Pseudomorphic and partially relaxed layers of corundum phase (Al, Ga)2O3 epilayers on (01.2)-oriented Al2O3 fabricated by pulsed laser deposition (PLD) are investigated. An exact analytical (continuum elasticity) strain theory for rhombohedral heterostructures as a function of the two substrate orientation angles fits the strain state of pseudomorphic and relaxed samples very well. From reciprocal space maps and a quantitative analysis of x-ray diffraction peaks and tilts using the strain theory, it is concluded that in the present samples grown below 800 °C, plastic strain relaxation above the critical thickness occurs first through slip on the prismatic a-plane glide system and subsequently via the basal c-plane system. We also present a general PLD stoichiometry transfer model simultaneously explaining the epilayer alloy composition and growth rate in the entire composition range.

Pseudomorphic epitaxial strain has been discussed for a number of heterostructures from non-cubic materials, including hexagonal (wurtzite: GaN and ZnO),1,2 rhombohedral (corundum: Al2O3),3,4 and monoclinic (β-Ga2O3)3,5 crystals.

Here, we discuss epitaxy of α-phase (corundum) (Al, Ga)2O3 alloys since the sesquioxide in this (non-equilibrium) phase is a promising material for power transistors6,7 and can be used to build dielectric heterostructures. The conduction band offsets in the corundum (Al, Ga)2O3 system have been predicted to be larger than for the monoclinic phase8 with small valence band offsets as confirmed experimentally.9 

We focus on epitaxy on the r-plane.4 It is well known that wurtzite ZnO10 and GaN11,12 grow in a-plane orientation on r-plane alumina. Here, the sesquioxide epilayer has the same crystallographic orientation as the substrate.4,13–15

We will distinguish the (01.2) r-plane [or equivalently (11¯.2) or (1¯0.2)], its backside, the (01¯.2¯) r¯-plane, and the (10.2) r′-plane as shown in Fig. 1. One should exercise care, since in the literature, through a different choice of lattice vectors than the most usual one [cf. Eq. (1)], the r-plane is also sometimes labeled (01¯.2).16 

FIG. 1.

Hexagonal unit cell of the rhombohedral material. The (01.2) r-plane (ϕ = 90°) and the (10.2) r′-plane (ϕ = 30°) are indicated.

FIG. 1.

Hexagonal unit cell of the rhombohedral material. The (01.2) r-plane (ϕ = 90°) and the (10.2) r′-plane (ϕ = 30°) are indicated.

Close modal

As stated in Ref. 17, the r- and r¯-planes are identical (as is true for the m/m¯- and c/c¯-planes). Since alumina is not a polar crystal, the r-orientation of alumina shall not be denoted as “semipolar” as it would be in the nomenclature for wurtzite materials. However, due to the threefold symmetry around the c-axis, the r- and r′-planes are not equivalent. This is irrelevant for the popular case of epitaxy of wurtzite materials on (00.1) alumina (c-plane sapphire). However, when strained corundum layers are deposited on alumina, this aspect becomes relevant. We remark that peculiarly in scratch tests of the c-oriented surface of alumina reported in Ref. 18, a threefold symmetry was not found.

Finally, we are interested in the mechanisms of anisotropic strain relaxation for layers beyond the pseudomorphic limit. Pseudomorphic and some relaxed layers of α-(Al, Ga)2O3 on α-Ga2O3 were reported in Ref. 14, but no details about the mechanisms of plastic relaxation were given. The α-(Al, Ga)2O3 on Al2O3 layers of different Al-concentrations reported in Ref. 15 was found to vary between almost pseudomorphic and fully relaxed, but also further information on the plastic relaxation was not reported.

Al2O3 has a rhombohedral lattice (space group R3¯c) and a threefold symmetry around its c-axis. Thus, the symmetry is lower than that of hexagonal wurtzite materials such as group-III nitrides or ZnO. This is expressed by the fact that the C14 component of the elastic constants is non-zero.3,19,20 Some intricacies of the corundum lattice have only recently paid more attention to. It has been pointed out that the front and back of an a-plane sapphire wafer [(11.0) or (21¯.0) orientation] are not equivalent,17 which also has consequences for epitaxy and device fabrication.21 

The (pseudo-)hexagonal unit cell of the orthorhombic material is spanned by the vectors a1 = T (1, 0, 0)T, a2 = T (0, 1, 0)T, and a3 = T (0, 0, 1)T with the transformation matrix (ζ = c/a),

T=a11/2003/2000ζ.
(1)

The orientation of the epitaxial plane (hk.l) is expressed via the inclination angle θ relative to the c-axis,

θ=arccosl/(4/3)ζ2h2+hk+k2+l2,
(2)

and the azimuthal orientation ϕ, ϕ = 0 denoting the direction of a1 (Fig. 1). Note that θ depends on ζ, which is, in general, different for the epilayer and substrate. We also introduce ξ = ζF/ζS = (cF/aF)/(cS/aS) denoting the c/a ratios of the epilayer and substrate, and the indices denoting the substrate and film property (see Fig. 2).

FIG. 2.

Strains ϵa and ϵc as defined in text and ϵxx according to Eq. (7) for the r-plane as a function of the Al-concentration x for (AlxGa1−x)2O3 on Al2O3. ξ − 1 is also depicted.

FIG. 2.

Strains ϵa and ϵc as defined in text and ϵxx according to Eq. (7) for the r-plane as a function of the Al-concentration x for (AlxGa1−x)2O3 on Al2O3. ξ − 1 is also depicted.

Close modal

The material parameters we use are given in Tables I and II. We emphasize that the sign of C14 for Ga2O3 is positive in our coordinate system since the coordinate system in Ref. 22 is chosen differently, which leads to a reversal of sign. In the calculus of Ref. 22, C14 for Ga2O3 and Al2O3 has the same sign.23 

TABLE I.

Hexagonal lattice parameters of rhombohedral (R3¯c) alumina and gallia.

Material a (nm)c (nm) ζ = c/aReference
α-Al2O3 0.475 9 1.2991 2.7298 25  
α-Ga2O3 0.498 25 1.3433 2.6960 26  
Material a (nm)c (nm) ζ = c/aReference
α-Al2O3 0.475 9 1.2991 2.7298 25  
α-Ga2O3 0.498 25 1.3433 2.6960 26  
TABLE II.

Elastic constants of rhombohedral (R3¯c) alumina (experimental values) and gallia (theoretical values) (in units of 1011 Pa).

Material C11C12C13C33C14C44Reference
α-Al2O3 4.97 1.63 1.16 5.01 0.22 1.47 27  
α-Ga2O3 3.815 1.736 1.26 3.458 0.173 0.797 22 and 23  
Material C11C12C13C33C14C44Reference
α-Al2O3 4.97 1.63 1.16 5.01 0.22 1.47 27  
α-Ga2O3 3.815 1.736 1.26 3.458 0.173 0.797 22 and 23  

Note that ζ ≈ 2.7 here for the sesquioxides is much larger than ζ ≈ 1.6 for hexagonal wurtzite GaN- or ZnO-based systems. Values for alloys are linearly interpolated. Vegard’s law for α-(Al, Ga)2O3 has been tested,24 although not with high precision.

We introduce the strains ϵa = aS/aF − 1 and ϵc = cS/cF − 1 = ξ aS/aF − 1 that are depicted in Fig. 2.

It is well known3 that the out-of-plane strain for pseudomorphic layers on the a-plane is given by

ϵzz,a=C12ϵa+C13ϵcC11.
(3)

Equation (3) holds for wurtzite crystals not only for the a-plane but also for the m-plane (as well as for any other azimuth ϕ).2 For the rhombohedral crystal with C14 ≠ 0, we find (by simplifying involved analytical formulas for arbitrary azimuths) the out-of-plane strain on the m-plane to be

ϵzz,m=(C12+C̃)ϵa+C13ϵcC11C̃,
(4)

with C̃=C142/C44. Obviously, for the hexagonal case with C14 = 0, Eq. (4) degenerates to Eq. (3). In a numerical example of (Al0.9Ga0.1)2O3 on Al2O3, C̃0.03 is small compared to C11, C12, or C13, all being larger than 1 (in units of 1011 Pa). Thus, the difference for a- and m-planes is fairly small. The out-of-plane strain of planes perpendicular to the c-plane is shown in Fig. 3 for all azimuthal directions. In addition, it does not depend on the sign of C14.

FIG. 3.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of azimuthal angle ϕ for θ = π/2. The orientation of the m- and a-planes is indicated.

FIG. 3.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of azimuthal angle ϕ for θ = π/2. The orientation of the m- and a-planes is indicated.

Close modal

The difference in out-of-plane strain for various azimuthal directions becomes larger for other inclinations and is dependent on the sign of C14, namely, for the r-type planes. In Fig. 4, the out-of-plane strain is shown for various azimuths for the inclination angle of the r-plane (θS = 57.6° for Al2O3). Notably, the r-plane and the r′-plane behave differently.

FIG. 4.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of azimuthal angle ϕ for θ = 57.6° (inclination angle of the r-plane). The (11¯.2)r-plane is at ϕ = −30° and the (10.2) r′-plane at ϕ = +30°. The value for the a-plane (ϕ = −60°, 0°, and 60°) is shown as the horizontal dashed line.

FIG. 4.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of azimuthal angle ϕ for θ = 57.6° (inclination angle of the r-plane). The (11¯.2)r-plane is at ϕ = −30° and the (10.2) r′-plane at ϕ = +30°. The value for the a-plane (ϕ = −60°, 0°, and 60°) is shown as the horizontal dashed line.

Close modal

In Fig. 5, the out-of-plane strain is depicted as a function of the inclination angle θ with respect to the c-axis for three different azimuths, namely, (01.l) including the r- and m-planes [ϕ = 90° or equivalently the (11¯.l) at ϕ = −30°], the (11.l) (ϕ = 0°) including the a-plane, and the (10·l) including the r′- and m-planes ϕ = +30°. The maximum splitting between the strain for ϕ = ±30° is close to the r-plane inclination. If C14 is set to zero, the black dashed curve in Fig. 5 is obtained. The maximum splitting for ϕ = ±30° amounts to about 30% of the average value and thus represents a sizable effect. We note that the result for the r′-plane can also be obtained by calculating for the r-plane and reversing the sign of C14. The out-of-plane lattice mismatch [divided by (1 − x)] is depicted in Fig. 6 for various epitaxial planes. Although the in-plane strains depend linearly on the concentration (Fig. 2), the out-of-plane strain depends non-linearly on x due to the change of elastic constants.

FIG. 5.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of inclination angle θ for three different azimuths, namely, (11¯.l) (ϕ = −30°, blue curve), (11.l) (ϕ = 0°, black dashed curve), and (10.l) (ϕ = +30°, blue dashed–dotted curve). The angular positions of the c-, m-, a-, r-, and r′-planes are labeled. The inclination angle of the r-plane is shown as the vertical dashed line. The black solid curve represents the in-plane isotropic result when setting C14 = 0.

FIG. 5.

Out-of-plane strain of the pseudomorphic (Al0.9Ga0.1)2O3/Al2O3 epilayer as a function of inclination angle θ for three different azimuths, namely, (11¯.l) (ϕ = −30°, blue curve), (11.l) (ϕ = 0°, black dashed curve), and (10.l) (ϕ = +30°, blue dashed–dotted curve). The angular positions of the c-, m-, a-, r-, and r′-planes are labeled. The inclination angle of the r-plane is shown as the vertical dashed line. The black solid curve represents the in-plane isotropic result when setting C14 = 0.

Close modal
FIG. 6.

Out-of-plane lattice mismatch divided by (1 − x) of pseudomorphic (AlxGa1−x)2O3/Al2O3 epilayers (30k PLD pulses) as a function of Al-concentration x for various substrate orientations. The blue squares are from samples A–I (relaxed samples are depicted with open symbols) and the blue circles from samples reported in Ref. 4. The error bars are for an absolute error in x of ±0.003. The purple data points are for another pair of r/r¯ samples. The black diamond is from Ref. 37 for 1 nm α-Ga2O3 on a-plane Al2O3. The lines are the theoretical dependencies.

FIG. 6.

Out-of-plane lattice mismatch divided by (1 − x) of pseudomorphic (AlxGa1−x)2O3/Al2O3 epilayers (30k PLD pulses) as a function of Al-concentration x for various substrate orientations. The blue squares are from samples A–I (relaxed samples are depicted with open symbols) and the blue circles from samples reported in Ref. 4. The error bars are for an absolute error in x of ±0.003. The purple data points are for another pair of r/r¯ samples. The black diamond is from Ref. 37 for 1 nm α-Ga2O3 on a-plane Al2O3. The lines are the theoretical dependencies.

Close modal

In the following, we partly reformulate the strain theory from Ref. 3. We note that in a remark ibid (Ref. 16 of Ref. 3), the different rotation angles of the crystal for the substrate and epilayer have been discussed. The exact treatment of the displacement (in laboratory coordinates) involves the matrix RSTSTF1RFT1, where RS and RF denote the rotation matrices with the different inclination angles of the same (hk.l)-plane of substrate θS and film material θF, respectively (due to different c/a-ratios). The matrices T denote the transformation for the crystal lattice as in (1). The exact relation of θF and θSfor the same (hk.l) interface plane is (for any angle)

θF=arccoscotθS/ξ2+cot2θS.
(5)

The difference θFθS is depicted in Fig. 7. For θS = n π/2, θF = θS. For 0 ≤ θSπ/2, the formula is equivalent to θF=arccos(1/ξ2tan2θS+1); the extremum of θFθS is at θSe=arccotξ(π+1ξ)/4+(1ξ)2/8, the approximation being valid for values of ξ close to 1. The extremal value θF(θSe)θSe is 2arctan(ξ)π/2(1ξ)/2(1ξ)2/4. For the present case of (Al, Ga)2O3/Al2O3 structures, even the linear approximations are very good since 0.9876 ≤ ξ ≤ 1 (cf. Fig. 2).

FIG. 7.

Difference of the inclination angle θF of (AlxGa1−x)2O3 and θS of Al2O3 for the same lattice plane (for x = 0 and x = 0.5 as labeled). The angular position of the r-plane is indicated.

FIG. 7.

Difference of the inclination angle θF of (AlxGa1−x)2O3 and θS of Al2O3 for the same lattice plane (for x = 0 and x = 0.5 as labeled). The angular position of the r-plane is indicated.

Close modal

The difference θFθS being in the 0.1° range does not seem large, but one should keep in mind that compared to the involved lattice plane tilts Δω from XRD measurements (Table III) and the tilt β of the (01.2) lattice plane in relaxed samples (Table VI), it is quite significant and must be considered.

TABLE III.

X-ray diffraction peak separation Δ2θ (in degrees) and tilt between the normal of epilayer and substrate planes Δω (in degrees) for six asymmetric reflections for (Al, Ga)2O3/Al2O3 samples A–I (30k PLD pulses). |Δ2θ|¯ denotes the average of the (absolute value of the) experimental peak separation for the N = 12 reflections. n.m.: not measured with sufficient precision. See Figs. S1 and S2 of the supplementary material for some of the raw data.

Δ2θ (hk.l)A (r)B (r¯)CDEFGHI
(02.4) 0.5712 0.5515 0.6204 0.3742 0.9749 0.8567 1.0638 1.2805 1.4601 
(04.8) 2.1665 2.1173 2.3733 1.3984 3.7031 3.2694 3.9984 4.8059 5.3894 
(00.6) 0.1040 0.1000 0.112 0.064 0.174 0.156 0.180 0.216 0.354 
(00.12) 0.2675 0.2662 0.2915 0.1723 0.468 0.416 0.470 0.553 0.962 
(1¯4¯.6) 1.0600 1.0500 1.160 0.680 1.845 1.620 1.950 n.m. n.m. 
(4¯1¯.6) 0.3600 0.3520 0.392 0.224 0.689 0.585 0.845 1.12 n.m. 
(14.6) 2.6993 2.6303 2.9259 1.7634 4.5613 4.0490 4.9358 5.9407 6.6703 
(41.6) 1.4651 1.4258 1.5831 0.9440 2.5074 2.2124 2.9012 3.7377 4.5366 
(51¯.6) 0.5990 0.5991 0.6382 0.3731 1.1002 0.9625 1.4446 2.1830 3.0215 
(10.16) 1.3277 1.2982 1.4556 0.8556 2.2423 1.9866 2.3702 2.9216 4.0166 
(40.10) 1.5145 1.4752 1.6325 0.9835 2.5865 2.3112 2.9902 4.0345 5.0108 
(11.9) 0.5989 0.5792 0.6381 0.3729 1.0210 0.8836 1.1096 1.3653 1.5341 
|Δ2θ|¯ 1.061 1.037 1.152 0.684 1.823 1.609 2.022 2.560 3.296 
Δ2θ (hk.l)A (r)B (r¯)CDEFGHI
(02.4) 0.5712 0.5515 0.6204 0.3742 0.9749 0.8567 1.0638 1.2805 1.4601 
(04.8) 2.1665 2.1173 2.3733 1.3984 3.7031 3.2694 3.9984 4.8059 5.3894 
(00.6) 0.1040 0.1000 0.112 0.064 0.174 0.156 0.180 0.216 0.354 
(00.12) 0.2675 0.2662 0.2915 0.1723 0.468 0.416 0.470 0.553 0.962 
(1¯4¯.6) 1.0600 1.0500 1.160 0.680 1.845 1.620 1.950 n.m. n.m. 
(4¯1¯.6) 0.3600 0.3520 0.392 0.224 0.689 0.585 0.845 1.12 n.m. 
(14.6) 2.6993 2.6303 2.9259 1.7634 4.5613 4.0490 4.9358 5.9407 6.6703 
(41.6) 1.4651 1.4258 1.5831 0.9440 2.5074 2.2124 2.9012 3.7377 4.5366 
(51¯.6) 0.5990 0.5991 0.6382 0.3731 1.1002 0.9625 1.4446 2.1830 3.0215 
(10.16) 1.3277 1.2982 1.4556 0.8556 2.2423 1.9866 2.3702 2.9216 4.0166 
(40.10) 1.5145 1.4752 1.6325 0.9835 2.5865 2.3112 2.9902 4.0345 5.0108 
(11.9) 0.5989 0.5792 0.6381 0.3729 1.0210 0.8836 1.1096 1.3653 1.5341 
|Δ2θ|¯ 1.061 1.037 1.152 0.684 1.823 1.609 2.022 2.560 3.296 
Δω (hk.l)ABCDEFGHI
(14.6) 0.137 0.135 0.150 0.090 0.237 0.21 0.28 0.35 0.45 
(41.6) 0.292 0.282 0.315 0.190 0.392 0.44 0.505 0.58 0.43 
(51¯.6) 0.222 0.212 0.252 0.147 0.36 0.337 0.39 0.41 0.23 
(10.16) 0.232 0.227 0.252 0.152 0.39 0.355 0.415 0.41 0.16 
(40.10) 0.267 0.262 0.297 0.172 0.455 0.407 0.47 0.42 0.22 
(11.9) 0.242 0.237 0.272 0.167 0.42 0.375 0.425 0.46 0.28 
Δω (hk.l)ABCDEFGHI
(14.6) 0.137 0.135 0.150 0.090 0.237 0.21 0.28 0.35 0.45 
(41.6) 0.292 0.282 0.315 0.190 0.392 0.44 0.505 0.58 0.43 
(51¯.6) 0.222 0.212 0.252 0.147 0.36 0.337 0.39 0.41 0.23 
(10.16) 0.232 0.227 0.252 0.152 0.39 0.355 0.415 0.41 0.16 
(40.10) 0.267 0.262 0.297 0.172 0.455 0.407 0.47 0.42 0.22 
(11.9) 0.242 0.237 0.272 0.167 0.42 0.375 0.425 0.46 0.28 
TABLE IV.

Fitted Al-concentration x for the x-ray diffraction data (peak separations of Table III) of (AlxGa1−x)2O3 epilayers on (01.2)/Al2O3 for two sets of r/r¯-samples A and B and two further samples C and D. σ2θ=χmin2/N denotes the standard deviation (in 0.01°) of the least squares fit to the 2θ separation between substrate and epilayer of N = 12 reflections (as listed in Table III). σω denotes the standard deviation of the difference between experiment and simulation for the tilt Δω between 6 lattice planes (see Table III) for the given fit parameters.

No. xEDXxfitσ2θ (°) σω (°)
0.859 0.866 0.015 0.0035 
0.863 0.870 0.017 0.0047 
0.846 0.855 0.015 0.0031 
0.919 0.914 0.009 0.0042 
No. xEDXxfitσ2θ (°) σω (°)
0.859 0.866 0.015 0.0035 
0.863 0.870 0.017 0.0047 
0.846 0.855 0.015 0.0031 
0.919 0.914 0.009 0.0042 

We note that Eq. (7) of Ref. 2,

ϵxx=(cosθ/(1+ϵa))2+(sinθ/(1+ϵc))21/2,
(6)

is exact for θ = θS [also Eq. (7) of Ref. 1 is exact for θ = θS]. Equation (6) is equivalent to (7), a new exact formula we found that is linear in the lattice constants (a, c) or the strains (ϵa, ϵc) and is also fully symmetric in substrate and film lattice properties,

ϵxx=aSaFcosθScosθF+cScFsinθSsinθF1=(1+ϵa)cosθScosθF+(1+ϵc)sinθSsinθF1.
(7)

In first order of (1 − ξ), ϵxxϵa + (1 − ξ) (1 + ϵa) sin2θS (for 0 ≤ θSπ/2); For ξ → 1, ϵxxϵyy = ϵa and the in-plane strain becomes isotropic and is independent of the inclination angle. All calculations here28 are made taking into account the difference of θ for the epilayer and substrate due to their different c/a ratios exactly.

All epitaxial layers29 were deposited using pulsed laser deposition30 (PLD) on Al2O3 substrates sourced from CrysTec company. The r-plane wafer orientation was specified within ±0.3°. Experimental details for the epitaxy are given in Ref. 4. The laser targets were sintered pellets pressed from high purity (4N7) Al2O3 and (5N) Ga2O3 powders provided by Alfa Aesar. The targets were ablated using a 248 nm KrF Coherent LPX PRO 305 F excimer laser. The substrates were heated by using a resistive heater to an approximate growth temperature of 750 °C. The oxygen partial pressure was set to 0.001 hPa. Samples A–I were fabricated with 30k PLD pulses from targets of different Al/Ga-ratios; sample J is similar to sample I but made with 10k PLD pulses. For two target compositions, samples of different thicknesses (1.1k, 3.3k, 10k, and 30k PLD pulses) were grown.

The structural properties of the heterostructures were investigated by x-ray diffraction using a PANalytical X’pert PRO materials research diffractometer employing the line focus of a copper x-ray tube. A parabolic mirror (Kα-radiation) was used as incident beam optics for measurements of totally 8 symmetric and asymmetric reflections. For skew-symmetric peaks such as (00.6), the mirror was combined with a 4× Ge(220) monochromator (Kα1-radiation). A 2-bounce hybrid monochromator including mirror was used for higher intensities of the (00.6) reciprocal space maps (RSMs). Accordingly, the detection was performed with a PIXcel3D array detector or a Ge secondary monochromator with proportional counter (triple-axis configuration). Fast RSMs were measured with the frame-based option of the PIXcel3D detector. With that, the measurement time is reduced by a factor of about 5 in relation to conventionally recorded RSMs. As an example, such measurements are shown for sample C in Fig. S1 of the supplementary material.

Electron microscopy imaging and energy-dispersive x-ray (EDX) analysis were performed with a FEI Nova 200 Nano Lab dual beam electron microscope.

Our experimental results discussed below include the Al/Ga cation ratio in the epilayers as a function of the PLD target composition. We denote the Al-concentration in the film and target as x and xt, respectively. The experimental data for x vs xt are depicted in Fig. 8. We develop a simple model of this dependency. The Al/Ga-ratio arriving at the growth surface shall be pxt/(1 − xt), taking into account possible different transfers of Al and Ga from the target to the growth surface due to different scattering of species between target and substrate. Then, there is a competition for cation lattice sites with gallium typically exhibiting the higher desorption rate due to the formation of volatile gallium suboxides as observed in MBE31 and PLD.32 The ratio of probabilities of Ga or Al being incorporated (once arrived) shall be denoted by r (r < 1 here). Thus, the Al-concentration in the epilayer x is the ratio of the cation sites with Al (∝pxt) to all filled sites [∝pxt + r(1 − xt)],

x=xtq+(1q)xt  or  1x=qxt+1q,
(8)

with q = r/p. Naturally, this function fulfills the obvious restrictions x(xt = 1) = 1 and x(xt = 0) = 0. Stoichiometric transfer, i.e., x = xt, requires q = 1 (but not necessarily r = p = 1). We note that the derivative is equal to +1 at the intersection with 1 − xt at xt=(qq)/(1q) and the function value there is x=1/(1+q). The incorporation rate of Ga for a small Ga target content is q; the incorporation rate of Al for a small Al target content is 1/q. For given q, the necessary target concentration xt to achieve x in the epilayer is xt = qx/[1 −(1 − q)x]. We note that the model presented here is similar to a model invoked for the modeling of ablation from segmented PLD targets33 for the purpose of combinatorial PLD.34,35

FIG. 8.

Al-concentration in epilayer x vs the Al-concentration in the PLD target xt. The squares are from this work and the small circles from Ref. 4 (films grown at 800 °C). The dashed line is a fit of the present data (squares) with Eq. (8) for q = 0.25. Further dotted lines are shown for other values of q as labeled. The gray solid lines denote the error margin of ±0.03 for q (q = 0.22 and q = 0.28).

FIG. 8.

Al-concentration in epilayer x vs the Al-concentration in the PLD target xt. The squares are from this work and the small circles from Ref. 4 (films grown at 800 °C). The dashed line is a fit of the present data (squares) with Eq. (8) for q = 0.25. Further dotted lines are shown for other values of q as labeled. The gray solid lines denote the error margin of ±0.03 for q (q = 0.22 and q = 0.28).

Close modal

We fit our data by calculating q from all experimental (xt, x) pairs,

q=xtx1x1xt,
(9)

and using the average q = 0.25(3). The fit is shown as a dashed line in Fig. 8 and reproduces the present experimental data (and two previously determined values under similar conditions in a different growth chamber4) quite accurately. For comparison, the dependencies for other values of q are included in Fig. 8 as dotted lines.

The growth rate g follows directly from the above stoichiometry transfer model. Keeping in mind that the rate of Ga incorporation is q times that of Al, the growth rate for a target with composition xt is given by

g=ĝ(xt)[xt+q(1xt)]=ĝ(xt)xtx.
(10)

The proportionality factor in Eq. (10) is the ablation rate of the target ĝ(xt). We have determined this quantity via deposition at room temperature (no substrate heating), assuming that all ablated material is incorporated into the (amorphous) thin film. The result is shown in Fig. 9 as magenta circles. We attribute the increase in ablation rate to the increase in absorption of the laser in the target due to the decrease in bandgap with decreasing Al-concentration; for Ga2O3, the bandgap is close to the PLD laser photon energy (5.0 eV).

FIG. 9.

Growth rates as a function of Al-concentration in the PLD target xt. Experimental data of growth rate g for the heated substrate (black squares from x-ray methods and diamonds from spectroscopic ellipsometry) are shown together with theoretical expectation from Eq. (10) (black dashed line) using the stoichiometry transfer model with q = 0.25. The growth rate for the substrate at room temperature (=ablation rate ĝ) is shown as magenta circles (dashed line is a guide to the eyes). The blue dashed line is the stoichiometry transfer model for q = 0.17.

FIG. 9.

Growth rates as a function of Al-concentration in the PLD target xt. Experimental data of growth rate g for the heated substrate (black squares from x-ray methods and diamonds from spectroscopic ellipsometry) are shown together with theoretical expectation from Eq. (10) (black dashed line) using the stoichiometry transfer model with q = 0.25. The growth rate for the substrate at room temperature (=ablation rate ĝ) is shown as magenta circles (dashed line is a guide to the eyes). The blue dashed line is the stoichiometry transfer model for q = 0.17.

Close modal

The expected growth rate for the heated substrate, according to Eq. (10) with q = 0.25, is depicted in Fig. 9 as the black dashed line. The experimentally determined dependence of the growth rate (from thickness determination using x-ray Pendellösung oscillations, x-ray reflectivity measurements, and ellipsometry) on the nominal target Al-concentration xt (black symbols in Fig. 9) is consistent with our stoichiometry transfer model. However, the best fit to the growth rate data is obtained with q = 0.17 (blue dashed line in Fig. 9).

Thus, two effects on the growth rate almost cancel: With decreasing Al-concentration in the target the ablation rate increases. Since Ga is incorporated in the layer at a lower rate than Al (q < 1), the incorporation rate decreases when the Al-concentration decreases. The good fit with g(0) = ĝ(0) means that all ablated Al is incorporated in the thin film both at room and at growth temperature.

Next, we investigate the equivalence of the r- and r¯-planes as it is expected theoretically. For this purpose, pseudomorphic (AlxGa1−x)2O3 epilayers were deposited on substrates obtained from the front and back surface of the same double-sided polished r-Al2O3 wafer (samples A and B). Using x-ray diffraction (high-resolution for the skew-symmetric peaks), the peak separations of the following reflections were measured: (02.4), (04.8), (00.6), (00.12) (1¯4¯.6), (4¯1¯.6), (14.6), (41.6), (51¯.6), (10.16), (40.10), and (11.9), similar to the investigation reported in Ref. 4. See Fig. S1 of the supplementary material for an overview on the experimental RSMs around these reflections for sample C.

Then, using the material parameters given in Table I, we have fitted the Al-concentration x from the least square sum of deviations between experimental and theoretical Bragg peak separations.36 Two more samples, C and D, have been evaluated in a similar fashion with their fit values included in Table IV. From the fit parameters obtained from the fitting of the 2θ separation of the 14 reflexes, the tilt of lattice planes (as listed in Table IV) can be independently reproduced very well. As shown in the last column of Table IV, the standard deviation of predicted and experimental tilts is very small (σtilt < 0.005°). These samples are also on the curve for pseudomorphic lattice mismatch on the r-plane (Fig. 6). The Al-concentration from the fits is close to experimental values from standard-free EDX using 3 kV or 4 kV acceleration voltage (an analysis involving more samples will be made below and is given in Table V and shown in Fig. 12).

TABLE V.

Fit parameters (Al-concentration x, relaxation parameters ρx, ρy, and tilt angles βx, βy) for the x-ray diffraction data of (AlxGa1−x)2O3 epilayers on (01.2) Al2O3 for various samples. xEDX lists the Al-concentrations as experimentally determined with EDX. σ=χmin2/N denotes the standard deviation of the least squares fits to N ≤ 18 x-ray diffraction data (peak separations and tilts).

No.xEDXxfitρxρyβx (°)βy (°)σ (°)
0.859 0.866 0.991 0.981 0.001 0.001 0.008 
0.863 0.870 0.983 0.977 0.003 0.000 0.007 
0.846 0.855 0.983 0.992 0.005 0.006 0.010 
0.919 0.914 1.002 0.992 0.004 0.004 0.008 
0.763 0.770 0.981 0.953 0.008 0.060 0.020 
0.786 0.797 0.986 0.957 0.005 0.011 0.011 
0.731 0.745 1.018 0.853 0.010 0.029 0.022 
0.670 0.681 1.030 0.727 0.114 0.020 0.060 
0.588 0.608 0.758 0.575 0.285 0.179 0.044 
No.xEDXxfitρxρyβx (°)βy (°)σ (°)
0.859 0.866 0.991 0.981 0.001 0.001 0.008 
0.863 0.870 0.983 0.977 0.003 0.000 0.007 
0.846 0.855 0.983 0.992 0.005 0.006 0.010 
0.919 0.914 1.002 0.992 0.004 0.004 0.008 
0.763 0.770 0.981 0.953 0.008 0.060 0.020 
0.786 0.797 0.986 0.957 0.005 0.011 0.011 
0.731 0.745 1.018 0.853 0.010 0.029 0.022 
0.670 0.681 1.030 0.727 0.114 0.020 0.060 
0.588 0.608 0.758 0.575 0.285 0.179 0.044 

The only other quantitative strain value for corundum (Al, Ga)2O3/Al2O3 heterostructures we could find in the literature is shown in Fig. 6 as black diamond, representing an ultrathin (1 nm) α-Ga2O3 layer on a-plane Al2O3; the given related experimental value for the out-of-plane strain is ϵzz,a = 0.07437 and agrees well with Eq. (3) and the material parameters of Tables I and II, yielding ϵzz,a = 0.0738.

Several epilayers (samples E–I) have been grown exceeding the pseudomorphic limit, exhibiting partial relaxation of the lattice. In Fig. S2 of the supplementary material, the RSMs of the (14.6) and (41.6) reflexes are shown for samples A–I.

Therefore, all samples are (re-)evaluated, introducing strain relaxation within the interface plane. The strain relaxation in the x′- and y′-directions of the interface plane [within the laboratory coordinate system as depicted in Fig. 10(a)] from pseudomorphic values ϵ′ to partially relaxed values ϵ̃ is parameterized by independent parameters ρx and ρy, ρ = 1 meaning pseudomorphic growth and ρ = 0 meaning full relaxation,

ϵ̃xx=ϵxxρx=ϵxxΔϵxx,
(11)
ϵ̃yy=ϵyyρy=ϵyyΔϵyy.
(12)
FIG. 10.

(a) r-plane with two crystallographic directions and x′- and y′-directions of the laboratory system. The blue and red lines indicate the possible dislocation lines of the a-plane prismatic slip system. The blue and red arrows indicate the directions of the edge component of the associated 1¯101-type Burgers vectors; the solid (dashed) arrow indicates upward (downward) tilt component. (b) 3D view of the geometry of the r-plane with c-plane basal and two prismatic a-plane slip systems.

FIG. 10.

(a) r-plane with two crystallographic directions and x′- and y′-directions of the laboratory system. The blue and red lines indicate the possible dislocation lines of the a-plane prismatic slip system. The blue and red arrows indicate the directions of the edge component of the associated 1¯101-type Burgers vectors; the solid (dashed) arrow indicates upward (downward) tilt component. (b) 3D view of the geometry of the r-plane with c-plane basal and two prismatic a-plane slip systems.

Close modal

Now, we fit the 2θ separations and the tilts together with the relaxation model; the fit parameters are shown in Table V. For samples A–D, the relaxation parameters are close to 1 within the (statistical) error of typically ±0.01. For strongly relaxed samples, it turns out to be necessary to introduce a global tilt of the (01.2) epilayer plane against that of the substrate. Thus, we introduce the tilt angles βx and βy for the tilt along the x′- and y′-direction, respectively. βx is related to ρx as detailed below. βy turns out to be small (βy < 0.1°) for all layers but one. The partially relaxed samples, of course, deviate from the curve for pseudomorphic lattice mismatch on the r-plane as depicted in Fig. 6.

The relaxation parameters ρ for all samples are visualized in Fig. 11. Obviously, first the relaxation occurs in the y′-direction. This means that misfit dislocations exhibit Burgers vector edge components along this direction. Finally, in the strongest relaxed samples, relaxation occurs in both x′- and y′-directions. The fitted Al-concentration agrees for all samples very well with the Al-concentration determined from EDX apart from a systematic deviation (Fig. 12).

FIG. 11.

Fitted relaxation parameters of (AlxGa1−x)2O3 epilayers on (01.2) Al2O3 (samples A–I and a few other). The plot shows ρy vs ρx, (1, 1) representing pseudomorphic layers, and (0, 0) denoting fully relaxed layers. The dashed line is a guide to the eyes. The black symbols represent the parameters from a fit (of ρx and ρy) to peak separations and tilts (as given in Table V) for samples A–I (black squares) and other layers of various Al-concentrations and smaller thicknesses (blue symbols).

FIG. 11.

Fitted relaxation parameters of (AlxGa1−x)2O3 epilayers on (01.2) Al2O3 (samples A–I and a few other). The plot shows ρy vs ρx, (1, 1) representing pseudomorphic layers, and (0, 0) denoting fully relaxed layers. The dashed line is a guide to the eyes. The black symbols represent the parameters from a fit (of ρx and ρy) to peak separations and tilts (as given in Table V) for samples A–I (black squares) and other layers of various Al-concentrations and smaller thicknesses (blue symbols).

Close modal
FIG. 12.

Al-concentration of (AlxGa1−x)2O3 epilayers determined via EDX (xEDX) vs the concentration determined from fitting x-ray diffraction data (xfit). Squares are for samples A–I (pseudomorphic and partially relaxed layers); circles are for the samples discussed in Ref. 4. The solid line visualizes xEDX = xfit. xEDX deviates from xfit systematically as if the Ga-signal is proportional to about 7% too large (dashed line).

FIG. 12.

Al-concentration of (AlxGa1−x)2O3 epilayers determined via EDX (xEDX) vs the concentration determined from fitting x-ray diffraction data (xfit). Squares are for samples A–I (pseudomorphic and partially relaxed layers); circles are for the samples discussed in Ref. 4. The solid line visualizes xEDX = xfit. xEDX deviates from xfit systematically as if the Ga-signal is proportional to about 7% too large (dashed line).

Close modal

We have also investigated the introduction of further shear strains as additional fit parameters. As can be expected from a fit with more free parameters, the standard deviation is further reduced, but all such shear strains remain small.

The described scenario is supported by reciprocal space maps (RSMs). In Fig. 13, RSMs around the (00.6) reflex are shown with the x-ray beam pointing along the x′-direction. For pseudomorphic samples (A and C), the film peak is at Qx = 0. For samples G and H that are partially relaxed but only along y′, the epilayer peak becomes broader and weaker, but Qx remains centered around zero. For sample I, exhibiting relaxation in both in-plane directions, Qx of the film peak is non-zero.

FIG. 13.

Reciprocal space maps (using a hybrid monochromator) around the (00.6) reflection for samples A, C, G, H, and I (from left to right). The x-ray beam is along the x′-direction; relaxation (with the basal glide system) is only visible for sample I. See Figs. S1S3 of the supplementary material for more RSMs.

FIG. 13.

Reciprocal space maps (using a hybrid monochromator) around the (00.6) reflection for samples A, C, G, H, and I (from left to right). The x-ray beam is along the x′-direction; relaxation (with the basal glide system) is only visible for sample I. See Figs. S1S3 of the supplementary material for more RSMs.

Close modal

The critical thickness for one Al-concentration can be extracted from Fig. 14(a), where the reduction of the relaxation parameter ρxρy from its pseudomorphic value of 1 is tracked for a constant number of PLD pulses (30k). Relaxation begins at x = 0.86 (ϵa = −0.69%) at a thickness of these samples of 780(40) nm. Two series of samples with similar composition but different thicknesses have also been fabricated. For x ≈ 0.61 (x = 0.53), the critical thickness is about 60 nm ± 10 nm (40 nm ± 10 nm) [Fig. 14(b)].

FIG. 14.

(a) Relaxation parameter ρxρy vs Al-concentration x for samples with similar thickness (30k PLD pulses). The gray area visualizes pseudomorphic layers. (b) Relaxation parameter vs layer thickness d for samples with similar Al-concentration x as labeled.

FIG. 14.

(a) Relaxation parameter ρxρy vs Al-concentration x for samples with similar thickness (30k PLD pulses). The gray area visualizes pseudomorphic layers. (b) Relaxation parameter vs layer thickness d for samples with similar Al-concentration x as labeled.

Close modal

From Fig. 14, we can thus deduct three (x, dcrit) value pairs. These are shown together in Fig. 15 with dependence on ∝ ϵ3/2. The exponent thus lies in between dependence on forces ∝ ϵ and on strain energy ∝ ϵ2. A more detailed model for critical thickness goes beyond the scope of this paper and will be subject to further experimental and theoretical investigation.

FIG. 15.

Critical thickness dcrit for (01.2) (AlxGa1−x)2O3/Al2O3 as a function of Al-concentration x. Symbols represent experimental data points of this work. The dashed line follows ∝(1 − x)3/2.

FIG. 15.

Critical thickness dcrit for (01.2) (AlxGa1−x)2O3/Al2O3 as a function of Al-concentration x. Symbols represent experimental data points of this work. The dashed line follows ∝(1 − x)3/2.

Close modal

In order to elucidate microscopic reasons for the anisotropy of the relaxation, we discuss the slip systems and possible Burgers vectors for relaxation. The two shortest Burgers vectors b of Al2O338 are 1/3112¯0 (b = a = 0.476 nm) and 1/31¯101 (b=a3+ζ2/3=0.513 nm). The two prevalent slip systems are the basal (c-) plane {0001}1/3112¯0 and the prismatic (a-) plane {112¯0}1/311¯01 systems39,40 [Fig. 10(b)].

The basal system intersects with the r-plane along the y′-direction and allows relaxation along the x′-direction. There are two Burgers vectors that relax strain in the x′-direction, 1/3[1¯1¯20] and 1/3[12¯10], which possess the same edge bc, and the same tilt bc, components but opposite screw component bc, (in the (x′, y′, z′) coordinate system),

bc=bc,bc,bc,=a23/3+ζ2±13/1+3/ζ2.
(13)

For Al2O3, bc = (0.221, ±0.238, 0.348) nm. The ratio of tilt and edge components is given by bc,/bc,=ζ/3. Two other possible Burgers vectors (±1/3[21¯1¯0]) are pure screw-type and will not lead to strain relaxation.

For the a-plane slip system, the 1/3[1¯101] direction is shown on the (112¯0) plane in Fig. 10(b). The angle α that the dislocation line (intersection of the a- and r-planes) forms with the x′-direction is αa=arctan(3/3+ζ2)=42.9 for Al2O3. The (in-plane) edge, tilt, and screw components of Burgers vector are (for Al2O3 lattice parameters) ba, = 0.373 nm, ba,=bc, = 0.348 nm, and ba, = 0.052 nm, respectively41 [another possible Burgers vector (1/311¯01) is about 90% screw type and expected to contribute little to strain relaxation]. The edge components (within the epitaxial r-plane and perpendicular to the dislocation line) of two Burgers vectors for relaxation of compressive strain from the two a-plane slip systems are shown in Fig. 10(a). The tilt components of the two vectors point in opposite directions. When dislocations arise in equal numbers with their Burgers vectors having up- and downward tilt components, the tilts compensate; in total, such a pair of Burgers vectors adds up to ba′ = (0, a, 0), effectively allowing strain relaxation along the y′-direction.

Through the components of Burgers vector bc′, the in-plane relaxation along x′, cf. (11),

Δϵxx=ϵxxϵ^xx=ϵxx(1ρx),
(14)

and the subsequent tilt βx of the r-plane of the epilayer against that of the substrate is related42 via

βx=Δϵxxbc,bc,=(1ρx)ζFϵxx3.
(15)

The lattice tilt of the r-plane has been measured directly from the RSM around the (02.4) reflection under different azimuths, ϕ = 0 denoting the x-ray beam along the x′-direction (toward the c-direction), as depicted in Fig. 16. In Fig. S3 of the supplementary material, the RSM for sample J is depicted for various azimuthal positions. A quantitative fit with a harmonic function yields the tilts in x′- and y′-directions as listed in Table VI for a few relaxed samples together with the parameters from fitting the 2θ separations and tilt for various reflexes. First of all, the data confirm that the tilt is mostly along the x′-direction and due to the basal slip system. For all samples with (or without) relaxation along y′, no or very small tilt was found in this direction (βy ≪ 0.1°). This means that lattice tilts along x′ due to the prismatic slip systems cancel on average. We note that another possibility could be the occurrence of two distinct (opposite) tilt values as reported in Ref. 43 for the (Al, Ga)N/GaN system, but such an effect is not observed for the present case. In strongly relaxed samples I and J, the tilt is largest, about 0.37°, mostly along the x′-direction. Once the basal slip system is active for strong relaxation, it allows relaxation along the x′-direction and tilt components are summed along this direction, producing significant values for βx.

FIG. 16.

Tilt of the r-plane of the epilayer vs the substrate, measured from the (02.4) reflex under different azimuths, ϕ = 0 denoting the x-ray beam along the −x′-direction for samples H, I, and J (J is similar to I but with 10k PLD pulses). Fits with a 2π-periodic harmonic function are shown as dashed lines; fit parameters are given in Table VI.

FIG. 16.

Tilt of the r-plane of the epilayer vs the substrate, measured from the (02.4) reflex under different azimuths, ϕ = 0 denoting the x-ray beam along the −x′-direction for samples H, I, and J (J is similar to I but with 10k PLD pulses). Fits with a 2π-periodic harmonic function are shown as dashed lines; fit parameters are given in Table VI.

Close modal
TABLE VI.

In-plane relaxation parameters ρx and ρy and tilt of the (01.2) plane in the x′- and y′ directions for strongly relaxed samples. Fit parameters βx, βy, and β=βx2+βy2 to peak separations and tilts. Parameters βxϕ, βyϕ, and βϕ from direct tilt measurement from (02.4) RSM for different azimuths ϕ (fit parameters from Fig. 16 and errors ±0.025°). See Fig. S3 of the supplementary material for the raw data of sample J.

No.ρxρyβx (°)βy (°)β (°)βxϕ (°)βyϕ (°)βϕ (°)
1.018 0.853 0.029 0.022 0.037 0.037 0.013 0.039 
1.030 0.727 0.112 0.020 0.114 0.105 0.051 0.117 
0.758 0.575 0.285 0.179 0.337 0.361 0.112 0.377 
0.753 0.685 0.344 0.012 0.344 0.340 0.015 0.364 
No.ρxρyβx (°)βy (°)β (°)βxϕ (°)βyϕ (°)βϕ (°)
1.018 0.853 0.029 0.022 0.037 0.037 0.013 0.039 
1.030 0.727 0.112 0.020 0.114 0.105 0.051 0.117 
0.758 0.575 0.285 0.179 0.337 0.361 0.112 0.377 
0.753 0.685 0.344 0.012 0.344 0.340 0.015 0.364 

A direct visual proof of the presence of the prismatic a-plane slip system is provided by the crosshatch pattern observed on a (partially) relaxed sample Y (xEDX = 0.47), as depicted in Fig. 17. We attribute the corrugation to slip lines. We find the line direction α, as defined previously, produced by the a- and m-plane prismatic slip on an epitaxial plane inclined by θ (for any azimuth) to be given as

αa=±arctan3cosθ,
(16)
αm=±arctancosθ/3.
(17)

For the basal system, always αc = π/2. It is remarkable that Eqs. (16) and (17) depend only on the interface inclination angle θ and thus are universal and material independent. However, for a given (hk.l) plane, the inclination θ, of course, depends on c/a. The angles αa and αm are depicted in Fig. 18 as a function of θ. On the c-plane, for θ = 0, the expected angles of αa = 60° and αm = 30° are recovered. The ratio αa/αm varies between 2 for θ = 0 and 3 for θπ/2.

FIG. 17.

SEM image of the (partially) relaxed (Al0.47Ga0.53)2O3 epilayer on the r-plane Al2O3. The blue and red lines indicate the evaluated angle of the crosshatch pattern. The inset is the Fourier transform of the image from which αa = 40.2(5)° is measured.

FIG. 17.

SEM image of the (partially) relaxed (Al0.47Ga0.53)2O3 epilayer on the r-plane Al2O3. The blue and red lines indicate the evaluated angle of the crosshatch pattern. The inset is the Fourier transform of the image from which αa = 40.2(5)° is measured.

Close modal
FIG. 18.

Dislocation line orientation angle α (vs the projected c-axis direction, as defined in Fig. 10) for the a-plane (solid line) and the m-plane (dashed line) prismatic glide systems as a function of interface inclination angle θ. Experimental data (cf. Table VII) are shown for this work (black square) and from epitaxy of wurtzites in semipolar orientations, (Al, Ga)N/GaN42 (diamond), (In, Ga)N/GaN47,49–51 (circles), GaN/Si52 (hexagon), and (Mg, Zn)O/ZnO53 (star).

FIG. 18.

Dislocation line orientation angle α (vs the projected c-axis direction, as defined in Fig. 10) for the a-plane (solid line) and the m-plane (dashed line) prismatic glide systems as a function of interface inclination angle θ. Experimental data (cf. Table VII) are shown for this work (black square) and from epitaxy of wurtzites in semipolar orientations, (Al, Ga)N/GaN42 (diamond), (In, Ga)N/GaN47,49–51 (circles), GaN/Si52 (hexagon), and (Mg, Zn)O/ZnO53 (star).

Close modal

The observed crosshatch angle from SEM images of sample Y (recorded for the stage at 0° and 90° giving the same experimental values, best extracted from the Fourier images) is αexp = 40.2(5)°. Our data point is entered into Fig. 18 and obviously fits to the a-plane type glide system. This value is close to the expected value of αa = 42.9°. The reason for the surprisingly large discrepancy of about 2.5° is unknown at this point. Our simulations show that effects of strain relaxation and/or tilt of the epitaxial plane have a small influence but cannot be invoked for a quantitative explanation unless huge amounts of additional screw dislocations are invoked. A 3.2° off-cut from the exact r-plane could also explain this difference but is definitely not present. However, in Sec. VI, it is shown that deviations of several degrees are also found in other systems and seem typical. Dislocation line orientations for a possible m-plane glide system [αm=arctan(1/3+ζ2)17] or any other glide system listed in Ref. 38 do not agree at all with the present situation for sure.

The resolved shear stresses for the basal (τc) and the two prismatic (112¯0)[1¯100] and (12¯10)[101¯0] (τa) slip systems are

τc=34(σyyσxx)+12σyz34(σyyσxx),  
(18)
τa=τm=32σxz+12σyz32σxz.
(19)

Note that these shear stresses σ refer to the crystal system. The present pseudomorphic strain situation forces σyz = σxy = 0, recovering the formulas given in Ref. 1.

For the r-plane, τa is little larger than τc, as shown in Fig. 19. For the third a-plane (21¯1¯0)[011¯0] slip system, τa = 0 since this plane is perpendicular to the r-plane. In Fig. 19, the resolved shear stress has also been calculated for lattice parameters at 800° close to (growth) temperature, using the temperature dependent lattice parameters from Refs. 44 and 45. The situation τaτc seems to prevail, however, that the same room temperature elastic constants were used since their temperature dependence is known for Al2O346 but not for α-Ga2O3.

FIG. 19.

Resolved shear strain in pseudomorphic (AlxGa1−x)2O3 epilayers on Al2O3 for the prismatic (τa, dashed line) and basal (τc, solid line) slip systems as a function of (a) the interface orientation for x = 0.9 [room temperature (black lines) and 800 °C (blue lines)] and (b) as a function of the Al-concentration for the r-plane at room temperature. The vertical dashed line in panel (a) indicates the inclination angle of the r-plane.

FIG. 19.

Resolved shear strain in pseudomorphic (AlxGa1−x)2O3 epilayers on Al2O3 for the prismatic (τa, dashed line) and basal (τc, solid line) slip systems as a function of (a) the interface orientation for x = 0.9 [room temperature (black lines) and 800 °C (blue lines)] and (b) as a function of the Al-concentration for the r-plane at room temperature. The vertical dashed line in panel (a) indicates the inclination angle of the r-plane.

Close modal

The critical resolved shear stress at a temperature of about 800° has been found to be 0.4–0.5 GPa as reported in Ref. 39 from studies on bulk Al2O3. At this temperature, it is similar for the basal and prismatic slip systems; for lower (higher) temperatures, the prismatic (basal) slip system dominates. Since our growth temperature is rather bit lower, the prismatic slip system is expected to be activated first under the present conditions. It seems presently favored by its higher resolved shear stress and its lower critical resolved shear stress.

The situation found here for corundum (Al, Ga)2O3 on Al2O3 is quite different from the wurtzite (Al, In, Ga)N on GaN systems that have been investigated for basal and prismatic slip in various works. For the nitride systems, first and foremost the basal slip system is active for growth on polar and semipolar substrates (the dislocation direction in this case is αc = π/2 for any θ).

Only for semipolar orientations θ > 70° and sufficient epilayer thickness, relaxation with a prismatic slip system becomes activated as reported in Refs. 42 and 47. In the report on (Al, Ga)N/GaN,42 it is stated that the m-plane type prismatic slip system (11¯00)1/3112¯0 is activated. Our analysis of the dislocation line directions visible in panchromatic cathodoluminescence images yields dislocation orientation angles in this report of α = 15.5° for epitaxy on (303¯1) [from Fig. 4(c) of Ref. 42]. This value (error about 0.5°) is shown as diamond in Fig. 18. The experimental dislocation orientation clearly favors the a-plane slip system. Both the a- and m-plane types are possible slip systems in the wurtzite structure.48 The various material systems discussed here are summarized in Table VII.

TABLE VII.

Dislocation angle α for various material systems and epitaxial planes (hk.l) and their inclination angle θ.

Material(hk.l) θ (°)α (°)Reference
(Al0.47Ga0.53)2O3/Al2O3 (01.2) 57.6 40.2 This work 
Al0.13Ga0.87N/GaN (30.1) 79.9 15.5 42  
In0.1Ga0.9N/GaN (20.1) 79.9 21.3 47  
In0.1Ga0.9N/GaN (20.1) 79.9 22.3 47  
In0.1Ga0.9N/GaN (30.1) 75.1 15.5 47  
In0.1Ga0.9N/GaN (30.1) 75.1 18.0 47  
In0.06Ga0.94N/GaN (20.1) 79.9 5.0 49  
In0.06Ga0.94N/GaN (20.1) 75.1 9.6 49  
In0.07Ga0.93N/GaN (11.2) 58.4 20, 42 50  
In0.126Ga0.874N/GaN (11.2) 58.4 45 51  
GaN/Si(001) 7° off (10.1) 62.0 14.9 52  
Mg0.18Zn0.82O/ZnO (01.2) 42.7 24.5 53  
Material(hk.l) θ (°)α (°)Reference
(Al0.47Ga0.53)2O3/Al2O3 (01.2) 57.6 40.2 This work 
Al0.13Ga0.87N/GaN (30.1) 79.9 15.5 42  
In0.1Ga0.9N/GaN (20.1) 79.9 21.3 47  
In0.1Ga0.9N/GaN (20.1) 79.9 22.3 47  
In0.1Ga0.9N/GaN (30.1) 75.1 15.5 47  
In0.1Ga0.9N/GaN (30.1) 75.1 18.0 47  
In0.06Ga0.94N/GaN (20.1) 79.9 5.0 49  
In0.06Ga0.94N/GaN (20.1) 75.1 9.6 49  
In0.07Ga0.93N/GaN (11.2) 58.4 20, 42 50  
In0.126Ga0.874N/GaN (11.2) 58.4 45 51  
GaN/Si(001) 7° off (10.1) 62.0 14.9 52  
Mg0.18Zn0.82O/ZnO (01.2) 42.7 24.5 53  

For (In, Ga)N/GaN, we find α = 18.0° and 15.5° on (303¯1¯) [Figs. 3(f) and 3(g) of Ref. 47] and α = 21.3° and 22.3° on (202¯1¯) [Figs. 3(c) and 3(d) of Ref. 47], also suggesting the a-plane slip system. The m-plane slip system has correctly been assigned in an analysis of (In, Ga)N on semipolar (202¯1) and (303¯1) GaN49 (cf. circles in Fig. 18). For (In, Ga)N on GaN (112¯2), coexisting a-plane and m-plane slip system dislocations were also observed in Ref. 50 (cf. circles in Fig. 18). Another work on (In, Ga)N on (112¯2) GaN51 exhibits angles consistent with a-plane orientation, but m-plane slip was discussed in this report. The non-basal plane dislocations in GaN on patterned silicon52 seemingly stem from m-type glide planes (cf. hexagon in Fig. 18). In (Mg, Zn)O on r-plane ZnO, the m-plane slip system was also observed53 (cf. star in Fig. 18). We emphasize that Eqs. (16) and (17) indeed describe results from the literature for various material systems quite well.

In summary, we have quantitatively analyzed the strain state of pseudomorphic and partially relaxed sesquioxide α-(Al, Ga)2O3 epilayers on (01.2) r-plane Al2O3, which have been prepared by pulsed laser deposition. A model has been put forward that simultaneously explains the observed PLD target to epilayer stoichiometry transfer and the dependence of growth rate on the cation ratio.

An exact continuum elasticity model has been established, formulated linearly in the lattice constants and the interface inclination angle of the substrate. The differences between the rhombohedral (threefold, sesquioxide) and the hexagonal (sixfold, wurtzite) strain situation have been pointed out.

In general, for a number of pseudomorphic and partially relaxed samples of different Al-concentrations and thicknesses, the x-ray diffraction data can be fitted to the theory with high precision. The relaxation is found to be anisotropic due to basal and prismatic slip. Under the present growth conditions below 800 °C, first, the prismatic slip system is activated. The prismatic slip system has been identified as being an a-plane type from the orientation of slip lines. The strain relaxation with the prismatic slip system is not connected with a tilt of the epitaxial plane due to the cancellation of the tilt components of Burgers vectors of the two equivalent systems.

For larger strain relaxation beyond about 30% with the prismatic slip system, the basal slip system is activated. As expected, the tilt components of two possible Burgers vectors are identical and thus such strain relaxation leads to a global tilt of the epitaxial plane.

Universal formulas have been derived for the dislocation line orientation as a function of θ and also compared to various results in the literature on wurtzite nitrides and oxides.

See the supplementary material for various reciprocal space maps (RSMs).

We would like to thank M. Hahn for PLD target preparation, H. Hochmuth and M. Kneiß for PLD growth, C. Sturm for thickness determination with ellipsometry, and J. Lenzner for EDX measurements and SEM thickness determination. Software for displaying RSMs was kindly provided by E. Rose. We are grateful for fruitful discussions with J. Zúñiga-Pérez, P. Vénneguès, and J.-M. Chauveau as well as the kind hospitality at CRNS-CRHEA, Valbonne. Communications with J. Furthmüller, O. Bierwagen, and M. Hanke are also appreciated. This work was supported by La Fédération Wolfgang Doeblin, Nice, the European Social Fund within the Young Investigator Group “Oxide Heterostructures” (Grant No. SAB 100310460) and by Universität Leipzig in the framework of research profile area “Complex Matter.” The x-ray diffractometer had been granted by Deutsche Forschungsgemeinschaft and the Federal State of Saxony (Grant No. INST 268/284-1).

1.
A. E.
Romanov
,
T. J.
Baker
,
S.
Nakamura
, and
J. S.
Speck
, “
Strain-induced polarization in wurtzite III-nitride semipolar layers
,”
J. Appl. Phys.
100
,
023522
(
2006
).
2.
M.
Grundmann
and
J.
Zúñiga-Pérez
, “
Pseudomorphic ZnO-based heterostructures: From polar through all semipolar to nonpolar orientations
,”
Phys. Status Solidi B
253
,
351
360
(
2016
).
3.
M.
Grundmann
, “
Elastic theory of pseudomorphic monoclinic and rhombohedral heterostructures
,”
J. Appl. Phys.
124
,
185302-1
185302-10
(
2018
).
4.
M.
Lorenz
,
S.
Hohenberger
,
E.
Rose
, and
M.
Grundmann
, “
Atomically stepped, pseudomorphic, corundum-phase (Al1−xGax)2O3 thin films (0 ≤ x < 0.08) grown on R-plane sapphire
,”
Appl. Phys. Lett.
113
,
231902-1
231902-5
(
2018
).
5.
M.
Grundmann
, “
Strain in pseudomorphic monoclinic Ga2O3-based heterostructures
,”
Phys. Status Solidi B
254
,
1700134
(
2017
).
6.
D.
Shinohara
and
S.
Fujita
, “
Heteroepitaxy of corundum-structured α-Ga2O3 thin Films on α-Al2O3 Substrates by ultrasonic mist chemical vapor deposition
,”
Jpn. J. Appl. Phys.
47
,
7311
7313
(
2008
).
7.
K.
Kaneko
,
S.
Fujita
, and
T.
Hitora
, “
A power device material of corundum-structured α-Ga2O3 fabricated by MIST EPITAXY® technique
,”
Jpn. J. Appl. Phys.
57
,
02CB18-1
02CB18-5
(
2018
).
8.
T.
Wang
,
W.
Li
,
C.
Ni
, and
A.
Janotti
, “
Band gap and band offset of Ga2O3 and (AlxGa1?x)2O3 alloys
,”
Phys. Rev. Appl.
10
,
011003
(
2018
).
9.
T.
Uchida
,
R.
Jinno
,
S.
Takemoto
,
K.
Kaneko
, and
S.
Fujita
, “
Characterization of band offset in α-(AlxGa1?x)2O3/α-Ga2O3 heterostructures
,” in
Proceedingsof 2016 Compound Semiconductor Week (CSW)
(
IEEE
,
Piscataway
,
2016
), ThC1-5.
10.
Y. J.
Kim
,
Y. T.
Kim
,
H. K.
Yang
,
J. C.
Park
,
J. I.
Han
,
Y. E.
Lee
, and
H. J.
Kim
, “
Epitaxial growth of ZnO thin films on R-plane sapphire substrate by radio frequency magnetron sputtering
,”
J. Vac. Sci. Technol., A
15
,
1103
1107
(
1997
).
11.
M. D.
Craven
,
S. H.
Lim
,
F.
Wu
,
J. S.
Speck
, and
S. P.
DenBaars
, “
Structural characterization of nonpolar (112¯0) a-plane GaN thin films grown on (11¯02) r-plane sapphire
,”
Appl. Phys. Lett.
81
,
469
471
(
2002
).
12.
L.
Liu
and
J. H.
Edgar
, “
Substrates for gallium nitride epitaxy
,”
Mater. Sci. Eng.
37
,
61
127
(
2002
).
13.
H.
Nishinaka
,
D.
Tahara
,
S.
Morimoto
, and
M.
Yoshimoto
, “
Epitaxial growth of α-Ga2O3 thin films on a-, m-, and r-plane sapphire substrates by mist chemical vapor deposition using α-Fe2O3 buffer layers
,”
Mater. Lett.
205
,
28
31
(
2017
).
14.
S.
Fujita
,
M.
Oda
,
K.
Kaneko
, and
T.
Hitora
, “
Evolution of corundum-structured III-oxide semiconductors: Growth, properties, and devices
,”
Jpn. J. Appl. Phys.
55
,
1202A3-1
1202A3-9
(
2016
).
15.
R.
Kumaran
,
T.
Tiedje
,
S. E.
Webster
,
S.
Penson
, and
W.
Li
, “
Epitaxial Nd-doped α-(Al1−xGax)2O3 films on sapphire for solid-state waveguide lasers
,”
Opt. Lett.
35
,
3793
3795
(
2010
).
16.
J.
Wang
,
B.
Guo
,
Q.
Zhao
,
C.
Zhang
,
Q.
Zhang
,
H.
Chen
, and
J.
Sun
, “
Dependence of material removal on crystal orientation of sapphire under cross scratching
,”
J. Eur. Ceram. Soc.
37
,
2465
2472
(
2017
).
17.
P. A.
Yunin
and
Y. N.
Drozdov
, “
How to distinguish between opposite faces of an a-plane sapphire wafer
,”
J. Appl. Crystallogr.
51
,
549
551
(
2018
).
18.
Y.
Mizumoto
,
P.
Maas
,
Y.
Kakinuma
, and
S.
Min
, “
Investigation of the cutting mechanisms and the anisotropic ductility of monocrystalline sapphire
,”
CIRP Ann.
66
,
89
92
(
2017
).
19.
W.
Voigt
,
Lehrbuch der Kristallphysik
(
Springer
,
Wiesbaden
,
1966
), reproduction of the 1928 edition of the 1910 original textbook.
20.
J. F.
Nye
,
Physical Properties of Crystals
(
Clarendon Press
,
Oxford
,
2012
).
21.
P. A.
Yunin
,
Yu. N.
Drozdov
,
O. I.
Khrykin
, and
V. A.
Grigoryev
, “
Investigation of the anisotropy of the structural properties of GaN(0001) layers grown by MOVPE on a-plane (112¯0) sapphire
,”
Semiconductors
52
,
1412
1415
(
2018
).
22.
J.
Furthmüller
and
F.
Bechstedt
, “
Quasiparticle bands and spectra of Ga2O3 polymorphs
,”
Phys. Rev. B
93
,
115204
(
2016
).
23.
J.
Furthmüller
, private communication (
2019
).
24.
G. T.
Dang
,
T.
Yasuoka
,
Y.
Tagashira
,
T.
Tadokoro
,
W.
Theiss
, and
T.
Kawaharamura
, “
Bandgap engineering of α-(AlxGa1−x)2O3 by a mist chemical vapor deposition two-chamber system and verification of Vegard’s Law
,”
Appl. Phys. Lett.
113
,
062102-1
062102-5
(
2018
).
25.
E. R.
Dobrovinskaya
,
L. A.
Lytvynov
, and
V.
Pishchik
,
Sapphire: Material, Manufacturing, Applications
(
Springer Science+Business Media
,
2009
).
26.
M.
Marezioo
and
J. R.
Remeika
, “
Bond Lengths in the α-Ga2O3 Structure and the high-pressure Phase of Ga2−xFexO3
,”
J. Chem. Phys.
46
,
1862
(
1967
).
27.
D. B.
Hovis
,
A.
Reddy
, and
A. H.
Heuer
, “
X-ray elastic constants for α-Al2O3
,”
Appl. Phys. Lett.
88
,
131910
(
2006
).
28.
All symbolic and numerical calculations were made with computer code Mathematica, Version 12.0.0.0,
Wolfram Research
,
Champaign, IL
,
2019
.
29.

The internal codes of the samples discussed in this paper in detail are the following: A: W4957; B: W4958; C: W4816; D: W4820; E: W4749; F: W4778; G: W4777; H: W4748; I: W4671; J: W5118; and Y:W4658.

30.
M.
Lorenz
,
Pulsed Laser Deposition
, Digital Encyclopedia of Applied Physics Eap810 (
Wiley-VCH Verlag GmbH & Co. KGaA
,
Weinheim
,
2019
).
31.
P.
Vogt
and
O.
Bierwagen
, “
The competing oxide and sub-oxide formation in metal-oxide molecular beam epitaxy
,”
Appl. Phys. Lett.
106
,
081910-1
081910-4
(
2015
).
32.
A.
Hassa
,
H.
von Wenckstern
,
L.
Vines
, and
M.
Grundmann
, “
Influence of oxygen pressure on growth of Si-doped (AlxGa1−x)2O3 thin films on c-sapphire substrates by pulsed laser deposition
,”
ECS J. Solid State Sci. Technol.
8
,
Q3217
Q3220
(
2019
).
33.
H.
von Wenckstern
,
M.
Kneiß
,
P.
Storm
, and
M.
Grundmann
, “
A review of the segmented-target approach to combinatorial material synthesis by pulsed-laser deposition
,”
Phys. Status Solidi B
(published online).
34.
H.
von Wenckstern
,
Z.
Zhang
,
F.
Schmidt
,
J.
Lenzner
,
H.
Hochmuth
, and
M.
Grundmann
, “
Continuous composition spread using pulsed-laser deposition with a single, segmented target
,”
CrystEngComm
15
,
10020
10027
(
2013
).
35.
M.
Kneiß
,
P.
Storm
,
G.
Benndorf
,
M.
Grundmann
, and
H.
von Wenckstern
, “
Combinatorial material science and strain engineering enabled by pulsed laser deposition using radially segmented targets
,”
ACS Comb. Sci.
20
,
643
652
(
2018
).
36.
P. H.
Richter
, “
Estimating errors in least-squares fitting
,” TDA Progress Report No. 42-122, 1–137,
Jet Propulsion Laboratory
,
Pasadena
,
1995
.
37.
Z.
Cheng
,
M.
Hanke
,
P.
Vogt
,
O.
Bierwagen
, and
A.
Trampert
, “
Phase formation and strain relaxation of Ga2O3 on c-plane and a-plane sapphire substrates as studied by synchrotron-based x-ray diffraction
,”
Appl. Phys. Lett.
111
,
162104-1
162104-4
(
2017
).
38.
J. D.
Snow
and
A. H.
Heuer
, “
Slip Systems in Al2O3
,”
J. Am. Ceram. Soc.
56
,
153
157
(
1973
).
39.
J.
Castaing
,
J.
Cadoz
, and
S.
Kirby
, “
Deformation of Al2O3 single crystals between 25°C and 1800°C: Basal and prismatic slip
,”
J. Phys. Colloq.
42
(
C3
),
C3
43
(
1981
).
40.
J.
Castaing
,
J.
Cadoz
, and
S.
Kirby
, “
Prismatic slip of Al2O3 single crystals below 1000°C in compression under hydrostatic pressure
,”
J. Am. Ceram. Soc.
64
,
504
511
(
1981
).
41.

ba,=3a/2×ζ2/ζ4+15ζ2+36 and ba,=a/3×(ζ26)/ζ2+12.

42.
A. M.
Smirnov
,
E. C.
Young
,
V. E.
Bougrov
,
J. S.
Speck
, and
A. E.
Romanov
, “
Critical thickness for the formation of misfit dislocations originating from prismatic slip in semipolar and nonpolar III-nitride heterostructures
,”
APL Mater.
4
,
016105-1
016105-8
(
2016
).
43.
S.
Yoshida
,
T.
Yokogawa
,
Y.
Imai
,
S.
Kimura
, and
O.
Sakata
, “
Evidence of lattice tilt and slip in m-plane InGaN/GaN heterostructure
,”
Appl. Phys. Lett.
99
,
131909-1
131909-3
(
2011
).
44.
W. M.
Yim
and
R. J.
Paff
, “
Thermal expansion of AlN, sapphire, and silicon
,”
J. Appl. Phys.
45
,
1456
1457
(
1974
).
45.
L. J.
Eckert
and
R. C.
Bradt
, “
Thermal expansion of alpha Ga2O3
,”
J. Am. Ceram. Soc.
56
,
229
230
(
1973
).
46.
S.-L.
Shang
,
H.
Zhang
,
Y.
Wang
, and
Z.-K.
Liu
, “
Temperature-dependent elastic stiffness constants of α- and θ-Al2O3 from first-principles calculations
,”
J. Phys.: Condens. Matter
22
,
375403-1
375403-8
(
2010
).
47.
D. L.
Becerra
,
Y.
Zhao
,
S. H.
Oh
,
C. D.
Pynn
,
K.
Fujito
,
S. P.
DenBaars
, and
S.
Nakamura
, “
High-power low-droop violet semipolar (303¯1¯) InGaN/GaN light-emitting diodes with thick active layer design
,”
Appl. Phys. Lett.
105
,
171106-1
171106-4
(
2014
).
48.
Yu. A.
Osipiyan
and
I. S.
Smirnova
, “
Perfect dislocations in the wurtzite lattice
,”
Phys. Status Solidi B
30
,
19
29
(
1968
).
49.
M. T.
Hardy
,
P. S.
Hsu
,
F.
Wu
,
I. L.
Koslow
,
E. C.
Young
,
S.
Nakamura
,
A. E.
Romanov
,
S. P.
DenBaars
, and
J. S.
Speck
, “
Trace analysis of non-basal plane misfit stress relaxation in (202¯1) and (303¯1¯) semipolar InGaN/GaN heterostructures
,”
Appl. Phys. Lett.
100
,
202103-1
202103-4
(
2012
).
50.
F.
Wu
,
E. C.
Young
,
I.
Koslow
,
M. T.
Hardy
,
P. S.
Hsu
,
A. E.
Romanov
,
S.
Nakamura
,
S. P.
DenBaars
, and
J. S.
Speck
, “
Observation of non-basal slip in semipolar InxGa1−xN/GaN heterostructures
,”
Appl. Phys. Lett.
99
,
251909-1
251909-4
(
2011
).
51.
I. L.
Koslow
,
M. T.
Hardy
,
P. S.
Hsu
,
F.
Wu
,
A. E.
Romanov
,
E. C.
Young
,
S.
Nakamura
,
S. P.
DenBaars
, and
J. S.
Speck
, “
Onset of plastic relaxation in semipolar (112¯2)InxGa1xN/GaN heterostructures
,”
J. Crystal Growth
388
,
48
53
(
2014
).
52.
M.
Khoury
,
P.
Vennéguès
,
M.
Leroux
,
V.
Delaye
,
G.
Feuillet
, and
J.
Zúñiga-Pérez
, “
Defect blocking via laterally induced growth of semipolar (101¯1) GaN on patterned substrates
,”
J. Phys. D: Appl. Phys.
49
,
475104-1
475104-8
(
2016
).
53.
N.
Le Biavan
,
J.-M.
Chauveau
, private communication, and
N.
Le Biavan
, “
Vers un laser à cascade quantique à base d’oxyde de zinc
,” Ph.D. thesis,
Université Côte d’Azur
(
2019
).

Supplementary Material