Noncentrosymmetric, transverse structural modulation in SrAl4 , and elucidation of its origin in the BaAl4 family of compounds

At ambient conditions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>SrAl</a:mi><a:mn>4</a:mn></a:msub></a:math> adopts the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>BaAl</b:mi><b:mn>4</b:mn></b:msub></b:math> structure type with space group <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>I</c:mi><c:mn>4</c:mn><c:mo>/</c:mo><c:mi>m</c:mi><c:mi>m</c:mi><c:mi>m</c:mi></c:mrow></c:math>. It undergoes a charge-density-wave (CDW) transition at <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>T</d:mi><d:mi>CDW</d:mi></d:msub><d:mo>=</d:mo><d:mn>243</d:mn></d:mrow></d:math> K, followed by a structural transition at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>T</e:mi><e:mi>S</e:mi></e:msub><e:mo>=</e:mo><e:mn>87</e:mn></e:mrow></e:math> K. Temperature-dependent single-crystal x-ray diffraction (SXRD) leads to the observation of incommensurate superlattice reflections at <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi mathvariant="bold">q</f:mi><f:mo>=</f:mo><f:mi>σ</f:mi><f:msup><f:mi mathvariant="bold">c</f:mi><f:mo>*</f:mo></f:msup></f:mrow></f:math> with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>σ</i:mi><i:mo>=</i:mo><i:mn>0.1116</i:mn></i:mrow></i:math> at 200 K. The CDW has orthorhombic symmetry with the noncentrosymmetric superspace group <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>F</j:mi><j:mn>222</j:mn><j:mo>(</j:mo><j:mn>0</j:mn><j:mspace width="0.16em"/><j:mn>0</j:mn><j:mspace width="0.16em"/><j:mi>σ</j:mi><j:mo>)</j:mo><j:mn>00</j:mn><j:mi>s</j:mi></j:mrow></j:math>, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>F</m:mi><m:mn>222</m:mn></m:mrow></m:math> is a subgroup of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mi>F</n:mi><n:mi>m</n:mi><n:mi>m</n:mi><n:mi>m</n:mi></n:mrow></n:math> as well as of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mi>I</o:mi><o:mn>4</o:mn><o:mo>/</o:mo><o:mi>m</o:mi><o:mi>m</o:mi><o:mi>m</o:mi></o:mrow></o:math>. Atomic displacements mainly represent a transverse wave, with displacements that are 90 deg out of phase between the two diagonal directions of the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mi>I</p:mi></p:math>-centered unit cell, resulting in a helical wave. Small longitudinal displacements are provided by the second harmonic modulation. The orthorhombic phase realized in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:msub><q:mi>SrAl</q:mi><q:mn>4</q:mn></q:msub></q:math> is similar to that found in <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:msub><r:mi>EuAl</r:mi><r:mn>4</r:mn></r:msub></r:math>, except that no second harmonic could be determined for the latter compound. Electronic structure calculations and phonon calculations by density functional theory (DFT) have failed to reveal the mechanism of CDW formation. No clear Fermi surface nesting, electron-phonon coupling, or involvement of Dirac points could be established. However, DFT reveals that Al atoms dominate the density of states near the Fermi level, thus corroborating the SXRD measurements. <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:msub><s:mi>SrAl</s:mi><s:mn>4</s:mn></s:msub></s:math> remains incommensurately modulated at the structural transition, where the symmetry lowers from orthorhombic to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mi mathvariant="bold">b</t:mi></t:math>-unique monoclinic. The present work draws a comparison on the modulated structures of nonmagnetic <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:msub><v:mi>SrAl</v:mi><v:mn>4</v:mn></v:msub></v:math> and magnetic <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"><w:msub><w:mi>EuAl</w:mi><w:mn>4</w:mn></w:msub></w:math> elucidating their similarities and differences, and firmly establishing that although substitution of Eu to Sr plays little to no role in the structure, the transition temperatures are affected by the atomic sizes. We have identified a simple criterion that correlates the presence of a phase transition with the interatomic distances. Only those compounds <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"><x:mrow><x:mi>X</x:mi><x:msub><x:mi>Al</x:mi><x:mrow><x:mn>4</x:mn><x:mo>−</x:mo><x:mi>x</x:mi></x:mrow></x:msub><x:msub><x:mi>Ga</x:mi><x:mi>x</x:mi></x:msub></x:mrow></x:math> <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mrow><y:mo>(</y:mo><y:mi>X</y:mi><y:mo>=</y:mo><y:mi>Ba</y:mi><y:mo>,</y:mo><y:mo> </y:mo><y:mi>Eu</y:mi><y:mo>,</y:mo><y:mo> </y:mo><y:mi>Sr</y:mi><y:mo>,</y:mo><y:mo> </y:mo><y:mi>Ca</y:mi><y:mo>;</y:mo><y:mo> </y:mo><y:mn>0</y:mn><y:mo><</y:mo><y:mi>x</y:mi><y:mo><</y:mo><y:mn>4</y:mn><y:mo>)</y:mo></y:mrow></y:math> undergo phase transitions, for which the ratio <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:mrow><z:mi>c</z:mi><z:mo>/</z:mo><z:mi>a</z:mi></z:mrow></z:math> falls within the narrow range <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:mrow><ab:mn>2.51</ab:mn><ab:mo><</ab:mo><ab:mi>c</ab:mi><ab:mo>/</ab:mo><ab:mi>a</ab:mi><ab:mo><</ab:mo><ab:mn>2.54</ab:mn></ab:mrow></ab:math>. Published by the American Physical Society 2024