Damping of coherent oscillations in a quantum dot photodiode

Damping of coherent oscillations in a quantum dot photodiode

ARTICLE IN PRESS Physica E 26 (2005) 337–341 www.elsevier.com/locate/physe Damping of coherent oscillations in a quantum dot photodiode J.M. Villas-...

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Physica E 26 (2005) 337–341 www.elsevier.com/locate/physe

Damping of coherent oscillations in a quantum dot photodiode J.M. Villas-Boˆasa,b,, Sergio E. Ulloaa, Alexander O. Govorova a

Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979, USA b Departamento de Fı´sica, Universidade Federal de Sa˜o Carlos, 13565-905, Sa˜o Carlos, Sa˜o Paulo, Brazil Available online 24 November 2004

Abstract In this letter, we develop a model to describe the Rabi oscillations observed in a quantum-dot photodiode. Using a multi-level density matrix formulation, which includes multi-exciton and single particle states, we show that the damping observed in recent experiments is the result of a non-resonant excitation from or to the continuum of the wetting layer states. r 2004 Elsevier B.V. All rights reserved. PACS: 78.67.Hc; 42.50.Hz; 81.07.Ta Keywords: Tunneling; Quantum dots; Rabi oscillation

Recent advances in fabrication, manipulation and probing techniques have made semiconductor quantum dots (QDs) a promising candidate for applications in the field of quantum computation and information processing. Using these advances, several groups have successfully demonstrated the coherent control of the temporal evolution of the exciton population in a single QD [1–4]. They use a strong resonant electromagnetic pulse and different probing techniques. The coherent phenomena observed in these experiments, known as Rabi Corresponding author. Department of Physics and Astron-

omy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979, USA.

oscillations, is the first step in the quest for quantum computation. Zrenner et al. [4] have developed a single self-assembled QD photodiode in which the population inversion induced by a strong and carefully tuned optical pulse is probed by the photocurrent signal. In their device the pulse generates an electron–hole pair and assisted by an external gate voltage, the electron and hole tunnel out of the dot into nearby contacts. This process generates a photocurrent signal that is used to monitor the coherent state of the system. Their results show Rabi oscillations that are damped with the area of the pulse for a fixed pulse duration (1 ps). However, the mechanisms that produce this decoherence were not fully reviewed.

1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.073

ARTICLE IN PRESS J.M. Villas-Boˆas et al. / Physica E 26 (2005) 337–341


Here we address the decoherence processes that induce this damping. We use a density matrix approach that incorporates dipole coupling, multiexciton and single particle states, and an offresonant excitation to the wetting layer. The electron and hole tunneling processes are introduced via a microscopic model of the structure in a WKB approximation [5]. Our model shows that for short pulses (of the order of picoseconds) the two-level system approximation fails for a p-pulse (the pulse necessary to invert the exciton population) and other states should be included. This results in a frequency shift of the measured photocurrent oscillations, but does not induce additional damping. We find that inclusion of excitations to wetting layer (WL) states is essential to understanding the decoherence observed in experiments. This explanation also gives a natural description for the background photocurrent signal observed [6]. In Fig. 1 we show the system and level configuration taken into account by our model Hamiltonian where the two-way (up and down) arrows indicates dipole coupling between levels and single-way arrows indicate incoherent processes, where down indicates decay by recombination, down left by electron tunneling, and down right by hole tunneling. Using a unitary transformation [7] we can remove the fast timedependent part of the Hamiltonian which then becomes H ¼ dx jxihxj þ db jbihbj þ dx jx ihx j þ


1 2½Ox ðtÞj0ihxj

þ dxþ jx ihx j  þ Ob ðtÞjxihbj  þ þ Ox ðtÞjeihx j þ Oxþ ðtÞjhihx j þ h:c: : ð1Þ Here dx ¼ ex  _o accounts for the exciton detuning with the laser energy _o; db ¼ eb  2_o is the two-photon biexciton detuning, dx ¼ ex  ee  _o and dxþ ¼ exþ  eh  _o are the laser detuning ~ with charged excitons, Ox ðtÞ ¼ h0j~ m EðtÞjxi=_;  ~ ~ Ob ðtÞ ¼ hxj~ m EðtÞjbi=_; Ox ðtÞ ¼ hej~ m EðtÞjx i=_; þ ~ Oxþ ðtÞ ¼ hhj~ m EðtÞjx i=_; where the electric dipole moment ~ m describes the coupling of the ~ excitonic transition to the radiation field, and EðtÞ is the pulse amplitude which we assume to have a Gaussian shape, with tp full-width at half-maximum (FWHM), or ‘‘pulse duration’’.

The dynamics of our system is computed using a density matrix formalism of the form dr i ¼  ½H; r þ LðrÞ; dt _


where the first term in the right yields the unitary evolution of the quantum system and LðrÞ is the dissipative part of the evolution, and it is assumed to be linear in r: We use Gi to describe all population decay processes of the level i; (the decay of the diagonal part of the density matrix), while gij ¼ Gi =2 þ gdep is the dephasing (the decay of the off-diagonal part of the density matrix) which has a contribution from the population decay and may also have additional pure dephasing gdep that comes from other elastic scattering channels. We consider two types of population decay, one due to the spontaneous decay Grec i given by the recombination rate (for the exciton state the spontaneous decay time is known to be trec ¼ 1=Grec x ’ 1 ns for this kind of QD) and other Gtun which result when the particles (electrons and i holes) leave the system by tunneling. We can estimate the tunneling rate Gtun for a s single particle s (electron or hole) using the tunneling Hamiltonian [8,9]. The result probability of 0D to 3D tunneling for this single particle s can be expressed as [5] " pffiffiffiffiffiffiffiffiffi # ffi rffiffiffiffiffiffiffiffiffiffi 4 2m s 16V 2s L m s tun 3=2 Gs ¼ ; (3) jE s j exp  3_eF 2jE s j p2 _ 2 where m s and E s is the effective mass and energy of the particle s (electron or hole) in the dot, L and V s are the width and depth of the corresponding square well, respectively, and F is the electric field provided by the gate voltage. In a first approximation, we consider the electron–hole interaction as a shift (the exciton binding energy) in the single particle energy level E s in the quantum dot and use Eq. (3) to obtain the rates with and without electron–hole interaction. Using parameters from the Ref. [10] and the assumption that V e ’ 3V h ; we obtain that the tunneling time for electron or hole is similar when there is electron–hole interaction tun (1=Gtun x ¼ tx ’ 15 ps), and different when there is tun no interaction (ttun e ’ 6 ps, th ’ 10 ps). Of course

ARTICLE IN PRESS J.M. Villas-Boˆas et al. / Physica E 26 (2005) 337–341


|b〉 +

|x 〉


|x 〉



hω |h〉






Fig. 1. Schematic band structure and level configuration of a single QD photodiode. (a) An electromagnetic pulse creates an excitation (exciton) in the dot, then, with a gate voltage applied, the electron and hole tunnel out generating the measured photocurrent signal. (b) Schematic representation of the processes and levels involved in this system. Two-way arrows are to indicate dipole coupling between level, single-way arrows are the incoherent process.

this depends on which kind of QD we are considering, and what is the real profile of the QD with respect to the valence and conduction band. After one of the particle tunnels, the remaining particle tunnels out faster, as it does not experience the electron–hole interaction. In our model we include all possible level configuration, so the rates we consider are the tunneling rates for individual single particles states given by Eq. (3). As the photocurrent is a signal induced by the particles that tunnel out, when the next pulse arrives the system is already in the vacuum state j0i (meaning that they are different processes that account for a statistical average, as required for a quantum mechanical measurement), we can write the expression for the photocurrent as Z 1 X I PC ¼ fq Gtun r ðtÞ dt ; (4) ii i i


where the summation is over all states with Gtun i being the electron tunneling from the state i (or correspondingly hole tunneling1), f is the repetition frequency of the pulse sequence (we use 1 The photocurrent signal is given by the tunneling out of one of the carriers.

f ¼ 82 MHz as in Zrenner’s experiment [4]), and q is the electronic charge. Our results show that for a strong pulse in resonance with the exciton energy (dx ¼ 0), a biexciton binding energy of D ¼ 3 meV is relatively low detuning to ignore the biexciton contribution in the dynamics, as can be seen in Fig. 2(a). There we show the average occupation (during the pulse) of the states of the R 1 system as a function of the pulse area Y ¼ 1 OðtÞ dt for a pulse length tp ¼ 1 ps. Mostly the exciton and biexciton present significant contributions during the pulse, while all other states have negligible average occupation. One would have expected a sizeable contribution from the charged excitons, but this is not the case due to their relatively large detuning (’ 4:6 meV). We assume that the biexciton and charged excitons have the same dipole moment of the exciton transition ½Ob ðtÞ ¼ Ox ðtÞ ¼ Oxþ ðtÞ ¼ Ox ðtÞ ¼ OðtÞ : The photocurrent can then be computed using Eq. (4). The result is shown in Fig. 2(b) by triangles where we see a decay of the oscillation amplitude, but also an increase with the pulse area, which is not consistent with the experimental observations. The experimental results from Ref. [4] are represented by circles and exhibit a decay of

ARTICLE IN PRESS J.M. Villas-Boˆas et al. / Physica E 26 (2005) 337–341


light pulse. As shown in Ref. [11] it is possible to excite electrons from a bound state in the QD (valence band) to the continuum of the WL (conduction) and from the continuum (valence) to bound (conduction). The WL has a broad continuous distribution of levels, and once the electron or hole is excited there it goes away quickly. We describe this excitation as an incoherent pump excitation with rate GWL connecting the ground states j0i to the WL. This rate can be written as


Average occupation


Exciton Biexciton Others states

0.4 0.3 0.2 0.1 0 14 (b)


Photocurrent (pA)

12 10

2 ~ ~ m EðtÞ j0i dðE n  _oÞ; hnj 2


where the summation is over all n levels that compose the WL. This summation (integration) results in

8 6 4


Simulation Simulation with WL Experiment [4]

2 0

2p X ¼ _ n



2 3 4 Pulse Area Θ (π)



Fig. 2. (a) Average occupation of the states of the system as a function of the pulse area Y for a pulse duration tp ¼ 1 ps. (b) Photocurrent signal derived from Eq. (4) as a function of the pulse area Y: Triangles show results using model Hamiltonian (Fig. 1 and Eq. (1)). Squares are results including an offresonance excitation to wetting layer levels. Circles are the experiment results of Ref. [4].

the oscillation with increasing pulse area, which cannot be obtained by our model so far. Notice that any kind of additional constant decay or dephasing cannot explain the experimental trace either, since increasing the pulse area makes this quantity smaller compared with the pulse intensity, and the Rabi oscillation is enhanced overall. This behavior suggests the presence of other levels that contribute to dephasing. It has been recently demonstrated that the continuum of the wetting layer (WL) levels plays an important role in the background absorption in self-assembled QDs [11]. In connection to this situation, we include the WL continuum in our simulation, which in essence is being populated non-resonantly by the

p r m2 EðtÞ2 ; 2_ WL w


where rWL is the WL density of states, and mw is the effective dipole moment connecting the ground state to the WL. The result for this simulation is represented in Fig. 2(b) by squares, where we use the density of states rWL ¼ Gw =2p=G2w =4 þ d2w ; with dw being the WL detuning with the laser energy, and Gw is the broadening of the WL levels. Notice this incoherent channel description naturally results in an intensity dependent decoherence, as described recently [12]. In conclusion we have presented a model to understand the damping observed in the Rabi oscillations in a single QD photodiode in the presence of pulsed light. Our model includes exciton, biexciton, single particles and WL states, and is based on the density matrix formalism. After a systematic study of the Rabi oscillation damping, including several possible mechanisms in our simulation, we conclude that the damping with increasing pulse area (and fixed pulse duration) is due to an off-resonant excitation to the WL levels. This work was partially supported by FAPESP, the US DOE Grant no. DE-FG02-91ER45334, and the Indiana 21st Century Fund.

ARTICLE IN PRESS J.M. Villas-Boˆas et al. / Physica E 26 (2005) 337–341

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