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

: 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 -LiAlO 2 substrate, which induces the anisotropic strain during epi-growth.


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 x Ga 1−x N/GaN or GaN/In x Ga 1−x N 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 x Ga 1−x N/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 x Ga 1−x N QWs, and hence degrade its optical performance, limiting the application of high-indium alloyed GaN/In x Ga 1−x N QWs [6][7][8]. This obstacle limits the application of GaN/In x Ga 1−x N QW for a full-spectrum LED display. A possible workaround has been proposed by using blue light from the GaN/In x Ga 1−x N (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 x Ga 1−x N epi-film, while the other is a direct epi-growth of GaN/In x Ga 1−x N 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-LiAlO 2 (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 x Ga 1−x N/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 x Ga 1−x N/GaN QWs, a full-spectrum RGB micro-LED display can be achieved simply by engineering the indium-content of the In x Ga 1−x N/GaN QW. By doing so, the existence of lattice-mismatched strain allows for easier manufacture of high-indium content In x Ga 1−x N/GaN micro-LEDs but requires theoretical re-calculation of the In x Ga 1−x N/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.

Materials and Methods
Three c-plane (0001) GaN microdisk samples were grown on 1 × 1 cm 2 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 [1100] direction, as shown by the dashed lines in Figure 1a [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 {0001} planes during the epi-growth by PAMBE, finally leaving dangling bonds on the oblique surfaces of (1101) and the top (0001) 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 [1100] direction, shown in Figure 1c,f,i. The lattice constants were then calculated from the microscopic TEM images.

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 [11 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

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 [1100] 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 [1100] direction for dM-spacing and c-axis for dc-spacing, as marked by yellow lines in   The lateral relaxation at the top surface was also evaluated for HR07-HR10, as shown in Figure 3a-d, respectively. The lateral relaxation at the top surface was also evaluated for HR07-HR10, as shown in Figure  3a-d, respectively.  Table 1. 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.  Table 1. 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 Figures 5 and 6. Table 2 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.   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 Figures 5 and 6. Table 2 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.  Table 1, Table 2, and Table 3 versus stacked height (z) along c-axis for samples A, B, and C.     Tables 1-3 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 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 (0001) microdisk gowth [14].

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 {0001 } 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 x Ga 1−x N/GaN QW micro-LEDs.   By controlling growth parameters, the 3D c-plane GaN (0001) 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 (a 0 = 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.

Conclusions
We studied the lattice relaxation of self-assembled GaN microdisks grown by PAMBE on a γ-LiAlO 2 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) = c 0 + c 1 z, with c 0 = 5.240 Å and c 1 = 3.213 × 10 −6 ; and a quadratic variation for lateral strain of lattice constant a: f(z) = a 0 + a 1 z + a 2 z 2 , with a 0 = 3.163 Å, a 1 = 5.114 × 10 −5 and a 2 = −1.427 × 10 −8 . The lattice mismatch is about 0.94% at the interface between GaN microdisks and the γ-LiAlO 2 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 {0001} 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 x Ga 1−x N/GaN QW micro-LEDs.