A Theoretical Study of Light Absorption in Self Assembled Quantum Dots

Self assembled quantum dots have shown a great promise as a leading candidate for infrared detection at room temperature. In this paper, a theoretical model of the absorption coefficient of quantum dot devices is presented. Both of bound to bound absorption and bound to continuum absorption are taken into consideration in this model. This model is based on the effective mass theory and the Non Equilibrium Greens Function (NEGF) formalism. NEGF formalism is used to calculate the bound to continuum absorption coefficient. The results of the model have been compared with a published experimental work and a good agreement is obtained. Based on the presented model, the bound to bound absorption coefficient component is compared to the bound to continuum absorption coefficient component. In addition, the effects of the dot dimensions and electron filling on the bound to continuum absorption coefficient are also investigated. In general, increasing the dot filling increases the absorption and decreasing the dots dimensions will increase the absorption and move the absorption peak towards longer wavelengths.


Introduction
Self assembled quantum dots have attracted the attention due to the promise to improve the performance of many applications, like quantum dot infrared photodetectors (QDIP) [1], [2], and intermediate band solar cells (IBSC) [3].For the QDIPs, it is reported that due to 3-dimensional confinement of the electrons in the quantum dots, QDIPs should have lower dark current than the conventional photodetectors at the same operating temperature, thus QDIPs can operate at higher temperature with the same signal to noise ratio [1].Also QDIPs are very sensitive to the normal incidence unlike the QWIP [1].While for the IBSC it should have higher efficiency than the conventional solar cell [4].The proposed model is based on the non equilibrium Green's function formulation (NEGF).NEGF formalism provides an approach to study the transport in quantum systems in the presence of open boundary conditions via the concept of self energy [5].The NEGF formalism was used before for modeling the quantum dot-in-a-well (DWELL) structure [6][7][8].This model calculated the responsivity not the absorption coefficient of DWELL structure and in arbitrary units.For calculating the bound to bound absorption coefficient of a self assembled quantum dot, a model was presented for the InAS/GaAs QDIP [9].This model has led to an important result: for QDIP the in-plane polarized absorption is dominant as long as the dot height is not very small compared to its base radius.This result was confirmed experimentally [10].There are many models that calculate the bound to bound absorption [9,11], but very little work has been done for the bound to continuum absorption.The intraband absorption coefficient α is an important parameter for the design of different quantum dot applications.
In our model, the effective mass theory is used to build a hermitian Hamiltonian matrix for an isolated self assembled quantum dot.Diagonalizing this matrix gives the bound states and energies.Then, this Hamiltonian matrix with a nonhermitian self energy matrix are used to get the NEGF.Using the NEGF, the continuum states of the system have been calculated.Then, the bound to bound and bound to continuum absorption coefficients are both calculated.A comparison of our model and experimental data of [12] has been done, showing a good agreement.

Theoretical Model
A schematic of self assembled quantum dot islands is shown in Figure 1.The following assumptions are made: • The self assembled quantum dot islands are assumed to have ideal conical shape with uniform size.• Both the effects of the wetting layer and the coupling between neighboring dots are neglected.
These assumptions are justified from the scanning probe microscopy done to the self assembled dots [13].Modeling the absorption coefficient α for quantum dots in previous models [9,11] considered only the bound to bound absorption, where the ground state is the one filled with electrons and the excited states are weakly bound and unoccupied by electrons.These models do not include the bound to continuum absorption.For an efficient and accurate model for α, both the bound to bound and the bound to continuum absorption must be considered.The absorption coefficient α can be expressed as where  is the photon angular frequency, dots is the number of dots per unit volume, the refractive index of the material is , the speed of light in free space is C, the free space permittivity is 0 , where and n i n  f stand for initial and final states of possible transitions.For the degeneracy of the initial states we multiply by i , and for the degeneracy of the final states we multiply by F is the probability of having an electron in an energy state, ê is the polarization of the incident light, and fi d is the first order dipole moment and is given by For the bound to continuum absorption, the initial states i  are the bound states of the dots, and the final states f  are the continuum states in the conduction band.
For the bound to bound absorption, the initial states are the filled bound states of the dot, and the final states are the empty bound states in the dot.The density of states at f E is taken to be Gaussian function to add the effect of inhomogeneous broadening resulting from the variation in the dot dimensions, and the bound to bound α can be expressed as this is equivalent to the expressions in [9], [11].
As shown in Figure 1, the potential U of the dot in the sample has cylindrical symmetry inside the dot and outside the dot in the barrier region.The effective mass Hamiltonian takes the following form in the cylindrical coordinates ) ,  add positive infinite s a simal imaginary part to the energy [14].The self energy matrix R  is given by [5] where t is the coupling coefficient between an elem the basic cell on the boundary and the adjacent el in the lead as shown in Figure 2.
is the average of ent in ement  (1) ( ( )) sin( ) sin( ) )) )) where l is an integer, The continuum wave function of the device is, at disn points that are adjacent to t e lead.
points, is the ex fu ) For each mode, the number of continuum states in the mode is determined from the density of states (D Th ( OS). e DOS in the continuum spectrum have negligible dependence on the quantum dot.Thus, the number of states f N with energy f E in the mode l is calcu- lated in the absence of the dot potential.The device shape becomes a solid disk made from the barrier material with radius L r and height b h .

Results and Conclusion
The model can be used to calculate α for any self assemits band parameters, s bled quantum dot structure given doping, dot dimensions d r , d h , and the polarization of the incident light ê .The ability to absorb normally incident radiation is an important property to the quantum dot structure, unlike the quantum well structure that is insensitive to the in-plane polarized light.Experimental results have indicated that the normal incidence absorption is more dominant in the quantum dot, unless its height is very small compared to the lateral dimension [10].So, we will make the calculations for the normally incidence case in this paper.
Calculations have been done for InAs/GaAs quantum dot with dimensions of 9.8 at different values of filling.As shown in this figure, the bound to continuum absorption increases when the excited states are not empty, as the dipole moment from excited states is much greater than the dipole moment from the ground state.Also the bound to bound absorption has greater peak and smaller band width than the bound to continuum absorption.The greater the variations in the dot dimensions, the greater the broadening in the bound to bound absorption.For the bound to bound absorption to occur the dot dimensions must be large enough to have at least one excited state.To see the effect of changing the dot dimensions, calculations are made for different dot dimensions with 2 electrons per dot.To check the validity of our model, we have calculated the bound to continuum absorption coefficient α for an InGaAs/GaAs uncoupled self assembled quantum dots that is measured experimentally in [10].The calculated results show a very good agreement with the experimental measurements.However the measurements are normalized and the paper doesn't state the number of dots in the studied sample, we made the calculations assuming number of dots Figure 6 shows the measured and the final analysis s will increase the abo h will be no bou .calculated α together.In , the bound to continuum absorption coefficient becomes much greater as the electron filling in the dot increases.lso decreasing the dot dimension A s rption and move the absorption peak towards greater wavelengths with higher peaks.On the other hand, the bound to bound absorption occurs at the energy difference between the bound states and it is inhomogeneously broadened by variations in the dot dimensions.The bound to bound is more peaky and larger in value than the bound to continuum absorption coefficient.But if there are no excited states or the excited states are filled with electrons, t ere nd to bound absorption.

Figure 1 .
Figure 1.A schematic of the self assembled quantum dot structure.where r m is the effective mass in the lateral direction, and z m is the effective mass in the growth direction, nd e a n is th  quantum number.The discretization of Equatio 4) gives the hermitian Hamiltonian matrix n ( H . Diag lizing this matrix gives the bound state wave functions and energies.For the Continuum states, NEGF is used to include the effects of open boundary conditio .The retarded Green's function ona an isolated lead.A closed form solution for retarded Green' ction in the semi-infinite lead is obtained by solving uation (5) assuming zero ndary conditions.The result is a series of harmonics and Bessel functions,

Figure 2 .
Figure 2. Discretization of the basic cell in the cylindrical coordinates.


the source vector which is zeros except cretizatio in the lead close to the interface points i p[5].The wave function L  is obtained analytically in the isolated lead with zero boun y conditions.For the infinite e of the lead at ( r   ), another bounda point L r   is added a d (

Figure 3
shows the calculated α for the system of InAs/GaAs material with the number of dot unit

Figure 4 Figure 4 .Figure 5 .Figure 6 .
Figure 3. Absorption coefficient α for es of electron filling calculated for dot with h d = 9.8 nm calculated for normal incidence.diffe d = 6.rent cas 3 nm, r