress tensor. Based on the model of an elastic continuum, the theory of the spectrum and acoustic modes of shear acoustic phonons arising in two-well-RTS with GaAsquantum wells and AlAs -quantum barriers was developed in the proposed work. Direct calculations of the spectrum of acoustic phonons were performed for the three-barrier RTS as the active band of QCD. It is shown that the developed theory can be the basis for studying the interaction of electrons with a shear acoustic phonon in multilayer RTS.

EQUATIONS FOR SHEAR ACOUSTIC PHONON MODES IN NANOSTRUCTURE. DISPERSION EQUATIONS FOR THE DETERMINATION OF THE ACOUSTIC PHONON SPECTRUM
We will explore the shear acoustic phonons arising in the RTS, consisting of two In1-xGaxAs potential quantum wells and internal In1-xAlxAs potential barrier. The geometrical scheme of the nanosystem is presented in Fig. 1.
The 0x3 axis is directed perpendicular to the interfaces of the nanosystem media. In view of this and the designations in Fig. 1, the density of the nanostructure material medium can be represented as  Using the elastic continuum model, the equation of motion for the elastic displacement vector in the isotropic case looks like: is the stress tensor. Equation (2) can be reduced to the form: ,  (4) are the elastic constants for the corresponding p-th RTS layer.
In the case of shear acoustic phonons where the vector 3 () ux has two non-zero components: Vectors 13 33 ( ), ( ) xx u x u x can be represented in the form: where for the components () x ux of shear acoustic phonons, the following conditions are satisfied: Solutions of equation (3) taking into account (5), (6) will be found in the form: Then equation (3) within the p-th RTS layer splits into two equations: is the propagation velocity of longitudinal and transverse waves, respectively. Solutions of equations (8) are as follows: ;.
then from (11) and (12), respectively, we get: Accounting of (7) gives: 01019-3 Now the solutions of equations (8) within the RTS can be represented as: In expressions (15) it is taken into account that from a physical point of view deformation can not grow infinitely when 3 x   , which requires enforcement of condition: Condition (16) is ensured by equating to zero the coefficients in expressions (15) at ,; x xx    are used, that is: where in ratios (17) where, taking into account (15), (17), the matrix and using the phonon amplitude normalization condition [16]

DISCUSSION OF THE RESULTS
Using the theory developed above, we carried out calculations of the shear acoustic phonons spectrum on the wave vector q and the geometric parameters of the studied RTS with GaAs -quantum wells and AlAsquantum barriers. The geometric parameters of the three-barrier RTS were chosen as follows: quantum well widths 24 2 nm dd  , potential barrier widths should be noted that the dependences for the I and III groups of the spectrum have a similar behavior, consisting in the approximation of the curves for the adjacent levels () nq q  and 1 () nq q   , however, for the III group, the adjacent level except for the approximation, as can be seen from Fig. 2 tend to merge with increasing q. By their physical nature, phonons of the I group belong to the transverse displacements of the medium and of the III group -to the longitudinal displacements.  As can be seen from Fig. 4, an increase in the thickness of the internal barrier at constant other geometrical parameters of the nanosystem leads to a steady decrease the values of the phonon energy levels energies. In this case, there is a gradual convergence of the first and second phonon energy levels in each group.

CONCLUSIONS
Using the model of an elastic continuum, the theory of the energy spectrum and phonon modes of shear acoustic phonons arising in a two-dimensional plane nanostructure is developed. It is shown how the acoustic modes can be normalized for the studied type of acoustic phonons. Using the developed theory on the example of a double-well nanosystem with GaAsquantum wells and AlAs -quantum barriers, we cal-01019-6 culated the spectrum of shear acoustic phonons. The properties of the shear acoustic phonons spectrum arising in the nanostructure on its geometrical parameters are established. The developed theory can be used as a basis for further investigation of the interaction processes of electrons with shear acoustic phonons in multilayer arsenide semiconductor nanosystems.