Ultrafast carrier capture in charged InAs quantum dots

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Ultrafast carrier capture in charged InAs quantum dots

K.W. Sun

*

_{, A. Kechiantz}

1

Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Da Hseuh Road, Hsinchu, Taiwan

Available online 16 May 2006

Abstract

We report theoretical and experimental results of our investigation on carrier capture and relaxation processes in undoped and mod-ulation-doped InAs/GaAs self-assembled quantum dots (QDs). We find that carrier capture and relaxation in the ground state is faster in the modulation-doped quantum dots compared to the case in neutral dots at an excitation level as low as one electron–hole pair per dot. The ultrafast photoluminescence (PL) transient rise time observed in the charged dots is attributed to the relaxing of strained field induced by the presence of cold carriers in the dots. The Hamiltonian of electron’s interaction with local vibrating field and carrier cap-ture time are also calculated.

PACS: 61.46.+w; 78.47.+p; 78.55.m

Keywords: Laser–matter interactions; Optical spectroscopy; Quantum wells, wires and dots; Luminescence; Ultrafast processes and measurements; Upconversion; Time resolved measurements

1. Introduction

Quantum dots (QDs) are the subject of a rapidly devel-oping area in semiconductor research. Many groups have reported the fabrication of InAs on GaAs by methods of self-organized growth. One of the prominent fabrication methods for QDs is the Stranski–Krastanov (S–K) process which uses the lattice mismatch between the substrate and the over layers [1–3]. Formation of ordered pyramidal-shaped QDs was observed on a residual two-dimensional layer above a nominal coverage of two monolayer (ML) for InAs on GaAs (0 0 1) substrates by Ruvimov and co-workers [4]. Segregation of InAs wetting layers (WLs) was investigated by Oﬀermans et al. using cross-section scanning tunneling microscopy [5]. It was concluded that the formation of WLs is a delicate interplay between

sur-face migration, strain-driven segregation, and the dissolu-tion of QDs during growth. Calculadissolu-tions also indicated that the WL considerably aﬀects both the single-particle energy and wave function [6]. Recently, a new self-orga-nized growth method using droplet epitaxy was reported for direct formation of QD systems without WLs [7,8].

The study of carrier relaxation and capture in InAs/ GaAs QDs[9–12]has attracted much attention due to their unique optical and electrical properties for potential device applications. It has been shown that the removal of WL does not aﬀect the carrier capture and relaxation in QDs

[13]. Recently, Yuan et al.[14,15]studied the carrier dynam-ics in InAs/GaAs QDs. Their observation on systematically longer PL risetimes in the higher excited states was inter-preted in the frame work of sequential state ﬁlling, resulting from fast trapping and intra-dot relaxation. However, few experiments have examined carrier dynamics in charged QDs. In a recent work done by Gu¨ndog˘du et al.[16,17], car-rier capture and relaxation to the ground state was found to be much faster in the highly charged dots compared to the neutral dots. Their results for p-doped QDs reveal a three-fold decrease in the room temperature electron capture and

*

Corresponding author. Tel.: +886 3 5712121x56582; fax: +886 3 572 3764.

E-mail address:[email protected](K.W. Sun).

1 _{Address: Scientiﬁc Research Division, State Engineering University of} Armenia, Yerevan, Armenia.

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relaxation time relative to corresponding undoped QDs. The enhancement of carrier capture and relaxation was attributed to the rapid electron–hole scattering involving the built-in carrier population.

In this paper, we investigate the carrier capture and relaxation process in undoped, lightly n-doped and p-doped InAs/GaAs QDs using time-resolved luminescence up-conversion spectroscopy techniques. We report obser-vation of ultrafast carrier capture and relaxation in the charged QDs’ ground states at very low doping concentra-tions and at low excitation levels. We have proposed a model for interpreting our experimental results. A compar-ison between experiments and calculations allows us to conclude that the presence of the few cold carriers has a sig-niﬁcant inﬂuence on the capture and relaxation in the charged QDs.

2. Experiments and results

The InAs QD samples were grown by using a solid source molecular beam epitaxy (MBE) machine [18]. The growth procedure and condition are described as follows. After native oxide desorpted under As ﬂux at 610C, a 2500 A˚ GaAs buﬀer layer was deposited at 570 C to recover the substrate surface. 300 A˚ Al0.3Ga0.7As and

1400 A˚ GaAs were deposited at a growth rate of 1 lm/h at the same temperature. Then, 100 A˚ GaAs was grown at a growth rate of 0.3 lm/h. In the meanwhile, the sub-strate temperature is lowered down (520C) for InAs depo-sition, and the desired As ﬂux for QDs growth was achieved by adjusting the needle valve of the As cracker cell. Afterwards, 10 nm GaAs capped layer was deposited at the same temperature of QDs growth with a growth rate of 1 lm/h. Then the substrate temperature and As ﬂux were raised to the original values for GaAs and Al0.3Ga0.7As growth. Finally, a layer of uncapped QDs

was grown with the same conditions, and then the substrate cooled down under As ﬂux immediately. The n-doped (p-doped) samples contain a Si-delta (Be-delta) doping layer 2 nm below the QD layer with nominal densities of about 2· 1010

cm2. The average sheet density of the QDs is about 2· 1010

cm2. Images from atomic force microscopy (AFM) measurements reveal circular shaped QDs with an average dot size of approximately 20 nm in base width and a height of about 5 nm. Free carriers from the doped layer accumulated in the low energy states within the InAs QDs. In the lightly doped samples, most of the dots only contain a single electron or hole.

The measurements of carrier dynamics were performed by time-resolved photoluminescence with time-resolution of about 200 fs. The self-mode-locked Ti:sapphire laser was operated at 788 nm with spectral width of about 18 meV (full width at half maximum) to excited carriers in the GaAs barrier. According to the focus spot size and the absorption depth at the photoexcitation wavelength, the laser pumping power was adjusted to give injected car-rier densities from 2· 1010

cm2 (low) to 5· 1011

cm2

(high), respectively. The accuracy of the excitation density is approximately a factor of two. The room temperature steady state PL spectra of the undoped QD sample taken at an excitation wavelength of 788 nm and intensity of 1· 103W/cm2are shown inFig. 1. The spectral lines cen-tered at 872 nm and 930 nm are attributed to the bandgap energies of the GaAs and WLs (as shown in Fig. 1(a)), respectively. Three spectrally well-separated PL lines n = 1 (ground state), n = 2 and n = 3 (excited states) at longer wavelengths (as shown inFig. 1(b)) are due to elec-tron–hole recombination between distinct QD states of the conduction and valence bands. Only the n = 1 peak is observed at low excitation intensity and it is assigned as the QD ground electron to ground hole transition. The cen-ter wavelengths of the spectral peak n = 1 to n = 3 are 1235 nm, 1160 nm and 1085 nm, respectively. The energy separations (also shown inFig. 1(b)) between the QD con-ﬁned states are 65 and 74 meV. In the PL studies on mod-ulation-doped QDs, due to the lightly doping, we did not observe signiﬁcant changes in the spectra lineshape or

shift-Fig. 1. Room temperature PL spectra of InAs/GaAs self-assembled QDs, displaying GaAs barrier, wetting layer and excited state radiative recombination in wavelength range from (a) 820 to 950 nm and (b) 950 to 1300 nm. Spectra were excited at 788 nm with a self-mode-locked Ti:sapphire laser.

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ing of ground-state optical transition. Carrier capture and relaxation to the QD’s ground level are examined as a func-tion of excitafunc-tion density and temperature by measuring PL rise times at the energy of QD ground state identiﬁed in the steady state PL spectra. The time evolution of the PL signal then follows from the analysis of the rate equations

IðtÞ / A ½expðt=srÞ expðt=sdÞ=ðsr sdÞ;

where sr and sdare the PL rise and decay time constants,

respectively.

InFig. 2we show PL transients detected at the ground state of the undoped QDs for the first 20 ps at low temper-ature. Time-resolved PL measured at the same energy but at three different excitation levels (low, moderate and high) is displayed in parallel for comparison. The PL rise times of the ground state in the QDs accelerate as the excitation power increases and reach a value of less than 1 ps at a photoexcited carrier density P1· 1011cm2. It is believed that the carrier density dependence of the ultrafast relaxa-tion is due to Auger-like carrier–carrier scattering [9]. Experiments on our modulation-doped QDs allow the relaxation dynamics of electrons and holes to be investi-gated separately. At low temperature, low excitation densi-ties, the photo-generated carriers do not significantly perturb the well-defined Fermi distribution of doped cold carriers. Therefore, the luminescence dynamics is domi-nated by the electron (hole) dynamics in p (n) modula-tion-doped QDs. For the modulamodula-tion-doped samples with doping density of only 2· 1010

cm2, only the lowest elec-tron (n-doped QDs) or hole (p-doped QDs) level is occu-pied prior to the optical excitation. The initial transient

PL at the ground-state optical transition in the modula-tion-doped QDs at low temperature is shown in Fig. 3. The results in Fig. 3 correspond to an optical excitation level of only one electron–hole pair per dot. Therefore, the major diﬀerence between the time-resolved PL experi-ments on the charged and uncharged QDs is the presence of small built-in electron (hole) population in the n-doped (p-doped) QDs prior to optical excitation. However, the ﬁts of the PL transients in Fig. 3 indicate capture times of less then 2 ps for charged QDs which are more rapid than in the undoped QDs, regardless of the species of cold carrier involved. The time-resolved PL rise times also show no discernible temperature dependence for both the charged and uncharged QDs.

In contrast to the earlier report on carrier capture and relaxation in highly charged QDs [16,17], the total carrier densities in our experiments are only on the order of 2 · 1010

cm2. According to the results reported in Ref.

[19], for a plasma density of 1010

cm2 and a quantum dot lateral size of 20 nm, the calculated Auger scattering rate was only on the order of 2 · 1010s1. Therefore, it is unlikely that the Coulomb scattering within the elec-tron–hole plasma is responsible for the ultrafast carrier capture and relaxation observed in our charged QD exper-iments. The observation of no discernible temperature dependence on the PL transient rise time also indicates that phonon scattering was not responsible for the accelerated carrier capture observed in our experiments. In the follow-ing section, a theoretical model is proposed to interpret our experimental results.

3. Theoretical model

We model a circular shaped InAs QD with a base diameter of 20 nm and a height of 2 nm. In response to

Fig. 2. Time-resolved photoluminescence intensity measured at the energy of the undoped QDs’ ground states at three diﬀerent excitation levels and at low temperature. The solid curves indicate ﬁts of the PL rise to a single exponential.

Fig. 3. Results of time-resolved PL experiments at 77 K for p-doped (circles), n-doped (squares) and undoped QDs (triangles).

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the localized (doped) charges in the dots, a polarization ﬁeld ~Pð~RÞ ¼ e0r~RVð~RÞ (where V ð~RÞ ¼ eð1 1=esÞ=4pe0R

and es is the static dielectric constant of semiconductor)

must be induced around the QDs. During the scattering, electron (or hole) must encounter two electric fields of opposite signs. Those fields are positive (negative) field of a bare hole (electron) confined within the dot and screening field of local polarizations around the dot. When the elec-tric field inducing the strain around the dots suddenly dis-appears due to the capture of an electron or a hole into the QDs, local strain of crystal lattice must trigger vibrations of ions and bond electrons around the dot to relax the strain and to release the energy.

When a mobile electron was scattered on the polarization field produced by positively charged dots, by Fourier trans-forming this polarization field, the electron energy in the polarization field Ekccan be expressed by the sum of partial

contributions from local strains with wave number q,

Eev¼ e2_{ð1 1=e} sÞ 2p2_e 0 Z _sinðqRÞ qR dq: ð1Þ

Assuming that the system of ions and bond electrons is lin-ear during interaction, the decay of locally induced vibra-tions is given by the Bessel function J0(xmt), where xm is

the maximum value of phonon frequency[20]. For simplic-ity we assume that polarization induces constant potential, V(R0), within the dot, i.e. the volume of the dot is never

polarized, where R0 is the average radius of quantum

dot. Then for electron interacting with local vibrations trig-gered within the dot at the time t0, the time-dependent

Hamiltonian can be written as

Hevðt t0Þ ¼ e2_{ð1 1=e} sÞ 2p2_e 0 Z _sinðqR 0Þ qR₀ J0½xmðt t0Þ qR0ð1 R=R0Þdq; ð2Þ

where qR0(1 R/R0)/xm is the time that vibration with

wave number q takes to arrive to a point R within the dot. The Hamiltonian Hev(t t0) brings short-time

pertur-bation to the electron scattering processes and it can result in the capture of photoexcited carriers into the QD’s con-ﬁned states. Due to low photoexcitation densities, number of excitons created in the sample was no more than one exciton per dot. Distance between QDs was far enough so that the Coulomb interaction was negligible. Therefore, the Coulomb interaction was not included in the Hamiltonian.

Denote n the density of mobile electrons and X the vol-ume of quantum dot. In the frame of time-dependant per-turbation theory, if the wave function of mobile statesjki is normalized to the ﬂow of one electron per second, the rate W ¼ jRhcjHevðt t0Þjkidtj

2 ffiffiffiffiffiffi

nX p

=h2Dt of the variation-assisted transitions of electrons from mobile into conﬁned electronic statejci can be written as

W ¼e 4_{ð1 1=e} sÞ 2 n1=2 16p2_e2 0h 2 xmR 1=6 0 FðDtÞjKðkÞj2; ð3Þ where FðDtÞ ¼xm Dt Z Dt 0 J0ðxmtÞ exp½xkctdt 2 ; ð4Þ KðkÞ ¼ð12=pÞ 1=4 p R3=20 Z Z _sinðqR 0Þ q exp½i~k~R exp ixkc xm qð1 rÞ B DtqR0ð1 rÞ xm w_cdq dR3; ð5Þ and B½v ¼ Z v 0 J0ðxmzÞ expðixkczÞdz Z Dt 0 J0ðxmzÞ exp½ixkczdz: ð6Þ

Here ~kis the electron wave vector in mobile state, w_c is the electron wave function in conﬁned state, hxkcis the energy

separation between conﬁned and mobile electronic states in quantum dot and r = R/R0. Vibrations around the

quan-tum dot decay as cylindrical Bessel function J0(xmt). By

approximating the conﬁned electronic states with the spherical Bessel function, Wc¼

ffiffiffiffiffiffiffiffi 2=3 p p j0ðp rÞ= ffiffiffiffi X p , Eq.

(5)can be further reduced to

KðkÞ ¼4 ffiffiffi 3 4 p p3=4 Z xmDt 0 dqsin qR0 q Z 1 0 dr exp ixkcqR0ð1 rÞ xm sinðrpÞsinðkR0rÞ kR0 : ð7Þ

Therefore, the rate of carrier capture into the QDs s = 1/W is calculated to be s¼16p 2_e2 0hm ðx 2 kc x 2 mÞR 5=2 0 e4_{ð1 1=e} sÞ 2 n1=2_jKðkÞj2 : ð8Þ

Fig. 4. Dependence of oscillation frequency F(Dt) on xmDt(dash curve) for xkc/xm= 4.286. When xmDt> 1, F(Dt) can be reduced to_ðx2xm

kcx2mÞDt

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The rate s is mainly decided by the energy separation of conﬁned states hxkc, the duration of perturbation Dt and

the function F(Dt). The dependence of F(Dt) on xmDt is

plotted in Fig. 4 for InAs/GaAs quantum dots with

hxkc= 150 meV and hxm= 36 meV. For the given

param-eters in our p-doped QD experiments, Eq.(8)gives a cap-ture rate of s = 1.7 ps. This calculated value is in good agreement with our experimental results. In the case of holes been captured in an n-doped QD, the calculations should taking into account the warping of valence bands due to the quantum conﬁnement and strain.

4. Conclusion

In conclusion, we have investigated carrier capture and relaxation in the ground state of undoped and modula-tion-doped InAs/GaAs QDs. We observe faster capture and relaxation processes in the charged QDs in comparing to the undoped dots even in the one electron–hole pair per dot regime. Our results suggest that, under low excitation intensity and low doping level, the relaxing of polarization field induced by the confined charge in the quantum dot is the dominant factor for the acceleration of carrier capture. Our calculations also show that carriers interact with decaying field results in ultrafast capture into confined electronic states. We have performed calculations repro-ducing the experimental conditions used in our experiments and a capture time of 1.7 ps was obtained from our calculations.

Acknowledgement

This work was supported by the National Science Coun-cil of Republic of China under contract No. NSC 93-2112-M-259-009.

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