Competition between Electron–Phonon and Spin–Phonon Interaction on the Band Gap and Phonon Spectrum in Magnetic Semiconductors

Using the microscopic s-f model and Green’s function theory, we study the temperature dependence of the band gap energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula> and the phonon energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> and damping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> of ferro- and antiferromagnetic semiconductors, i.e., with different signs of the s-f interaction constant <i>I</i>. The band gap is a fundamental quantity which affects various optical, electronic and energy applications of the materials. In the temperature dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula> and the phonon spectrum, there is a kink at the phase transition temperature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>C</mi></msub></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>N</mi></msub></semantics></math></inline-formula> due to the anharmonic spin–phonon interaction (SPI) <i>R</i>. Moreover, the effect of the SPI <i>R</i> and electron–phonon interaction (EPI) <i>A</i> on these properties is discussed. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula> decreases with increasing SPI and EPI, whereas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo><</mo><mn>0</mn><mo>,</mo><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, there is a competition; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula> increases with raising the EPI and decreases for enhanced SPI. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, in both cases, the SPI and EPI reduce <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula>. The magnetic field dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>g</mi></msub></semantics></math></inline-formula> for the two signs of <i>I</i> and <i>R</i> is discussed. The SPI and EPI lead to reducing the energy of the phonon mode <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> = 445 cm⁻¹ in EuO (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>), whereas <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> = 151 cm⁻¹ in EuSe (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>) is enhanced with increasing EPI and reduced with SPI. Both the SPI and EPI lead to an increasing of the phonon damping in EuO and EuSe. The results are compared with the existing experimental data.