Reexamination of spin decoherence in semiconductor quantum dots from
equation-of-motion approach
J. H. Jiang,1, 2 Y. Y. Wang,2 and M. W. Wu1, 2, ∗
Hefei National Laboratory for Physical Sciences at Microscale,
University of Science and Technology of China, Hefei, Anhui, 230026, China
Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
(Dated: November 2, 2018)
The longitudinal and transversal spin decoherence times, T1 and T2, in semiconductor quantum
dots are investigated from equation-of-motion approach for different magnetic fields, quantum dot
sizes, and temperatures. Various mechanisms, such as the hyperfine interaction with the surrounding
nuclei, the Dresselhaus spin-orbit coupling together with the electron–bulk-phonon interaction, the
g-factor fluctuations, the direct spin-phonon coupling due to the phonon-induced strain, and the
coaction of the electron–bulk/surface-phonon interaction together with the hyperfine interaction
are included. The relative contributions from these spin decoherence mechanisms are compared in
detail. In our calculation, the spin-orbit coupling is included in each mechanism and is shown to
have marked effect in most cases. The equation-of-motion approach is applied in studying both the
spin relaxation time T1 and the spin dephasing time T2, either in Markovian or in non-Markovian
limit. When many levels are involved at finite temperature, we demonstrate how to obtain the
spin relaxation time from the Fermi Golden rule in the limit of weak spin-orbit coupling. However,
at high temperature and/or for large spin-orbit coupling, one has to use the equation-of-motion
approach when many levels are involved. Moreover, spin dephasing can be much more efficient
than spin relaxation at high temperature, though the two only differs by a factor of two at low
temperature.
PACS numbers: 72.25.Rb, 73.21.La,71.70.Ej
I. INTRODUCTION
One of the most important issues in the growing
field of spintronics is quantum information processing
in quantum dots (QDs) using electron spin.1,2,3,4,5 A
main obstacle is that the electron spin is unavoidably
coupled to the environment (such as, the lattice) which
leads to considerable spin decoherence (including lon-
gitudinal and transversal spin decoherences).6,7 Vari-
ous mechanisms, such as, the hyperfine interaction with
the surrounding nuclei,8,9 the Dresselhaus/Rashba spin-
orbit coupling (SOC)10,11 together with the electron-
phonon interaction, g-factor fluctuations,12 the direct
spin-phonon coupling due to the phonon-induced strain,9
and the coaction of the hyperfine interaction and the
electron-phonon interaction can lead to the spin deco-
herence. There are quite a lot of theoretical works
on spin decoherence in QD. Specifically, Khaetskii and
Nazarov analyzed the spin-flip transition rate using a
perturbative approach due to the SOC together with the
electron-phonon interaction, g-factor fluctuations, the di-
rect spin-phonon coupling due to the phonon-induced
strain qualitatively.13,14,15 After that, the longitudinal
spin decoherence time T1 due to the Dresslhaus and/or
the Rashba SOC together with the electron-phonon in-
teraction were studied quantitatively in Refs. 16,17,18,19,
20,21,22,23,24,25,26. Among these works, Cheng et al.18
developed an exact diagonalization method and showed
that due to the strong SOC, the previous perturbation
method14,15,16 is inadequate in describing T1. Further-
more, they also showed that, the perturbation method
previously used missed an important second-order en-
ergy correction and would yield qualitatively wrong re-
sults if the energy correction is correctly included and
only the lowest few states are kept as those in Refs.
14,15,16. These results were later confirmed by Deste-
fani and Ulloa.21 The contribution of the coaction of the
hyperfine interaction and the electron-phonon interac-
tion to longitudinal spin decoherence was calculated in
Refs. 27 and 28. In contrast to the longitudinal spin de-
coherence time, there are relatively fewer works on the
transversal spin decoherence time, T2, also referred to as
the spin dephasing time (while the longitudinal spin de-
coherence time is referred to as the spin relaxation time
for short). The spin dephasing time due to the Dressel-
haus and/or the Rashba SOC together with the electron-
phonon interaction was studied by Semenov and Kim29
and by Golovach et al..20 The contributions of the hyper-
fine interaction and the g-factor fluctuation were studied
in Refs. 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44 and
in Ref. 45 respectively. However, a quantitative calcula-
tion of electron spin decoherence induced by the direct
spin-phonon coupling due to phonon-induced strain in
QDs is still missing. This is one of the issues we are
going to present in this paper. In brief, the spin re-
laxation/dephasing due to various mechanisms has been
studied previously in many theoretical works. However,
almost all of these works only focus individually on one
mechanism. Khaetskii and Nazarov discussed the ef-
fects of different mechanisms on the spin relaxation time.
Nevertheless, their results are only qualitative and there
is no comparison of the relative importance of the dif-
http://arxiv.org/abs/0704.0148v5
ferent mechanisms.13,14,15 Recently, Semenov and Kim
discussed various mechanisms contributed to the spin
dephasing,46 where they gave a “phase diagram” to in-
dicate the most important spin dephasing mechanism in
Si QD where the SOC is not important. However, the
SOC is very important in GaAs QDs. To fully under-
stand the microscopic mechanisms of spin relaxation and
dephasing, and to achieve control over the spin coherence
in QDs,47,48,49 one needs to gain insight into the relative
importance of each mechanism to T1 and T2 under vari-
ous conditions. This is one of the main purposes of this
paper.
Another issue we are going to address relates to differ-
ent approaches used in the study of the spin relaxation
time. The Fermi-Golden-rule approach, which is widely
used in the literature, can be used in calculation of the re-
laxation time τi→f between any initial state |i〉 and final
state |f〉.12,13,14,15,16,17,18,19,21,23,24,25,27,28,50,51,52 How-
ever, the problem is that when the process of the spin
relaxation relates to many states, (e.g., when tempera-
ture is high, the electron can distribute over many states),
one should find a proper way to average over the relax-
ation times (τi→f ) of the involved processes to give the
total spin relaxation time (T1). What makes it difficult
in GaAs QDs, is that all the states are impure spin states
with different expectation values of spin. In the existing
literature, spin relaxation time is given by the average of
the relaxation times of processes from the initial state |i〉
to the final state |f〉 (with opposite majority spin of |i〉)
weighted by the distribution of the initial states fi,
18,51,52
i.e.,
T−11 =
i→f . (1)
This is a good approximation in the limit of small SOC
as each state only carries a small amount of minority
spin. However, when the SOC is very strong which hap-
pens at high levels, it is difficult to find the proper way
to perform the average. We will show that Eq. (1) is
not adequate any more. Thus, to investigate both T1
and T2 at finite temperature for arbitrary strength of
SOC, we develop an equation-of-motion approach for the
many-level system via projection operator technique56 in
the Born approximation. With the rotating wave ap-
proximation, we obtain a formal solution to the equation
of motion. By assuming a proper initial distribution,
we can calculate the evolution of the expectation value
of spin. We thus obtain the spin relaxation/dephasing
time by the 1/e decay of the expectation value of spin
operator 〈Sz〉 or |〈S+〉| (to its equilibrium value), with
S+ ≡ Sx+ iSy. With this approach, we are able to study
spin relaxation/dephasing for various temperature, SOC
strength, and magnetic field.
For quantum information processing based on electron
spin in QDs, the quantum phase coherence is very im-
portant. Thus, the spin dephasing time is a more rel-
evant quantity. There are two kinds of spin dephasing
times: the ensemble spin dephasing time T ∗2 and the
irreversible spin dephasing time T2. For a direct mea-
surement of an ensemble of QDs58 or an average over
many measurements at different times where the config-
urations of the environment have been changed,59,60,61
it gives the ensemble spin dephasing time T ∗2 . The ir-
reversible spin dephasing time T2 can be obtained by
spin echo measurement.60,61 A widely discussed source
which leads to both T ∗2 and T2 is the hyperfine interac-
tion between the electron spin and the nuclear spins of
the lattice. As it has been found that T ∗2 is around 10 ns,
which is too short and makes a practical quantum infor-
mation processing difficult in electron spin based qubits
in QDs. Thus a spin echo technique is needed to remove
the free induction decay and to elongate the spin dephas-
ing time. Fortunately, this technique has been achieved
first by Petta et al. for two electron triplet-singlet system
and then by Koppens et al. for a single electron spin sys-
tem. The achieved spin dephasing time is ∼ 1 µs, which
is much longer than T ∗2 . We therefore discuss only the ir-
reversible spin dephasing time T2 throughout the paper,
i.e., we do not consider the free induction decay in the
hyperfine-interaction-induced spin dephasing.
It is further noticed that Golovach et al. have shown
that the spin dephasing time T2 is two times the spin
relaxation time T1.
20 However, as temperature increases,
this relation does not hold. Semenov and Kim on the
other hand reported that the spin dephasing time is much
smaller than the spin relaxation time.29 In this paper,
we calculate the temperature dependence of the ratio of
the spin relaxation time to the spin dephasing time and
analyze the underlying physics.
This paper is organized as follows: In Sec. II, we
present our model and formalism of the equation-of-
motion approach. We also briefly introduce all the spin
decoherence mechanisms considered in our calculations.
In Sec. III we present our numerical results to indicate
the contribution of each spin decoherence mechanism to
spin relaxation/dephasing time under various conditions
based on the equation-of-motion approach. Then we
study the problem of how to obtain the spin relaxation
time from the Fermi Golden rule when many levels are
involved in Sec. IV. The temperature dependence of T1
and T2 is investigated in Sec. V. We conclude in Sec. VI.
II. MODEL AND FORMALISM
A. Model and Hamiltonian
We consider a QD system, where the QD is confined
by a parabolic potential Vc(x, y) =
m∗ω20(x
2 + y2) in
the quantum well plane. The width of the quantum well
is a. The external magnetic field B is along z direction,
except in Sec. IV. The total Hamiltonian of the system
of electron together with the lattice is:
HT = He +HL +HeL , (2)
where He, HL, HeL are the Hamiltonians of the elec-
tron, the lattice and their interaction, respectively. The
electron Hamiltonian is given by
+ Vc(r) +HZ +HSO (3)
where P = −i~∇+ e
A with A = (B⊥/2)(−y,x) (B⊥ is
the magnetic field along z direction), HZ =
gµBB · σ
is the Zeeman energy with µB the Bohr magneton,
and HSO is the Hamiltonian of SOC. In GaAs, when
the quantum well width is small or the gate-voltage
along the growth direction is small, the Rashba SOC is
unimportant.53 Therefore, only the Dresselhaus term10
contributes to HSO. When the quantum well width is
smaller than the QD radius, the dominant term in the
Dresselhaus SOC reads
Hso =
〈P 2z 〉λ0(−Pxσx + Pyσy) , (4)
with γ0 denoting the Dresselhaus coefficient, λ0 being
the quantum well subband index of the lowest one and
〈P 2z 〉λ ≡ −~2
ψ∗zλ(z)∂
2/∂z2ψzλ(z)dz. The Hamiltonian
of the lattice consists of two parts HL = Hph +Hnuclei,
where Hph =
qη ~ωqηa
qηaqη (a
†/a is the phonon cre-
ation/annihilation operator) describes the vibration of
the lattice and Hnuclei =
j γIB · Ij (γI is the gyro-
magnetic ratios of the nuclei and Ij is the spin of the j-
th nucleus) describes the precession of the nuclear spins
of the lattice in the external magnetic field. We focus
on the spin dynamics due to hyperfine interaction at a
time scale much smaller than the nuclear dipole-dipole
correlation time (10−4 s in GaAs33,40), where the nuclear
dipole-dipole interaction can be ignored. Under this ap-
proximation, the equation of motion for the reduced elec-
tron system can be obtained which only depends on the
initial distribution of the nuclear spin bath.33 The in-
teraction between the electron and the lattice also has
two parts HeL = HeI + He−ph, where HeI is the hy-
perfine interaction between the electron and nuclei and
He−ph represents the electron-phonon interaction which
is further composed of the electron–bulk-phonon (BP)
interaction Hep, the direct spin-phonon coupling due to
the phonon-induced strain Hstrain and phonon-induced
g-factor fluctuation Hg.
B. Equation-of-motion approach
The equations of motion can describe both the co-
herent and the dissipative dynamics of the electron sys-
tem. When the quasi-particles of the bath relax much
faster than the electron system, the Markovian approx-
imation can be made; otherwise the kinetics is the non-
Markovian. For electron-phonon coupling, due to the
fast relaxation of the phonon bath and the weak electron-
phonon scattering, the kinetics of the electron is Marko-
vian. Nevertheless, as the nuclear spin bath relaxes much
slower than the electron spin, the kinetics due to the
coupling with nuclei is of non-Markovian type.30,32,33 It
is further noted that there is also a contribution from
the coaction of the electron-phonon and electron-nuclei
couplings, which is a fourth order coupling to the bath.
For this contribution, the decoherence of spin is mainly
controlled by the electron-phonon scattering while the
hyperfine (Overhauser) field54 acts as a static magnetic
field. Thus, this fourth order coupling is also Markovian.
Finally, since the electron orbit relaxation is much faster
than the electron spin relaxation,55 we always assume a
thermo-equilibrium initial distribution of the orbital de-
grees of freedom.
Generally, the interaction between the electron and the
quasi-particle of the bath is weak. Therefore the first
Born approximation is adequate in the treatment of the
interaction. Under this approximation, the equation of
motion for the electron system coupled to the lattice envi-
ronment can be obtained with the help of the projection
operator technique.56 We then assume a sudden approxi-
mation so that the initial distribution of the whole system
is ρ(t = 0) = ρe(0) ⊗ ρL(0), where ρe, ρL is the density
matrix of the system and bath respectively. This approx-
imation corresponds to a sudden injection of the electron
into the quantum dot, which is reasonable for genuine
experimental setup.33 As the initial distribution of the
the lattice ρL(0) commutates with the Hamiltonian of
the lattice HL, the equation of motion can be written as
dρe(t)
= − i
[He +TrL(HeLρ
L(0)), ρe(t)]
dτTrL[HeL, U0(τ)(P̂ [HeL, ρe(t− τ)
⊗ρL(0)])U †0 (τ)] , (5)
where ρe(t) is the density operator of the electron sys-
tem at time t, TrL stands for the trace over the lattice
degree of freedom, and U0(τ) = e
−i(HL+He)τ is time-
evolution operator without HeL. P̂ = 1̂ − ρL(0) ⊗ TrL
is the projection operator. The initial distribution of the
phonon system is chosen to be the thermo-equilibrium
distribution.20 It has been shown by previous theoretical
studies that the initial state of the nuclear spin bath is
crucial to the spin dephasing and relaxation.30,32,33 Al-
though it may take a long time (e.g., seconds) for the
nuclear spin system to relax to its thermo-equilibrium
state, one can still assume that its initial state is the
thermo-equilibrium one. This assumption corresponds
to the genuine case of enough long waiting time during
every individual measurement. For a typical setup at
above 10 mK and with about 10 T external magnetic
field, the thermo-equilibrium distribution is a distribu-
tion with equal probability on every state. For these ini-
tial distributions of phonons and nuclear spins, the term
TrL(HeLρ
L(0)) is zero. Thus,
P̂ [HeL, ρe(t−τ)⊗ρL(0)] = [HeL, ρe(t−τ)⊗ρL(0)] . (6)
The equation of motion is then simplified to,
dρe(t)
= − i
[He, ρ
e(t)] − 1
dτTrL[HeL, [H
eL(−τ),
Ue0 (t)ρ
Ie(t− τ)Ue0
(t)ρL(0)]] , (7)
where HIeL and ρ
Ie are the corresponding operators (HeL
and ρe ) in the interaction picture, and Ue0 (t) = e
−iHet
is the time-evolution operator of He. It should be fur-
ther noted that the first Born approximation can not
fully account for the non-Markovian dynamics due to the
hyperfine interaction with nuclear spins.33,57 Only when
the Zeeman splitting is much larger than the fluctuating
Overhauser shift, the first Born approximation is ade-
quate. For GaAs QDs, this requires B ≫ 3.5 T.33 In this
paper, we focus on the study of spin dephasing for the
high magnetic field regime of B > 3.5 T under the first
Born approximation, where the second Born approxima-
tion only affects the long-time behavior.33 Later we will
argue that this correction of long time dynamics changes
the spin dephasing time very little.
1. Markovian kinetics
The kinetics due to the coupling with phonons can be
investigated within the Markovian approximation, where
the equation of motion reduces to,
dρe(t)
= − i
[He, ρ
e(t)]− 1
dτTrph[He−ph,
[HIe−ph(−τ), ρe(t)⊗ ρph(0)]] . (8)
Here Trph is the trace over phonon degrees of freedom
and ρph(0) is the initial distribution of the phonon bath.
Within the basis of the eigen-states of the electron Hamil-
tonian, {|ℓ〉}, the above equation reads,
ρeℓ1ℓ2= −i
(εℓ1 − εℓ2)
ρeℓ1ℓ2
Trp(H
I e−ph
ρeℓ4ℓ2 ⊗ ρ
−HI e−phℓ1ℓ3 ρ
⊗ ρpeqH
) +H.c.
. (9)
Here H
= 〈ℓ1|He−ph|ℓ3〉 and HI e−phℓ1ℓ3 =
〈ℓ1|HIe−ph(−τ)|ℓ3〉. A general form of the electron-
phonon interaction reads
He−ph =
Φqη(aqη + a
−qη)Xqη(r,σ) . (10)
Here, η represents the phonon branch index; Φqη is the
matrix element of the electron-phonon interaction; aqη
is the phonon annihilation operator; Xqη(r,σ) denotes
a function of electron position and spin. Substituting
this into Eq. (9), we obtain, after integration within the
Markovian approximation,49
ρeℓ1ℓ2 = i
(εℓ1 − εℓ2)
ρeℓ1ℓ2
|Φqη|2{Xqηℓ1ℓ3X
ρeℓ4ℓ2
×Cqη(εℓ4 − εℓ3)−X
ρeℓ3ℓ4
×Cqη(εℓ3 − εℓ1)} +H.c.
in which X
= 〈ℓ1|Xqη(r,σ)|ℓ2〉, and Cqη(∆ε) =
n̄(ωqη)δ(∆ε+ωqη)+[n̄(ωqη)+1]δ(∆ε−ωqη). Here n̄(ωqη)
represents the Bose distribution function. Equation (11)
can be written in a more compact form
ρeℓ1ℓ2 = −
Λℓ1ℓ2ℓ3ℓ4ρ
, (12)
which is a linear differential equation. This equation can
be solved by diagonalizing Λ. Given an initial distribu-
tion ρeℓ1ℓ2(0), the density matrix ρ
(t) and the expec-
tation value of any physical quantity 〈O〉t = Tr(Ôρe(t))
at time t can be obtained:49
〈O〉t = Tr(Ôρe)
ℓ1···ℓ6
〈ℓ2|Ô|ℓ1〉P(ℓ1ℓ2)(ℓ3ℓ4)
× e−Γ(ℓ3ℓ4)tP−1
(ℓ3ℓ4)(ℓ5ℓ6)
ρeℓ5ℓ6(0) (13)
with Γ = P−1ΛP being the diagonal matrix and P repre-
senting the transformation matrix. To study spin dynam-
ics, we calculate 〈Sz〉t (|〈S+〉t|) and define the spin relax-
ation (dephasing) time as the time when 〈Sz〉t (|〈S+〉t|)
decays to 1/e of its initial value (to its equilibrium value).
2. Non-Markovian kinetics
Experiments have already shown that for a large en-
semble of quantum dots or for an ensemble of many mea-
surements on the same quantum dot at different times,
the spin dephasing time due to hyperfine interaction is
quite short, ∼ 10 ns.58,59,60,61 This rapid spin dephas-
ing is caused by the ensemble broadening of the preces-
sion frequency due to the hyperfine fields.40 When the
external magnetic field is much larger than the random
Overhauser field, the rotation due to the Overhauser field
perpendicular to the magnetic field is blocked. Only
the broadening of the Overhauser field parallel to the
magnetic field contribute to the spin dephasing. To de-
scribe this free induction decay for this high magnetic
field case, we write the hyperfine interaction into two
parts: HeI = h · S = HeI1 +HeI2. Here h = (hx, hy, hz)
and S = (Sx, Sy, Sz) are the Overhauser field and the
electron spin respectively. HeI1 = hzSz and HeI2 =
(h+S− + h−S+) with h± = hx ± ihy. The longitudi-
nal part HeI1 is responsible for the free induction de-
cay, while the transversal part HeI2 is responsible for
high order irreversible decay. As the rapid free induction
decay can be removed by spin echo,60,61 elongating the
spin dephasing time to ∼ 1 µs which is more favorable
for quantum computation and quantum information pro-
cessing, we then discuss only the irreversible decay. We
first classify the states of the nuclear spin system with its
polarization. Then we reconstruct the states within the
same class to make it spatially uniform. These uniformly
polarized pure states, |n〉’s, are eigen-states of hz. They
also form a complete-orthogonal basis of the nuclear spin
system. A formal expression of |n〉 is33
|n〉 =
m1···mN
αnm1···mN
|I,mj〉 . (14)
Here |I,mj〉 denotes the eigen-state of the z-component
of the j-th nuclear spin Ijz with the eigenvalue ~mj. N
denotes the number of the nuclei. The equation of motion
for the case with initial nuclear spin state ρns1 (0) = |n〉〈n|
is given by33
dρe(t)
[He +Trns(HeIρ
1 (0)), ρ
e(t)]
dτTrns[HeI2, U
0 (τ)
×[HeI2, ρe(t− τ)⊗ ρns1 (0)]UeI0
(τ)] . (15)
As in traditional projection operator technique, the dy-
namics of the nuclear spin subsystem is incorporated
self-consistently in the last term.33,56 Here Trns is the
trace over nuclear spin degrees of freedom. UeI0 (τ) =
exp[−iτ(He+HI +HeI1)]. The Overhauser field is given
by h =
j Av0Ijδ(r − Rj), where the constants A and
v0 are given later. Ij and Rj are the spin and posi-
tion of j-th nucleus respectively. As mentioned above,
the initial state of the nuclear spin bath is chosen to be
a state with equal probability of each state, therefore
ρns(0) =
n 1/Nw|n〉〈n|, with Nw =
n 1 being the
number of states of the basis {|n〉}. To quantify the ir-
reversible decay, we calculate the time evolution of S
for every case with initial nuclear spin state |n〉. We then
sum over n and obtain
||〈S+〉t|| =
|〈S(n)+ 〉t|. (16)
It is noted that the summation is performed after the
absolute value of 〈S(n)+ 〉t. Therefore, the destructive in-
terference due to the difference in precession frequency
ωzn, which originates from the longitudinal part of the
hyperfine interaction (HeI1), is removed. We thus use
1/e decay of the envelope of ||〈S+〉t|| to describe the irre-
versible spin dephasing time T2. Similar description has
been used in the irreversible spin dephasing in semicon-
ductor quantum wells62 and the irreversible inter-band
optical dephasing in semiconductors.63,64
Expanding Eq. (15) in the basis of {|n〉}, one obtains,
ρeℓ1ℓ2= −
(εℓ1δℓ1ℓ3 +H
nℓ1;nℓ3
)ρeℓ3ℓ2
−ρeℓ1ℓ3(εℓ3δℓ3ℓ2 +H
nℓ3;nℓ2
[HeI2nℓ1;n1ℓ3H
I eI2
n1ℓ3;nℓ4
ρeℓ4ℓ2(t− τ)
−HI eI2nℓ1;n1ℓ3ρ
(t− τ)HeI2n1ℓ4;nℓ2 ] +H.c.
. (17)
Here HeI2nℓ1;n1ℓ3 = 〈nℓ1|HeI2|n1ℓ3〉 and H
I eI2
nℓ1;n1ℓ3
〈nℓ1|HIeI2(−τ)|n1ℓ3〉. For simplicity, we neglect the
terms concerning different orbital wavefunctions which
are much smaller. For small spin mixing, assuming an
equilibrium distribution in orbital degree of freedom, un-
der rotating wave approximation, and trace over the or-
bital degree of freedom, we finally arrive at
〈S(n)+ 〉t = iωzn〈S
+ 〉t −
fk([h+]knn′
× [h−]kn′n + [h−]knn′ [h+]kn′n)
× exp[iτ(ωkn − ωkn′)]}〈S(n)+ 〉t−τ . (18)
Here ωzn =
k fk(Ezk/~ + ωkn) with Ezk representing
the electron Zeeman splitting of the k-th orbital level.
[hi]knn′ = 〈n|〈k|hi|k〉|n′〉 (i = ±, z). ωkn = [hz]knn + ǫnz
with ǫnz denoting the nuclear Zeeman splitting which is
very small and can be neglected. By solving the above
equation, we obtain |〈S(n)+ 〉t| for a given |n〉. We then sum
over n and determine the irreversible spin dephasing time
T2 as 1/e decay of the envelop of ||〈S+〉t||. By noting that
only the polarization of nuclear spin state |n〉 determines
the evolution of |〈S(n)+ 〉t|, the summation over n is then
reduced to summation over polarization which becomes a
integration for large N . This integration can be handled
numerically.
In the limiting case of zero SOC and very low tem-
perature, only the lowest two Zeeman sublevels are con-
cerned. The equation for 〈S+〉t with initial nuclear spin
state ρns1 (0) = |n〉〈n| reduce to
〈S+〉t = iωzn〈S+〉t −
([h+]nn′
× [h−]n′n + [h−]nn′ [h+]n′n) exp[iτ(ωn − ωn′)]}〈S+〉t−τ
= iωz〈S+〉t −
dτΣ(τ)〈S+〉t−τ . (19)
In this equation ωzn = (gµBB + [hz]nn′)/~, [hξ]nn′ =
〈n|〈ψ1|hξ|ψ1〉|n′〉 (ξ = ±, z and ψ1 is the orbital quantum
number of the ground state), and ωn = [hz]nn. Similar
equation has been obtained by Coish and Loss,33 and
later by Deng and Hu35 at very low temperature such
that only the lowest two Zeeman sublevels are considered.
Coish and Loss also presented an efficient way to evaluate
Σ(τ) in terms of their Laplace transformations, Σ(s) =
dτe−sτΣ(τ). They gave,
Σ(s) =
([h+]nn′ [h−]n′n
+ [h−]nn′ [h+]n′n)/(s− iδωnn′) , (20)
with δωnn′ =
(ωn − ωn′). With the help of this tech-
nique, we are able to investigate the spin dephasing due
to the hyperfine interaction.
C. Spin decoherence mechanisms
In this subsection we briefly summarize all the spin de-
coherence mechanisms. It is noted that the SOC modifies
all the mechanisms. This is because the SOC modifies the
Zeeman splitting18 and the spin-resolved eigen-states of
the electron Hamiltonian, it hence greatly changes the
effect of the electron-BP scattering.18 These two modifi-
cations, especially the modification of the Zeeman split-
ting, also change the effect of other mechanisms, such
as, the direct spin-phonon coupling due to the phonon-
induced strain, the g-factor fluctuation, the coaction of
the electron-phonon interaction and the hyperfine inter-
action. In the literature, except for the electron-BP scat-
tering, the effects from the SOC are neglected except for
the work by Woods et al.16 in which the spin relaxation
time between the two Zeeman sub-levels of the lowest
electronic state due to the phonon-induced strain is in-
vestigated. However, the perturbation method they used
does not include the important second-order energy cor-
rection. In our investigation, the effects of the SOC are
included in all the mechanisms and we will show that
they lead to marked effects in most cases.
1. SOC together with electron-phonon scattering
As the SOC mixes different spins, the electron-BP scat-
tering can induce spin relaxation and dephasing. The
electron-BP coupling is given by
Hep =
Mqη(aqη + a
−qη)e
iq·r , (21)
where Mqη is the matrix element of the electron-phonon
interaction. In the general form of the electron phonon
interaction He−ph, Φqη = Mqη and Xqη(r,σ) = e
iq·r.
|Mqsl|2 = ~Ξ2q/2ρvslV for the electron-BP coupling due
to the deformation potential. For the piezoelectric cou-
pling, |Mqpl|2 = (32~π2e2e214/κ2ρvslV )[(3qxqyqz)2/q7]
for the longitudinal phonon mode and
j=1,2 |Mqptj |2 =
[32~π2e2e214/(κ
2ρvstq
5V )][q2xq
y + q
z + q
(3qxqyqz)
2/q2] for the two transverse modes. Here
Ξ stands for the acoustic deformation potential; ρ is the
GaAs volume density; V is the volume of the lattice;
e14 is the piezoelectric constant and κ denotes the
static dielectric constant. The acoustic phonon spectra
ωqql = vslq for the longitudinal mode and ωqpt = vstq
for the transverse mode with vsl and vst representing
the corresponding sound velocities.
Besides the electron-BP scattering, electron also cou-
ples to vibrations of the confining potential, i.e., the
surface-phonons,28
δV (r) = −
2ρωqηV
(aqη + a
−qη)ǫqη · ∇rVc(r) ,
in which ǫqη is the polarization vector of a phonon mode
with wave-vector q in branch η. However, this contri-
bution is much smaller than the electron-BP coupling.
Compared to the coupling due to the deformation poten-
tial for example, the ratio of the two coupling strengths is
≈ ~ω0/Ξql0 , where l0 is the characteristic length of the
quantum dot and ~ω0 is the orbital level splitting. The
phonon wave-vector q is determined by the energy differ-
ence between the final and initial states of the transition.
Typically phonon transitions between Zeeman sublevels
and different orbital levels, ql0 ranges from 0.1 to 10.
Bearing in mind that ~ω0 is about 1 meV while Ξ = 7 eV
in GaAs, ~ω0/Ξql0 is about 10
−3. The piezoelectric cou-
pling is of the same order as the deformation potential.
Therefore spin decoherence due to the electron–surface-
phonon coupling is negligible.
2. Direct spin-phonon coupling due to phonon-induced
strain
The direct spin-phonon coupling due to the phonon-
induced strain is given by65
Hstrain =
s(p) · σ , (23)
where hsx = −Dpx(ǫyy − ǫzz), hsy = −Dpy(ǫzz − ǫxx) and
hsz = −Dpz(ǫxx− ǫyy) with p = (px, py, pz) = −i~∇ and
D being the material strain constant. ǫij (i, j = x, y, z)
can be expressed by the phonon creation and annihilation
operators:
ǫij =
qη=l,t1,t2
2ρωqηV
(aq,η + a
−q,η)(ξiηqj
+ ξjηqi)e
iq·r , (24)
in which ξil = qi/q for the longitudinal phonon
mode and (ξxt1 , ξyt1 , ξzt1) = (qxqz, qyqz,−q2‖)/qq‖,
(ξxt2 , ξyt2 , ξzt2) = (qy,−qx, 0)/q‖ for the two trans-
verse phonon modes with q‖ =
q2x + q
y . There-
fore, in the general form of electron-phonon interaction
He−ph, Φqη = −iD
~/(32ρωqηV ) and Xqη(r,σ) =
ijk ǫijk(ξjηqj − ξkηqk)pieiq·rσi with ǫijk denoting the
Levi-Civita tensor.
3. g-factor fluctuation
The spin-lattice interaction via phonon modulation of
the g-factor is given by12
ijkl=x,y,z
AijklµBBiσjǫkl , (25)
where ǫkl is given in Eq. (24) and Aijkl is a
tensor determined by the material. Therefore in
He−ph, Φqη = i
~/(32ρωqηV ) and Xqη(r,σ) =
i,j,k,l Ai,j,k,lµBBi(ξkηqk − ξlηql)σjeiq·r. Due to the
axial symmetry with respect to the z-axis, and keep-
ing in mind that the external magnetic field is along
the z direction, the only finite element of Hg is Hg =
[(A33−A31)ǫzz+A31
i ǫii]~µBBσz/2 with A33 = Azzzz ,
A31 = Azzxx and A66 = Axyxy. A33 + 2A31 = 0.
4. Hyperfine interaction
The hyperfine interaction between the electron and nu-
clear spins is66
HeI(r) =
Av0S · Ijδ(r−Rj) , (26)
where S = ~σ/2 and Ij are the electron and nucleus
spins respectively, v0 = a
0 is the volume of the unit cell
with a0 representing the crystal lattice parameter, and r
(Rj) denotes the position of the electron (the j-th nu-
cleus). A = 4µ0µBµI/(3Iv0) is the hyperfine coupling
constant with µ0, µB and µI representing the perme-
ability of vacuum, the Bohr magneton and the nuclear
magneton separately.
As the Zeeman splitting of the electron is much larger
(three orders of magnitude larger) than that of the nu-
cleus spin, to conserve the energy for the spin relax-
ation processes, there must be phonon-assisted transi-
tions when considering the spin-flip processes. Tak-
ing into account directly the BP induced motion of nu-
clei spin of the lattice leads to a new spin relaxation
mechanism:28
eI−ph(r) = −
Av0S · Ij(u(R0j) ·∇r)δ(r−Rj) , (27)
where u(R0j) =
~/(2ρωqηv0)(aqη + a
qη)ǫqηe
iq·R0
is the lattice displacement vector. Therefore using the
notation of Eq. (10), Φ =
~/(2ρV ωqη) and Xqη =
j Av0S·Ij∇rδ(r−Rj). The second-order process of the
surface phonon and the BP together with the hyperfine
interaction also leads to spin relaxation:
eI−ph(r) = |ℓ2〉
m 6=ℓ1
〈ℓ2|δVc(r)|m〉〈m|HeI (r)|ℓ1〉
εℓ1 − εm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|δVc(r)|ℓ1〉
εℓ2 − εm
〈ℓ1| , (28)
eI−ph = |ℓ2〉
m 6=ℓ1
〈ℓ2|Hep|m〉〈m|HeI(r)|ℓ1〉
εℓ1 − εm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|Hep|ℓ1〉
εℓ2 − εm
〈ℓ1| , (29)
in which |ℓ1〉 and |ℓ2〉 are the eigen states ofHe. By using
the notations in He−ph, Φqη =
~/(2ρωqηv0) and
Xqη = |ℓ2〉ǫqη ·
m 6=ℓ1
εℓ1 − εm
〈ℓ2|[He,P]|m〉
× 〈m|S · Ijδ(r−Rj)|ℓ1〉+
m 6=ℓ2
εℓ2 − εm
〈m|[He,P]|ℓ1〉
Av0〈ℓ2|S · Ijδ(r−Rj)|m〉
〈ℓ1| (30)
for V
eI−ph. Similarly Φqη =Mqη and
Xqη = |ℓ2〉
m 6=ℓ1
〈ℓ2|eiq·r|m〉
εℓ1 − εm
Av0〈m|S · Ij
× δ(r−Rj)|ℓ1〉+
m 6=ℓ2
εℓ2 − εm
〈m|eiq·r|ℓ1〉
Av0〈ℓ2|S · Ijδ(r−Rj)|m〉
〈ℓ1| (31)
for V
eI−ph. Again as the contribution from the surface
phonon is much smaller than that of the BP, V
eI−ph can
be neglected. It is noted that, the direct spin-phonon
coupling due to the phonon-induced strain together with
the hyperfine interaction gives a fourth-order scattering
and hence induces a spin relaxation/dephasing. The in-
teraction is
eI−ph = |ℓ2〉
m 6=ℓ1
〈ℓ2|Hzstrain|m〉〈m|HeI(r)|ℓ1〉
εℓ1 − ǫm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|Hzstrain|ℓ1〉
ǫℓ2 − ǫm
〈ℓ1| , (32)
with Hzstrain = h
sσz/2 only changing the electron energy
but conserving the spin polarization. It can be written
hzs = −
2ρωq,ηV
(ξyηqy−ξzηqz)qzeiq·r . (33)
Comparing this to the electron-BP interaction Eq. (21),
the ratio is ≈ ~Dq/Ξ, which is about 10−3. Therefore,
the second-order term of the direct spin-phonon coupling
due to the phonon-induced strain together with the hy-
perfine interaction is very small and can be neglected.
Also the coaction of the g-factor fluctuation and the hy-
perfine interaction is very small compared to that of the
electron-BP interaction jointly with the hyperfine inter-
action as µBB/Ξ is around 10
−5 when B = 1 T. There-
fore it can also be neglected. In the following, we only
retain the first and the third order terms V
eI−ph and
eI−ph in calculating the spin relaxation time.
The spin dephasing time induced by the hyperfine in-
teraction can be calculated from the non-Markovian ki-
netic Eq. (18), for unpolarized initial nuclear spin state
|n0〉, resulting in
〈S(n0)+ 〉t ∝
dr|ψk(r)|4 cos(
|ψk(r)|2t) ,
where fk is the thermo-equilibrium distribution of the
orbital degree of freedom. When only the lowest two
Zeeman sublevels are considered, assuming a simple
form of the wavefunction, |Ψ(r)|2 = 1
exp(−r2
/d20)
with d‖/az representing the QD diameter/quantum well
width, and r‖ = x
2 + y2, the integration can be carried
〈S(n0)+ 〉t ∝
cos(t/t0)− 1
(t/t0)2
sin(t/t0)
. (35)
Here, t0 = (2πazd
‖)/(Av0) determines the spin dephas-
ing time. Note that t0 is proportional to the factor azd
where az/d
is the characteristic length/area of the QD
along z direction / in the quantum well plane. By solving
Eq. (18) for various n, and summing over n, we obtain
||〈S+〉t|| =
n |〈S
+ 〉t|. We then define the time when
the envelop of ||〈S+〉t|| decays to 1/e of its initial value
as the spin dephasing time T2. As mentioned above the
hyperfine interaction can not transfer an energy of the
order of the Zeeman splitting, thus the hyperfine inter-
action alone can not lead to any spin relaxation.43
In the above discussion, the nuclear spin dipole-dipole
interaction is neglected. Recently, more careful exami-
nations based on quantum cluster expansion method or
pair correlationmethod have been performed.41,42,43,47 In
these works, the nuclear spin dipole-dipole interaction is
also included. This interaction together with the hyper-
fine mediated nuclear spin-spin interaction is the origin
of the fluctuation of the nuclear spin bath. To the lowest
order, the fluctuation is dominated by nuclear spin pair
flips.41,42,43,47 This fluctuation provides the source of the
electron spin dephasing, as the electron spin is coupled
to the nuclear spin system via hyperfine interaction. Our
method used here includes only the hyperfine interaction
to the second order in scattering. However, it is found
that the dipole-dipole-interaction–induced spin dephas-
ing is much weaker than the hyperfine interaction for a
QD with a = 2.8 nm and d0 = 27 nm until the par-
allel magnetic field is larger than ∼ 20 T.42 Therefore,
for the situation in this paper, the nuclear dipole-dipole-
interaction–induced spin dephasing can be ignored.67
III. SPIN DECOHERENCE DUE TO VARIOUS
MECHANISMS
Following the equation-of-motion approach developed
in Sec. II, we perform a numerical calculation of the spin
relaxation and dephasing times in GaAs QDs. Two mag-
netic field configurations are considered: i.e., the mag-
netic fields perpendicular and parallel to the well plane
(along x-axis). The temperature is taken to be T = 4
K unless otherwise specified. For all the cases we con-
sidered in this manuscript, the orbital level splitting is
larger than an energy corresponding to 40 K. Therefore,
the lowest Zeeman sublevels are mainly responsible for
the spin decoherence. When calculating T1, the initial
distribution is taken to be in the spin majority down
state of the eigen-state of the Hamiltonian He with a
Maxwell-Boltzmann distribution fk = C exp[−ǫk/(kBT )]
for different orbital levels (C is the normalization con-
stant). For the calculation of T2, we assign the same
distribution between different orbital levels, but with a
superposition of the two spin states within the same or-
bital level. The parameters used in the calculation are
listed in Table I.8,68,69
TABLE I: Parameters used in the calculation
ρ 5.3× 103 kg/m3 κ 12.9
vst 2.48 × 10
3 m/s g −0.44
vsl 5.29 × 10
3 m/s Ξ 7.0 eV
e14 1.41 × 10
9 V/m m∗ 0.067m0
A 90 µeV A33 19.6
γ0 27.5 Å
3·eV I 3
D 1.59 × 104 m/s a0 5.6534 Å
A. Spin Relaxation Time T1
We now study the spin relaxation time and show how
it changes with the well width a, the magnetic field B
and the effective diameter d0 =
~π/m∗ω0. We also
compare the relative contributions from each relaxation
mechanism.
1. Well width dependence
In Fig. 1(a) and (b), the spin relaxation times induced
by different mechanisms are plotted as function of the
width of the quantum well in which the QD is confined
for perpendicular magnetic field B⊥ = 0.5 T and parallel
magnetic field B‖ = 0.5 T respectively. We first concen-
trate on the perpendicular magnetic field case. In Fig.
1(a), the calculation indicates that the spin relaxation
due to each mechanism decreases with the increase of well
g-factor
strain
eI−ph
eI−ph
B⊥ = 0.5 T
a (nm)
1098765432
10−10
B‖ = 0.5 T
a (nm)
1098765432
10−10
FIG. 1: (Color online) T−11 induced by different mechanisms
vs. the well width for (a): perpendicular magnetic field B⊥ =
0.5 T with (solid curves) and without (dashed curves) the
SOC; (b) parallel magnetic field B‖ = 0.5 T with the SOC.
The effective diameter d0 = 20 nm, and temperature T = 4 K.
Curves with � — T−11 induced by the electron-BP scattering
together with the SOC; Curves with • — T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
width. Particularly the electron-BP scattering mecha-
nism decreases much faster than the other mechanisms.
It is indicated in the figures that when the well width is
small (smaller than 7 nm in the present case), the spin re-
laxation time is determined by the electron-BP scattering
together with the SOC. However, for wider well widths,
the direct spin-phonon coupling due to phonon-induced
strain and the first-order process of hyperfine interac-
tion combined with the electron-BP scattering becomes
more important. The decrease of spin relaxation due to
each mechanism is mainly caused by the decrease of the
SOC which is proportional to a−2. The SOC has two
effects which are crucial. First, in the second order per-
turbation the SOC contributes a finite correction to the
Zeeman splitting which determines the absorbed/emitted
phonon frequency and wave-vector.18 Second, it leads to
spin mixing. The decrease of the SOC thus leads to the
decrease of Zeeman splitting and spin mixing. The for-
mer leads to small phonon wave-vector and small phonon
absorption/emission efficiency.18 Therefore the electron-
BP mechanism decreases rapidly with increasing a. On
the other hand, the other two largest mechanisms can flip
spin without the help of the SOC. The spin relaxations
due to these two mechanisms decrease in a relatively mild
way. It is further confirmed that without SOC they de-
creases in a much milder way with increasing a (dashed
curves in Fig. 1). It is also noted that the spin relaxation
rate due to the g-factor fluctuation is at least six orders
of magnitude smaller than that due to the leading spin
decoherence mechanisms and can therefore be neglected.
It is noted that in the calculation, the SOC is always
included as it has large effect on the eigen-energy and
eigen-wavefunction of the electron.18 We also show the
spin relaxation times induced by the hyperfine interac-
tions (V
eI−ph and V
eI−ph) and the direct spin-phonon
coupling due to the phonon-induced strain but without
the SOC as in the literature.27,28,45 It can be seen clearly
that the spin relaxation that includes the SOC is much
larger than that without the SOC. For example, the spin
relaxation induced by the second-order process of the hy-
perfine interaction together with the BP (V
eI−ph) is at
least one order of magnitude larger when the SOC is in-
cluded than that when the SOC is neglected. This is
because when the SOC is neglected, 〈m|HeI(r)|ℓ1〉 and
〈ℓ2|HeI(r)|m〉 in Eq. (29) are small as the matrix el-
ements of HeI(r) between different orbital energy lev-
els are very small. However, when the SOC is taken
into account, the spin-up and -down levels with differ-
ent orbital quantum numbers are mixed and therefore
|ℓ〉 and |m〉 include the components with the same or-
bital quantum number. Consequently the matrix ele-
ments of 〈m|HeI(r)|ℓ1〉 and 〈ℓ2|HeI(r)|m〉 become much
larger. Therefore, spin relaxation induced by this mech-
anism depends crucially on the SOC.
It is emphasized from the above discussion that the
SOC should be included in each spin relaxation mecha-
nism. In the following calculations it is always included
unless otherwise specified. In particular in reference to
the mechanism of electron-BP interaction, we always con-
sider it together with the SOC.
We further discuss the parallel magnetic field case. In
Fig. 1(b) the spin relaxation times due to different mech-
anisms are plotted as function of the quantum well width
for same parameters as Fig. 1(a), but with a parallel mag-
netic field B‖ = 0.5 T. It is noted that the spin relaxation
rate due to each mechanism becomes much smaller for
small a compared with the perpendicular case. Another
feature is that the spin relaxation due to each mecha-
nism decrease in a much slower rate with increasing a.
The electron-BP mechanism is dominant even at a = 10
nm but decrease faster than other mechanisms with a. It
is expected to be less effective than the V
eI−ph mecha-
nism or V
eI−ph mechanism or the direct spin-phonon cou-
pling due to phonon-induced strain mechanism for large
enough a. The g-factor fluctuation mechanism is negli-
gible again. These features can be explained as follows.
For parallel magnetic field the contribution of the SOC to
Zeeman splitting is much less than in the perpendicular
magnetic field geometry.21 Moreover, this contribution is
negative which makes Zeeman splitting smaller.21 There-
fore, the phonon absorption/emission efficiency becomes
much smaller for small a, i.e., large SOC. When a in-
creases, the Zeeman splitting increases. However, the
spin mixing decreases. The former effect is weak, and
only cancels part of the latter, thus the spin relaxation
due to each mechanism decrease slowly with a.
g-factor
strain
eI−ph
eI−ph
a = 5 nm
B⊥ (T)
543210
a = 10 nm
B⊥ (T)
543210
FIG. 2: (Color online) T−11 induced by different mechanisms
vs. the perpendicular magnetic field B⊥ for d0 = 20 nm and
(a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � —
T−11 induced by the electron-BP scattering; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N
— T−11 induced by the first-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with H
— T−11 induced by the direct spin-phonon coupling due to
phonon-induced strain; Curves with � — T−11 induced by the
g-factor fluctuation.
g-factor
strain
eI−ph
eI−ph
a = 5 nm
B‖ (T)
543210
10−10
a = 10 nm
B‖ (T)
543210
FIG. 3: (Color online) T−11 induced by different mechanisms
vs. the parallel magnetic field B‖ for d0 = 20 nm and (a)
a = 5 nm and (b) 10 nm. T = 4 K. Curves with � —
T−11 induced by the electron-BP scattering; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N
— T−11 induced by the first-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with H
— T−11 induced by the direct spin-phonon coupling due to
phonon-induced strain; Curves with � — T−11 induced by the
g-factor fluctuation.
2. Magnetic Field Dependence
We first study the perpendicular-magnetic-field case.
The magnetic field dependence of T1 for two different
well widths are shown in Fig. 2(a) and Fig. 2(b). In the
calculation, d0 = 20 nm. It can be seen that the ef-
fect of each mechanism increases with the magnetic field.
Particularly the electron-BP mechanism increases much
faster than other ones and becomes dominant at high
magnetic fields. For small well width (5 nm in Fig. 2a),
the spin relaxation induced by the electron-BP scattering
is dominant except at very low magnetic fields (0.1 T in
the figure) where contributions from the first-order pro-
cess of hyperfine interaction together with the electron-
BP scattering and the direct spin-phonon coupling due to
phonon-induced strain also contribute. It is interesting
to see that when a is increased to 10 nm, the electron-BP
scattering is the largest spin relaxation mechanism only
at high magnetic fields (>1.1 T). For 0.4 T < B⊥ < 1.1
T (B⊥ < 0.4 T), the direct spin-phonon coupling due
to the phonon-induced strain (the first order hyperfine
interaction together with the BP ) becomes the largest
relaxation mechanism. It is also noted that there is no
single mechanism which dominates the whole spin relax-
ation. Two or three mechanisms are jointly responsible
for the spin relaxation. It is indicated that the spin relax-
ations induced by different mechanisms all increase with
B⊥. This can be understood from a perturbation theory:
when the magnetic field is small the spin relaxation be-
tween two Zeeman split states for each mechanism is pro-
portional to n̄(∆E)(∆E)m (∆E is the Zeeman splitting)
with m = 7 for electron-BP scattering due to the de-
formation potential18,25 and for the second-order process
of the hyperfine interaction together with the electron-
BP scattering due to the deformation potential V
eI−ph;
m = 5 for electron-BP scattering due to the piezoelec-
tric coupling15,18,25 and for the second-order process of
the hyperfine interaction together with the electron-BP
scattering due to the piezoelectric coupling V
eI−ph;
27 and
m = 5 for the direct spin-phonon coupling due to phonon-
induced strain;15 m = 1 for the first-order process of the
hyperfine interaction together with the BP V
eI−ph. The
spin relaxation induced by the g-factor fluctuation is pro-
portional to n̄(∆E)(∆E)5B2⊥. For most of the cases stud-
ied, ∆E is smaller than kBT , hence n̄(∆E) ∼ kBT/∆E,
and n̄(∆E)(∆E)m ∼ (∆E)m−1. m > 1 hold for all mech-
anism except the V
eI−ph mechanism, therefore the spin
relaxation due to these mechanisms increases with in-
creasing B⊥. However, from Eq. (27) one can see that
it has a term with ∇r, which indicates that the effect of
this mechanism is proportional to 1/d0. As the vector
potential of the magnetic field increases the confinement
of the QD and gives rise to smaller effective diameter d0,
this mechanism also increases with the magnetic field in
the perpendicular magnetic field geometry.
We then study the case with the magnetic field par-
allel to the quantum well plane. In Fig. 3 the spin re-
laxation induced by different mechanisms are plotted as
function of the parallel magnetic field B‖ for two dif-
ferent well widths. In the calculation, d0 = 20 nm. It
can be seen that, similar to the case with perpendicular
magnetic field, the effects of most mechanisms increase
with the magnetic field. Also the electron-BP mechanism
increases much faster than the other ones and becomes
dominant at high magnetic fields. However, without the
orbital effect of the magnetic field in the present con-
figuration, the effect of V
eI−ph changes very little with
the magnetic field. For both small (5 nm in Fig. 3(a))
and large (10 nm in Fig. 3(b)) well widths, the electron-
BP scattering is dominant except at very low magnetic
field (0.1 T in the figure), where the first-order process of
the hyperfine interaction together with the electron-BP
interaction V
eI−ph also contributes.
a = 5 nm
d0 (nm)
3025201510
g-factor
strain
eI−ph
eI−ph
a = 10 nm
d0 (nm)
3025201510
10−10
FIG. 4: (Color online) T−11 induced by different mechanisms
vs. the effective diameter d0 for B⊥ = 0.5 T and (a) a = 5 nm
and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by
the electron-BP scattering; Curves with • – T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
3. Diameter Dependence
We now turn to the investigation of the diameter de-
pendence of the spin relaxation. We first concentrate on
the perpendicular magnetic field geometry. The spin re-
laxation rate due to each mechanism is shown in Fig. 4a
for a small (a = 5 nm) and Fig. 4b for a large (a = 10
nm) well widths respectively with a fixed perpendicu-
lar magnetic field B⊥ = 0.5 T. In the figure, the spin
relaxation rate due each mechanism except V
eI−ph in-
creases with the effective diameter. Specifically, the ef-
fect of the electron-BP mechanism increases very fast,
while the effect of the direct spin-phonon coupling due to
phonon-induced strain mechanism increases very mildly.
The V
eI−ph decreases with d0 slowly. Other mechanisms
are unimportant. The electron-BP mechanism eventu-
ally dominates spin relaxation when the diameter is large
a = 5 nm
d0 (nm)
3025201510
10−10
g-factor
strain
eI−ph
eI−ph
a = 10 nm
d0 (nm)
3025201510
FIG. 5: (Color online) T−11 induced by different mechanisms
vs. the effect diameter d0 with B‖ = 0.5 T and (a) a = 5 nm
and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by
the electron-BP scattering; Curves with • — T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
enough. The threshold increases from 12 nm to 26 nm
when the well width increases from 5 nm to 10 nm. For
small diameter the V
eI−ph and the direct spin-phonon
coupling due to phonon-induced strain mechanism dom-
inate the spin relaxation. The increase/decrease of the
spin relaxation due to these mechanisms can be under-
stood from the following. The effect of the SOC on the
Zeeman splitting is proportional to d20 for small magnetic
field.18 The increase of d0 thus leads to a increase of Zee-
man splitting, therefore the efficiency of the phonon ab-
sorption/emission increases. Another effect is that the in-
crease of d0 will increase the phonon absorption/emission
efficiency due to the increase of the form factor.18 Thus
the spin relaxation increases. Moreover, the spin mix-
ing is also proportional to d0 also.
18 This leads to much
faster increasing of the effect of the electron-BP mecha-
nism and the V
eI−ph mechanism. However, the spin re-
laxation due to V
eI−ph decreases with the diameter. This
is because V
eI−ph contains a term ∇r [Eq. (27)] which
decreases with the increase of d0. Physically speaking,
the decrease of the effect of V
eI−ph is due to the fact
that the spin mixing due to the hyperfine interaction de-
creases with the increase of the number of nuclei within
the dot N as the random Overhauser field is proportional
to 1/
N . The spin relaxation induced by the g-factor is
also negligible here for both small and large well width.
We then turn to the parallel magnetic field case. In
the calculation, B‖ = 0.5 T. The results are shown for
both small well width (a = 5 nm in Fig. 5(a)) and large
well width (a = 10 nm in Fig. 5(b)) respectively. Simi-
lar to the perpendicular magnetic field case, the effect of
every mechanism except the V
eI−ph mechanism increases
with increasing diameter. The effect of the electron-BP
mechanism increases fastest and becomes dominant for
d0 > 12 nm for both small and large well width. For
d0 < 12 nm for the two cases the first-order process
of the V
eI−ph mechanism becomes dominant. The ef-
fect of the V
eI−ph mechanism become larger than that of
the direct spin-phonon coupling due to phonon-induced
strain mechanism. However, these two mechanism are
still unimportant and becomes more and more unimpor-
tant for larger d0. Here, the spin relaxation induced by
the g-factor is negligible.
4. Comparison with Experiment
In this subsection, we apply our analysis to experiment
data in Ref. 7. We first show that our calculation is in
good agreement with the experimental results. Then we
compare contributions from different mechanisms to spin
relaxation as function of the magnetic field. In the cal-
culation we choose the quantum dot diameter d0 = 56
nm (~ω0 = 1.1 meV as in experiment). The quantum
well is taken to be an infinite-depth well with a = 13
nm. The Dresselhaus SOC parameter γ0〈k2z〉 is taken
to be 4.5 meV·Åand the Rashba SOC parameter is 3.3
meV·Å. T = 0 K as kBT ≪ gµBB in the experiment.
The magnetic field is applied parallel to the well plane in
[110]-direction. The Dresselhaus cubic term is also taken
into consideration. All these parameters are the same
with (or close to) those used in Ref. 24 in which a cal-
culation based on the electron-BP scattering mechanism
agrees well with the experimental results. For this mech-
anism, we reproduce their results. The spin relaxation
time measured by the experiments (black dots with er-
ror bar in the figure) almost coincide with the calculated
spin relaxation time due to the electron-BP scattering
mechanism (curves with � in the figure).71 It is noted
from the figure that other mechanisms are unimportant
for small magnetic field. However, for large magnetic
field the effect of the direct-spin phonon coupling due
to phonon-induced strain becomes comparable with that
of the electron-BP mechanism. At B‖ = 10 T, the two
differs by a factor of ∼ 5.
g-factor
strain
eI−ph
eI−ph
B‖ (T)
15129630
10−10
10−12
FIG. 6: (Color online) T−11 induced by different mechanisms
vs. the parallel magnetic field B‖ in the [110] direction for
d0 = 56 nm and a = 13 nm with both the Rashba and Dres-
selhaus SOCs. T = 0 K. The black dots with error bar is the
experimental results in Ref. 7. Curves with � — T−11 induced
by the electron-BP scattering; Curves with • — T−11 induced
by the second-order process of the hyperfine interaction to-
gether with the BP (V
eI−ph); Curves with N — T
1 induced
by the first-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with H — T
1 induced by the
direct spin-phonon coupling due to phonon-induced strain;
Curves with � — T−11 induced by the g-factor fluctuation.
B. Spin Dephasing Time T2
In this subsection, we investigate the spin dephasing
time for different well widths, magnetic fields and QD
diameters. As in the previous subsection, the contribu-
tions of the different mechanisms to spin dephasing are
compared.70 To justify the first Born approximation in
studying the hyperfine interaction induced spin dephas-
ing, we focus mainly on the high magnetic field regime
of B > 3.5 T. A typical magnetic field is 4 T. We also
demonstrate via extrapolation that in the low magnetic
field regime spin dephasing is dominated by the hyperfine
interaction.
1. Well Width Dependence
In Fig. 7 the well width dependence of the spin dephas-
ing induced by different mechanisms is presented under
the perpendicular (a) and parallel (b) magnetic fields. In
the calculations B⊥ = 4 T/B‖ = 4 T and d0 = 20 nm. It
can be seen in both figures that the spin dephasing due to
each mechanism decreases with a. Moreover, the spin de-
phasing due to the electron-BP scattering decreases much
faster than that due to the hyperfine interaction. These
features can be understood as following. The spin de-
phasing due to electron-BP scattering depends crucially
on the SOC. As the SOC is proportional to a−2, the spin
dephasing decreases fast with a. For the hyperfine inter-
action, from Eq. (35) one can deduce that the decay rate
of ||〈S+〉t|| is mainly determined by the factor 1/(azd2‖)
(here az = a), which thus decreases with a, but in a very
mild way. The fast decrease of the electron-BP mecha-
nism makes it eventually unimportant. For the present
perpendicular-magnetic-field case the threshold is around
2 nm. For parallel magnetic field it is even smaller. A
higher temperature may enhance the electron-BP mech-
anism (see discussion in Sec. V) and make it more im-
portant than the hyperfine mechanism. It is noted that
other mechanisms contribute very little to the spin de-
phasing. Thus, in the following discussion, we do not con-
sider these mechanisms. Comparing Figs 7(a) and (b),
one finds that a main difference is that the electron-BP
mechanism is less effective for the parallel-magnetic-field
case. As has been discussed in the previous subsection,
the spin mixing and the Zeeman splitting in the parallel
filed case is smaller than those in the perpendicular field
case. Therefore, the electron-BP mechanism is weakened
markedly.
Similar to Fig. 1, the SOC is always included in the
computation as it has large effect on the eigen-energy and
eigen-wavefunction of the electrons. The spin dephasings
calculated without the SOC for the hyperfine interaction,
the direct spin-phonon coupling due to phonon-induced
strain and the g-factor fluctuation are also shown in Fig.
7(a) as dashed curves. It can be seen from the figure
that for the spin dephasings induced by the direct spin-
phonon coupling due to phonon-induced strain and by
the g-factor fluctuation, the contributions with the SOC
are much larger than those without. This is because when
the SOC is included, the fluctuation of the effective field
induced by both mechanisms becomes much stronger and
more scattering channels are opened. However, what
should be emphasized is that the spin dephasings induced
by the hyperfine interaction with and without the SOC
are nearly the same (the solid and the dashed curves
nearly coincide). That is because the change of the wave-
function Ψ(r) due to the SOC is very small (less than 1
% in our condition) and therefore the factor 1/(azd
‖) is
almost unchanged when the SOC is neglected. Thus the
spin dephasing rate is almost unchanged.
In the inset of Fig. 7(a), the time evolution of ||〈S+〉t||
induced by the hyperfine interaction is shown, with a = 2
nm. It can be seen that ||〈S+〉t|| decays very fast and de-
creases to less than 10 % of its initial value within the
first two oscillating periods. Therefore, T2 is determined
by the first two or three periods of ||〈S+〉t||. Thus the
correction of the long time dynamics due to higher or-
der scattering33 contributes little to the spin dephasing
time. For quantum computation and quantum informa-
tion processing, the initial, e.g., 1 % decay of ||〈S+〉t||
may be more important than the 1/e decay.42,43 Indeed,
the spin dephasing time defined by the exponential fitting
of 1 % decay is short than that defined by the 1/e decay.
However, the two differs less than 5 times. For a rough
B⊥ = 4 T
t (µs)
6543210
a (nm)
1098765432
eI−ph
eI−ph
g-factor
strain
hyperfine
B‖ = 4 T
a (nm)
1098765432
10−10
10−15
10−20
10−25
FIG. 7: (Color online) T−12 induced by different mechanisms
vs. the well width for d0 = 20 nm. T = 4 K. (a): B⊥ = 4 T
with (solid curves) and without (dashed curves) the SOC; (b):
B‖ = 4 T only with the SOC. Curve with � — T
2 induced
by the electron-BP interaction; Curves with • — T−12 induced
by the hyperfine interaction; Curves with H — T−12 induced
by the direct spin-phonon coupling due to phonon-induced
strain; Curves with � — T−12 induced by g-factor fluctuation;
N — T−12 induced by the second-order process of the hyper-
fine interaction together with the BP (V
eI−ph); Curves with �
— T−12 induced by the first-order process of the hyperfine in-
teraction together with the BP (V
eI−ph). The time evolution
of ||〈S+〉t|| induced by the hyperfine interaction with a = 2
nm is shown in the inset of (a).
comparison of contributions from different mechanisms to
spin dephasing where only the order-of-magnitude differ-
ence is concerned (see Figs. 7-9), this difference due to
the definition does not jeopardize our conclusions.
2. Magnetic Field Dependence
We then investigate the magnetic field dependence of
the spin dephasing induced by the electron-BP scatter-
ing and by the hyperfine interaction for two different well
widths (a = 3 nm and a = 5 nm) with both perpendicular
and parallel magnetic field. From Fig. 8(a) and (b) one
hyperfine
B⊥ (T)
87.576.565.554.543.5
B‖ (T)
87.576.565.554.543.5
FIG. 8: (Color online) T−12 induced by the electron-BP scat-
tering and the hyperfine interaction vs. (a): the perpendic-
ular magnetic field B⊥ ; (b): the parallel magnetic field B‖
for a = 3 nm (solid curves) and 5 nm (dashed curves). T = 4
K, and d0 = 20 nm. Curves with � — T
2 induced by the
electron-BP interaction; Curves with • — T−12 induced by the
hyperfine interaction.
can see that the spin dephasing due to the electron-BP
scattering increases with magnetic field, whereas that due
to the hyperfine interaction decreases with magnetic field.
Thus, the electron-BP mechanism eventually dominates
the spin dephasing for high enough magnetic field. The
threshold is Bc⊥ = 4 T / B
= 7 T for a = 3 nm with per-
pendicular/parallel magnetic field. For larger well width,
e.g., a = 5 nm with parallel magnetic field or perpendicu-
lar magnetic field, the threshold magnetic fields increase
to larger than 8 T. The different magnetic field depen-
dences above can be understood as following. Besides
spin relaxation, the spin-flip scattering also contributes
to spin dephasing.20 As has been demonstrated in Sec.
IIIA, the electron-BP scattering induced spin-flip tran-
sition rate increases with the magnetic field. Therefore
the spin dephasing rate increases with the magnetic field
also. In contrast, spin dephasing induced by the hyper-
fine interaction decreases with the magnetic field. This
is because when the magnetic field becomes larger, the
fluctuation of the effective magnetic field due to the sur-
rounding nuclei becomes insignificant compared. There-
fore, the hyperfine-interaction-induced spin dephasing is
reduced. Similar results have been obtained by Deng and
Hu.44
hyperfine
B⊥ = 4 T
d0 (nm)
3025201510
B‖ = 4 T (b)
d0 (nm)
3025201510
FIG. 9: (Color online) T−12 induced by the electron-BP scat-
tering and the hyperfine interaction vs. the effective diameter
d0 T = 4 K. (a): B⊥ = 4 T ; (b): B‖ = 4 T for a = 3 nm
(solid curves) and 5 nm (dashed curves). Curves with � —
T−12 induced by the electron-BP interaction; Curves with •
— T−12 induced by the hyperfine interaction.
3. Diameter Dependence
In Fig. 9 the spin dephasing times induced by the
electron-BP scattering and the hyperfine interaction are
plotted as function of the diameter d0 for a small (a = 3
nm) and a large (a = 5 nm) well widths. In the cal-
culation, B⊥ = 4 T in Fig. 9(a) and B‖ = 4 T in (b).
It is noted that the effect of the electron-BP mechanism
increases rapidly with d0, whereas the effect of the hy-
perfine mechanism decreases slowly. Consequently, the
electron-BP mechanism eventually dominates the spin
dephasing for large enough d0. The threshold is d
0 = 19
(27) nm for a = 3 (5) nm case with the perpendicular
magnetic field and dc0 = 26 (30) nm for a = 3 (5) nm
case under the parallel magnetic field. As has been dis-
cussed in Sec. IIIA, both the effect of the SOC and the ef-
ficiency of the phonon absorption/emission increase with
d0. Therefore, the spin dephasing due to the electron-BP
mechanism increases rapidly with d0.
18,21 The decrease
of the effect of the hyperfine interaction is due to the de-
crease of the factor 1/(azd
) [Eq. 35] with the diameter
IV. SPIN RELAXATION TIMES FROM FERMI
GOLDEN RULE AND FROM EQUATION OF
MOTION
In this section, we will try to find a proper method to
average over the transition rates from the Fermi Golden
rule, τ−1i→f , to give the spin relaxation time T1. In the
limit of small SOC, we rederive Eq. (1) from the equation
of motion. We further show that Eq. (1) fails for large
SOC where a full calculation from the equation of motion
is needed.
T = 12 K
10987654321
γ = γ0
T (K)
4035302520151050
FIG. 10: (Color online) Spin relaxation time T1 calculated
from the equation-of-motion approach (�) v.s. that obtained
from Eq. (1) (•) as function of (a): the strength of the SOC
for T=12 K; (b): the temperature for γ = γ0. The well width
a = 5 nm, perpendicular magnetic field B⊥ = 0.5 T, QD
diameter d0 = 30 nm. The ratio of the two R is also plotted
in the figure. Note the scale of T−11 is at the right hand side
of the frame.
We first rederive Eq. (1) for small SOC from the equa-
tion of motion. In QDs, the orbital level splitting is
usually much larger than the Zeeman splitting. Each
Zeeman sublevel has two states: one with majority up-
spin, the other with majority down-spin. We call the
former “minus state” (as it corresponds to a lower en-
ergy) while the latter “plus state”. For small SOC,
the spin mixing is small. Thus we neglect the much
smaller contribution from the off-diagonal terms of the
density matrix to Sz. Therefore Sz(t) =
z fi±(t)
where i± denotes the plus/minus state of the i-th or-
bital state. For small SOC, the spin relaxation is
much slower than the orbital relaxation.25,55 This im-
plies that the time takes to establish equilibrium within
the plus/minus states is much smaller than the spin relax-
ation time. Thus we can assume a equilibrium (Maxwell-
Boltzmann) distribution between the plus/minus states
at any time. The distribution function is therefore given
by fi±(t) = N±(t) exp(−εi±/kBT )/Z±. Here N±(t) =
i fi±(t) is the total probability of the plus/minus states
with N+(t) + N−(t) = 1 for single electron in QD and
i exp(−εi±/kBT ) is the partition function for
the plus/minus state. At equilibrium, N± = N
± . The
equation for Sz(t) is hence,
Sz(t) =
[Sz(t)− Seqz ]
Si±z exp(−εi±/kBT )/Z±
δN±(t) , (36)
with δN±(t) = N±(t) − Neq± . As the orbital level
splitting is usually much larger than the Zeeman split-
ting, the factor exp(−εi±/kBT )/Z± can be approximated
by exp(−εi0/kBT )/Z0 with εi0 = 12 (εi+ + εi−) and
i exp[−εi0/kBT ]. Further using the particle-
conservation relation
± δN±(t) = 0, one has
Sz(t) = [
(Si+z −Si−z ) exp(−εi0/kBT )/Z0]
δN+(t) .
As Sz(t) − Seqz = [δN+(t)/Z0]
Si−z ) exp(−εi0/kBT ), one finds that the spin relax-
ation time is nothing but the relaxation time of N+.
The next step is to derive the equation of d
δN+(t),
which is given in our previous work:49
δN+(t) =
δfi+(t)
[τ−1i+→f−δfi+(t)− τ
i−→f+δfi−(t)]
[τ−1i+→f− + τ
i−→f+]
e−εi0/kBT
δN+(t) .(38)
Thus spin relaxation time is given by,
(τ−1i+→f− + τ
i−→f+)
e−εi0/kBT
. (39)
Furthermore, substituting e−εi0/kBT /Z0 by f
exp(−εi±/kBT )/Z±, we have
(τ−1i+→f−f
i+ + τ
i−→f+f
i−) . (40)
This is exactly Eq. (1).
T1/T2
T (K)
10−10
10−11
10−12
10−13
2520151050
FIG. 11: (Color online) Spin relaxation time T1, spin dephas-
ing time T2 and T1/T2 against temperature T . B⊥ = 4 T,
a = 5 nm and d0 = 30 nm. Note the scale of T1 and T2 is at
the right hand side of the frame.
For large SOC, or large spin mixing due to anticrossing
of different spin states,19,25 the spin relaxation rate be-
comes comparable with the orbital relaxation rate. Fur-
thermore, the decay of the off-diagonal term of the den-
sity matrix should contribute to the decay of Sz. There-
fore, the above analysis does not hold. In this case, it is
difficult to obtain such a formula, and a full calculation
from the equation-of-motion is needed.
In Fig. 10(a), we show that (for T = 12 K, a = 5
nm, B⊥ = 0.5 T, d0 = 30 nm) the spin relaxation times
T1 calculated from equation-of-motion approach and that
obtained from Eq. (40). Here, for simplicity and without
loss of generality, we consider only the electron-BP scat-
tering mechanism. The discrepancy of T1 obtained from
the two approaches increases with γ. At γ = 10γ0, the
ratio of the two becomes as large as ∼ 3. In Fig. 10(b),
we plot the spin relaxation times obtained via the two
approaches as function of temperature for γ = γ0 with
other parameters remaining unchanged. It is noted that
the discrepancy of T1 obtained from the two approaches
increases with temperature. For high temperature, the
higher levels are involved in the spin dynamics where the
SOC becomes larger. At 40 K, the discrepancy is as large
as 60 %. The ratio increases very slowly for T < 20 K
where only the lowest two Zeeman sublevels are involved
in the dynamics.
g-factor
strain
eI−ph
eI−ph
B⊥ = 0.5 T
T (K)
2520151050
10−10
10−15
B⊥ = 0.9 T
T (K)
2520151050
FIG. 12: (Color online) Spin relaxation time T1 against tem-
perature T for (a): B⊥ = 0.5 T; (b): B⊥ = 0.9 T. a = 10
nm and d0 = 20 nm. Curves with � — T
1 induced by the
electron-BP scattering together with the SOC; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N —
T−11 induced by the first-order process of the hyperfine inter-
action together with the BP (V
eI−ph); Curves with H — T
induced by the direct spin-phonon coupling due to phonon-
induced strain; Curves with � — T−11 induced by the g-factor
fluctuation.
V. TEMPERATURE DEPENDENCE OF SPIN
RELAXATION TIME T1 AND SPIN DEPHASING
TIME T2
We first study the relative magnitude of the spin re-
laxation time T1 and the spin dephasing time T2. We
consider a QD with d0 = 30 nm and a = 5 nm at B⊥ = 4
T where the largest contribution to both spin relaxation
and dephasing comes from the electron-BP scattering
(see Fig. 4(a) and Fig. 9(a), we have checked that the
electron-BP scattering mechanism is dominant through-
out the temperature range). From Fig. 11, one finds that
when the temperature is low (T < 5 K in the figure),
T2 = 2T1, which is in agreement with the discussion in
Ref. 20. However, T1/T2 increases very quickly with T
and for T = 20 K, T1/T2 ∼ 2 × 102. This is understood
from the fact that when T is low, the electron mostly dis-
tributes in the lowest two Zeeman sublevels. For small
SOC, Golovach et al. have shown via perturbation the-
ory that phonon induces only the spin-flip noise in the
leading order. Consequently, T2 = 2T1.
20 When the tem-
perature becomes comparable with the orbital level split-
ting ~ω0, the distribution over the upper orbital levels is
not negligible any more. As mentioned previously, the
SOC contributes a non-trivial part to the Zeeman split-
ting. Specifically, the second order energy correction due
to the SOC contributes to the Zeeman splitting. The
energy correction for different orbital levels is generally
unequal (always larger for higher levels). When the elec-
tron is scattered by phonons randomly from one orbital
state to another one with the same major spin polariza-
tion, the frequency of its precession around z direction
changes. Continuous scattering leads to random fluctu-
ation of the precession frequency and thus leads to spin
dephasing.29,46 Note that this fluctuation only leads to a
phase randomization of S+, but not flips the z compo-
nent spin Sz, i.e., not leads to spin relaxation. There-
fore, the spin dephasing becomes stronger than the spin
relaxation for high temperatures. Moreover, this effect
increases with temperature rapidly as the distribution
over higher levels and the phonon numbers both increase
with temperature.
We further study the temperature dependence of spin
relaxation for lower magnetic field and larger quantum
well width where other mechanisms may be more im-
portant than the electron-BP mechanism. In Fig. 12(a),
the spin relaxation time is plotted as function of tem-
perature for B⊥ = 0.5 T, a = 10 nm and d0 = 20 nm.
It is seen from the figure that the direct spin-phonon
coupling due to phonon-induced strain mechanism dom-
inates the spin relaxation throughout the temperature
range. It is also noted that for T ≤ 4 K the spin relax-
ation rates induced by different mechanisms all increase
with temperature according to the phonon number fac-
tor 2n̄(Ez1) + 1 with Ez1 being the Zeeman splitting of
the lowest Zeeman sublevels. However, for T > 4 K, the
spin relaxation rates induced by the direct spin-phonon
coupling due to phonon-induced strain and the electron-
BP interaction increase rapidly with temperature, while
the spin relaxation rates induced by V
eI−ph and V
eI−ph
increase mildly according to 2n̄(Ez1) + 1 throughout the
temperature range. These features can be understood as
what follows. For T ≤ 4 K, the distribution over the high
levels is negligible. Only the lowest two Zeeman sublevels
involve in the spin dynamics. The spin relaxation rates
thus increase with 2n̄(Ez1) + 1 and the relative impor-
tance of each mechanism does not change. Therefore, our
previous analysis on comparison of relative importance of
different spin decoherence mechanisms at 4 K holds true
for the range 0 ≤ T ≤ 4 K. When the temperature gets
higher, the contribution from higher levels becomes more
important. Although the distribution at the higher lev-
els is still very small, for the direct spin-phonon coupling
mechanism, the transition rates between the higher lev-
els and that between higher levels and the lowest two
sublevels are very large. For the electron-BP mechanism
the transition rates between the higher levels are very
large due to the large SOC in these levels. Therefore, the
contribution from the higher levels becomes larger than
that from the lowest two sublevels. Consequently, the in-
crease of temperature leads to rapid increase of the spin
relaxation rates. However, for the two hyperfine mech-
anisms: the V
eI−ph and the V
eI−ph, the spin relaxation
rates does not change much when the higher levels are
involved. They thus increase by the phonon number fac-
In Fig. 12(b) we show the temperature dependence of
the spin relaxation time for the same condition but with
B = 0.9 T. It is noted that the spin relaxation rate due
to the electron-BP mechanism catches up with that in-
duced by the direct spin-phonon coupling due to phonon-
induced strain at T = 9 K and becomes larger for higher
temperature. This indicates that the temperature depen-
dence of the two mechanisms are quite different.
In Fig. 13 we show the spin dephasing induced by
electron-BP scattering and the hyperfine interaction as
function of temperature for B⊥ = 4 T, a = 10 nm and
d0 = 20 nm. We choose the conditions so that the spin
dephasing is dominated by the hyperfine interaction at
low temperature. However, the effect of the electron-BP
mechanism increases with temperature quickly while that
of the hyperfine interaction remains nearly unchanged.
The fast increase of the effect from the electron-BP scat-
tering is due to three factors: 1) the increase of the
phonon number; 2) the increase of scattering channels;
and 3) the increase of the SOC induced spin mixing in
higher levels. On the other hand, from Eq. 35, one can
deduce that the spin dephasing rate of the hyperfine in-
teraction depends mainly on the factor 1/(azd
) with
is the characteristic length/area along the z di-
rection / in the quantum well plane. For higher levels,
the d2‖ is larger, but only about a factor smaller than 10.
Thus the effect of the hyperfine interaction increases very
slowly with temperature.
It should be noted that in the above discussion, we
neglected the two-phonon scattering mechanism,15,46,50
which may be important at high temperature. The con-
tribution of this mechanism should be calculated via
the equation-of-motion approach developed in this paper,
and compared with the contribution of other mechanisms
showed here.
VI. CONCLUSION
In conclusion, we have investigated the longitudinal
and transversal spin decoherence times T1 and T2, called
spin relaxation time and spin dephasing time, in differ-
ent conditions in GaAs QDs from the equation-of-motion
approach. Various mechanisms, including the electron-
BP scattering, the hyperfine interaction, the direct spin-
hyperfine
T (K)
2520151050
FIG. 13: (Color online) Spin relaxation time T1 against tem-
perature T . B⊥ = 4 T, a = 10 nm and d0 = 20 nm. Curves
with � — T−12 induced by the electron-BP scattering together
with the SOC; Curves with • — T−12 induced by the hyperfine
interaction
phonon coupling due to phonon-induced strain and the
g-factor fluctuation are considered. Their relative im-
portance is compared. There is no doubt that for spin
decoherence induced by electron-BP scattering, the SOC
must be included. However, for spin decoherence induced
by the hyperfine interaction, the direct spin-phonon cou-
pling due to phonon-induced strain, g-factor fluctua-
tion, and hyperfine interaction combined with electron-
phonon scattering, the SOC is neglected in the existing
literature.27,28,45 Our calculations have shown that, as
the SOC has marked effect on the eigen-energy and the
eigen-wavefunction of the electron, the spin decoherence
induced by these mechanisms with the SOC is larger than
that without it. Especially, the decoherence from the
second-order process of hyperfine interaction combined
with the electron-BP interaction increases at least one
order of magnitude when the SOC is included. Our cal-
culations show that, with the SOC, in some conditions
some of these mechanisms (except g-factor fluctuation
mechanism) can even dominate the spin decoherence.
There is no single mechanism which dominates spin re-
laxation or spin dephasing in all parameter regimes. The
relative importance of each mechanism varies with the
well width, magnetic field and QD diameter. In particu-
lar, the electron-BP scattering mechanism has the largest
contribution to spin relaxation and spin dephasing for
small well width and/or high magnetic field and/or large
QD diameter. However, for other parameters the hyper-
fine interaction, the first-order process of the hyperfine
interaction combined with electron-BP scattering, and
the direct spin-phonon coupling due to phonon-induced
strain can be more important. It is noted that the g-
factor fluctuation always has very little contribution to
spin relaxation and spin dephasing which can thus be
neglected all the time. For spin dephasing, the electron-
BP scattering mechanism and the hyperfine interaction
mechanism are more important than other mechanisms
for magnetic field higher than 3.5 T. For this regime,
other mechanisms can thus be neglected. It is also shown
that spin dephasing induced by the electron-BP mecha-
nism increases rapidly with temperature. Extrapolated
from our calculation, the hyperfine interaction mecha-
nism is believed to be dominant for small magnetic field.
We also discussed the problem of finding a proper
method to average over the transition rates τ−1i→f obtained
from the Fermi Golden rule, to give the spin relaxation
time T1 at finite temperature. For small SOC, we red-
erived the formula for T1 at finite temperature used in
the existing literature18,51,52 from the equation of mo-
tion. We further demonstrated that this formula is in-
adequate at high temperature and/or for large SOC. For
such cases, a full calculation from the equation-of-motion
approach is needed. The equation-of-motion approach
provides an easy and powerful way to calculate the spin
decoherence at any temperature and SOC.
We also studied the temperature dependence of spin re-
laxation T1 and dephasing T2. We show that for very low
temperature if the electron only distributes on the low-
est two Zeeman sublevels, T2 = 2T1. However, for higher
temperatures, the electron spin dephasing increases with
temperature much faster than the spin relaxation. Con-
sequently T1 ≫ T2. The spin relaxation and dephasing
due to different mechanisms are also compared.
Acknowledgments
This work was supported by the Natural Science
Foundation of China under Grant Nos. 10574120 and
10725417, the National Basic Research Program of China
under Grant No. 2006CB922005 and the Innovation
Project of Chinese Academy of Sciences. Y.Y.W. would
like to thank J. L. Cheng for valuable discussions.
∗ Author to whom correspondence should be addressed;
Electronic address: mwwu@ustc.edu.cn
† Mailing Address
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70 It should be mentioned that one effect is not included :
when electron is scattered by phonon from one orbital state
to another, it feels a difference in the spin precession fre-
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http://arxiv.org/abs/0711.0479

  The longitudinal and transversal spin decoherence times, $T_1$ and $T_2$, in
semiconductor quantum dots are investigated from equation-of-motion approach
for different magnetic fields, quantum dot sizes, and temperatures. Various
mechanisms, such as the hyperfine interaction with the surrounding nuclei, the
Dresselhaus spin-orbit coupling together with the electron--bulk-phonon
interaction, the $g$-factor fluctuations, the direct spin-phonon coupling due
to the phonon-induced strain, and the coaction of the
electron--bulk/surface-phonon interaction together with the hyperfine
interaction are included. The relative contributions from these spin
decoherence mechanisms are compared in detail. In our calculation, the
spin-orbit coupling is included in each mechanism and is shown to have marked
effect in most cases. The equation-of-motion approach is applied in studying
both the spin relaxation time $T_1$ and the spin dephasing time $T_2$, either
in Markovian or in non-Markovian limit. When many levels are involved at finite
temperature, we demonstrate how to obtain the spin relaxation time from the
Fermi Golden rule in the limit of weak spin-orbit coupling. However, at high
temperature and/or for large spin-orbit coupling, one has to use the
equation-of-motion approach when many levels are involved. Moreover, spin
dephasing can be much more efficient than spin relaxation at high temperature,
though the two only differs by a factor of two at low temperature.

Reexamination of spin decoherence in semiconductor quantum dots from
equation-of-motion approach
J. H. Jiang,1, 2 Y. Y. Wang,2 and M. W. Wu1, 2, ∗
Hefei National Laboratory for Physical Sciences at Microscale,
University of Science and Technology of China, Hefei, Anhui, 230026, China
Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
(Dated: November 2, 2018)
The longitudinal and transversal spin decoherence times, T1 and T2, in semiconductor quantum
dots are investigated from equation-of-motion approach for different magnetic fields, quantum dot
sizes, and temperatures. Various mechanisms, such as the hyperfine interaction with the surrounding
nuclei, the Dresselhaus spin-orbit coupling together with the electron–bulk-phonon interaction, the
g-factor fluctuations, the direct spin-phonon coupling due to the phonon-induced strain, and the
coaction of the electron–bulk/surface-phonon interaction together with the hyperfine interaction
are included. The relative contributions from these spin decoherence mechanisms are compared in
detail. In our calculation, the spin-orbit coupling is included in each mechanism and is shown to
have marked effect in most cases. The equation-of-motion approach is applied in studying both the
spin relaxation time T1 and the spin dephasing time T2, either in Markovian or in non-Markovian
limit. When many levels are involved at finite temperature, we demonstrate how to obtain the
spin relaxation time from the Fermi Golden rule in the limit of weak spin-orbit coupling. However,
at high temperature and/or for large spin-orbit coupling, one has to use the equation-of-motion
approach when many levels are involved. Moreover, spin dephasing can be much more efficient
than spin relaxation at high temperature, though the two only differs by a factor of two at low
temperature.
PACS numbers: 72.25.Rb, 73.21.La,71.70.Ej
I. INTRODUCTION
One of the most important issues in the growing
field of spintronics is quantum information processing
in quantum dots (QDs) using electron spin.1,2,3,4,5 A
main obstacle is that the electron spin is unavoidably
coupled to the environment (such as, the lattice) which
leads to considerable spin decoherence (including lon-
gitudinal and transversal spin decoherences).6,7 Vari-
ous mechanisms, such as, the hyperfine interaction with
the surrounding nuclei,8,9 the Dresselhaus/Rashba spin-
orbit coupling (SOC)10,11 together with the electron-
phonon interaction, g-factor fluctuations,12 the direct
spin-phonon coupling due to the phonon-induced strain,9
and the coaction of the hyperfine interaction and the
electron-phonon interaction can lead to the spin deco-
herence. There are quite a lot of theoretical works
on spin decoherence in QD. Specifically, Khaetskii and
Nazarov analyzed the spin-flip transition rate using a
perturbative approach due to the SOC together with the
electron-phonon interaction, g-factor fluctuations, the di-
rect spin-phonon coupling due to the phonon-induced
strain qualitatively.13,14,15 After that, the longitudinal
spin decoherence time T1 due to the Dresslhaus and/or
the Rashba SOC together with the electron-phonon in-
teraction were studied quantitatively in Refs. 16,17,18,19,
20,21,22,23,24,25,26. Among these works, Cheng et al.18
developed an exact diagonalization method and showed
that due to the strong SOC, the previous perturbation
method14,15,16 is inadequate in describing T1. Further-
more, they also showed that, the perturbation method
previously used missed an important second-order en-
ergy correction and would yield qualitatively wrong re-
sults if the energy correction is correctly included and
only the lowest few states are kept as those in Refs.
14,15,16. These results were later confirmed by Deste-
fani and Ulloa.21 The contribution of the coaction of the
hyperfine interaction and the electron-phonon interac-
tion to longitudinal spin decoherence was calculated in
Refs. 27 and 28. In contrast to the longitudinal spin de-
coherence time, there are relatively fewer works on the
transversal spin decoherence time, T2, also referred to as
the spin dephasing time (while the longitudinal spin de-
coherence time is referred to as the spin relaxation time
for short). The spin dephasing time due to the Dressel-
haus and/or the Rashba SOC together with the electron-
phonon interaction was studied by Semenov and Kim29
and by Golovach et al..20 The contributions of the hyper-
fine interaction and the g-factor fluctuation were studied
in Refs. 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44 and
in Ref. 45 respectively. However, a quantitative calcula-
tion of electron spin decoherence induced by the direct
spin-phonon coupling due to phonon-induced strain in
QDs is still missing. This is one of the issues we are
going to present in this paper. In brief, the spin re-
laxation/dephasing due to various mechanisms has been
studied previously in many theoretical works. However,
almost all of these works only focus individually on one
mechanism. Khaetskii and Nazarov discussed the ef-
fects of different mechanisms on the spin relaxation time.
Nevertheless, their results are only qualitative and there
is no comparison of the relative importance of the dif-
http://arxiv.org/abs/0704.0148v5
ferent mechanisms.13,14,15 Recently, Semenov and Kim
discussed various mechanisms contributed to the spin
dephasing,46 where they gave a “phase diagram” to in-
dicate the most important spin dephasing mechanism in
Si QD where the SOC is not important. However, the
SOC is very important in GaAs QDs. To fully under-
stand the microscopic mechanisms of spin relaxation and
dephasing, and to achieve control over the spin coherence
in QDs,47,48,49 one needs to gain insight into the relative
importance of each mechanism to T1 and T2 under vari-
ous conditions. This is one of the main purposes of this
paper.
Another issue we are going to address relates to differ-
ent approaches used in the study of the spin relaxation
time. The Fermi-Golden-rule approach, which is widely
used in the literature, can be used in calculation of the re-
laxation time τi→f between any initial state |i〉 and final
state |f〉.12,13,14,15,16,17,18,19,21,23,24,25,27,28,50,51,52 How-
ever, the problem is that when the process of the spin
relaxation relates to many states, (e.g., when tempera-
ture is high, the electron can distribute over many states),
one should find a proper way to average over the relax-
ation times (τi→f ) of the involved processes to give the
total spin relaxation time (T1). What makes it difficult
in GaAs QDs, is that all the states are impure spin states
with different expectation values of spin. In the existing
literature, spin relaxation time is given by the average of
the relaxation times of processes from the initial state |i〉
to the final state |f〉 (with opposite majority spin of |i〉)
weighted by the distribution of the initial states fi,
18,51,52
i.e.,
T−11 =
i→f . (1)
This is a good approximation in the limit of small SOC
as each state only carries a small amount of minority
spin. However, when the SOC is very strong which hap-
pens at high levels, it is difficult to find the proper way
to perform the average. We will show that Eq. (1) is
not adequate any more. Thus, to investigate both T1
and T2 at finite temperature for arbitrary strength of
SOC, we develop an equation-of-motion approach for the
many-level system via projection operator technique56 in
the Born approximation. With the rotating wave ap-
proximation, we obtain a formal solution to the equation
of motion. By assuming a proper initial distribution,
we can calculate the evolution of the expectation value
of spin. We thus obtain the spin relaxation/dephasing
time by the 1/e decay of the expectation value of spin
operator 〈Sz〉 or |〈S+〉| (to its equilibrium value), with
S+ ≡ Sx+ iSy. With this approach, we are able to study
spin relaxation/dephasing for various temperature, SOC
strength, and magnetic field.
For quantum information processing based on electron
spin in QDs, the quantum phase coherence is very im-
portant. Thus, the spin dephasing time is a more rel-
evant quantity. There are two kinds of spin dephasing
times: the ensemble spin dephasing time T ∗2 and the
irreversible spin dephasing time T2. For a direct mea-
surement of an ensemble of QDs58 or an average over
many measurements at different times where the config-
urations of the environment have been changed,59,60,61
it gives the ensemble spin dephasing time T ∗2 . The ir-
reversible spin dephasing time T2 can be obtained by
spin echo measurement.60,61 A widely discussed source
which leads to both T ∗2 and T2 is the hyperfine interac-
tion between the electron spin and the nuclear spins of
the lattice. As it has been found that T ∗2 is around 10 ns,
which is too short and makes a practical quantum infor-
mation processing difficult in electron spin based qubits
in QDs. Thus a spin echo technique is needed to remove
the free induction decay and to elongate the spin dephas-
ing time. Fortunately, this technique has been achieved
first by Petta et al. for two electron triplet-singlet system
and then by Koppens et al. for a single electron spin sys-
tem. The achieved spin dephasing time is ∼ 1 µs, which
is much longer than T ∗2 . We therefore discuss only the ir-
reversible spin dephasing time T2 throughout the paper,
i.e., we do not consider the free induction decay in the
hyperfine-interaction-induced spin dephasing.
It is further noticed that Golovach et al. have shown
that the spin dephasing time T2 is two times the spin
relaxation time T1.
20 However, as temperature increases,
this relation does not hold. Semenov and Kim on the
other hand reported that the spin dephasing time is much
smaller than the spin relaxation time.29 In this paper,
we calculate the temperature dependence of the ratio of
the spin relaxation time to the spin dephasing time and
analyze the underlying physics.
This paper is organized as follows: In Sec. II, we
present our model and formalism of the equation-of-
motion approach. We also briefly introduce all the spin
decoherence mechanisms considered in our calculations.
In Sec. III we present our numerical results to indicate
the contribution of each spin decoherence mechanism to
spin relaxation/dephasing time under various conditions
based on the equation-of-motion approach. Then we
study the problem of how to obtain the spin relaxation
time from the Fermi Golden rule when many levels are
involved in Sec. IV. The temperature dependence of T1
and T2 is investigated in Sec. V. We conclude in Sec. VI.
II. MODEL AND FORMALISM
A. Model and Hamiltonian
We consider a QD system, where the QD is confined
by a parabolic potential Vc(x, y) =
m∗ω20(x
2 + y2) in
the quantum well plane. The width of the quantum well
is a. The external magnetic field B is along z direction,
except in Sec. IV. The total Hamiltonian of the system
of electron together with the lattice is:
HT = He +HL +HeL , (2)
where He, HL, HeL are the Hamiltonians of the elec-
tron, the lattice and their interaction, respectively. The
electron Hamiltonian is given by
+ Vc(r) +HZ +HSO (3)
where P = −i~∇+ e
A with A = (B⊥/2)(−y,x) (B⊥ is
the magnetic field along z direction), HZ =
gµBB · σ
is the Zeeman energy with µB the Bohr magneton,
and HSO is the Hamiltonian of SOC. In GaAs, when
the quantum well width is small or the gate-voltage
along the growth direction is small, the Rashba SOC is
unimportant.53 Therefore, only the Dresselhaus term10
contributes to HSO. When the quantum well width is
smaller than the QD radius, the dominant term in the
Dresselhaus SOC reads
Hso =
〈P 2z 〉λ0(−Pxσx + Pyσy) , (4)
with γ0 denoting the Dresselhaus coefficient, λ0 being
the quantum well subband index of the lowest one and
〈P 2z 〉λ ≡ −~2
ψ∗zλ(z)∂
2/∂z2ψzλ(z)dz. The Hamiltonian
of the lattice consists of two parts HL = Hph +Hnuclei,
where Hph =
qη ~ωqηa
qηaqη (a
†/a is the phonon cre-
ation/annihilation operator) describes the vibration of
the lattice and Hnuclei =
j γIB · Ij (γI is the gyro-
magnetic ratios of the nuclei and Ij is the spin of the j-
th nucleus) describes the precession of the nuclear spins
of the lattice in the external magnetic field. We focus
on the spin dynamics due to hyperfine interaction at a
time scale much smaller than the nuclear dipole-dipole
correlation time (10−4 s in GaAs33,40), where the nuclear
dipole-dipole interaction can be ignored. Under this ap-
proximation, the equation of motion for the reduced elec-
tron system can be obtained which only depends on the
initial distribution of the nuclear spin bath.33 The in-
teraction between the electron and the lattice also has
two parts HeL = HeI + He−ph, where HeI is the hy-
perfine interaction between the electron and nuclei and
He−ph represents the electron-phonon interaction which
is further composed of the electron–bulk-phonon (BP)
interaction Hep, the direct spin-phonon coupling due to
the phonon-induced strain Hstrain and phonon-induced
g-factor fluctuation Hg.
B. Equation-of-motion approach
The equations of motion can describe both the co-
herent and the dissipative dynamics of the electron sys-
tem. When the quasi-particles of the bath relax much
faster than the electron system, the Markovian approx-
imation can be made; otherwise the kinetics is the non-
Markovian. For electron-phonon coupling, due to the
fast relaxation of the phonon bath and the weak electron-
phonon scattering, the kinetics of the electron is Marko-
vian. Nevertheless, as the nuclear spin bath relaxes much
slower than the electron spin, the kinetics due to the
coupling with nuclei is of non-Markovian type.30,32,33 It
is further noted that there is also a contribution from
the coaction of the electron-phonon and electron-nuclei
couplings, which is a fourth order coupling to the bath.
For this contribution, the decoherence of spin is mainly
controlled by the electron-phonon scattering while the
hyperfine (Overhauser) field54 acts as a static magnetic
field. Thus, this fourth order coupling is also Markovian.
Finally, since the electron orbit relaxation is much faster
than the electron spin relaxation,55 we always assume a
thermo-equilibrium initial distribution of the orbital de-
grees of freedom.
Generally, the interaction between the electron and the
quasi-particle of the bath is weak. Therefore the first
Born approximation is adequate in the treatment of the
interaction. Under this approximation, the equation of
motion for the electron system coupled to the lattice envi-
ronment can be obtained with the help of the projection
operator technique.56 We then assume a sudden approxi-
mation so that the initial distribution of the whole system
is ρ(t = 0) = ρe(0) ⊗ ρL(0), where ρe, ρL is the density
matrix of the system and bath respectively. This approx-
imation corresponds to a sudden injection of the electron
into the quantum dot, which is reasonable for genuine
experimental setup.33 As the initial distribution of the
the lattice ρL(0) commutates with the Hamiltonian of
the lattice HL, the equation of motion can be written as
dρe(t)
= − i
[He +TrL(HeLρ
L(0)), ρe(t)]
dτTrL[HeL, U0(τ)(P̂ [HeL, ρe(t− τ)
⊗ρL(0)])U †0 (τ)] , (5)
where ρe(t) is the density operator of the electron sys-
tem at time t, TrL stands for the trace over the lattice
degree of freedom, and U0(τ) = e
−i(HL+He)τ is time-
evolution operator without HeL. P̂ = 1̂ − ρL(0) ⊗ TrL
is the projection operator. The initial distribution of the
phonon system is chosen to be the thermo-equilibrium
distribution.20 It has been shown by previous theoretical
studies that the initial state of the nuclear spin bath is
crucial to the spin dephasing and relaxation.30,32,33 Al-
though it may take a long time (e.g., seconds) for the
nuclear spin system to relax to its thermo-equilibrium
state, one can still assume that its initial state is the
thermo-equilibrium one. This assumption corresponds
to the genuine case of enough long waiting time during
every individual measurement. For a typical setup at
above 10 mK and with about 10 T external magnetic
field, the thermo-equilibrium distribution is a distribu-
tion with equal probability on every state. For these ini-
tial distributions of phonons and nuclear spins, the term
TrL(HeLρ
L(0)) is zero. Thus,
P̂ [HeL, ρe(t−τ)⊗ρL(0)] = [HeL, ρe(t−τ)⊗ρL(0)] . (6)
The equation of motion is then simplified to,
dρe(t)
= − i
[He, ρ
e(t)] − 1
dτTrL[HeL, [H
eL(−τ),
Ue0 (t)ρ
Ie(t− τ)Ue0
(t)ρL(0)]] , (7)
where HIeL and ρ
Ie are the corresponding operators (HeL
and ρe ) in the interaction picture, and Ue0 (t) = e
−iHet
is the time-evolution operator of He. It should be fur-
ther noted that the first Born approximation can not
fully account for the non-Markovian dynamics due to the
hyperfine interaction with nuclear spins.33,57 Only when
the Zeeman splitting is much larger than the fluctuating
Overhauser shift, the first Born approximation is ade-
quate. For GaAs QDs, this requires B ≫ 3.5 T.33 In this
paper, we focus on the study of spin dephasing for the
high magnetic field regime of B > 3.5 T under the first
Born approximation, where the second Born approxima-
tion only affects the long-time behavior.33 Later we will
argue that this correction of long time dynamics changes
the spin dephasing time very little.
1. Markovian kinetics
The kinetics due to the coupling with phonons can be
investigated within the Markovian approximation, where
the equation of motion reduces to,
dρe(t)
= − i
[He, ρ
e(t)]− 1
dτTrph[He−ph,
[HIe−ph(−τ), ρe(t)⊗ ρph(0)]] . (8)
Here Trph is the trace over phonon degrees of freedom
and ρph(0) is the initial distribution of the phonon bath.
Within the basis of the eigen-states of the electron Hamil-
tonian, {|ℓ〉}, the above equation reads,
ρeℓ1ℓ2= −i
(εℓ1 − εℓ2)
ρeℓ1ℓ2
Trp(H
I e−ph
ρeℓ4ℓ2 ⊗ ρ
−HI e−phℓ1ℓ3 ρ
⊗ ρpeqH
) +H.c.
. (9)
Here H
= 〈ℓ1|He−ph|ℓ3〉 and HI e−phℓ1ℓ3 =
〈ℓ1|HIe−ph(−τ)|ℓ3〉. A general form of the electron-
phonon interaction reads
He−ph =
Φqη(aqη + a
−qη)Xqη(r,σ) . (10)
Here, η represents the phonon branch index; Φqη is the
matrix element of the electron-phonon interaction; aqη
is the phonon annihilation operator; Xqη(r,σ) denotes
a function of electron position and spin. Substituting
this into Eq. (9), we obtain, after integration within the
Markovian approximation,49
ρeℓ1ℓ2 = i
(εℓ1 − εℓ2)
ρeℓ1ℓ2
|Φqη|2{Xqηℓ1ℓ3X
ρeℓ4ℓ2
×Cqη(εℓ4 − εℓ3)−X
ρeℓ3ℓ4
×Cqη(εℓ3 − εℓ1)} +H.c.
in which X
= 〈ℓ1|Xqη(r,σ)|ℓ2〉, and Cqη(∆ε) =
n̄(ωqη)δ(∆ε+ωqη)+[n̄(ωqη)+1]δ(∆ε−ωqη). Here n̄(ωqη)
represents the Bose distribution function. Equation (11)
can be written in a more compact form
ρeℓ1ℓ2 = −
Λℓ1ℓ2ℓ3ℓ4ρ
, (12)
which is a linear differential equation. This equation can
be solved by diagonalizing Λ. Given an initial distribu-
tion ρeℓ1ℓ2(0), the density matrix ρ
(t) and the expec-
tation value of any physical quantity 〈O〉t = Tr(Ôρe(t))
at time t can be obtained:49
〈O〉t = Tr(Ôρe)
ℓ1···ℓ6
〈ℓ2|Ô|ℓ1〉P(ℓ1ℓ2)(ℓ3ℓ4)
× e−Γ(ℓ3ℓ4)tP−1
(ℓ3ℓ4)(ℓ5ℓ6)
ρeℓ5ℓ6(0) (13)
with Γ = P−1ΛP being the diagonal matrix and P repre-
senting the transformation matrix. To study spin dynam-
ics, we calculate 〈Sz〉t (|〈S+〉t|) and define the spin relax-
ation (dephasing) time as the time when 〈Sz〉t (|〈S+〉t|)
decays to 1/e of its initial value (to its equilibrium value).
2. Non-Markovian kinetics
Experiments have already shown that for a large en-
semble of quantum dots or for an ensemble of many mea-
surements on the same quantum dot at different times,
the spin dephasing time due to hyperfine interaction is
quite short, ∼ 10 ns.58,59,60,61 This rapid spin dephas-
ing is caused by the ensemble broadening of the preces-
sion frequency due to the hyperfine fields.40 When the
external magnetic field is much larger than the random
Overhauser field, the rotation due to the Overhauser field
perpendicular to the magnetic field is blocked. Only
the broadening of the Overhauser field parallel to the
magnetic field contribute to the spin dephasing. To de-
scribe this free induction decay for this high magnetic
field case, we write the hyperfine interaction into two
parts: HeI = h · S = HeI1 +HeI2. Here h = (hx, hy, hz)
and S = (Sx, Sy, Sz) are the Overhauser field and the
electron spin respectively. HeI1 = hzSz and HeI2 =
(h+S− + h−S+) with h± = hx ± ihy. The longitudi-
nal part HeI1 is responsible for the free induction de-
cay, while the transversal part HeI2 is responsible for
high order irreversible decay. As the rapid free induction
decay can be removed by spin echo,60,61 elongating the
spin dephasing time to ∼ 1 µs which is more favorable
for quantum computation and quantum information pro-
cessing, we then discuss only the irreversible decay. We
first classify the states of the nuclear spin system with its
polarization. Then we reconstruct the states within the
same class to make it spatially uniform. These uniformly
polarized pure states, |n〉’s, are eigen-states of hz. They
also form a complete-orthogonal basis of the nuclear spin
system. A formal expression of |n〉 is33
|n〉 =
m1···mN
αnm1···mN
|I,mj〉 . (14)
Here |I,mj〉 denotes the eigen-state of the z-component
of the j-th nuclear spin Ijz with the eigenvalue ~mj. N
denotes the number of the nuclei. The equation of motion
for the case with initial nuclear spin state ρns1 (0) = |n〉〈n|
is given by33
dρe(t)
[He +Trns(HeIρ
1 (0)), ρ
e(t)]
dτTrns[HeI2, U
0 (τ)
×[HeI2, ρe(t− τ)⊗ ρns1 (0)]UeI0
(τ)] . (15)
As in traditional projection operator technique, the dy-
namics of the nuclear spin subsystem is incorporated
self-consistently in the last term.33,56 Here Trns is the
trace over nuclear spin degrees of freedom. UeI0 (τ) =
exp[−iτ(He+HI +HeI1)]. The Overhauser field is given
by h =
j Av0Ijδ(r − Rj), where the constants A and
v0 are given later. Ij and Rj are the spin and posi-
tion of j-th nucleus respectively. As mentioned above,
the initial state of the nuclear spin bath is chosen to be
a state with equal probability of each state, therefore
ρns(0) =
n 1/Nw|n〉〈n|, with Nw =
n 1 being the
number of states of the basis {|n〉}. To quantify the ir-
reversible decay, we calculate the time evolution of S
for every case with initial nuclear spin state |n〉. We then
sum over n and obtain
||〈S+〉t|| =
|〈S(n)+ 〉t|. (16)
It is noted that the summation is performed after the
absolute value of 〈S(n)+ 〉t. Therefore, the destructive in-
terference due to the difference in precession frequency
ωzn, which originates from the longitudinal part of the
hyperfine interaction (HeI1), is removed. We thus use
1/e decay of the envelope of ||〈S+〉t|| to describe the irre-
versible spin dephasing time T2. Similar description has
been used in the irreversible spin dephasing in semicon-
ductor quantum wells62 and the irreversible inter-band
optical dephasing in semiconductors.63,64
Expanding Eq. (15) in the basis of {|n〉}, one obtains,
ρeℓ1ℓ2= −
(εℓ1δℓ1ℓ3 +H
nℓ1;nℓ3
)ρeℓ3ℓ2
−ρeℓ1ℓ3(εℓ3δℓ3ℓ2 +H
nℓ3;nℓ2
[HeI2nℓ1;n1ℓ3H
I eI2
n1ℓ3;nℓ4
ρeℓ4ℓ2(t− τ)
−HI eI2nℓ1;n1ℓ3ρ
(t− τ)HeI2n1ℓ4;nℓ2 ] +H.c.
. (17)
Here HeI2nℓ1;n1ℓ3 = 〈nℓ1|HeI2|n1ℓ3〉 and H
I eI2
nℓ1;n1ℓ3
〈nℓ1|HIeI2(−τ)|n1ℓ3〉. For simplicity, we neglect the
terms concerning different orbital wavefunctions which
are much smaller. For small spin mixing, assuming an
equilibrium distribution in orbital degree of freedom, un-
der rotating wave approximation, and trace over the or-
bital degree of freedom, we finally arrive at
〈S(n)+ 〉t = iωzn〈S
+ 〉t −
fk([h+]knn′
× [h−]kn′n + [h−]knn′ [h+]kn′n)
× exp[iτ(ωkn − ωkn′)]}〈S(n)+ 〉t−τ . (18)
Here ωzn =
k fk(Ezk/~ + ωkn) with Ezk representing
the electron Zeeman splitting of the k-th orbital level.
[hi]knn′ = 〈n|〈k|hi|k〉|n′〉 (i = ±, z). ωkn = [hz]knn + ǫnz
with ǫnz denoting the nuclear Zeeman splitting which is
very small and can be neglected. By solving the above
equation, we obtain |〈S(n)+ 〉t| for a given |n〉. We then sum
over n and determine the irreversible spin dephasing time
T2 as 1/e decay of the envelop of ||〈S+〉t||. By noting that
only the polarization of nuclear spin state |n〉 determines
the evolution of |〈S(n)+ 〉t|, the summation over n is then
reduced to summation over polarization which becomes a
integration for large N . This integration can be handled
numerically.
In the limiting case of zero SOC and very low tem-
perature, only the lowest two Zeeman sublevels are con-
cerned. The equation for 〈S+〉t with initial nuclear spin
state ρns1 (0) = |n〉〈n| reduce to
〈S+〉t = iωzn〈S+〉t −
([h+]nn′
× [h−]n′n + [h−]nn′ [h+]n′n) exp[iτ(ωn − ωn′)]}〈S+〉t−τ
= iωz〈S+〉t −
dτΣ(τ)〈S+〉t−τ . (19)
In this equation ωzn = (gµBB + [hz]nn′)/~, [hξ]nn′ =
〈n|〈ψ1|hξ|ψ1〉|n′〉 (ξ = ±, z and ψ1 is the orbital quantum
number of the ground state), and ωn = [hz]nn. Similar
equation has been obtained by Coish and Loss,33 and
later by Deng and Hu35 at very low temperature such
that only the lowest two Zeeman sublevels are considered.
Coish and Loss also presented an efficient way to evaluate
Σ(τ) in terms of their Laplace transformations, Σ(s) =
dτe−sτΣ(τ). They gave,
Σ(s) =
([h+]nn′ [h−]n′n
+ [h−]nn′ [h+]n′n)/(s− iδωnn′) , (20)
with δωnn′ =
(ωn − ωn′). With the help of this tech-
nique, we are able to investigate the spin dephasing due
to the hyperfine interaction.
C. Spin decoherence mechanisms
In this subsection we briefly summarize all the spin de-
coherence mechanisms. It is noted that the SOC modifies
all the mechanisms. This is because the SOC modifies the
Zeeman splitting18 and the spin-resolved eigen-states of
the electron Hamiltonian, it hence greatly changes the
effect of the electron-BP scattering.18 These two modifi-
cations, especially the modification of the Zeeman split-
ting, also change the effect of other mechanisms, such
as, the direct spin-phonon coupling due to the phonon-
induced strain, the g-factor fluctuation, the coaction of
the electron-phonon interaction and the hyperfine inter-
action. In the literature, except for the electron-BP scat-
tering, the effects from the SOC are neglected except for
the work by Woods et al.16 in which the spin relaxation
time between the two Zeeman sub-levels of the lowest
electronic state due to the phonon-induced strain is in-
vestigated. However, the perturbation method they used
does not include the important second-order energy cor-
rection. In our investigation, the effects of the SOC are
included in all the mechanisms and we will show that
they lead to marked effects in most cases.
1. SOC together with electron-phonon scattering
As the SOC mixes different spins, the electron-BP scat-
tering can induce spin relaxation and dephasing. The
electron-BP coupling is given by
Hep =
Mqη(aqη + a
−qη)e
iq·r , (21)
where Mqη is the matrix element of the electron-phonon
interaction. In the general form of the electron phonon
interaction He−ph, Φqη = Mqη and Xqη(r,σ) = e
iq·r.
|Mqsl|2 = ~Ξ2q/2ρvslV for the electron-BP coupling due
to the deformation potential. For the piezoelectric cou-
pling, |Mqpl|2 = (32~π2e2e214/κ2ρvslV )[(3qxqyqz)2/q7]
for the longitudinal phonon mode and
j=1,2 |Mqptj |2 =
[32~π2e2e214/(κ
2ρvstq
5V )][q2xq
y + q
z + q
(3qxqyqz)
2/q2] for the two transverse modes. Here
Ξ stands for the acoustic deformation potential; ρ is the
GaAs volume density; V is the volume of the lattice;
e14 is the piezoelectric constant and κ denotes the
static dielectric constant. The acoustic phonon spectra
ωqql = vslq for the longitudinal mode and ωqpt = vstq
for the transverse mode with vsl and vst representing
the corresponding sound velocities.
Besides the electron-BP scattering, electron also cou-
ples to vibrations of the confining potential, i.e., the
surface-phonons,28
δV (r) = −
2ρωqηV
(aqη + a
−qη)ǫqη · ∇rVc(r) ,
in which ǫqη is the polarization vector of a phonon mode
with wave-vector q in branch η. However, this contri-
bution is much smaller than the electron-BP coupling.
Compared to the coupling due to the deformation poten-
tial for example, the ratio of the two coupling strengths is
≈ ~ω0/Ξql0 , where l0 is the characteristic length of the
quantum dot and ~ω0 is the orbital level splitting. The
phonon wave-vector q is determined by the energy differ-
ence between the final and initial states of the transition.
Typically phonon transitions between Zeeman sublevels
and different orbital levels, ql0 ranges from 0.1 to 10.
Bearing in mind that ~ω0 is about 1 meV while Ξ = 7 eV
in GaAs, ~ω0/Ξql0 is about 10
−3. The piezoelectric cou-
pling is of the same order as the deformation potential.
Therefore spin decoherence due to the electron–surface-
phonon coupling is negligible.
2. Direct spin-phonon coupling due to phonon-induced
strain
The direct spin-phonon coupling due to the phonon-
induced strain is given by65
Hstrain =
s(p) · σ , (23)
where hsx = −Dpx(ǫyy − ǫzz), hsy = −Dpy(ǫzz − ǫxx) and
hsz = −Dpz(ǫxx− ǫyy) with p = (px, py, pz) = −i~∇ and
D being the material strain constant. ǫij (i, j = x, y, z)
can be expressed by the phonon creation and annihilation
operators:
ǫij =
qη=l,t1,t2
2ρωqηV
(aq,η + a
−q,η)(ξiηqj
+ ξjηqi)e
iq·r , (24)
in which ξil = qi/q for the longitudinal phonon
mode and (ξxt1 , ξyt1 , ξzt1) = (qxqz, qyqz,−q2‖)/qq‖,
(ξxt2 , ξyt2 , ξzt2) = (qy,−qx, 0)/q‖ for the two trans-
verse phonon modes with q‖ =
q2x + q
y . There-
fore, in the general form of electron-phonon interaction
He−ph, Φqη = −iD
~/(32ρωqηV ) and Xqη(r,σ) =
ijk ǫijk(ξjηqj − ξkηqk)pieiq·rσi with ǫijk denoting the
Levi-Civita tensor.
3. g-factor fluctuation
The spin-lattice interaction via phonon modulation of
the g-factor is given by12
ijkl=x,y,z
AijklµBBiσjǫkl , (25)
where ǫkl is given in Eq. (24) and Aijkl is a
tensor determined by the material. Therefore in
He−ph, Φqη = i
~/(32ρωqηV ) and Xqη(r,σ) =
i,j,k,l Ai,j,k,lµBBi(ξkηqk − ξlηql)σjeiq·r. Due to the
axial symmetry with respect to the z-axis, and keep-
ing in mind that the external magnetic field is along
the z direction, the only finite element of Hg is Hg =
[(A33−A31)ǫzz+A31
i ǫii]~µBBσz/2 with A33 = Azzzz ,
A31 = Azzxx and A66 = Axyxy. A33 + 2A31 = 0.
4. Hyperfine interaction
The hyperfine interaction between the electron and nu-
clear spins is66
HeI(r) =
Av0S · Ijδ(r−Rj) , (26)
where S = ~σ/2 and Ij are the electron and nucleus
spins respectively, v0 = a
0 is the volume of the unit cell
with a0 representing the crystal lattice parameter, and r
(Rj) denotes the position of the electron (the j-th nu-
cleus). A = 4µ0µBµI/(3Iv0) is the hyperfine coupling
constant with µ0, µB and µI representing the perme-
ability of vacuum, the Bohr magneton and the nuclear
magneton separately.
As the Zeeman splitting of the electron is much larger
(three orders of magnitude larger) than that of the nu-
cleus spin, to conserve the energy for the spin relax-
ation processes, there must be phonon-assisted transi-
tions when considering the spin-flip processes. Tak-
ing into account directly the BP induced motion of nu-
clei spin of the lattice leads to a new spin relaxation
mechanism:28
eI−ph(r) = −
Av0S · Ij(u(R0j) ·∇r)δ(r−Rj) , (27)
where u(R0j) =
~/(2ρωqηv0)(aqη + a
qη)ǫqηe
iq·R0
is the lattice displacement vector. Therefore using the
notation of Eq. (10), Φ =
~/(2ρV ωqη) and Xqη =
j Av0S·Ij∇rδ(r−Rj). The second-order process of the
surface phonon and the BP together with the hyperfine
interaction also leads to spin relaxation:
eI−ph(r) = |ℓ2〉
m 6=ℓ1
〈ℓ2|δVc(r)|m〉〈m|HeI (r)|ℓ1〉
εℓ1 − εm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|δVc(r)|ℓ1〉
εℓ2 − εm
〈ℓ1| , (28)
eI−ph = |ℓ2〉
m 6=ℓ1
〈ℓ2|Hep|m〉〈m|HeI(r)|ℓ1〉
εℓ1 − εm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|Hep|ℓ1〉
εℓ2 − εm
〈ℓ1| , (29)
in which |ℓ1〉 and |ℓ2〉 are the eigen states ofHe. By using
the notations in He−ph, Φqη =
~/(2ρωqηv0) and
Xqη = |ℓ2〉ǫqη ·
m 6=ℓ1
εℓ1 − εm
〈ℓ2|[He,P]|m〉
× 〈m|S · Ijδ(r−Rj)|ℓ1〉+
m 6=ℓ2
εℓ2 − εm
〈m|[He,P]|ℓ1〉
Av0〈ℓ2|S · Ijδ(r−Rj)|m〉
〈ℓ1| (30)
for V
eI−ph. Similarly Φqη =Mqη and
Xqη = |ℓ2〉
m 6=ℓ1
〈ℓ2|eiq·r|m〉
εℓ1 − εm
Av0〈m|S · Ij
× δ(r−Rj)|ℓ1〉+
m 6=ℓ2
εℓ2 − εm
〈m|eiq·r|ℓ1〉
Av0〈ℓ2|S · Ijδ(r−Rj)|m〉
〈ℓ1| (31)
for V
eI−ph. Again as the contribution from the surface
phonon is much smaller than that of the BP, V
eI−ph can
be neglected. It is noted that, the direct spin-phonon
coupling due to the phonon-induced strain together with
the hyperfine interaction gives a fourth-order scattering
and hence induces a spin relaxation/dephasing. The in-
teraction is
eI−ph = |ℓ2〉
m 6=ℓ1
〈ℓ2|Hzstrain|m〉〈m|HeI(r)|ℓ1〉
εℓ1 − ǫm
m 6=ℓ2
〈ℓ2|HeI(r)|m〉〈m|Hzstrain|ℓ1〉
ǫℓ2 − ǫm
〈ℓ1| , (32)
with Hzstrain = h
sσz/2 only changing the electron energy
but conserving the spin polarization. It can be written
hzs = −
2ρωq,ηV
(ξyηqy−ξzηqz)qzeiq·r . (33)
Comparing this to the electron-BP interaction Eq. (21),
the ratio is ≈ ~Dq/Ξ, which is about 10−3. Therefore,
the second-order term of the direct spin-phonon coupling
due to the phonon-induced strain together with the hy-
perfine interaction is very small and can be neglected.
Also the coaction of the g-factor fluctuation and the hy-
perfine interaction is very small compared to that of the
electron-BP interaction jointly with the hyperfine inter-
action as µBB/Ξ is around 10
−5 when B = 1 T. There-
fore it can also be neglected. In the following, we only
retain the first and the third order terms V
eI−ph and
eI−ph in calculating the spin relaxation time.
The spin dephasing time induced by the hyperfine in-
teraction can be calculated from the non-Markovian ki-
netic Eq. (18), for unpolarized initial nuclear spin state
|n0〉, resulting in
〈S(n0)+ 〉t ∝
dr|ψk(r)|4 cos(
|ψk(r)|2t) ,
where fk is the thermo-equilibrium distribution of the
orbital degree of freedom. When only the lowest two
Zeeman sublevels are considered, assuming a simple
form of the wavefunction, |Ψ(r)|2 = 1
exp(−r2
/d20)
with d‖/az representing the QD diameter/quantum well
width, and r‖ = x
2 + y2, the integration can be carried
〈S(n0)+ 〉t ∝
cos(t/t0)− 1
(t/t0)2
sin(t/t0)
. (35)
Here, t0 = (2πazd
‖)/(Av0) determines the spin dephas-
ing time. Note that t0 is proportional to the factor azd
where az/d
is the characteristic length/area of the QD
along z direction / in the quantum well plane. By solving
Eq. (18) for various n, and summing over n, we obtain
||〈S+〉t|| =
n |〈S
+ 〉t|. We then define the time when
the envelop of ||〈S+〉t|| decays to 1/e of its initial value
as the spin dephasing time T2. As mentioned above the
hyperfine interaction can not transfer an energy of the
order of the Zeeman splitting, thus the hyperfine inter-
action alone can not lead to any spin relaxation.43
In the above discussion, the nuclear spin dipole-dipole
interaction is neglected. Recently, more careful exami-
nations based on quantum cluster expansion method or
pair correlationmethod have been performed.41,42,43,47 In
these works, the nuclear spin dipole-dipole interaction is
also included. This interaction together with the hyper-
fine mediated nuclear spin-spin interaction is the origin
of the fluctuation of the nuclear spin bath. To the lowest
order, the fluctuation is dominated by nuclear spin pair
flips.41,42,43,47 This fluctuation provides the source of the
electron spin dephasing, as the electron spin is coupled
to the nuclear spin system via hyperfine interaction. Our
method used here includes only the hyperfine interaction
to the second order in scattering. However, it is found
that the dipole-dipole-interaction–induced spin dephas-
ing is much weaker than the hyperfine interaction for a
QD with a = 2.8 nm and d0 = 27 nm until the par-
allel magnetic field is larger than ∼ 20 T.42 Therefore,
for the situation in this paper, the nuclear dipole-dipole-
interaction–induced spin dephasing can be ignored.67
III. SPIN DECOHERENCE DUE TO VARIOUS
MECHANISMS
Following the equation-of-motion approach developed
in Sec. II, we perform a numerical calculation of the spin
relaxation and dephasing times in GaAs QDs. Two mag-
netic field configurations are considered: i.e., the mag-
netic fields perpendicular and parallel to the well plane
(along x-axis). The temperature is taken to be T = 4
K unless otherwise specified. For all the cases we con-
sidered in this manuscript, the orbital level splitting is
larger than an energy corresponding to 40 K. Therefore,
the lowest Zeeman sublevels are mainly responsible for
the spin decoherence. When calculating T1, the initial
distribution is taken to be in the spin majority down
state of the eigen-state of the Hamiltonian He with a
Maxwell-Boltzmann distribution fk = C exp[−ǫk/(kBT )]
for different orbital levels (C is the normalization con-
stant). For the calculation of T2, we assign the same
distribution between different orbital levels, but with a
superposition of the two spin states within the same or-
bital level. The parameters used in the calculation are
listed in Table I.8,68,69
TABLE I: Parameters used in the calculation
ρ 5.3× 103 kg/m3 κ 12.9
vst 2.48 × 10
3 m/s g −0.44
vsl 5.29 × 10
3 m/s Ξ 7.0 eV
e14 1.41 × 10
9 V/m m∗ 0.067m0
A 90 µeV A33 19.6
γ0 27.5 Å
3·eV I 3
D 1.59 × 104 m/s a0 5.6534 Å
A. Spin Relaxation Time T1
We now study the spin relaxation time and show how
it changes with the well width a, the magnetic field B
and the effective diameter d0 =
~π/m∗ω0. We also
compare the relative contributions from each relaxation
mechanism.
1. Well width dependence
In Fig. 1(a) and (b), the spin relaxation times induced
by different mechanisms are plotted as function of the
width of the quantum well in which the QD is confined
for perpendicular magnetic field B⊥ = 0.5 T and parallel
magnetic field B‖ = 0.5 T respectively. We first concen-
trate on the perpendicular magnetic field case. In Fig.
1(a), the calculation indicates that the spin relaxation
due to each mechanism decreases with the increase of well
g-factor
strain
eI−ph
eI−ph
B⊥ = 0.5 T
a (nm)
1098765432
10−10
B‖ = 0.5 T
a (nm)
1098765432
10−10
FIG. 1: (Color online) T−11 induced by different mechanisms
vs. the well width for (a): perpendicular magnetic field B⊥ =
0.5 T with (solid curves) and without (dashed curves) the
SOC; (b) parallel magnetic field B‖ = 0.5 T with the SOC.
The effective diameter d0 = 20 nm, and temperature T = 4 K.
Curves with � — T−11 induced by the electron-BP scattering
together with the SOC; Curves with • — T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
width. Particularly the electron-BP scattering mecha-
nism decreases much faster than the other mechanisms.
It is indicated in the figures that when the well width is
small (smaller than 7 nm in the present case), the spin re-
laxation time is determined by the electron-BP scattering
together with the SOC. However, for wider well widths,
the direct spin-phonon coupling due to phonon-induced
strain and the first-order process of hyperfine interac-
tion combined with the electron-BP scattering becomes
more important. The decrease of spin relaxation due to
each mechanism is mainly caused by the decrease of the
SOC which is proportional to a−2. The SOC has two
effects which are crucial. First, in the second order per-
turbation the SOC contributes a finite correction to the
Zeeman splitting which determines the absorbed/emitted
phonon frequency and wave-vector.18 Second, it leads to
spin mixing. The decrease of the SOC thus leads to the
decrease of Zeeman splitting and spin mixing. The for-
mer leads to small phonon wave-vector and small phonon
absorption/emission efficiency.18 Therefore the electron-
BP mechanism decreases rapidly with increasing a. On
the other hand, the other two largest mechanisms can flip
spin without the help of the SOC. The spin relaxations
due to these two mechanisms decrease in a relatively mild
way. It is further confirmed that without SOC they de-
creases in a much milder way with increasing a (dashed
curves in Fig. 1). It is also noted that the spin relaxation
rate due to the g-factor fluctuation is at least six orders
of magnitude smaller than that due to the leading spin
decoherence mechanisms and can therefore be neglected.
It is noted that in the calculation, the SOC is always
included as it has large effect on the eigen-energy and
eigen-wavefunction of the electron.18 We also show the
spin relaxation times induced by the hyperfine interac-
tions (V
eI−ph and V
eI−ph) and the direct spin-phonon
coupling due to the phonon-induced strain but without
the SOC as in the literature.27,28,45 It can be seen clearly
that the spin relaxation that includes the SOC is much
larger than that without the SOC. For example, the spin
relaxation induced by the second-order process of the hy-
perfine interaction together with the BP (V
eI−ph) is at
least one order of magnitude larger when the SOC is in-
cluded than that when the SOC is neglected. This is
because when the SOC is neglected, 〈m|HeI(r)|ℓ1〉 and
〈ℓ2|HeI(r)|m〉 in Eq. (29) are small as the matrix el-
ements of HeI(r) between different orbital energy lev-
els are very small. However, when the SOC is taken
into account, the spin-up and -down levels with differ-
ent orbital quantum numbers are mixed and therefore
|ℓ〉 and |m〉 include the components with the same or-
bital quantum number. Consequently the matrix ele-
ments of 〈m|HeI(r)|ℓ1〉 and 〈ℓ2|HeI(r)|m〉 become much
larger. Therefore, spin relaxation induced by this mech-
anism depends crucially on the SOC.
It is emphasized from the above discussion that the
SOC should be included in each spin relaxation mecha-
nism. In the following calculations it is always included
unless otherwise specified. In particular in reference to
the mechanism of electron-BP interaction, we always con-
sider it together with the SOC.
We further discuss the parallel magnetic field case. In
Fig. 1(b) the spin relaxation times due to different mech-
anisms are plotted as function of the quantum well width
for same parameters as Fig. 1(a), but with a parallel mag-
netic field B‖ = 0.5 T. It is noted that the spin relaxation
rate due to each mechanism becomes much smaller for
small a compared with the perpendicular case. Another
feature is that the spin relaxation due to each mecha-
nism decrease in a much slower rate with increasing a.
The electron-BP mechanism is dominant even at a = 10
nm but decrease faster than other mechanisms with a. It
is expected to be less effective than the V
eI−ph mecha-
nism or V
eI−ph mechanism or the direct spin-phonon cou-
pling due to phonon-induced strain mechanism for large
enough a. The g-factor fluctuation mechanism is negli-
gible again. These features can be explained as follows.
For parallel magnetic field the contribution of the SOC to
Zeeman splitting is much less than in the perpendicular
magnetic field geometry.21 Moreover, this contribution is
negative which makes Zeeman splitting smaller.21 There-
fore, the phonon absorption/emission efficiency becomes
much smaller for small a, i.e., large SOC. When a in-
creases, the Zeeman splitting increases. However, the
spin mixing decreases. The former effect is weak, and
only cancels part of the latter, thus the spin relaxation
due to each mechanism decrease slowly with a.
g-factor
strain
eI−ph
eI−ph
a = 5 nm
B⊥ (T)
543210
a = 10 nm
B⊥ (T)
543210
FIG. 2: (Color online) T−11 induced by different mechanisms
vs. the perpendicular magnetic field B⊥ for d0 = 20 nm and
(a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � —
T−11 induced by the electron-BP scattering; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N
— T−11 induced by the first-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with H
— T−11 induced by the direct spin-phonon coupling due to
phonon-induced strain; Curves with � — T−11 induced by the
g-factor fluctuation.
g-factor
strain
eI−ph
eI−ph
a = 5 nm
B‖ (T)
543210
10−10
a = 10 nm
B‖ (T)
543210
FIG. 3: (Color online) T−11 induced by different mechanisms
vs. the parallel magnetic field B‖ for d0 = 20 nm and (a)
a = 5 nm and (b) 10 nm. T = 4 K. Curves with � —
T−11 induced by the electron-BP scattering; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N
— T−11 induced by the first-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with H
— T−11 induced by the direct spin-phonon coupling due to
phonon-induced strain; Curves with � — T−11 induced by the
g-factor fluctuation.
2. Magnetic Field Dependence
We first study the perpendicular-magnetic-field case.
The magnetic field dependence of T1 for two different
well widths are shown in Fig. 2(a) and Fig. 2(b). In the
calculation, d0 = 20 nm. It can be seen that the ef-
fect of each mechanism increases with the magnetic field.
Particularly the electron-BP mechanism increases much
faster than other ones and becomes dominant at high
magnetic fields. For small well width (5 nm in Fig. 2a),
the spin relaxation induced by the electron-BP scattering
is dominant except at very low magnetic fields (0.1 T in
the figure) where contributions from the first-order pro-
cess of hyperfine interaction together with the electron-
BP scattering and the direct spin-phonon coupling due to
phonon-induced strain also contribute. It is interesting
to see that when a is increased to 10 nm, the electron-BP
scattering is the largest spin relaxation mechanism only
at high magnetic fields (>1.1 T). For 0.4 T < B⊥ < 1.1
T (B⊥ < 0.4 T), the direct spin-phonon coupling due
to the phonon-induced strain (the first order hyperfine
interaction together with the BP ) becomes the largest
relaxation mechanism. It is also noted that there is no
single mechanism which dominates the whole spin relax-
ation. Two or three mechanisms are jointly responsible
for the spin relaxation. It is indicated that the spin relax-
ations induced by different mechanisms all increase with
B⊥. This can be understood from a perturbation theory:
when the magnetic field is small the spin relaxation be-
tween two Zeeman split states for each mechanism is pro-
portional to n̄(∆E)(∆E)m (∆E is the Zeeman splitting)
with m = 7 for electron-BP scattering due to the de-
formation potential18,25 and for the second-order process
of the hyperfine interaction together with the electron-
BP scattering due to the deformation potential V
eI−ph;
m = 5 for electron-BP scattering due to the piezoelec-
tric coupling15,18,25 and for the second-order process of
the hyperfine interaction together with the electron-BP
scattering due to the piezoelectric coupling V
eI−ph;
27 and
m = 5 for the direct spin-phonon coupling due to phonon-
induced strain;15 m = 1 for the first-order process of the
hyperfine interaction together with the BP V
eI−ph. The
spin relaxation induced by the g-factor fluctuation is pro-
portional to n̄(∆E)(∆E)5B2⊥. For most of the cases stud-
ied, ∆E is smaller than kBT , hence n̄(∆E) ∼ kBT/∆E,
and n̄(∆E)(∆E)m ∼ (∆E)m−1. m > 1 hold for all mech-
anism except the V
eI−ph mechanism, therefore the spin
relaxation due to these mechanisms increases with in-
creasing B⊥. However, from Eq. (27) one can see that
it has a term with ∇r, which indicates that the effect of
this mechanism is proportional to 1/d0. As the vector
potential of the magnetic field increases the confinement
of the QD and gives rise to smaller effective diameter d0,
this mechanism also increases with the magnetic field in
the perpendicular magnetic field geometry.
We then study the case with the magnetic field par-
allel to the quantum well plane. In Fig. 3 the spin re-
laxation induced by different mechanisms are plotted as
function of the parallel magnetic field B‖ for two dif-
ferent well widths. In the calculation, d0 = 20 nm. It
can be seen that, similar to the case with perpendicular
magnetic field, the effects of most mechanisms increase
with the magnetic field. Also the electron-BP mechanism
increases much faster than the other ones and becomes
dominant at high magnetic fields. However, without the
orbital effect of the magnetic field in the present con-
figuration, the effect of V
eI−ph changes very little with
the magnetic field. For both small (5 nm in Fig. 3(a))
and large (10 nm in Fig. 3(b)) well widths, the electron-
BP scattering is dominant except at very low magnetic
field (0.1 T in the figure), where the first-order process of
the hyperfine interaction together with the electron-BP
interaction V
eI−ph also contributes.
a = 5 nm
d0 (nm)
3025201510
g-factor
strain
eI−ph
eI−ph
a = 10 nm
d0 (nm)
3025201510
10−10
FIG. 4: (Color online) T−11 induced by different mechanisms
vs. the effective diameter d0 for B⊥ = 0.5 T and (a) a = 5 nm
and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by
the electron-BP scattering; Curves with • – T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
3. Diameter Dependence
We now turn to the investigation of the diameter de-
pendence of the spin relaxation. We first concentrate on
the perpendicular magnetic field geometry. The spin re-
laxation rate due to each mechanism is shown in Fig. 4a
for a small (a = 5 nm) and Fig. 4b for a large (a = 10
nm) well widths respectively with a fixed perpendicu-
lar magnetic field B⊥ = 0.5 T. In the figure, the spin
relaxation rate due each mechanism except V
eI−ph in-
creases with the effective diameter. Specifically, the ef-
fect of the electron-BP mechanism increases very fast,
while the effect of the direct spin-phonon coupling due to
phonon-induced strain mechanism increases very mildly.
The V
eI−ph decreases with d0 slowly. Other mechanisms
are unimportant. The electron-BP mechanism eventu-
ally dominates spin relaxation when the diameter is large
a = 5 nm
d0 (nm)
3025201510
10−10
g-factor
strain
eI−ph
eI−ph
a = 10 nm
d0 (nm)
3025201510
FIG. 5: (Color online) T−11 induced by different mechanisms
vs. the effect diameter d0 with B‖ = 0.5 T and (a) a = 5 nm
and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by
the electron-BP scattering; Curves with • — T−11 induced by
the second-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with N — T
1 induced by the
first-order process of the hyperfine interaction together with
the BP (V
eI−ph); Curves with H — T
1 induced by the direct
spin-phonon coupling due to phonon-induced strain; Curves
with � — T−11 induced by the g-factor fluctuation.
enough. The threshold increases from 12 nm to 26 nm
when the well width increases from 5 nm to 10 nm. For
small diameter the V
eI−ph and the direct spin-phonon
coupling due to phonon-induced strain mechanism dom-
inate the spin relaxation. The increase/decrease of the
spin relaxation due to these mechanisms can be under-
stood from the following. The effect of the SOC on the
Zeeman splitting is proportional to d20 for small magnetic
field.18 The increase of d0 thus leads to a increase of Zee-
man splitting, therefore the efficiency of the phonon ab-
sorption/emission increases. Another effect is that the in-
crease of d0 will increase the phonon absorption/emission
efficiency due to the increase of the form factor.18 Thus
the spin relaxation increases. Moreover, the spin mix-
ing is also proportional to d0 also.
18 This leads to much
faster increasing of the effect of the electron-BP mecha-
nism and the V
eI−ph mechanism. However, the spin re-
laxation due to V
eI−ph decreases with the diameter. This
is because V
eI−ph contains a term ∇r [Eq. (27)] which
decreases with the increase of d0. Physically speaking,
the decrease of the effect of V
eI−ph is due to the fact
that the spin mixing due to the hyperfine interaction de-
creases with the increase of the number of nuclei within
the dot N as the random Overhauser field is proportional
to 1/
N . The spin relaxation induced by the g-factor is
also negligible here for both small and large well width.
We then turn to the parallel magnetic field case. In
the calculation, B‖ = 0.5 T. The results are shown for
both small well width (a = 5 nm in Fig. 5(a)) and large
well width (a = 10 nm in Fig. 5(b)) respectively. Simi-
lar to the perpendicular magnetic field case, the effect of
every mechanism except the V
eI−ph mechanism increases
with increasing diameter. The effect of the electron-BP
mechanism increases fastest and becomes dominant for
d0 > 12 nm for both small and large well width. For
d0 < 12 nm for the two cases the first-order process
of the V
eI−ph mechanism becomes dominant. The ef-
fect of the V
eI−ph mechanism become larger than that of
the direct spin-phonon coupling due to phonon-induced
strain mechanism. However, these two mechanism are
still unimportant and becomes more and more unimpor-
tant for larger d0. Here, the spin relaxation induced by
the g-factor is negligible.
4. Comparison with Experiment
In this subsection, we apply our analysis to experiment
data in Ref. 7. We first show that our calculation is in
good agreement with the experimental results. Then we
compare contributions from different mechanisms to spin
relaxation as function of the magnetic field. In the cal-
culation we choose the quantum dot diameter d0 = 56
nm (~ω0 = 1.1 meV as in experiment). The quantum
well is taken to be an infinite-depth well with a = 13
nm. The Dresselhaus SOC parameter γ0〈k2z〉 is taken
to be 4.5 meV·Åand the Rashba SOC parameter is 3.3
meV·Å. T = 0 K as kBT ≪ gµBB in the experiment.
The magnetic field is applied parallel to the well plane in
[110]-direction. The Dresselhaus cubic term is also taken
into consideration. All these parameters are the same
with (or close to) those used in Ref. 24 in which a cal-
culation based on the electron-BP scattering mechanism
agrees well with the experimental results. For this mech-
anism, we reproduce their results. The spin relaxation
time measured by the experiments (black dots with er-
ror bar in the figure) almost coincide with the calculated
spin relaxation time due to the electron-BP scattering
mechanism (curves with � in the figure).71 It is noted
from the figure that other mechanisms are unimportant
for small magnetic field. However, for large magnetic
field the effect of the direct-spin phonon coupling due
to phonon-induced strain becomes comparable with that
of the electron-BP mechanism. At B‖ = 10 T, the two
differs by a factor of ∼ 5.
g-factor
strain
eI−ph
eI−ph
B‖ (T)
15129630
10−10
10−12
FIG. 6: (Color online) T−11 induced by different mechanisms
vs. the parallel magnetic field B‖ in the [110] direction for
d0 = 56 nm and a = 13 nm with both the Rashba and Dres-
selhaus SOCs. T = 0 K. The black dots with error bar is the
experimental results in Ref. 7. Curves with � — T−11 induced
by the electron-BP scattering; Curves with • — T−11 induced
by the second-order process of the hyperfine interaction to-
gether with the BP (V
eI−ph); Curves with N — T
1 induced
by the first-order process of the hyperfine interaction together
with the BP (V
eI−ph); Curves with H — T
1 induced by the
direct spin-phonon coupling due to phonon-induced strain;
Curves with � — T−11 induced by the g-factor fluctuation.
B. Spin Dephasing Time T2
In this subsection, we investigate the spin dephasing
time for different well widths, magnetic fields and QD
diameters. As in the previous subsection, the contribu-
tions of the different mechanisms to spin dephasing are
compared.70 To justify the first Born approximation in
studying the hyperfine interaction induced spin dephas-
ing, we focus mainly on the high magnetic field regime
of B > 3.5 T. A typical magnetic field is 4 T. We also
demonstrate via extrapolation that in the low magnetic
field regime spin dephasing is dominated by the hyperfine
interaction.
1. Well Width Dependence
In Fig. 7 the well width dependence of the spin dephas-
ing induced by different mechanisms is presented under
the perpendicular (a) and parallel (b) magnetic fields. In
the calculations B⊥ = 4 T/B‖ = 4 T and d0 = 20 nm. It
can be seen in both figures that the spin dephasing due to
each mechanism decreases with a. Moreover, the spin de-
phasing due to the electron-BP scattering decreases much
faster than that due to the hyperfine interaction. These
features can be understood as following. The spin de-
phasing due to electron-BP scattering depends crucially
on the SOC. As the SOC is proportional to a−2, the spin
dephasing decreases fast with a. For the hyperfine inter-
action, from Eq. (35) one can deduce that the decay rate
of ||〈S+〉t|| is mainly determined by the factor 1/(azd2‖)
(here az = a), which thus decreases with a, but in a very
mild way. The fast decrease of the electron-BP mecha-
nism makes it eventually unimportant. For the present
perpendicular-magnetic-field case the threshold is around
2 nm. For parallel magnetic field it is even smaller. A
higher temperature may enhance the electron-BP mech-
anism (see discussion in Sec. V) and make it more im-
portant than the hyperfine mechanism. It is noted that
other mechanisms contribute very little to the spin de-
phasing. Thus, in the following discussion, we do not con-
sider these mechanisms. Comparing Figs 7(a) and (b),
one finds that a main difference is that the electron-BP
mechanism is less effective for the parallel-magnetic-field
case. As has been discussed in the previous subsection,
the spin mixing and the Zeeman splitting in the parallel
filed case is smaller than those in the perpendicular field
case. Therefore, the electron-BP mechanism is weakened
markedly.
Similar to Fig. 1, the SOC is always included in the
computation as it has large effect on the eigen-energy and
eigen-wavefunction of the electrons. The spin dephasings
calculated without the SOC for the hyperfine interaction,
the direct spin-phonon coupling due to phonon-induced
strain and the g-factor fluctuation are also shown in Fig.
7(a) as dashed curves. It can be seen from the figure
that for the spin dephasings induced by the direct spin-
phonon coupling due to phonon-induced strain and by
the g-factor fluctuation, the contributions with the SOC
are much larger than those without. This is because when
the SOC is included, the fluctuation of the effective field
induced by both mechanisms becomes much stronger and
more scattering channels are opened. However, what
should be emphasized is that the spin dephasings induced
by the hyperfine interaction with and without the SOC
are nearly the same (the solid and the dashed curves
nearly coincide). That is because the change of the wave-
function Ψ(r) due to the SOC is very small (less than 1
% in our condition) and therefore the factor 1/(azd
‖) is
almost unchanged when the SOC is neglected. Thus the
spin dephasing rate is almost unchanged.
In the inset of Fig. 7(a), the time evolution of ||〈S+〉t||
induced by the hyperfine interaction is shown, with a = 2
nm. It can be seen that ||〈S+〉t|| decays very fast and de-
creases to less than 10 % of its initial value within the
first two oscillating periods. Therefore, T2 is determined
by the first two or three periods of ||〈S+〉t||. Thus the
correction of the long time dynamics due to higher or-
der scattering33 contributes little to the spin dephasing
time. For quantum computation and quantum informa-
tion processing, the initial, e.g., 1 % decay of ||〈S+〉t||
may be more important than the 1/e decay.42,43 Indeed,
the spin dephasing time defined by the exponential fitting
of 1 % decay is short than that defined by the 1/e decay.
However, the two differs less than 5 times. For a rough
B⊥ = 4 T
t (µs)
6543210
a (nm)
1098765432
eI−ph
eI−ph
g-factor
strain
hyperfine
B‖ = 4 T
a (nm)
1098765432
10−10
10−15
10−20
10−25
FIG. 7: (Color online) T−12 induced by different mechanisms
vs. the well width for d0 = 20 nm. T = 4 K. (a): B⊥ = 4 T
with (solid curves) and without (dashed curves) the SOC; (b):
B‖ = 4 T only with the SOC. Curve with � — T
2 induced
by the electron-BP interaction; Curves with • — T−12 induced
by the hyperfine interaction; Curves with H — T−12 induced
by the direct spin-phonon coupling due to phonon-induced
strain; Curves with � — T−12 induced by g-factor fluctuation;
N — T−12 induced by the second-order process of the hyper-
fine interaction together with the BP (V
eI−ph); Curves with �
— T−12 induced by the first-order process of the hyperfine in-
teraction together with the BP (V
eI−ph). The time evolution
of ||〈S+〉t|| induced by the hyperfine interaction with a = 2
nm is shown in the inset of (a).
comparison of contributions from different mechanisms to
spin dephasing where only the order-of-magnitude differ-
ence is concerned (see Figs. 7-9), this difference due to
the definition does not jeopardize our conclusions.
2. Magnetic Field Dependence
We then investigate the magnetic field dependence of
the spin dephasing induced by the electron-BP scatter-
ing and by the hyperfine interaction for two different well
widths (a = 3 nm and a = 5 nm) with both perpendicular
and parallel magnetic field. From Fig. 8(a) and (b) one
hyperfine
B⊥ (T)
87.576.565.554.543.5
B‖ (T)
87.576.565.554.543.5
FIG. 8: (Color online) T−12 induced by the electron-BP scat-
tering and the hyperfine interaction vs. (a): the perpendic-
ular magnetic field B⊥ ; (b): the parallel magnetic field B‖
for a = 3 nm (solid curves) and 5 nm (dashed curves). T = 4
K, and d0 = 20 nm. Curves with � — T
2 induced by the
electron-BP interaction; Curves with • — T−12 induced by the
hyperfine interaction.
can see that the spin dephasing due to the electron-BP
scattering increases with magnetic field, whereas that due
to the hyperfine interaction decreases with magnetic field.
Thus, the electron-BP mechanism eventually dominates
the spin dephasing for high enough magnetic field. The
threshold is Bc⊥ = 4 T / B
= 7 T for a = 3 nm with per-
pendicular/parallel magnetic field. For larger well width,
e.g., a = 5 nm with parallel magnetic field or perpendicu-
lar magnetic field, the threshold magnetic fields increase
to larger than 8 T. The different magnetic field depen-
dences above can be understood as following. Besides
spin relaxation, the spin-flip scattering also contributes
to spin dephasing.20 As has been demonstrated in Sec.
IIIA, the electron-BP scattering induced spin-flip tran-
sition rate increases with the magnetic field. Therefore
the spin dephasing rate increases with the magnetic field
also. In contrast, spin dephasing induced by the hyper-
fine interaction decreases with the magnetic field. This
is because when the magnetic field becomes larger, the
fluctuation of the effective magnetic field due to the sur-
rounding nuclei becomes insignificant compared. There-
fore, the hyperfine-interaction-induced spin dephasing is
reduced. Similar results have been obtained by Deng and
Hu.44
hyperfine
B⊥ = 4 T
d0 (nm)
3025201510
B‖ = 4 T (b)
d0 (nm)
3025201510
FIG. 9: (Color online) T−12 induced by the electron-BP scat-
tering and the hyperfine interaction vs. the effective diameter
d0 T = 4 K. (a): B⊥ = 4 T ; (b): B‖ = 4 T for a = 3 nm
(solid curves) and 5 nm (dashed curves). Curves with � —
T−12 induced by the electron-BP interaction; Curves with •
— T−12 induced by the hyperfine interaction.
3. Diameter Dependence
In Fig. 9 the spin dephasing times induced by the
electron-BP scattering and the hyperfine interaction are
plotted as function of the diameter d0 for a small (a = 3
nm) and a large (a = 5 nm) well widths. In the cal-
culation, B⊥ = 4 T in Fig. 9(a) and B‖ = 4 T in (b).
It is noted that the effect of the electron-BP mechanism
increases rapidly with d0, whereas the effect of the hy-
perfine mechanism decreases slowly. Consequently, the
electron-BP mechanism eventually dominates the spin
dephasing for large enough d0. The threshold is d
0 = 19
(27) nm for a = 3 (5) nm case with the perpendicular
magnetic field and dc0 = 26 (30) nm for a = 3 (5) nm
case under the parallel magnetic field. As has been dis-
cussed in Sec. IIIA, both the effect of the SOC and the ef-
ficiency of the phonon absorption/emission increase with
d0. Therefore, the spin dephasing due to the electron-BP
mechanism increases rapidly with d0.
18,21 The decrease
of the effect of the hyperfine interaction is due to the de-
crease of the factor 1/(azd
) [Eq. 35] with the diameter
IV. SPIN RELAXATION TIMES FROM FERMI
GOLDEN RULE AND FROM EQUATION OF
MOTION
In this section, we will try to find a proper method to
average over the transition rates from the Fermi Golden
rule, τ−1i→f , to give the spin relaxation time T1. In the
limit of small SOC, we rederive Eq. (1) from the equation
of motion. We further show that Eq. (1) fails for large
SOC where a full calculation from the equation of motion
is needed.
T = 12 K
10987654321
γ = γ0
T (K)
4035302520151050
FIG. 10: (Color online) Spin relaxation time T1 calculated
from the equation-of-motion approach (�) v.s. that obtained
from Eq. (1) (•) as function of (a): the strength of the SOC
for T=12 K; (b): the temperature for γ = γ0. The well width
a = 5 nm, perpendicular magnetic field B⊥ = 0.5 T, QD
diameter d0 = 30 nm. The ratio of the two R is also plotted
in the figure. Note the scale of T−11 is at the right hand side
of the frame.
We first rederive Eq. (1) for small SOC from the equa-
tion of motion. In QDs, the orbital level splitting is
usually much larger than the Zeeman splitting. Each
Zeeman sublevel has two states: one with majority up-
spin, the other with majority down-spin. We call the
former “minus state” (as it corresponds to a lower en-
ergy) while the latter “plus state”. For small SOC,
the spin mixing is small. Thus we neglect the much
smaller contribution from the off-diagonal terms of the
density matrix to Sz. Therefore Sz(t) =
z fi±(t)
where i± denotes the plus/minus state of the i-th or-
bital state. For small SOC, the spin relaxation is
much slower than the orbital relaxation.25,55 This im-
plies that the time takes to establish equilibrium within
the plus/minus states is much smaller than the spin relax-
ation time. Thus we can assume a equilibrium (Maxwell-
Boltzmann) distribution between the plus/minus states
at any time. The distribution function is therefore given
by fi±(t) = N±(t) exp(−εi±/kBT )/Z±. Here N±(t) =
i fi±(t) is the total probability of the plus/minus states
with N+(t) + N−(t) = 1 for single electron in QD and
i exp(−εi±/kBT ) is the partition function for
the plus/minus state. At equilibrium, N± = N
± . The
equation for Sz(t) is hence,
Sz(t) =
[Sz(t)− Seqz ]
Si±z exp(−εi±/kBT )/Z±
δN±(t) , (36)
with δN±(t) = N±(t) − Neq± . As the orbital level
splitting is usually much larger than the Zeeman split-
ting, the factor exp(−εi±/kBT )/Z± can be approximated
by exp(−εi0/kBT )/Z0 with εi0 = 12 (εi+ + εi−) and
i exp[−εi0/kBT ]. Further using the particle-
conservation relation
± δN±(t) = 0, one has
Sz(t) = [
(Si+z −Si−z ) exp(−εi0/kBT )/Z0]
δN+(t) .
As Sz(t) − Seqz = [δN+(t)/Z0]
Si−z ) exp(−εi0/kBT ), one finds that the spin relax-
ation time is nothing but the relaxation time of N+.
The next step is to derive the equation of d
δN+(t),
which is given in our previous work:49
δN+(t) =
δfi+(t)
[τ−1i+→f−δfi+(t)− τ
i−→f+δfi−(t)]
[τ−1i+→f− + τ
i−→f+]
e−εi0/kBT
δN+(t) .(38)
Thus spin relaxation time is given by,
(τ−1i+→f− + τ
i−→f+)
e−εi0/kBT
. (39)
Furthermore, substituting e−εi0/kBT /Z0 by f
exp(−εi±/kBT )/Z±, we have
(τ−1i+→f−f
i+ + τ
i−→f+f
i−) . (40)
This is exactly Eq. (1).
T1/T2
T (K)
10−10
10−11
10−12
10−13
2520151050
FIG. 11: (Color online) Spin relaxation time T1, spin dephas-
ing time T2 and T1/T2 against temperature T . B⊥ = 4 T,
a = 5 nm and d0 = 30 nm. Note the scale of T1 and T2 is at
the right hand side of the frame.
For large SOC, or large spin mixing due to anticrossing
of different spin states,19,25 the spin relaxation rate be-
comes comparable with the orbital relaxation rate. Fur-
thermore, the decay of the off-diagonal term of the den-
sity matrix should contribute to the decay of Sz. There-
fore, the above analysis does not hold. In this case, it is
difficult to obtain such a formula, and a full calculation
from the equation-of-motion is needed.
In Fig. 10(a), we show that (for T = 12 K, a = 5
nm, B⊥ = 0.5 T, d0 = 30 nm) the spin relaxation times
T1 calculated from equation-of-motion approach and that
obtained from Eq. (40). Here, for simplicity and without
loss of generality, we consider only the electron-BP scat-
tering mechanism. The discrepancy of T1 obtained from
the two approaches increases with γ. At γ = 10γ0, the
ratio of the two becomes as large as ∼ 3. In Fig. 10(b),
we plot the spin relaxation times obtained via the two
approaches as function of temperature for γ = γ0 with
other parameters remaining unchanged. It is noted that
the discrepancy of T1 obtained from the two approaches
increases with temperature. For high temperature, the
higher levels are involved in the spin dynamics where the
SOC becomes larger. At 40 K, the discrepancy is as large
as 60 %. The ratio increases very slowly for T < 20 K
where only the lowest two Zeeman sublevels are involved
in the dynamics.
g-factor
strain
eI−ph
eI−ph
B⊥ = 0.5 T
T (K)
2520151050
10−10
10−15
B⊥ = 0.9 T
T (K)
2520151050
FIG. 12: (Color online) Spin relaxation time T1 against tem-
perature T for (a): B⊥ = 0.5 T; (b): B⊥ = 0.9 T. a = 10
nm and d0 = 20 nm. Curves with � — T
1 induced by the
electron-BP scattering together with the SOC; Curves with •
— T−11 induced by the second-order process of the hyperfine
interaction together with the BP (V
eI−ph); Curves with N —
T−11 induced by the first-order process of the hyperfine inter-
action together with the BP (V
eI−ph); Curves with H — T
induced by the direct spin-phonon coupling due to phonon-
induced strain; Curves with � — T−11 induced by the g-factor
fluctuation.
V. TEMPERATURE DEPENDENCE OF SPIN
RELAXATION TIME T1 AND SPIN DEPHASING
TIME T2
We first study the relative magnitude of the spin re-
laxation time T1 and the spin dephasing time T2. We
consider a QD with d0 = 30 nm and a = 5 nm at B⊥ = 4
T where the largest contribution to both spin relaxation
and dephasing comes from the electron-BP scattering
(see Fig. 4(a) and Fig. 9(a), we have checked that the
electron-BP scattering mechanism is dominant through-
out the temperature range). From Fig. 11, one finds that
when the temperature is low (T < 5 K in the figure),
T2 = 2T1, which is in agreement with the discussion in
Ref. 20. However, T1/T2 increases very quickly with T
and for T = 20 K, T1/T2 ∼ 2 × 102. This is understood
from the fact that when T is low, the electron mostly dis-
tributes in the lowest two Zeeman sublevels. For small
SOC, Golovach et al. have shown via perturbation the-
ory that phonon induces only the spin-flip noise in the
leading order. Consequently, T2 = 2T1.
20 When the tem-
perature becomes comparable with the orbital level split-
ting ~ω0, the distribution over the upper orbital levels is
not negligible any more. As mentioned previously, the
SOC contributes a non-trivial part to the Zeeman split-
ting. Specifically, the second order energy correction due
to the SOC contributes to the Zeeman splitting. The
energy correction for different orbital levels is generally
unequal (always larger for higher levels). When the elec-
tron is scattered by phonons randomly from one orbital
state to another one with the same major spin polariza-
tion, the frequency of its precession around z direction
changes. Continuous scattering leads to random fluctu-
ation of the precession frequency and thus leads to spin
dephasing.29,46 Note that this fluctuation only leads to a
phase randomization of S+, but not flips the z compo-
nent spin Sz, i.e., not leads to spin relaxation. There-
fore, the spin dephasing becomes stronger than the spin
relaxation for high temperatures. Moreover, this effect
increases with temperature rapidly as the distribution
over higher levels and the phonon numbers both increase
with temperature.
We further study the temperature dependence of spin
relaxation for lower magnetic field and larger quantum
well width where other mechanisms may be more im-
portant than the electron-BP mechanism. In Fig. 12(a),
the spin relaxation time is plotted as function of tem-
perature for B⊥ = 0.5 T, a = 10 nm and d0 = 20 nm.
It is seen from the figure that the direct spin-phonon
coupling due to phonon-induced strain mechanism dom-
inates the spin relaxation throughout the temperature
range. It is also noted that for T ≤ 4 K the spin relax-
ation rates induced by different mechanisms all increase
with temperature according to the phonon number fac-
tor 2n̄(Ez1) + 1 with Ez1 being the Zeeman splitting of
the lowest Zeeman sublevels. However, for T > 4 K, the
spin relaxation rates induced by the direct spin-phonon
coupling due to phonon-induced strain and the electron-
BP interaction increase rapidly with temperature, while
the spin relaxation rates induced by V
eI−ph and V
eI−ph
increase mildly according to 2n̄(Ez1) + 1 throughout the
temperature range. These features can be understood as
what follows. For T ≤ 4 K, the distribution over the high
levels is negligible. Only the lowest two Zeeman sublevels
involve in the spin dynamics. The spin relaxation rates
thus increase with 2n̄(Ez1) + 1 and the relative impor-
tance of each mechanism does not change. Therefore, our
previous analysis on comparison of relative importance of
different spin decoherence mechanisms at 4 K holds true
for the range 0 ≤ T ≤ 4 K. When the temperature gets
higher, the contribution from higher levels becomes more
important. Although the distribution at the higher lev-
els is still very small, for the direct spin-phonon coupling
mechanism, the transition rates between the higher lev-
els and that between higher levels and the lowest two
sublevels are very large. For the electron-BP mechanism
the transition rates between the higher levels are very
large due to the large SOC in these levels. Therefore, the
contribution from the higher levels becomes larger than
that from the lowest two sublevels. Consequently, the in-
crease of temperature leads to rapid increase of the spin
relaxation rates. However, for the two hyperfine mech-
anisms: the V
eI−ph and the V
eI−ph, the spin relaxation
rates does not change much when the higher levels are
involved. They thus increase by the phonon number fac-
In Fig. 12(b) we show the temperature dependence of
the spin relaxation time for the same condition but with
B = 0.9 T. It is noted that the spin relaxation rate due
to the electron-BP mechanism catches up with that in-
duced by the direct spin-phonon coupling due to phonon-
induced strain at T = 9 K and becomes larger for higher
temperature. This indicates that the temperature depen-
dence of the two mechanisms are quite different.
In Fig. 13 we show the spin dephasing induced by
electron-BP scattering and the hyperfine interaction as
function of temperature for B⊥ = 4 T, a = 10 nm and
d0 = 20 nm. We choose the conditions so that the spin
dephasing is dominated by the hyperfine interaction at
low temperature. However, the effect of the electron-BP
mechanism increases with temperature quickly while that
of the hyperfine interaction remains nearly unchanged.
The fast increase of the effect from the electron-BP scat-
tering is due to three factors: 1) the increase of the
phonon number; 2) the increase of scattering channels;
and 3) the increase of the SOC induced spin mixing in
higher levels. On the other hand, from Eq. 35, one can
deduce that the spin dephasing rate of the hyperfine in-
teraction depends mainly on the factor 1/(azd
) with
is the characteristic length/area along the z di-
rection / in the quantum well plane. For higher levels,
the d2‖ is larger, but only about a factor smaller than 10.
Thus the effect of the hyperfine interaction increases very
slowly with temperature.
It should be noted that in the above discussion, we
neglected the two-phonon scattering mechanism,15,46,50
which may be important at high temperature. The con-
tribution of this mechanism should be calculated via
the equation-of-motion approach developed in this paper,
and compared with the contribution of other mechanisms
showed here.
VI. CONCLUSION
In conclusion, we have investigated the longitudinal
and transversal spin decoherence times T1 and T2, called
spin relaxation time and spin dephasing time, in differ-
ent conditions in GaAs QDs from the equation-of-motion
approach. Various mechanisms, including the electron-
BP scattering, the hyperfine interaction, the direct spin-
hyperfine
T (K)
2520151050
FIG. 13: (Color online) Spin relaxation time T1 against tem-
perature T . B⊥ = 4 T, a = 10 nm and d0 = 20 nm. Curves
with � — T−12 induced by the electron-BP scattering together
with the SOC; Curves with • — T−12 induced by the hyperfine
interaction
phonon coupling due to phonon-induced strain and the
g-factor fluctuation are considered. Their relative im-
portance is compared. There is no doubt that for spin
decoherence induced by electron-BP scattering, the SOC
must be included. However, for spin decoherence induced
by the hyperfine interaction, the direct spin-phonon cou-
pling due to phonon-induced strain, g-factor fluctua-
tion, and hyperfine interaction combined with electron-
phonon scattering, the SOC is neglected in the existing
literature.27,28,45 Our calculations have shown that, as
the SOC has marked effect on the eigen-energy and the
eigen-wavefunction of the electron, the spin decoherence
induced by these mechanisms with the SOC is larger than
that without it. Especially, the decoherence from the
second-order process of hyperfine interaction combined
with the electron-BP interaction increases at least one
order of magnitude when the SOC is included. Our cal-
culations show that, with the SOC, in some conditions
some of these mechanisms (except g-factor fluctuation
mechanism) can even dominate the spin decoherence.
There is no single mechanism which dominates spin re-
laxation or spin dephasing in all parameter regimes. The
relative importance of each mechanism varies with the
well width, magnetic field and QD diameter. In particu-
lar, the electron-BP scattering mechanism has the largest
contribution to spin relaxation and spin dephasing for
small well width and/or high magnetic field and/or large
QD diameter. However, for other parameters the hyper-
fine interaction, the first-order process of the hyperfine
interaction combined with electron-BP scattering, and
the direct spin-phonon coupling due to phonon-induced
strain can be more important. It is noted that the g-
factor fluctuation always has very little contribution to
spin relaxation and spin dephasing which can thus be
neglected all the time. For spin dephasing, the electron-
BP scattering mechanism and the hyperfine interaction
mechanism are more important than other mechanisms
for magnetic field higher than 3.5 T. For this regime,
other mechanisms can thus be neglected. It is also shown
that spin dephasing induced by the electron-BP mecha-
nism increases rapidly with temperature. Extrapolated
from our calculation, the hyperfine interaction mecha-
nism is believed to be dominant for small magnetic field.
We also discussed the problem of finding a proper
method to average over the transition rates τ−1i→f obtained
from the Fermi Golden rule, to give the spin relaxation
time T1 at finite temperature. For small SOC, we red-
erived the formula for T1 at finite temperature used in
the existing literature18,51,52 from the equation of mo-
tion. We further demonstrated that this formula is in-
adequate at high temperature and/or for large SOC. For
such cases, a full calculation from the equation-of-motion
approach is needed. The equation-of-motion approach
provides an easy and powerful way to calculate the spin
decoherence at any temperature and SOC.
We also studied the temperature dependence of spin re-
laxation T1 and dephasing T2. We show that for very low
temperature if the electron only distributes on the low-
est two Zeeman sublevels, T2 = 2T1. However, for higher
temperatures, the electron spin dephasing increases with
temperature much faster than the spin relaxation. Con-
sequently T1 ≫ T2. The spin relaxation and dephasing
due to different mechanisms are also compared.
Acknowledgments
This work was supported by the Natural Science
Foundation of China under Grant Nos. 10574120 and
10725417, the National Basic Research Program of China
under Grant No. 2006CB922005 and the Innovation
Project of Chinese Academy of Sciences. Y.Y.W. would
like to thank J. L. Cheng for valuable discussions.
∗ Author to whom correspondence should be addressed;
Electronic address: mwwu@ustc.edu.cn
† Mailing Address
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70 It should be mentioned that one effect is not included :
when electron is scattered by phonon from one orbital state
to another, it feels a difference in the spin precession fre-
quency since the strength of longitudinal (along the exter-
nal magnetic field) component of Overhauser field differs
with orbital states. This effect randomizes the spin preces-
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this effect is negligible in our manuscript.
71 The deviation of our calculation from the experiment data
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http://arxiv.org/abs/0711.0479
