Structural Chemistry & Crystallography Communication

Keywords

Triple-axes X-ray diffractometry; Reciprocal space mapping; Elastic strain; Step-graded buffer; Spatial strain distribution; Work hardening

Introduction

Hetero structures applied to ultrahigh frequency (UHF) electronics devices (for example, high electron mobility transistors–HEMTs) are created, as a rule, on a single crystal substrate GaAs of (001) orientation and consist of a metamorphic (MM) buffer aiming to remove a mismatch between the substrate and device active layers including a quantum well (QW). The MM-buffer may have a different design, for example, it may be step-graded [1- 3] or linear graded with an increasing value of the lattice misfit [2,3]. Very often, the MM-buffer has such additional elements as a healing layer or an inverse step [4]. During the successive growth of MM-buffer layers, the strain relief occurs, which is accompanied by the generation of misfit dislocations and the propagation of threading dislocations into heterostructure top layers [5]. The MM-buffer should prevent the penetration of threading dislocations into the device active layers. The creation of MM-buffer is based on possibility of the system to form a dislocation free layer, which is, in its turn, a platform for following the healing layer or the inverse step. Тhe theoretical prediction of the formation of such a dislocation free layer was done in Ref. [6]. This model considered MM-buffer with a continuous increasing lattice misfit due to the increase of the concentration of an alloying element. When moving this system to the equilibrium, the formation of dislocation free layer at the top region of buffer gives an energetic gain for the system [3-6]. Details concerning the mechanism of strain relief are currently revealed for single heteroepitaxial layers. It was established that there are three stages of strain relief [7]. In the first one, the process of relaxation is slow because only the bending of the dislocations penetrating from the substrate to the epitaxial layer takes place. The second stage of relaxation is the fast stage due to the multiplication of misfit dislocations; this stage occurs if the thickness of layer is more than 100 nm. The third stage manifests an inhibition of dislocation multiplication and occurs for much thicker layers due to the work hardening. For the fast stage of strain relief, there are some numerical relations between a residual strain and a thickness of single heteroepitaxial layer. Thus, Dunstan with collaborators in the series of articles [8-10] gave some evidence concerning the existence of an inverse proportion between the residual compressive strain and the layer thickness of 100- 1000 nm. Such a relation between strains and thicknesses was explained in the framework of a “geometrical approach” taking into account the character of the spatial distribution of misfit dislocations in the region between the substrate and the growing epitaxial layer. By contrary, in Ref. [11] there was established that the residual compressive strain varies in an inverse proportion to the root square of layer thickness. It means that there is an energetic limit for an elastic strained thin film, which governs the strain relief in single layer hetero structures. Both these approaches may be described numerically and each of them is characterised by its own phenomenological constant. Such a description of strain relief in single layer hetero structures may be considered as a phenomenological approach. This article is aimed to reveal possibility of the application of the phenomenological approach to describe the strain relief in multilayer systems, particularly, in MM step-graded buffers, and to find some numerical criterions for the formation of dislocation free layer. We believe that the comparative analysis of the structural parameters of step-graded MM-buffers of different design is a key to solve this problem. Two step-graded MMbuffers of different design are the subjects of this investigation. The principle difference between two MM-buffers concluded in their final constructive elements: first of them was terminated by the healing layer and second one had in its structure the inverse step. Moreover, the buffers had different thicknesses of their steps. The epitaxial layers of both step-graded MM-buffers were based on In_xAl_1-xAs ternary solid solutions. X-ray reciprocal space mapping and transmission electron microscopy (TEM) were involved in this study. Below, the hetero structures with MMbuffers we will mark as metamorphic HEMTs–MHEMTs.

Experimental

Sample preparation

Molecular beam epitaxy (MBE) was employed to create two MHEMTs with the step-graded buffers of different design. A Riber 32 MBE system was used for the fabrication of MHEMTs. MHEMTs consisted of InAlAs/InGaAs/InAlAs active layers and six-layered MM-buffers. The layers covering up the upper steps of MM-buffers, the healing layer (MHEMT 1) or the inverse step (MHEMT 2), had close values of the molar fraction of In. In these constructive elements of MHEMTs the molar fraction of In, X_In was equal to 0.39 and 0.394, correspondingly. MBE was performed at a constant temperature of substrates. MHEMT 1 was grown on a standard semi-isolated (001) GaAs substrate, while MHEMT 2 was grown on the vicinal surface of GaAs substrate with a deviation angle of 2° from (001) plane. The step-to-step change in the molar fraction of In in In_x Al_1-xAs ternary solutions for the first five steps of MM-buffers was achieved in the process of noninterrupted epitaxial growth at a constant Al-source temperature. At this stage of heterostructure growth, the thickness of each layer was equal to 0.1 μm for MHEMT 1 and 0.2 μm for MHEMT 2. During growth, the temperature of substrates was equal to 380°C for MHEMT 1 and 400°C for MHEMT 2. The barriers layers of both MHEMTs were grown at the higher temperatures of the substrates: 480 and 500°C, correspondingly. The growth rate of epitaxial layers was equal to 0.5 μm/h. The layer by layer growth was interrupted for two minutes prior to the growth of the healing layer (MHEMT 1) or the inverse step (MHEMT 2). The growth was also interrupted for five minutes when transiting to the regime of high temperature growth. The composition of the epitaxial layers was controlled by regulating the temperature of In, Al, Ga, as and Si molecular sources based on the calibrated temperature dependence of molecular fluxes. The common feature of two MHEMTs was the equality of thicknesses of the final constructive elements: the healing layer (MHEMT 1) and the inverse step (MHEMT 2). The thicknesses of the barriers layers in both MHEMTs were also equal. The details concerning the process of epitaxial growth and the technological characteristics of the constructive elements of MHEMTs are presented in Table 1. X-ray diffraction measurements were performed with a Smart Lab 9 kW X-ray diffractometer in the three-axial configuration. The diffractometer operated in the step-by-step mode of X-ray recording using the Cu K_α(1) irradiation. A Ge single crystal with (002) orientation was employed as an analyzer crystal. Two reflections, 004 and 224 (at glancing exit geometry), were recorded in the regime of the so-called ω-20 scanning, which implies the use of Bragg – Brentano technique with the variation of sample position relatively the Bragg maximum of substrate. During recording, X-ray reflected radiation was detected along the scattering vector H_hkl: H₀₀₁ (for symmetric recording) and H₂₂₄ (for asymmetric recording). The initial position for 004 symmetric recording corresponded to the Bragg maximum of the substrate (ω=0), where ω is a deviation of the substrate from the Bragg position). Such a mode of recording allowed us to reveal minor X-ray maxima for all layers of MM-buffers. For asymmetric recording, the scanning along the scattering vector H₂₂₄ is achieved by the variation of hkl values that gives the decomposition of the scattering vector H₂₂₄ along the directions [001] and [110] of the reciprocal space. On the basis of the ω-20 scanning, it is possible to plot the so-called reciprocal space maps, which represent the positions of minor X-ray maxima in the reciprocal space. The use of the technique of reciprocal space mapping have some advantages in comparison with the conventional technique of rocking curves because the latter is invalid in the case of the spatial misorientation of epitaxial layers relative the substrate. The effect of spatial misorientation between an epitaxial layer grown on the vicinal surface of substrate and the corresponding substrate plane is a well known fact [12]. The angular parameters of spatial misorientation of epitaxial layers in heterosructures grown on the vicinal surface of (001) GaAs substrate were presented in Ref. [12-14]. The actual axes of reciprocal space for epitaxial films grown on (001) GaAs substrates are the axis [110] (axis Y) and the axis [001] (axis Z). The antinodes of iso-concentration contours revealed in the reciprocal space maps correspond to the interference maxima of X-ray radiation. The coordinates of the antinodes (X-ray maxima) are expressed by the vectors q₁₁₀ and q₀₀₁, which are deviations (along the corresponding reciprocal space axes) of minor X-ray maxima from the major X-ray maximum H⁰_wd corresponding to the substrate. The knowledge of q₁₁₀ and q₀₀₁ allows us to determine the vertical and lateral lattice parameters for all epitaxial layers.

Table 1: The main parameters of the growth of MHEMT epitaxial layers.

Characterization of MHEMTs microstructure

The structural investigation of MHEMTs was performed by TEM with a Jeol JEM-2100 operating at an accelerating voltage of 200 kV. Figure 1 shows cross-sectional bright-field electron microscopy images obtained for two investigated MHEMTs. It is seen that, despite the interruption of growth, layers 5, 6 and 7 in MHEMT 1 do not have interphase boundaries and we can consider them as a single phase. This combined layer of MHEMT 1 does not have threading dislocations and, consequently, the elastic strain developed in it should be sufficiently higher than that in the internal layers of MHEMT 1. Similar situation, the formation of dislocation free layer, is realized for layer 5 in MHEMT 2. The inverse step and the barrier layer of this MHEMT (layers 6 and 7) are the special elements of heterostructure design; similar layer 5 these layers do not have threading dislocations. The role of these constructive elements in the formation of strain fields in MHEMT 2 will be discussed below. It is important that, the electron microscopy image taken from MHEMT 2 demonstrates the inverse step and the barrier layer as a single structural layer.

Figure 1: The cross-sectional bright field images for (a) MHEMT 1 and (b) MHEMT 2.

Reciprocal space maps and their processing

Reciprocal space maps for MHEMTs under study plotted on the basis of 004 and 224 reflections are presented in Figures 2 (MHEMT 1) and 3 (MHEMT 2). The maps for MHEMT 1 manifest five minor X-ray maxima whose values of Q_z(in ascending order) correspond to layers 1, 2, 3, 4 and 5 (6, 7). Attention is drawn to the fact that layers 5, 6 and 7 are characterized by a single reflex that indicates the affinity of the structural parameters of these layers. The similar situation is realized for layers 6 (the inverse step) and 7 (the barrier layer) of MHEMT 2; these layers are also characterized by a single X-ray maximum. So, the reciprocal space maps are in agreement with the MHEMTs microstructure revealed by TEM. The arrangement of minor X-ray maxima in the reciprocal space maps for MHEMT 2 is more complicated than that for MHEMT 1 due to their partial overlap and absence of sharp X-ray maximum corresponding to layer 4. To find all positions of interference maxima in the map obtained for 004 reflection we use the procedure of the modelling of X-ray sans by a set of Gaussians [14]. Taking into account that for MHEMT 2 there is no a difference in microstructure between the inverse step and the barrier layer (Figure 1b), we considered the strong central X-ray peak at the reciprocal space maps for MHEMT 2 as the peak corresponding simultaneously to both constructive elements. This circumstance facilitates the processing of the map plotted on the basis of 004 reflection for MHEMT 2 (the expansion of X-ray scan along the axis Z on six Gaussians) and allows us to determine the values of q_Z⁰⁰⁴ and q_y⁰⁰⁴ for layers 3, 4, and 6(7). For all maps the coordinates of distinct defined X-ray maxima, the values of q_Y and q_Z were determined by searching for the maximal value of X-ray reflected radiation counts using a special option of the Origin 15 software. The q_Z⁰⁰⁴, q_y⁰⁰⁴, and q_y²²⁴ values (excepting the value of q_y²²⁴ for layer 4 in MHEMT 2) are listed in Table 2. The complete characterization of the structural state of layer 4 in MHEMT 2 was done using the specific features of the MHEMT 2 arrangement and will be described below.

Figure 2: The reciprocal space maps plotted for MHEMT 1 on the basis of (a) 004 reflection and (b) 224 reflection. The numerals indicate the number of layer responsible for the appearance of given X-ray maximum.

Figure 3: The reciprocal space maps plotted for MHEMT 2 on the basis of (a) 004 reflection and (b) 224 reflection. The numerals indicate the number of layer responsible for the appearance of given X-ray maximum.

Table 2: Reciprocal space vectors for the maps obtained on the basis of 004 and 224 reflections.

Strain fields in epitaxial layers of MHEMTs

The vectors q_Z⁰⁰⁴, q_y⁰⁰⁴, q_y²²⁴, and allow us to determine the so called “total” strains (relative to GaAs substrate) of MHEMTs epitaxial layers: [(a₁-a_s)]/a_a]_[001] (based on 004 reflection) and [(a₁-a_s)]/a_a]_[110] (based on 224 reflection), where a₁ and a_o are, respectively, the vertical and lateral lattice parameters of layer and aS is the substrate lattice parameters. In accordance with Ref. [15], for the epitaxial layers of MHEMTs created on GaAs substrates the “total” strains are determined by the following relationships.

(1)

(2)

where

(3)

Eq. (3) takes into account the effect of spatial misorientation of epitaxial layers relative to the substrate. The quantities [(a₁- a_s)]/a_a]_[001] and [(a₁-a_s)]/a_a]_[110] correspond to the total strains ε33 and ε₁₁, which are directed along the main crystallographic axes. (Note, when rotating a cubic lattice by an angle of 45o about the axis [001, the required operation for transition from the axis [110] to the axis [100], the elity between the quantities [(a₁- a_s)]/a_a] _[110] and ε₁₁ will be achieved, if the distortion tensor does not have off-diagonal components and the elastic strains e₁₁ and ε₂₂ are equal. Generally, this condition is accepted default). The knowledge of the values of E₃₃ and e₁₁ gives us possibility to calculate the lattice misfit E₀=(a_r-a_s)/a_s, where a_r is the lattice parameter of fully relaxed lattice, and the residual compressive elastic strain ε₁₁ on the basis of Hooke law using the linear theory of elasticity. The concept according to which the residual strain in epitaxial layers is elastic is conventional and generally accepted in calculations of the lattice misfit ε₀. According to this concept, the stress σ₃₃ in the [001] direction is assumed to be zero due to the plastic deformation near the interphase boundaries oriented perpendicular to the [001] axis. On the basis of the equality of lateral stresses σ₁₁ and σ₂₂, it follows from the Hook law for the crystals of cubic system, elastic strains e₁₁ and e₃₃ are subjected to the next relation

e₃₃/e₁₁=-2c₁₂/c₁₁ (4)

where C₁₁ and C₁₂ are the elastic stiffness coefficients. The relationship between the total strains ε_ii (measured directly in an experiment) and the elastic strain e₁₁ is expressed by the following relationship [16]

e₁₁=ε_ii- ε₀  (5)

Taking into consideration Eq. (4) and Eq. (5) we arrive to

(6)

The results of calculation of ε₀ ⁽ⁿ⁾and ε₁₁ ⁽ⁿ⁾ (n is the layer number) based on the measured q_z ⁰⁰⁴, q_y⁰⁰⁴ and q_y²²⁴ values are presented in Table 3. The GaAs lattice parameter was assumed to be 0.565321 nm [17]. The elastic stiffness coefficients C₁₁ and C₁₂ of In_xAl_1-x As ternary solutions, which must be known to calculate ε₀ ⁽ⁿ⁾ and ε₁₁ ⁽ⁿ⁾, were obtained on the basis of the Vegard law proceeding from the corresponding data for AlAs and InAs [18]. For MHEMT 2 the values of ε₀⁽⁴⁾ and ε₁₁⁽⁴⁾, were determined on the basis of a linear dependence of ε₀ on the layer position in space and using Eq. (6). Results presented in Table 3 allows us to conclude that the design of MM-buffer in MHEMT 2 satisfies the philosophy of the inverse step creation [4] because the strain in this constructive element of MM-buffer is practically equal to zero. The existence of the inverse step in MM-buffer of MHEMT 2 does not influence the strain field in the internal layers of the buffer. Because layers 6 and 7 of MHEMT 2 do not contribute the strains into the total strain field, they were excluded from the following analysis. The lattice misfit spatial profiles and strain spatial profiles in both MHEMTs are shown in Figures 4 and 5 correspondingly. The residual strain profiles in MHEMTs are presented in the form of a function of e₁₁ on ε₀ Both dependences are similar and manifest non-zero strains for the internal layers of MM-buffers that is in contradiction with the models predicting the full strain relief in internal layers [19,20]. Such a situation is realized due to the work hardening, which may appear in multilayer systems at sufficiently.

Table 3: The lattice misfit, the residual strain and the thickness of the constructive elements of MM-buffers.

Figure 4: The lattice misfit spatial profiles in the epitaxial layers of MM-buffers for: (a) MHEMT 1 and (b) MHEMT 2.

Figure 5: The strain spatial distributions plotted in the form of a function of E₁₁ on E_o for MHEMTs.

Discussion

On the basis of our experimental data, it is possible to show that for two MHEMTs the following relationship is performed

(7)

where h_dfl is the thickness of the dislocation free layer. For MHEMT 1 h_dfl=0.5 μm; for MHEMT 2 h_dfl =0.2 μm. Eq. (7) characterises numerically the process of strain relief in the final constructive elements of MM-buffer in MHEMTs.

The product (E₁₁^(5-6-7)- E⁽⁴⁾₁₁)2 equals 0.0038 nm while the product (E ⁽⁵⁾₁₁-E⁽⁴⁾₁₁)² h_dft is equal to 0.0036 nm. The average value of two products equals 0.0037 nm. In a general form Eq. (7) can be written as

(8)

where E^dft₁₁ the residual strain in the dislocation free layer, E^ial₁₁ is the residual strain in the internal adjacent layer and k is a phenomenological constant. Attention is drawn to the fact that Eq. (8) is very close to the equation describing the strain relief in a single layer heterostructure. As was shown in Ref. [11], the strain relief of epitaxial layer of In Ga As grown on (001) GaAs substrate is governed by the equation

e₁₁²=K/H (9)

where h is the thickness of epitaxial layer. The phenomenological constant k in Eq. (9) is equal to 0.0037 nm. Eq. (8) and Eq. (9) operate equal values of k that indicates the common mechanism of strain relief in both cases. We may consider Eq. (8) as an extension of Eq. (9) to two layer thin film system where the dislocation free layer plays the role of a substrate. It should be pointed out that, in such a two layers system the strain relief occurs in the internal layer while the dislocation free layer exhibits strong compression. Because the dislocation free layer is sufficiently thinner than a real massive substrate in a single layer heterostructure, it gives a possibility to perform its pseudomorphic growth on the platform of internal adjacent layer. Both layers, the dislocation free layer and the internal adjacent layer, are, in fact, the single equalized system. The determination of interrelations between residual strains in all the layers of multilayer system requires additional investigation.

Conclusion

The performed study gives some evidence concerning the possibility of numerical description of a value of strain in the dislocation free layer of the step-graded MM-buffer in the framework of the phenomenological approach developed for single layer hetero structures. This description has a quadratic form as an inverse proportion between the residual compressive stain in the dislocation free layer and the root square of the layer thickness. The value of the phenomenological constant governing the process of the strain relief in the layers of metamorphic stepgraded buffers based on ternary In_xAl_1-xAs solutions coincides with that which controls the strain relief in single layer hetero structures. The numerical expression takes into account a correction for the work hardening in the internal layer adjacent to the dislocation free layer.

Acknowledgements

The reported study was partially funded by Russian Funding for Basic Research according to the research projects.

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