Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures

Using multiband $\vec{k}\cdot \vec{p}$ theory we study the size and geometry dependence on the slow light properties of conical semiconductor quantum dots. We find the $V$-type scheme for electromagnetically induced transparency (EIT) to be most favorable and identify an optimal height and size for efficient EIT operation. In case of the ladder scheme, the existence of additional dipole allowed intraband transitions along with an almost equidistant energy-level spacing adds additional decay pathways, which significantly impairs the EIT effect. We further study the influence of strain and band mixing comparing four different $\vec{k}\cdot \vec{p}$ band-structure models. In addition to the separation of the heavy and light holes due to the biaxial-strain component, we observe a general reduction in the transition strengths due to energy crossings in the valence bands caused by strain and band-mixing effects. We furthermore find a nontrivial quantum dot size dependence of the dipole moments directly related to the biaxial-strain component. Due to the separation of the heavy and light holes the optical transition strengths between the lower conduction and upper most valence-band states computed using one-band model and eight-band model show general qualitative agreement, with exceptions relevant for EIT operation.

Figure 1

The three generic EIT schemes. ${\omega }_p$ and ${\omega }_c$ refer to the frequency of the probe and coupling transitions, respectively.

Figure 2

The shape of the quantum dots under consideration.

Figure 3

(Color online) Energy levels and the most relevant dipole moments for EIT (top) and values of the eight strongest interband dipole moments (bottom) for one-band (left) and eight-band (right) model without strain for a dot with $h=7.5\text{nm}$.

Figure 4

(Color online) Energy levels and the most relevant dipole moments for EIT (top) and values of the eight strongest interband dipole moments (bottom) for one-band (left) and eight-band (right) model with strain for a dot with $h=7.5\text{nm}$.

Figure 5

(Color online) Probability density ${\left|\psi \right|}^2$ of the valence-band ground state for the eight-band model without (up) and with (down) strain. The dimensions of the quantum dot are $h=11.5\text{nm}$ and $r=23\text{nm}$.

Figure 6

Interband ${\mu }_{11}$ dipole moments as a function of $h$ for the four different models.

Figure 7

Energies of the six considered levels in conduction (top) and valence (bottom) band as a function of $h$ for eight-band model with strain. The onset of continuous WL states is also indicated (dashed-horizontal line).

Figure 8

Interband dipole moments $\mu$ (top) and relative band energy difference $\Delta E$ (bottom) as a function of the height $h$ with fixed aspect ratio $r=2h$.

Figure 9

Interband dipole moments $\mu$ (top) and relative band energy difference $\Delta E$ (bottom) as a function of the aspect ratio at constant volume.

Figure 10

Example of susceptibility calculated for a ladder scheme using the three levels or a full multilevel approach based on either the eight-band strained or one-band unstrained model. The results are presented for dots of height 7.5 nm (top panel) and 9 nm (bottom panel), both $A_{sp}=2$. The reason for the larger features in the one-band calculations is that the strength of the probe transition is larger within this model (see Fig. 6).

Figure 11

EIT absorption spectrum calculated using ${\mu }_{41}$ as probe and ${\mu }_{46}$ as coupling transition for different models: one-band unstrained (full line), one-band strained (dashed line), and eight-band strained (dotted line). The additional feature in the eight-band strained calculation is due to the nearby lying ${\mu }_{55}$ resonance.

Figure 12

(Color online) Slowdown versus coupling beam intensity calculated for three different dot sizes. The black curve refers to the $V$ scheme whereas the red curve is for $\Lambda$ configuration.