Microscopic study of relaxation oscillations in quantum-dot VCSELs

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Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 337–344 www.elsevier.com/locate/photonics

Microscopic study of relaxation oscillations in quantum-dot VCSELs Jeong Eun Kim a,*, Matthias-Ren´e Dachner a, Alexander Wilms b, Marten Richter a, Ermin Malic a a

Institut ¨fur Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany b Weierstraß-Institut ¨fur Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany Received 28 January 2011; received in revised form 6 April 2011; accepted 8 April 2011 Available online 16 April 2011

Abstract We propose a theoretical model to investigate the switch-on dynamics of electrically pumped quantum dot vertical-cavity surface-emitting lasers. The model is based on the self-consistently combined quantum dot-wetting layer Maxwell–Bloch equations incorporating microscopically calculated Coulomb and phonon-assisted scattering processes between the quantum dot and the quantum dot-embedding wetting layer states. Our approach allows the calculation of the time delay before lasing as well as of the frequency and damping rate of the appearing relaxation oscillation. We study their dependence on the strength of the injection current density and on the number of QD-layers and DBR-layers by using the finite-difference time-domain method. The results show that switch-on delay time is in inversely proportion to the injection current density while the frequency and the damping rate of relaxation oscillation are proportional to the current density. If we increase the numbers of QD-layers and DBR-layers the delay time becomes shorter, and the frequency and the damping rate become larger. # 2011 Elsevier B.V. All rights reserved. Keywords: Quantum-dot vertical-cavity surface-emitting laser; Maxwell–Bloch equations; Coulomb scattering rates; Electron–phonon scattering rates; Finite-difference time-domain method

1. Introduction Vertical-cavity surface-emitting lasers (VCSELs) have emerged as promising devices for use in optical communications due to their intrinsic advantages. The most important features are low threshold current and single-mode operation due to a small active volume as well as high temperature stability [1]. Furthermore, due to their geometry on-wafer testing and efficient

* Corresponding author. Tel.: +49 30 314 24858; fax: +49 30 314 21130. E-mail address: [email protected] (J.E. Kim). 1569-4410/$ – see front matter # 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2011.04.004

coupling to optical fibers are available. With quantum dots (QDs) as a gain medium, it shows even lower threshold and higher output power due to 3-dimensional confinement. Their emission wavelength is tunable by varying the size and the composition of QDs [2,3]. Due to their high performance, QD-VCSELs are predicted to have high speed modulation response which strongly depends on the frequency and the damping rate of the relaxation oscillations [4]. The latter are significantly affected by the carrier dynamics, which needs to be understood on a microscopical level. Therefore, in this paper we propose a model for electrically pumped VCSEL, which includes selfassembled QDs within the Stranski–Krastanow growth

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mode. The model is based on the Maxwell–Bloch equations derived for the description of spatiotemporal dynamics of semiconductor lasers and nonlinear optical properties of semiconductor quantum well microcavities [5,6]. In this work, we apply the approach to describe the generation of relaxation oscillations in QDVCSELs. We extend the model including the microscopic carrier dynamics between the QD and the WL states via Coulomb and electron-phonon scattering processes. Within this model, we study the switch on dynamics to investigate the internal time scales (e.g. switch-on delay time and frequency and damping rate of the relaxation) in dependence on the strength of the injection current, the Bragg mirror reflectivity, and the number of QD-layers. Our aim is to find optimal conditions for high speed modulation. The numeric calculations for the switch-on dynamics are performed by using the finite-difference time-domain (FDTD) method with a restriction to one dimension in the direction of the vertical axis, assuming that the fundamental TEM00 transverse mode propagates along this axis. To simplify the model, we only treat QD ground states of electron and hole as two-level system, which is valid for small in-plane size of QDs which have one electron state and two close lying hole states [7]. Furthermore, the energetically lowest electron and hole levels in the QDs contribute crucially to the emission dynamics at high WL carrier densities [8]. The paper is organized as follows: In Section 2, a theoretical approach to the system for the switch-on dynamics is introduced. In Section 3, QD-VCSEL model structures are presented. In Section 4, the microscopic description of the switch-on dynamics is shown, and the switch-on delay time and the frequency

and the damping rate of relaxation oscillations are calculated. Finally, we summarize the results in Section 5. 2. Theoretical model In order to study the switch-on dynamics of QDVCSEL, we treat the laser field with Maxwell equations and carriers with semiconductor Bloch equations [9]. Fig. 1 shows a schematic diagram of the self-consistent theoretical model. The Maxwell’s equations for field dynamics and the Bloch equations for the carrier dynamics are combined via the macroscopic polarization of the gain medium. The polarization (P1(z, t) and P2(z, t): real and imaginary component of the polarization, respectively) interacts with charge carriers via the driving field 0 emission Ex ðz; tÞ. Furthermore, the Coulomb and electron–phonon scattering processes between QDs and WLs are microscopically considered while the carrier transfer from bulk to WLs is phenomenologically taken into account. Note that at this step, we assume that the electron-phonon and hole-phonon scattering rates are equal. The inset of Fig. 1 depicts a simplified energy scheme with an electrical pumping. QDs have discrete electronic states due to the three-dimensional confinement while the WL has quasi-continuous states. When the external current J(t) is applied to the system, the number of carriers in the WL increases. Then, the carriers in the WL can be captured into the QD states in (Sin e for electrons and Sh for holes) and scattered out from the QD into the WL states (Sout e for electrons and Sout for holes) by Coulomb and electron-phonon h

Fig. 1. A schematic diagram of the self-consistent treatment of the field interacting with the polarizations and the QD- and WL-densities. Inset: in Scheme of the Coulomb scattering dynamics between discrete QD states and quasi-continuous WL states. Sin e and Sh describe in-scattering rates for out out electrons and holes, respectively, while Se and Sh stand for the out-scattering of charge carriers from QD into WL states.

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scattering processes. Finally the carriers can relax radiatively. The scattering rates are microscopically calculated within a second order screened Born–Markov approximation with the Hamiltonian H ¼ H 0 þ H cc þ H cph þ H eL :

(1)

The free carriers are described by X y X y e i ai ai þ hvph  H0 ¼ q bq bq ;

(2)

describes the coupling between carriers in the QD ground state and the classical field E(t). Here, di,j is the dipole matrix element between i- and j-state. By applying the Heisenberg’s equations of motion and by summing over all QDs in the normalization area, we obtain the equation of motions of the charge carrier density and the polarization for QD [13]. The equation of motion of the WL carrier density is obtained by assuming the number of carriers to be conserved within the QD–WL system [14]. The microscopic calculations for the Coulomb and electron–phonon scattering rates are given in detail in Refs. [15] and [16], respectively. Fig. 2 shows the combined Coulomb and electronphonon scattering rates at room temperature with binding energies of ee = 120 meV and eh = 40 meV defined with respect to the edge of the WL conduction and valence band for electron and hole, respectively [7]. At room temperature, the scattering is dominated by electron–phonon processes. When the WL carrier density is low, the in-scattering rates is increasing, while the out-scattering rate is decreasing. Then, both of in- and out-scattering rates saturate at high WL carrier density due to Paul-blocking. Note that out-scattering process when there is no WL carrier density is provided by electron–phonon interaction [16]. For the numerical evaluations for switch-on dynamics, we implement the scattering rates into the self-consistently combined QD-WL Maxwell–Bloch equations as a function of the WL carrier density. A detailed derivation of the QD-WL Maxwell–Bloch equations is found in Ref. [15]. They consist of the QD/ QD=WL WL population density nb for electrons (b = e) and holes (b = h), the real- and imaginary part of the polarization P1 and P2, and the electric and the magnetic

q

i

where ayi and byq (ai and bq) denote creation (annihilation) operators for electrons and phonons within second quantization. ei is the energy of the electron and hole states i of QDs, and hvph q is the energy of a phonon with wave vector q. The Coulomb interaction energy is given as H cc ¼

1 X ab y y V a a ad ac : 2 abcd cd a b

(3)

ab is defined as Here, the Coulomb matrix element Vcd 0 0 0 ¼ hFa ðrÞFb ðr ÞjVðr  r ÞjFc ðr ÞFd ðrÞi with compound indices a, b, c, and d denoting QD or WL charge carrier states [10,11]. The carrier–phonon interaction energy is given by XX q y H cph ¼ gi; j ai a j ðbq þ byq Þ (4) ab Vcd

i; j

339

q

with the Fro¨hlich coupling element of gqi; j , which describes the coupling of electrons to optical phonons [12]. The electron–light interaction energy X H eL ¼  d i; j  EðtÞayi a j (5) i 6¼ j

Combined scattering rate (ps-1)

8

Sein

Seout in

Sh

6

out Sh

γ2

4

2

0 0

0.5

1

1.5

2 12

2.5

-2

WL carrier density (10 cm ) in out out Fig. 2. Scattering rates as a function of WL carrier density. Sin e and Sh are in-scattering rates for electrons and holes and Se and Sh are outscattering rates for electrons and holes, respectively. g2 is the rate, which contributes to the dephasing.

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field Ex and Hy: n˙ QD b n˙ WL b P˙ 1 P˙ 2 E˙ x ˙y H

d eh in QD Ex P2  g 1;b nQD b þ Sb N h   A˜ JðtÞ in QD ¼ 1  nWL þ g 1;b nQD b  Sb N C b e0 ¼ v0 P2  g 2 P1 deh QD QD Ex ðnQD ¼ v 0 P 1  g 2 P2 þ 2 Þ e þ nh  N h  1 @H y deh v0 g deh þ P 2 þ 2 P1 ¼ e @z Le Le 1 @Ex : ¼ m @z (6) ¼

NQD is the QD density, deh is the dipole moment, and L is the QD layer thickness. v0 is the transition frequency between the QD ground states for electrons and holes. g1,b is the population relaxation rate defined out as g 1;b ¼ Sin b þ Sb and g2 is the polarization decay rate which includes both of the population-decay and the population-conserving (i.e. pure dephasing) scattering WL ˜ processes. The term ð1  ðA=CÞn b Þ accounts for blocking of the injection due to the Pauli exclusion principle [17], so that carriers can not be injected anymore into the WL, when all states C are filled. C refers to the total number of active states in the effective area for a QD A˜ ¼ A=N with N as the number of QDs in the area A: Z kmax X A˜ A˜ 2 k : C¼ 1¼ 2pkdk ¼ (7) 2 4p max ð2pÞ 0 k

Refractive index, |E|2

3.5

Assuming that electrons are occupied up to the energy of 300 meV, the maximum wavenumber of the optical active sates kmax obtains 0.56 nm1. For the numeric evaluation of the equations, the FDTD method is used for the whole QD-VCSEL structure. Furthermore, Gaussian random noise X obtained by Box–Mu¨ller transform [18] is implemented to the electric field at each time step within the QD regions to simulate spontaneous emission, where X ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2jE ln acosð2pbÞ with variance jE. a and b are the uniformly distributed random numbers which have zero mean value [19,20]. Now, microscopic descriptions of the electromagnetic field propagation and the carrier dynamics in QDs and the QD-embedding WL are available in spatial and in temporal domain. 3. Model structure In order to calculate the frequency and the damping rate in dependence on the QD-VCSEL structure designed for 1.3 mm, we have varied the number of QD and DBR layers. As shown in Fig. 3, 2l GaAs cavity is composed of multi stacks of InAs/GaAs QD layers placed at the antinodes of the standing wave for a strong light-matter coupling. First, the numbers of the QD layers are changed to 8, 10, and 12 by keeping the number of GaAs/AlAs DBRs as 20/31 pairs on the left/right side. For the positions of the varied QD layers, see the shaded area in Fig. 3. Each QD layer is 10 nm thick and separated by 30 nm within the stack. The QD density NQD is 1  1011 cm2, and

(a)

8 QD layers, W L

(b)

10 QD layers, W L

(c)

12 QD layers, W L

3

3.5 3

3.5 3 4

4.5

5

5.5

Distance ( μm ) Fig. 3. Refractive index profile and the spatial distribution of the intensity in the center region of InAs/GaAs QD-VCSELs as changing the number of QD layers: (a) 8 QD layers (b) 10 QD layers (c) 12 QD layers. The four stacks of the QD layers are placed within 2l-long GaAs cavity (l = 1.3 mm). The DBRs are composed of alternating l/4 GaAs/AlAs layers, and the numbers are set to 20/31 pairs on the left/right side for 12 QD layered structure.

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Table 1 Applied amplitudes of the injection current density J0 and the estimated carrier transport time ttrans in accordance with Ref. [16].

2

J0 (kA/cm ) ttrans(ns)

Value

Description

0.4 0.774

0.8 0.387

1.28 0.24

1.6 0.1935

the dipole moment d is 0.6e0 nm corresponding to experimental values [21–24]. Second, the number of DBR layers is varied to 9/29, 20/31, and 25/35 pairs on the left/right side of the cavity for a structure, which has 12 QD layers. Both cases are calculated for different amplitudes of the injection current density, which is described as J(t) = J0exp[ ((t  t0)/ttrans)2] for t < t0 and J(t) = J0 for t  t0. Here, ttrans is considered as the carrier transport time from the bulk to the WL, and it has to

1.2

QD

ne nhQD J (t)

8

1 0.8

6

4

2

(b)

3

0.6

2

0.4

neWL

1

nh

0 0

Injection current density Transport time

be varied for different amplitude of the injection current density, since the transport time is proportional to the inverse of the amplitude. As consequence, we can estimate the carrier transport time ttrans from Ref. [16], which shows that at J0 = 0.069 kA/cm2 4.5 ns are necessary to reach the stationary WL carrier density. The time offset is set to t0 = 1.03ttrans to ensure that the current is not suddenly applied to the system. The values are summarized in Table 1.

(a)

WL Carrier density 12 -2 (10 cm )

QD Carrier density (1010cm-2)

10

3.2 0.09675

0.5

1

WL

1.5

2

2.5

0.2

3

Time (ns) 0

Injection current density (kA/cm-2)

Symbol (Unit)

0 0

0.5

1

1.5

2

2.5

3

Time (ns) Fig. 4. Time evolution of the charge carrier densities for the 8 QD layers and 20/31 pairs of DBRs. (a) QDs and (b) WL carrier densities as response to the current injection with J0 = 5  108 nm2.

(a)

7

(b)

6

1 / fRO

8

5 6

4 3

∝ exp(-γ t)

4

2 2

τdelay

1

0 0

Light output (a.u.)

Output Intensity (kW/cm2)

10

1

Time (ns)

2

1.32

1.322

0 1.324

Wavelength (μm)

Fig. 5. (a) The output intensity corresponding to Fig. 4 showing a pronounced relaxation oscillation. tdelay is the switch-on delay time and fRO and gRO are the frequency and the damping rate of the relaxation oscillation, respectively.

342

12 8 QD layers 10 QD layers 12 QD layers

10

fRO (GHz)

τdelay (ns)

2

60 8 QD layers 10 QD layers 12 QD layers

b

1.5

1

0.5

0

8 6

0.5

1

1.5

2

2.5

3

30 20

2

10

3.5

0 0

0.5

1

J0 (kA/cm2)

1.5

2

2.5

3

3.5

0

0.5

1

J0 (kA/cm2)

2.5

1.5

2

2.5

3

3.5

3

3.5

J0 (kA/cm2)

15 19/29 20/31 25/35

fRO (GHz)

2

19/29 20/31 25/35

e

1.5

1

10

19/29 20/31 25/35

f

80

γRO (GHz)

d

τdelay (ns)

40

4

0 0

8 QD layers 10 QD layers 12 QD layers

c 50

γRO (GHz)

a

60

40

5 20

0.5

0

0 0

0.5

1

1.5

2

J0 (kA/cm2)

2.5

3

3.5

0 0

0.5

1

1.5

2

J0 (kA/cm2)

2.5

3

3.5

0

0.5

1

1.5

2

2.5

J0 (kA/cm2)

Fig. 6. Switch-on delay time and frequency and damping rate of relaxation oscillation as a function of the injection current density for the various number of the QD-layers for the number of DBRlayers 20/31 ((a),(b), and (c)) and the DBRs for 12 QD-layers((d), (e), and (f)).

J.E. Kim et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 337–344

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J.E. Kim et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 337–344

4. Results and discussion Evaluating Eq. (6), we are able to describe the charge carrier dynamics and the field propagation in the QDVCSELs. First, we have calculated the switch-on dynamics for a structure which has 8 QD layers and 20/31 pairs of DBRs on the left/right side of the cavity. Fig. 4 shows the time evolution of the QD and WL carrier densities as response to the injection current density. The amplitude of the current density is chosen as J0 = 0.8 kA/cm2, so that the corresponding transport time is set to ttrans = 0.387 ns, see Table 1. When the injection is turned on, the WL carrier density starts to increase. Due to enhanced probability of the in-scattering process from the WL into the QD states, also the QD carrier density is gradually increasing. About 1 ns later, both of the QD and WL carrier densities saturate due to the Pauli blocking at high carrier densities, and in- and out-scattering rates start to equilibrate with the radiative processes in the confined dot states. Therefore, the carrier densities reach their stationary values after a few cycles of oscillations. Fig. 5 (a) shows the final output intensity through the structure. Comparing to Fig. 4, we can see that the exponentially increased field in time induces a reduction in the population inversion (i.e. QD QD nQD ) due to the increased stimulated e þ nh  N emission, so that the field decreases. However, the population inversion recovers from the carrier injection. This mechanism is repeated for a while showing the oscillations, and finally the output intensity reaches its steady state. The switch-on delay time tdelay is about 1.2 ns and the frequency f RO and the damping rate gRO of the relaxation oscillation are about 3 GHz and 8 GHz, respectively. When we apply higher injection currents, the switch-on delay time reduces while the frequency and the damping rate increase [25,26]. By Fourier transforming the time dependent field in the steady state, we obtain the emission spectrum shown in Fig. 5(b). It shows the lasing wavelength of 1.32 mm, which is slightly longer than the designed wavelength of 1.3 mm.This can be ascribed to the effective cavity length being longer than the defined length due to field penetration into the DBRs. In Fig. 6, we show the dependence of those characteristic values on the number of QD- and DBR-layers, respectively. When the number of QDlayers is increased (Fig. 6 (a)–(c)), the switch-on delay time decreases due to the higher gain. As a

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consequence, the lasing occurs already at small WL carrier densities leading to higher population relaxation rates (c.f. Fig. 2). Since the relaxation is caused by the interaction between carriers and the laser field, both the frequency and the damping rate of the relaxation oscillation increase due to increased relaxation rate. This is in agreement with experimental data and the analytical solutions in Refs. [26–28]. Increasing the number of DBR-layers (Fig. 6 (d)– (f)), the switch-on delay time decreases, while the frequency and the damping rates of the relaxation oscillations increase as shown in Fig. 6. The reason is that if the laser has higher reflectivity due to higher number of DBR-layers, the field can stay longer in the cavity to interact with excited electron. As a result, the amplitude of the field increases inducing a higher probability for stimulated emission. Therefore, the lasing occurs at smaller WL carrier densities. Due to the same effect, enhanced scattering rates increase the frequency and the damping rate of the observed relaxation oscillations. 5. Conclusion We have proposed a model, which gives a microscopic description of the carrier dynamics and of the propagation of the electric field during the laser switch-on dynamics. The model is based on QD-WL Maxwell–Bloch equations including Coulomb and phonon-assisted scattering rates. Using the FDTD method, the switch-on delay time and the frequency and damping rate of the relaxation oscillations are calculated in dependence on the amplitude of the injection current density, the number of QD-layers, and the number of DBR-layers within the full structure of QD-VCSELs. The switch-on delay time decreases proportionally to the amplitude, when the amplitude of the injection current is increased. In contrast, the frequency and the damping rate of the relaxation oscillations are proportional to the current density. The increase of the QD- or DBR-layers, leads to short delay times and a higher frequency and damping rate. Our results are in agreement with experimental data and analytical solutions [26,27]. The proposed model can be used to optimize the internal time scales important for high-speed optical devices. Acknowledgments The authors gratefully acknowledge Professor A. Knorr for his fruitful discussions. This work was supported by the research center Sfb (Sonder-

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