Multiphoton exciton absorption in a semiconductor superlattice in a dc electric field

An analytical approach to the problem of the multiphoton exciton absorption in biased narrow-well superlattices (SLs) induced by the optical transitions to the localized resonant exciton states is developed. Both the ac electric field of the intense optical wave and the dc electric field are directed parallel to the SL axis. The SL is formed by a periodic sequence of quantum wells (QWs) whose widths are taken to be much less than the exciton Bohr radius. The model of the SL potential employs a limiting form of the Kronig-Penney potential, i.e., a periodic chain of QWs separated by $\delta$-function-type barriers. A sufficiently strong dc electric field provides the localization of the carriers within one period of the SL. Analytical dependencies of the coefficient of the multiphoton exciton absorption on the characteristics of the dc and ac electric fields and on the parameters of the SL in the approximation of both isolated and interacting Wannier-Stark levels are obtained in the nearest-neighbor tight-binding approximations. Our analytical results correlate well with those obtained in numerical investigations. Estimates of the expected experimental values are performed for the parameters of a $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ SL.

Figure 1

The dependence of the oscillator strength $I_0^{\left(1\right)}$ [Eq. (59)] of one-photon transitions $(l=1)$ to the ground exciton state $n_0=1$, $\sigma =0$ and of the binding energy ${\mathcal{E}}_b^{\left(0\right)}$ [Eq. (58)] on the dimensionless dc electric field $E∕E_0$ $(E_0=\frac{\Delta }{2ed})$ for different QW widths $d_w$ scaled to the exciton Bohr radius $a_0$ $(s=d_w∕a_0=0.26,0.35,0.43)$ corresponding to the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ SLs studied in Refs. 11, 22, 18, respectively.

Figure 2

The schematic form of the spectrum of the two-photon exciton absorption as a function of the shift of the photon energy $2\hslash \omega$ with respect to the peak position ${\mathcal{E}}_0^{\left(0\right)}\left(1\right)$ [Eq. (38)] scaled to the exciton Rydberg constant Ry for different dc electric fields $E$ $(k=\frac{eEd}{\mathrm{Ry}})$.

Figure 3

The dependence of the oscillator strength $I_{\sigma }^{\left(1\right)}$ [Eq. (60)] of two-photon transitions $(l=2)$ to the ground exciton state $n_0=1$ adjacent to the WSLs $\sigma =\pm 1$ $\left(I_{\pm 1}^{\left(2\right)}\right)$ and $\sigma =0$ $\left(I_0^{\left(2\right)}\right)$ on the dimensionless dc electric field $E∕E_0$ $(E_0=\frac{\Delta }{2ed})$ for the parameters of the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ SLs (Ref. 43): $d_w=19\mathrm{\AA }$, $d=36\mathrm{\AA }$, ${\Delta }_e=193\mathrm{meV}$, and ${\Delta }_h=50\mathrm{meV}$.

Figure 4

The width ${\Gamma }_0\left(1\right)$ [Eq. (52)] of the ground exciton state $n_0=1$, $\sigma =0$ scaled to the exciton Rydberg constant Ry versus the dimensionless dc electric field $E∕E_0$ $(E_0=\frac{\Delta }{2ed})$ for different QW widths $d_w$, $s=\frac{d_w}{a_0}=0.30,0.35,0.43$ ($a_0$ is the exciton Bohr radius) relevant to the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ SLs studied in Refs. 19, 22, 18, respectively.

Figure 5

The dimensionless width ${\Gamma }_0\left(1\right)∕\mathrm{Ry}$ [Eq. (52)] (Ry is the exciton Rydberg constant) of the ground exciton state $n_0=1$, $\sigma =0$ as a function of the period of the SL $d$ scaled to the exciton Bohr radius $a_0$ for different parameters $p=E∕E_{ex}$ $(E_{ex}=\frac{{\Delta }_0}{2e{a}_0})$, where ${\Delta }_0=8.0\mathrm{meV}$ calculated from the parameters of the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ SL (Ref. 19): $d=51\mathrm{\AA }$, ${\Delta }_e=94.5\mathrm{meV}$, and ${\Delta }_h=5.5\mathrm{meV}$.

Figure 6

Isowidth curves ${\Gamma }_0\left(1\right)∕\mathrm{Ry}=q=\mathrm{const}$, where ${\Gamma }_0\left(1\right)$ [Eq. (52)] is the width of the ground exciton state $n_0=1$, $\sigma =0$, Ry and $a_0$ are the Rydberg constant and the Bohr radius of the exciton, respectively, $d$ is the SL period, and $E$ is the dc electric field scaled to the field $E_{ex}=\frac{{\Delta }_0}{2e{a}_0}$, ${\Delta }_0=8.0\mathrm{meV}$ (see Fig. 5 caption for details).