Abstract <p>In this study, we calculate the diamagnetic susceptibility of p-Sn<sub>1-x</sub>Eu<sub>x</sub>Te using a two-band model within the framework of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\vec {k} \cdot \vec {\pi }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>k</mi> <mo stretchy="false">→</mo> </mover> <mo>·</mo> <mover accent="true"> <mi>π</mi> <mo stretchy="false">→</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> formalism for the host <i>SnTe</i>. The band edge states <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_{62}^{-}\alpha (\beta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mn>62</mn> </mrow> <mo>-</mo> </msubsup> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L_{61}^{+}\alpha (\beta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mn>61</mn> </mrow> <mo>+</mo> </msubsup> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are exactly diagonalized to get the energy eigenvalue and wave functions. The introduction of <i>Eu</i> impurity in <i>SnTe</i> significantly modulates the band structure. The <i>Eu</i> ion at x=0.02 gives band inversion and sets an upper limit for susceptibility measurement. The susceptibility of p<InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(-Sn_{1-x}Eu_{x}Te\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>S</mi> <msub> <mi>n</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>x</mi> </mrow> </msub> <mi>E</mi> <msub> <mi>u</mi> <mi>x</mi> </msub> <mi>T</mi> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation> remains negative, ranging from <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(-4.13 \times 10^{-7} emu/gm\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>4.13</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </msup> <mi>e</mi> <mi>m</mi> <mi>u</mi> <mo stretchy="false">/</mo> <mi>g</mi> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(-4.12 \times 10^{-7} emu/gm\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>4.12</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </msup> <mi>e</mi> <mi>m</mi> <mi>u</mi> <mo stretchy="false">/</mo> <mi>g</mi> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, reflecting the influence of strong spin–orbit interaction and band inversion. However, the change in susceptibility is minimal and remains nearly constant at T=300K. The calculated value of diamagnetic susceptibility, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\chi _{dia}=-4.12 \times 10^{-7} emu/gm\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mrow> <mi mathvariant="italic">dia</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mn>4.12</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </msup> <mi>e</mi> <mi>m</mi> <mi>u</mi> <mo stretchy="false">/</mo> <mi>g</mi> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(Sn_{1-x}Eu_{x}Te\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>n</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>x</mi> </mrow> </msub> <mi>E</mi> <msub> <mi>u</mi> <mi>x</mi> </msub> <mi>T</mi> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation>, is in excellent agreement with similar materials like <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(Sn_{1-x}Gd_{x}Te\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>n</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>x</mi> </mrow> </msub> <mi>G</mi> <msub> <mi>d</mi> <mi>x</mi> </msub> <mi>T</mi> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\chi _{dia}=-4.0 \times 10^{-7} emu/gm\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mrow> <mi mathvariant="italic">dia</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mn>4.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </msup> <mi>e</mi> <mi>m</mi> <mi>u</mi> <mo stretchy="false">/</mo> <mi>g</mi> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>.</p> Graphic Abstract <p></p>

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Anisotropic lattice diamagnetic susceptibility in p-type Sn1−xEuxTe

  • Sashi S. Behera,
  • Saptarshi Nayak,
  • Rajiba L. Hota

摘要

Abstract

In this study, we calculate the diamagnetic susceptibility of p-Sn1-xEuxTe using a two-band model within the framework of the \(\vec {k} \cdot \vec {\pi }\) k · π formalism for the host SnTe. The band edge states \(L_{62}^{-}\alpha (\beta )\) L 62 - α ( β ) and \(L_{61}^{+}\alpha (\beta )\) L 61 + α ( β ) are exactly diagonalized to get the energy eigenvalue and wave functions. The introduction of Eu impurity in SnTe significantly modulates the band structure. The Eu ion at x=0.02 gives band inversion and sets an upper limit for susceptibility measurement. The susceptibility of p \(-Sn_{1-x}Eu_{x}Te\) - S n 1 - x E u x T e remains negative, ranging from \(-4.13 \times 10^{-7} emu/gm\) - 4.13 × 10 - 7 e m u / g m to \(-4.12 \times 10^{-7} emu/gm\) - 4.12 × 10 - 7 e m u / g m , reflecting the influence of strong spin–orbit interaction and band inversion. However, the change in susceptibility is minimal and remains nearly constant at T=300K. The calculated value of diamagnetic susceptibility, \(\chi _{dia}=-4.12 \times 10^{-7} emu/gm\) χ dia = - 4.12 × 10 - 7 e m u / g m for \(Sn_{1-x}Eu_{x}Te\) S n 1 - x E u x T e , is in excellent agreement with similar materials like \(Sn_{1-x}Gd_{x}Te\) S n 1 - x G d x T e , where \(\chi _{dia}=-4.0 \times 10^{-7} emu/gm\) χ dia = - 4.0 × 10 - 7 e m u / g m .

Graphic Abstract