Excitonic and impurity-related optical transitions in Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ multiple quantum wells: Fractional-dimensional space approach

We have investigated the optical transitions in Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ multiple quantum wells with various width and doping levels. The fractional dimensionality model was extended to describe free-electron–acceptor (free hole-donor) transitions in a quantum well. The measured photoluminescence spectra from the samples were interpreted within the framework of this model, and acceptor-impurity induced effects in the photoluminescence line shapes from multiple quantum wells of different widths were demonstrated.

Figure 1

The PL spectra of the Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ samples for three different $L_W=10\mathrm{nm}$ $(N_{\mathrm{Be}}=5\times {10}^{10}{\mathrm{cm}}^{-2})$, $15\mathrm{nm}$ $(2.5\times {10}^{12}{\mathrm{cm}}^{-2})$, and $20\mathrm{nm}$ $(2.5\times {10}^{12}{\mathrm{cm}}^{-2})$ MQWs at room temperature. Arrows indicate $E_{e1\text{−}\mathit{hh}1},E_{e2\text{−}\mathit{hh}2},E_{e1\text{−}\mathit{lh}1}$ calculated energy differences from first heavy-hole to first electron, the second heavy-hole to second electron, and the first light-hole to first electron energy levels, respectively. For the sake of clarity, the spectra are shifted vertically.

Figure 2

The PL spectra of Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ samples with three different $L_W=10\mathrm{nm}$ $(N_{\mathrm{Be}}=5\times {10}^{10}{\mathrm{cm}}^{-2})$, $15\mathrm{nm}$ $(2.5\times {10}^{12}{\mathrm{cm}}^{-2})$, and $20\mathrm{nm}$ $(2.5\times {10}^{12}{\mathrm{cm}}^{-2})$ MQWs at liquid nitrogen temperature. The symbols $X_{e1\text{−}\mathit{hh}1}$ and $X_{e1\text{−}\mathit{lh}1}$ indicate heavy-hole and light-hole excitonic transitions, [Be, $X$]–to Be acceptor–bound exciton, $e\text{−}\mathrm{Be}$–free-electron–neutral Be acceptor transitions. For the sake of clarity, the spectra are shifted vertically.

Figure 3

The PL spectra of the Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ $L_W=20\mathrm{nm}$ samples for two different $N_{\mathrm{Be}}=5\times {10}^{10}{\mathrm{cm}}^{-2}$ (open circles) and $2.5\times {10}^{12}{\mathrm{cm}}^{-2}$ (closed circles) MQWs at liquid nitrogen temperature. $X_{e1\text{−}\mathit{hh}1}$ and $X_{e1\text{−}\mathit{lh}1}$ indicate heavy-hole and light-hole excitonic transitions, [Be, $X$]–to Be acceptor–bound exciton, $e\text{−}\mathrm{Be}$–free-electron-neutral Be acceptor transitions. The inset depicts an enlarged region of the $e\text{−}\mathrm{Be}$ emission spectra; the weakly doped MQWs PL intensity has been multiplied by a factor of 1.93.

Figure 4

The PL spectra of the $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ MQWs with $L_W=15\mathrm{nm}$ $(N_{\mathrm{Be}}=2.5\times {10}^{12}{\mathrm{cm}}^{-2})$ at liquid nitrogen temperature. $X_{e1\text{−}\mathit{hh}1}$ and $X_{e1\text{−}\mathit{lh}1}$ indicate heavy-hole and light-hole excitonic transitions, [Be, $X$] labels exciton bound-to-Be acceptor, $e\text{−}\mathrm{Be}$ designates electron-neutral Be acceptor transitions, and $e1\text{−}\mathit{hh}1$ indicate electron-hole transitions. Dashed lines depict theoretical calculations; solid lines represent the sum of all recombination mechanisms. The dimensionality, full width at half maximum of the Lorentzian or Gaussian convolution (given in parentheses) and relative weightings in the integrated PL spectra for various transitions are as follows: $e\text{−}\mathrm{Be}$ $(\alpha =2.87,w_G=6.4\mathrm{meV})-0.23$; [Be, $X$] $(w_G=4.7\mathrm{meV})-0.15$; $X_{e1\text{−}\mathit{hh}1}$ $(\alpha =2.3,w_L=3.5\mathrm{meV})-0.5$; $X_{e1\text{−}\mathit{lh}1}$ $(\alpha =2.2,w_L=3.5\mathrm{meV})-0.1$; $e1\text{−}\mathit{hh}1$ $(\alpha =2.3,w_L=3.5\mathrm{meV})-0.02$.

Figure 5

The PL spectra of the Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ $L_W=10\mathrm{nm}$ MQWs at room temperature. Symbols $X_{e1\text{−}\mathit{hh}1}$ and $X_{e1\text{−}\mathit{lh}1}$ indicate heavy-hole and light-hole excitonic transitions and $e1\text{−}\mathit{hh}1$ depict electron-hole transitions. Dashed lines are the theory, the solid line shows the sum of both excitonic and $e1\text{−}\mathit{hh}1$ transitions. The dimensionality (given in parentheses) and relative weightings in the integrated PL spectra for various transitions are as follows: $X_{e1\text{−}\mathit{hh}1}$ $(\alpha =2.2)-0.66$; $X_{e1\text{−}\mathit{lh}1}$ $(\alpha =2.06)-0.17$; $e1\text{−}\mathit{hh}1$ $(\alpha =2.2)-0.12$. For all the peaks the full width at half maximum of the Lorentzian convolution was found to be $w_L=11\mathrm{meV}$.

Figure 6

The PL spectra of the Be $\delta$-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ $L_W=15\mathrm{nm}$ MQWs at room temperature. Symbols $X_{e1\text{−}\mathit{hh}1}$, $X_{e2\text{−}\mathit{hh}2}$, and $X_{e1\text{−}\mathit{lh}1}$ indicate heavy-hole and light-hole excitonic transitions. Dashed lines are the theory, solid lines represent the sum of excitonic and both $e1\text{−}\mathit{hh}1$, $e2\text{−}\mathit{hh}2$ transitions. The dimensionality (given in parentheses) and relative weightings in the integrated PL spectra for various transitions are as follows: $X_{e1\text{−}\mathit{hh}1}$ $(\alpha =2.3)-0.49$; $X_{e1\text{−}\mathit{lh}1}$ $(\alpha =2.2)-0.27$; $X_{e2\text{−}\mathit{hh}2}$ $(\alpha =2.5)-0.03$; $e1\text{−}\mathit{hh}1$ $(\alpha =2.3)$, and $e2\text{−}\mathit{hh}2$ $(\alpha =2.5)-0.21$. For all curves the full width at half maximum of the Lorentzian convolution was found to be $w_L=9\mathrm{meV}$.

Figure 7

The spectral shape of the electron-acceptor transition absorption coefficient. The quantum well dimensionality is given as a parameter. The energetic scale is relative and shifted by the forbidden quantum well energy minus the acceptor binding energy for all of the dimensionality cases.

Figure 8

The calculated line shape of the free-electron–acceptor photoluminescence intensity on the QW dimensionality at liquid nitrogen temperature. The energetic scale is relative and shifted by the forbidden QW energy minus acceptor binding energy for all the dimensionality cases.

Figure 9

The spectral shape of the free-electron–acceptor transition in $L_W=10\mathrm{nm}$ width MQWs $(N_{\mathrm{Be}}=5\times {10}^{10}{\mathrm{cm}}^{-2})$ at liquid nitrogen temperature. Points show experimental data, lines denote theory: the dotted line is without and solid line is with the Gaussian convolution; the full width at half maximum is $w_G=6\mathrm{meV}$. Dimensionality $\alpha =2.8$. The energetic scale is relatively and shifted by the forbidden quantum well energy minus the acceptor binding energy.