Strain-stress relationships for coherent in-plane strain in heterostructures with monoclinic crystal systems: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> - <mml:math xmlns:mml="http://www.

In this work, we derive the relationship between the states of strain and stress in pseudomorphic epitaxial layers on a substrate with differing equilibrium lattice parameters under symmetry-conserving conditions for the case of monoclinic crystal symmetry. We compare surface vectors across the template-epitaxial layer interface and impose conditions of a stress-free epitaxial layer. As a result, we demonstrate the theoretical existence of up to three possible unit cells that can establish on a given template. We demonstrate this approach for the special case of symmetry-conserving strain in a class of templates with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo stretchy="false">(</a:mo> <a:mi>h</a:mi> <a:mn>0</a:mn> <a:mi>l</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> surface orientations. We use <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>β</e:mi> </e:math> - <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo stretchy="false">(</g:mo> <g:msub> <g:mi>Al</g:mi> <g:mi>x</g:mi> </g:msub> <g:msub> <g:mi>Ga</g:mi> <g:mrow> <g:mn>1</g:mn> <g:mo>−</g:mo> <g:mi>x</g:mi> </g:mrow> </g:msub> <g:msub> <g:mo stretchy="false">)</g:mo> <g:mn>2</g:mn> </g:msub> <g:msub> <g:mi mathvariant="normal">O</g:mi> <g:mn>3</g:mn> </g:msub> </g:math> on <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mo stretchy="false">(</l:mo> <l:mi>h</l:mi> <l:mn>0</l:mn> <l:mi>l</l:mi> <l:mo stretchy="false">)</l:mo> <l:mspace width="0.2em"/> <l:mi>β</l:mi> </l:math> - <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>Ga</q:mi> <q:mn>2</q:mn> </q:msub> <q:msub> <q:mi mathvariant="normal">O</q:mi> <q:mn>3</q:mn> </q:msub> </q:math> as an example. We discuss the effects of the composition <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>x</t:mi> </t:math> and surface orientation on the formation of three elastically stable unit cells, their strain and stress tensors, unit-cell axes, unit-cell volumes, lattice spacings, elastic potential energies, and stress-free directions. The previous paradigm for epitaxial layer growth, where the only stress-free direction is perpendicular to the growing surface, is not generally valid for low-symmetry materials. In the example here, we find that, in addition to the stress-free growth direction for crystallographic planes with low Miller indices, two possible competing domains can exist with stress-free directions oblique to the surface of the template for almost all planes <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mo stretchy="false">(</v:mo> <v:mi>h</v:mi> <v:mn>0</v:mn> <v:mi>l</v:mi> <v:mo stretchy="false">)</v:mo> </v:math> . We further demonstrate the effect of the orientation-dependent differently strained domains and calculate the strain-shifted anisotropic band-to-band transitions for <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>β</z:mi> </z:math> - <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mo stretchy="false">(</bb:mo> <bb:msub> <bb:mi>Al</bb:mi> <bb:mrow> <bb:mn>0.1</bb:mn> </bb:mrow> </bb:msub> <bb:msub> <bb:mi>Ga</bb:mi> <bb:mrow> <bb:mn>0.9</bb:mn> </bb:mrow> </bb:msub> <bb:msub> <bb:mo stretchy="false">)</bb:mo> <bb:mn>2</bb:mn> </bb:msub> <bb:msub> <bb:mi mathvariant="normal">O</bb:mi> <bb:mn>3</bb:mn> </bb:msub> </bb:math> on <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mo stretchy="false">(</gb:mo> <gb:mi>h</gb:mi> <gb:mn>0</gb:mn> <gb:mi>l</gb:mi> <gb:mo stretchy="false">)</gb:mo> <gb:mspace width="0.2em"/> <gb:mi>β</gb:mi> </gb:math> - <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:msub> <lb:mi>Ga</lb:mi> <lb:mn>2</lb:mn> </lb:msub> <lb:msub> <lb:mi mathvariant="normal">O</lb:mi> <lb:mn>3</lb:mn> </lb:msub> </lb:math> . We also provide examples of x-ray diffraction analyses, which may help guide the search for the occurrence of such differently strained unit cells in epitaxial layers grown by experiment, here taking the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mo stretchy="false">(</ob:mo> <ob:mrow> <ob:mover> <ob:mn>2</ob:mn> <ob:mo stretchy="false">¯</ob:mo> </ob:mover> </ob:mrow> <ob:mn>01</ob:mn> <ob:mo stretchy="false">)</ob:mo> </ob:math> surface orientation as case for demonstration.