Magnetic field-engineered optical nonlinearity in germanene nanotubes

Abstract The linear and nonlinear optical properties of germanene nanotubes (GeNTs) under the influence of magnetic fields are investigated through theoretical analysis. By employing a tight-binding model beyond the Dirac cone approximation, the electronic band structure, density of states, dipole matrix elements, and third-order nonlinear optical susceptibilities are calculated. The application of a magnetic field induces a Zeeman splitting of the energy levels, breaking the band degeneracy. Consequently, the dipole matrix elements exhibit a magnetic field dependence, altering the intensity and number of allowed transitions. In the infrared and ultraviolet energy regions, the linear optical absorption spectrum displays the splitting of peaks into dual neighboring peaks, with the splitting distance linearly dependent on the magnetic field strength. The third-order nonlinear optical susceptibilities, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext>THG</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext>IDIR</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , reveal the emergence of multiple resonance peaks in different energy regions due to two-photon and three-photon absorption processes. The magnetic field significantly influences the number, position, and height of these optical peaks, enabling effective control over the nonlinear optical response. By investigating the scaled energy ( E / E _g) dependence, a universal behavior is observed for the three- and two photon resonance peaks, where their positions remain constant, but their peak intensities are significantly enhanced under the influence of the magnetic field. These findings demonstrate the potential of GeNTs as tunable nonlinear optical materials, with the magnetic field serving as an effective parameter for engineering their optical properties for various applications in optoelectronics, photonics, and related fields.