Thermal and magnetoelastic properties of the van der Waals ferromagnet Fe3−δGeTe2 : Anisotropic spontaneous magnetostriction and ferromagnetic magnon excitations

By determining the lattice parameters as a function of temperature of the hexagonal van der Waals ferromagnet <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>Fe</a:mi><a:mrow><a:mn>2.92</a:mn><a:mo>(</a:mo><a:mn>1</a:mn><a:mo>)</a:mo></a:mrow></a:msub><a:msub><a:mi>Ge</a:mi><a:mrow><a:mn>1.02</a:mn><a:mo>(</a:mo><a:mn>3</a:mn><a:mo>)</a:mo></a:mrow></a:msub><a:msub><a:mi>Te</a:mi><a:mn>2</a:mn></a:msub></a:mrow></a:math> we obtain the temperature dependence of the spontaneous in-plane magnetostriction in the ferromagnetic and the linear thermal expansion coefficients in the paramagnetic state. The spontaneous magnetostriction is clearly seen in the temperature dependence of the in-plane lattice parameter <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>a</b:mi><b:mo>(</b:mo><b:mi>T</b:mi><b:mo>)</b:mo></b:mrow></b:math>, but less well pronounced perpendicular to the planes along <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>c</c:mi></c:math>. Below <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>T</d:mi><d:mi mathvariant="normal">C</d:mi></d:msub></d:math> the spontaneous magnetostriction follows a power law <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>M</f:mi><f:msup><f:mrow><f:mo>(</f:mo><f:mi>T</f:mi><f:mo>)</f:mo></f:mrow><f:mrow><f:mn>3</f:mn><f:mo>/</f:mo><f:mn>2</f:mn></f:mrow></f:msup></f:mrow></f:math> of the magnetization and leads to an expansion of the hexagonal layers. Extrapolating to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>T</g:mi><g:mo>→</g:mo></g:mrow></g:math> 0 K we obtain a spontaneous in-plane saturation magnetostriction of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msub><h:mi>λ</h:mi><h:mrow><h:mi>sp</h:mi><h:mo>,</h:mo><h:mi>a</h:mi></h:mrow></h:msub><h:mrow><h:mo>(</h:mo><h:mi>T</h:mi><h:mo>→</h:mo><h:mn>0</h:mn><h:mo>)</h:mo></h:mrow><h:mo>≈</h:mo><h:mo>−</h:mo><h:mn>220</h:mn><h:mspace width="0.16em"/><h:mo>×</h:mo><h:mspace width="0.16em"/><h:msup><h:mn>10</h:mn><h:mrow><h:mo>−</h:mo><h:mn>6</h:mn></h:mrow></h:msup></h:mrow></h:math>. In the paramagnetic state the linear thermal expansion coefficients amount to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mn>13.9</k:mn><k:mrow><k:mo>(</k:mo><k:mn>1</k:mn><k:mo>)</k:mo></k:mrow><k:mo>×</k:mo><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>6</k:mn></k:mrow></k:msup><k:mspace width="4pt"/><k:msup><k:mrow><k:mi mathvariant="normal">K</k:mi></k:mrow><k:mrow><k:mo>−</k:mo><k:mn>1</k:mn></k:mrow></k:msup></k:mrow></k:math> and to <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mn>23.2</n:mn><n:mrow><n:mo>(</n:mo><n:mn>2</n:mn><n:mo>)</n:mo></n:mrow><n:mo>×</n:mo><n:msup><n:mn>10</n:mn><n:mrow><n:mo>−</n:mo><n:mn>6</n:mn></n:mrow></n:msup><n:mspace width="4pt"/><n:msup><n:mrow><n:mi mathvariant="normal">K</n:mi></n:mrow><n:mrow><n:mo>−</n:mo><n:mn>1</n:mn></n:mrow></n:msup></n:mrow></n:math> for the in-plane and out-of-plane direction, respectively, indicating a linear volume thermal expansion coefficient of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mn>50.8</q:mn><q:mrow><q:mo>(</q:mo><q:mn>4</q:mn><q:mo>)</q:mo></q:mrow><q:mo>×</q:mo><q:msup><q:mn>10</q:mn><q:mrow><q:mo>−</q:mo><q:mn>6</q:mn></q:mrow></q:msup><q:mspace width="0.28em"/><q:mspace width="4pt"/><q:msup><q:mrow><q:mi mathvariant="normal">K</q:mi></q:mrow><q:mrow><q:mo>−</q:mo><q:mn>1</q:mn></q:mrow></q:msup></q:mrow></q:math> which we use to estimate the volume thermal expansion contribution to the heat capacity determined at constant pressure. A Sommerfeld-type linear term in the low-temperature heat capacities can be quantitatively ascribed to two-dimensional (2D) ferromagnetic magnon excitations.