Modelling a Quantum Dot Ensemble

Overview: In the following, we show how an inhomogenously broadened ensemble of QDs can be modelled using a spatially resolved statistical approach. The effect of QD broadening on material gain and carrier induced refractive index is discussed.

1  Modelling an Ensemble of Quantum Dots
    1.1  Spatially Resolved Description of an Ensemble of Quantum Dots
    1.1  Material Gain of an Inhomogeneously Broadened QD Ensemble
    1.1  Carrier Induced Refractive Index of QD Media

Currently, the most promising fabrication methods of QDs involve the use of self-organised epitaxial growth. A common method of producing self-organised quantum dots is based on Stranski-Krastanow (SK) growth mode [7]. SK growth mode can be applied for material systems like InAs/GaAs, InGaAs/GaAs or InP/GaInP. All these semiconductor pairs show a substantial lattice mismatch. If InAs is grown on a GaAs substrate, the nucleation process starts at a critical thickness, leading to the formation of small islands. The growth process of a self-assembled semiconductor quantum dot is shown in Fig. 1.1.

Fig. 1.1: Animation showing the growth of an In_0.6Ga_0.4As QD on an In_0.1Ga_0.9As substrate. The QD is overgrown with a layer of In_0.1Ga_0.9As forming a quantum well with a height of 6.5 nm. Note: To start the animation hover the mouse pointer over the figure content.

The size of SOQDs is remarkably uniform, the fluctuations being in the range of only 10 % [3]. The S-K growth method has been successfully applied using molecular beam epitaxy and metal-organic chemical vapour deposition.

In general it is desired to produce QD arrays with high dot density in the range of 10¹¹/cm² . A further increase of the number of QDs can be achieved by stacking layers of QDs. The thickness of the barriers between the layers determines the degree of vertical alignment and electronic coupling [5]. Due to the strain field induced by the QDs in the first layer the islands in the next layer tend to form just above the QDs in the bottom layer [3].

An optoelectronic device based on QDs (leaving out micro-cavity structures with a low number of QDs) has a size of the order of 10⁴ μm² and contains a large number of QDs (of the order of 10⁶ − 10⁸) that are excited simultaneously [8]. Equations (1.13), (1.14), and (1.15) describing the time evolution of the occupation probability of the bound QD states and the microscopic polarisation refer to a single QD. In order to simulate a QDSOA, it is therefore essential to make suitable approximations.

In our approach, the total number of QDs within a device is described by a statistically representative subgroup of QDs. The representative subset of QDs is formed by randomly choosing QDs from an ensemble assuming that they follow a normal distribution.

The principle of sampling is demonstrated in Fig. 1.2. For this purpose, we randomly draw values from a Gaussian distribution centered at 0 and with a standard deviation of 2. Each random value is represented by a square (shown in the inset). We start with a sampling size of 100 points and increase the sampling set up to 400000 points. As the number of points in the sampling set increases, the corresponding histogram (red curve) approaches the statistical distribution (green curve) the values are following.

Fig 1.2: The red graph represents a histogram (normalised frequency count) of values randomly drawn from a set of points that are normally distributed. The green curve is a plot of the Gaussian distribution centered at 0 and with standard deviation 2. The inset shows the set of values that have been drawn, each square representing one specific value (visualised as color shade).

To obtain a statistically representative subgroup of QDs, we assume that the mean of the Gaussian distribution is given by the centre energy of the homogeneously broadened spectral lines. A typical gain spectrum of a single InGaAs QD is represented by the green curve in the Fig. 1.3. For a single QD, the transitions are clearly visible (ground state transition at 1.09meV) and the broadening of the lines is given by ∆E_hom^FWHM. (FWHM stands for Full Width at Half Measure.)

An animation can be started by hovering the mouse cursor over the figure. It shows the average material gain of an ensemble of QDs as a function of energy. The inhomogeneously broadened gain profile is obtained by successively averaging over an increasing number of non-identical QDs. The averaging leads to a broadening of the gain profile due to the inhomogeneous broadening. We notice that the averaged gain profile does not change significantly once the sampling rate exceeds approximately 10³ QDs.

Fig. 1.3: Material gain of an inhomogeneously broadened ensemble of InGaAs QDs. Inhomogeneous broadening QD ground state: 30meV. Inhomogeneous broadening QD excited states: 55meV. Homogeneous broadening QD ground state: 8meV, Homogeneous broadening QD excited states: 12meV.

The standard deviation of the normal distribution describing the inhomongeneous broadening is: σ=∆E_FWHM·[2√{2ln2}]⁻¹, where ∆E_FWHM is the FWHM of the inhomogeneously broadened spectral line.

In order to integrate the two dimensional wave-equation (1.34) numerically, the area of the device is discretised in approximately 10⁴ grid-points (depending on the dimensions of the device and the wave-length of the emitted radiation). Assigning one representative QD per numerical grid-point is a way to obtain the statistically representative subgroup of QDs. In this way a group of e.g. 10⁸ QDs is represented by a subgroup of 10⁴ QDs. Physical properties described by parameters that vary from dot to dot (like confinement energies, scattering rates, and the coefficients describing spectral broadening and shift) are calculated and assigned for each representative QD before starting the numerical integration. In this way, the inhomogeneous broadening, a characteristic property of an ensemble of QDs with varying sizes, is incorporated into the numerical model.

Physical properties that depend on the whole ensemble of QDs can be calculated by statistical averaging over the total number N_sim of representative QDs. For example the gain spectrum of the inhomogeneously broadened QDSOA can be calculated using [6]:

g(ω) = k02 β Γ 2 ħ ϵ0 nQD Nl da Nsim ∑ ξ  ⎧ ⎨ ⎩ ∑ i,j  ex(∆ijξ ⊗∆ijξ ∗ )ex [ ne,ξi + nhξj − 1 ]L(ω,ωj,iξ,γj,iξ) ⎫ ⎬ ⎭ , (1.1)

where d_a is the width of the active area and N_l is the number of QD layers. The effective propagation constant β is defined here, the confinement factor Γ is defined here, and the QD level electron and hole occcupation probabilities n^e,ξ_i and n^h,ξ_j, respectively, are defined here.

The index ξ runs over all QDs in the statistical ensemble, whereas i and j labels QD electron and hole levels, respectively. We have assumed that the transverse electric field is polarised in x-direction (compare with Fig. ). As a result the dipole matrix element tensor is multiplied with the unit vector in x-direction e_x. Additionally, the following definitions have been used:

γj,iξ = γj,i + ∆γj,iξ, (1.2) ωj,iξ = ωj,ie,h;ξ + ∆ωj,iξ, (1.3) L(ω,ωj,iξ,γj,iξ) = γj,iξ [ (ω− ωj,iξ)2 + (γj,iξ)2] (1.4)

where γ_j,i = [1/(2ħ )]∆E_hom^FWHM and ∆E_hom^FWHM is the FWHM of the homogeneously broadened QD transition i ↔ j. The quantity ω_j,i^e,h;ξ is the central frequency of the transition i ↔ j of sample QD ξ, whereas L(ω,ω_j,i^ξ,γ_j,i^ξ) is a non-normalised Lorentzian function characterizing the line-shape of the transition.

Figure 1.4 shows the transient gain and emission spectrum recorded during the first 15ps after the startup of a QD laser. At timepoint 0ps the gain is negative (absorbtive medium) due to the low carrier occupation probability of the QD states. The material gain then gradually increases due to carrier capture into the QDs. Note that the gain spectrum differs from the one presented in Fig. 1.3 since in this case only the ground state and the first excited state have been included. Moreover, the gain is negative at energies around 1.12eV indicating that the excited state has carrier occupation probability below 0.5.

Fig.1.4: Transient gain and emission spectrum recorded during the startup of a QD laser with a cavity length of 600μm and an inhomogeneous broadening of 30meV. Recording time 15ps.

Using the definitions introduced in (1.2) - (1.4), the average change of refractive index due to carriers confined to QDs can be calculated using [6]:

δn(ω) = − ΓnQD Nl 2 ħ ϵ0 da Nsim ∑ ξ  ⎧ ⎨ ⎩ ∑ i,j  ex(∆ijξ ⊗∆ijξ ∗ )ex [ ne  ,ξi + nh  , ξj − 1 ] ω− ωj,iξ γj,iξ L(ω,ωj,iξ,γj,iξ) ⎫ ⎬ ⎭ (1.5)

The carrier induced refractive index during the startup of a QDSOA is shown in Fig. 1.5. The QDSOA contains QDs with an inhomogeneous broadening of 30meV and and areal density of 10¹¹cm^-2. After applying a voltage of 1.20V the carrier concentration in the 2-D layer adjacent to the QDs increases. This leads to a increased scattering of carriers into the QDs inducing a change of the refractive index.

Fig. 1.5: Carrier induced refractive index during the startup of a QDSOA. A voltage is applied at timepoint 0 ps. The concentration of electrons and holes in the QDs increases gradually leading to a change of the refractive index.

[1]

P. Bhattacharaya. Properties of Lattice-Matched and Strained Indium Gallium Arsenide. Inspec, 1993.

[2]

P. Bhattacharaya . Properties of III-V Quantum Wells and Superlattices. Inspec, The Institution of Electrical Engineers, London, United Kingdom, 1996.

[3]

D. Bimberg, M. Grundmann, and N. N. Ledentsov. Quantum Dot Heterostructures. John Wiley and Sons, Chichester, 1999.

[4]

A.R. Gourlay and J. L. Morris. Hopscotch difference methods for nonlinear hyperbolic systems. IBM Journal of Research and Development, 16(4):349, 1972.

[5]

M. Grundmann, O. Stier, and D. Bimberg. InAs/GaAs pyramidal quantum dots: strain distribution, optical phonons, and electronic structure. Physical Review B (Condensed Matter), 52(16):11969 - 81, 1995/10/15.

[6]

H. Haug and S. W. Koch. Quantum Theory Of The Optical And Electronic Properties Of Semiconductors. Singapore, World Scientific, 1998.

[7]

H. Ishikawa, H. Shoji, Y. Nakata, K. Mukai, M. Sugawara, M. Egawa, N. Otsuka, Y. Sugiyama, T. Futatsugi, and N. Yokohama. Self-organized quantum dots and quatum dot lasers (invited). J. Vac. Sci. Technol. A, 16(2):794-800, 1998.

[8]

M.S. Skolnik and D.J. Mowbray. Self-assembled semiconductor quantum dots: Fundamental physics and device applications. Annu. Rev. Mater. Res., 34:181-218, 2004.

[9]

O. Stier . Electronic and Optical Properties of Quantum Dots and Wires. Wissenschaft und Technik Verlag, Berlin, 2001.

[10]

O. Stier, M. Grundmann, and D. Bimberg. Electronic and optical properties of strained quantum dots modeled by 8-band k·p theory. Physical Review B, 59(8):5688 - 5701, 1999.

[11]

V. I. Zubkov, M. A. Melnik, A. V. Solomonov, E. O. Tsvelev, F. Bugge, M. Weyers, and G. Trankle. Precision determination of band offsets in strained In_xGa_{1 − x}As/GaAs quantum wells by capacitance-voltage profiling and Schrödinger-Poisson self-consistent simulation. Physical Review B, 70(7):075312, 2004.

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