PHYSICAL REVIEW B, VOLUME 65, 113305

Anharmonicity-induced polaron relaxation in GaAsÕInAs quantum dots

L. Jacak,1 J. Krasnyj,2,* D. Jacak,3 and P. Machnikowski1

1Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wrocław, Poland2Institute of Mathematics, University of Opole, Oleska 48, 45-051 Opole, Poland

3Institute of Mathematics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wrocław, Poland~Received 30 June 2001; revised manuscript received 11 October 2001; published 28 February 2002!

The anharmonicity-induced relaxation of a polaron in a quantum dot is analyzed using the Davydov diago-nalization method, including the coherent renormalization of the relevant third-order phonon interaction. Theresulting relaxation time for a small GaAs/InAs self-assembled quantum dot turns out to be a few times longerthan that found previously by a perturbative method.

DOI: 10.1103/PhysRevB.65.113305 PACS number~s!: 73.21.La, 71.38.2k, 03.67.Lx

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The recent proposal of a fully optically driven quantugate for a scalable quantum computer in a system of qutum dots employing excitonic states1 is one of the mostpromising ideas for quantum information processing, duethe possibility of the application of ultrafast resonant optimethods.2 However the feasibility of this idea depends crcially on the decoherence ratio, i.e., the ratio of the charteristic decoherence time to the time needed for elemenoperations. The error correction schemes require thatratio not be smaller than 105. As optical techniques allow foan even femtosecond scale for one-qubit operations,nanosecond scale of the typical exciton lifetime in quantdots was originally recognized as a promising opportunHowever serious objection appeared when the electphonon interaction in GaAs/InAs quantum dots3 was identi-fied to be much stronger in comparison to busemiconductor,4 and to lead to a dressing of electrons~holes!with longitudinal optical phonons, i.e., to the creation of efective quasiparticles, polarons, even at energies relatifar from the resonant energy.5,6

These polarons turn out not to be everlasting, but dedue to a scattering of the optical phonons via their anhmonic interaction with the continuum of acoustical phonoExperimental investigations7 indicated that for bulk GaAsthe most efficient anharmonic process~of picosecond timescale! is zone-center LO-phonon decay into a zone-edgephonon and a TO phonon with the opposite momentumtheoretical analysis based on a phenomenological appro8

leads to a similar, picosecond lifetime for confined Lphonons: for GaAs/InAs dots with sizes larger than 15 nthe anharmonic decay lifetime of the confined LO phonowas estimated to be 7–2.5 ps for temperatures from 0 toK.8

The LO-TA phonon decay in GaAs gives rise toanharmonicity-induced relaxation of a polaron in seassembled GaAs/InAs quantum dots. The correspondinglaxation time, treated as the upper limit for the decoheretime of the polaron state, was estimated to be of the orde3 ps for a dot with a'10-nm radius.5

In Refs. 4 and 5 the polarons were analyzed by a dir~numerical! diagonalization of the Hamiltonian containinterms for confined electrons in a quantum dot, free phonoand the Fro¨hlich electron-phonon interaction. An anharmonthird-order term~LO-TA!, was further included perturba

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tively. The question is whether including the coherent effof phononic dressing in the anharmonic term changesrelaxation time or not. The purpose of this Brief Report isverify this possibility via the application of an analyticamethod of the Davydov canonical transformation, i.e., byapproximate but explicit diagonalization of the Fro¨hlich in-teraction. This method allows for a simultaneous accountof the anharmonic phonon interaction, including its coherrenormalization.

It is convenient to divide the Hamiltonian of the systeinto two parts:

H5H01H1 . ~1!

H0 includes the free-electron term, the free-phonon term,the Frohlich interaction,

H05(n

Enan†an1(

k\Vbk

†bk1(s,q

\vs,qcs,q† cs,q

11

AN(

n1 ,n2 ,kFn1 ,n2

(o) ~k!an1

† an2~bk1b2k

† !, ~2!

wherean are fermionic operators for the electron in thenthlocalized state in the dot (n comprises all the relevant quantum numbers!, En are corresponding energies,V denotes thedispersionless~for simplicity! frequency of the LO phononsvs,q5Csq is the frequency of acoustical phonons (Cs is thesound velocity for thes branch, wheres denotes the polar-ization: t, transversal, andl, longitudinal!, N is the number ofcells in the crystal. The electron–LO-phonon coupling is dscribed by the function

Fn1 ,n2

(o) ~k!52\VA4pa

vQ0

1

kE fn1* ~r!eik•rfn2

~r!d3r,

~3!

wherefn* (r) is the wave function of electron in the dot,v isthe volume of the elementary cell,Q05A2m* V/\, anda isthe dimensionless Fro¨hlich constant.

The second part of the Hamiltonian,H1, describes theinteraction of the LA phonons with electrons via the defomation potential and the anharmonic third-order LO-TA ph

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BRIEF REPORTS PHYSICAL REVIEW B 65 113305

non interaction~in accordance with the dominating LOphonon decay channel determined experimentallyGaAs-bulk7!:

H151

AN(

n1 ,n2 ,qFn1 ,n2

(a) ~q!an1

† an2~cl ,q1cl ,2q

† !

1 (k1 ,k2 ,q

W~k1 ,k2 ,q!dk1 ,k21qbk1

† bk2~ctq1c2tq

† !.

~4!

The electron–LA-phonon coupling is described by the fution

Fn1 ,n2

(a) ~q!5sA \q

2MClE fn1

* ~r!eiq•rfn2~r!d3r, ~5!

where s denotes the deformation potential parameter~forGaAs, s56.8 eV), M is the total mass of ions in the eementary cell, andW(k1 ,k2 ,q) is the energy of the anharmonic LO-TA phonon interaction.

Now applying the Davydov canonical transformation9 onecan approximately diagonalizeH0 via the transformation ofthe operatorsa andb into a andb ~quasiparticle annihilationoperators!, according to the formulas:

a5er(a,b)ae2r(a,b), b5er(a,b)be2r(a,b), ~6!

wherer(a,b) is the anti-Hermitian operator,

r~a,b!5 (n1 ,n2 ,k

Fn1 ,n2~k!an1

† an2~bk2b2k

† !, ~7!

andFn1 ,n2(k) is a scalar function.

Upon the suitable choice of a functionFn1 ,n2(k), the

HamiltonianH0, expressed in terms of the operatorsa andb, is diagonal~neglecting residual multipolaron and mutiphonon terms, and assuming a single electron in thesimilarly to the bulk case9!,

H05(n

Enan†an1(

k\Vbk

†bk1(s,q

\vs,qcs,q† cs,q ,

~8!

with the polaron energiesEn defined by the equation in thform9

En2En51

N (n8

I nn8

En2En82\V, ~9!

whereI nn85@v/(2p)3#* uFnn8(o) (k)u2d3k.

If we confine ourselves to only the ground state~denotedby n50) and the first excited state~doubly degenerate,n561), the above equation can be solved in an elemenmanner, and leads to a typical splitting of an excited polastate near the resonance with a LO phonon. This behavipresented in the inset on Fig. 1. We focus on three modea quantum dot: a flat dot with parabolic confinement~P2D!,a dot with parabolic confinement and a finite height equa1/10 of its radius~P3D!, and a flat dot with a square-bo

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confinement~S2D!. The values of the factorsI nn8 for thesemodels are given byI nn85gnn8a(\V)3/2ADE, whereDE isthe separation between the ground and excited state.factorsgnn85gn8n are given below.

P2D P3D S2D

I 00 Ap/2'0.89 0.74 0.87I 01 Ap/8'0.22 0.16 0.27I 115I 2121 11Ap/32'0.61 0.54 0.78I 121 3Ap/32'0.17 0.11 0.13.

We verified that the canonical transformation approareproduces the spectrum obtained via the exact numediagonalization10 with high accuracy. This is connected witthe fact that in our case the Fro¨hlich constanta!1 and theterms omitted in the analytical diagonalization are ofhigher order ina, and thus are small.

Simultaneously with the diagonalization ofH0, the H1part of the Hamiltonian attains the form

H151

AN(

n1 ,n2 ,qFn1 ,n2

(a) ~q!an1

† an2~cl ,q1cl ,2q

† !

1 (k1 ,k2 ,q

W~k1 ,k2 ,q!dk1 ,k21qbk1

† bk2~ct,q1ct,2q

† !

1 (n1 ,n2q,k,s

Wn1 ,n2

s ~q,k!an1

† an2bk~cs,q1cs,q

† !1H.c.,

~10!

where

Wn1 ,n2

t ~q,k!521

AN

Fn1 ,n2

(o) ~k1q!W~k1q,k,q!

En22En1

1\V~11!

and

FIG. 1. Lifetime of the polaron with respect to the LO–Tchannel as a function ofDE—the separation between the bare eletron levels in the dot for three different dot models as describedthe text. Inset: energy difference between the ground and firstcited polaron levels as a function of the bare electron level seption in the dot. The dotted lines show the energy interval whrelaxation atT50 K is possible.

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BRIEF REPORTS PHYSICAL REVIEW B 65 113305

Wn1 ,n2

l ~q,k!521

N (n3

FFn1 ,n3

(o) ~k!Fn3 ,n2

(a) ~q!

En32En1

1\V

2Fn3 ,n2

(o) ~k!Fn1 ,n3

(a) ~q!

En22En3

1\VG . ~12!

The last term inH1 @Eq. ~10!# describes relaxation of thpolaron. The LO-TA anharmonicity-induced relaxation chanel corresponds to Eq.~11!, and the LO-LA channel to Eq~12!. Both these channels lead to a change of the the polastate accompanied by the creation or annihilation of a paiphonons: an optical phonon and acoustical phonon. Therefour possibilities for this process, with probabilities~accord-ing to the Fermi golden rule!

wn1 ,n2

x,y ~q,k!52p

\uWs~n1 ,n2 ,~x•y!q,k!u2

3~Nk1hx!~ns,q1hy!

3d~En12En2

2x\V2y\vs,q!, ~13!

wherex ~y! 56 and1 correspond to the emission and2 tothe absorption of an optical~acoustical! phonon, respec-tively, andh151, h250 (Nk is the number of LO phononin thek state, andnq,s is the number of acoustical phononsthe stateq and polarizations).

At sufficiently low temperatures@e.g., for GaAs forT,11 K ~Ref. 7!#, the phonon occupation numbers are neligible, and the only contribution is from the process ofpolaron transition with simultaneous emissions of twphonons; the corresponding probability iswn1,n2

11 (q,k). Therelaxation probability for this process is the sum

wn1→n25 (

k,q,swn1 ,n2

11 ~q,k!. ~14!

For the third-order phonon–phonon coupling strengwith one long-wavelength acoustical phonon involved,have11

uW~k1q,k,q!u25g2

Nq. ~15!

Thus, assuming thatg is independent ofk ~ask are near theG point5!, the relaxation probability via the LO-TA channemay be written as

wn1→n2

(t) 52

p

I n1n2g2qtv

\4Ct3

, ~16!

whereqt5(En12En2

2\V)/\Ct ~limited by the Debye fre-quency!.

For the LO-LA channel, the probability of relaxation hathe form @retaining only the largest terms inWn1 ,n2

l (q,k)#

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wn1→n2

(l) 5s2I n1n2

4p%\3Cl4ql

Jn1n2, ~17!

whereql5(En12En2

2\V)/\Cl ~limited by the Debye fre-

quency!, Jn1n25*d3qu f n2n2

(q)2 f n1n1(q)u2d(q2ql) , and

f nn8(q)5*fn* (r)exp(iq•r)fn8* (r)d3r.

The Frohlich constant~included in I n1n2) is given bya

5e2/ eAm* /(2\3V), where 1/e51/e`21/e0. For bulkGaAs, a'0.07 ~as e0512.9, e`510.9, m* 50.067me and\V536 meV). For a self-assembled GaAs/InAs dot withradius of ;10 nm, the recent far-infrared attenuatiomeasurements4 suggested, however, thata'0.15. An en-hancement of the electron–LO-phonon interaction in qutum dots also manifested itself via a significant increasethe Huang-Rhys parameter for LO-phonon-assisted lumincence features.12–14An explanation of this phenomenon canot be achieved only bye-h separation~even extremal!,12,14

and piezoelectricity in pyramid-shaped III-V dots12 and nonadiabatic effects in spherical II-VI dots13 were invoked. Inthe following, we perform calculations for botha50.01 and0.15, in order to describe the influence of the possiblecrease of the Fro¨hlich coupling.

Let us now estimate the polaron relaxation rate forLO-TA channel in a case similar to that discussed in Refbut for the quantum dot models introduced above. ForLO-TA process, at low temperatures, when only phonemission is possible, the polaron may relax only from tupper branch of the split excited level~shown in the inset ofFig. 1!. Energy conservation restricts this process to a cerenergy range, related to the maximum energy of the TA pnon, ;8 meV ~indicated by the dotted lines in the Fig. 1!.Thus this channel of polaron relaxation is ineffective inour models forDE greater than;37 and;40 meV fora50.15 and 0.07, respectively.

For a quantitative estimate of the corresponding relation rate, the value of the anharmonic phonon-phonon cpling constant is needed; this can be fitted using the expmental data for bulk GaAs.7 For GaAs, the dominananharmonic process involves TA phonons withq in the vi-cinity of the L point in the Brillouin zone.7 At low tempera-tures, for bulk GaAs, for the LO-phonon lifetime we find

tLO5w215p\2c

g2vq03m

, ~18!

where the factorm accounts for anisotropy effects, andq0corresponds to the vicinity of theL point. From the experi-mental data for phonon dispersions,15 one can note that theenergy conservation needed for the considered channeLO-phonon decay is satisfied along theL-W line on the hex-agonal zone wall, but is violated toward theS line. It there-fore seems reasonable to assume thatm'0.4. From the pho-non dispersion curves it also results that, on theG-L line nearthe L point, one hasc'0.6Ct . Using the lifetime tLO59.2 ps atT56 K, reported in Ref. 7, one can thus esmate theg factor. Using this value, the lifetime for the polaron in a GaAs self-assembled quantum dot with respec

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BRIEF REPORTS PHYSICAL REVIEW B 65 113305

the LO-TA relaxation channel can be estimated@at T50 K,according to Eq.~16!#, and this time versus the bare confinement energy in the dot is presented in Fig. 1. Note, howevthat the LO-LA channel@Eq. ~17!# can lead to an even stronger polaron relaxation rate, especially close to the resonDE energy.

The relaxation time for the LO-TA channel is on averaglonger by one order than the corresponding time foundthe numerical perturbation approach5 ~a similar proportion isalso maintained for nonzero temperatures!. For a bulk valueof a50.07, the relaxation time is even longer (;4 times!than in the case ofa50.15. The differences between thparabolic and square-box models do not exceed 40%. Moover, more realistic~finite height! models yield longer life-times. Therefore, the results may be expected to be not vsensitive to the variation of the quantum dot model.

The enhancement of the polaron relaxation time, treaas the growth of the upper limit for the decoherence timepolarons, even though significant, is too small for the neederror correction procedures for fully optically driven gatesGaAs/InAs quantum dot systems, unless the windows offectiveness of the LO-TA~and LO-LA! channels are

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avoided. As the technology for self-assembled dots impostrong restrictions on the dot dimensions, other quantumwould be more convenient for the implementation of extonic quantum information processing. Electric-field-defindots in ultranarrow quantum wells3 might be promising candidates. They offer much more flexible electronic structuthan self-assembled dots, including dots with greaterd, i.e.,with less strong phonon-electron coupling. Moreover,metastable excitonic states in electric-field-defined dwould allow for different methods of manipulation by manetic field.3 However, the problem consists of technologicdifficulties in the preparation of sufficiently small electrosystems for such structures.

Summarizing, we applied the Davydov transformationanalyze the anharmonic LO-TA channel of polaron relation in three different models of a quantum dot, explicincluding coherent polaronic effects. We indicated that ininterval of the confinement energy where this process mtake place, the corresponding relaxation time is a few timlonger than that found by the previous numerical approa

This work was supported by E.E.C. Project No. IST-1911311~SQID! and KBN Project No. PB 2 PO3B 055 18.

D.

N.

ys.

F.

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