Pseudomorphic and partially relaxed layers of corundum phase (Al, Ga)_{2}O_{3} epilayers on (01.2)-oriented Al_{2}O_{3} 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.

## I. INTRODUCTION

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: Al_{2}O_{3}),^{3,4} and monoclinic (*β*-Ga_{2}O_{3})^{3,5} crystals.

Here, we discuss epitaxy of *α*-phase (corundum) (Al, Ga)_{2}O_{3} alloys since the sesquioxide in this (non-equilibrium) phase is a promising material for power transistors^{6,7} and can be used to build dielectric heterostructures. The conduction band offsets in the corundum (Al, Ga)_{2}O_{3} system have been predicted to be larger than for the monoclinic phase^{8} with small valence band offsets as confirmed experimentally.^{9}

We focus on epitaxy on the *r*-plane.^{4} It is well known that wurtzite ZnO^{10} and GaN^{11,12} 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 ($11\xaf.2$) or ($1\xaf0.2$)], its backside, the ($01\xaf.2\xaf$) $r\xaf$-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 $(01\xaf.2)$.^{16}

As stated in Ref. 17, the *r*- and $r\xaf$-planes are identical (as is true for the *m*/$m\xaf$- and *c*/$c\xaf$-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)_{2}O_{3} on *α*-Ga_{2}O_{3} were reported in Ref. 14, but no details about the mechanisms of plastic relaxation were given. The *α*-(Al, Ga)_{2}O_{3} on Al_{2}O_{3} 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.

## II. THEORY

### A. Definitions and material parameters

Al_{2}O_{3} has a rhombohedral lattice (space group R$3\xaf$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 C_{14} 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 ($21\xaf.0$) 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 *a*_{1} = ** T** (1, 0, 0)

^{T},

*a*_{2}=

**(0, 1, 0)**

*T*^{T}, and

*a*_{3}=

**(0, 0, 1)**

*T*^{T}with the transformation matrix (

*ζ*=

*c*/

*a*),

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

and the azimuthal orientation *ϕ*, *ϕ* = 0 denoting the direction of *a*_{1} (Fig. 1). Note that *θ* depends on *ζ*, which is, in general, different for the epilayer and substrate. We also introduce *ξ* = *ζ*_{F}/*ζ*_{S} = (*c*_{F}/*a*_{F})/(*c*_{S}/*a*_{S}) denoting the *c*/*a* ratios of the epilayer and substrate, and the indices denoting the substrate and film property (see Fig. 2).

The material parameters we use are given in Tables I and II. We emphasize that the sign of C_{14} for Ga_{2}O_{3} 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, C_{14} for Ga_{2}O_{3} and Al_{2}O_{3} has the same sign.^{23}

Material . | C_{11}
. | C_{12}
. | C_{13}
. | C_{33}
. | C_{14}
. | C_{44}
. | Reference . |
---|---|---|---|---|---|---|---|

α-Al_{2}O_{3} | 4.97 | 1.63 | 1.16 | 5.01 | 0.22 | 1.47 | 27 |

α-Ga_{2}O_{3} | 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)_{2}O_{3} has been tested,^{24} although not with high precision.

We introduce the strains *ϵ*_{a} = *a*_{S}/*a*_{F} − 1 and *ϵ*_{c} = *c*_{S}/*c*_{F} − 1 = *ξ a*_{S}/*a*_{F} − 1 that are depicted in Fig. 2.

### B. Specificities of the rhombohedral/trigonal system

It is well known^{3} that the out-of-plane strain for pseudomorphic layers on the *a*-plane is given by

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 C_{14} ≠ 0, we find (by simplifying involved analytical formulas for arbitrary azimuths) the out-of-plane strain on the *m*-plane to be

with $C\u0303=C142/C44$. Obviously, for the hexagonal case with C_{14} = 0, Eq. (4) degenerates to Eq. (3). In a numerical example of (Al_{0.9}Ga_{0.1})_{2}O_{3} on Al_{2}O_{3}, $C\u0303\u22480.03$ is small compared to C_{11}, C_{12}, or C_{13}, all being larger than 1 (in units of 10^{11} 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 C_{14}.

The difference in out-of-plane strain for various azimuthal directions becomes larger for other inclinations and is dependent on the sign of C_{14}, 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 (*θ*_{S} = 57.6° for Al_{2}O_{3}). Notably, the *r*-plane and the *r*′-plane behave differently.

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 $(11\xaf.l)$ 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 C_{14} 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 C_{14}. 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.

### C. Exact formulation considering different angles $\theta $ for substrate and epilayer

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 $RS\u2009TS\u2009TF\u22121\u2009RFT\u22121$, where *R*_{S} and *R*_{F} denote the rotation matrices with the different inclination angles of the same (*hk*.*l*)-plane of substrate *θ*_{S} and film material *θ*_{F}, respectively (due to different *c*/*a*-ratios). The matrices ** T** denote the transformation for the crystal lattice as in (1). The exact relation of

*θ*

_{F}and

*θ*

_{S}

*for the same*(

*hk*.

*l*)

*interface plane*is (for any angle)

The difference *θ*_{F} − *θ*_{S} is depicted in Fig. 7. For *θ*_{S} = *n π*/2, *θ*_{F} = *θ*_{S}. For 0 ≤ *θ*_{S} ≤ *π*/2, the formula is equivalent to $\theta F=arccos(1/\xi 2\u2009tan2\theta S+1)$; the extremum of *θ*_{F} − *θ*_{S} is at $\theta Se=arccot\u2009\xi \u2248(\pi +1\u2212\xi )/4+(1\u2212\xi )2/8$, the approximation being valid for values of *ξ* close to 1. The extremal value $\theta F(\theta Se)\u2212\theta Se$ is $2\u2061arctan(\xi )\u2212\pi /2\u2248\u2212(1\u2212\xi )/2\u2212(1\u2212\xi )2/4$. For the present case of (Al, Ga)_{2}O_{3}/Al_{2}O_{3} structures, even the linear approximations are very good since 0.9876 ≤ *ξ* ≤ 1 (cf. Fig. 2).

The difference *θ*_{F} − *θ*_{S} being in the 0.1° range does not seem large, but one should keep in mind that compared to the involved lattice plane tilts Δ*ω* 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.

Δ2θ (hk.l)
. | A (r)
. | B ($r\xaf$) . | C . | D . | E . | F . | G . | H . | I . |
---|---|---|---|---|---|---|---|---|---|

(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 |

$(1\xaf4\xaf.6)$ | 1.0600 | 1.0500 | 1.160 | 0.680 | 1.845 | 1.620 | 1.950 | n.m. | n.m. |

$(4\xaf1\xaf.6)$ | 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 |

$(51\xaf.6)$ | 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 |

$|\Delta \u20092\theta |\xaf$ | 1.061 | 1.037 | 1.152 | 0.684 | 1.823 | 1.609 | 2.022 | 2.560 | 3.296 |

Δ2θ (hk.l)
. | A (r)
. | B ($r\xaf$) . | C . | D . | E . | F . | G . | H . | I . |
---|---|---|---|---|---|---|---|---|---|

(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 |

$(1\xaf4\xaf.6)$ | 1.0600 | 1.0500 | 1.160 | 0.680 | 1.845 | 1.620 | 1.950 | n.m. | n.m. |

$(4\xaf1\xaf.6)$ | 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 |

$(51\xaf.6)$ | 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 |

$|\Delta \u20092\theta |\xaf$ | 1.061 | 1.037 | 1.152 | 0.684 | 1.823 | 1.609 | 2.022 | 2.560 | 3.296 |

Δω (hk.l)
. | A . | B . | C . | D . | E . | F . | G . | H . | I . |
---|---|---|---|---|---|---|---|---|---|

(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 |

$(51\xaf.6)$ | 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)
. | A . | B . | C . | D . | E . | F . | G . | H . | I . |
---|---|---|---|---|---|---|---|---|---|

(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 |

$(51\xaf.6)$ | 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 |

No. . | x_{EDX}
. | x_{fit}
. | σ_{2θ} (°)
. | σ_{ω} (°)
. |
---|---|---|---|---|

A | 0.859 | 0.866 | 0.015 | 0.0035 |

B | 0.863 | 0.870 | 0.017 | 0.0047 |

C | 0.846 | 0.855 | 0.015 | 0.0031 |

D | 0.919 | 0.914 | 0.009 | 0.0042 |

No. . | x_{EDX}
. | x_{fit}
. | σ_{2θ} (°)
. | σ_{ω} (°)
. |
---|---|---|---|---|

A | 0.859 | 0.866 | 0.015 | 0.0035 |

B | 0.863 | 0.870 | 0.017 | 0.0047 |

C | 0.846 | 0.855 | 0.015 | 0.0031 |

D | 0.919 | 0.914 | 0.009 | 0.0042 |

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

In first order of (1 − *ξ*), $\u03f5xx\u2032$ ≈ *ϵ*_{a} + (1 − *ξ*) (1 + *ϵ*_{a}) sin^{2}*θ*_{S} (for 0 ≤ *θ*_{S} ≤ *π*/2); For *ξ* → 1, $\u03f5xx\u2032$ → $\u03f5yy\u2032$ = *ϵ*_{a} and the in-plane strain becomes isotropic and is independent of the inclination angle. All calculations here^{28} are made taking into account the difference of *θ* for the epilayer and substrate due to their different *c*/*a* ratios exactly.

## III. EXPERIMENTAL METHODS

All epitaxial layers^{29} were deposited using pulsed laser deposition^{30} (PLD) on Al_{2}O_{3} 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) Al_{2}O_{3} and (5N) Ga_{2}O_{3} 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 (K_{α}-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 ($K\alpha 1$-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 PIXcel^{3D} array detector or a Ge secondary monochromator with proportional counter (triple-axis configuration). Fast RSMs were measured with the frame-based option of the PIXcel^{3D} 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.

## IV. EXPERIMENTAL RESULTS AND ANALYSIS

### A. Stoichiometry transfer

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 *x*_{t}, respectively. The experimental data for *x* vs *x*_{t} are depicted in Fig. 8. We develop a simple model of this dependency. The Al/Ga-ratio arriving at the growth surface shall be *px*_{t}/(1 − *x*_{t}), 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 MBE^{31} 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 (∝*px*_{t}) to all filled sites [∝*px*_{t} + *r*(1 − *x*_{t})],

with *q* = *r*/*p*. Naturally, this function fulfills the obvious restrictions *x*(*x*_{t} = 1) = 1 and *x*(*x*_{t} = 0) = 0. Stoichiometric transfer, i.e., *x* = *x*_{t}, requires *q* = 1 (but not necessarily *r* = *p* = 1). We note that the derivative is equal to +1 at the intersection with 1 − *x*_{t} at $xt=(q\u2212q)/(1\u2212q)$ 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 *x*_{t} to achieve *x* in the epilayer is *x*_{t} = *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 targets^{33} for the purpose of combinatorial PLD.^{34,35}

We fit our data by calculating *q* from all experimental (*x*_{t}, *x*) pairs,

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 chamber^{4}) 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 *x*_{t} is given by

The proportionality factor in Eq. (10) is the ablation rate of the target *ĝ*(*x*_{t}). 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 Ga_{2}O_{3}, the bandgap is close to the PLD laser photon energy (5.0 eV).

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 *x*_{t} (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.

### B. Equivalence of *r*- and $r\xaf$-planes

Next, we investigate the equivalence of the *r*- and $r\xaf$-planes as it is expected theoretically. For this purpose, pseudomorphic (Al_{x}Ga_{1−x})_{2}O_{3} epilayers were deposited on substrates obtained from the front and back surface of the *same* double-sided polished *r*-Al_{2}O_{3} 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) ($1\xaf\u20094\xaf.6$), ($4\xaf\u20091\xaf.6$), (14.6), (41.6), ($51\xaf.6$), (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 2*θ* 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 (*σ*_{tilt} < 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).

No. . | x_{EDX}
. | x_{fit}
. | ρ_{x}
. | ρ_{y}
. | β_{x} (°)
. | β_{y} (°)
. | σ (°)
. |
---|---|---|---|---|---|---|---|

A | 0.859 | 0.866 | 0.991 | 0.981 | 0.001 | 0.001 | 0.008 |

B | 0.863 | 0.870 | 0.983 | 0.977 | 0.003 | 0.000 | 0.007 |

C | 0.846 | 0.855 | 0.983 | 0.992 | 0.005 | 0.006 | 0.010 |

D | 0.919 | 0.914 | 1.002 | 0.992 | 0.004 | 0.004 | 0.008 |

E | 0.763 | 0.770 | 0.981 | 0.953 | 0.008 | 0.060 | 0.020 |

F | 0.786 | 0.797 | 0.986 | 0.957 | 0.005 | 0.011 | 0.011 |

G | 0.731 | 0.745 | 1.018 | 0.853 | 0.010 | 0.029 | 0.022 |

H | 0.670 | 0.681 | 1.030 | 0.727 | 0.114 | 0.020 | 0.060 |

I | 0.588 | 0.608 | 0.758 | 0.575 | 0.285 | 0.179 | 0.044 |

No. . | x_{EDX}
. | x_{fit}
. | ρ_{x}
. | ρ_{y}
. | β_{x} (°)
. | β_{y} (°)
. | σ (°)
. |
---|---|---|---|---|---|---|---|

A | 0.859 | 0.866 | 0.991 | 0.981 | 0.001 | 0.001 | 0.008 |

B | 0.863 | 0.870 | 0.983 | 0.977 | 0.003 | 0.000 | 0.007 |

C | 0.846 | 0.855 | 0.983 | 0.992 | 0.005 | 0.006 | 0.010 |

D | 0.919 | 0.914 | 1.002 | 0.992 | 0.004 | 0.004 | 0.008 |

E | 0.763 | 0.770 | 0.981 | 0.953 | 0.008 | 0.060 | 0.020 |

F | 0.786 | 0.797 | 0.986 | 0.957 | 0.005 | 0.011 | 0.011 |

G | 0.731 | 0.745 | 1.018 | 0.853 | 0.010 | 0.029 | 0.022 |

H | 0.670 | 0.681 | 1.030 | 0.727 | 0.114 | 0.020 | 0.060 |

I | 0.588 | 0.608 | 0.758 | 0.575 | 0.285 | 0.179 | 0.044 |

The only other *quantitative* strain value for corundum (Al, Ga)_{2}O_{3}/Al_{2}O_{3} heterostructures we could find in the literature is shown in Fig. 6 as black diamond, representing an ultrathin (1 nm) *α*-Ga_{2}O_{3} layer on *a*-plane Al_{2}O_{3}; the given related experimental value for the out-of-plane strain is $\u03f5zz\u2032$_{,a} = 0.074^{37} and agrees well with Eq. (3) and the material parameters of Tables I and II, yielding $\u03f5zz\u2032$_{,a} = 0.0738.

### C. Layers beyond the critical thickness

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 $\u03f5\u0303\u2032$ is parameterized by independent parameters *ρ*_{x} and *ρ*_{y}, *ρ* = 1 meaning pseudomorphic growth and *ρ* = 0 meaning full relaxation,

Now, we fit the 2*θ* 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 *β*_{x} and *β*_{y} for the tilt along the *x*′- and *y*′-direction, respectively. *β*_{x} is related to *ρ*_{x} as detailed below. *β*_{y} turns out to be small (*β*_{y} < 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).

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 *Q*_{x} = 0. For samples G and H that are partially relaxed but only along *y*′, the epilayer peak becomes broader and weaker, but *Q*_{x} remains centered around zero. For sample I, exhibiting relaxation in both in-plane directions, *Q*_{x} of the film peak is non-zero.

### D. Critical thickness

The critical thickness for one Al-concentration can be extracted from Fig. 14(a), where the reduction of the relaxation parameter $\rho x\u2009\rho y$ from its pseudomorphic value of 1 is tracked for a constant number of PLD pulses (30k). Relaxation begins at *x* = 0.86 (*ϵ*_{a} = −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)].

From Fig. 14, we can thus deduct three (*x*, *d*_{crit}) value pairs. These are shown together in Fig. 15 with dependence on ∝ *ϵ*^{3/2}. The exponent thus lies in between dependence on forces ∝ *ϵ* and on strain energy ∝ *ϵ*^{2}. A more detailed model for critical thickness goes beyond the scope of this paper and will be subject to further experimental and theoretical investigation.

## V. MICROSCOPIC RELAXATION MODEL

### A. Basal and prismatic slip systems

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 Al

_{2}O

_{3}

^{38}are $1/3\u27e8112\xaf0\u27e9$ (

*b*=

*a*= 0.476 nm) and $1/3\u27e81\xaf101\u27e9$ ($b=a\u20093+\zeta 2/3=0.513$ nm). The two prevalent slip systems are the basal (

*c*-) plane ${0001}\u20091/3\u27e8112\xaf0\u27e9$ and the prismatic (

*a*-) plane ${112\xaf0}\u20091/3\u27e811\xaf01\u27e9$ systems

^{39,40}[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, $1/3[1\xaf\u20091\xaf20]$ and $1/3[12\xaf10]$, which possess the same edge $bc,\u2225\u2032$ and the *same* tilt $bc,\u22a5\u2032$ components but opposite screw component $bc,\u25cb\u2032$ (in the (*x*′, *y*′, *z*′) coordinate system),

For Al_{2}O_{3}, $bc\u2032$ = (0.221, ±0.238, 0.348) nm. The ratio of tilt and edge components is given by $bc,\u22a5\u2032/bc,\u2225\u2032=\zeta /3$. Two other possible Burgers vectors ($\xb11/3[21\xaf\u20091\xaf0]$) are pure screw-type and will not lead to strain relaxation.

For the *a*-plane slip system, the $1/3\u2009[1\xaf101]$ direction is shown on the $(112\xaf0)$ 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 $\alpha a=arctan(3/3+\zeta 2)=42.9\u25cb$ for Al_{2}O_{3}. The (in-plane) edge, tilt, and screw components of Burgers vector are (for Al_{2}O_{3} lattice parameters) $ba,\u2225\u2032$ = 0.373 nm, $ba,\u22a5\u2032=bc,\u22a5\u2032$ = 0.348 nm, and $ba,\u25cb\u2032$ = 0.052 nm, respectively^{41} [another possible Burgers vector ($1/3\u27e811\xaf01\u27e9$) 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 *b*_{a}′ = (0, *a*, 0), effectively allowing strain relaxation along the *y*′-direction.

### B. Tilt of epitaxial plane

Through the components of Burgers vector *b*_{c}′, the in-plane relaxation along *x*′, cf. (11),

and the subsequent tilt *β*_{x} of the *r*-plane of the epilayer against that of the substrate is related^{42} via

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 2*θ* 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 (*β*_{y} ≪ 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 *β*_{x}.

No. . | ρ_{x}
. | ρ_{y}
. | β_{x} (°)
. | β_{y} (°)
. | β (°)
. | $\beta x\varphi $ (°) . | $\beta y\varphi $ (°) . | β^{ϕ} (°)
. |
---|---|---|---|---|---|---|---|---|

G | 1.018 | 0.853 | 0.029 | 0.022 | 0.037 | 0.037 | 0.013 | 0.039 |

H | 1.030 | 0.727 | 0.112 | 0.020 | 0.114 | 0.105 | 0.051 | 0.117 |

I | 0.758 | 0.575 | 0.285 | 0.179 | 0.337 | 0.361 | 0.112 | 0.377 |

J | 0.753 | 0.685 | 0.344 | 0.012 | 0.344 | 0.340 | 0.015 | 0.364 |

No. . | ρ_{x}
. | ρ_{y}
. | β_{x} (°)
. | β_{y} (°)
. | β (°)
. | $\beta x\varphi $ (°) . | $\beta y\varphi $ (°) . | β^{ϕ} (°)
. |
---|---|---|---|---|---|---|---|---|

G | 1.018 | 0.853 | 0.029 | 0.022 | 0.037 | 0.037 | 0.013 | 0.039 |

H | 1.030 | 0.727 | 0.112 | 0.020 | 0.114 | 0.105 | 0.051 | 0.117 |

I | 0.758 | 0.575 | 0.285 | 0.179 | 0.337 | 0.361 | 0.112 | 0.377 |

J | 0.753 | 0.685 | 0.344 | 0.012 | 0.344 | 0.340 | 0.015 | 0.364 |

### C. Dislocation line orientation

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 (*x*_{EDX} = 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

For the basal system, always *α*_{c} = *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 *α*_{a} and *α*_{m} are depicted in Fig. 18 as a function of *θ*. On the *c*-plane, for *θ* = 0, the expected angles of *α*_{a} = 60° and *α*_{m} = 30° are recovered. The ratio *α*_{a}/*α*_{m} varies between 2 for *θ* = 0 and 3 for *θ* → *π*/2.

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 *α*_{exp} = 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 *α*_{a} = 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 [$\alpha m=arctan(1/3+\zeta 2)\u224817\u25cb$] or any other glide system listed in Ref. 38 do not agree at all with the present situation for sure.

### D. Critical resolved shear stress

The resolved shear stresses for the basal (*τ*_{c}) and the two prismatic $(112\xaf0)\u2009[1\xaf100]$ and $(12\xaf10)\u2009[101\xaf0]$ (*τ*_{a}) slip systems are

Note that these shear stresses *σ* refer to the crystal system. The present pseudomorphic strain situation forces *σ*_{yz} = *σ*_{xy} = 0, recovering the formulas given in Ref. 1.

For the *r*-plane, *τ*_{a} is little larger than *τ*_{c}, as shown in Fig. 19. For the third *a*-plane $(21\xaf\u20091\xaf0)\u2009[011\xaf0]$ slip system, *τ*_{a} = 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 *τ*_{a} ≳ *τ*_{c} seems to prevail, however, that the same room temperature elastic constants were used since their temperature dependence is known for Al_{2}O_{3}^{46} but not for *α*-Ga_{2}O_{3}.

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 Al_{2}O_{3}. 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.

## VI. COMPARISON TO WURTZITE SYSTEMS

The situation found here for corundum (Al, Ga)_{2}O_{3} on Al_{2}O_{3} 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 *α*_{c} = *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 $(11\xaf00)\u20091/3\u27e8112\xaf0\u27e9$ 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 $(303\xaf1)$ [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.

Material . | (hk.l)
. | θ (°)
. | α (°)
. | Reference . |
---|---|---|---|---|

(Al_{0.47}Ga_{0.53})_{2}O_{3}/Al_{2}O_{3} | (01.2) | 57.6 | 40.2 | This work |

Al_{0.13}Ga_{0.87}N/GaN | (30.1) | 79.9 | 15.5 | 42 |

In_{0.1}Ga_{0.9}N/GaN | (20.1) | 79.9 | 21.3 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (20.1) | 79.9 | 22.3 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (30.1) | 75.1 | 15.5 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (30.1) | 75.1 | 18.0 | 47 |

In_{0.06}Ga_{0.94}N/GaN | (20.1) | 79.9 | 5.0 | 49 |

In_{0.06}Ga_{0.94}N/GaN | (20.1) | 75.1 | 9.6 | 49 |

In_{0.07}Ga_{0.93}N/GaN | (11.2) | 58.4 | 20, 42 | 50 |

In_{0.126}Ga_{0.874}N/GaN | (11.2) | 58.4 | 45 | 51 |

GaN/Si(001) 7° off | (10.1) | 62.0 | 14.9 | 52 |

Mg_{0.18}Zn_{0.82}O/ZnO | (01.2) | 42.7 | 24.5 | 53 |

Material . | (hk.l)
. | θ (°)
. | α (°)
. | Reference . |
---|---|---|---|---|

(Al_{0.47}Ga_{0.53})_{2}O_{3}/Al_{2}O_{3} | (01.2) | 57.6 | 40.2 | This work |

Al_{0.13}Ga_{0.87}N/GaN | (30.1) | 79.9 | 15.5 | 42 |

In_{0.1}Ga_{0.9}N/GaN | (20.1) | 79.9 | 21.3 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (20.1) | 79.9 | 22.3 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (30.1) | 75.1 | 15.5 | 47 |

In_{0.1}Ga_{0.9}N/GaN | (30.1) | 75.1 | 18.0 | 47 |

In_{0.06}Ga_{0.94}N/GaN | (20.1) | 79.9 | 5.0 | 49 |

In_{0.06}Ga_{0.94}N/GaN | (20.1) | 75.1 | 9.6 | 49 |

In_{0.07}Ga_{0.93}N/GaN | (11.2) | 58.4 | 20, 42 | 50 |

In_{0.126}Ga_{0.874}N/GaN | (11.2) | 58.4 | 45 | 51 |

GaN/Si(001) 7° off | (10.1) | 62.0 | 14.9 | 52 |

Mg_{0.18}Zn_{0.82}O/ZnO | (01.2) | 42.7 | 24.5 | 53 |

For (In, Ga)N/GaN, we find *α* = 18.0° and 15.5° on $(303\xaf\u20091\xaf)$ [Figs. 3(f) and 3(g) of Ref. 47] and *α* = 21.3° and 22.3° on $(202\xaf\u20091\xaf)$ [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 $(202\xaf1)$ and $(303\xaf1)$ GaN^{49} (cf. circles in Fig. 18). For (In, Ga)N on GaN $(112\xaf2)$, 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 $(112\xaf2)$ GaN^{51} 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 silicon^{52} 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 observed^{53} (cf. star in Fig. 18). We emphasize that Eqs. (16) and (17) indeed describe results from the literature for various material systems quite well.

## VII. CONCLUSION

In summary, we have quantitatively analyzed the strain state of pseudomorphic and partially relaxed sesquioxide *α*-(Al, Ga)_{2}O_{3} epilayers on (01.2) *r*-plane Al_{2}O_{3}, 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.

## SUPPLEMENTARY MATERIAL

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

## ACKNOWLEDGMENTS

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).

## REFERENCES

_{1−x}Ga

_{x})

_{2}O

_{3}thin films (0 ≤ x < 0.08) grown on R-plane sapphire

_{2}O

_{3}-based heterostructures

_{2}O

_{3}thin Films on α-Al

_{2}O

_{3}Substrates by ultrasonic mist chemical vapor deposition

_{2}O

_{3}fabricated by MIST EPITAXY® technique

_{2}O

_{3}and (Al

_{x}Ga

_{1?x})

_{2}O

_{3}alloys

_{x}Ga

_{1?x})

_{2}O

_{3}/α-Ga

_{2}O

_{3}heterostructures

_{2}O

_{3}thin films on a-, m-, and r-plane sapphire substrates by mist chemical vapor deposition using α-Fe

_{2}O

_{3}buffer layers

_{1−x}Ga

_{x})

_{2}O

_{3}films on sapphire for solid-state waveguide lasers

_{2}O

_{3}polymorphs

_{x}Ga

_{1−x})

_{2}O

_{3}by a mist chemical vapor deposition two-chamber system and verification of Vegard’s Law

_{2}O

_{3}Structure and the high-pressure Phase of Ga

_{2−x}Fe

_{x}O

_{3}

_{2}O

_{3}

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.

_{x}Ga

_{1−x})

_{2}O

_{3}thin films on c-sapphire substrates by pulsed laser deposition

_{2}O

_{3}on c-plane and a-plane sapphire substrates as studied by synchrotron-based x-ray diffraction

_{2}O

_{3}

_{2}O

_{3}single crystals between 25°C and 1800°C: Basal and prismatic slip

_{2}O

_{3}single crystals below 1000°C in compression under hydrostatic pressure

$ba,\u2225\u2032=3a/2\u2009\xd7\u2009\zeta 2/\zeta 4+15\zeta 2+36$ and $ba,\u25cb\u2032=a/3\u2009\xd7\u2009(\zeta 2\u22126)/\zeta 2+12$.

_{2}O

_{3}

_{2}O

_{3}from first-principles calculations

_{x}Ga

_{1−x}N/GaN heterostructures