Table Of ContentElectron-nuclei spin relaxation through phonon-assisted hyperfine interaction
in a quantum dot
Veniamin A. Abalmassov∗
Institute of Semiconductor Physics and Novosibirsk State University,
SB RAS, 630090 Novosibirsk, Russia
4
0 Florian Marquardt†
0 Department of Physics, Yale University, New Haven CT, 06511, USA
2
(Dated: February 2, 2008)
n
We investigate the inelastic spin-flip rate for electrons in a quantum dot due to their contact
a
J hyperfine interaction with lattice nuclei. In contrast to other works, we obtain a spin-phonon
coupling term from this interaction by taking directly into account the motion of nuclei in the
5
vibratinglattice. Inthecalculation of thetransition ratetheinterferenceoffirst andsecond orders
of perturbation theory turns out to be essential. It leads to a suppression of relaxation at long
]
l phonon wavelengths, when the confining potential moves together with the nuclei embedded in
l
a the lattice. At higher frequencies (or for a fixed confining potential), the zero-temperature rate
h is proportional to the frequency of the emitted phonon. We address both the transition between
- Zeeman sublevels of a single electron ground state as well as the triplet-singlet transition, and we
s
provide numerical estimates for realistic system parameters. The mechanism turns out to be less
e
m efficient than electron-nuclei spin relaxation involving piezoelectric electron-phonon coupling in a
GaAs quantumdot.
.
t
a PACSnumbers: 71.23.La,71.70.Jp
m
-
d I. INTRODUCTION wasconsideredfor impurity-bound electronsin silicon16,
n
where the authors investigated the nuclear polarization
o
inthe Overhausereffect. The processthathasbeencon-
c Electronspindynamicsinmesoscopicdeviceshasbeen
[ attracting a lot of attention recently in the context of sideredinthatpaperisassociatedwithacrystaldilation
spintronics1 andquantumcomputation2,3. Acrucialfea- and a corresponding adiabatic change in the electron ef-
1
fective mass and, as a consequence, in the electronenve-
v ture of this dynamics is the relaxation of the electron’s
lope wave function and the hyperfine coupling constant.
9 spin due to the interaction with an environment. Gen-
Recently, electron spin relaxation due to the hyperfine
2 erally speaking, the coherence of an electronic spin state
0 vanishes during the time T , which limits the possibil- contact interaction has been readdressed with an appli-
1 2 cation to GaAs QDs18,19. The transition amplitude was
ity of coherent manipulation of qubits, while relaxation
0 calculated in second-order perturbation theory, describ-
to thermal equilibrium occurs during another time T ,
4 which is usually larger than T 4. 1 ing the action of the hyperfine contact interaction and a
0 2
spin-independentpiezoelectricelectron-phononcoupling.
/ Several types of spin-dependent interactions can give
t
a rise to electron spin relaxation, e.g. the electron spin- In this paper we will analyze another spin relaxation
m orbit interaction5,6,7,8,9,10,11,12,13,14 and the electron- mechanism provided by the combination of hyperfine
- nuclei hyperfine interaction4,15,16,17,18,19,20,21. Their ac- contact interaction and the influence of phonons on the
d
tion depends essentially on the dimension of the system. electron inside a QD. In our approach we take into ac-
n
Systems of zero dimension, i.e. quantum dots (QDs), countdirectlythephonon-inducedmotionofnucleiwhich
o
c arecharacterizedbyadiscreteelectronenergyspectrum. arecoupledtotheelectronspinthroughthehyperfinein-
: In this case energy conservation in the spin-flip process teraction. The electron-phonon interaction appears via
v
usuallycanbefulfilledonlybytransferringtheenergyto thedisplacementfieldshiftingthepositionsofthenuclei,
i
X another subsystem, e.g. phonons. The energy transfer andthereforeisindependentofthepiezoelectriccoupling
r includes both the Zeeman energy of the electron spin in thatappliesonlyto crystalswithout inversionsymmetry
a an external field and possibly the energy of an orbital (suchasGaAs,butnotSi). Moreover,thismechanismal-
transition. A discussionof other electronspin relaxation lows the electron-nucleus spin flip-flop with a simultane-
mechanisms not mentioned above and relevant to QDs ousemissionofa phononto appearalreadyinfirst-order
can be found in Ref. 10, for example. Electron spin re- perturbation theory. Nevertheless, it will turn out that
laxation in a QD due to hyperfine interaction alone, in it is necessary to keep as well the second-orderterms as-
the absence of an external magnetic field, was investi- sociatedwiththemotionoftheelectronconfiningpoten-
gated recently in Refs. 22,23. tial,sincetheyleadtoacrucialcancellationoffirst-order
Many yearsago,the phonon-assistedelectronspin-flip terms at low frequencies of the emitted phonon, thereby
transition between Zeeman sublevels due to hyperfine suppressing the relaxation rate. The physical reason be-
interaction with an impurity nucleus and lattice nuclei hind this is the following: Long-wavelengthphonons dis-
2
place both the lattice nuclei and the electron’s confining has the form [4]:
potential in the same way. However, it is only the rela-
tivemotionoftheelectronwithrespecttothenucleithat VˆHF(ri) = Vˆhf(ri Rj)
−
enables a transition. Therefore, the influence of long- j
X
wavelengthphononsissuppressed. If,ontheotherhand, = Av Sˆ ˆI δ(r Rˆ ), (2)
0 i j i j
the confining potential can be considered as fixed or it · −
j
moves independently from the lattice nuclei, this sup- X
pression does not apply any more, and the destructive where the hyperfine coupling constant is determined as
interference between first-order and second-order terms
4µ µ µ
is broken. A= 0 B I η, (3)
3I v
The present article is organized as follows: In Section 0
II, we introduce the Hamiltonian of our model, includ- withη = u(0)2 beingthesquareoftheBlochamplitude
ing the effect of lattice vibrations on the hyperfine cou- at the site| of t|he nucleus15, µ >0 the Bohr magneton,
B
pling and the confining potential. After that, we derive µ the nuclear magnetic moment, and v the size of the
I 0
the electron spin flip transition amplitudes due to this unit cell. The factor η, which is usually of the order of
perturbation,discussingthepartialcancellationofterms 102 to 103, depending on the material, makes the Fermi
andthe dominatingcontribution. We calculatethe tran- contacttermmuchmoreefficientthantheothertermsof
sition rate, Eq. (23), for the case of Zeeman-split sub- the electron-nucleus hyperfine interaction4. For a more
levelsoftheelectrongroundstate(Sec. III),andperform complicatedlattice structure the hyperfine constantA is
an analogous derivation for the case of a triplet-singlet defined as a sum of the individual constants A over all
j
transition (Sec. IV). Finally, in Section V we look at the nuclei in the unit cell, taking into account different
numerical estimates for realistic system parameters. values of spins and parameters η for different nuclei15.
The positions of the nuclei deviate slightly from equi-
librium,due to the lattice vibrations: Rˆ =R0+uˆ(R0).
j j j
II. THE MODEL The lattice displacement field uˆ(R0) is described via the
j
phonon creation-annihilation operators24,
In our model, we will assume the displacement of the
QdiDsplcaocnemfineinntgfipeoltdenetviaallutaotebdeadtestchreibecdenbteyrtohfetphheondoont uˆ(R0j)= k,λs2ρωk¯h,λVph(ˆbk,λ+ˆb†−k,λ)ǫk,λ eik·R0j, (4)
(which we take to be the origin). Note that this is anal- X
ogous to the case of an impurity-bound electron. Any where ǫk,λ is the polarization vector of a phonon with
more detailed description (e.g. allowing for a distortion wave vector k in branch λ, ωk,λ is its frequency, Vph is
ofthepotential)wouldrequirefurtherspecificationscon- the volume of the crystal in which phonon modes are
cerning the way this potential is applied to the dot, but quantized, and ρ is the crystal mass density.
would not add significantly to the realism of the present As a consequence, the total Hamiltonian acquires the
model. following term which can lead to a nucleus-electron spin
In the effective mass approximation the Hamiltonian flip-flopprocesscombinedwiththeemissionofaphonon:
of the system of electrons and phonons in the QD has
the following form, if the perturbations due to electron- VˆHF−ph(ri)=−Av0 Sˆi·ˆIj uˆ(R0j)·∇ri δ(ri−R0j).
phonon and hyperfine interactions are excluded: j
X (cid:0) (cid:1)
(5)
Pˆ2 In the following, we will omit the index 0 for nuclear
Hˆ = i +V(r )+g∗µ Sˆ B equilibrium positions.
0 "2m∗ i B i· # Likewise, the vibrations of the confining potential are
i
X
1 1 described by
+ 2Xi6=i′Ve−e(ri−ri′)+Xk,λ¯hωk,λ(cid:20)ˆb†k,λˆbk,λ+ 2(cid:21),(1) δVˆ(ri)=− (uˆ(0)·∇ri)V(ri)
i i
X X
twuhmereopPˆeir=ato−r.ih¯∇Trhii+seHAa(mrii)l/tocnisiatnhedkeisncermibaetsicianltemroamcteinng- =uˆ(0)[H0, ¯hiPˆi]=−m¯h2∗uˆ(0) [H0,[H0,ri]].(6)
electronswitheffectivemassm∗ andeffectiveg-factorg∗, Xi Xi
localized in a static external potential V(r), in the pres- We note that the total electron momentum commutes
ence of a magnetic field B (with a corresponding vector with the electron-electron interaction potential.
potential A(r)), and free phonons. We neglect the Zee- Thus, the total perturbation to the Hamiltonian Hˆ ,
0
mansplittingofnuclearspinstates,giventhesmallvalue Eq. (1), is given by the three terms described above:
of the nuclear magnetic moment.
The hyperfine contact interaction of electrons of spin δHˆ = δVˆ(ri)+VˆHF(ri)+VˆHF−ph(ri) . (7)
Sˆi at positions ri with nuclei of spinˆIj at positions Rj Xi h i
3
III. TRANSITIONS BETWEEN ZEEMAN where the sum is over all intermediate states which dif-
SUBLEVELS fer from the initial and final states, and E(e) refers to
electron energies only. In writing down the energy de-
At first, we will consider the transition of a single nominatorswehaveusedthefactthatthehyperfineper-
electron between the Zeeman-split spin levels of the QD turbation only changes the electronic energies, and that
ground state. initialandfinaltotalenergieswillbethesame. Notethat
The initial state of the system is given by the direct the complete electronic energy includes the Zeeman en-
product of electron, nuclear and phonon states: ergy as well, and the difference between initial and final
electron energies is accounted for by the energy of the
i = ie iN iph . (8) emitted phonon, h¯ω =E(e) E(e).
| i | i⊗| i⊗| i i − f
Let n denote the direction of magnetic field. The initial Now we employ the expressions for the perturbation
electronstate i = ψ(r) n isgivenbytheproduct δVˆ(r), Eq. (6), in order to rewrite the previous formula
e −
ofthespinwav|eifunc|tionin⊗| ,wihichisaneigenfunction in a form that displays the relation to the first order
−
of the equation (n Sˆ)n | =i 1 n , and the properly amplitude:
· | ±i ±2| ±i
normalized ground state coordinate wave function
f δHˆ i (2) = fph uˆ(0)iph fe,N rVˆHF(r)ie,N
1 1 h | | i h | | i h |∇ | i
ψ(r)= √V ϕ0(r), V Z d3r|ϕ0(r)|2 =1. (9) m∗ω2 ′ hfe,N|r|me,Nnihme,N|VˆHF(r)|ie,Ni
HereV istheeffectivevolumeofthedot. InGaAs,where − Xm " Ei(e)−Em(e)
the electron g-factor is negative, the state n− corre- f Vˆ (r)m m r i
sponds to the maximum of energy. The init|ialinuclear + h e,N| HF | e,Nih e,N| | e,Ni , (13)
spin state iN is a direct product of states of all indi- Ef(e)−Em(e) #)
| i
vidual nuclei. We will average over the initial phonon
field state in the end, by inserting mean phonon occu- We have used the expressions for δVˆ(r) in terms of the
pation numbers given by the Bose distribution function commutatorofHˆ withboththekinematicalmomentum
0
nω =1/(eh¯ω/kBT −1). and the position operator, as well as the fact that VˆHF
In the final state commutes with the position.
Thetotalamplitude ofthe transitionis the sumofthe
f = f f f , (10)
| i | ei⊗| Ni⊗| phi terms (11) and (13). We will regroup it into two parts.
the electronic spin points into the opposite direction, The firstone consistsofEq. (11)andacontributionofa
f = ψ(r) n . Nuclear, f , and phonon, f , similar form, the first term of Eq. (13),
e + N ph
| i | i⊗| i | i | i
final states are determined by the action of the pertur-
bation potential VˆHF−ph which changes the electron and A1 = hfph|uˆ(0)−uˆ(Rj)|iphihfe,N|∇rVˆhf(r−Rj)|ie,Ni.
nuclear spin states while conserving the total spin of the j
X
electron-nuclei system (flip-flop process) and creates a (14)
phonon with energy corresponding to the energy differ- It contains the difference between the phonon displace-
ence between electron initial and final states. ment fields evaluated at the origin and at the nucleus
The corresponding transition matrix element is pro- position, respectively, which is analogous to taking the
vided by first order perturbation theory in the potential divergenceofthedisplacementfield. Thisdifferencevan-
VˆHF−ph(r): ishesas|kRj|forlong-wavelengthphononsanddescribes
theimportantcancellationwhichisonlyfoundiffirstand
f δHˆ i (1) secondordersofperturbationtheoryarecombinedprop-
h | | i
= fph uˆ(Rj)iph fe,N rVˆhf(r Rj)ie,N .(11) erly.
− h | | ih |∇ − | i The second part,
j
X
It is important that an alternative process is possible A = f uˆ(0)i
2 ph ph
−h | | i
for the transition between the same two states, where
the remaining two terms in δHˆ (namely δVˆ and VˆHF) m∗ω2 ′ hfe,N|r|me,Nihme,N|VˆHF(r)|ie,Ni
contribute in second order perturbation theory, yielding × Xm " Ei(e)−Em(e)
an amplitude that is of the same order of magnitude as f Vˆ (r)m m r i
Eq. (11): + h e,N| HF | e,Nih e,N| | e,Ni , (15)
′ f δVˆ(r)m mVˆ (r)i Ef(e)−Em(e) #)
f δHˆ i (2) = h | | ih | HF | i
h | | i Xm " Ei(e)−Em(e) ctroonntaminaskainsgumaotvraenrsinittieornmbeedtiwateeenstathtees.ZFeeomraansinsuglbeleevleecls-
+ hf|VˆHF(r)|mihm|δVˆ(r)|ii , (12) ofitsgroundstate,wecansetEi(e)−Em(e) =ε0−(εm−¯hω)
Ef(e)−Em(e) # inthefirstsummandandEf(e)−Em(e) =(ε0−¯hω)−εm in
4
2
the second. Note that in the two sums in Eq. (15) inter- z ω
0
1 for kl 1. (17)
mediate states with the same orbital energies εm differ l (cid:18)ω0(cid:19) ≪ ≫
by the Zeeman energy.
In order to render the following discussion concrete, Taking this into account we will neglect A2 in what fol-
we will now specify the confining potential explicitly. lows.
We consider a QD which is formed in a two-dimensional Now we rewrite the total transition amplitude in the
electrongas(2DEG)byanexternalsymmetricparabolic following, more explicit form (retaining only the main
potential. The confining potential in z-direction is usu- contribution):
ally modelled by a square well in vertical QDs and by
a triangular-shaped potential in lateral QDs. We ne- v ¯h
f δHˆ i (total) = A 0 n +1
glect the contributions from higher excited states in the h | | i V s2ρωk,λVph ωk,λ
z-potential, given their large energetic separation, and p
restrict our discussion to the ground state χ0(z). In the × eik·Rj −1 Sˆ+α−hfN|Iˆjα|iNi
presence of an external magnetic field perpendicular to j
X(cid:0) (cid:1)
ethraelxd−imyen-psliaonne,bethcoemeelectthreonDwarawveinf-uFnocctkiosnosluitniotnhse(lsaete- × (ǫk,λ·∇Rj)ϕ20(Rj), (18)
e.g. Ref. 25), with the effective confining frequency where Sˆα = n Sˆα n (a sum over α in Eq. (18) is
ω0 = Ω20+ωc2/4, where Ω0 is the strength of the assumed+).− h +| | −i
parabolic potential and ωc = eB⊥/(m∗c) the cyclotron By means of Fermi’s golden rule we obtain the follow-
p
frequency in an external magnetic field with the compo- ing expression for the transition rate (including a sum
nentB⊥perpendiculartothelayer. Theenergyspectrum overfinalstatesandaproperaverageoverinitialstates):
for these states is ε =(2n+ λ +1)h¯ω λ¯hω /2.
nλ 0 c
| | −
In a harmonic oscillator the coordinate vector induces W˙ = 2π A2 d3k δ(h¯ωk ¯hω)
transitionsonlybetweennearestorbitallevels. Thus,the ¯h N2 (2π)3 −
Z
sum in Eq. (15) reduces to two terms only correspond- ¯h
ing to the transition from the ground state φ00 to states × 2ρωk (nωk +1) F(Rj,Rj′)Gjj′
φ , with n = 0 and λ = 1 being the radial and the j,j′
nλ X
±
aexnagmulpalre,mionmceynlitnudmricqaulacnotourmdinnautmesbweres,rerewsrpietecttivheelys.caFloarr × ∇Rjϕ20(Rj)·∇Rj′ϕ20(Rj′) , (19)
product of the phonon polarization vector and the coor- h i
dinate vector as ǫr = (ǫ ρe−iϕ +ǫ ρeiϕ)/√2 and note where N = V/v0 is the number of unit cells inside the
+ −
dotvolumeV. In(19)wehaveemployedthesoundwave
that φ ρe±iϕ/√2 = lφ , where l = ¯h/(m∗ω ) is
00 0±1 0 dispersionlawintheformωk =ks,i.e. wehaveneglected
the length scale which determines the spatial extent of
p the difference in the transverse and longitudinal sound
the electron wave function in the parabolic well in the
velocities in summation over phonon polarizations. This
presence of an external perpendicular magnetic field.
simplifiesourformulasbutshouldnotchangeappreciably
We now compare the two parts of the total amplitude our numerical results.
thataregivenbyEqs. (14)and(15). First,wenotethat For clarity, in Eq. (19) we have combined some expo-
the matrix element fe,N rVˆhf(r Rj)ie,N in the ex- nential factors from Eq. (18) into the following expres-
gprraesdsiieonntfionrAth1eisz-pdriorhpecotritoin|o∇ncaalntob|e∇−eRsjtφim20|(aRtejd)i|,awsher1e/tzhe, sion:
with z0 the transverse dimension of the dot. Gr∼adient0s F(Rj,Rj′)=4eik·Rj−2Rj′ sink·Rj sink·Rj′. (20)
in the lateraldirectionsare smallerby a factor z /l 1. 2 2
0
The Zeeman splitting, h¯ω = g∗µ B , is much less ≪than
| B | We have also separated all spin-related factors into the
the orbitalenergysplitting,h¯ω ¯hω ,intypicalexperi-
≪ 0 correlationfunction (cf. Refs. 18,19):
ments onspinrelaxationinQDs. Thus,upto acommon
prefactor, we can use the following estimates for the ex-
pressions of Eqs. (14) and (15): Gjj′ =Sˆ−α+Sˆ+β−hiN|IˆjαIˆjβ′|iNiav, (21)
where the subscript, av, indicates averaging over initial
min[kl,1] min[ωl/s,1] m∗ω2l ω2 nuclear spin states (in our case over a thermal distribu-
A , A ,
| 1|∝ z ∼ z | 2|∝ ¯hω ∼ ω2l tion).
0 0 0 0
(16) At temperatures much higher than 10−7K, which
∼
wheres is the meansoundvelocity. Finally, we findthat is the order of magnitude of the nuclear spin-spin inter-
the ratio A2/A1 is suppressed by a factor whose form action, the nuclear spins are not correlated, i.e. Gjj′ =
| |
depends on the emitted phonon wave-length: Gδjj′. We suppose that there are no other sources of
averagenuclearpolarizationeither. This means, inturn,
z ω s that the interference terms in Eq. (19), stemming from
0
l ω ω l ≪1 for kl≪1 different nuclei, j =j′, vanish.
0 0 6
5
With the help of the usual commutation rules for spin notethatthereisnoterminthetotalHamiltonianwhich
components and the equality n Sˆ n = 1n, we ob- couplesdirectlystateswithλ= 1. Wewriteforthespin
h −| | −i −2 ±
tain for each nuclear spin the formula part (as in Ref. 18)
Sˆ−α+Sˆ+β−IˆαIˆβ = 41 ˆI2−(ˆI·n)2+(ˆI·n) , (22) |Tri=−νx√−2iνy|1,+1i+ νx√+2iνy|1,−1i+ νz|1,0i,
h i (27)
which results in a correlation function G = 1I(I +1), where the coefficients ν determine the initial su-
6 x,y,z
provided the averagenuclear spin is zero. perposition of degenerate states S,m with different z-
Takingallofthisintoaccount,thefollowingexpression components m = 1,0 of the to|tal sipin S = 1 of two
describesourmainresultfortherateoftheelectron-spin electrons. ±
relaxationbetweenZeemansublevelsofthegroundstate, For the final spin-singlet state we have
duetothehyperfine-phononmechanismconsideredhere:
fSi = ψSi(r ,r ) Si , (28)
1 A2 n +1 | e i | 1 2 i⊗| i
ω
= ω I(I +1) where Si denotes the singlet spin state (S = 0), and
T1 N 6π¯hρs3 the coo|rdiinate wave function is given by the expression:
sin(k R′ )
× 1− k R|′ | [∇R′ϕ20(R′)]2. (23) ψSi(r ,r )= ϕ0(r1)ϕ0(r2). (29)
(cid:20) | | (cid:21) 1 2 V
The overbar in Eq. (23) indicates an average over the
Again, the transition amplitude is given by Eqs. (14)
positions R′ of allnucleiin the dot. If the electronenve-
and (15), where we haveto introduce sums overelectron
lope wavefunction changes little on the scale of distance
betweenthenuclei,thenthisisjustaspatialaverageover coordinates: e.g. ∇rVˆhf(r−Rj) → i∇riVˆhf(ri−Rj),
etc.
anelectronlocalizationvolume. InderivingEq. (23),we P
In the presentcase,the energy of the emitted phonon,
have used the identity
¯hω, is equal to the single-particle energy splitting, h¯ω .
0
sin(2k ξ ) Inthe expressionforA2,Eq. (15), the contributionfrom
dΩk sin2(k ξ)=2π 1 | | . (24) the spin-singletintermediate statedominates,due to the
· − 2k ξ
Z (cid:20) | | (cid:21) small denominator [18] given by the exchange splitting
We note that the relaxation rate for an individual nu- δST. However, although δST is smaller than h¯ω0, it
clear spin is obtained by dividing Eq. (23) by N. still has the same order of magnitude18, and the ratio
A /A (z /l)(ω /δ ) is still much less than unity
2 1 0 0 ST
| | ∼
(here we have kl 1). Therefore we can once more
≫
IV. TRIPLET-SINGLET TRANSITION neglect A .
2
We obtain for the triplet-singlet transition rate:
We proceed in the same way in order to calculate the 2π A2 d3k
transitionratefortwoelectronsthatinitiallyresideinthe W˙ ST= δ(h¯ωk ¯hω)
¯h N2 (2π)3 −
lowest-lying triplet state and decay towards the ground- Z
¯h
sdtuacteedsibnygleextt.eWrnealsuanpdp/oosertnhuactletahremZaeegmneatnicsfipeliltdtsin)gca(nprboe- × 2ρωk (nωk +1) F(Rj,Rj′)GSjjT′
j,j′
neglectedascomparedtothe orbitalenergyspacingthat X
1
defines the transition energy for this process. × 2 ∇Rj(ϕ0(Rj)ϕ1(Rj))·∇Rj′(ϕ0(Rj′)ϕ1(Rj′)) ,
The wave function corresponding to the initial spin-
h (30i)
triplet electron state is
where the correlationfunction is:
iTr = ψTr(r ,r ) Tr , (25)
| e i | 1 2 i⊗| i
where the coordinate wave function is assumed to be GSjjT′ = hT|Sˆ1α−Sˆ2α|SihS|Sˆ1β −Sˆ2β|TihiN|Iˆjα·Iˆjβ′|iNiav
1
given by a Slater determinant (i.e. neglecting correla- = ν∗ανβhiN|Iˆjα·Iˆjβ|iNiavδjj′ = 3I(I+1)δjj′.(31)
tions):
Again, the nuclear spin state is averaged over a com-
ϕ (r )ϕ (r ) ϕ (r )ϕ (r )
ψTr(r ,r )= 0 1 1 2 − 0 2 1 1 . (26) pletely unpolarized thermal distribution.
1 2
V√2 Finally, the relaxation rate in this case is:
Here ϕ (r) is the wave function of the first excited sin- 1 A2 n +1
1 = ω ω I(I +1)
gleelectronorbitalstatenormalizedaccordingtoEq. (9). TST N 6π¯hρs3
1
Forconcreteness,wechooseittocorrespondtothequan-
sin(k R′ )
tfiuemldnthuemebneerrsgny=of0thainsdstλat=e i1s(leinssatnhaenxtεernal)m, aangndewtiec × 1− k R|′ | [∇R′(ϕ0(R′)ϕ1(R′))]2. (32)
0,−1 (cid:20) | | (cid:21)
6
V. NUMERICAL ESTIMATES AND
DISCUSSION €€1€€€€€@´10-4s-1D
T
1
Inordertoestimate the ratesinbothofthe casesthat 7
have been considered above, we take into account realis- Temperature
tic dimensions of typical quantum dots. Usually the lat- 6
4 K
eral length of the dot is much larger than its transverse
5 4 K
dimension, l z , and we use the following approxima-
0
tion: [∇R′(ϕ≫0(R′)ϕ1(R′))]2 ≃ ∇R′ϕ20(R′) 2 ≃ (z0/12)2 4 0.1 K
(whichwesupposeholdsforthe averageovernuclearpo- 0.1 K
sitions R′). For QDs with a di(cid:2)sc shape, in(cid:3)the typical 3
limit kz < 1, we can obtain a simple analytical expres-
0 2
sion for the average,
1
sin(k R′ ) 2(1 cos(kl))
f(kl)≡ 1− k R|′ | ≃1− −(kl)2 , (33) 2 4 6 8 10 B@TD
(cid:20) | | (cid:21)
whichcanbewellapproximatedbymin (kl)2/12,1 . We
{ } FIG.1: ThesingleelectronspinrelaxationrateinaGaAsQD
remark that this factor, which is present in the case of
versusmagnetic field(according toEq. (35)),derivedforthe
the co-moving confining potential, is absent when the hyperfine-phononmechanismdiscussedinthepresentarticle.
confining potential position is fixed. This is the only The solid line corresponds to the case when the confining
difference in the results for the relaxation rates in these potentialvibrates togetherwith thelattice, while thedashed
two cases, and it becomes important only in the limit of line refers tothe case of a fixedconfining potential.
small phonon frequencies, kl 1.
≪
WecanwritetherelaxationratebetweenZeemansub-
levels and triplet and singlet electron states using the of transverse sound waves, s s 3 103 m/s26. The
t
∼ ≃ ×
same approximate expression (the difference lies in the typicaltransversedimensionofaquantumdotisz0 10
≃
energy scale ω): nm and its lateral size is l 100 nm. The dot contains
about N 105 unit cells (≃8 nuclei in each). Hence, we
1 2 A2 ω(nω +1) canwrite∼fortherelaxationratetheapproximateexpres-
I(I+1)f(kl). (34)
T ≃ 3 N π¯hρs3z2 sion:
1 0
The linear dependence of the relaxation rate on the 1 1 10−15 ω f(kl). (35)
phononfrequency(forphononwavelengthssmallerthan T1 ≃ × 1−exp(−kh¯BωT)
the size of the QD, i.e. for sufficiently strong magnetic
fields in the case of Zeeman sublevels relaxation) sets For the transition between Zeeman energy sublevels
our mechanism apart from those considered earlier16,19, the phonon energy is equal to the Zeeman splitting,
where a cubic phonon frequency dependence is expected g∗µ B , which corresponds to 0.025 meV T−1 in GaAs
B
for low temperatures, kBT ¯hω (and a quadratic one w| here g|∗ = 0.44. At the same time the cyclotron en-
in the opposite limit). In a≪ddition, our result does not ergy, h¯ω , is−as much as 1.76 meV T−1 due to the small
c
depend either on the proximity of the nearest level, in effective mass in GaAs, m∗ = 0.067m . Consequently,
e
contrast to the spin relaxation rates calculated in Refs. the condition ω ω is always satisfied when a perpen-
0
18,19. dicular magnetic≪field is applied. For in-plane magnetic
The result for the electron-nuclei flip-flop transition fields ω ω when B 10 T in a lateral and B 100
0
rate obtained in Ref. 16 is larger than our result, Eq. T in a v≪ertical QD (the≪single particle energy spac≪ing in
(34), by a factor (γz0/l)2 (provided kl 1), which a lateral QD is 100 - 300 µeV, and in a vertical QD the
∼ ≪
is of the order of 10 (γ 50) for typical QD sizes cited confining energy of an approximate 2D harmonic poten-
≃
below, but can be smaller for large QDs. We should be tial is 4 meV25,27). The condition kl 1 corresponds
cautious, however, in applying directly the reasoning of to a m∼agnetic field of the order of 1 T.∼Thus, for mag-
Ref. 16 to the triplet-singlet transition, when the emit- neticfieldslargerthan 1Ttherelaxationrateislinear
ted phonon energy corresponds to the electron binding in the emitted phonon∼frequency and as a consequence
energyandhencetheconditionofadiabaticelectronmo- it is linear in the strength of the magnetic field (see Fig.
tionina vibratinglattice (usedinthatwork)willnotbe (1) which is plotted for the limit kz < 1 corresponding
0
fulfilled. to B < 10 T). For the fixed or ”independent” confining
In GaAs all nuclei have spin I = 3, and A is the sum potential we would have f(kl) 1 and the dependence
2
ofAj overallthenucleiintheunitcell: A= jAj ≃90 of the rate on the emitted pho≡non frequency would be
µeV15. The mass density is ρ 5.32 103 kg/m3, and linear even for small magnetic fields, i.e. when kl 1
≃ × P ≪
we approximate the mean sound velocity by the velocity (apartfromthe temperature-dependentfactor(n +1)).
ω
7
As to the orderofmagnitude,the mechanismthathas QD only a lower bound is available: T > 50 µs for an
1
been considered in this paper gives a rate of the order in-plane magnetic field B=7.5 T at T=2∼0 mK30. Both
of 10−4 s−1 for a temperature of about 1 K and a mag- of these measurement results were obtained by means of
netic field of about 1 T. It therefore appears to be much transient current spectroscopy28.
less efficient than the piezoelectric coupling mechanism On the other hand, hyperfine coupling mechanisms
considered in Ref. 19, where the rate is about 1 s−1 for (such as the one considered in this paper) may be par-
comparable values of temperature, 4 K, and magnetic ticularly relevant as far as effects like dynamic nuclear
field, 0.5 T. polarizationareconcerned,wherethe electron-nucleihy-
For the triplet-singlet transition in a lateral dot we perfine interaction plays a crucial role4,15,16. Recently, a
find a rate of the order of 10−4 s−1 for temperatures up hyperfine nuclear spin relaxationtime on the orderof 10
to 1K (h¯ω 100µeV), and in a vertical QD we have min was measured in a single GaAs QD, at a bath tem-
1/T∼1 10−3 ∼s−1 for temperatures up to 10 K and perature of 40 mK and a magnetic field up to 0.5 T31.
∼ ∼
¯hω 1 meV (we note that in both cases kl 1). The However, this experiment dealt with a nonequilibrium
latte∼r result should be compared with 1/T1 ≫2 10−2 transportsituation, with a resultingspin-flip mechanism
s−1 calculated in Ref. 18. ≈ × whose rate turns out to be orders of magnitude larger
Let us now turn to spin relaxation in silicon. Taking than the one discussed in the present article (see Ref.
into account the natural abundance of 29Si nuclei with 20).
non-zero spin n = 4.68%, their magnetic moment
I=1/2
µ = 0.56µ , the lattice constant a =5.43 ˚A and the
I N
−
electronicdensityatthe positionofthe nucleusη 1864
we find the effective hyperfine coupling constant≃to be VI. CONCLUSIONS
A 5 µeV. This is far smaller than in GaAs, due to
≃
the smaller η andsmaller percentage of nuclei with spin.
In summary, we have considered a specific mechanism
Inserting the mass density of Si, ρ 2.3 103 kg/m3
and the transverse sound velocity s≃ 5.×4 103m/s, for inelastic electron spin relaxationin a QD induced by
we obtain a prefactor of the order oft10≃−20 in×Eq. (35). the electron-nuclei hyperfine interaction in combination
with lattice vibrations. We have found that the inter-
This corresponds to a very small relaxation rate 1/T
10−9 s−1 between Zeeman sublevels in a magnetic fi1el∼d ference between first and second orders of perturbation
theoryisessentialforacorrectdescriptionofthesuppres-
of B = 1 T, for temperatures up to 1 K (g = 2 in Si).
sion of relaxation at small transition frequencies. The
Note that the spin-orbit related electron spin relaxation
relaxation rate has been calculated both for the transi-
time in lateral Si QDs14, for Si:P bound electrons and
tionbetweenZeemansublevelsoftheorbitalgroundstate
for QDs in SiGe heterostructures13 has been predicted
and for the triplet-singlet transition. We have obtained
recently to be on the order of several minutes for the
estimatesbasedonthesegeneralresultsandrealisticsys-
same values of magnetic field and temperature. In the
temparameters. The estimates demonstratethatthe re-
latter case, however,the relaxationrate may be strongly
laxation rate due to this particular mechanism is very
decreased,belowthe valuesfoundforourmechanism,by
small: For the spin relaxationbetweenZeemansublevels
application of uniaxial compressive strain13.
it is much less than the rate calculated earlier in second
In general, electron spin relaxation induced by
order perturbation theory with an emission of a phonon
electron-nucleihyperfine interactionin a quantum dot is
throughpiezoelectricelectron-phononcouplinginaGaAs
notasefficientasrelaxationduetospin-orbitinteraction QD19, and it is less than (or at most comparable with)
for typical values of system parameters9,10,11,12,13,14,18.
the relaxation rate due to the change in the localized
In many experiments, the hyperfine related rate would
electron effective mass induced by the lattice dilation in
be masked by the spin-orbit mechanism (However, see silicon16. For the triplet-singlet transition, the rate in
Ref. 19 for a case where the hyperfine mechanism
GaAsQDs is stillsmallerby anorderofmagnitude than
may dominate). The present experimental data for the
the corresponding rate of the piezoelectric mechanism.
triplet-singlet transition rate in a GaAs vertical QD is
T 200µs at temperatures up to 0.5 K and triplet- We would like to thank B. L. Altshuler, D. V. Averin,
1
sing≈let energy splitting ε 0.6 meV29. For the re- C.Bruder,E.V.SukhorukovandA.V.Chaplikforuseful
S−T
∼
laxationtimebetweenZeemansublevelsinalateralGaAs discussions.
∗ Electronic address: [email protected] 3 D. P.DiVincenzo, Fortschr. Phys. 48, 771 (2000).
† Electronic address: [email protected] 4 A.Abragam,Theprinciplesofnuclearmagnetism(Oxford
1 S. A.Wolf et al.,Science 294, 1488 (2001). University Press 1961), Chapters VI,IX.
2 M. A. Nielsen and I. L. Chuang, Quantum Computation 5 G. E. Picus and A. N. Titkov, in Optical Orientation,
and Quantum Information (Cambridge University Press, editedbyF.MeierandB.P.Zakharchenya(Elsevier,Am-
Cambridge, New York,2000). sterdam, 1984).
8
6 H. Hasegawa, Phys.Rev.118, 1523 (1960). Rev.B 64, 195306 (2001).
7 L. M. Roth, Phys.Rev. 118, 1534 (1960). 19 S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 66,
8 D. M. Frenkel, Phys.Rev.B 43, 14228 (1991). 155327 (2002).
9 A.V.KhaetskiiandY.V.Nazarov,Phys.Rev.B61,12639 20 Y. B. Lyanda-Geller, I. L. Aleiner, and B. L. Altshuler,
(2000). Phys. Rev.Lett. 89, 107602 (2002).
10 A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 64, 21 R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 115322
125316 (2001). (2003).
11 L.M.Woods,T.L.Reinecke,andY.Lyanda-Geller,Phys. 22 I. A.Merkulov, Al. L. Efros, and M. Rosen, Phys.Rev. B
Rev.B 66, 161318(R) (2002). 65, 205309 (2002).
12 D. Mozyrsky, S. Kogan, V. N. Gorshkov, and G. P. 23 J. Schliemann, A. Khaetskii, and D. Loss, J. Phys.: Con-
Berman, Phys.Rev. B65, 245213 (2002). dens. Matter 15, R1809 (2003).
13 Ch. Tahan, M. Friesen, and R. Joynt, Phys. Rev. B 66, 24 L. D. Landau and E. M. Lifshitz, Statistical Physics, Part
035314 (2002). 1 (Pergamon Press Ltd 1980).
14 B. A. Glavin and K. W. Kim, Phys. Rev. B 68, 045308 25 L. P. Kouwenhoven, D. J. Austing, and S. Tarucha, Rep.
(2003). Prog. Phys. 64(3), 701 (2001).
15 M. I. Dyakonov and V. I. Perel, in Optical Orientation, 26 S. Adachi,J. Appl.Phys.58(3), R1(1985).
editedbyF.MeierandB.P.Zakharchenya(Elsevier,Am- 27 T. Fujisawa et al.,Physica B 314, 224 (2002).
sterdam, 1984). 28 T. Fujisawa, Y. Tokura, and Y. Hirayama, Phys. Rev. B
16 D. Pines, J. Bardeen, and C. P. Slichter, Phys. Rev. 106, 63, 081304(R) (2001).
489 (1957). 29 T. Fujisawa et al.,Nature(London) 419, 278 (2002).
17 J. H. Kim, I. D. Vagner, and L. Xing, Phys. Rev. B 49, 30 R.Hanson et al., Phys.Rev.Lett. 91, 196802 (2003).
16777 (1994). 31 A. K.Hu¨ttel et al., cond-mat/0308243.
18 S. I. Erlingsson, Y. V. Nazarov, and V. I. Fal’ko, Phys.
