Phonon- and Auger-assisted tunneling from a quantum well to a quantum dot

We investigate phonon- and Auger-assisted tunnelings from an adjacent quantum well coupled to a quantum dot by Fermi’s golden rule. The filling of quantum-well states is included, and we obtain an average tunneling lifetime depending on the concentration in the quantum well. Depending on the barrier width between the quantum dot and quantum well, both mechanisms can result in a tunneling time of a few to several hundred picoseconds. For the condition of high concentration of carriers in a quantum well, the typical time scale of the net increase of carriers in a quantum dot due to phonon-assisted tunneling can be in sub-picosecond range, which agrees with the recent experimental observation.

Figure 1

(a) The QW to QD LO-phonon-assisted tunneling. The carriers in the QW emit an LO phonon and tunnel into the QD. (b) The QW to QD Auger-assisted tunneling. Two electrons 1 and 2 in the QW interact by Coulomb repulsion. One makes a transition to a QD state labeled as $2^{\prime }$ while the other is excited to a higher QW-energy state labeled as $1^{\prime }$.

Figure 2

(a) The net capture rate due to phonon- and Auger-assisted processes, $W_{w\to \overline{N}}^{\text{phonon}}$ and $W_{w\to \overline{N}}^{\text{Auger}}$ using Eqs. (33, 34), respectively, as a function of the QD energy level. (b) The average tunneling time constants, ${\tau }_{\mathrm{Av}}^{\text{phonon}}\left(n_w\right)$ and ${\tau }_{\mathrm{Av}}^{\text{Auger}}\left(n_w\right)$ using Eqs. (35, 36), respectively. (c) The Auger coefficient $C^{\text{Auger}}\left(n_w\right)$. The parameters used are $d=50\mathrm{\AA }$; $h=100\mathrm{\AA }$; barrier $\text{width}=25\mathrm{\AA }$; ${\rho }_0=100\mathrm{\AA }$; $N_D={10}^{10}{\mathrm{cm}}^{-2}$; $n_w=7.3\times {10}^{11}{\mathrm{cm}}^{-2}$; and $T=300\mathrm{K}$.

Figure 3

(a) The net capture rate due to phonon- and Auger-assisted processes, $W_{w\to \overline{N}}^{\text{phonon}}$ and $W_{w\to \overline{N}}^{\text{Auger}}$ using Eqs. (33, 34), as a function of QW surface carrier density. (b) The average tunneling time constants ${\tau }_{\mathrm{Av}}^{\text{phonon}}\left(n_w\right)$ and ${\tau }_{\mathrm{Av}}^{\text{Auger}}\left(n_w\right)$ using Eqs. (35, 36), respectively, and (c) the Auger coefficient $C^{\text{Auger}}\left(n_w\right)$. The parameters used are $d=50\mathrm{\AA }$; $h=100\mathrm{\AA }$; barrier $\text{width}=25\mathrm{\AA }$; ${\rho }_0=100\mathrm{\AA }$; $E_w-E_{\overline{N}}\approx 35.9\mathrm{meV}$; $N_D={10}^{10}{\mathrm{cm}}^{-2}$; and $T=300\mathrm{K}$.

Figure 4

(a) The net capture rate due to the phonon-assisted process $W_{w\to \overline{N}}^{\text{phonon}}$ using Eq. (33), as a function of QW surface carrier density for different barrier widths. (b) The net capture rate due to the Auger-assisted process, $W_{w\to \overline{N}}^{\text{Auger}}$ using Eq. (34), as a function of the QW surface carrier density for different barrier widths. The parameters used are $d=50\mathrm{\AA }$; $h=100\mathrm{\AA }$; ${\rho }_0=100\mathrm{\AA }$; $E_w-E_{\overline{N}}\approx 35.9\mathrm{meV}$; $N_D={10}^{10}{\mathrm{cm}}^{-2}$; and $T=300\mathrm{K}$.

Figure 5

(a) The average tunneling rate of the QW carriers due to the phonon-assisted process $1∕{\tau }_{\mathrm{Av}}^{\text{phonon}}\left(n_w\right)$ using Eq. (35), as a function of the QW surface carrier density for different QD densities. (b) The average tunneling rate of the QW carriers due to the Auger-assisted process $1∕{\tau }_{\mathrm{Av}}^{\text{Auger}}\left(n_w\right)$ using Eq. (36), as a function of the QW surface carrier density for different QD densities. The parameters used are $d=50\mathrm{\AA }$; $h=100\mathrm{\AA }$; barrier $\text{width}=25\mathrm{\AA }$; ${\rho }_0=100\mathrm{\AA }$; $E_w-E_{\overline{N}}\approx 35.9\mathrm{meV}$; $N_D={10}^{10}{\mathrm{cm}}^{-2}$; and $T=300\mathrm{K}$.

Figure 6

(a) The net capture rate due to the phonon-assisted process $W_{w\to \overline{N}}^{\text{phonon}}$ using Eq. (33), as a function of the QW surface carrier density for different temperatures. (b) The net capture rate due to the Auger-assisted process $W_{w\to \overline{N}}^{\text{Auger}}$ using Eq. (34), as a function of the QW surface carrier density for different temperatures. The parameters used are $d=50\mathrm{\AA }$; $h=100\mathrm{\AA }$; barrier $\text{width}=25\mathrm{\AA }$; ${\rho }_0=100\mathrm{\AA }$; $E_w-E_{\overline{N}}\approx 35.9\mathrm{meV}$; and $N_D={10}^{10}{\mathrm{cm}}^{-2}$.