Physica E 12 (2002) 823–826www.elsevier.com/locate/physe

Hyper�ne-mediated spin-"ip transitions in GaAs quantum dots

Sigurdur I. Erlingssona ; ∗, Yuli V. Nazarova, Vladimir I. Fal’kob

aDepartment of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, NetherlandsbSchool of Physics and Chemistry, Lancaster University, Lancaster LA1 4YB, UK

Abstract

Spin-"ip rates in GaAs quantum dots can be quite slow, thus opening up the possibilities to manipulate spin states in thedots. We present here estimations of inelastic spin-"ip rates mediated by hyper�ne interaction with nuclei. Under generalassumptions, the nucleus mediated rate is proportional to the phonon relaxation rate for the corresponding non-spin-"iptransitions. The rate can be accelerated in the vicinity of a singlet-triplet excited states crossing. We compare our resultswith known mechanisms of spin-"ip in GaAs quantum dot. ? 2002 Elsevier Science B.V. All rights reserved.

PACS: 71.70.Ej; 71.70.Jp; 85.35.Be

Keywords: Quantum dots; Hyper�ne interaction; Spin-"ip rates

Introduction

The electron spin states in bulk semiconductor andheterostructures have attracted much attention in re-cent years. Experiments indicate very long spin deco-herence times and small transition rates between statesof di?erent spin [1]. These promising results have mo-tivated proposals for information processing based onelectron spins in quantum dots, which might lead to arealization of a quantum computer [2].

A quantum dot is a region where electrons arecon�ned. The energy spectrum is discrete, due to thesmall size, and can display atomic like properties[3]. Here, we will consider quantum dots in GaAs–AlGaAs heterostructures. There are two main typesof gate controlled dots in these systems, so-calledvertical and lateral dots [4], characterized by di?erent

∗ Corresponding author. Fax: +31-15-278-12-03.E-mail address: serlingsson@mailhost.tn.tudelft.nl

(S.I. Erlingsson).

transverse con�nement. Calculations of transitionrates between di?erent spin states in quantum dotsdue to phonon-assisted spin-"ip process mediated byspin–orbit coupling have given surprisingly low rates[5,6]. An alternative mechanism of spin relaxationin quantum dots is caused by hyper�ne coupling ofnuclear spins to those of the electrons.

Since the parameters of hyper�ne interaction be-tween conduction band electrons and underlyingnuclei in GaAs have been extensively investigated,including the Overhauser e?ect and spin-relaxationin GaAs=AlGaAs heterostructures [7,8] we are ablenow to predict the typical time scale for spin-"ip dueto hyper�ne interactions in particular quantum dotgeometries.

1. Model and assumptions

The ground state of a quantum dot is a many elec-tron singlet |Sg〉, for suHciently low magnetic �elds.

1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(01)00446 -5

824 S.I. Erlingsson et al. / Physica E 12 (2002) 823–826

S’

T’

Sg

Higher energystates.

δST

Γph HF-phΓε

E

Fig. 1. The lowest lying states of the quantum dot. The energyseparation of the ground state singlet and the �rst triplet is denotedby � and the exchange splitting by �ST . The two rates indicatedare the phonon rate �ph and the combined hyper�ne and phononrate �HF–ph.

We assume that the system is in a regime of mag-netic �eld so that the lowest lying states are orderedas shown in Fig. 1. The relevant energy scales used inthe following analysis are given by the energy di?er-ence between the triplet state (we assume small Zee-man splitting) and the ground state �= ET ′ − Eg, andexchange splitting �ST = ES′ − ET ′ between the �rstexcited singlet and the triplet.

The hyper�ne interaction can be described by thecontact interaction Hamiltonian

HHF = A∑i; k

Si · Ik �(ri − Rk); (1)

where Si (Ik) and ri (Rk) denote the spin and positionof the ith electron (kth nuclei). This coupling "ipsthe electron spin and simultaneously lowers=raises thez-component of a nuclear spin, which mixes spin statesand provides a possibility for the relaxation.

But the hyper�ne interaction alone does not guaran-tee that transitions between the above-described statesoccurs, since the nuclear spin "ip cannot relax the ex-cessive initial state energy. Therefore, the spin relax-ation process in a dot also requires taking into accountthe electron coupling to the lattice vibrations. The ex-cess energy from the quantum dot can be emitted inthe form of a phonon. Since the electron–phonon in-teraction, Hph, does not contain any spin operatorsand thus does not couple di?erent spin states, one has

to employ second-order perturbation theory which re-sults in transitions via virtual states. The amplitude ofsuch a transition between the triplet state |T ′〉 and theground state |Sg〉 is

〈T ′|Sg〉=∑t

〈T ′|Hph|t〉〈t|HHF|Sg〉ET ′ − (Et + ˜!q)

+∑s

〈T ′|HHF|s〉〈s|Hph|Sg〉ET ′ − (Es + ˜!n)

; (2)

where ˜!q is the energy of the emitted phonon and˜!n ≈ 0 is the energy changed by raising=lowering asingle nuclear spin.

It is natural to assume that the exchange splitting issmaller than the single-particle level splitting, so thatthe dominating contribution to the amplitude 〈T ′|Sg〉comes from the term describing the virtual state |s〉=|S ′〉, due to a small denominator. All other terms canbe ignored and the approximate amplitude takes theform

〈T ′|Sg〉 ≈ 〈T ′|HHF|S ′〉〈S ′|Hph|Sg〉ET ′ − ES′

: (3)

The justi�cation for this assumption is that we aim atobtaining estimates of the rates, and including higherstates would not a?ect the order of magnitude, even ifthe exchange splitting is substantial.

The transition rate from |T ′〉 to |Sg〉 is given byFermi’s golden rule

�̃HF–ph =2�˜

∑N ′q ;�′

|〈T ′|Sg〉|2�(Ei − Ef ); (4)

where N ′q and �′ are the �nal phonon and nuclear

states, respectively, and Ei and Ef stand for the initialand �nal energies. Inserting Eq. (3) into Eq. (4) andaveraging over initial nuclear states with probabilityP(�), we obtain an approximate equation for the nu-cleus mediated transition rate

�HF–ph =2�˜

∑N ′q

|〈S ′;N ′q|Hph|Sg;Nq〉|2�(˜cq− �)

∑�′ ;�

P(�)|〈T′; �′|HHF|S′; �〉|2(ET ′ − ES′)2 (5)

=�ph(�)∑�′ ;�

P(�)|〈T′; �′|HHF|S′; �〉|2(ET ′ − ES′)2 ;

(6)

S.I. Erlingsson et al. / Physica E 12 (2002) 823–826 825

where �ph is the non-spin-"ip phonon rate as a func-tion of the relaxed energy � = ET ′ − Eg.

We will approximate the many-body orbital wavefunctions by symmetric, |�S〉, and antisymmetric,|�T 〉, Slater determinants corresponding to the singletand triplet states, respectively. The singlet and tripletwave functions can be decomposed into orbital andspin parts: |T ′〉 = |�T 〉|T 〉 and |S ′〉 = |�S〉|S〉.

The hyper�ne contribution to the rate becomes∑�′�

P(�)|〈T′; �|Hn|S′; �′〉|2

=A2

4Gcorr

∑k

(|�1(Rk)|2 − |�2(Rk)|2)2; (7)

where �1;2 are the wave functions of the lowest en-ergy states and Gcorr contains the nuclear correlationfunctions

Gcorr =∑

�;�=x;y;z

!�!∗�

(NG�� +

12i���"〈I"〉

); (8)

where NG�� is the symmetric part of the nuclear corre-lation tensor, ���" is the totally antisymmetric tensorand the !�’s are the coeHcients in the triplet state ex-pansion.

Let us now introduce the length scales ‘ and z0which are the spatial extent of the electron wave func-tion in the lateral direction and the dot thickness, re-spectively. Let Cn denote concentration of nuclei withnon-zero spin. The e?ective number of nuclei con-tained within the quantum dot is Ne? = Cn‘2z0. InGaAs, Ne?�1 and the sum over the nuclei in Eq. (7)can be replaced by Cn

∫d3Rk and we de�ne the di-

mensionless quantity

�int = ‘2z0

∫d3Rk(|�1(Rk)|2 − |�2(Rk)|2)2: (9)

To relate the hyper�ne constant A (which has dimen-sion energy × volume) to a more convenient parame-ter we note that the splitting of spin-up and spin-downstates at maximum nuclear polarization is En = ACnI ,where I = 3

2 is the nuclear spin. Thus, the hyper�nemediated transition rate is

�HF–ph = �ph(�)(En�ST

)2 Gcorr�int

(2I)2Ne?: (10)

Note that the rate is inversely proportional to the num-ber of nuclei Ne? in the quantum dot and depends on

the inverse square of �ST , which are both possible tovary in experiments [9,10]. The nuclear correlationfunctions in Gcorr may also be manipulated by opticalorientation of the nuclear system [7].

2. Rate estimates and comparison

We now consider Eq. (10) for a speci�c quantumdot structure. It is assumed that the lateral con�nementis parabolic and that the total potential can be splitinto a lateral and transverse part. The wave functionsin the lateral direction are the Darwin–Fock solutions&n;l(x; y) with radial quantum number n and angularmomentum l. The single particle states correspondingto (n; l) = (0; 0) and (0;±1) are used to construct theSlater determinant for a two-electron quantum dot. Inthe case of these states, the factor �int in Eq. (9) isthen �int = 0:12 for a vertical dot and �int = 0:045 fora lateral one.

One property of the Darwin–Fock solution is the

relation ‘−2 = ˜(m∗=˜2, where ( =√(2

0 + !2c=4 is

the e?ective con�ning frequency and !c = eB=m∗ thecyclotron frequency. Inserting it into Eq. (10), the ratefor parabolic quantum dots becomes

�HF–ph = �ph(�)(En�ST

)2 Gcorr�int

(2I)2

(m∗˜(˜2Cnz0

):

(11)

Spin relaxation due to spin–orbit-related mecha-nisms in GaAs quantum dots were investigated byKhaetskii and Nazarov in Refs. [5,6]. They found thatthe dominating scattering mechanism is due to the ab-sence of inversion symmetry [5]. For the dots withcon�ning energies around 2–6 meV, the dominatingrate is given by

�2 = �ph(�)724

(m∗,2)(˜()E2z

: (12)

Here, m∗,2 is determined by the transverse con-�nement and band structure parameters and Ez =〈p2

z 〉=(2m∗). For vertical dots of thickness z0 =15 nm,then m∗,2 ≈ 4 × 10−3 meV, but one should be cau-tious when considering a di?erent thicknesses, sincem∗,2 ˙ z−4

0 , so the rates are sensitive to variationsin z0.

The rates given by Eqs. (11) and (12) have the same˜( dependence and they are both proportional to the

826 S.I. Erlingsson et al. / Physica E 12 (2002) 823–826

phonon transition rate �ph. It is thus suHcient to com-pare only the proportionality coeHcients, which wedenote as �HF–ph and �2. For clarity we will considera speci�c system [10], i.e. a vertical quantum dot inzero applied magnetic �eld with ˜(0 = 5:5 meV; �=2:7 meV and �ST =2:84 meV. For this case, �2 =2:8×10−6 and �HF–ph=6:5×10−10, so the hyper�ne medi-ated rate is much smaller than spin–orbit one. At suchhigh relaxation energies, �, the dominating phononscattering mechanism is due to the deformation poten-tial, giving �ph =7:2×107 s−1. The numerical valuesused for the hyper�ne rate in Eq. (10) are the follow-ing: En = 0:13 meV [7], and z0 = 15 nm.

An application of a magnetic �eld to the dot mayresult in two e?ects. At high enough �elds, a di?erentspin–orbit rate, which is proportional to Zeeman split-ting squared, becomes larger than �2. More impor-tantly, the exchange splitting �ST can vanish in somecases and the hyper�ne rate in Eq. (10) will domi-nate. Since the rates considered here are all linear inthe phonon rate, the divergence of �ph at the singlet–triplet transition does not a?ect the ratio of the rates.

In summary, we have calculated the nucleus medi-ated spin-"ip transition rate in GaAs quantum dots.The comparison of our results to those previously ob-tained for spin–orbit scattering mechanism indicatesthat the rates we obtained here are relatively low,due to the discrete spectrum, so that we believe thathyper�ne interaction would not cause problems forspin-coherent manipulation with GaAs quantum dots.Nevertheless, the hyper�ne rate, which was found tobe lower than the spin–orbit rates at small magnetic�eld, may diverge and become dominant at certain

values of magnetic �eld corresponding to the reso-nance between triplet and singlet excited states in thedot.

Acknowledgements

This work is a part of the research program of the“Stichting vor Fundementeel Onderzoek der Materie(FOM)”, EPSRC and INTAS. One of authors (VF)acknowledges support from NATO CLG and thanksR. Haug and I. Aleiner for discussions.

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