Suppression of acoustic-phonon-induced electron transitions in coupled quantum dots

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Physica E 26 (2005) 427–431 www.elsevier.com/locate/physe

Suppression of acoustic-phonon-induced electron transitions in coupled quantum dots A. Bertonia,, M. Rontania, G. Goldonia,b, F. Troiania,b, E. Molinaria,b a

INFM, National Research Center on nano-Structures and bio-Systems at Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy b Dipartimento di Fisica, Universita` di Modena e Reggio Emilia, via campi 213/A, Modena 41100, Italy Available online 25 November 2004

Abstract We calculate the longitudinal-acoustic phonon scattering rate for a vertical double quantum dot (DQD) system and show that a strong modulation of the single-electron excited states lifetime can be induced by an external magnetic field. The results are obtained for typical realistic devices using a Fermi golden rule approach and a three-dimensional description of the electronic quantum states. The DQDs considered are characterized by a weak lateral confinement and the longitudinal-phonon scattering represents the dominant source of decoherence, its tunable suppression can be a valuable tool for an experimental measure of electron-states lifetimes and serve as a signature for coherent delocalization of electrons in the DQD. r 2004 Elsevier B.V. All rights reserved. PACS: 73.21.La; 73.61.Ey; 72.10.Di Keywords: Coupled quantum dots; Acoustic phonons; Decoherence

1. Introduction Semiconductor quantum dots are the key structures in many proposals for solid-state quantum gates [1] and play an important role in future single-electron devices as well. The onset of the interactions with the environment can jeoparCorresponding author. Dipartimento de Fisica, Universita di Modena e Reggio Emilia, Via Campi 213/A, Modena 41100, Italy Fax: +39-05-9367488. E-mail address: [email protected] (A. Bertoni).

dize the functionality of such devices, in particular, the scattering of carriers by longitudinal acoustic (LA) phonons represents the main source of decoherence for GaAs-based dots with a level separation not exceeding 8–10 meV [2–4]. For a single quantum dot (SQD), it is well known that the electron-LA phonons scattering rate is substantially decreased, due to the effects of confinement, when the wavelength of the emitted phonon is very different from the dot dimension (bottleneck effect) [5]. Recent experiments based on electrical pump and probe technique [6] on single

1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.093

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A. Bertoni et al. / Physica E 26 (2005) 427–431

disk-shaped dots were able to reach a good agreement with a Fermi golden rule calculation of the emission rate of acoustic phonons from the first excited electron level. In the case of a double quantum dot (DQD) structure, the behavior of the scattering rate is complicated by the fact that two length scales, characteristic of the structure, come into play, i.e., the dimension of each dot and the dimension of the double dot system as a whole. In this work, we analyze the effect of an external field on the scattering rate of LA-phonons in a DQD, using a Fermi golden rule approach. We show that a strong oscillatory behavior of the scattering rate can be obtained by applying a vertical magnetic field, with a scattering rate that can be reduced up to four orders of magnitude with respect to the SQD case.

2. The double-dot system We consider a vertically DQD structure formed by two identical GaAs/AlGaAs dots with cylindrical shape. The confining potential has been modeled as a double well in the growth direction z, formed by the heterostructure band-offset (243 meV), while in the xy plane a 2D parabolic confinement is considered. The three-dimensional single-particle electronic quantum states are computed within the familiar envelope function approximation. We consider here only the electron–LA phonon scattering rate since this process is a more effective source of dephasing in the structures under consideration. In particular, the energy available in the transitions considered in the following is an order of magnitude lower than the optical-phonon energy [3,4], thus no efficient coupling is possible via Fro¨lich Hamiltonian. Furthermore, the contribution of the piezoelectric interaction via the transversal-acoustic (TA) phonon scattering mechanism is usually weak in GaAs/AlGaAs systems. It is worth noting, however, that the coupling Hamiltonian of the latter scattering mechanism has the same functional dependence on the electron coordinate as the deformation-potential Hamiltonian that we use for the electron–LA phonon coupling. As a consequence, the results obtained for LA-phonons

can be qualitatively extended to TA-phonon scattering mechanism. The electron–LAPphonon interaction Hamiltonian reads He2p ¼ q F ðqÞðbq eiqr þ byq eiqr Þ; where bq and byq are the annihilation and creation operators for q a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LA-phononffi with wave vector q and F ðqÞ ¼ jqj _D2 =ð2roq Þ: D is the deformation potential, r is the crystal density. A linear dispersion approximation oq ¼ vjqj is used for the LA branch, with v longitudinal sound speed. Using a Fermi golden rule approach to compute the electron–LA phonon scattering rate for the transition from the initial state jci i with energy E i to the final state jcf i with energy E f one obtains [7] t1 ¼

q20 F ðq0 Þ ½N q0 ðTÞ þ 1

vð2p_Þ2 Z p=2 Z 2p dj dy sin yjM ðzÞ  fi ðq0 cos yÞ 0 M ðxyÞ fi ðq0

0

cos j sin y; q0 sin j sin yÞj2 ;

ð1Þ

where q0 ¼ ðE i  E f Þ=ð_vÞ and N q ðTÞ is the Bose–Einstein distribution function at temperature T. M ðzÞ and M ðxyÞ represent the growth-direction fi fi and in-plane components of the structure factor M fi ðqÞ ¼ hcf jeiqr jci i; respectively. We first show, in Fig. 1, the results obtained without external applied fields and compare the scattering rates for three different DQDs and for a SQD at zero temperature. The transition considered is, as throughout this work, the one from the first excited to the ground level. For the system of Fig. 1 (see the figure caption for details), it implies a transition 0 ! 1 of the angular quantum number of the Fock–Darwin state in the xy plane [8]. t1 is computed as a function of the parabolic xy confining energy. The curves have a similar limiting behavior: when the confining potential and, as a consequence, the energy of the emitted phonon is very low, t1 decreases due to the q2 prefactor ensuing from the low-energy LA-phonon density of states in Eq. (1) and the decreasing value of M ðxyÞ fi ðqÞ due to orthogonality of the states. In the opposite limit (i.e., when the confinement energy is much larger than a characteristic energy related to the transition involved and the structure considered), the phonon oscillation length becomes much smaller than the typical length-scale of the electron

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dots, occur repeatedly as the confinement energy is varied; the two cases correspond to a maximum and a minimum of the matrix element M ðzÞ fi ; respectively.

(1/ns)

1.2 1 0.6

τ −1(1/ns)

τ

1

0.2

0.8

0

1_

2

hω0(meV)

0.6

3. Vertical magnetic field

0.4 0.2 0

429

1

_

hω0(meV)

2

3

Fig. 1. LA-phonon scattering rate for two DQDs and a SQD for the lower single-electron transition (see text) as a function of the in-plane parabolic confinement energy _o0 : The single dot (grey line) and the dots of each DQD system have a vertical confinement given by a 10 nm GaAs=Al :3 Ga:7 As quantum well. The two DQDs have a coupling barrier of 8 nm (dashed line) and 3 nm (solid line) respectively. A limit case of quasiindependent dots (40 nm barrier) is shown in the inset together with the SQD sample.

wave function, as a consequence M fi ðqÞ vanishes since the initial and final electron states do not posses the proper Fourier component able to trigger the phonon oscillation. In the intermediate regime, while the SQD curve shows a single welldefined maximum, the DQD curve oscillates. The oscillation pattern depends on the geometry of the system [9]. If the distance between the dots is increased, the energy distance between the maxima decreases and the number of oscillations increases, with peak intensities such that the DQD curve is enveloped by the SQD curve. The SQD limit is correctly recovered for nearly independent DQDs (Fig. 1, inset). Obviously, the pattern of rapid oscillations are the result of the perfectly coherent description of the states, in realistic situations, oscillations would be smeared out by coupling with external degrees of freedom during the state preparation. The confining energy-dependent oscillations of t1 ; which cover several orders of magnitude, have their origin in the phase relation between the phonon wave vector corresponding to the electronic transition considered and the electron wavefunction delocalized between the two QDs: two limiting conditions, one with equal phonon phase, the other with opposite phonon phase over the two

The modulation of phonon-induced scattering can also be obtained by introducing a magnetic field parallel to the growth direction. In fact, such an external field produces a non-uniform shift in the energies E i and E f (see Eq. (1)) and, consequently, a shift in the emitted phonon wave vector q0 : Furthermore, it increases the wavefunctions localization and modifies the in-plane scattering 1 matrix M ðxyÞ is thus similar fi : The influence on t to a modulation in the confining energy _o0 ; and the oscillations of the scattering rate with the field, shown in Fig. 2, have the same origin as the oscillations of Fig. 1. Our simulations show that for a typical DQD with a 2 meV confining energy, the application of a 1 T magnetic field reduces the LA-phonon scattering rate by four orders of magnitude. To understand better the single contributions to the scattering rate coming from the different components of the scattering matrix, we report, in Fig. 3, the plots of M ðxyÞ and M ðzÞ fi fi as a function of the applied magnetic field and of the emission angle; here, W ¼ 0 corresponds to the z direction. In both cases the emission probability vanishes for high values of the corresponding component of the phonon wave vector (q0 sin W and q0 cos W; respectively). This happens for large positive B and large W in the case of M ðxyÞ fi ; for large positive B and small W in the case of M ðzÞ fi : Fig. 3 also shows that the oscillatory behavior of t1 comes from the oscillations of the z component of the scattering matrix. In fact, the two components are multiplied [Eq. (1)] and integrated over the emission angle, the only contributions that survive correspond to low W (emission around the z-direction). The cycle between in-phase and out-of-phase conditions of the LA-phonon with respect to the electron state in the DQD, obtained by changing the magnetic field strength, can also be modulated by varying the distance between the dots. In the

ARTICLE IN PRESS A. Bertoni et al. / Physica E 26 (2005) 427–431

430

_ 1 10 10

_

hω0= 1 meV

hω0= 2 meV L= 4 nm

L= 3 nm

-2

-4

τ −1(1/ns)

_

10 10

_

hω0= 2 meV

1

hω0= 2 meV L= 5 nm

L= 3 nm

-2

-4

_

10 10

_

hω0= 3 meV

1

hω0= 2 meV

L= 3 nm

L= 6 nm

-2

-4

-3

-2

-1

0

1

2

B (T)

-2

-1

0

1

2

3

B (T)

Fig. 2. Scattering rate (for the lower single-electron transition) in six DQD samples with different parabolic lateral confinement and inter-dot barrier (indicated in the figures) as a function of the applied magnetic field. As the external vertical field B is modulated, the scattering rate shows strong oscillations (see text). The dotted lines are a guide to the eyes to follow the spectrum shift and modifications.

Fig. 3. In-plane (left) and orthogonal (right) components of the scattering matrix for the same sample and transition as in Fig. 2 with _o0 ¼ 1 meV: The darker regions correspond to higher values (in a logarithmic scale). W ¼ 0 corresponds to the z (double well) direction. M ðzÞ fi shows an oscillatory behavior that leads to the strong modulations of Fig. 2 through Eq. (1). The energy DE ¼ E i  E f of the emitted phonon for a given magnetic field is reported in the right axis for reference.

right part of Fig. 2, we report the scattering rate as a function of the external magnetic field for three DQD samples with different inter-dot barrier length. The increase of the barrier produces two effects. First, the two lobes of the electron wavefunction separates and the length that must

be matched by the phonon is increased; as a consequence the t1 ðBÞ curve is shifted to the left, i.e., toward a lower energy (higher wavelength) of the emitted phonon. Second, there is a change in the Fourier components that are modified during the transition and the shape of the curve also

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changes significantly, with the appearance of new local maxima.

4. Conclusions We have shown that the introduction of an external magnetic field is able to strongly reduce the decoherence process due to LA-phonons emission up to four order of magnitude for vertical DQD with weak lateral confinement. Due to the strong dependence of the scattering rate by the geometrical parameters we believe that an a priori engineering of a DQD system in order to reduce the LA-phonon scattering rate should be hardly feasible. On the other hand the a posteriori tuning of an external magnetic field should allow to control the decoherence rate of DQDs. We stress here that in our simulations, the electron wavefunction is coherently delocalized in both dots, while the experimental preparation of such a coherent state is hard to achieve and feasible only for strongly coupled dots, as the samples addressed in this work. As a consequence, the field-dependent oscillation of the transition time should also serve as a signature for the coherent delocalization of electron states in DQDs.

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Acknowledgements Work partially supported by projects MIURFIRB RBAU01ZEML, MIUR-COFIN 2003020984, INFM Calcolo Parallelo 2004, and MAE, Dir.Gen. Promozione Cooperazione Culturale. References [1] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120; E. Boiatti, R. Iotti, P. Zanardi, F. Rossi, Phys. Rev. Lett. 85 (2000) 5647; F. Troiani, U. Hohenester, E. Molinari, Phys. Rev. Lett. 90 (2003) 206802. [2] In smaller dots, the level spacing can be of the order of the dispersionless optical-phonon energy: in this case a full diagonalization of the Fro¨lich Hamiltonian is in order to well describe the polaron formation, as shown in Refs. [5,6]. [3] S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaıˆ tre, J.M. Ge´rard, Phys. Rev. Lett. 83 (1999) 4152. [4] L. Jacak, P. Machnikowski, J. Krasnyj, P. Zoller, Eur. Phys. J. D 22 (2003) 319. [5] T. Inoshita, H. Sakaki, Physica B 227 (1996) 373. [6] T. Fujisawa, D. Austing, Y. Tokura, Y. Hirayama, S. Tarucha, Nature 419 (2002) 278. [7] U. Bockelmann, Phys. Rev. B 50 (1994) 17271. [8] L. Jacak, P. Hawrylak, A. Wo´js, Quantum Dots, Springer, Berlin, 1998. [9] P. Zanardi, F. Rossi, Phys. Rev. Lett. 81 (1998) 4752.