Effect of electron-electron interaction on the phonon-mediated spin relaxation in
quantum dots
Juan I. Climente,1, ∗ Andrea Bertoni,1 Guido Goldoni,1, 2 Massimo Rontani,1 and Elisa Molinari1, 2
1CNR-INFM National Center on nanoStructures and bioSystems at Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy
2Dipartimento di Fisica, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy
(Dated: October 21, 2018)
We estimate the spin relaxation rate due to spin-orbit coupling and acoustic phonon scattering
in weakly-confined quantum dots with up to five interacting electrons. The Full Configuration
Interaction approach is used to account for the inter-electron repulsion, and Rashba and Dresselhaus
spin-orbit couplings are exactly diagonalized. We show that electron-electron interaction strongly
affects spin-orbit admixture in the sample. Consequently, relaxation rates strongly depend on the
number of carriers confined in the dot. We identify the mechanisms which may lead to improved
spin stability in few electron (> 2) quantum dots as compared to the usual one and two electron
devices. Finally, we discuss recent experiments on triplet-singlet transitions in GaAs dots subject
to external magnetic fields. Our simulations are in good agreement with the experimental findings,
and support the interpretation of the observed spin relaxation as being due to spin-orbit coupling
assisted by acoustic phonon emission.
PACS numbers: 73.21.La,71.70.Ej,72.10.Di,73.22.Lp
I. INTRODUCTION
There is currently interest in manipulating electron
spins in quantum dots (QDs) for quantum information
and quantum computing purposes.1,2,3 A major goal in
this research line is to optimize the spin relaxation time
(T1), which sets the upper limit of the spin coherence
time (T2): T2 ≤ 2T1.4 Therefore, designing two-level
spin systems with long spin relaxation times is an im-
portant step towards the realization of coherent quantum
operations and read-out measuraments. Up to date, spin
relaxation has been investigated almost exclusively in
single-electron4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 and two-
electron20,21,22,23,24,25,26,27,28,29,30 QDs. Spin relaxation
in QDs with a larger number of electrons has seldom been
considered28,31, even though Coulomb blockade makes
it possible to control the exact number of carriers con-
fined in a QD.32 Yet, recent theoretical works suggest
that Coulomb interaction renders few-electron charge de-
grees of freedom more stable than single-electron ones33,
which leads to the question of whether similar findings
hold for spin degrees of freedom. Moreover, in weakly-
confined QDs, acoustic phonon emission assisted by spin-
orbit (SO) interaction has been identified as the domi-
nant spin relaxation mechanism when cotunneling and
nuclei-mediated relaxation are reduced.6,8,31 The com-
bined effect of Coulomb interaction and SO coupling has
been shown to influence the energy spectrum of few-
electron QDs profoundly,34,35,36 but the consequences on
the spin relaxation remain largely unexplored.37
In Ref. 28 we investigated the effect of a magnetic
field on the triplet-singlet (TS) spin relaxation in two
and four-electron QDs with SO coupling, so as to under-
stand related experimental works. Motivated by the very
different response observed for different number of con-
fined particles, in this work we shall focus on the role of
electron-electron interaction in spin relaxation processes,
extending our analysis to different number of carriers,
highlighting, in particular, the different physics involved
in even and odd number of confined electrons. Further-
more, we will explicitly compare the predictions of our
theoretical model with very recent experiments on spin
relaxation in two-electron GaAs QDs.29
We study theoretically the energy structure and spin
relaxation of N interacting electrons (N = 1 − 5) in
parabolic GaAs QDs with SO coupling, subject to axial
magnetic fields. Both Rashba38 and Dresselhaus39 SO
terms are considered, and the electron-electron repulsion
is accounted for via the Full Configuration Interaction
method.40,41 By focusing on the two lowest spin states,
two different classes of systems are distinguished. For N
odd (1,3,5) and weak magnetic fields, the ground state is
a doublet and then the two-level system is defined by the
Zeeman-split sublevels of the lowest orbital. For N even
(2,4), the two-level system is defined by a singlet and a
triplet. We analyze these two classes of systems sepa-
rately because, as we shall comment below, the physics
involved in the spin transition differs. Thus, we compare
the phonon-induced spin relaxation of N = 1, 3, 5 elec-
trons and that of N = 2, 4 separately. As a general rule,
the larger the number of confined carriers, the stronger
the SO mixing, owing to the increasing density of elec-
tronic states. This would normally yield faster relax-
ation rates. However, we note that this is not necessarily
the case, and few-electron states may display compara-
ble or even slower relaxation than their single-electron
and two-electron counterparts. This is due to charac-
teristic features of the few-particle energy spectra which
tend to weaken the admixture between the initial and
final spin states. In N -odd systems, it is the presence
of low-energy quadruplets for N > 1 that reduces the
admixture between the Zeeman sublevels of the (dou-
blet) ground state, hence inhibiting the spin flipping. In
N -even systems, electronic correlations partially quench
http://arxiv.org/abs/0704.0868v2
phonon emission33, and the relaxation can be further sup-
pressed forN > 2 if one selects initial and final spin states
differing in more than one quantum of angular momen-
tum, which inhibits direct triplet-singlet SO mixing via
linear Rashba and Dresselhaus SO terms.28 Noteworthy,
all these effects are connected with Coulomb interaction
between confined carriers.
The paper is organized as follows. In Section II we give
details about the theoretical model we use. In Section
III we study the energy structure and spin relaxation of
a QD with an odd number of electrons (N = 1, 3, 5). In
Section IV we do the same for QDs with an even number
of electrons (N = 2, 4). In Section V we compare our
numerical simulations with experimental data recently
reported for N = 2 GaAs QDs. Finally, in Section VI we
present the conclusions of this work.
II. THEORY
We consider weakly-confined GaAs/AlGaAs QDs,
which are the kind of samples usually fabricated
by different groups to investigate spin relaxation
processes.7,8,20,22 In these structures, the dot and the sur-
rounding barrier have similar elastic properties, and the
lateral confinement (which we approximate as circular)
is much weaker than the vertical one. A number of use-
ful approximations can be made for such QDs. First,
since the weak lateral confinement gives inter-level spac-
ings within the range of few meV, only acoustic phonons
have significant interaction with bound carriers, while op-
tical phonons can be safely neglected. Second, the elasti-
cally homogeneous materials are not expected to induce
phonon confinement, which allows us to consider three-
dimensional bulk phonons. Finally, the different energy
scales of vertical and lateral electronic confinement allow
us to decouple vertical and lateral motion in the building
of single-electron spin-orbitals. Thus, we take a parabolic
confinement profile in the in-plane (x, y) direction, with
single-particle energy gaps h̄ω0, which yields the Fock-
Darwin states.42 In the vertical direction (z) the confine-
ment is provided by a rectangular quantum well of width
Lz and height determined by the band-offset between
the QD and barrier materials (the zero of energy is then
the bottom of the conduction band). The quantum well
eigenstates are derived numerically. In cylindrical coor-
dinates, the single-electron spin-orbitals can be written
ψµ(ρ, θ, z; sz) =
eimθ Rn,m(ρ) ξ0(z)χsz , (1)
where ξ0 is the lowest eigenstate of the quantum well,
χsz is the spinor eigenvector of the spin z-component
with eigenvalue sz, and Rn,m is the n−th Fock-Darwin
orbital with azimuthal angular momentum m,
Rn,m(ρ) =
(n+ |m|)!
0 L|m|n
In the above expression L|m|n denotes a generalized La-
guerre polynomial and l0 =
h̄/m∗ω0 is the effective
length scale, with m∗ standing for the electron effec-
tive mass. The energy of the single-particle Fock-Darwin
states is given by En,m = (2n + 1 + |m|)h̄Ωc + m2 h̄ωc,
where ωc =
is the cyclotron frequency and Ωc =
+ (ωc/2)2 is the total (spatial plus magnetic) con-
finement frequency.
With regard to Coulomb interaction, we need to go
beyond mean field approximations in order to properly
include electronic correlations, which play an important
role in determining the phonon-induced electron scatter-
ing rate.43 Moreover, since we are interested in the re-
laxation time of excited states, we need to know both
ground and excited states with comparable accuracy.
Our method of choice is the Full Configuration Interac-
tion approach: the few-electron wave functions are writ-
ten as linear combinations |Ψa〉 =
cai|Φi〉, where the
Slater determinants |Φi〉 = Πµic†µi |0〉 are obtained by
filling in the single-electron spin-orbitals µ with the N
electrons in all possible ways consistent with symmetry
requirements; here c†µ creates an electron in the level µ.
The fully interacting Hamiltonian is numerically diago-
nalized, exploiting orbital and spin symmetries.40,41 The
few-electron states can then be labeled by the total az-
imuthal angular momentumM = 0,±1,±2 . . ., total spin
S and its z-projection Sz.
The inclusion of SO terms is done following a similar
scheme to that of Ref. 44, although here we consider
not only Rashba but also linear Dresselhaus terms. For
a quantum well grown along the [001] direction, these
terms read:38,39
HR = α
(kysx − kxsy), (3)
HD = γc 〈k2z〉(kysy − kxsx), (4)
where α and γc are coupling constants, while sj and kj
are the j-th Cartesian projections of the electron spin and
canonical momentum, respectively, along the main crys-
talographic axes (〈k2z〉 = (π/Lz)2 for the lowest eigen-
state of the quantum well). The momentum operator
includes a magnetic field B applied along the vertical di-
rection z. Other SO terms may also be present in the
conduction band of a QD, such as the contribution aris-
ing from the system inversion asymmetry in the lateral
dimension or the cubic Dresselhaus term. However, in
GaAs QDs with strong vertical confinement, HR and HD
account for most of the SO interaction.36
We rewrite Eqs.(3,4) in terms of ladder operators as:
HR = α
(k+s− − k−s+), (5)
HD = β
(k+s+ + k−s−), (6)
where k± and s± change m and sz by one quantum,
respectively, and β = γc (π/Lz)
2 is the Dresselhaus in-
plane coupling constant. It is worth mentioning that
when only Rashba (Dresselhaus) coupling is present, the
total angular momentum j = m + sz (j = m − sz)
is conserved. However, in the general case, when both
coupling terms are present and α 6= β, all symmetries
are broken. Still, SO interaction in a large-gap semi-
conductor such as GaAs is rather weak, and the low-
lying states can be safely labelled by their approximate
quantum numbers (M,S, Sz) except in the vicinity of the
level anticrossings.11,26,45 Since the few-electron M and
Sz quantum numbers are given by the algebraic sum of
the single-particle states m and sz quantum numbers, it
is clear from Eqs. (5,6) that Rashba interaction mixes
(M,Sz) states with (M ± 1, Sz ∓ 1) ones, while Dressel-
haus interaction mixes (M,Sz) with (M ± 1, Sz ± 1).
The SO terms of Eqs. (5,6) can be spanned on a basis of
correlated few-electron states.46 The SO matrix elements
are then given by sums of single-particle contributions of
the form:
〈n′m′ s′z| HR +HD |nmsz〉 =
C∗R O+n′m′ nm δm′ m+1 δs′z sz−1+CR O
n′m′ nm δm′ m−1 δs′z sz+1+
C∗D O+n′m′ nm δm′ m+1 δs′z sz+1+CD O
n′m′ nm
δm′ m−1 δs′
sz−1.
Here CR = α and CD = −iβ are constans for the Rashba
and Dresselhaus interactions respectively, andO± are the
form factors:
Rnm(t),
Rnm(t),
with t = ρ2/l20. The above forms factors have analytical
expressions which depend on the set of quantum num-
bers {n′m′, nm}. The resulting SO-coupled eigenvec-
tors are then linear combinations of the correlated states,
|ΨSOA 〉 =
cAa|Ψa〉.
We assume zero temperature, which suffices to capture
the main features of one-phonon processes.9,16 Indeed, it
is one-phonon processes that account for most of the low-
temperature experimental observations in the SO cou-
pling regime.2,6,8,28,29,31 We evaluate the relaxation rate
between the initial (occupied) and final (empty) states
of the SO-coupled few-electron state, B and A, using the
Fermi Golden Rule:
τ−1B→A =
c∗BbcAa
c∗bicaj〈Φi|Vνq|Φj〉
δ(EB−EA−h̄ωq),
where the electron states |ΨSOK 〉 (K = A,B) have been
written explicitly as linear combinations of Slater deter-
minants, EK stands for the K electron state energy and
h̄ωq represents the phonon energy. Vνq is the interac-
tion operator of an electron with an acoustic phonon of
momentum q via the mechanism ν, which can be either
deformation potential or piezoelectric field interaction.
Details about the electron-phonon interaction matrix el-
ements can be found elsewhere.33
In this work we study a GaAs/Al0.3Ga0.7As QDs, us-
ing the following material parameters:47 electron effective
massm∗ = 0.067, band-offset Vc = 243 meV, crystal den-
sity d = 5310 kg/m3, acoustic deformation potential con-
stant D = 8.6 eV, effective dielectric constant ǫ = 12.9,
and piezoelectric constant h14 = 1.41 · 109 V/m. The
Landé factor is g = −0.44.5 As for GaAs sound speed, we
take cLA = 4.72 · 103 m/s for longitudinal phonon modes
and cTA = 3.34 · 103 m/s for transversal modes.48 Unless
otherwise stated, a lateral confinement of h̄ω0 = 4 meV
and a quantum well width of Lz = 10 nm are assumed
for the QD under study, and a Dressehlaus coupling pa-
rameter γc = 25.5 eV·Å3 is taken49, so that β ≈ 25
meV·Å. The value of the Rashba coupling constant can
be modulated externally e.g. with external electric fields.
Here we will investigate systems both with and without
Rashba interaction. When present, we shall mostly con-
sider α = 50 meV·Å, to represent the case where Rashba
effects prevail over Dresselhaus ones.
Few-body correlated states (M,S, Sz) are obtained us-
ing a basis set composed by the Slater determinants
(SDs) which result from all possible combinations of 42
single-electron spin-orbitals (i.e., from the six lowest en-
ergy shells of the Fock-Darwin spectrum at B = 0) filled
with N electrons. For N = 5, this means that the basis
rank may reach ∼ 2 · 105. The SO Hamiltonian is then
diagonalized in a basis of up to 56 few-electron states,
which grants a spin relaxation convergence error below
2%. Since SO terms break the spin and angular mo-
mentum symmetries, the SO-coupled states |ΨSOK 〉 are
described by a linear combination of SDs coming from
different (M,S, Sz) subspaces. Thus, for N = 5, the
states are described by up to ∼ 8.5 · 105 SDs. To evalu-
ate the electron-phonon interaction matrix elements, we
note that only a small percentage of the huge number
of possible pairs of SDs (∼ 7 · 1011 for N = 5) may
give non-zero matrix elements, owing to spin-orbital or-
thogonalities. We scan all pairs of SDs and filter those
which may give non-zero matrix elements writing the de-
terminants in binary representation and using efficient
bit-per-bit algorithms.40,41 The matrix elements of the
remaining pairs (∼ 2 ·106 for N = 5) are evaluated using
massive parallel computation.
 0 1 3
B (T)
FIG. 1: Low-lying energy levels in a QD with N = 1, 3, 5
interacting electrons, as a function of an axial magnetic field.
The SO interaction coefficients are α = 50 meV· Å and β =
25 meV· Å. The dot has h̄ω0 = 4 meV and Lz = 10 nm. Note
the increasing size of the SO-induced anticrossing gaps and
zero-field splittings with increasing N .
III. SPIN RELAXATION IN A QD WITH N ODD
A. Energy structure
When the number of electrons confined in the QD is
odd and the magnetic field is weak enough, the ground
and first excited states are usually the Zeeman sz = 1/2
and sz = −1/2 sublevels of a doublet [Fig. 1]. Since the
initial and final spin states belong to the same orbital,
∆M = 0 and SO mixing (which requires ∆M = ±1)
is only possible with higher-lying states. In addition,
the phonon energy (corresponding to the electron tran-
sition energy) is typically small (in the µeV scale). In
this case, the relaxation rate is determined essentially by
the phonon density, the strength and nature of the SO
interaction, and the proximity of higher-lying states.9,11
In order to gain some insight on the influence of these
factors, in Fig. 1 we compare the energy structure of a
QD with N = 1, 3, 5 vs. an axial magnetic field, in the
presence of Rashba and Dresselhaus interactions.55 One
can see that the increasing number of particles changes
the energy magneto-spectrum drastically. This is be-
cause the quantum numbers of the low-lying energy levels
change, resulting in a different field dependence, and be-
cause Coulomb interaction leads to an increased density
of electron states, as well as to a more complicated spec-
trum.
At first sight, the energy spectra of Fig. 1 closely resem-
ble those in the absence of SO effects. For instance, the
N = 1 spectrum is very similar to the pure Fock-Darwin
spectrum.42 Rashba and Dresselhaus interactions were
expected to split the degenerate |m| > 0 shells at B = 0,
shift the positions of the level crossings and turn them
into anticrossings36,52,53,54, but here such signatures are
hardly visible because SO interaction is weak in GaAs.
In fact, the magnitude of the SO-induced zero-field en-
ergy splittings and that of the anticrossing gaps is of very
few µeV, and SO effects simply add fine features to the
N = 1 spectrum.52
A significantly different picture arises in the N = 3
and N = 5 cases. Here, the increased density of elec-
tronic states enhances SO mixing as compared to the
single-electron case.56 As a result, the anticrossing gaps
can be as large as 30 µeV (N = 3) and 60 µeV (N = 5).
Moreover, unlike in the N = 1 case, where the ground
state orbital has m = 0, here it has |M | = 1. Therefore,
the Zeeman sublevels involved in the fundamental spin
transition are subject to SO-induced zero-field splittings.
To illustrate this point, in Fig. 2 we zoom in on the energy
spectrum of the four lowest states of N = 3 and N = 5
under weak magnetic fields, without (left panels) and
with (right panels) Rashba interaction. Clearly, the four-
fold degeneracy of |M | = 1 spin-orbitals at B = 0 has
been lifted by SO interaction.36 One can also see that the
order of the two lowest sublevels at B ∼ 0 changes when
Rashba interaction is switched on. Thus, for N = 3 and
α = 0, the two lowest sublevels are (M = −1, Sz = 1/2)
and (M = −1, Sz = −1/2), but this order is reversed
when α = 50 meV·Å. The opposite level order as a func-
tion of α is found for N = 5. This behavior constitutes a
qualitative difference with respect to the N = 1 case in
two aspects. First, the phonon energy (i.e., the energy
of the fundamental spin transition) is no longer given by
the bare Zeeman splitting. Instead, it has a more compli-
cated dependence on the magnetic field, and it is greatly
influenced by the particular values of α and β. This is
apparent in the N = 5 panels, where the energy splitting
between the two lowest states strongly differs depending
on the relative value of α and β. Second, it is possible
to find situations where the ground state at B ∼ 0 has
Sz = −1/2 and the first excited state has Sz = 1/2 (e.g.
N = 3 when α > β or N = 5 when α < β). In these
cases, the Zeeman splitting leads to a weak anticrossing
of the two sublevels (highlighted with dashed circles in
Fig. 2) which has no counterpart in single-electron sys-
tems. This kind of B-induced (i.e., not phonon-induced)
ground state spin mixing, also referred to as “intrinsic
spin mixing”, has been previously reported for singlet-
triplet transitions in N = 2 QDs.58 Here we show that
they may also exist in few-electron QDs with N odd.
(−2,1/2)(1,1/2)
(−1,1/2)
(1,1/2)
(1,1/2)
(−4,1/2)
(−2,1/2)
(−4,1/2)
(−1,1/2)
(−1,1/2)
(−1,1/2)
(1,1/2)
α = 50, β = 25α = 0, β = 25
N = 3
N = 5
 127.0
 127.5
 128.0
 128.5
0.5 1.0 0.0
 236.0
 236.5
 237.0
B (T)
 0.0 0.5 1.0
B (T)
FIG. 2: The four lowest energy levels in a QD with N = 3, 5
interacting electrons, as a function of an axial magnetic field,
without (left column) and with (right column) Rashba SO
interaction. The approximate quantum numbers (M,S) of
the levels are shown, with arrows denoting the spin projection
Sz = 1/2 (↑) and Sz = −1/2 (↓). The dashed circles highlight
the region of intrinsic spin mixing of the ground state.
Figure 1 puts forward yet another qualitative differ-
ence between SO coupling in single- and few-electron
QDs: while in the former low-energy anticrossings are
due to Rashba interaction11,36,52, in few-electron QDs,
when S = 3/2 states come into play, both Rashba and
Dresselhaus terms may induce anticrossings. For exam-
ple, the (M = −1, Sz = 1/2) sublevel couples directly to
both (M = −2, Sz = −1/2) and (M = −2, Sz = 3/2)
sublevels, via the Dresselhaus and Rashba interaction,
respectively. Coupling to S = 3/2 states is a characteris-
tic feature of N > 1 systems, which has important effects
on the spin relaxation rate, as we will discuss below.
B. Spin relaxation between Zeeman sublevels
In Fig. 3 we compare the magnetic field dependence of
the spin relaxation rate between the two lowest Zeeman
sublevels of N = 1, 3, 5. Dashed lines (solid lines) are
used for systems without (with) Rashba interaction.59
While for N = 1 the well-known exponential dependence
with B is found2,6,9, and the main effect of Rashba cou-
pling is to shift the curve upwards (i.e., to accelerate the
relaxation), for N = 3 and N = 5 the relaxation rate ex-
hibits complicated trends which strongly depend on the
values of the SO coupling parameters.
α = 50, β = 25
 α = 0, β = 25
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
B (T)
 0  1  2  3
FIG. 3: Spin relaxation rate in a QD with N = 1, 3, 5 inter-
acting electrons as a function of an axial magnetic field. Solid
(dashed) lines stand for the system with (without) Rashba
interaction. Note the strong influence of the SO interaction
in the shape of the relaxation curve for N > 1.
To understand this result, one has to bear in mind that
in spin relaxation processes two well-distinguished and
complementary ingredients are involved, namely SO in-
teraction and phonon emission. Phonon emission grants
the conservation of energy in the electron relaxation,
but phonons have zero spin and therefore cannot cou-
ple states with different spin. It is the SO interaction
that turns pure spin states into mixed ones, thus enabling
the phonon-induced transition. The overall efficiency of
the scattering event is then given by the combination
of the two phenomena: the phonon emission efficiency
modulated by the extent of the SO mixing. The shape
of spin relaxation curves shown in Fig. 3 can be directly
related to the energy dispersion of the phonon, which cor-
responds to the splitting between the two lowest levels of
the electron spectrum. Thus, for N = 1, the phonon
energy is simply proportional to B through the Zeeman
splitting, but for N = 3 and N = 5 it has a non-trivial
dependence on B, as shown in Fig. 2. Actually, the relax-
ation minima in Fig. 3 are connected with the magnetic
field values where the two lowest levels anticross in Fig. 2.
In these magnetic field windows, in spite of the fact that
SO coupling is strong, the phonon density is so small that
the relaxation rate is greatly suppressed.28 Similarly, the
relaxation rate fluctuations of N = 3 at B ∼ 3 T are
signatures of the anticrossings with high-angular momen-
tum states. For larger fields (B > 3 T), the ground state
approaches the maximum density droplet configuration
and high-spin states are possible.44 In this work, how-
ever, we restrict ourselves to the magnetic field regime
where the ground state is a doublet.
eV∆    (µ     )
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 0 20 40 60 80
 10  
FIG. 4: (Color online). Spin relaxation rate in a QD with
N = 1, 3, 5 interacting electrons as a function of the energy
splitting between the two lowest spin states. Top panel: α =
0, β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25
meV·Å. The relaxation of N = 3 is slower than that of N = 1
for a wide range of ∆12. The irregular data distribution is
due to the irregular relaxation rates vs. magnetic field. For
example, the strongly deviated points of N = 3 come from
the peaks at B ∼ 3 in Fig. 3.
For a more direct comparison between the relaxation
rates of N = 1, 3, 5, in Fig. 4 we replot the data of Fig. 3
as a function of the energy splitting between the two
lowest states, ∆12, without (top panel) and with (bot-
tom panel) Rashba interaction. Since the phonon energy
is identical for all points with the same ∆12, differences
in the relaxation rate arise exclusively from the different
strength of SO interaction. ∆12 is also a relevant pa-
rameter from the experimental point of view, since it is
usually required that it be large enough for the states to
be resolvable. In this sense, it is worth noting that, even
if the inter-level splittings shown in Fig. 4 are fairly small,
a number of experiments have successfully addressed this
regime.5,8,21
A most striking feature observed in the figure is that,
for most values of ∆12, the N = 3 relaxation rate is
clearly slower than the N = 1 one. Likewise, N = 5
shows a similar (or slightly faster) relaxation rate than
N = 1. These are interesting results, for they suggest
that improved spin stability may be achieved using few-
electron QDs instead of the single-electron ones typically
employed up to date.8 At first sight the results are sur-
prising, because the higher density of states in the few-
electron systems implies smaller inter-level spacings, and
hence stronger SO mixing, which should translate into
enhanced relaxation. It then follows that another physi-
cal mechanism must be acting upon the few-electron sys-
tems, which reduces the transition probability between
the initial and final spin states, and may even make it
smaller than for N = 1. Here we propose that such
mechanism is the SO admixture with low-lying quadru-
plet (S = 3/2) states, which become available for N > 1.
By coupling to S = 3/2 levels, the projection of the dou-
blet Sz = 1/2 levels onto Sz = −1/2 ones is reduced,
and this partly inhibitis the transition between the low-
est doublet sublevels.
Let us explain this by comparing the spin transition for
N = 1 and N = 3. For N = 1, the spin configuration of
the initial and final states, in the absence of SO coupling,
is |Sz = −1/2〉 and |Sz = +1/2〉, respectively. The tran-
sition between these states is spin-forbidden. However,
when SO coupling is switched on, the two states become
admixed with higher-lying S = 1/2 states fulfilling the
∆Sz = ±1 condition. The transition between the initial
and final states can then be represented schematically as:
ca |Sz = −1/2〉+cb|Sz = +1/2〉 ⇒ cr |Sz = +1/2〉+cs|Sz = −1/2〉,
where ci are the admixture coefficients (in general ca ≫
cb and cr ≫ cs). Clearly now both spin configurations of
the initial state have a finite overlap with the final state,
and so the transition is possible. Let us next consider
the N = 3 case. In the absence of SO coupling, the
initial and final states are again the Sz = −1/2 and Sz =
+1/2 doublets, respectively, and the transition is spin-
forbidden. When we switch on SO coupling, we note that
the ∆Sz = ±1 condition allows for mixing not only with
Sz = ±1/2 states (either doublets or quadruplets) but
also with Sz = ±3/2 quadruplets, so that the transition
can be represented as:
ca|Sz = −1/2〉+ cb|Sz = +1/2〉+ cc|Sz = −3/2〉 ⇒
cr|Sz = +1/2〉+ cs|Sz = −1/2〉+ ct|Sz = +3/2〉,
where, in general, ca ≫ cb, cc, and cr ≫ cs, ct. In this
case, |Sz = −3/2〉 has no overlap with the final state con-
figurations. Likewise, |Sz = +3/2〉 has no overlap with
the initial state configurations. Therefore, these quadru-
plet configurations are inactive from the point of view
of the transition, and the more important they are (i.e.,
the stronger the SO coupling with quadruplet states), the
less likely the transition is.
To prove this argument quantitatively, in Fig. 5 we il-
lustrate the spin relaxation of N = 3 calculated by diag-
onalization of the SO Hamiltonian including and exclud-
ing the low-lying S = 3/2 states from the basis set. As
expected, when the quadruplets are not considered, the
transition is visibly faster. For N = 5, low-lying S = 3/2
levels are also available, but in this case they barely com-
pensate for the large density of electron states, so that
the overall scattering rate turns out to be comparable to
that of N = 1.
eV∆    (µ     )
without S=3/2
with S=3/2
 10  
 10  
 10  
 10  
 10  
 0 20 40 60 80
FIG. 5: (Color online). Spin relaxation rate in a QD with N =
3 interacting electrons as a function of the energy splitting
between the two lowest spin states. α = 0 and β = 25 meV·Å.
Symbol + (×) stands for SO Hamiltonian diagonalized in a
basis which includes (excludes) S = 3/2 states. Clearly, the
inclusion of S = 3/2 states slows down the relaxation.
To test the robustness of the few-electron spin states
stability predicted above, we also compare the relax-
ation rate of N = 1 and N = 3 in a QD with dif-
ferent confinement, namely h̄ω0 = 6 meV, in Fig. 6.
Since the lateral confinement of the dot is now stronger,
(M = −1, S = 1/2) is the N = 3 ground state up to large
values of the magnetic field (B ∼ 5 T). This allows us
to investigate larger Zeeman splittings (i.e., larger ∆12),
which may be easier to resolve experimentally. As seen
in the figure, the relaxation rate of N = 3 is again slower
than that of N = 1 for a wide range of ∆12, the behavior
being very similar to that of Fig. 4, albeit extended to-
wards larger inter-level spacings. The crossing between
N = 3 and N = 1 relaxation rates at large ∆12 val-
ues, both in Fig. 4 and Fig. 6, is due to the proximity
of high-angular momentum levels coming down in energy
for N = 3 when the magnetic field (and hence the Zee-
man splitting) is large. Such levels bring about strong
SO admixture and thus fast relaxation (see middle panel
of Fig. 3 at B ∼ 3 T).
IV. SPIN RELAXATION IN A QD WITH N
A. Energy structure
When the number of electrons confined in the QD is
even and the magnetic field is not very strong, the ground
and first excited states are usually a singlet (S = 0)
and a triplet (S = 1) with three Zeeman sublevels
eV∆    (µ     )
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 0 40 80 120
FIG. 6: (Color online). Spin relaxation rate in a QD with
N = 1, 3 interacting electrons as a function of the energy
splitting between the two lowest spin states. The QD has
h̄ω0 = 6 meV. Top panel: α = 0, β = 25 meV·Å. Bottom
panel: α = 50 meV·Å, β = 25 meV·Å. As for the weaker-
confined dot of Fig. 4, the relaxation of N = 3 is slower than
that of N = 1 for a wide range of ∆12.
(Sz = +1, 0,−1). Unlike in the previous section, here the
initial and final states of the spin transition may have dif-
ferent orbital quantum numbers, and the inter-level split-
ting ∆12 may be significantly larger (in the meV scale).
Under these conditions, the phonon emission efficiency no
longer exhibits a simple proportionality with the phonon
density, but it further depends on the ratio between the
phonon wavelength and the QD dimensions.50,51 More-
over, SO interaction is sensitive to the quantum numbers
of the initial and final electron states.26,28 Therefore, in
this class of spin transitions the details of the energy
structure are also relevant to determine the relaxation
rate.
In Fig. 7 we plot the energy levels vs. magnetic field for
a QD with N = 2, 4 in the presence of Rashba and Dres-
selhaus interactions. The approximate quantum num-
bers (M,S) of the lowest-lying states are written between
parenthesis. For N = 2 and weak fields, the ground state
is the (M = 0, S = 0) singlet, and the first excited state
is the (M = −1, S = 1) triplet. As in the previous sec-
tion, SO interaction introduces small zero-field splittings
and anticrossings in the energy levels with |M | > 0.36
As a consequence, when α > β, the zero-field ordering of
the (M = −1, S = 1) Zeeman sublevels is such that they
anticross in the presence of an external magnetic field.
This anticrossing is highlighted in the figure by a dashed
circle. On the other hand, as B increases the singlet-
triplet energy spacing is gradually reduced, and then the
singlet experiences a series of weak anticrossings with all
(−1,1)
(−2,0)
(−3,1)
(0,1)
(0,0)
 0  1  2  3
B (T)
FIG. 7: Low-lying energy levels in a QD with N = 2, 4 in-
teracting electrons as a function of an axial magnetic field.
α = 50 meV· Å and β = 25 meV· Å. The approximate
quantum numbers (M,S) of the lowest states are shown. The
dashed circle in N = 2 highlights the anticrossing between
M = −1 Zeeman sublevels.
three Zeeman sublevels of the triplet. These anticross-
ings are due to the fact that (M = 0, S = 0, Sz = 0)
couples to the (M = −1, S = 1, Sz = −1) sublevel via
Dresselhaus interaction, to the (M = −1, S = 1, Sz =
+1) sublevel via Rashba interaction, and finally to the
(M = −1, S = 1, Sz = 0) sublevel indirectly through
higher-lying states.26,28
For N = 4, the density of electronic states is larger
than for N = 2, which again reflects in a larger magni-
tude of the anticrossings gaps due to the enhanced SO
interaction. The ground state at B = 0 is a triplet,
(M = 0, S = 1), but soon after it anticrosses with a
singlet, (M = −2, S = 0). After this, and before the
formation of Landau levels, two different branches of the
first excited state can be distinguished: when B < 1 T,
the first excited state is (M = 0, S = 1), and when B > 1
T it is (M = −3, S = 1). It is worth pointing out that
the complexity of the N = 4 spectrum, as compared to
the simple N = 2 one, implies a greater flexibility to
select initial and final spin states by means of external
fields. As we shall discuss below, this degree of freedom
has important consequences on the relaxation rate.
B. Triplet-singlet spin relaxation
In a recent work, we have investigated the magnetic
field dependence of the TS relaxation due to SO cou-
pling and phonon emission in N = 2 and N = 4 QDs.28.
Here we study this kind of transition from a different
perspective, namely we compare the spin relaxation of
two- and four-electron systems in order to highlight the
changes introduced by inter-electron repulsion. Increas-
ing the number of electrons confined in the QD has three
important consequences on the TS transition. First, it
increases the density of electronic states (and then the
SO mixing), leading to faster relaxation. Second, as
mentioned in the previous section, it introduces a wider
choice of orbital quantum numbers for the singlet and
triplet states. Third, it increases the strength of elec-
tronic correlations. Since now the initial and final spin
states have different orbital wave functions, the latter
factor effectively reduces phonon scattering, in a similar
fashion to charge relaxation processes33 (this effect has
been recently pointed out in Ref. 30 as well). To find out
the overall combined effect of these three factors, in this
section we analyze quantitative simulations of correlated
We focus on the magnetic field regions where the
ground state is a singlet and the excited state is a triplet.
A complete description of the TS transition should then
include spin relaxation between the Zeeman-split sub-
levels of the triplet. However, for the weak fields we con-
sider this relaxation is orders of magnitude slower than
the TS one (compare Figs. 3 and 8),60 the reason for
this being the small Zeeman energy and the fact that the
Zeeman sublevels are not directly coupled by Rashba and
Dresselhaus terms, as mentioned in Section III. There-
fore, it is a good approximation to assume that all three
triplet Zeeman sublevels are equally populated and they
relax directly to the singlet.26
α = 50, β = 25
 α = 0, β = 25
 10  
 10  
 10  
 10  
 10  
 10  
 0.5  1  1.5  2  2.5  3
B (T)
FIG. 8: Spin relaxation rate in a QD with N = 2, 4 interact-
ing electrons as a function of an axial magnetic field. Solid
(dashed) lines stand for the system with (without) Rashba
interaction. The relaxation of N = 4 when B < 1 T is slower
than that of N = 2.
Figure 8 represents the TS relaxation rate in a QD with
N = 2, 4, after averaging the relaxation from the three
triplet sublevels. Solid (dashed) lines stand for the case
with (without) Rashba interaction.59 The main effect of
Rashba and Dresselhaus interactions is to accelerate the
spin transition by shifting the relaxation curve upwards.
This is in contrast to the N -odd case, where these terms
may induce drastic changes in the shape of the relaxation
rate curve (see Fig. 3). Figure 8 also reveals a different
behavior of the N = 2 and N = 4 TS relaxation rates.
The former increases gradually with B and then drops
in the vicinity of the TS anticrossing, due to the small
phonon energies.28,29,30 Conversely, for N = 4 an addi-
tional feature is found, namely an abrupt step at B ∼ 1.
This is due to the change of angular momentum of the
excited triplet. For B < 1 T the triplet has M = 0, and
for B > 1 T it has M = −3. Since the ground state
is a singlet with M = −2, the M = 0 triplet does not
fulfill the ∆M = ±1 condition for linear SO coupling.
This inhibits direct spin mixing between initial and final
states and reduces the relaxation rate by about one order
of magnitude.28
meV∆    (         )
N=4, M=0
N=4, M=−3
  10  
 10  
 10  
 10  
 10  
  10  
  10  
 0  0.4  0.8  1.2
FIG. 9: (Color online). Spin relaxation rate in a QD with
N = 2, 4 interacting electrons as a function of the energy
spacing between the singlet and the triplet. Here M stands
for the angular momentum of the triplet. Top panel: α = 0,
β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25 meV·Å.
The relaxation of N = 4 is comparable to that of N = 2 when
the triplet has M = −3, and it is much smaller when M = 0.
Noteworthy, the choice of states differing in more than
one quantum of angular momentum is only possible for
N > 2 QDs. One may then wonder if it is more conve-
nient to use these systems instead of the N = 2 ones dom-
inating the experimental literature up to date20,21,29, i.e.
if it compensates for the increased density of electronic
states. Interestingly, Fig. 8 predicts slower relaxation for
the N = 4 QD with M = 0 triplet than for N = 2. To
verify that this arises from weakend SO coupling rather
than from different phonon energy values, in Fig. 9 we
replot the spin relaxation rate of N = 2, 4 as a function
of the TS energy splitting. In the figure, the upper and
bottom panels represent the situations without and with
Rashba interaction, respectively. While N = 4 shows
similar relaxation rate to N = 2 when the triplet has
M = −3, the relaxation is slower by about one order
of magnitude when the triplet has M = 0. This result
indicates that the weakening of SO mixing due to the
violation of the ∆M = ±1 condition clearly exceeds the
strengthening due to the higher density of states, con-
firming that N = 4 systems are more attractive than
N = 2 ones to obtain long triplet lifetimes. We also
point out that, in spite of the different density of states,
the relaxation rate of N = 2 and N = 4, M = −3 triplets
is quite similar. This can be ascribed to the phonon scat-
tering reduction by electronic correlations,33 which may
also explain the fact that experimentally resolved TS re-
laxation rates of N = 8 QDs and N = 2 QDs be quite
similar.20,31
V. COMPARISON WITH N = 2 EXPERIMENTS
Whereas, to our knowledge, no experiments have mea-
sured transitions between Zeeman-split sublevels in N >
1 systems yet, a number of works have dealt with TS re-
laxation in QDs with few interacting electrons. In Ref. 28
we showed that our model correctly predicts the trends
observed in experiments with N = 2 and N = 8 QDs
subject to axial magnetic fields.20,21,31 In this section, we
extend the comparison to new experiments available for
N = 2 TS relaxation in QDs,29 which for the first time
provide continuous measurements of the average triplet
lifetime against axial magnetic fields, from B = 0 to the
vicinity of the TS anticrossing. By using a simple model,
the authors of the experimental work showed that the
measuraments are in clear agreement with the behavior
expected from SO coupling plus acoustic phonon scatter-
ing. However, in such model: (i) the TS energy splitting
was a taken directly from the experimental data, (ii) the
SO coupling effect was accounted for by parametrizing
the admixture of the lowest singlet and triplet states only,
and (iii) the B-dependence of the SO-induced admix-
ture was neglected. Approximation (ii) may overlook the
correlation-induced reduction of phonon scattering,30,33
that we have shown above to be significant, and which
may have an important contribution from higher excited
states in weakly-confined QDs. In turn, approximation
(iii) may overlook the important influence of SO coupling
in the B-dependence of the triplet lifetime, as we had
anticipated in Ref. 28. Here we compare with the exper-
imental findings using our model, which includes these
effects properly. We assume a QD with an effective well
width Lz = 30 nm, as expected by Ref. 29 authors, and
a lateral confinement parabola of h̄ω0 = 2 meV which,
as we shall see next, fits well the position of the TS an-
ticrossing. Yet, the comparison is limited by the lack of
detailed information about the Rashba and Dresselhaus
interaction constants, and because we deal with circular
QDs instead of elliptical ones (the latter effect introduces
simple deviations from the circular case26). In addition,
in the experiment a tilted magnetic field of magnitude
B∗, forming an angle of 68◦ with the vertical direction
was used. Here we consider the vertical component of
the field (B = 0.37B∗), which is the main responsible
for the changes in the energy structure, and the effect of
the in-plane component enters via the Zeeman splitting
only.
Figure 10 illustrates the average triplet lifetime for
N = 2. The bottom axis shows the vertical magnetic
field B value, while the top axis shows the value to be
compared with the experiment B∗.59 As can be seen, the
triplet lifetime first decreases with the field and then it
abruptly increases in the vicinity of the TS anticross-
ing, due to the small phonon density.28 This behavior
is in clear agreement with the experiment (cf. Fig. 3 of
Ref. 29). The position of the anticrossing (B∗ ∼ 2.9 T) is
also close to the experimental value (B∗ ∼ 2.8 T), which
confirms that that h̄ω0 = 2 meV is similar to the mean
confinement frequency of the experimental sample. A
departure from the experimental trend appears at weak
fields (B < 0.5 T), where we observe a continuous in-
crease of T1 with decreasing B, while the experiment re-
ports a plateau. This is most likely due to the ellipticity
of the experimental sample, which renders the electron
states (and consequently the relaxation rate) insensitive
to the field in the B∗ = 0 − 0.5 T region (see Fig. 1a
in 29). In any case, Fig. 10 clearly confirms the role of
phonon-induced relaxation in the experiments, using a
realistic model for the description of correlated electron
states, SO admixture and phonon scattering.
A comment is worth here on the magnitude of the SO
coupling terms. In Fig. 10, we obtain good agreement
with the experimental relaxation times by using small
values of the SO coupling parameters. In particular, a
close fit is obtained using β = 1, α = 0.5 meV·Å, which
yields a spin-orbit length λSO = 48 µm. This value,
which coincides with the experimental guess (λSO ≈ 50
µm), indicates that SO coupling is several times weaker
than that reported for other GaAs QDs.8 Typical GaAs
parameters are often larger. For instance, measuraments
of the Rashba and Dresselhaus constants by analysis of
the weak antilocalization in clean GaAs/AlGaAs two-
dimensional gases revealed α = 4−5 meV·Å, and γc = 28
eV·Å3 (i.e, β = 3 meV·Å for our quantum well of Lz = 30
nm).61 To be sure, the small SO coupling parameters in
the experiment have a major influence on the lifetime
scale. Compare e.g. the β = 1 and β = 5 meV·Å curves
in Fig. 10. Actually, we note that accurate comparison
with the timescale reported for other GaAs samples31 is
also possible within our model, but assuming stronger
SO coupling constants.28 In Ref. 29, it was suspected
that the weak SO coupling inferred from the experimen-
tal data could be the result of the exclusion of higher
orbitals and the magnetic field dependence of SO ad-
α = 0.5 β = 1
β = 5
β = 2
β = 1 α = 0
B (T)
 0.2  0.4  0.6  0.8  1 0
 0.53 1.58 2.11 2.631.05
B (T)
FIG. 10: Average triplet lifetime in a QD with N = 2 elec-
trons as a function of an axial magnetic field. Only the field
region before the TS anticrossing is shown. α and β are in
meV·Å units. B is the applied axial magnetic field, and B∗
is the equivalent tilted magnetic field, for comparison with
Ref. 29 experiment.
mixture in their model (higher states reduce the effective
SO coupling constants by decreasing the phonon-induced
scattering30,33). Here we have considered both these ef-
fects and still small SO coupling constants are needed to
reproduce the experiment. Therefore, understanding the
origin of their small value remains as an open question.
One possibility could be that the particular direction of
the tilted magnetic field used in the experiment corre-
sponded to a reduced degree of SO admixture.30
VI. CONCLUSIONS
We have investigated theoretically the energy structure
and spin relaxation rate of weakly-confined QDs with
N = 1 − 5 interacting electrons, subject to axial mag-
netic fields, in the presence of linear Rashba and Dressel-
haus SO interactions. It has been shown that the num-
ber of electrons confined in the dot introduces changes
in the energy spectrum which significantly influence the
intensity of the SO admixture, and hence the spin re-
laxation. In general, the larger the number of confined
carriers, the higher the density of electronic states. This
decreases the energy splitting between consecutive lev-
els and then enhances SO admixture, which should lead
to faster spin relaxation. However, we find that this is
not necessarily the case, and slower relaxation rate may
be found for few-electron QDs as compared to the usual
single and two-electron QDs used up to date. The physi-
cal mechanisms responsible for this have been identified.
For N -odd systems, when the spin transition takes place
between Zeeman-split sublevels, it is the presence of low-
energy S = 3/2 states for N > 1 that reduces the pro-
jection of the doublet Sz = 1/2 sublevels into Sz = −1/2
ones, thus partly inhibiting the spin transition. For N -
even systems, when the spin transition takes place be-
tween triplet and singlet levels, there are two underlying
mechanisms. On the one hand, electronic correlations
tend to reduce phonon emission efficiency. On the other
hand, for N > 2 a magnetic field can be used to se-
lect a pair of singlet-triplet states which do not fulfill
the ∆M = ±1 condition of direct SO admixture, which
significantly weakens the SO mixing.
Last, we have compared our estimates with recent
experimental data for TS relaxation in N = 2 QDs.29
Our results support the interpretation of the experi-
ment in terms of SO admixture plus acoustic phonon
scattering, even though quantitative agreement with the
experiment requires assuming much weaker SO coupling
than that reported for similar GaAs structures.
Acknowledgments
We acknowledge support from the Italian Ministry
for University and Scientific Research under FIRB
RBIN04EY74, Cineca Calcolo parallelo 2006, and
Marie Curie IEF project NANO-CORR MEIF-CT-2006-
023797.
∗ Electronic address: climente@unimore.it;
URL: www.nanoscience.unimore.it
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44 P. Lucignano, B. Jouault, and A. Tagliacozzo, Phys. Rev.
B 69, 045314 (2004).
45 The (M,S, Sz) quantum numbers of few-electron states are
a good approximation for the lowest-lying states only. For
higher-lying states, the energy spectrum becomes denser
and the SO interaction becomes very strong even for GaAs,
which leads to important departures from the SO-free pic-
ture. This does not occur in single-electron parabolic QDs
because the energy levels are equally spaced.
46 The convenience of using exact diagonalization procedures,
instead of perturbational approaches, to account for the SO
coupling in GaAs QDs has been claimed in Ref. 10.
47 C.S. Ting (ed.), Physics of Hot Electron Transport in Semi-
conductors, (World Scientific, 1992).
48 Landolt-Börnstein: Numerical Data and Functional Rela-
tionships in Science and Technology, Vol. 17. Semiconduc-
tors, Group IV Elements and III-V Compounds, edited by
O. Madelung, (Springer-Verlag, 1982).
49 M. Cardona, N.E. Christensen, and G. Fasol, Phys. Rev.
B 38, 1806 (1988).
50 U. Bockelmann, Phys. Rev. B 50, 17271 (1994).
51 J.I. Climente, A. Bertoni, G. Goldoni, and E. Molinari,
Phys. Rev. B 74, 035313 (2006).
52 P. Stano, and J. Fabian, Phys. Rev. B 72, 155410 (2005).
53 O. Voskoboynikov, C.P. Lee, and O. Tretyak, Phys. Rev.
B 63, 165306 (2001).
54 W.H. Kuan, and C.S. Tang, J. Appl. Phys. 95, 6368 (2004).
55 The energy magneto-spectrum of GaAs parabolic QDs
with SO interaction and up to four interacting electrons
was also investigated in Ref. 35, but considering Rashba
interaction only.
56 Coulomb-enhanced SO interaction was previously pre-
dicted for higher-dimensional structures.57 Here we report
it for QDs.
57 G.H. Chen, and M.E. Raikh, Phys. Rev. B 60, 4826 (1999).
58 C.F. Destefani, S.E. Ulloa, and G.E. Marques, Phys. Rev.
B 69, 125302 (2004).
59 For simplicity of the discussion, in Figs. 3, 8 and 10, the
near vicinity of B = 0 T is not shown. In that range one
finds damped phonon-induced relaxation rates due to de-
generacies arising from the time-reversal symmetry and the
circular symmetry of the confinement we have assumed.
We do not expect these features to be observable in exper-
iments, because QDs are not perfectly circular and because
hyperfine interaction is expected to be the dominant spin
relaxation mechanism for very weak fields (see Refs. 18,23).
60 Greatly suppressed TS spin relaxation, comparable to that
of inter-Zeeman sublevels at very weak B, may be achieved
by means of geometrically or field-induced acoustic phonon
emission minima.27,28
61 J.B. Miller, D.M. Zumbühl, C.M. Marcus, Y.B. Lyanda-
Geller, D. Goldhaber-Gordon, K. Campman, and A.C.
Gossard, Phys. Rev. Lett. 90, 076807 (2003).
http://www.s3.infm.it/donrodrigo

  We estimate the spin relaxation rate due to spin-orbit coupling and acoustic
phonon scattering in weakly-confined quantum dots with up to five interacting
electrons. The Full Configuration Interaction approach is used to account for
the inter-electron repulsion, and Rashba and Dresselhaus spin-orbit couplings
are exactly diagonalized. We show that electron-electron interaction strongly
affects spin-orbit admixture in the sample. Consequently, relaxation rates
strongly depend on the number of carriers confined in the dot. We identify the
mechanisms which may lead to improved spin stability in few electron (>2)
quantum dots as compared to the usual one and two electron devices. Finally, we
discuss recent experiments on triplet-singlet transitions in GaAs dots subject
to external magnetic fields. Our simulations are in good agreement with the
experimental findings, and support the interpretation of the observed spin
relaxation as being due to spin-orbit coupling assisted by acoustic phonon
emission.

Effect of electron-electron interaction on the phonon-mediated spin relaxation in
quantum dots
Juan I. Climente,1, ∗ Andrea Bertoni,1 Guido Goldoni,1, 2 Massimo Rontani,1 and Elisa Molinari1, 2
1CNR-INFM National Center on nanoStructures and bioSystems at Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy
2Dipartimento di Fisica, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy
(Dated: October 21, 2018)
We estimate the spin relaxation rate due to spin-orbit coupling and acoustic phonon scattering
in weakly-confined quantum dots with up to five interacting electrons. The Full Configuration
Interaction approach is used to account for the inter-electron repulsion, and Rashba and Dresselhaus
spin-orbit couplings are exactly diagonalized. We show that electron-electron interaction strongly
affects spin-orbit admixture in the sample. Consequently, relaxation rates strongly depend on the
number of carriers confined in the dot. We identify the mechanisms which may lead to improved
spin stability in few electron (> 2) quantum dots as compared to the usual one and two electron
devices. Finally, we discuss recent experiments on triplet-singlet transitions in GaAs dots subject
to external magnetic fields. Our simulations are in good agreement with the experimental findings,
and support the interpretation of the observed spin relaxation as being due to spin-orbit coupling
assisted by acoustic phonon emission.
PACS numbers: 73.21.La,71.70.Ej,72.10.Di,73.22.Lp
I. INTRODUCTION
There is currently interest in manipulating electron
spins in quantum dots (QDs) for quantum information
and quantum computing purposes.1,2,3 A major goal in
this research line is to optimize the spin relaxation time
(T1), which sets the upper limit of the spin coherence
time (T2): T2 ≤ 2T1.4 Therefore, designing two-level
spin systems with long spin relaxation times is an im-
portant step towards the realization of coherent quantum
operations and read-out measuraments. Up to date, spin
relaxation has been investigated almost exclusively in
single-electron4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 and two-
electron20,21,22,23,24,25,26,27,28,29,30 QDs. Spin relaxation
in QDs with a larger number of electrons has seldom been
considered28,31, even though Coulomb blockade makes
it possible to control the exact number of carriers con-
fined in a QD.32 Yet, recent theoretical works suggest
that Coulomb interaction renders few-electron charge de-
grees of freedom more stable than single-electron ones33,
which leads to the question of whether similar findings
hold for spin degrees of freedom. Moreover, in weakly-
confined QDs, acoustic phonon emission assisted by spin-
orbit (SO) interaction has been identified as the domi-
nant spin relaxation mechanism when cotunneling and
nuclei-mediated relaxation are reduced.6,8,31 The com-
bined effect of Coulomb interaction and SO coupling has
been shown to influence the energy spectrum of few-
electron QDs profoundly,34,35,36 but the consequences on
the spin relaxation remain largely unexplored.37
In Ref. 28 we investigated the effect of a magnetic
field on the triplet-singlet (TS) spin relaxation in two
and four-electron QDs with SO coupling, so as to under-
stand related experimental works. Motivated by the very
different response observed for different number of con-
fined particles, in this work we shall focus on the role of
electron-electron interaction in spin relaxation processes,
extending our analysis to different number of carriers,
highlighting, in particular, the different physics involved
in even and odd number of confined electrons. Further-
more, we will explicitly compare the predictions of our
theoretical model with very recent experiments on spin
relaxation in two-electron GaAs QDs.29
We study theoretically the energy structure and spin
relaxation of N interacting electrons (N = 1 − 5) in
parabolic GaAs QDs with SO coupling, subject to axial
magnetic fields. Both Rashba38 and Dresselhaus39 SO
terms are considered, and the electron-electron repulsion
is accounted for via the Full Configuration Interaction
method.40,41 By focusing on the two lowest spin states,
two different classes of systems are distinguished. For N
odd (1,3,5) and weak magnetic fields, the ground state is
a doublet and then the two-level system is defined by the
Zeeman-split sublevels of the lowest orbital. For N even
(2,4), the two-level system is defined by a singlet and a
triplet. We analyze these two classes of systems sepa-
rately because, as we shall comment below, the physics
involved in the spin transition differs. Thus, we compare
the phonon-induced spin relaxation of N = 1, 3, 5 elec-
trons and that of N = 2, 4 separately. As a general rule,
the larger the number of confined carriers, the stronger
the SO mixing, owing to the increasing density of elec-
tronic states. This would normally yield faster relax-
ation rates. However, we note that this is not necessarily
the case, and few-electron states may display compara-
ble or even slower relaxation than their single-electron
and two-electron counterparts. This is due to charac-
teristic features of the few-particle energy spectra which
tend to weaken the admixture between the initial and
final spin states. In N -odd systems, it is the presence
of low-energy quadruplets for N > 1 that reduces the
admixture between the Zeeman sublevels of the (dou-
blet) ground state, hence inhibiting the spin flipping. In
N -even systems, electronic correlations partially quench
http://arxiv.org/abs/0704.0868v2
phonon emission33, and the relaxation can be further sup-
pressed forN > 2 if one selects initial and final spin states
differing in more than one quantum of angular momen-
tum, which inhibits direct triplet-singlet SO mixing via
linear Rashba and Dresselhaus SO terms.28 Noteworthy,
all these effects are connected with Coulomb interaction
between confined carriers.
The paper is organized as follows. In Section II we give
details about the theoretical model we use. In Section
III we study the energy structure and spin relaxation of
a QD with an odd number of electrons (N = 1, 3, 5). In
Section IV we do the same for QDs with an even number
of electrons (N = 2, 4). In Section V we compare our
numerical simulations with experimental data recently
reported for N = 2 GaAs QDs. Finally, in Section VI we
present the conclusions of this work.
II. THEORY
We consider weakly-confined GaAs/AlGaAs QDs,
which are the kind of samples usually fabricated
by different groups to investigate spin relaxation
processes.7,8,20,22 In these structures, the dot and the sur-
rounding barrier have similar elastic properties, and the
lateral confinement (which we approximate as circular)
is much weaker than the vertical one. A number of use-
ful approximations can be made for such QDs. First,
since the weak lateral confinement gives inter-level spac-
ings within the range of few meV, only acoustic phonons
have significant interaction with bound carriers, while op-
tical phonons can be safely neglected. Second, the elasti-
cally homogeneous materials are not expected to induce
phonon confinement, which allows us to consider three-
dimensional bulk phonons. Finally, the different energy
scales of vertical and lateral electronic confinement allow
us to decouple vertical and lateral motion in the building
of single-electron spin-orbitals. Thus, we take a parabolic
confinement profile in the in-plane (x, y) direction, with
single-particle energy gaps h̄ω0, which yields the Fock-
Darwin states.42 In the vertical direction (z) the confine-
ment is provided by a rectangular quantum well of width
Lz and height determined by the band-offset between
the QD and barrier materials (the zero of energy is then
the bottom of the conduction band). The quantum well
eigenstates are derived numerically. In cylindrical coor-
dinates, the single-electron spin-orbitals can be written
ψµ(ρ, θ, z; sz) =
eimθ Rn,m(ρ) ξ0(z)χsz , (1)
where ξ0 is the lowest eigenstate of the quantum well,
χsz is the spinor eigenvector of the spin z-component
with eigenvalue sz, and Rn,m is the n−th Fock-Darwin
orbital with azimuthal angular momentum m,
Rn,m(ρ) =
(n+ |m|)!
0 L|m|n
In the above expression L|m|n denotes a generalized La-
guerre polynomial and l0 =
h̄/m∗ω0 is the effective
length scale, with m∗ standing for the electron effec-
tive mass. The energy of the single-particle Fock-Darwin
states is given by En,m = (2n + 1 + |m|)h̄Ωc + m2 h̄ωc,
where ωc =
is the cyclotron frequency and Ωc =
+ (ωc/2)2 is the total (spatial plus magnetic) con-
finement frequency.
With regard to Coulomb interaction, we need to go
beyond mean field approximations in order to properly
include electronic correlations, which play an important
role in determining the phonon-induced electron scatter-
ing rate.43 Moreover, since we are interested in the re-
laxation time of excited states, we need to know both
ground and excited states with comparable accuracy.
Our method of choice is the Full Configuration Interac-
tion approach: the few-electron wave functions are writ-
ten as linear combinations |Ψa〉 =
cai|Φi〉, where the
Slater determinants |Φi〉 = Πµic†µi |0〉 are obtained by
filling in the single-electron spin-orbitals µ with the N
electrons in all possible ways consistent with symmetry
requirements; here c†µ creates an electron in the level µ.
The fully interacting Hamiltonian is numerically diago-
nalized, exploiting orbital and spin symmetries.40,41 The
few-electron states can then be labeled by the total az-
imuthal angular momentumM = 0,±1,±2 . . ., total spin
S and its z-projection Sz.
The inclusion of SO terms is done following a similar
scheme to that of Ref. 44, although here we consider
not only Rashba but also linear Dresselhaus terms. For
a quantum well grown along the [001] direction, these
terms read:38,39
HR = α
(kysx − kxsy), (3)
HD = γc 〈k2z〉(kysy − kxsx), (4)
where α and γc are coupling constants, while sj and kj
are the j-th Cartesian projections of the electron spin and
canonical momentum, respectively, along the main crys-
talographic axes (〈k2z〉 = (π/Lz)2 for the lowest eigen-
state of the quantum well). The momentum operator
includes a magnetic field B applied along the vertical di-
rection z. Other SO terms may also be present in the
conduction band of a QD, such as the contribution aris-
ing from the system inversion asymmetry in the lateral
dimension or the cubic Dresselhaus term. However, in
GaAs QDs with strong vertical confinement, HR and HD
account for most of the SO interaction.36
We rewrite Eqs.(3,4) in terms of ladder operators as:
HR = α
(k+s− − k−s+), (5)
HD = β
(k+s+ + k−s−), (6)
where k± and s± change m and sz by one quantum,
respectively, and β = γc (π/Lz)
2 is the Dresselhaus in-
plane coupling constant. It is worth mentioning that
when only Rashba (Dresselhaus) coupling is present, the
total angular momentum j = m + sz (j = m − sz)
is conserved. However, in the general case, when both
coupling terms are present and α 6= β, all symmetries
are broken. Still, SO interaction in a large-gap semi-
conductor such as GaAs is rather weak, and the low-
lying states can be safely labelled by their approximate
quantum numbers (M,S, Sz) except in the vicinity of the
level anticrossings.11,26,45 Since the few-electron M and
Sz quantum numbers are given by the algebraic sum of
the single-particle states m and sz quantum numbers, it
is clear from Eqs. (5,6) that Rashba interaction mixes
(M,Sz) states with (M ± 1, Sz ∓ 1) ones, while Dressel-
haus interaction mixes (M,Sz) with (M ± 1, Sz ± 1).
The SO terms of Eqs. (5,6) can be spanned on a basis of
correlated few-electron states.46 The SO matrix elements
are then given by sums of single-particle contributions of
the form:
〈n′m′ s′z| HR +HD |nmsz〉 =
C∗R O+n′m′ nm δm′ m+1 δs′z sz−1+CR O
n′m′ nm δm′ m−1 δs′z sz+1+
C∗D O+n′m′ nm δm′ m+1 δs′z sz+1+CD O
n′m′ nm
δm′ m−1 δs′
sz−1.
Here CR = α and CD = −iβ are constans for the Rashba
and Dresselhaus interactions respectively, andO± are the
form factors:
Rnm(t),
Rnm(t),
with t = ρ2/l20. The above forms factors have analytical
expressions which depend on the set of quantum num-
bers {n′m′, nm}. The resulting SO-coupled eigenvec-
tors are then linear combinations of the correlated states,
|ΨSOA 〉 =
cAa|Ψa〉.
We assume zero temperature, which suffices to capture
the main features of one-phonon processes.9,16 Indeed, it
is one-phonon processes that account for most of the low-
temperature experimental observations in the SO cou-
pling regime.2,6,8,28,29,31 We evaluate the relaxation rate
between the initial (occupied) and final (empty) states
of the SO-coupled few-electron state, B and A, using the
Fermi Golden Rule:
τ−1B→A =
c∗BbcAa
c∗bicaj〈Φi|Vνq|Φj〉
δ(EB−EA−h̄ωq),
where the electron states |ΨSOK 〉 (K = A,B) have been
written explicitly as linear combinations of Slater deter-
minants, EK stands for the K electron state energy and
h̄ωq represents the phonon energy. Vνq is the interac-
tion operator of an electron with an acoustic phonon of
momentum q via the mechanism ν, which can be either
deformation potential or piezoelectric field interaction.
Details about the electron-phonon interaction matrix el-
ements can be found elsewhere.33
In this work we study a GaAs/Al0.3Ga0.7As QDs, us-
ing the following material parameters:47 electron effective
massm∗ = 0.067, band-offset Vc = 243 meV, crystal den-
sity d = 5310 kg/m3, acoustic deformation potential con-
stant D = 8.6 eV, effective dielectric constant ǫ = 12.9,
and piezoelectric constant h14 = 1.41 · 109 V/m. The
Landé factor is g = −0.44.5 As for GaAs sound speed, we
take cLA = 4.72 · 103 m/s for longitudinal phonon modes
and cTA = 3.34 · 103 m/s for transversal modes.48 Unless
otherwise stated, a lateral confinement of h̄ω0 = 4 meV
and a quantum well width of Lz = 10 nm are assumed
for the QD under study, and a Dressehlaus coupling pa-
rameter γc = 25.5 eV·Å3 is taken49, so that β ≈ 25
meV·Å. The value of the Rashba coupling constant can
be modulated externally e.g. with external electric fields.
Here we will investigate systems both with and without
Rashba interaction. When present, we shall mostly con-
sider α = 50 meV·Å, to represent the case where Rashba
effects prevail over Dresselhaus ones.
Few-body correlated states (M,S, Sz) are obtained us-
ing a basis set composed by the Slater determinants
(SDs) which result from all possible combinations of 42
single-electron spin-orbitals (i.e., from the six lowest en-
ergy shells of the Fock-Darwin spectrum at B = 0) filled
with N electrons. For N = 5, this means that the basis
rank may reach ∼ 2 · 105. The SO Hamiltonian is then
diagonalized in a basis of up to 56 few-electron states,
which grants a spin relaxation convergence error below
2%. Since SO terms break the spin and angular mo-
mentum symmetries, the SO-coupled states |ΨSOK 〉 are
described by a linear combination of SDs coming from
different (M,S, Sz) subspaces. Thus, for N = 5, the
states are described by up to ∼ 8.5 · 105 SDs. To evalu-
ate the electron-phonon interaction matrix elements, we
note that only a small percentage of the huge number
of possible pairs of SDs (∼ 7 · 1011 for N = 5) may
give non-zero matrix elements, owing to spin-orbital or-
thogonalities. We scan all pairs of SDs and filter those
which may give non-zero matrix elements writing the de-
terminants in binary representation and using efficient
bit-per-bit algorithms.40,41 The matrix elements of the
remaining pairs (∼ 2 ·106 for N = 5) are evaluated using
massive parallel computation.
 0 1 3
B (T)
FIG. 1: Low-lying energy levels in a QD with N = 1, 3, 5
interacting electrons, as a function of an axial magnetic field.
The SO interaction coefficients are α = 50 meV· Å and β =
25 meV· Å. The dot has h̄ω0 = 4 meV and Lz = 10 nm. Note
the increasing size of the SO-induced anticrossing gaps and
zero-field splittings with increasing N .
III. SPIN RELAXATION IN A QD WITH N ODD
A. Energy structure
When the number of electrons confined in the QD is
odd and the magnetic field is weak enough, the ground
and first excited states are usually the Zeeman sz = 1/2
and sz = −1/2 sublevels of a doublet [Fig. 1]. Since the
initial and final spin states belong to the same orbital,
∆M = 0 and SO mixing (which requires ∆M = ±1)
is only possible with higher-lying states. In addition,
the phonon energy (corresponding to the electron tran-
sition energy) is typically small (in the µeV scale). In
this case, the relaxation rate is determined essentially by
the phonon density, the strength and nature of the SO
interaction, and the proximity of higher-lying states.9,11
In order to gain some insight on the influence of these
factors, in Fig. 1 we compare the energy structure of a
QD with N = 1, 3, 5 vs. an axial magnetic field, in the
presence of Rashba and Dresselhaus interactions.55 One
can see that the increasing number of particles changes
the energy magneto-spectrum drastically. This is be-
cause the quantum numbers of the low-lying energy levels
change, resulting in a different field dependence, and be-
cause Coulomb interaction leads to an increased density
of electron states, as well as to a more complicated spec-
trum.
At first sight, the energy spectra of Fig. 1 closely resem-
ble those in the absence of SO effects. For instance, the
N = 1 spectrum is very similar to the pure Fock-Darwin
spectrum.42 Rashba and Dresselhaus interactions were
expected to split the degenerate |m| > 0 shells at B = 0,
shift the positions of the level crossings and turn them
into anticrossings36,52,53,54, but here such signatures are
hardly visible because SO interaction is weak in GaAs.
In fact, the magnitude of the SO-induced zero-field en-
ergy splittings and that of the anticrossing gaps is of very
few µeV, and SO effects simply add fine features to the
N = 1 spectrum.52
A significantly different picture arises in the N = 3
and N = 5 cases. Here, the increased density of elec-
tronic states enhances SO mixing as compared to the
single-electron case.56 As a result, the anticrossing gaps
can be as large as 30 µeV (N = 3) and 60 µeV (N = 5).
Moreover, unlike in the N = 1 case, where the ground
state orbital has m = 0, here it has |M | = 1. Therefore,
the Zeeman sublevels involved in the fundamental spin
transition are subject to SO-induced zero-field splittings.
To illustrate this point, in Fig. 2 we zoom in on the energy
spectrum of the four lowest states of N = 3 and N = 5
under weak magnetic fields, without (left panels) and
with (right panels) Rashba interaction. Clearly, the four-
fold degeneracy of |M | = 1 spin-orbitals at B = 0 has
been lifted by SO interaction.36 One can also see that the
order of the two lowest sublevels at B ∼ 0 changes when
Rashba interaction is switched on. Thus, for N = 3 and
α = 0, the two lowest sublevels are (M = −1, Sz = 1/2)
and (M = −1, Sz = −1/2), but this order is reversed
when α = 50 meV·Å. The opposite level order as a func-
tion of α is found for N = 5. This behavior constitutes a
qualitative difference with respect to the N = 1 case in
two aspects. First, the phonon energy (i.e., the energy
of the fundamental spin transition) is no longer given by
the bare Zeeman splitting. Instead, it has a more compli-
cated dependence on the magnetic field, and it is greatly
influenced by the particular values of α and β. This is
apparent in the N = 5 panels, where the energy splitting
between the two lowest states strongly differs depending
on the relative value of α and β. Second, it is possible
to find situations where the ground state at B ∼ 0 has
Sz = −1/2 and the first excited state has Sz = 1/2 (e.g.
N = 3 when α > β or N = 5 when α < β). In these
cases, the Zeeman splitting leads to a weak anticrossing
of the two sublevels (highlighted with dashed circles in
Fig. 2) which has no counterpart in single-electron sys-
tems. This kind of B-induced (i.e., not phonon-induced)
ground state spin mixing, also referred to as “intrinsic
spin mixing”, has been previously reported for singlet-
triplet transitions in N = 2 QDs.58 Here we show that
they may also exist in few-electron QDs with N odd.
(−2,1/2)(1,1/2)
(−1,1/2)
(1,1/2)
(1,1/2)
(−4,1/2)
(−2,1/2)
(−4,1/2)
(−1,1/2)
(−1,1/2)
(−1,1/2)
(1,1/2)
α = 50, β = 25α = 0, β = 25
N = 3
N = 5
 127.0
 127.5
 128.0
 128.5
0.5 1.0 0.0
 236.0
 236.5
 237.0
B (T)
 0.0 0.5 1.0
B (T)
FIG. 2: The four lowest energy levels in a QD with N = 3, 5
interacting electrons, as a function of an axial magnetic field,
without (left column) and with (right column) Rashba SO
interaction. The approximate quantum numbers (M,S) of
the levels are shown, with arrows denoting the spin projection
Sz = 1/2 (↑) and Sz = −1/2 (↓). The dashed circles highlight
the region of intrinsic spin mixing of the ground state.
Figure 1 puts forward yet another qualitative differ-
ence between SO coupling in single- and few-electron
QDs: while in the former low-energy anticrossings are
due to Rashba interaction11,36,52, in few-electron QDs,
when S = 3/2 states come into play, both Rashba and
Dresselhaus terms may induce anticrossings. For exam-
ple, the (M = −1, Sz = 1/2) sublevel couples directly to
both (M = −2, Sz = −1/2) and (M = −2, Sz = 3/2)
sublevels, via the Dresselhaus and Rashba interaction,
respectively. Coupling to S = 3/2 states is a characteris-
tic feature of N > 1 systems, which has important effects
on the spin relaxation rate, as we will discuss below.
B. Spin relaxation between Zeeman sublevels
In Fig. 3 we compare the magnetic field dependence of
the spin relaxation rate between the two lowest Zeeman
sublevels of N = 1, 3, 5. Dashed lines (solid lines) are
used for systems without (with) Rashba interaction.59
While for N = 1 the well-known exponential dependence
with B is found2,6,9, and the main effect of Rashba cou-
pling is to shift the curve upwards (i.e., to accelerate the
relaxation), for N = 3 and N = 5 the relaxation rate ex-
hibits complicated trends which strongly depend on the
values of the SO coupling parameters.
α = 50, β = 25
 α = 0, β = 25
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
B (T)
 0  1  2  3
FIG. 3: Spin relaxation rate in a QD with N = 1, 3, 5 inter-
acting electrons as a function of an axial magnetic field. Solid
(dashed) lines stand for the system with (without) Rashba
interaction. Note the strong influence of the SO interaction
in the shape of the relaxation curve for N > 1.
To understand this result, one has to bear in mind that
in spin relaxation processes two well-distinguished and
complementary ingredients are involved, namely SO in-
teraction and phonon emission. Phonon emission grants
the conservation of energy in the electron relaxation,
but phonons have zero spin and therefore cannot cou-
ple states with different spin. It is the SO interaction
that turns pure spin states into mixed ones, thus enabling
the phonon-induced transition. The overall efficiency of
the scattering event is then given by the combination
of the two phenomena: the phonon emission efficiency
modulated by the extent of the SO mixing. The shape
of spin relaxation curves shown in Fig. 3 can be directly
related to the energy dispersion of the phonon, which cor-
responds to the splitting between the two lowest levels of
the electron spectrum. Thus, for N = 1, the phonon
energy is simply proportional to B through the Zeeman
splitting, but for N = 3 and N = 5 it has a non-trivial
dependence on B, as shown in Fig. 2. Actually, the relax-
ation minima in Fig. 3 are connected with the magnetic
field values where the two lowest levels anticross in Fig. 2.
In these magnetic field windows, in spite of the fact that
SO coupling is strong, the phonon density is so small that
the relaxation rate is greatly suppressed.28 Similarly, the
relaxation rate fluctuations of N = 3 at B ∼ 3 T are
signatures of the anticrossings with high-angular momen-
tum states. For larger fields (B > 3 T), the ground state
approaches the maximum density droplet configuration
and high-spin states are possible.44 In this work, how-
ever, we restrict ourselves to the magnetic field regime
where the ground state is a doublet.
eV∆    (µ     )
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 0 20 40 60 80
 10  
FIG. 4: (Color online). Spin relaxation rate in a QD with
N = 1, 3, 5 interacting electrons as a function of the energy
splitting between the two lowest spin states. Top panel: α =
0, β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25
meV·Å. The relaxation of N = 3 is slower than that of N = 1
for a wide range of ∆12. The irregular data distribution is
due to the irregular relaxation rates vs. magnetic field. For
example, the strongly deviated points of N = 3 come from
the peaks at B ∼ 3 in Fig. 3.
For a more direct comparison between the relaxation
rates of N = 1, 3, 5, in Fig. 4 we replot the data of Fig. 3
as a function of the energy splitting between the two
lowest states, ∆12, without (top panel) and with (bot-
tom panel) Rashba interaction. Since the phonon energy
is identical for all points with the same ∆12, differences
in the relaxation rate arise exclusively from the different
strength of SO interaction. ∆12 is also a relevant pa-
rameter from the experimental point of view, since it is
usually required that it be large enough for the states to
be resolvable. In this sense, it is worth noting that, even
if the inter-level splittings shown in Fig. 4 are fairly small,
a number of experiments have successfully addressed this
regime.5,8,21
A most striking feature observed in the figure is that,
for most values of ∆12, the N = 3 relaxation rate is
clearly slower than the N = 1 one. Likewise, N = 5
shows a similar (or slightly faster) relaxation rate than
N = 1. These are interesting results, for they suggest
that improved spin stability may be achieved using few-
electron QDs instead of the single-electron ones typically
employed up to date.8 At first sight the results are sur-
prising, because the higher density of states in the few-
electron systems implies smaller inter-level spacings, and
hence stronger SO mixing, which should translate into
enhanced relaxation. It then follows that another physi-
cal mechanism must be acting upon the few-electron sys-
tems, which reduces the transition probability between
the initial and final spin states, and may even make it
smaller than for N = 1. Here we propose that such
mechanism is the SO admixture with low-lying quadru-
plet (S = 3/2) states, which become available for N > 1.
By coupling to S = 3/2 levels, the projection of the dou-
blet Sz = 1/2 levels onto Sz = −1/2 ones is reduced,
and this partly inhibitis the transition between the low-
est doublet sublevels.
Let us explain this by comparing the spin transition for
N = 1 and N = 3. For N = 1, the spin configuration of
the initial and final states, in the absence of SO coupling,
is |Sz = −1/2〉 and |Sz = +1/2〉, respectively. The tran-
sition between these states is spin-forbidden. However,
when SO coupling is switched on, the two states become
admixed with higher-lying S = 1/2 states fulfilling the
∆Sz = ±1 condition. The transition between the initial
and final states can then be represented schematically as:
ca |Sz = −1/2〉+cb|Sz = +1/2〉 ⇒ cr |Sz = +1/2〉+cs|Sz = −1/2〉,
where ci are the admixture coefficients (in general ca ≫
cb and cr ≫ cs). Clearly now both spin configurations of
the initial state have a finite overlap with the final state,
and so the transition is possible. Let us next consider
the N = 3 case. In the absence of SO coupling, the
initial and final states are again the Sz = −1/2 and Sz =
+1/2 doublets, respectively, and the transition is spin-
forbidden. When we switch on SO coupling, we note that
the ∆Sz = ±1 condition allows for mixing not only with
Sz = ±1/2 states (either doublets or quadruplets) but
also with Sz = ±3/2 quadruplets, so that the transition
can be represented as:
ca|Sz = −1/2〉+ cb|Sz = +1/2〉+ cc|Sz = −3/2〉 ⇒
cr|Sz = +1/2〉+ cs|Sz = −1/2〉+ ct|Sz = +3/2〉,
where, in general, ca ≫ cb, cc, and cr ≫ cs, ct. In this
case, |Sz = −3/2〉 has no overlap with the final state con-
figurations. Likewise, |Sz = +3/2〉 has no overlap with
the initial state configurations. Therefore, these quadru-
plet configurations are inactive from the point of view
of the transition, and the more important they are (i.e.,
the stronger the SO coupling with quadruplet states), the
less likely the transition is.
To prove this argument quantitatively, in Fig. 5 we il-
lustrate the spin relaxation of N = 3 calculated by diag-
onalization of the SO Hamiltonian including and exclud-
ing the low-lying S = 3/2 states from the basis set. As
expected, when the quadruplets are not considered, the
transition is visibly faster. For N = 5, low-lying S = 3/2
levels are also available, but in this case they barely com-
pensate for the large density of electron states, so that
the overall scattering rate turns out to be comparable to
that of N = 1.
eV∆    (µ     )
without S=3/2
with S=3/2
 10  
 10  
 10  
 10  
 10  
 0 20 40 60 80
FIG. 5: (Color online). Spin relaxation rate in a QD with N =
3 interacting electrons as a function of the energy splitting
between the two lowest spin states. α = 0 and β = 25 meV·Å.
Symbol + (×) stands for SO Hamiltonian diagonalized in a
basis which includes (excludes) S = 3/2 states. Clearly, the
inclusion of S = 3/2 states slows down the relaxation.
To test the robustness of the few-electron spin states
stability predicted above, we also compare the relax-
ation rate of N = 1 and N = 3 in a QD with dif-
ferent confinement, namely h̄ω0 = 6 meV, in Fig. 6.
Since the lateral confinement of the dot is now stronger,
(M = −1, S = 1/2) is the N = 3 ground state up to large
values of the magnetic field (B ∼ 5 T). This allows us
to investigate larger Zeeman splittings (i.e., larger ∆12),
which may be easier to resolve experimentally. As seen
in the figure, the relaxation rate of N = 3 is again slower
than that of N = 1 for a wide range of ∆12, the behavior
being very similar to that of Fig. 4, albeit extended to-
wards larger inter-level spacings. The crossing between
N = 3 and N = 1 relaxation rates at large ∆12 val-
ues, both in Fig. 4 and Fig. 6, is due to the proximity
of high-angular momentum levels coming down in energy
for N = 3 when the magnetic field (and hence the Zee-
man splitting) is large. Such levels bring about strong
SO admixture and thus fast relaxation (see middle panel
of Fig. 3 at B ∼ 3 T).
IV. SPIN RELAXATION IN A QD WITH N
A. Energy structure
When the number of electrons confined in the QD is
even and the magnetic field is not very strong, the ground
and first excited states are usually a singlet (S = 0)
and a triplet (S = 1) with three Zeeman sublevels
eV∆    (µ     )
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 10  
 0 40 80 120
FIG. 6: (Color online). Spin relaxation rate in a QD with
N = 1, 3 interacting electrons as a function of the energy
splitting between the two lowest spin states. The QD has
h̄ω0 = 6 meV. Top panel: α = 0, β = 25 meV·Å. Bottom
panel: α = 50 meV·Å, β = 25 meV·Å. As for the weaker-
confined dot of Fig. 4, the relaxation of N = 3 is slower than
that of N = 1 for a wide range of ∆12.
(Sz = +1, 0,−1). Unlike in the previous section, here the
initial and final states of the spin transition may have dif-
ferent orbital quantum numbers, and the inter-level split-
ting ∆12 may be significantly larger (in the meV scale).
Under these conditions, the phonon emission efficiency no
longer exhibits a simple proportionality with the phonon
density, but it further depends on the ratio between the
phonon wavelength and the QD dimensions.50,51 More-
over, SO interaction is sensitive to the quantum numbers
of the initial and final electron states.26,28 Therefore, in
this class of spin transitions the details of the energy
structure are also relevant to determine the relaxation
rate.
In Fig. 7 we plot the energy levels vs. magnetic field for
a QD with N = 2, 4 in the presence of Rashba and Dres-
selhaus interactions. The approximate quantum num-
bers (M,S) of the lowest-lying states are written between
parenthesis. For N = 2 and weak fields, the ground state
is the (M = 0, S = 0) singlet, and the first excited state
is the (M = −1, S = 1) triplet. As in the previous sec-
tion, SO interaction introduces small zero-field splittings
and anticrossings in the energy levels with |M | > 0.36
As a consequence, when α > β, the zero-field ordering of
the (M = −1, S = 1) Zeeman sublevels is such that they
anticross in the presence of an external magnetic field.
This anticrossing is highlighted in the figure by a dashed
circle. On the other hand, as B increases the singlet-
triplet energy spacing is gradually reduced, and then the
singlet experiences a series of weak anticrossings with all
(−1,1)
(−2,0)
(−3,1)
(0,1)
(0,0)
 0  1  2  3
B (T)
FIG. 7: Low-lying energy levels in a QD with N = 2, 4 in-
teracting electrons as a function of an axial magnetic field.
α = 50 meV· Å and β = 25 meV· Å. The approximate
quantum numbers (M,S) of the lowest states are shown. The
dashed circle in N = 2 highlights the anticrossing between
M = −1 Zeeman sublevels.
three Zeeman sublevels of the triplet. These anticross-
ings are due to the fact that (M = 0, S = 0, Sz = 0)
couples to the (M = −1, S = 1, Sz = −1) sublevel via
Dresselhaus interaction, to the (M = −1, S = 1, Sz =
+1) sublevel via Rashba interaction, and finally to the
(M = −1, S = 1, Sz = 0) sublevel indirectly through
higher-lying states.26,28
For N = 4, the density of electronic states is larger
than for N = 2, which again reflects in a larger magni-
tude of the anticrossings gaps due to the enhanced SO
interaction. The ground state at B = 0 is a triplet,
(M = 0, S = 1), but soon after it anticrosses with a
singlet, (M = −2, S = 0). After this, and before the
formation of Landau levels, two different branches of the
first excited state can be distinguished: when B < 1 T,
the first excited state is (M = 0, S = 1), and when B > 1
T it is (M = −3, S = 1). It is worth pointing out that
the complexity of the N = 4 spectrum, as compared to
the simple N = 2 one, implies a greater flexibility to
select initial and final spin states by means of external
fields. As we shall discuss below, this degree of freedom
has important consequences on the relaxation rate.
B. Triplet-singlet spin relaxation
In a recent work, we have investigated the magnetic
field dependence of the TS relaxation due to SO cou-
pling and phonon emission in N = 2 and N = 4 QDs.28.
Here we study this kind of transition from a different
perspective, namely we compare the spin relaxation of
two- and four-electron systems in order to highlight the
changes introduced by inter-electron repulsion. Increas-
ing the number of electrons confined in the QD has three
important consequences on the TS transition. First, it
increases the density of electronic states (and then the
SO mixing), leading to faster relaxation. Second, as
mentioned in the previous section, it introduces a wider
choice of orbital quantum numbers for the singlet and
triplet states. Third, it increases the strength of elec-
tronic correlations. Since now the initial and final spin
states have different orbital wave functions, the latter
factor effectively reduces phonon scattering, in a similar
fashion to charge relaxation processes33 (this effect has
been recently pointed out in Ref. 30 as well). To find out
the overall combined effect of these three factors, in this
section we analyze quantitative simulations of correlated
We focus on the magnetic field regions where the
ground state is a singlet and the excited state is a triplet.
A complete description of the TS transition should then
include spin relaxation between the Zeeman-split sub-
levels of the triplet. However, for the weak fields we con-
sider this relaxation is orders of magnitude slower than
the TS one (compare Figs. 3 and 8),60 the reason for
this being the small Zeeman energy and the fact that the
Zeeman sublevels are not directly coupled by Rashba and
Dresselhaus terms, as mentioned in Section III. There-
fore, it is a good approximation to assume that all three
triplet Zeeman sublevels are equally populated and they
relax directly to the singlet.26
α = 50, β = 25
 α = 0, β = 25
 10  
 10  
 10  
 10  
 10  
 10  
 0.5  1  1.5  2  2.5  3
B (T)
FIG. 8: Spin relaxation rate in a QD with N = 2, 4 interact-
ing electrons as a function of an axial magnetic field. Solid
(dashed) lines stand for the system with (without) Rashba
interaction. The relaxation of N = 4 when B < 1 T is slower
than that of N = 2.
Figure 8 represents the TS relaxation rate in a QD with
N = 2, 4, after averaging the relaxation from the three
triplet sublevels. Solid (dashed) lines stand for the case
with (without) Rashba interaction.59 The main effect of
Rashba and Dresselhaus interactions is to accelerate the
spin transition by shifting the relaxation curve upwards.
This is in contrast to the N -odd case, where these terms
may induce drastic changes in the shape of the relaxation
rate curve (see Fig. 3). Figure 8 also reveals a different
behavior of the N = 2 and N = 4 TS relaxation rates.
The former increases gradually with B and then drops
in the vicinity of the TS anticrossing, due to the small
phonon energies.28,29,30 Conversely, for N = 4 an addi-
tional feature is found, namely an abrupt step at B ∼ 1.
This is due to the change of angular momentum of the
excited triplet. For B < 1 T the triplet has M = 0, and
for B > 1 T it has M = −3. Since the ground state
is a singlet with M = −2, the M = 0 triplet does not
fulfill the ∆M = ±1 condition for linear SO coupling.
This inhibits direct spin mixing between initial and final
states and reduces the relaxation rate by about one order
of magnitude.28
meV∆    (         )
N=4, M=0
N=4, M=−3
  10  
 10  
 10  
 10  
 10  
  10  
  10  
 0  0.4  0.8  1.2
FIG. 9: (Color online). Spin relaxation rate in a QD with
N = 2, 4 interacting electrons as a function of the energy
spacing between the singlet and the triplet. Here M stands
for the angular momentum of the triplet. Top panel: α = 0,
β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25 meV·Å.
The relaxation of N = 4 is comparable to that of N = 2 when
the triplet has M = −3, and it is much smaller when M = 0.
Noteworthy, the choice of states differing in more than
one quantum of angular momentum is only possible for
N > 2 QDs. One may then wonder if it is more conve-
nient to use these systems instead of the N = 2 ones dom-
inating the experimental literature up to date20,21,29, i.e.
if it compensates for the increased density of electronic
states. Interestingly, Fig. 8 predicts slower relaxation for
the N = 4 QD with M = 0 triplet than for N = 2. To
verify that this arises from weakend SO coupling rather
than from different phonon energy values, in Fig. 9 we
replot the spin relaxation rate of N = 2, 4 as a function
of the TS energy splitting. In the figure, the upper and
bottom panels represent the situations without and with
Rashba interaction, respectively. While N = 4 shows
similar relaxation rate to N = 2 when the triplet has
M = −3, the relaxation is slower by about one order
of magnitude when the triplet has M = 0. This result
indicates that the weakening of SO mixing due to the
violation of the ∆M = ±1 condition clearly exceeds the
strengthening due to the higher density of states, con-
firming that N = 4 systems are more attractive than
N = 2 ones to obtain long triplet lifetimes. We also
point out that, in spite of the different density of states,
the relaxation rate of N = 2 and N = 4, M = −3 triplets
is quite similar. This can be ascribed to the phonon scat-
tering reduction by electronic correlations,33 which may
also explain the fact that experimentally resolved TS re-
laxation rates of N = 8 QDs and N = 2 QDs be quite
similar.20,31
V. COMPARISON WITH N = 2 EXPERIMENTS
Whereas, to our knowledge, no experiments have mea-
sured transitions between Zeeman-split sublevels in N >
1 systems yet, a number of works have dealt with TS re-
laxation in QDs with few interacting electrons. In Ref. 28
we showed that our model correctly predicts the trends
observed in experiments with N = 2 and N = 8 QDs
subject to axial magnetic fields.20,21,31 In this section, we
extend the comparison to new experiments available for
N = 2 TS relaxation in QDs,29 which for the first time
provide continuous measurements of the average triplet
lifetime against axial magnetic fields, from B = 0 to the
vicinity of the TS anticrossing. By using a simple model,
the authors of the experimental work showed that the
measuraments are in clear agreement with the behavior
expected from SO coupling plus acoustic phonon scatter-
ing. However, in such model: (i) the TS energy splitting
was a taken directly from the experimental data, (ii) the
SO coupling effect was accounted for by parametrizing
the admixture of the lowest singlet and triplet states only,
and (iii) the B-dependence of the SO-induced admix-
ture was neglected. Approximation (ii) may overlook the
correlation-induced reduction of phonon scattering,30,33
that we have shown above to be significant, and which
may have an important contribution from higher excited
states in weakly-confined QDs. In turn, approximation
(iii) may overlook the important influence of SO coupling
in the B-dependence of the triplet lifetime, as we had
anticipated in Ref. 28. Here we compare with the exper-
imental findings using our model, which includes these
effects properly. We assume a QD with an effective well
width Lz = 30 nm, as expected by Ref. 29 authors, and
a lateral confinement parabola of h̄ω0 = 2 meV which,
as we shall see next, fits well the position of the TS an-
ticrossing. Yet, the comparison is limited by the lack of
detailed information about the Rashba and Dresselhaus
interaction constants, and because we deal with circular
QDs instead of elliptical ones (the latter effect introduces
simple deviations from the circular case26). In addition,
in the experiment a tilted magnetic field of magnitude
B∗, forming an angle of 68◦ with the vertical direction
was used. Here we consider the vertical component of
the field (B = 0.37B∗), which is the main responsible
for the changes in the energy structure, and the effect of
the in-plane component enters via the Zeeman splitting
only.
Figure 10 illustrates the average triplet lifetime for
N = 2. The bottom axis shows the vertical magnetic
field B value, while the top axis shows the value to be
compared with the experiment B∗.59 As can be seen, the
triplet lifetime first decreases with the field and then it
abruptly increases in the vicinity of the TS anticross-
ing, due to the small phonon density.28 This behavior
is in clear agreement with the experiment (cf. Fig. 3 of
Ref. 29). The position of the anticrossing (B∗ ∼ 2.9 T) is
also close to the experimental value (B∗ ∼ 2.8 T), which
confirms that that h̄ω0 = 2 meV is similar to the mean
confinement frequency of the experimental sample. A
departure from the experimental trend appears at weak
fields (B < 0.5 T), where we observe a continuous in-
crease of T1 with decreasing B, while the experiment re-
ports a plateau. This is most likely due to the ellipticity
of the experimental sample, which renders the electron
states (and consequently the relaxation rate) insensitive
to the field in the B∗ = 0 − 0.5 T region (see Fig. 1a
in 29). In any case, Fig. 10 clearly confirms the role of
phonon-induced relaxation in the experiments, using a
realistic model for the description of correlated electron
states, SO admixture and phonon scattering.
A comment is worth here on the magnitude of the SO
coupling terms. In Fig. 10, we obtain good agreement
with the experimental relaxation times by using small
values of the SO coupling parameters. In particular, a
close fit is obtained using β = 1, α = 0.5 meV·Å, which
yields a spin-orbit length λSO = 48 µm. This value,
which coincides with the experimental guess (λSO ≈ 50
µm), indicates that SO coupling is several times weaker
than that reported for other GaAs QDs.8 Typical GaAs
parameters are often larger. For instance, measuraments
of the Rashba and Dresselhaus constants by analysis of
the weak antilocalization in clean GaAs/AlGaAs two-
dimensional gases revealed α = 4−5 meV·Å, and γc = 28
eV·Å3 (i.e, β = 3 meV·Å for our quantum well of Lz = 30
nm).61 To be sure, the small SO coupling parameters in
the experiment have a major influence on the lifetime
scale. Compare e.g. the β = 1 and β = 5 meV·Å curves
in Fig. 10. Actually, we note that accurate comparison
with the timescale reported for other GaAs samples31 is
also possible within our model, but assuming stronger
SO coupling constants.28 In Ref. 29, it was suspected
that the weak SO coupling inferred from the experimen-
tal data could be the result of the exclusion of higher
orbitals and the magnetic field dependence of SO ad-
α = 0.5 β = 1
β = 5
β = 2
β = 1 α = 0
B (T)
 0.2  0.4  0.6  0.8  1 0
 0.53 1.58 2.11 2.631.05
B (T)
FIG. 10: Average triplet lifetime in a QD with N = 2 elec-
trons as a function of an axial magnetic field. Only the field
region before the TS anticrossing is shown. α and β are in
meV·Å units. B is the applied axial magnetic field, and B∗
is the equivalent tilted magnetic field, for comparison with
Ref. 29 experiment.
mixture in their model (higher states reduce the effective
SO coupling constants by decreasing the phonon-induced
scattering30,33). Here we have considered both these ef-
fects and still small SO coupling constants are needed to
reproduce the experiment. Therefore, understanding the
origin of their small value remains as an open question.
One possibility could be that the particular direction of
the tilted magnetic field used in the experiment corre-
sponded to a reduced degree of SO admixture.30
VI. CONCLUSIONS
We have investigated theoretically the energy structure
and spin relaxation rate of weakly-confined QDs with
N = 1 − 5 interacting electrons, subject to axial mag-
netic fields, in the presence of linear Rashba and Dressel-
haus SO interactions. It has been shown that the num-
ber of electrons confined in the dot introduces changes
in the energy spectrum which significantly influence the
intensity of the SO admixture, and hence the spin re-
laxation. In general, the larger the number of confined
carriers, the higher the density of electronic states. This
decreases the energy splitting between consecutive lev-
els and then enhances SO admixture, which should lead
to faster spin relaxation. However, we find that this is
not necessarily the case, and slower relaxation rate may
be found for few-electron QDs as compared to the usual
single and two-electron QDs used up to date. The physi-
cal mechanisms responsible for this have been identified.
For N -odd systems, when the spin transition takes place
between Zeeman-split sublevels, it is the presence of low-
energy S = 3/2 states for N > 1 that reduces the pro-
jection of the doublet Sz = 1/2 sublevels into Sz = −1/2
ones, thus partly inhibiting the spin transition. For N -
even systems, when the spin transition takes place be-
tween triplet and singlet levels, there are two underlying
mechanisms. On the one hand, electronic correlations
tend to reduce phonon emission efficiency. On the other
hand, for N > 2 a magnetic field can be used to se-
lect a pair of singlet-triplet states which do not fulfill
the ∆M = ±1 condition of direct SO admixture, which
significantly weakens the SO mixing.
Last, we have compared our estimates with recent
experimental data for TS relaxation in N = 2 QDs.29
Our results support the interpretation of the experi-
ment in terms of SO admixture plus acoustic phonon
scattering, even though quantitative agreement with the
experiment requires assuming much weaker SO coupling
than that reported for similar GaAs structures.
Acknowledgments
We acknowledge support from the Italian Ministry
for University and Scientific Research under FIRB
RBIN04EY74, Cineca Calcolo parallelo 2006, and
Marie Curie IEF project NANO-CORR MEIF-CT-2006-
023797.
∗ Electronic address: climente@unimore.it;
URL: www.nanoscience.unimore.it
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http://www.s3.infm.it/donrodrigo
