Table Of ContentPhonon modulation of the spin-orbit interaction as a spin relaxation mechanism in
quantum dots
C. L. Romano∗
Departamento de F´ısica, Universidad de Buenos Aires, C1428EHA Buenos Aires, Argentina
G. E. Marques
Departamento de F´ısica, Universidade Federal de S˜ao Carlos, 13565-905 S˜ao Carlos SP, Brazil
8
L. Sanz and A. M. Alcalde
0
0 Instituto de F´ısica, Universidade Federal de Uberlˆandia, 38400-902 Uberlˆandia MG, Brazil
2 (Dated: February 2, 2008)
n We calculate the spin relaxation rates in a parabolic InSb quantum dots due to the spin in-
a teraction with acoustical phonons. We considered the deformation potential mechanism as the
J dominant electron-phonon coupling in the Pavlov-Firsov spin-phonon Hamiltonian. By studying
1 suitable choices of magnetic field and lateral dot size, we determine regions where the spin relax-
1 ation ratescanbepractically suppressed. Weanalyzethebehaviorofthespin relaxation ratesasa
functionofan externalmagneticfieldand meanquantumdotradius. Effectsof thespinadmixture
] dueto Dresselhaus contribution tospin-orbit interaction are also discussed.
l
l
a
PACSnumbers: PACS:73.20.Dx,63.20.Kr,71.38.+i
h
-
s
e I. INTRODUCTION open subject. Extensive theoretical works in QD sys-
m tems have studied the main phonon mediated spin-flip
. mechanisms, including admixture processes due to spin-
t
a orbit coupling [6] and phonon coupling due to interface
The ability to manipulate and control processes that
m motion (ripple mechanism) [7]. Spin relaxationrates de-
involvetransitionsbetweenspinstatesis,atthemoment,
- of extreme importance due to the recent applications pend strongly on the dot size, magnetic field strength,
d
and temperature, as reported by several authors [6, 8].
in polarized spin electronics and quantum computation.
n
Itwasshownthatthequantumconfinementproduces,in
o Spin dephasing is the most critical aspect that should
general, a strong reduction of the QD relaxation rates.
c be considered in the elaboration of proposals of quan-
[ tum computation based in single spin states as qubits in Inthiswork,wecalculatethespin-fliptransitionrates,
quantum dots (QDs) [1]. Currently, QDs of diverse ge- considering the phonon modulation by the spin-orbit
1
interaction. For this purpose we will use the spin-
v ometries are considered as promising candidates for im-
phononinteractionHamiltonianproposedbyPavlovand
9 plementation of quantum computation devices because
9 the electronic, magnetic and optical properties can be Firsov[9, 10]. In this model, the Hamiltoniandescribing
6 controlled through the modern grown and nanofabrica- the transitions with spin reversal, due to the scattering
1 ofelectronsbyphonons,canbewritteninageneralform
tion techniques. Due to the long electron spin dephas-
1. ing time experimentally reported [2, 3, 4], the spin of an as
0 electronlocalizedinaQDhasbeensuggestedtorealizea
8 quantumbit. Whileforbulkandfor2Dsystemsthespin Hph =Vph+γ[σ×∇Vph]·(p+e/cA), (1)
0
relaxationprocesseshavebeenstudiedinsomedetail,the
: where V is the phonon operator, σ is the Pauli spin
v problem for QD’s still require deeper and further discus- ph
i sions. Several processes that can induce spin relaxation operator, γ is related with the strength of the electron-
X phonon interaction, p is the linear momentum operator
in semiconductors have been identified and were stud-
r andAisthevectorialpotentialrelatedwiththeexternal
a ied. Atthemomentremainsindiscussionwhich,between
magneticfieldB. Thismodelhastheadvantageofbeing
theseprocesses,isdominantinzero-dimensionalsystems.
easily adapted to the study of other interaction mecha-
Some experimental results have shown good agreement
nisms with phonons.
withthetheoreticalpredictionsfor2Dsystems[5]but,in
general,theidentificationoftheprocessesthroughdirect
comparisonwith the experimental results may become a
II. THEORY
formidable task. This problem is more critical for QDs,
since few experimental results exist and the theoretical
discussion of the spin relaxation mechanisms is still an Basedontheeffectivemasstheoryappliedtotheprob-
lem of the interaction of an electron with lattice vibra-
tions,includingthespin-orbitinteractionandinpresence
of an external magnetic field, Pavlov and Firsov [9, 10]
∗[email protected] have obtained the spin-phonon Hamiltonian that de-
2
scribes the transitions with spin reversal of the conduc- lattice vibrations as
tion band electrons due to scattering with longitudinal
~ 1/2 0 nˆ−׈e p eA
H =d(q) eiq·rb q + +q +h.c , (2)
ph (cid:18)ρMVvq(cid:19) (cid:26) q(cid:20) nˆ+׈eq 0 (cid:21)(cid:18)~ ~c (cid:19) (cid:27)
where, b (b†) are annihilation (creation) phonon opera- can be written as
q q
tors, the magnetic vector potential A is obtained in the
sBymormieenttreicdgaaluongeg ctohneszidearxiinsg. anˆn±e=xtexˆrn±aliyˆm,awgnheetriecxˆfi,elyˆd fn,M,σ =Cn,Maρ|M|M|+|1e−2ρa22eiMϕLn|M| ρ2/a2 χ(σ), (4)
(cid:0) (cid:1)
are unitary vectors along the x and y axis. ˆe is a unit
q
vector in the direction of the phonon polarization, q is where Cn,M = n!/[π(n+|M|)!], Ln|M| is the Laguerre
the phonon wavevector,p is the momentum operator,v polynomial,n (Mp)is the principal(azimuthal)quantum
is the average sound velocity, ρ is the mass density, V number, and χ(σ) is the spin wave function for the spin
M
isthesystemvolumeandd(q) isadeformationpotential variable σ. The corresponding eigenenergies are
electron-phonon coupling constant [10]
E =(2n+|M|+1)~Ω+(M/2)~ω +(σ/2)gµ B,
n,M,σ c B
(5)
~2q2E m∗ E
d(q)= 0 1− f g , (3) where Ω = (ω02 +ωc2/4)1/2, µB is the Bohr magneton,
2m∗ Eg (cid:18) m0(cid:19) (cid:18) ∆(cid:19) a = (~/mΩ)1/2 is the effective length, σ = ±1 for spin
up and down respectively, and ω =eB/m.
c
whereE =0.17eVisdebandgapforInSb,m∗=0.013m , In our model, we also consider the effects of the Dres-
g 0
E =7.0 eV is the deformation potential, ∆=0.81 eV is selhaus contribution that provides additional admixture
0
the spin-orbit splitting of the valence band, and f(x) = between spin states. For 2D systems, the linear Dressel-
(1+2x)/[(1+ 3x)(1+x)]. haus Hamiltonian can be written as
2
According to Eq. (2), we will restrict ourselves to the
β
case of bulk phonon modes. The longitudinal acous- HD = ~ (σxpx−σypy), (6)
tic phonons are described by plane-waves of the form
u0ˆeqexpiq·r,whereu0 isthenormalizationconstant. We where pi = −i~∇i+(e/c)Ai with i = x,y and β is the
consider the phonon energy dispersion in the Debye ap- Dresselhaus coupling parameter for this confinement. If
proximation: Eq = ~vq, for InSb we use v = 3.4×103 theconfinementpotentialinthez-directionisconsidered
m/s. It is important to point out that the theoretical highly symmetrical, then ∇V ∼ 0 and the Rashba con-
z
treatment of the spin-phonon interaction developed by tribution can be safely ignored.
Pavlov and Firsov [9, 10] can be easily adapted to sev- The spin relaxation rates (W) between the electronic
eralelectron-phononinteractionprocesses. Anappropri- states: (n,M,↑ (↓)) → (n′,M′,↓ (↑)), with emission
ate choice of the electron-phonon coupling d(q) allows of one acoustic phonon, are calculated from the Fermi
us to adapt the matrix element (2) for piezoacoustic or golden rule.
LO-optical spin scattering.
In the Hamiltonian (2), we only consider the deforma-
It has been assumed that the confinement along the z tionpotentialelectron-phononcoupling,thisisduetothe
axisismuchstrongerthanthelateralconfinement. Thus, large g-factor in narrow gap InSb (|g| ∼ 51), the domi-
the lateralmotionis decoupledfromthe one alongz and nant electron-phonon coupling for spin relaxation is the
the envelope functions separate ψ(r)=f(x,y)φ(z). The deformation potential mechanism [12]. The piezoelectric
z-dependent part of ψ(r) is an eigenfunction of a sym- couplinggovernsthe spin relaxationprocessesinwide or
metric quantum well of width L. In lens-shaped quasi- intermediate gap semiconductors. In the transition ma-
twodimensionalselfassembledQDs, the boundstatesof trixelements calculation,wenotonly considerthe linear
bothelectronsandvalence-bandholescanbeunderstood term iq·r in the expansion of exp(iq·r) [6], but also
by assuming a lateral spatial confinement modeled by a the integral representation of Bessel function is used in
parabolicpotentialwithrotationalsymmetryinthex−y the evaluation of electron-phonon overlapintegrals. The
plane [11], V(ρ) = 1mω2ρ2, where ~ω is the character- linear approximation of exp(iq·r) may be valid for spin
2 0 0
istic confinement energy, and ρ is the radial coordinate. inversion transitions in the spin polarized ground-states
Byusing the one-bandeffective massapproximationand of GaAs based QDs where, due to the small value of the
considering an external magnetic field B applied normal electron g-factor, only long wavelength phonons are in-
totheplaneoftheQD,theelectronlateralwavefunction volved.
3
2.0
III. RESULTS AND DISCUSSION
(0,0,› ) fi (0,0,fl ) a)
InSb W (1/ns)
The calculations were performed for a parabolic InSb
QD at T ∼ 0 K. The material parameters for the InSb 1.5 1
system are listed in Ref. [13]. We only have considered
electron transitions between ground state electron Zee- 6
man levels (0,0,↑) → (0,0,↓) and (0,1,↓) → (0,1,↑). )
T
The temperature dependence for one-phonon emission ( 1.0
B 36
rate is determined from W = W (n + 1), where n
0 B B
is the Bose-Einstein distribution function and W is the
0
rate at T = 0 K. In the temperature regime T ≤10 K, 1587 220
0.5
we obtain n +1 ≈ 1 and W ≈ W . For temperatures
B 0
larger than few Kelvin degrees (T > 10 K), two-phonon 1326
315
processes become the dominant spin relaxation mecha-
36
nism [7]. These types of processes have not been consid-
8000
ered in the present calculation. 60 80 100 120 140 160 180 200
r (Å)
0
2.0
r=50 Å 1284 (0,1,fl ) fi (0,1,› ) b)
104 0 InSb W (1/ns)
1.5 100
r=200 Å
0
s) 101 3006 413
n T)
W (1/ B ( 1.0 51248 1705
10-2 10771 7039
0.5
(0,0,› ) fi (0,0,fl ) 10771 29063
(0,1,fl ) fi (0,1,› ) InSb 1284 3006
10-5 120000
0.5 1.0 1.5 2.0 60 80 100 120 140 160 180 200
r (Å)
B (T)
0
FIG. 2: (Color online) Contour plot of the spin relaxation
FIG. 1: (Color online) Spin relaxation rates, W, for a
rates as a function of magnetic field B and lateral size r0 for
parabolicInSbQDconsideringthedeformationpotentialcou-
transitions: a) (0,0,↑)→(0,0,↓) and b)(0,1,↓)→(0,1,↑).
pling mechanism as a function of the magnetic field B. We
consider the transitions: (0,0,↑) → (0,0,↓) and (0,1,↓) →
(0,1,↑) and several lateral dot radius r0 = 50, 75, 100, 125,
150, 175, and 200 ˚A(same r0 ordering for both transitions) The position of this main maximum is defined from the
transition energy conservation: Enlσ′ −En′l′σ′ =~vq.
In the Fig. 1 we show the spin relaxation rates due to ii) The oscillatory behavior of the rates, observed for
deformation potential electron-phonon mechanism, as a B >0.7T are mainly produced by complex interplay be-
functionoftheexternalmagneticfieldB andconsidering tween the spin admixture and electron-phonon overlap
sometypicalvaluesfortheeffectivelateralQDsize,r = integrals. The phonon modulated Rashba interaction,
0
~/mω . given in Eqs. (1) and (2) produce level admixture ac-
0
pSome interesting facts about these results should be cording to the selection rules: (n,M,↑)→(n,M +1,↓),
pointed out: (n,M,↓) → (n,M −1,↑) and (n,M,↑ (↓)) → (n,M,↓
i) The rates show a strong dependence with the mag- (↑)). The last condition, which hybridize states with
netic field. This fact can be explained from the depen- different spin orientations at the same level M, is the
denceoftherateswiththetransitionenergy∆E. Ingen- main responsible for the rates oscillations at B > 0.7T.
eral,weobtainthatW ∼[g∗µ B]n =(∆E)n,nbeingan For∆M =0transitions,theDresselhausSOinteraction,
B
integernumberthatdependsontheelectron-phononcou- given in (6), is not able to produce spin admixture.
plingprocessandg∗theeffectiveg-factor. Ascanbeseen As it is shown in Fig. 1, the g∗-factor effects are par-
in Fig. 1, when the magnetic field increases, the rates ticulary important for the ground-state Zeeman transi-
also increase until reaching a maximum near B ∼0.5 T. tion. For small magnetic fields, g∗ → g and we may
bulk
4
neglect the spin admixture effects. Therefore, the spin regimeisdefinedapproximatelybyB >1Tandr >100
0
relaxation shows no oscillations and becomes almost in- ˚A. However,the relaxationtimes, in this region, are two
dependent of r . This small QD size dependence is in orders of magnitude faster than for (0,0,↑) → (0,0,↓)
0
agreement with the experimental observations of Gupta transition.
and Kikkawa [14]. Inconclusion,inthisworkwepresentedthecalculation
iii) The rates dependence with the lateral QD size r0, of spin relaxation rates via electron-phonon interaction.
are related to the interplay effects between the spatial The model used in our work, allow us to estimate the
and magnetic confinements. This competing effects are magnitude of the scattering rates considering the most
contained in the electron-phonon overlap integral, I ∝ important contributions to the spin relaxation through
fn∗′,l′,σ′(ρ)exp(iq · r)fn,l,σ(ρ)dr. For large fields, the acousticdeformationpotentialinteraction. Theratesex-
mR agnetic confinement causes a gradual decrease in the hibit a strong dependence with the magnetic field and
overlap integral as the r0 increases. For small magnetic with the lateral QD size. We identify regimes of strong
fields, the spatial confinement is dominant. Thus, when spin coherence, for magnetic field, B > 1T, and for lat-
r0 diminishes the wave functions become more localized eral sizes, r0 > 100 ˚A, we obtain relaxation times ap-
and the overlap integral should increase. This effects proximately of 10−2 ns to 1.0 ns. The Zeeman ground
explain the behavior of the spin transition (0,1,↓) → state transition (0,0,↑) → (0,0,↓), exhibits longer spin
(0,1,↑)showedinFig.1(redlines). TheZeemanground- coherence times than the high energy transitions, which
state rates (black lines) are strongly dependent on ∆E isimportantforspinqubitapplications. Themagnitudes
and, for small B, the rates are weakly dependent on I. of the calculated rates are compatible with those ob-
iv) The same rates calculated for GaAs (not showed tained experimentally. For a more rigorous analysis of
here),areingeneral,oneorderofmagnitudesmallerthan the spinrelaxationprocess,manyothereffects shouldbe
InSbrates. As weexpected, the relaxationviapiezoelec- considered, like the anisotropy of the electron g-factor
tric coupling is more efficient than via the deformation and the effects of the spatial confinement on the acous-
potential phonon processes. tic phonons. These two issues, are still under intense
In Fig. 2 a) we have plotted the spin relaxation rates discussion and should be considered for a more precise
for the ground-state Zeeman transition as a function of determination of the spin relaxation times.
r and B. We clearly identify a region of strong spin
0
coherence, defined by B > 1 T and r > 100 ˚A. In this
0
regime,therelaxationtimesτ areinthensorderandthis
isanimportantfeatureforspinqubitengineering. Inthe
Acknowledgments
B <0.1Tregime,therelaxationtimesareapproximately
of few µs. This spin frozen region is not robust against
thetemperatureandwilldisappearwheneverthethermal ThisworkhasbeensupportedbyFundac¸a˜odeAmparo
energyislargerthanthespintransitionenergy. Theplot `a Pesquisa do Estado de Minas Gerais (FAPEMIG) and
in Fig. 2 b) shows the spin rates for (0,1,↓) → (0,1,↑) by Conselho Nacional de Desenvolvimento Cient´ıfico e
transition. As in the previous case, the strong coherence Tecnol´ogico (CNPq).
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