Dynamics of quantum dot light-emitting diode with filtered optical feedback

Abstract

We study a quantum dot light-emitting diodes (QD-LEDs) subject to filtered optical feedback, where the filter is characterized by a mean frequency Ω _m and a filter width λ. In the limit of a narrow filter (λ = 0), the QD-LED equations reduce under some conditions to the equations for a QD-LED with optical feedback, whereas they become the Lang Kobayashi equations in the limit of an unbounded filter width (λ > 0). Through simulations based on the rate equations for a QD-LED with filtered external optical feedback modes, we show that the output’s nonlinear dynamical system attractors can be controlled through the filter parameters: the filter’s spectral width λ and its central frequency Ω _m. This is illustrated for a filter-induced global bistability.

Keywords

Quantum dot optical feedback control

Introduction

In nanotechnology, nanoparticles are classified by their size (in the range of 1–100 nm) and properties. Nanocrystals are described as having at least one dimension of less than or equal to100 nm and single crystalline. Quantum dots (QDs), also known as nanocrystals, are a nontraditional type of semiconductor with limitless applications as an enabling material across many industries.¹ The quantum dots, which can also be called artificial atoms, are nanometer-scale “boxes” that selectively hold or release electrons. These semiconductors range in size from 2 nm to 10 nm in diameter, which consist of 10–50 atoms. From a few hundred to a few hundred thousand atoms, QDs bridge the gap between single atoms and solid state, and because of this, they exhibit a combination of atomic and solid-state properties. The emission wavelength, or the color emitted, of QDs depends on the size, and using simple chemistry with semiconductor nanocrystals, the color can be precisely controlled. Light-emitting diodes (LEDs) have been created and produced in various colors from QDs.²

Semiconductor LEDs are very efficient in transforming electrical energy to incoherent light. Incoherent light is created in the LED by recombination of electron–hole pairs, which are generated by an electrical pump current. This recombination results in spontaneous emission of photons (light) and is amplified by multi emission. These photons should be allowed to escape from the device without being reabsorbed. On the down side, quantum dot light-emitting diodes (QD-LEDs) are known to be very sensitive to optical influences,³ especially in the form of external optical feedback from other optical components (such as mirrors and lenses) and via coupling to other systems. Depending on the exact situation, optical feedback may lead to many different kinds of output dynamics, from increased stability^4,5 all the way to complicated dynamics; for example, a period doubling cascade to chaos,⁶ torus break-up,⁷ and a boundary crisis⁸ have been identified. See the study of Fischer et al.^9,10 as entry points to the extensive literature on the possible dynamics of output with optical feedback.

A main concern in this article is to achieve stable, and possibly tunable, QD-LED operation. One way of achieving this has been to use filtered optical feedback (FOF) where the reflected light is spectrally filtered before it reenters the QD. As in any optical feedback system, important parameters are the delay time and the feedback strength. Moreover, FOF is a form of coherent feedback, meaning that the phase relationship between outgoing and returning light is also an important parameter.

The interest in the FOF QD-LED is due to the fact that filtering of the reflected light allows additional control over the behavior of the output of the system by means of choosing the spectral width of the filter and its detuning from the light frequency. The basic idea is that the FOF QD-LED produces stable output at the central frequency of the filter, which is of interest, for example, for achieving stable frequency tuning of QD-LEDs for the telecommunications applications.

QD-LED model with FOF

Depending on the model of Marino et al.¹¹ of LED, the QD-LED with external FOF is modeled, here, mathematically by four equations. Before going into the model, one must refer that this type modeling was accepted since we go into dimensionless modeling and it gives a behavior similar to an experimental one.¹² A new modeling for QD-LED appear in our recent work¹³ may accepted at any case.

Depending on the model of Marino et al.,¹¹ the rate equations for the (complex) electric field E and the (real) number of carriers in QDs n _QD and in the Wetting Layer (WL) n _wl can be written as:

E^{\cdot} = - \frac{1}{2} (1 + i α_{e}) α E - γ_{s} \frac{E}{2} + \frac{K}{2} E_{τ} e^{- i Θ} + E_{sp} (t),

n_{QD}^{\cdot} = γ_{c} n_{wl} (1 - \frac{n_{QD}}{2 N_{d}}) - γ_{r_{QD}} n_{QD} - (E_{sp} (t) - d {| E |}^{2}),

n_{wl}^{\cdot} = \frac{I}{e} - γ_{r_{wl}} n_{wl} - γ_{c} n_{wl} (1 - \frac{n_{QD}}{2 N_{d}}) .

Here, $E (t) = \sqrt{S} e^{- i Φ (t)}$ is the normalized slowly varying complex amplitude of the electrical field given in polar coordinates by the photon number S and the phase Φ, and time t is scaled with $ω_{r}^{- 1}$ , where ω_r is the frequency of the relaxation oscillation, an intrinsic resonance of the optical mode. The parameter α_e is the linewidth enhancement factor, ω_o is the solitary optical mode frequency, the parameter K measures the injected field strength. The phase shift of the light during one round trip in the external cavity ( $τ = 2 l / c$ ) is given by $Θ = ω_{o} τ$ , c is the speed of light. The field labeled by the subscript τ, E_τ , and therewith φ_τ, are the electric field amplitude and the optical phase taken at the delayed time t − τ. $γ_{r_{QD}}$ and $γ_{r_{wl}}$ are the nonradiative decay rates of the number of carriers in the QD and WL, respectively; N_d is the total number of QDs; I is the injection current;e is the elementary charge; γ_c is the capture rate from WL into the dot; and α and γ_s are the absorption and output coupling rate of photons in the optical mode, respectively.

For a three-level atomic system where the transition is homogeneously broadened, it can be shown from the Einstein relation that:¹⁴

α = A Γ n_{o} .

Here, A is the spontaneous emission rate into the optical mode, n_o is the initial carrier occupation number of the QDs, and Γ is the optical confinement factor. In such a case, the spontaneous emission coefficient and absorption coefficient possess identical lineshapes. For realistic QD material system, both the QD and WL states can be inhomogenously broadened. Population distributions in both WL and QD are taken into account explicitly in order to determine the correct relation between absorption and spontaneous emission spectra. E _sp (t) is the stochastic function corresponding to the zero-mean random field for spontaneous emissions. The field has the relation of $〈 E_{sp} (t) E^{*} (t) 〉 = R_{sp} / 2$ .¹⁵ The term R _sp is usually used for the effect of spontaneous emission in the photon number equation and is given by:¹⁶

R_{sp} = A n_{QD} and A = \frac{β D (ω)}{V_{D} p (ω)},

where β is the spontaneous emission factor, V_D the normalization QD volume, D(w) the normalized lineshape function, and p(w) the density of photon for the nonuniform QD.

Furthermore, when we investigate the fundamental dynamics of instability and chaos in nonlinear systems, we can treat the deterministic terms with considering statistical noises.

A straightforward way to let a QD-LED operate in a single longitudinal mode is by filtering the feedback light. Light with certain frequencies can pass the filter, while other frequencies are filtered (Figure 1). In equations (1) to (3), a delay term $\frac{K}{2} E (t - τ) e^{- i w_{o} τ}$ appears, modeling feedback from an external flat mirror. To incorporate the filter, this term is replaced by a term ζ F(t). It is the (complex) field amplitude reentering from the active region.

Figure 1.

Schematic setup of the considered QD-LED device with filtered optical feedback. QD-LED: quantum dot light emitting diode.

For simplicity, suppose that we build a new function $F (t) = F_{c} (t) + i F_{s} (t), F_{c} (t) and F_{s} (t)$ are the even and odd Fourier transform functions, respectively. The response function is assumed to be given by a simple Lorentzian frequency filter. Indeed, the spectral form of the transfer function induced by optical feedback from a grating or Fabry–Perot filter. Under this assumption, one obtains the differential equation for the filtered electric field F(t) satisfies:

\frac{d F_{c}}{d t} + Λ F_{c} + w_{m} F_{s} = Λ \sqrt{S S_{τ}} cos (- w_{o} τ),

\frac{d F_{s}}{d t} + Λ F_{s} - w_{m} F_{c} = Λ \sqrt{S_{τ} / S} sin (- w_{o} τ),

where Λ is the half width at half-maximum (HWHM) of the spectrum and $ω_{m} = ω_{f} - ω_{o}$ , ω_m is the central frequency and ω_f is the full (optical) frequency of the Lorentzian component. Realization of the FOF QD-LED in Figure 1 can be written in the full model as:

S^{\cdot} = - α S - γ_{s} S + A n_{QD} + ζ F_{c},

Φ^{\cdot} = \frac{1}{2} α_{e} α - ζ F_{s},

n_{QD}^{\cdot} = γ_{c} n_{wl} (1 - \frac{n_{QD}}{2 N_{d}}) - γ_{r_{QD}} n_{QD} - (A n_{QD} - α S),

n_{wl}^{\cdot} = \frac{I}{e} - γ_{r_{wl}} n_{wl} - γ_{c} n_{wl} (1 - \frac{n_{QD}}{2 N_{d}}),

F_{c}^{\cdot} = Λ \sqrt{S S_{τ}} cos (- w_{o} τ) - Λ F_{c} - w_{m} F_{s},

F_{s}^{\cdot} = Λ \sqrt{S_{τ} / S} sin (- w_{o} τ) - Λ F_{s} + w_{m} F_{c} .

Here $ζ = k / 2$ . See the study of Fischer et al.¹⁷ and references therein for an explanation, a derivation, and numerical studies of the resulting system. These first studies were followed by a series of numerical and experimental studies,^18

–21 in which various types of behavior were found and studied.However, analytically, even the simplest behavior of the QD-LED with FOF is still hardly understood, and since the injection is still obtained through feedback, the feedback delay appears to play a role in the selection of frequency and phase shift of the electric field E of the optical mode.

For numerical purposes, it is useful to rewrite equations (8) to (13) in dimensionless form. To this end, we introduce the new variables:

x = S, Φ \equiv Φ, y = \frac{A}{γ_{s}} (n_{QD} - Γ n_{o} S),

F \equiv F, w = \frac{n_{wl} γ_{c}}{A}, Ω_{o} = \frac{ω_{o}}{γ_{r_{wl}}},

γ = \frac{γ_{s}}{γ_{r_{wl}}}, γ_{1} = \frac{A}{γ_{r_{wl}}}, γ_{2} = \frac{A}{γ_{s}},

γ_{3} = \frac{γ_{r_{QD}}}{γ_{r_{wl}}}, γ_{4} = \frac{γ_{c}}{γ_{r_{wl}}},

$N_{d} \equiv a, Γ n_{o} \equiv b, δ_{o} = \frac{I}{A e}, θ = γ_{r_{wl}} τ,$ and the time scale $t^{'} = γ_{r_{wl}} t$ . The rate:

x^{\cdot} = γ (y - x) + ∊ F_{c},

Φ^{\cdot} = \frac{α_{e} b γ_{1}}{2} - ∊ F_{s},

y^{\cdot} = γ_{1} γ_{2} w (1 - b x / a) - γ_{1} y (1 + w / a) - γ_{1} b (x - y) - γ_{3} (y + γ_{2} b x),

w_{}^{\cdot} = γ_{4} δ_{o} - w (1 - γ_{4} y / a γ_{2}) - γ_{4} w (1 - b x / a),

F_{c}^{\cdot} = λ \sqrt{x x_{τ}} cos (- Ω_{o} θ) - λ F_{c} - Ω_{m} F_{s},

F_{s}^{\cdot} = λ \sqrt{x_{τ} / x} sin (- Ω_{o} θ) - λ F_{s} + Ω_{m} F_{c},

where $λ = Λ / γ_{r_{wl}}, Ω_{m} = w_{m} / γ_{r_{wl}}, and ∊ = ζ / γ_{r_{wl}}$ . The well-established assumptions here are that the delay times τ is larger than the light roundtrip time inside the active region and that the filters have a Lorentzian transmittance profile (Figure 1); see the studies of Al Husseini et al., Yousefi et al., and Diekman et al.^14,21,22 for more details. More specifically, one obtains equations (14) to (19) as an extension of the rate equations model of the single FOF laser (Al Naimee et al.,¹³ equations (1) to –(3)), with additional separate equations for the field and filtered term to the polar coordinates.

Equation (14) describes the time evolution of the real-valued slowly varying electric field amplitude and equation (15) describes the phase of the emission light. Equations (16) and (17) describe normalized number of carriers within the QD and WL. In equation (15) the material properties of the QD-LED are described by the linewidth enhancement factor α (which quantifies the amplitude-phase coupling or frequency shift under changes in number of carriers²³), the ratio between the carrier injection and the photon nonradiative decay rates, and the dimensionless pump parameter δ_o . Time is measured in units of the inverse photon nonradiative decay rates of 10⁻¹¹ s. Throughout, we use values of the QD-LED parameters given in Table 1.

Table 1.

Numerical parameters used in the simulation unless stated otherwise.

Parameters	Value	Parameters	Value
x_o	0.066	γ ₂	0.27
Φ_o	0.066	γ ₃	0.08
y_o	0.99	γ ₄	0.143
w_o	0.0049	α_e	2
F_s	0.1	a	2.988
F_c	0.1	b	2.821
γ	0.057	Ɛ ₁ = Ɛ ₂	0.03
γ ₁	0.12	δ_o	0.6

The FOF loops enter equations (14) and (15) as feedback terms Ɛ ₁ F_c (t) and Ɛ ₂ F_s (t) with normalized feedback strengths Ɛ ₁ and Ɛ ₂ of the normalized filter terms F_c (t) and F_s (t). In general, the presence of a filter in the system gives rise to an integral equation for the filter field. However, in the case of a Lorentzian transmittance profile as assumed here, derivation of the respective integral equation yields the description of the filter fields by a delay differential equations (18) and (19); see the study of Yousefi and Lenstra¹⁹ for more details. The filter loop is characterized by a number of parameters. As for any coherent feedback, we have the feedback strength Ɛ_i , the delay time θ, and the feedback phase C_p of the filter field, which is accumulated by the light during its travel through the feedback loop. Hence, C_p = Ω ₀θ. Owing to the large difference in time scales between the optical period 2π/Ω₀ and the delay time θ, one generally considers θ and C_p as independent parameters.

To investigate the dynamics, numerical integration of the rate equations is performed using a modified Runge–Kutta method of the fourth order. Throughout the simulations, the internal QD-LED parameters were fixed, while the solitary frequency was biased around 0.2/π, τ = 0.05, and central frequency Ω_m = 0.02/π. The results are presented in terms of bifurcation diagram where the behavior of the output of the QD-LED system is depicted as a function of the HWHM of the spectrum, as shown in Figure 2. In order to allow quick overview of all the different types of dynamics that may occur during a full scan of the HWHM of the spectrum, a visualization technique very similar to that of the time series, attractor’s section representation is employed at inter-spike interval. A plane is defined in a three-dimensional phase space, chosen such that all the possible to it as shown in Figure 3(a) to (d).

Figure 2.

Bifurcation diagram summarizing different dynamics that occur when HWHM is varying. HWHM: half width at half-maximum.

Figure 3.

Panel portraits (a), (b), and (c) for various time series, attractors, and ISI, respectively, of Figure 2. ISI: inter-spike interval. (d) Panel portrait [D] for time series of Figure 2.

In summary, as the HWHM is increased, we observe the transition between the steady state and oscillatory states at smaller bias value, the system fires less spikes and tends to exhibit small amplitude.

In Figure 4, the system shows multi-chaotic behavior with one-stable region between the frequencies which are about −0.09 to −0.06 not apart. The one-stable region is centered on the filter center and the solitary optical mode island of fixed points, respectively. In Figure 4, the bifurcation diagram for the filter-induced global bistable interval is shown. Going from right to left in Figure 4 (increasing solitary optical mode), the system is initially in a state of chaotic attractors and then it passes through a cascade of period doubled with small amplitude, self-oscillations (limit cycles), achieving the steady state before reprising itself again. When the locking range of the filter is reached, the global bistability appears.

Figure 4.

Bifurcation diagram of the FOF system in the intensity dynamics and the solitary optical mode frequency plane. HWHM is 0.03, Ω_f = −0.24/π (−0.0764), τ = 0.05, and Ɛ = 0.02; similar parameters are used in Figures 6 and 7. FOF: filtered optical feedback; HWHM: half width at half-maximum.

External filtered modes

We restrict ourselves to the study of “fixed points” or the so-called external filtered modes (EFMs) and the bifurcations associated with these states, which occur when the filter parameter λ(Λ) is varied. Although the equations for the fixed points have been written many times before,¹⁹ they haven’t been analyzed for QDs in a concise way yet. In this section, we analyze the QD rate equations of the EFM.

The basic solution of rate equations is the trivial solution (E; n _QD; n _wl; F) = (0; 0; P; 0) of equation (8). When this basic solution is stable, there is no optical mode activity; the output is off. Increasing the value of the pump P, it becomes unstable at the light threshold P = 0: the output starts optical mode and the so-called EFMs are formed. Such solutions have constant intensities and population, and a phase that depends linearly on time:

\begin{array}{l} E (t^{'}) = S_{s} e^{i Ω_{s} t^{'}}, F (t^{'}) = F_{s} e^{i (Ω_{s} t^{'} + φ)}, \\ n_{QD} (t^{'}) = n_{QDs} and n_{wl} (t^{'}) = n_{wls} \end{array}

Note here, that Ω_s is the detuning of the frequencies of E and F, where $S_{s}, F_{s}, n_{QDs}, n_{wls}, and φ$ are all real constants. They are often referred to as “fixed points.” Since the electrical fields E and F are optically related, both have the same frequency, possibly with a phase shift φ. These solutions are easily studied in a polar coordinate setting, with $Ω_{s} t^{'}$ defined as the linear part of the phase F. To find the EFMs, we substitute equation (20) into equations (14) to (19), separating real and imaginary parts then gives the equation:

Ω_{s} θ - Ω_{s} = 0.

where:

Ω_{s} θ = \frac{α Γ θ}{2} - \sqrt{1 + α^{2}} (\frac{∊ λ sin (φ - {tan}^{- 1} (α))}{\sqrt{λ^{2} + {(Ω_{s} - Ω_{m})}^{2}}})

and

φ = - Ω_{s} θ - C_{p} + {tan}^{- 1} (\frac{Ω_{s} - Ω_{m}}{λ}) .

The final form of EFMs is equation (22); this equation is similar to the EFMs of a conventional laser, with the addition of the first term in the denominator and the dependence on two sets of parameters. Equation (21) is a transcendental and implicit equation that allows one to determine all possible frequencies Ω_s of the EFMs for a given set of filter parameters. More specifically, the sought frequency values Ω_s of the FOF QD-LED can be determined from equation (21) numerically by root finding; for example, by Newton’s method in combination with numerical continuation. The advantage of the formulation of equation (21) is that it has a nice geometric interpretation: Ω_sθ is a function of Ω_s that oscillates about fixed value. More precisely, when Ω_o is changed over 2π, the graph of Ω_sθ sweeps out over the area.

Once Ω_s is known, the corresponding values of the other state variables of the EFMs can be found from:

n_{QDs} = - 1 / Γ + \sqrt{1 + α^{2}} (\frac{∊ λ cos (φ)}{\sqrt{λ^{2} + {(Ω_{s} - Ω_{m})}^{2}}}),

F_{s} = \frac{E_{s} λ}{\sqrt{λ^{2} + {(Ω_{s} - Ω_{m})}^{2}}},

Note that in this manner each mode of single-frequency operation of the output corresponds to a fixed point. The solutions to equation (9) can be obtained by graphical methods, from which the intersection of the straight line Ω _sθ with the oscillating right-hand side of equation (9) can be found. Thus, the solutions of equation (9) correspond to the intersections of f (Ω _s ) and g(Ω _s ), and bifurcations of EFMs occur when both f (Ω _s ) = g(Ω_s ) and f Ω_S = gΩ _S . A combination of these two requirements will lead to our bifurcation results. In Figure 5, the functions f and g are plotted as Ω_f is increased.

Figure 5.

Plot of f and g as functions of Ω_f as the parameter is increased. (a) Only one EFM exists. (b) Two EFMs are formed in a saddle-node bifurcation. (c) There are three EFMs. EFM: external filtered mode.

Figure 5 shows an example of the solutions of equation (8) as intersection points (blue line) between the oscillatory function Ω_S (ω_s ) and the diagonal (the straight line through the origin with slope 1); see also the study of Yousefi and Lenstra.¹⁹ This means that an EFM is, in fact, uniquely determined by its value of ω_s . Furthermore, Figure 5 represents all the relevant geometric information needed to determine and classify EFMs. Notice that in this specific example, the EFMs are separated into three groups. This geometric picture is very similar to that of the single FOF laser,¹⁹ but there is an important difference. For the FOF QD-LED, considering the attractor of the carrier’s number function as the detuning of the frequencies in equation (11). Figure 6 shows a small cycle in Figure 6(a), (b), and (d) at the same index of frequency and different indexes for large cycles.

Figure 6.

Fixed-point attractor solution. Corresponding with Figure 5.

The results of the fixed-point analysis are presented in terms of bifurcation maps where the behavior of the instantaneous frequency of the QD-LED system is depicted as a function of the solitary output frequency. Figure 7 shows the fixed points in the (Ω_S (Ω_O ), Ω_O )-plane, where Ωs is the frequency of the compound QD-LED system. It is precisely in these intervals that one should expect filter-induced bistability. This can be studied by investigating the dynamical behavior of the system. Note, in this respect, that nothing has been said yet about the stability of the fixed points. In fact, it is well known that feedback has a large destabilizing influence on the system in general, but in the case of filtering, one may expect some type of stabilization to occur.

Figure 7.

The fixed points in the (Ω_S (Ω_O ), Ω_O ) plane. The dash line Ω_O = Ω_f indicates the filter center. The fixed points are created and annihilated in saddle node bifurcations as the solitary QD-LED frequency is changed. QD-LED: quantum dot light emitting diode.

Conclusion

The main conclusion we draw from our simulations is that the filter detuning can be used for the controlled access to the various different dynamics. For the bifurcation diagram parameters studied, most of the dynamics show, sometimes in limit cycles and double periodic, large regions of chaotic dynamics. We have theoretically analyzed the influence of a relatively narrow frequency filter (HWHM = 0.03) while the solitary frequency is biased around 0.2/π, τ = 0.05, and central frequency Ω_m = 0.02/π on the deterministic nonlinear dynamics of a semiconductor QD-LED with filtered external optical feedback.

It should be mentioned that in our numerous experiments, we have observed many more interesting dynamical attractors, like limit cycles, quasi-periodicity, and strange attractors, which are not addressed in this article.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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