Acoustic phonons in multilayer nitride-based AlN/GaN resonant tunneling structures

The study of physical processes associated with acoustic phonons in nitride-based nanosystems is of great importance for the effective operation of modern nanoscale devices. In this paper, a consistent theory of acoustic phonons arising in multilayer nitride-based semiconductor resonant tunneling structures, that can function as a separate cascade of a quantum cascade laser or detector is proposed. Using the physical and geometric parameters of a typical nanostructure, the spectrum of various types of acoustic phonons and the corresponding normalized components of the elastic displacement vector are calculated. It has been established that the spectrum of acoustic phonons of a multilayer nanostructure consists of two groups of the shear phonons dependencies and three groups of dependencies for a mixed spectrum of flexural and dilatational phonons. The dependencies of the acoustic phonons spectrum of the nanostructure and the components of the elastic displacement vector on its geometric parameters are studied. It has been established that for the components of the displacement vector u2 for shear phonons have a decrease in the absolute values of their maxima with increasing of energy level number. The components u1 and u3 of flexural and dilatational phonons behave respectively as symmetric and antisymmetric functions relatively the center of an separate selected layer of the nanostructure. The proposed theory can be further applied to study the interaction of electrons with acoustic phonons in multilayer resonant tunneling structures.


Introduction
Modern quantum cascade lasers (QCL) [1][2][3] and detectors (QCD) [4][5][6] of the near and mid infrared ranges of electromagnetic waves use plane multilayer nanostructures based on double GaN, AlN and triple AlGaN compounds of nitride semiconductor materials as their active elements. The characteristic feature of the mentioned materials is the possibility of nanodevices operation based on them in a wide temperature range and the generation of a constant electric field caused by the piezoelectric effect in the nanostructure layers. For the nanostructures based on such semiconductors, phonon processes, especially processes associated with acoustic phonons and electron scattering on them, are rather poorly studied. Acoustic phonons propagating in the direction perpendicular to the multilayer superlattice layers were widely studied in long-standing related papers [7][8][9]. Theoretical results on the study of the acoustic phonons spectra in single-well nitride semiconductor nanostructures and nanowires, which were obtained in [10][11][12][13][14][15][16], cannot be applied to multilayer resonant tunneling structures (RTS) for the following reason: single-well nanostructures in [10][11][12] were considered to be placed in the external environment of the sapphire substrate, which made it possible to take the components of the elastic displacement vector u and the components of the stress tensor σ ij , i, j=1, 2, 3 equal to zero at the external boundaries of the nanostructure. Taking into account the fact, that the multilayer QCL cascades and QCD should be consistent with each other to ensure coherent tunneling mode [17], the above conditions cannot be used for the boundaries of the RTS with the external environment, and at the boundaries of its internal layers.
For an adequate description of physical processes in resonant tunneling structures, they should be considered to be placed in an external semiconductor medium. So, the active use of nanodevices, such as QCL and QCD, causes considerable interest in the physical processes that occur in their precision elements, this direction is the study of acoustic phonons in particular. By virtue of the facts indicated by us, features of the functioning the nanodevice active bands and cascades determine the formulation of a different model for the study of acoustic phonons. The problem, which is associated with the application of nonzero boundary conditions on the nanostructures heteroboundaries, as far as it is known, is not still an unresolved problem.
In the proposed paper the components of the displacement field u=(u 1 , u 2 , u 3 ), spectrum, phonon modes for shear, flexural and dilatational phonons are obtained by solving the equation of motion for the elastic continuum, using the boundary conditions for the elastic displacement components u 1 , u 2 , u 3 and components of the stress tensor σ ij , i, j=1, 2, 3 on all heteroboundaries of the nanosystem. Using the developed theory for the geometric and physical parameters of a typical nitride-based RTS, which can function as an active zone of the near infrared QCD, the spectrum and components of the elastic displacement for acoustic phonons were studied. Their dependence on the geometric parameters of the investigated RTS is established.

Analytical solutions of equations for the elastic displacement of a nanosystem medium
In the statement of the problem we assume that the investigated RTS, which contains N semiconductor layers being in the external AlN semiconductor medium, is placed so, that the Oz axis is perpendicular to the interfaces of the layers of the nanosystem (figure 1). Taking into account the notation introduced in figure 1, the density ρ(z) and elastic coefficients C iklm (z) of the RTS layers are dependent on the coordinate z and can be presented correspondingly as follows: where θ(z) is a unit step function. Acoustic phonon modes arising in a multilayer AlN/GaN RTS are obtained by solving the equation of motion for the elastic displacement of the medium of the RTS layers, which, taking into account (1), (2), can be presented as:  For the case of the investigated RTS, we assume that the propagation of acoustic phonons occurs within the Ox axis. Since for an arbitrarily chosen RTS p-th layer, it is uniform in the plane Oxy, and the elastic displacement vector u i (r, t) is independent on coordinate y, this makes it possible to search for solutions of equation (    and, as it can be seen from the equations (9) and (11), they form a system relatively u z . Formally, following the terminology for the classification introduced in [7,8], the types of phonons arising in a separate p-th layer of the nanosystem are as follows. The solutions of equation (10) describe shear (SH) phonons. The solutions of the system of equations (9) and (11) determine the flexural (FL) and dilatational (DL) acoustic phonons, which are defined respectively as: , where the indices 'S' and 'A' denote the symmetric and antisymmetric functions of z correspondingly.
First lets find solutions for equation (10) that describe shear (SH) acoustic phonons. They look like: In relation (12), it is taken into account, that = = , since the displacements u 2 (q, ω , z) cannot be increased indefinitely in a semiconductor medium AlN to the left and right of the RTS, according to the condition: Solutions u 1 (z) and u 3 (z) of the system of equations (9) and (11) are obtained as follows. If the following notation is introduced, then the system of equations u 1 (z), u 3 (z) is equivalent to the following differential equation: (15) in following form

Searching for solutions of equation
we obtain an equation, that makes it possible to determine the eigenvalues λ and functions a a , From expression (17) the biquadratic equation for the eigenvalues is obtained: its solutions, taking into account the notation in the equations (9), (11): Therefore, the general solutions of system (9) where λ 1 , λ 2 -roots of the characteristic equation of system (9), (11). Besides, there should be , according to the condition similar to condition (13), i.e.:

Theory of the acoustic phonons spectrum in multilayered nanostructures
The relation between the coefficients A B , Using the transfer matrix method [15] for conditions (25), the coefficients in solutions (12) can be sequentially expressed for RTS layers from the left to right: Since the transfer matrix w From conditions (29), using the transfer matrix method [15], we obtain: The flexural and dilatational acoustic phonons spectrum dependencies W q FL DL , ( ) ( ) are determined from the dispersion equation: Further the following coefficients are introduced:

For each nanosystem layer the coefficients
) are likely to be found using the boundary conditions (25) for relations (12), (26). Thus, the coefficients for relations (23), (30), (31). The coefficients , , ) are expressed in terms of the coefficient A 1 0 ( ) , which is now found from the normalization conditions for flexural and dilatational acoustic phonons [19]: x y which allows to determine the displacement components u 1 (q, ω, z) and u 3 (q, ω, z) uniquely. Besides, in expressions (36), (37) l x , l y are the geometric dimensions of the studied nanosystem cross-sectional area by the xy plane.

Results and discussion
The spectrum of various types of acoustic phonons arising in the studied multilayered nanostructure and the corresponding normalized components of the elastic displacement were calculated using the theory developed above. The calculations were carried out on the example of a two-well AlN/GaN nanosystem -active band of quantum cascade detector. Geometric parameters used for calculation are as follows: the widths of the potential wells are = = d d 5 nm; 5 nm 1 2 , the width of the potential barrier is -b=3 nm. The values of the geometric parameters of the cross section of the nanosystem along the Ox and Oy axes were chosen such as in the paper .The values of the nanosystem physical parameters used in the calculations, taken from papers [21,22], are presented on table 1.
The value of the spectrum energy of acoustic phonons within the proposed theory is limited to the first Brillouin zone and, as it is known from the papers [23][24][25], reaches a maximum value of the order of 25meV. With this in mind and to compare the numerical results of our calculations with other papers [10][11][12] the maximum value of the energy range of acoustic phonons, the calculation of which is given below, it is advisable to choose a value of 20 meV.

Spectrum characteristics and displacement field components for shear acoustic phonons
In figures 2(a), (b) are shown the dependencies of the shear acoustic phonon spectrum energy levels W n SH (Figure 2(a)) on the wave vector q, as well as the dependences of this spectrum ( figure 2(b)) on the position of the internal potential barrier relatively the external boundaries of the nanosystem (0dd 1 +d 2 ), calculated at q=39/(d 1 +d 2 +b). As it is seen from figure 2(a) [10-12, 15, 16]. It can be seen from the figure, that the spectrum dependencies W q n SH ( ) are formed at energy values W = W q AlN SH ( )), forming dependency branches, which consist of two dependencies with close energies values. Besides, as it can be seen from the callout in figure 2(a), the distances between the initial energies for neighboring branches are almost equidistant. When q values increase, the values of the W q n SH ( ) energies grow quasi-quadratically at first, and then, they grow in such a way, that the curves W q n SH ( ) become actually parallel to curve W q GaN SH ( ). As it can be seen from figure 2(b), where the dependencies of the energies W n SH on the value of d are shown, with the increase of d, a certain number of maxima and minima are formed in them, associated with number of the level n. That is, as it can be seen from the figure, each level of dependence W d n SH ( ) is characterized by the presence of n−1 minima and n maxima. It is also worth noting the occurrence of an anticrossing effect between adjacent energy levels W d n SH ( ) and W + d n SH 1 ( ). In figure 3 the dependences of the displacement field components u 2 (q, ω , z) on the geometric dimensions of the nanostructure, calculated for all values of the shear phonon spectrum W n SH at q=39/(d 1 +d 2 +b) and normalized with the condition (36), are presented. As it can be seen from the above dependences, the maximum values of the function u 2 (q, ω , z) are formed for odd values of the energy level number n in the left potential well, and for even values of n: in the right potential well. Besides, as it can be seen from the figure, the absolute values of the maxima w u q z max , , 2 | ( )|for even values of n are much larger than the corresponding values for odd n. It should also be noted, that with an increase in the number n, the maxima for odd n decreases slowly, while, for example, for n=2 and n=10, the values w u q z max , , 2 | ( )|actually differ by two orders.  (Figure 4(a)) are presented, as well as the dependences of this spectrum ( figure 4(b)) on the position of the internal potential barrier relatively the external boundaries of the nanosystem (0dd 1 +d 2 ), calculated at a fixed value q=26/(d 1 +d 2 +b). Here, the value of the wave vector differs from its value used in the calculations for shear acoustic phonons. It means, that we want the number of given dependences u 1 (q, ω, z) , u 3 (q, ω, z) and u 2 (q, ω, z) to be approximately the same, and since the is inherently a mixed spectrum of dilatational and flexural acoustic phonons modes of a nanosystem, and as it can be seen from figure 4(a) [10,11,26]. This means, that the values of the calculated energies for spectrum of dilatational and flexural acoustic phonons are in the same range as in papers [10-12, 15, 16].
As it can be seen from figure 4(a), the dependence branches W q n FL DL ,    It should also be noted, that successive energy levels for odd and even values of n, starting from the values of wave vector q=2 nm −1 , become equidistant to each other. As it can be seen from figure 4( . The energy dependencies for the first group (I) are characterized by the presence of n maxima and n−1 minima for each number n of the energy level as it was established above for shear acoustic phonons.
In the second group (II) of dependencies the following features arise: the first energy level of this group is formed by dependencies, expressed only in four intervals of change of d, that is: T values. The next two dependencies of this group form six maximums and five minimums, the dependence for the last level of this group has seven maximums and six minimums.
As it can be seen from the figure 4(b), the energy dependences for the third group (III) are symmetric relatively the position of the internal potential barrier in the total potential well d. Besides, the last level of this