**2. Theoretical Consideration**

The self-assembled GeSn QD has been considered to have a pyramidal shape with 1nm thick wetting layer (WL) embedded in Ge matrix which is one of the frequently observed shapes for semiconducting self-assembled QD [41] as illustrated by Figure 1a. Throughout this work, we set the tin composition at 30% and a QD height to base side length's (L) ratio of 1/3 (Figure 1b,c).

**Figure 1.** Schematic sketch of the pyramidal shaped self-assembled GeSn QD with 1 nm thick wetting layer (WL): (**a**) 3D projection of the pyramidal QD with wetting layer, (**b**) cross-sectional view (ZX) showing the QD height and the direction of the external electric field, (**c**) plane view (XY).

To evaluate the QD s- and p- like electrons' energy levels and associated wave functions in Γ-valley, single band 3D-Schrodinger equation (Equation (1)) is solved in Cartesian coordinates within the effective mass approximation by finite elements method offered by COMSOL multiphysics software (version 5.0, COMSOL Inc., Stockholm, Sweden) [42] for the strained pyramidal GeSn QD under vertical applied electric field (Figure 1).

$$-\frac{\hbar^2}{2}\nabla\left[\frac{1}{m^\*(\mathbf{x},y,z)}\nabla\mathcal{Q}(\mathbf{x},y,z)\right] + (V(\mathbf{x},y,z) + \epsilon\mathbf{Fz})\mathcal{Q}(\mathbf{x},y,z) = \epsilon\mathcal{Q}(\mathbf{x},y,z) \tag{1}$$

where , ∅, *V* and *m*\* represent the electron energy level, envelop wave function, potential barrier and effective mass, respectively. F is the external electric field and e the elementary charge. Further details can be found elsewhere [42,43]. The calculation of the transition angular frequency associated dipole moment is mandatory to evaluate the AC and RIC. Indeed, the angular frequency dependent total intersubband optical AC (α) and RIC ( <sup>δ</sup>*<sup>n</sup> nr* ) are given by [10,11]:

$$a(\omega) = a^{(1)}(\omega) + a^{(3)}(\omega, I) \tag{2}$$

$$\frac{\delta n(\omega)}{n\_r} = \frac{\delta n^{(1)}(\omega)}{n\_r} + \frac{\delta n^{(3)}(\omega)}{n\_r} \tag{3}$$

where α(1) and <sup>δ</sup>*n*(1) *nr* , denote the linear AC and RIC (Equations (4) and (6)). <sup>α</sup>(3) and <sup>δ</sup>*n*(3) *nr* represent the 3rd order nonlinear AC and RIC expressed respectively by Equations (5) and (7):

$$\alpha^{(1)}(\omega) = \frac{\omega}{\hbar} \sqrt{\frac{\mu}{\varepsilon\_r}} \frac{\sigma \left| \mathcal{M}\_{fi} \right|^2 \Gamma}{\left[ \left( \omega\_{fi} - \omega \right)^2 + \Gamma^2 \right]} \tag{4}$$

$$\begin{split} \alpha^{(3)}(\omega, I) &= - \quad \frac{\omega \upsilon I}{2\varepsilon\_0 \eta\_r \omega \hbar^3} \sqrt{\frac{\mu}{\epsilon\_r}} \frac{|M\_{fi}|^2 \Gamma}{\left[ \left( \omega\_{fi} - \omega \right)^2 + \Gamma^2 \right]^2} \\ &\times \left[ 4 \left| M\_{fi} \right|^2 - \frac{\left( M\_{fi} - M\_{ii} \right)^2 \left( 3\omega\_{fi}^2 - 4\omega\_{fi}\omega + (\omega^2 - \Gamma^2) \right)}{\omega\_{fi}^2 + \Gamma^2} \right] \end{split} \tag{5}$$

$$\frac{\delta n^{(1)}(\omega)}{n\_r} = \frac{\sigma \left| M\_{fi} \right|^2}{2n\_r^2 \varepsilon\_0 \hbar} \frac{\omega\_{fi} - \omega}{\left[ \left( \omega\_{fi} - \omega \right)^2 + \Gamma^2 \right]} \tag{6}$$

$$\begin{split} \frac{\delta n^{(3)}(\omega, I)}{n\_r} &= \frac{-\mu \alpha |M\_{fi}|^2}{4n\_r^3 \epsilon\_0 \hbar^3 \left[\left(\omega\_{fi} - \omega\right)^2 + \Gamma^2\right]^2} \\ &\times \left[4\left(\omega\_{fi} - \omega\right) \middle|M\_{fi}|\right]^2 \\ &- \frac{\left(M\_{fi} - M\_{\tilde{u}}\right)^2 \left[\left(\omega\_{fi} - \omega\right) \times \left[\omega\_{fi}\left(\omega\_{fi} - \omega\right) - \Gamma^2\right] - \Gamma^2 \left(2\omega\_{fi} - \omega\right)\right]}{\omega\_{fi}^2 + \Gamma^2} \end{split} \tag{7}$$

I is the incident in-plane polarized light intensity, σ denotes the electron density (one electron per QD) [12]. Γ = 10 ps−<sup>1</sup> is the relaxation rate and *nr* the GeSn material's refractive index deduced by linear interpolation [16]. ω*fi* is the p-to-s transition frequency and *Mfi* = # ∅*f ex* ∅*i* \$ denotes the corresponding dipole moment for in-plane X polarized incident radiation. The subscript f and i refer to the final and initial states (QD p- and s electron states in this study). The p states are doubly degenerated (identified as px and py). A selection rule making the allowed transition to arise only from px state can be done by considering the incident radiation to be polarized along X direction [7,16,44].
