phys. stat. sol. (c) 3, No. 11, 3707–3712 (2006) / DOI 10.1002/pssc.200671563

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Dephasing of orbital and spin degrees of freedom in semicon-ductor quantum dots due to phonons and magnons

W. Jacak1, J. Krasnyj2, J. Jacak1, R. Gonczarek1, and L. Jacak1

1 Institute of Physics, Wrocław University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław,Poland

2 Institute of Mathematics, University of Opole, Oleska 48, 45-051 Opole, Poland

Received 1 May 2006, revised 14 June 2006, accepted 18 July 2006Published online 24 November 2006

PACS 03.65.Yz, 63.20.Kr, 73.21.La, 73.22.–f, 75.50.Pp

Phonon-induced decoherence of orbital degrees of freedom in quantum dots (QDs) (GaAs/InAs) is studiedand the relevant time-scales are estimated versus dot dimension. Dephasing of excitons due to acousticphonons and optical phonons, including enhancement of the effective Frohlich constant caused by local-ization, is assessed for the state-of-art QDs. Temporal inefficiency of Pauli blocking in QDs due to latticeinertia is additionally predicted. For QD placed in a diluted magnetic semiconductor medium a magnoninduced dephasing of spin is estimated in accordance with experimental results for Zn(Mn)Se/CdSe.

1 Introduction The eigenstates of the QD exciton interacting with bulk phonons correspond to a com-posite quasiparticle of polaron type: the exciton accompanied by a coherent phonon cloud [1, 2]. Thephonon cloud corresponds to the energy-minimizing state and is characterized by a red-shift with respectto the original bare exciton energy. The excess energy is being transferred from the QD region to the restof the crystal – to the phonon subsystem. In a polar material (e.g. weakly polar GaAs) with dominatingcoupling of electrons with longitudinal optical (LO) phonons, the polarization interaction causes almostall composite quasiparticle energy red-shift, of order of a few meV [2, 3]. The deformation interactionof exciton with longitudinal acoustic (LA) phonons is a few orders weaker (in the GaAs case). Despiteof the small energy shift the dephasing due to LA phonons is however pronounced. The gapless, almostlinear (close to the point) and wide dispersion of LA phonons allows for effective (many LA phononscontribute) channel of deformation energy transfer from the step-by-step dressing exciton to LA phononsea. The characteristic time of dressing can be approximated as the ratio of a dot dimension and a phonongroup velocity. It corresponds to a time needed for transfer of an excess energy from the QED region to asurrounding crystal.

The dispersion of LO phonons is narrow and near the point of almost parabolic form (only the regionof small k is important due to the bottle-neck effect for QDs). It results in a longer time of dressing via theLO channel (hundreds of ps) in comparison to the LA channel (the group velocity of LO phonons scales asinverse of the dot dimension and thus the corresponding dressing time is proportional to the square of dotdiameter). The dressing with LO phonons is aided by anharmonicity (in particular decay of LO phononinto another LO phonon and transversal acoustic phonon (TA), the most important anharmonic channel inGaAs [4]), which allows for polarization energy transfer to the TA phonon sea – the characteristic time ofanharmonical accelerated LO(LO–TA) dephasing can be thus estimated as the decay time of LO phonons(several ps).

Corresponding author: e-mail: jacak@if.pwr.wroc.pl

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3708 W. Jacak et al.: Dephasing of orbital and spin degrees of freedom in semiconductor QDs

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

Picosecond time-scale of dephasing of charges (excitons) in QDs s inconveniently located between the

possible sub-picosecond operation time and the nanosecond exciton recombination time, which proba-

bly precludes feasibility of error correction implementation in QD-based only-by-light-controlled-gate for

quantum information processing (QIP).

The relatively slow dressing of electrons with phonon clouds (of picosecond time-scale) can be a source

of the temporal inefficiency of Pauli blocking important for possible spin-charge conversion schemes for

QIP [5]. It is caused by a fact that a quickly, i.e. nonadiabatically excited electron differs as a particle from

an electron stored in a QD and thus already dressed with phonons, which limits fidelity of spin exclusion

Pauli principle, unless the excitation is carried out adiabatically (thus much slower than picosecond scale).

In QDs manufactured in diluted magnetic semiconductors [6,7] (promising for the coherent spin control

due to giant gyromagnetic factor) the role of phonons is played by magnons. The dressing of a localized

spin with magnons also results in an inconvenient time scale of dephasing, of order of 500 ps [6, 7],

similarly as for LO phonons due to quadratic form of magnon dispersion.

2 Phonon induced dephasing of exciton in QD Hamiltonian describing a single exciton interacting with

bulk phonons has the form:

H =∑

n

Ena+n an +

∑q,s

hωs(q)c+q,scq,s +

1√N

∑q,n1,n2,s

Fs(n1, n2,q)a+n1

an2

(cq,s + c+

−q,s

), (1)

where interactions with LO (s = o) and LA (s = a) phonons are described by functions

Fo(n1, n2,q) = −e

q

√2πhΩ

vε

∫Φ∗

n1(Re,Rh)

(eiq·Re − eiq·Rh

)Φn2(Re,Rh)d3Red

3Rh (2)

and

Fa(n1, n2,q) = −√

hq

2MCa

∫Φ∗

n1(Re,Rh)

(σee

iq·Re − σheiq·Rh)Φn2(Re,Rh)d3Red

3Rh, (3)

here c(+)q,s is the bosonic annihilation (creation) operator for LO/LA phonon with quasi-momentum q and

with the frequency ωo = Ωq Ω − βq2 (Ω denotes the gap of LO phonons at the Γ point, β is the

curvature of LO phonon dispersion in vicinity of the Γ point) and ωa = Caq, Ca – the sound velocity for

LA phonons, M – the mass of ions in the elementary cell, σe,h – the deformation constant for electrons

and holes, respectively, v – the volume of the elementary cell, N – the number of cells in the crystal,

ε = (1/ε∞ − 1/ε0)−1 – the effective dielectric constant Re,Rh are coordinates of the electron and hole,

respectively, Φn is the exciton wave function and a(+)n – annihilation (creation) oparator of exciton.

The dressing process can be described by an exciton single-particle correlation function 〈an(t)a+n (0)〉

– the overlap of the excitonic state at time t with this state at the initial moment The Fourier transform

of the correlation function, In,n(ω) =∫ ∞−∞〈an(t)a+

n (0)〉eiωtdt, is usually called the spectral density

[8], and it can be expressed by the imaginary part of the retarded Green function [8]: ImGr(n, n, ω) =− 1

2hI(n, n, ω). This Green function and the correlation function can be found via standard Matsubara

Green function technique which leads to a Dyson equation with an appropriate mass operator [8].

The imaginary part of the mass operator (crucial for spectral density) is given by the equation [9]:

γn(ω) =π

N

∑k,n1

|Fo(n, n1,k)|2

[(1 + Nk,o)δ(hω − En1 − ∆n1 − hΩk) + Nk,oδ(hω − En1 − ∆n1 + hΩk)]

(4)+|Fa(n, n1,k)|2

[(1 + Nk,a)δ(hω − En1 − ∆n1 − hCak) + Nk,aδ(hω − En1 − ∆n1 + hCak)]

.

×

×

phys. stat. sol. (c) 3, No. 11 (2006) 3709

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Fig. 1 A: spectral intensity vs. energy for the averaged dot dimension l = 6 nm (GaAs/InAs) (upper); the temper-

ature evolution of LA phonon side band and satellite-emission LO phonon peak (lower) – only side band grows in

this temperature range; absorption processes are unimportant for LO phonons at low temperatures (left satellite peak

corresponding to LO phonon absorption is of a few orders smaller than that for emission, and of a similar size as the

emission-peak for T > 80 K); the satellite LO peak grows significantly with enhancement of the Fröhlich constant

in QD (b) in comparison to its bulk value (a); B: the typical shape of the modulus of the correlation function for LA

(upper), LO (middle) and LA+LO (lower) phonons (the oscillations correspond to the gap of LO phonons); C: dressing

time vs. averaged QD dimension l for LA channel – linear and for LO channel – quadratic.

The first term in Eq. (4) is related with the polarization energy transfer to LO phonon sea, while the second

one corresponds to the deformation energy transfer form gradually dressing exciton to LA phonon sea (∆is a real part of the mass operator) [9]. The correlation function (imaginary part of the retarded Green

function):.

ImGr(0, 0, ω) = −a−1πδ(x) − a−1γ(x)/x2

1 + (γ(x)/x)2, (5)

(x = hω − E0 − ∆, a – residuum in a pole).

For the model InAs/GaAs self-assembled QD we assume hωe0 = 20 meV, hωh

0 = 3.5 meV, le =√h

m∗eωe

0= lh =

√h

m∗hωh

0= 7.5 nm, i.e. the same lateral dimension for noninteracting e and h; for

vertical confinement le(h)z 2 nm (suitably to appropriately chosen ω

e(h)z ). The approximate modelling

wavefunction which describes the ground state of the QD exciton (including Coulomb interaction) has the

form Φ0(re, rh) = 1(π)3/2

1LeLhLz

e− r2

e⊥2L2

e− r2

h⊥2L2

h− z2

e+z2h

L2z , where, for the parameters used here, Le = 6.6 nm

and Lh = 5.1 nm are found numerically [9], Lz = lz . The difference for e and h is due to the fact that the

Coulomb interaction energy is comparable to the inter-level distance for a heavier hole while the lowest

excited electron states are much higher in energy. For the above ground state exciton the form factors attain

the form: |Fo(0, 0,k)|2 πe2hΩk2

18vε (L2e − L2

h)2e−αk2, |Fa(0, 0,k)|2 hk

2MCa(σe − σh)2e−αk2

, where

α = l2/2, l is the dimension of the QD averaged over all directions. The exponential factor e−αk2reflects

the bottle-neck effect for QDs.

0 40 80 120

1LO

0.995

time [ps]

ab

sva

lue

ofc

orre

latio

nfu

nc

tio

n

102 6 14

1

3

5

LA

ps

0

200

400

600

800

2 6 10 140

LO

800

QD dimension [nm]l

tim

eo

fd

ep

ha

sin

g[p

s]

meV

-30 -20 -10 0 10 20 30

0

0.02

0.04

0.06

0.08

0.1

-30 0 30

0

0.1

LO

LAl = 6 nm

-4 -2 0 2 4

0

0.05

0.1

0.2

0.25

-4 0 4

0

0.25

10 K

5 K

3 K

0 Kl = 6 nm

LA - side band

36.3 36.4

0

0.6

LO - satellite peak

l = 6 nm a ( = 70.3 )

b ( = 10.1)

energy [meV]

spe

ctra

l inte

nsi

ty[1

/s]

~

~

0 2 4 6 8 10 12 14

0.98

0.985

0.99

0.995

1

0 4 8 12

0.98

1

LA+ LOl = 6 nm , T=0

O

10.1 (QD)~

A B CT=5K

0.75

0.8

0.85

0.9

0.95

1

0.75

1

l = 4 nm

50 K

20 K

10 K5 K0 K

0 4 8 12

LA

3710 W. Jacak et al.: Dephasing of orbital and spin degrees of freedom in semiconductor QDs

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

The spectral density and its inverse Fourier transform (the correlation function), calculated numerically,

are plotted in Fig. 1, for various temperatures and dot dimensions. The coincidence of the time behaviour

of correlation function with that observed experimentally for 0.2 ps pulse excitation in a small QD [10] is

attained at l ∼ 4 nm (cf. Fig. 1B – upper part). The LA channel of dressing is the most effective and it gives

for the typical QDs the ps scale of dressing. The LO channel is slower and accompanied by oscillations

(of ∼ 30 fs scale) related to the gap in the LO phonon dispersion. The dephasing caused by LO phonons

is significantly smaller than LA phonon dephasing, oppositely to corresponding energy shifts. Inclusion

of the LO channel does not modify significantly the simultaneous LO and LA dephasing in comparison

to LA channel solely (LO phonon contribution grows with renormalization of the Fröhlich constant). The

dressing time for LA and LO channels vs. dot dimension are plotted (C) – linear and quadratic for LA and

LO channels, respectively. This property corresponds to the qualitative picture of dressing with phonons,

when the dressing time can be estimated as the dot dimension ∼ l divided by group velocity of phonons

– Ca, for LA phonons, and 2βk ∼ 2β/l, for LO phonons. For LA channel it gives a linear dependence

∼ lCa

while for LO channel a quadratic one ∼ l2

2β .

For interaction with LO phonons important is the dimensionless Fröhlich constant αe = e2

ε

√m∗

2h3Ω,

where 1ε = 1

ε∞− 1

ε0. In GaAs-bulk αe 0.07, while for electrons confined in InAs/GaAs QD it has been

reported to be significantly greater [1, 9]. To account for the increase of αe in nanostructures let us remind

that the electron–LO phonon interaction in a polar material is caused only by an inertial part of local

polarization. The noninertial part of this polarization, accompanying the moving electron, is included into

the crystal field and thus already accounted via definition of appropriate electron states. For the confined

electron in a QD, the inertial part of the polarization is however greater in comparison to the free-moving

band electron – the quasiclassical velocity of an electron in QD with the diameter d, vd hm∗d , is greater

than conducting-band electron velocity (especially close to the Γ point), thus a smaller part of the local

polarization can accompany the more quickly moving electron in QD in comparison to band electron.

Within the linear approximation with respect to the small parameter ad , (a – -diameter of the elementary

cell), for the confined electron we obtain [11] the effective dielectric constant ε′,

1ε′

=1 − a/d

ε∞− 1

ε0+

a

d.

This formula leads to the renormalized Fröhlich constant with ε substituted by ε′. For GaAs/InAs QD with

a radius of order of 10 nm (i.e. d ≈ 20 nm), the renormalized Fröhlich constant ≈ 0.15 in agreement with

FIR spectroscopy measurement for such a dot [1] . The enhancement of electron-LO phonon interaction

for QDs manifests itself also via a significant increase of the Huang–Rhys factor [12] for satellite LO

phonon-assistant photoluminescence feature in QDs (GaAs/InAs) [13, 14].

3 Magnon induced decoherence in QD in diluted magnetic semiconductor In magnetically ordered

medium the role of phonons is played by magnons which can dress spin in QD, eg. in QDs placed in diluted

magnetic semiconductor (DMS) of the type III-V (Ga(Mn)As/InAs) or II-VI (Zn(Mn)Se/CdSe) [6, 7].

Similarly to phonon case, the crucial for QD spin dephasing in DMS medium is the spin waves dispersion.

Hamiltonian for the exciton in QD in DMS has the form

H = Hex + Hsd + Hpd, (6)

where the excitonic part Hex describes e-h pair located in QD, Hpd is the Hamiltonian of magnetic system

(holes-admixtures) and Hsd describes interaction of magnetic system with spins of localized e-h pair (QD

exciton). The Hamiltonian of the magnetic subsystem has the form

Hpd = −12

∑n,m

J(Rn − Rm)Sn · Sm + Hp − 2Np∑j=1

∑n

Ah(Rj − Rn)S(h)j · Sn, (7)

phys. stat. sol. (c) 3, No. 11 (2006) 3711

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6 10 14 18

200

400

800

1200

cb

aQ

DEM

Pfo

rma

tion

time

[ps]

QD dimension [nm]d

Zn1-x

MnxSe/CdSe

Hole-spin exchange enery -1.3 eVElectron-spin echange energy 0.26 eV

Spin of dopant (Mn) 5/2Separation factor B 0.10

p cxxi 025.0,25.0

bxx pi 025.0,05.0

axx pi 01.0,25.0

Fig. 2 Dephasing time of spin of exciton in DMS QD [Zn(Mn)Se/CdSe] due to magnons vs dot dimension d, for

various dopant and hole concentrations.

here Rj denotes the position of a hole, S(h)j is the spin operator of a hole, and Hp is free-band holes

Hamiltonian (Np – number of holes), Sn is the spin operator of the admixture placed at the point Rn, J and

Ah are exchange interactions between magnetic dopants and holes. Using Holstein–Primakoff approach

one can represent this Hamiltonian by Bose operators for elementary excitations in two spin subsystems: of

admixtures and holes. As in DMS concentration of magnetic dopants is small (few percent only [6, 7]), an

averaging over their random distribution is essential. After averaging, the spin part of Hpd can be rewritten

as [9] v0(2π)3

∫d3kHpd(k), where

Hpd(k) = εd(k)B+(k)B(k) + εp(k)b+(k)b(k) + γp(k)[b(k)B+(k) + b+(k)B(k)].

After diagonalization it can be expressed in terms of spin waves:

Hpd(k) = ε1(k)α1(k)†α1(k) + ε2(k)α2(k)†α2(k), (8)

where the operators α1(k), α2(k) are magnon (spin-waves) operators. In the zero external magnetic field

and in the small wave vectors regime (ka 1) the dispersion relations have the form [9]

ε1(k) = D0−Dk2, ε2(k) = Dk2, D0 = −Ah(0)(xp +2Sxi), D = −Ah(0)2Sxixpa

2B

xp + 2Sxi. (9)

(xi[p] – admixtures [holes] concentration, tilde over A and J indicates their Fourier picture). Hence, in

the DMS system there are two branches of spin waves: ε2(k) – gapless, ε1(k) – gapped, but both are of

quadratic-type dispersions.

Similarly to the dressing of a nonadiabatically created QD exciton with phonons, we deal with dressing

of the exciton spin with magnons. The resulting dephasing time (time of dressing of QD exciton spin with

magnons) is governed by the dispersion of magnons, and can be estimated as the ratio of the dot dimension

and the magnon group velocity. The time of dressing with magnons is similar to the dressing of exciton

charge with LO phonons, since both branches of magnons have a quadratic dispersion, and thus scales

as d2 (as τ ∼ d/(∂ε/∂k) ∼ d2) – it is illustrated in Fig. 2 for several hole and dopant concentrations

(entering via D parameter – cf. Eq. (9)). For typical dot dimensions it is relatively long – and indeed in

experiment the time-scale of spin dephasing of QD exciton is of order of 500 ps [6, 7].

Thus, we can state that even though the DMS medium of QDs allows for significant acceleration of

single-qubit control, this medium strongly enhances simultaneously decoherence of spin due to dressing

of QD spin degrees of freedom with spin waves.

3712 W. Jacak et al.: Dephasing of orbital and spin degrees of freedom in semiconductor QDs

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

Acknowledgements Supported by the Polish Ministry of Scientific Research and Information Technology under

the grant No. 2 PO3B 085 25 and No. PBZ-Min-008/PO3/03

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