Twofold gain enhancement by elongation of QDs in polarization preserving QD-SOAs [original]

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Greif, L. A. T., Mittelstädt, A., Jagsch, S. T., & Schliwa, A. (2019). Twofold gain enhancement by

elongation of QDs in polarization preserving QD-SOAs. Semiconductor Science and Technology, 34(7),

075003. https://doi.org/10.1088/1361-6641/ab1c06

Ludwig A. Th. Greif, Alexander Mittelstädt, Stefan T. Jagsch, Andrei

Schliwa

Twofold gain enhancement by elongation

of QDs in polarization preservin

QD-SOAs

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Twofold gain enhancement by elongation of QDs in polarization

preserving QD-SOAs

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Twofold Gain Enhancement by Elongation of

QDs in Polarization Preserving QD-SOAs

L A Th Greif,∗A Mittelstädt, S T Jagsch, and A Schliwa

Technische Universität Berlin, Institut für Festkörperphysik, Hardenbergstraße 36, 10623

Berlin, Germany

E-mail: [email protected]berlin.de

Abstract

The impact of quantum dot(QD) elongation

on key parameters of QD-based semiconductor

optical amplifiers(SOAs) is investigated using

a combination of 8-band k·p-theory including

strain and piezoelectricity up to second order

and a rate equation model describing the popu-

lation of QD ground, excited and wetting layer

states. By considering columnar QDs of se-

lected aspect ratios, we show that chip gain

and saturation gain can be enhanced by up to

+3.6dB via an increased elongation of the in-

dividual QDs while retaining polarization pre-

serving amplification and gain recovery times

below 700fs. Our results enable the optimiza-

tion of polarization preserving QD-SOA devices

which combine ultrafast gain recovery with high

gain and low power consumption.

Keywords

self-assembled quantum dots, semiconductor

optical amplifier, k·p-theory, gain enhance-

ment

1 Introduction

Since the first pioneering works predicted most

desirable electronic and dynamic properties for

quantum dots(QDs),1,2 they have been in the

focus of research for use in semiconductor op-

tical amplifiers(SOAs). Ultrafast gain recovery

and pronounced nonlinear effects3substantiate

the high potential of QD-SOAs in the fields

of high frequency signal amplification, all-op-

tical data processing and wavelength conver-

sion using four-wave-mixing(FWM)4–7. De-

vices with modulation frequencies exceeding

200GHz8and a 120nm broadband gain with

saturation output powers above 20dB facil-

itate wavelength division multiplexing9,10and

high efficiency symmetric FWM.11 Further-

more, several experimental and theoretical

works have shown that, using closely stacked

QDs, polarization preserving amplification can

be realized.12–18 A remaining drawback is the

lower chip gain of QD-SOAs when compared to

quantum well-based systems, requiring longer

or multiple gain devisions to achieve compara-

ble amplification.9

elongated [110]-axis

optical [110]-axis

Figure 1: Scheme of the active region of a QD-

SOA with closely stacked QDs, showing the axis

of elongation with respect to the optical axis.

In this paper, we study an approach to in-

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crease the chip gain of a QD-SOA while simul-

taneously preserving the input signal polariza-

tion. The gain enhancement is realized using

c-plane QDs that are elongated perpendicular

to the optical [1¯

10]-axis of the device (figure 1).

Several growth studies have demonstrated that

such elongated QDs can be realized with molec-

ular beam epitaxy(MBE) as well as metalor-

ganic vapor phase epitaxy(MOVPE) via ad-

justing the growth temperature or including an-

nealing steps.19–21 However, the formation of a

uniform elongation of all QDs within one stack

is still challenging.22

Due to the asymmetric confinement in the

(001)-plane the QD ground state transition

dipole moment is rotated into the (1¯

10)-plane,

optimizing the alignment with the electric field

of the optical signal. Thereby, the polarization

of the amplified signal is preserved in the (1¯

10)-

plane. The results are not limited to this unique

configuration of optical axis and QD-axis but

can be generalized as long as the axis of elonga-

tion is orientated perpendicular to the optical

axis of the QD-SOA.

Anisotropic transition dipole moments caused

by anisotropic nanostructures were already

used in InAs/InP quantum dash lasers and

SOAs.23–25 Quantum dashes represent an in-

termediate structure between QDs and quan-

tum wires with at least one dimension sig-

nificantly exceeding the exciton Bohr radius

and thus exhibit only two-dimensional quantum

confinement.26 Due to the bigger volume com-

pared to QDs, the threshold current is usually

slightly higher than in QD based structures27

and although polarization insensitive amplifica-

tion seems achievable28 to our best knowledge

it has not been demonstrated so far.

The paper is organized as follows. After intro-

ducing the device structure, a brief paragraph

discusses the method for calculating the spa-

tially resolved radiation intensities of a stack

of QDs. QD stacks with a vanishing degree of

polarization(DOP) in the (1¯

10)-plane are then

calculated for five different horizontal aspect ra-

tios ARh=d[110]/d[1¯

10](elongations), where di

is the length of the diagonal [i] in the basis

plane. Subsequently, the results are used to de-

termine the impact of the QD elongation on the

stimulated emission rate as well as other key

properties of the QD-SOA, such as chip gain,

saturation gain and gain recovery time(GRT).

2 Structure Details

Typical QD-SOA designs use the tilted ridge

waveguide geometry,5,29,30 where the cross sec-

tion provides refractive index nguiding along

the vertical and gain guiding in lateral direc-

tion (figure 2). Following Jayavel et al.,31 we

use Al0.5Ga0.5As for the cladding (n= 3.1) and

GaAs (n= 3.3) as matrix material in order to

realize the index guiding.

Figure 2: Schematic QD-SOA with pump cur-

rent I, a resulting current injection region of

width W, height of the active region LW, cav-

ity length Land nllayers of quantum dots, each

separated by a barrier of width wbarrier.

Figure 2 shows the active region composed

of seven layers of closely stacked QDs aligned

along the c-axis. The QDs are placed on a two-

monolayer(ML)-thick wetting layer (included

in the QD height). The barrier width wbarrier is

set to zero ML, as experimental findings suggest

that columnar QDs stay in contact with each

other along the growth direction.12,14,15 The in-

dividual QDs are modeled as truncated InAs-

pyramids with a side-wall angle of 30◦and di-

mensions given in table 1.

As shown in [13,32,33], elongation affects the

polarization properties of single and stacked

QDs. Thus, when changing the QD elongation,

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Table 1: Geometric specifications of the indi-

vidual QDs.

ARhQD height d[110] [nm]

1.0 1.13 12.39

1.5 1.13 11.19

2.0 1.41 12.39

2.5 1.70 12.39

3.0 1.13 7.59

adjustments in the geometry of the individual

QDs are necessary to preserve polarization in-

sensitivity of the stacks. Differences in the QD

volume are of no particular significance as only

the spatial radiation pattern enters the subse-

quent simulations of the QD-SOA dynamics as

discussed in section 3.2.

Device and operational parameters are

adapted from [34,35] and listed in table 2.

3 Method of Calculation

In order to connect macroscopic device prop-

erties like electric current and photon density

with the elongation of the zero dimensional

QDs, we use the rate equation model described

in [34–37] accompanied by numerically obtained

QD emission characteristics. In a first step, the

radiation pattern of the QD stacks with differ-

ent elongations are calculated using an 8-band

k·p-model including strain and strain-induced

piezoelectricity.38,39 Subsequently, the radiation

pattern is used to determine the ratio of pho-

tons spontaneously emitted into the waveguide,

known as β-factor

β=# spon. emitted along the waveguide

# all spon. emitted .

(1)

This figure of merit enters the rate equation

model and allows us to qualitatively study the

impact of the QD elongation on the device prop-

erties via its influence on the electron and hole

wavefunctions.

3.1 Beta-Factor and Spontaneous

Emission Rate

In order to obtain the rate of photons emitted

by the QDs within the waveguide structure, we

use Fermi’s golden rule

Γi→f()∝ |· hf|∇|ii |2.(2)

The polarization dependent transition rate

Γi→f()can be derived with the help of the 8-

band k·pelectron and hole wavefunctions |fi

and |ii, respectively. Following [40], Γi→f()

is transformed into a function of the propaga-

tion direction k. Only photons whose propa-

gation direction k(φk, θk)fulfills the following

equations41 are confined within the waveguide

θk≤arccos ˆn2

ˆn1=θlim,(3)

φk≤arctan W

2(L−x)=φlim.(4)

Here, ˆn1,ˆn2are the refractive index of the ma-

trix and cladding material, respectively, Wis

the width of the injection region, Lis the SOA

length and xis the position in the SOA along

the optical axis. We eliminate the spatial de-

pendence of (4) by averaging the limit angle

φlim over the active region

φlim =1

LZL

dx φlim(x).(5)

With the help of the calculated limit angles, the

β-factor can be derived as the ratio of the sum

of all photons emitted within the limit angles

to the total sum of emitted photons.

3.2 Rate Equation Model

The rate equation model is adapted from

[34,35,37] with corrections suggested by Li and

Li in [36]. The model describes the temporal

evolution of the charge carrier densities within

the wetting layer NW, the QD excited state

N2=˜

NQn2and ground state N1=˜

NQn1.

Here, n2and n1are the occupation proba-

bilities of excited state and ground state, re-

spectively. Occupation probabilities and carrier

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