Charge-carrier statistics at $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dots

The statistics of thermal electron emission from $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dots with base/height dimensions of $20∕10\left(\mathrm{nm}\right)$ are developed. The quantum dots considered are assumed to have two electron energy levels. For the electrons captured in the ground state, this gives the possibility of two different emission paths. Starting from a grand canonical ensemble and using an idea for “truncated cascade capture,” we derive “effective thermal emission rates” corresponding to experimental quantities. From experimental data of the capture cross sections, we demonstrate that the thermal emission path for electrons is shifted when the temperature is changed. In an Arrhenius plot for electron emission rates from the ground state, this is manifested as a transition region with varying slope which does not give any information about activation energies. The position on a temperature scale of this transition region depends on the internal relaxation time for electrons to go from the excited to the ground states. Due to limitations of experimental setups normally used for measuring activation energies, such measurements are done within a very limited temperature range. Erroneous interpretations of measured data therefore may occur if the possibility of a change in emission path is not taken into account. A method to avoid this problem in an experimental situation is pointed out in the discussion.

Figure 1

Energy scheme for an $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dot with base/height dimensions $20∕10\mathrm{nm}$. For larger dots, a level with $d$ character appears closer to the conduction band at $E_c$. Arrows (a) and (c)–(e) indicate the two possible thermal emission paths for $s$ electrons while arrows (b) and (d)–(f) show the corresponding capture paths.

Figure 2

Energy scheme illustrating the electron “effective emission rates” from a single-level two-electron system. The first electron leaves with a rate $e_{e,2}$ from the $s$ shell, occupied by two electrons, and turns the QD into a one-electron system (dashed arrow). This increases the concentration of one-electron systems before the second electron leaves with a rate $e_{e,1}$.

Figure 3

Sticking probability of the $(s,1)$ electron as a function of reciprocal temperature as obtained by Eq. (22) for different values of the time $t_1$ it takes for an electron to relax from the $p$ level to the $s$ level when the $p$ level is filled by one electron and the $s$ level is empty.

Figure 4

Thermal emission rate $e_{e,1}$ of an electron from the $s$ level when occupied by one electron as obtained by Eq. (13). For the lower temperatures, the two-step emission path across the $p$ level dominates the emission, while the direct process from the $s$ level to the conduction band dominates for higher temperatures. Between these two temperature regimes, there is a transition region, where the slope of the Arrhenius plot origins from a combination of the two emissions.

Figure 5

Emission transients for the first $(s,2)$ and second $(s,1)$ electrons leaving the quantum dot. The curves show the probabilities $P_{s,1}=n_{s,1}∕N_T$ and $P_{s,2}=n_{s,2}∕N_T$ that the $s$ shell is occupied by one and two electrons, respectively. Calculation has been done for $t_1=t_2={10}^{-10}\mathrm{s}$ and $1000∕T=12{\mathrm{mK}}^{-1}$

Figure 6

Logarithm of the charge $\Delta p_c$ in units of elementary charge, created per quantum dot in a semiconductor depletion region as it would appear in a DLTS measurement for $t_1=t_2={10}^{-10}\mathrm{s}$ and $1000∕T=12{\mathrm{mK}}^{-1}$ as in Fig. 5