Anisotropic Strain on GaN Microdisks Grown by Plasma-Assisted Molecular Beam Epitaxy

Abstract

Lattice relaxation on wurtzite GaN microdisks grown by plasma-assisted molecular beam epitaxy was systematically studied. The lattice constants of GaN microdisks were evaluated from high-resolution transmission electron microscopy, and the anisotropic strain was then analyzed by observing the microscopic atomic layers. We found that the vertical lattice strain along the c-axis followed a linear relationship, while the lateral lattice strain along the a-axis exhibited a quadratic deviation. The lattice mismatch is about 0.94% at the interface between the GaN microdisks and the γ-LiAlO₂ substrate, which induces the anisotropic strain during epi-growth.

1. Introduction

Group-III nitride compounds have been extensively investigated in recent years due to their promising applications in spintronic devices [1,2] and optoelectronic lighting sources [3,4]. Because the lattice constants are quite different among these binary compounds, i.e., AlN (a = 3.11 Å, c = 4.98 Å), GaN (a = 3.160 Å, c = 5.125 Å) and InN (a = 3.5446 Å, c = 5.7034 Å) [5]. The hetero-structured quantum wells (QW) made of their ternary alloys, such as Al_xGa_1−xN/GaN or GaN/In_xGa_1−xN QWs, always produce a strain at the hetero-interface due to the lattice mismatch. The lattice-mismatched strain can induce a piezoelectric field at the hetero-interface of the Al_xGa_1−xN/GaN QW to enhance the spin-orbital interaction for spintronic qubit applications [1,2]. However, the lattice-mismatched strain will also degrade the quality of GaN/In_xGa_1−xN QWs, and hence degrade its optical performance, limiting the application of high-indium alloyed GaN/In_xGa_1−xN QWs [6,7,8]. This obstacle limits the application of GaN/In_xGa_1−xN QW for a full-spectrum LED display. A possible workaround has been proposed by using blue light from the GaN/In_xGa_1−xN (x~0.13) QW light-emitting diode (LED) and mixing with phosphor to produce a white lighting source [3,4]. Currently, the trend of LED displays the reduction in red-green-blue (RGB) LED pixel size from mini-meters down to micro-meters. There are two approaches to manufacturing the RGB micro-LED display: one is a direct photo-lithographic mini-meter/micro-meter process on GaN/In_xGa_1−xN epi-film, while the other is a direct epi-growth of GaN/In_xGa_1−xN microdisk, previously developed by our group [9,10,11]. In the photo-lithographic processing approach, one limitation is that the smaller the mini/micro-LED lithographed from epi-film, the worse the edge damage on the mini/micro-LED that occurs. Therefore, Lo et al. developed a 3-dimensional (3D), low-temperature, self-assembling, epi-growth of hexagonal c-plane GaN microdisks on γ-LiAlO₂ (LAO) substrates by plasma-assisted molecular beam epitaxy (PAMBE) [9]. The 3D GaN microdisk can be treated as a nearly free-standing GaN substrate for the further self-assembling growth of In_xGa_1−xN/GaN QW [10,11] and RGB LED [12]. Because the size of a 3D GaN microdisk can be controlled to 1~4 μm with high-quality In_xGa_1−xN/GaN QWs, a full-spectrum RGB micro-LED display can be achieved simply by engineering the indium-content of the In_xGa_1−xN/GaN QW. By doing so, the existence of lattice-mismatched strain allows for easier manufacture of high-indium content In_xGa_1−xN/GaN micro-LEDs but requires theoretical re-calculation of the In_xGa_1−xN/GaN QW band structure. In this paper, we study the lattice relaxation of the microstructure on wurtzite GaN microdisks grown by PAMBE. The strain effect on the GaN microdisk is evaluated from the lattice relaxation by high-resolution transmission electron microscopy (TEM), and the microscopic atomic layers are analyzed.

2. Materials and Methods

Three c-plane (000$\overline{1}$) GaN microdisk samples were grown on 1 × 1 cm² γ-LAO substrates by PAMBE (Veeco Applied-GEN 930 system (White Bear Lake, Minnesota)) with standard effusion cells for Ga-evaporation and an rf-plasma cell with 450 W for the N2-plasma source. The LAO substrates were cleaned with acetone (5 min), isopropanol (5 min), phosphoric acid (1:30) (5 min), de-ionized water (5 sec), and then were sequentially dried by nitrogen gas. After the cleaning, the LAO substrates were mounted on a holder and out-gassed in an MBE chamber at 700 °C for 10 minutes. Thereafter, the substrate temperature was decreased to the growth temperature. The detail of the epi-growth can be accessed in our previous papers [9,10]. The Ga wetting layer was deposited on the LAO substrate for 5 minutes at, and then the GaN microdisk samples were grown at three temperatures: 620, 630 and 640 °C (denoted as samples A, B and C, respectively) with the same (N:Ga) flux ratio of (9.0 × 10⁻⁶ torr:6.5 × 10⁻⁸ torr) for 70 minutes. Because the surface diffusion of GaN epi-growth is a function of growth temperature, the effect of different temperature is then under investigation as well. Three GaN microdisks with a diameter of ~1.5 μm were selected from samples A, B, and C for the study of microscopic lattice relaxation. The scanning electron microscope (SEM) images from the top view and a tilted angle of the samples focused on the hexagonal GaN microdisks are shown in Figure 1.

The transmission electron microscope (TEM) specimens were prepared by a dual-beam focus ion beam (FIB) with a cleavage face along the [1$\overline{1}$00] direction, as shown by the dashed lines in Figure 1a,d,g. Based on the ball-stick model of GaN microdisk, as shown in the inset of Figure 1c [9], the lateral width in the [1$\overline{1}$00] direction per primitive cell is equal to the spacing between {1$\overline{1}$00} planes, d_M = $\sqrt{3}$a/2, and this spacing between {000$\overline{1}$} planes is just equal to lattice constant c, d_c = c, where a and c are the lattice constants of wurtzite GaN. The edge of oblique surface (1$\overline{1}$01) is then tilted off the c-axis by the angle = tan⁻¹(d_M/d_c) = tan⁻¹($\sqrt{3}$a/2c ) = 28.1° (we used a= 0.3160 nm, and c= 0.5125 nm for the ideal bulk GaN wurtzite) [5]. The angle of 28° can be easily checked on the TEM images of the three samples shown in Figure 1c,f,i. However, the 3D divergent self-assembling growth of the GaN microdisk will generate unbalanced stress between {000$\overline{1}$} planes during the epi-growth by PAMBE, finally leaving dangling bonds on the oblique surfaces of (1$\overline{1}$01) and the top (000$\overline{1}$) surface. On the epi-growth of the GaN microdisk, the unbalanced stress, therefore, yielded a lattice relaxation during the growth of the layer-by-layer self-assembly. In order to investigate the lattice relaxation, we performed the high-resolution field-emission TEM measurement (Tecnai F20G2 MAT S-TWIN) at the locations marked with vertical lines along the c-axis and the lateral line along the [1$\overline{1}$00] direction, shown in Figure 1c,f,i. The lattice constants were then calculated from the microscopic TEM images.

3. Results and Discussion

The high-resolution TEM measurement was performed for sample A on locations HR01–HR06 along the c-axis and lateral locations HR07–HR10 along the [1$\overline{1}$00] direction, shown in Figure 1c. The high-resolution TEM image of spot HR01 is shown in Figure 2a1. The d-spacing can be calculated from atomic layers by scanning the microscopic structure of this high-resolution TEM image using the internal software “DigitalMicrograph” provided by Tecnai, Inc. We scanned ten successive lattice lines along the [1$\overline{1}$00] direction for dM-spacing and c-axis for dc-spacing, as marked by yellow lines in Figure 2a1, and took an average value for the location (e.g., at height z). The examples of scanned lattice lines are shown in Figure 2a2 (horizontal) and Figure 2a3 (vertical) for d_M-spacing and d_c-spacing, respectively. We then obtained the averages d_M = 0.2742 nm and dc = 0.52408 nm at HR01, yielding the angle θ = tan⁻¹(d_M/d_c) = 27.62° near the initial nucleation of GaN atop the LAO substrate. Similar analyses were repeated for HR02–HR06, some of which are shown in Figure 2b for HR02, Figure 2c for HR04, and Figure 2d for HR06. For instance, we obtained the average d_M = 0.2748 nm and d_c = 0.5237 nm for HR02, giving θ = 27.69°. For HR04, we obtained average d_M = 0.2764 nm and d_c = 0.5241 nm, yielding θ = 27.81°. For HR06, we obtained average d_M = 0.2771 nm and d_c = 0.5241 nm, yielding θ = 27.87°. The lattice strain is relaxed along the c-axis with the angle spreading from 27.62 to 27.81° during the epi-growth from the substrate up to the top surface.

The lateral relaxation at the top surface was also evaluated for HR07–HR10, as shown in Figure 3a–d, respectively.

We obtained the average d_M = 0.2778 nm and d_c = 0.5240 nm for HR07, giving to θ = 27.93^o. For HR08, average d_M = 0.2772 nm and d_c = 0.5244 nm, and θ = 27.86^o. For HR09, average d_M = 0.2771 nm and d_c = 0.5241 nm, and θ = 27.87^o. For HR10, average d_M = 0.2766 nm and d_c = 0.5242 nm, while θ = 27.82^o. The lateral lattice relaxation of the 2D aggregation looks slighter than the vertical relaxation of 3D stacked growth along the c-axis. We converted the lattice constants in Å units (i.e., d_M = $\sqrt{3}$a/2 and d_c = c) of sample A (growth temperature = 620 °C) from the measured d-spacing with standard deviation in

Table 1. The lattice constants of sample A.

Sample A	Lattice Constant a (Å)	Standard Deviation of a (Å)	Lattice Constant c (Å)	Standard Deviation of c (Å)
HR01	3.168	0.005	5.241	0.015
HR02	3.174	0.004	5.237	0.014
HR03	3.185	0.006	5.242	0.014
HR04	3.191	0.006	5.241	0.008
HR05	3.196	0.007	5.238	0.013
HR06	3.200	0.005	5.241	0.002
HR07	3.208	0.005	5.240	0.018
HR08	3.201	0.009	5.244	0.010
HR09	3.200	0.009	5.242	0.012
HR10	3.194	0.013	5.241	0.011

. The vertical lattice relaxation as a function of stacked height (z) can be easily observed from the plot of constants a and c against stacked height (z) along the c-axis from HR01 to HR06, as shown by the black squares in Figure 4.

We also analyzed samples B (630^°) and C (640^°) in the same way to evaluate the effect of growth temperature on the stacked lattice relaxation of GaN microdisk. Some of the high-resolution TEM images and d-spacing measurements are shown in Figure 5 and Figure 6.

Table 2. The lattice constants of sample B.

Sample B	Lattice Constant a (Å)	Standard Deviation of a (Å)	Lattice Constant c (Å)	Standard Deviation of c (Å)
HR11	3.165	0.005	5.246	0.003
HR12	3.172	0.005	5.243	0.008
HR13	3.184	0.004	5.237	0.003
HR14	3.191	0.004	5.242	0.001
HR15	3.201	0.003	5.245	0.004
HR16	3.204	0.004	5.248	0.007
HR17	3.206	0.004	5.244	0.007

presents data for the obtained average d_M and d_c values, as well as those for θ. Again, we converted the lattice constants (a and c) of sample B (630^°) from the measured d-spacing, as well as for sample C (640^°). TEM Images corresponding to locations in the range HR11 to HR17 are also presented. Figure 5a1 shows location HR11 from Figure 1f. Figure 5b1 is HR13, Figure 5c1 is HR15, and Figure 5d1 is HR17.

Values shown in Table 1, Table 2 and

Table 3. The lattice constants of sample C.

Sample C	Lattice Constant a (Å)	Standard Deviation of a (Å)	Lattice Constant c (Å)	Standard Deviation of c (Å)
HR18	3.165	0.005	5.239	0.008
HR19	3.178	0.005	5.236	0.008
HR20	3.193	0.005	5.239	0.006
HR21	3.200	0.005	5.244	0.001
HR22	3.204	0.004	5.245	0.004
HR23	3.208	0.004	5.243	0.005
HR24	3.210	0.003	5.246	0.003

were used to create fitting lines in Figure 4 by plotting constants a and c against stacked height (z) along c-axis, as shown by blue triangles. Similarly, for sample C, the obtained d_M, d_c, and θ values are presented in Table 3, with corresponding TEM images in Figure 6a1 (HR18), Figure 6b1 (HR20), Figure 6c1 ( HR22), and Figure 6d1 (HR24). Lattice constants (a and c) for Sample C (640^o) are shown by red circles in Figure 4. From the best curve fitting results, we obtained a linear relationship for lattice constant c: f(z) = c₀ + c₁z, with c₀ = (5.240 ± 0.001)Å and c₁ = (3.213 ± 1.376) × 10⁻⁶; and a quadratic variation for lattice constant a: f(z) = a₀ + a₁z + a₂z², with a₀ = (3.163 ± 0.001)Å, a₁ = (5.114 ± 0.356) × 10⁻⁵ and a₂ = (−1.426 ± 0.222) × 10⁻⁸.

The datapoints for all three samples follow the fitting equations within the standard deviations marked by error bars. The strain (ε) against stacked height (z) can be defined as ε = [f(z) – f(0)]/f(0) during the epi-growth. This indicates that the awl-shaped hexagonal GaN microdisks were grown under an anisotropic strain and the effect of temperature (from 620 to 640 ^oC) results in minor fluctuations in strain during the epi-growth. From the extrapolation of fitting curves, we obtained the initial lattice constants of the GaN microdisk at nucleation interface on the LAO substrate (z = 0 nm): a₀ = 3.163 Å and c₀ = 5.240 Å. Because the substrate (γ-LiAlO₂) is a tetragonal structure with lattice constants a_LAO = b_LAO = 5.169 Å and c_LAO = 6.268 Å, it provides a nearly lattice-matched rectangular anionic basal plane for 2D M-plane GaN (1$\overline{1}$00) epi-growth, where a_LAO $\cong$ c_GaN (within 1.4%) and c_LAO $\cong$ 2a_GaN (within 0.9%), as shown in the inset of Figure 7 [13]. In addition, the substrate also provides a hexagonal anionic basal plane (i.e., oxygen sites in the inset of Figure 7) for 3D c-plane GaN (000$\overline{1}$) microdisk gowth [14].

By controlling growth parameters, the 3D c-plane GaN (000$\overline{1}$) microdisks can be easily achieved in our PAMBE system [9,10,11]. Therefore, the edge of the hexagonal basal plane is equal to (1/2)c_LAO = 3.134Å, which offers the hexagonal basis for the nucleation of the GaN microdisk (a₀ = 3.163Å) with a 0.94% mismatch. This mismatch was the origin of the stress for further GaN microdisk growth, resulting in the strain with a linear relationship to stacked height (z) in lattice constant c and a quadratic deviation in lattice constant a. We also checked the optical property of the samples by photoluminescence (PL) measurements at room temperature, as shown in Figure 7. We took PL measurements with a He-Cd 325 nm laser as a light source and focused the laser beam on center of microdisk. The detailed analyses can be seen in our previous study in reference [10]. The major peaks at 3.4 eV referred to the band edge transition of GaN, and exhibited no difference among samples, and only a slight change occurs in their full width at half maximum (FWHM). The other peaks (e.g., 2.23 and 3.22 eV) were attributed to the emission of excitons bound to the structured defects of microstructures [15,16,17] near the microdisks, as shown in Figure 1b,e,h.

4. Conclusions

We studied the lattice relaxation of self-assembled GaN microdisks grown by PAMBE on a γ-LiAlO₂ substrate. We obtained the anisotropic strain of the GaN microdisk against stacked height (z); a linear relationship for vertical strain of lattice constant c: f(z) = c₀ + c₁z, with c₀ = 5.240 Å and c₁ = 3.213 × 10⁻⁶; and a quadratic variation for lateral strain of lattice constant a: f(z) = a₀ + a₁z + a₂z₂, with a₀ = 3.163 Å, a₁ = 5.114 × 10⁻⁵ and a₂ = −1.427 × 10⁻⁸. The lattice mismatch is about 0.94% at the interface between GaN microdisks and the γ-LiAlO₂ substrate, which induces the anisotropic strain during the epi-growth. The 3D self-assembling 28^o growth of GaN microdisk created the unbalanced stress between {000$\overline{1}$} planes, leading to the anisotropic strain during the layer-by-layer self-assembly. The change in growth temperature (from 620 to 640 °C) only slightly influences the lattice relaxation at the hetero-interface between GaN and LAO substrates, as well as the microstructure of GaN microdisks. The strain relationship fitting provides important information for the theoretical band calculation, as well as for engineering In_xGa_1−xN/GaN QW micro-LEDs.