Spin relaxation in CdTe quantum dots with a single Mn atom

0.38 MB·English

by  Marko D. Petrović

Spin relaxation in CdTe quantum dots with a single Mn atom
Marko D. Petrovi´c1 and Nenad Vukmirovi´c1,∗
1Scientific Computing Laboratory, Institute of Physics Belgrade,
University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
Wehaveinvestigated spin relaxation timesinCdTequantumdotsdopedwith asingle Mn atom,
a prototype of a system where the interaction between a single charge carrier and a single spin
takesplace. A theoretical model that was used includes theelectron–Mn spin exchangeinteraction
2 responsible for mixing of the states of different spin in the basic Hamiltonian and electron–phonon
1 interaction as a perturbation responsible for transitions between the states. It was found that the
0 dominant electron–phonon interaction mechanism responsible for spin relaxation is the interaction
2 with acoustic phonons through deformation potential. Electron and Mn spin relaxation times at
room temperature take values in the range from microseconds at a magnetic field of 0.5 T down
n
to nanoseconds at a magnetic field of 10 T and become three orders of magnitude larger at cryo-
a
genic temperatures. It was found that electron spin-orbit interaction has a negligible effect on spin
J
relaxation times, while the changes in the position of the Mn atom within the dot and in the dot
6
dimensions can change thespin relaxation times by up to one order of magnitude.
2

]
l I. INTRODUCTION and electron spin in these dots. A theoretical analysis19
al of optical orientation experiments17 provided estimates
h Thepotentialofutilizingthespindegreeoffreedomas of hole and electron spin relaxation times necessary for
s- a classical bit in spintronic devices1–3 and as a quantum Mn spin orientationto occur. In a veryrecent work,Mn
e bitinpotentialquantuminformationprocessingdevices4 spinrelaxationtimesinquantumdotsinthepresenceofa
m has been largely recognized in the last two decades. Op- holeoranexcitonwerecalculatedbasedonamicroscopic
. eration of these devices crucially depends on the ability theory.20 Othertheoreticalstudies ofquantumdots with
at to manipulatethe spindegreeoffreedomofthe system.5 Mn atoms21–31 were focused on electronic, optical, mag-
m However, in all realistic open systems, undesirable spin netic or transport properties without any discussion of
- flips occur due to the interaction with environment. It spin lifetimes. Numerous studies of spin relaxation in
d is therefore of great importance to understand and be quantum dots were restricted to quantum dots without
n able to quantitatively describe the mechanisms of spin a Mn atom32–42 or on diluted magnetic semiconductor
o
c dephasing and spin relaxation. quantum dots with many Mn atoms.43
[ Physicalsystemswhichareexpectedtobeparticularly In this work, we calculate the relaxation times of Mn

suitableforaforementionedapplicationsarebasedonar- and electron spin caused by interaction with phonons in
1
chitectures that containquantum dots – artificialnanos- a singly negatively charged CdTe quantum dot contain-
v
4 tructures where charge carriers are confined in all three ing one Mn atom. Our calculation is based on a micro-
7 spatial directions.6,7 Due to quantum confinement effect scopic theory that links the system geometry with spin
6 the spectrum of electronic states in quantum dots is dis- relaxation times. In Sec. II we introduce the theoretical
5 creteandasaconsequencephasespaceforrelaxationand model used to describe the system at hand and relevant
1. dephasing processes is greatly reduced.8 Therefore, long spin relaxation times. In Sec. III we present the results
0 spin lifetimes of electrons confined in quantum dots are and analyze the effects of different electron–phonon in-
2 expected. teraction mechanisms, spin-orbit (SO) interaction, mag-
1 Inaddition,quantumdotsprovideaplaygroundwhere netic field, temperature, quantum dot dimensions and
:
v onecanstudyfundamentalinteractionsonasinglecarrier Mn atom position. In Sec. IV we compare our results to
i or spin level. Such a level of understanding is necessary theoretical results for spin relaxation in somewhat simi-
X
before one can proceed to understand more complicated larsystemsandexperiments,andanalyzethestrengthof
ar device structures. Quantum dots doped with a single other possible spin relaxation mechanisms not included
Mn atomhavedrawnparticularattentionfor fundamen- in our model.
tal studies in recent years. The manganese atom acts
effectively as an additional spin 5/2 degree of freedom
and therefore enables fundamental studies of interaction
II. THEORETICAL MODEL
between a single charge and a single spin in these dots.
Experimentally observed signature of this interaction is
the splitting of an exciton line in quantum dot photolu- In this section, we describe a theoretical model used
minescence spectrum.9–13 Predictions that it is possible to describe a CdTe quantum dot that contains one extra
to optically manipulate the state of a Mn spin in the electron and one Mn atom placed in its interior. Such
quantum dot12,14,15 were recently realized in several ex- a scenario is experimentally realized in a layer of self-
periments.16–18 assembledquantumdotsgrownatrelativelylowdotden-
However,verylittleisknownaboutthelifetimesofMn sity such that interdot interactions are negligible.
2
The Hamiltonian of the system reads In Eq. (refeq:jijr), R is the position of the Mn atom and
J (R) is the electron–Mn spin coupling strength, equal
ij
Hˆo =Hˆel+Hˆm-el+HˆB, (1) to Jcψi∗(R)ψj(R). Mˆz, Mˆ+ and Mˆ− are the Mn spin
operators whose properties are governedby the spin 5/2
where Hˆel is the electronic Hamiltonian, Hˆm-el describes algebra.
the interaction of electron with Mn spin and HˆB is the The last term of Hˆo, also known as the Zeeman term,
Zeeman term that describes the interaction of electron describes the interaction of the whole system with an
andMnspinwithexternalmagneticfield. ThisHamilto- external magnetic field B parallel to the z direction. It
nianactsintheHilbertspaceofthesystemwhichisgiven is given as
as the direct product of electron orbital space, electron
spin space and Mn spin space. Hˆ =−µ g BSˆ −µ g BMˆ , (6)
B B e z B Mn z
The first, electronic term reads
where µ is the Bohr magneton, and g (g ) is the
B e Mn
Hˆel = Ei,σcˆ†i,σcˆi,σ, (2) electron spin (Mn spin) g factor. Sˆz is the operator of
the z component of the electron spin.
i,σ
X
In Sec. IIIC, we also consider electronic SO cou-
withcˆ† andcˆ representingelectroncreationandanni- plingthatarisesinmaterialslackinginversionsymmetry
hilatioin,σoperatio,σrs. Typical self-assembled quantum dots (Dresselhaus SO coupling45)
aremuchlargerinthelateralplane(xyplane)thaninthe
growth (z) direction, while they can take various shapes Hˆso =γh·σ, (7)
–lenses,pyramids,truncatedpyramids,etc. Forthisrea-
where σ-s are Pauli matrices, h is the Dresselhaus effec-
son, we adopt the simplest possible quantum dot model
tive magnetic field
that captures all essential features of the single-particle
electronicspectrum–arectangularboxwithin-planedi-
h=[k (k2−k2),k (k2−k2),k (k2−k2)], (8)
mensions(L andL )muchlargerthanthedimensionin x y z y z x z x y
x y
the z direction (L ) with infinite potential barriers out-
z with k = −i ∇ and γ is the Dresselhaus SO coupling
side of the region 0 ≤ x ≤ L , 0 ≤ y ≤ L , 0 ≤ z ≤ L .
x y z strength.
The electronic states in the conduction band of semi-
After numerically solving the Hamiltonian eigenvalue
conductor nanostructures within first several hundreds
problem, we obtainthe eigenenergiesE and eigenstates
a
of meV are well described with envelope function effec-
|Ψ i
a
tive mass Hamiltonian. For our quantum dot model, the
conductionbandelectronenvelopefunctionisthengiven Hˆ |Ψ i=E |Ψ i,
o a a a
as
|Ψ i= ca |i,S ,M i. (9)
a iSzMz z z
ψ(x,y,z)=N sin nxπx sin nyπy sin nzπz i,SXz,Mz
c
L L L
(cid:18) x (cid:19) (cid:18) y (cid:19) (cid:18) z (cid:19) We use i and Sz to denote the orbital and spin state of
(3)
the electron and M for the Mn spin state.
z
whereN = 8/(L L L )isthenormalizationconstant
c x y z Electron–phonon interaction is considered to be the
andn ,n andn arepositiveintegersthatrepresentor-
x y z main mechanism responsible for the transitions between
p
bitalquantumnumbers ineachdirection. Single particle
the eigenstates |Ψ i and consequently spin relaxation.
a
energyE ofanelectronicstate i withorbitalquantum
i,σ We will show in Sec. IIIA that relevant transition ener-
numbers n , n and n is given as
x y z giesareoftheorderofmeV,whicharetypicalenergiesof
acoustic phonons. Because of the high energy of optical
E = ~2π2 n2x + n2y + n2z , (4) phononscomparedtotheacousticones,weconsideronly
i,σ 2m∗ L2 L2 L2 the interactionwith acoustic phonons. The Hamiltonian
x y z!
of electron–phonon interaction is given as7,44
where m∗ is the effective mass of the conduction band
electron in CdTe. Hˆe-ph = Mq,λ(ˆb†q,λ+ˆb−q,λ)eiqr, (10)
ThesecondtermHˆ describestheexchangeinterac- q,λ
m-el X
tion between an electronand a Mn atom, and it is given
by the spin impurity model Hamiltonian,44 a model well whereˆb† andˆb are phonon creation and annihilation op-
establishedinprevioustheoreticalstudiesofCdTequan- erators and q is the phonon wave vector. For acoustic
tum dots with a few Mn atoms21,23–25,30,31 phonons,alineardispersionrelationconnectsthephonon
wave vector and its energy ω =qv, where v is the sound
1
Hˆ =− J (R) (cˆ† cˆ −cˆ† cˆ )Mˆ velocity for a particular acoustic phonon branch in a
m-el 2 ij i,↑ j,↑ i,↓ j,↓ z given material.
Xi,j h
The scattering matrix M depends on the type of
+cˆ†i,↓cˆj,↑Mˆ++cˆ†i,↑cˆj,↓Mˆ− . (5) electron–phononinteraction.q,Fλortheinteractionthrough
i
3
deformation potential it is given as44 process of phonon emission, while the (−) sign corre-
sponds to phonon absorption.
~D2|q|
2 We will show in Sec. III that most of the eigenstates
|M | = , (11)
q,λ 2VρvLA of Hˆo have a well defined Mn and electron spin because
one of the c coefficients in Eq. (9) is typically sig-
and for the interaction through piezoelectric field it can iSzMz
nificantly larger than the others. Nevertheless, the re-
be represented as44
mainingcoefficients,thatcorrespondtobasisstateswith
~ξ (3q q q )2 other values of Sz and Mz, are nonzero. For this rea-
2 x y z
|Mq,λ| = v |q|7 (12) son, the transitions between the states with different
LA values of electron (or Mn) spin are allowed despite the
for longitudinal phonons and factthatthe electron–phononinteractionHamiltonianis
spin–independent.
~ξ q2q2+q2q2+q2q2 (3q q q )2 To obtain the average electron spin relaxation time
2 x y y z z x x y z
|M | = −
q,λ v |q|5 |q|7 one has to consider all possible transitions in the system
TA " #
where a particular change of spin occurs. For example,
(13)
the mean relaxation time for electron spin change from
for transversal ones. For transversal acoustic phonons
the initial spin S = 1/2 to the final spin S = −1/2 is
(TA) therearetwobranches(λ=2,3), whereasforlongi- z z
given as
tudinal acoustic (LA) phonons there is only one branch
(λ=1). IntheprecedingexpressionsD istheacousticde-
formationpotential,ρistheCdTematerialmassdensity, 1 = ifi fΓif, (15)
τ f
V is the volume of the system and ξ is given as P Pi i
32π2e2h2 where the sum over i includPes all possible initial states
ξ = 14, with S = 1/2 and the sum over f includes all possible
κVρ z
final states with S = −1/2, with f being the thermal
z i
where κ is the static dielectric constant and h14 is the weighting factor of state i at a temperature T, given as
piezoelectric constant. f =exp[−E /(k T)].
i i B
For the work presented here, we have used the follow-
ingparameters: J =15eV˚A3, m∗ =0.106m (Ref.23),
c 0
g =−1.67, g =2.02 (Ref. 25), γ =11.74eV˚A3 III. RESULTS
e Mn
(Ref. 46), v =3083m/s, v =1847m/s,
LA TA
h =3.94×108V/m (Ref. 43), D =5.1eV, A. Energy spectrum
14
ρ=4.85×103kg/m3, κ=9.6 (Ref. 47). Unless
otherwisestated, quantum dotdimensions weretakenas The energy spectrum of CdTe quantum dots with a
Lx =150˚A, Ly =140˚A, Lz =30˚A. Inourcalculations, single Mn atom has been studied in the past21,30,48 and
we consider first N =10 electron orbitals, which gives herewe only reviewthe mainfeatures,with anemphasis
the Hilbert space of Nmax =120 basis vectors. This on those that are relevant for our work.
ensures that the calculated quantities have converged to Relevant energies for our problem are the single-
their real values. particle electron orbital energies E [Eq. (4)], the
i,σ
Because of the fact that the electron–acoustic phonon electron–Mn spin exchange interaction energy J (R)
ij
interactiontermis muchsmaller comparedto the restof [Eq.(5)]andthe Zeemansplitting energyµ B [Eq.(6)].
B
the Hamiltonian,thistermcanbe treatedasaperturba- The separation between the first two orbital energies is
tion responsible for transitions between eigenstates |Ψai of the order of 50 meV and is much larger than the ex-
of Hˆ . If the system starts in aninitial state |Ψ i, it will changeinteractionandZeemansplittingenergywhichare
o i
make a transition to a final state |Ψ i due to electron– of the order of meV. As a consequence, the first twelve
f
phononscattering. Thescatteringrateforthisprocessis eigenstates of our system all originate from the ground
determined by Fermi’s golden rule orbital state and are well separated from higher excited
states,whoseaveragepopulations aremuchsmallereven
2π
Γ = |M |2|hΨ |eiqr|Ψ i|2 at room temperature. The dependence of their energies
if ~ q,λ f i
q,λ on magnetic field is shown in Fig. 1(a).
X
Looking at the structure of energy levels for electron–
1 1
× n¯q,λ+ ± δ(Ef −Ei±~ωq,λ), (14) MnatomsysteminFig. 1(a), severaldistinctive features
2 2
(cid:18) (cid:19) can be noticed. Spin of an electron is 1/2 and that of a
where E and E are the energies of the unperturbed Mn atom is 5/2. Combined, they will give two possible
i f
system in the initial and the final state, n¯ is the total spin numbers, F =2 and F =3, respectively with
q,λ
mean number of phonons at a given temperature and five and sevenspin projectionsalong the directionof the
|hΨ |eiqr|Ψ i| is the form factor for electron–phonon in- externalmagneticfield. Whenthereisnomagneticfield,
f i
teraction. The (+) sign in Eq. (14) corresponds to the total spin is a good quantum number and the presence
4
(see Fig. 1). Spin relaxation times calculated within our
3
430 approachshould be taken with caution in such cases.
2
429 1
) > B. Spin relaxation time
V MZ
( me 428 < 0 Since electron–phonon interaction Hamiltonian is in-
E -1
dependentofelectronandMnspin,phononscanonlyin-
427 (a) -2 (b) duce transitions between energy levels with the same to-
talspinprojectionnumber. Undertheseconditions,there
-3 are only five downhill (as well as five uphill) transitions
0 1 2 3 4 5 0 1 2 3 4 5
allowed. Each of these downhill transitions corresponds
B ( T ) B ( T ) toMnspin–flipfromM =F +1/2toM =F −1/2and
z z z z
Figure 1. Magnetic field dependence of the: (a) energies of in the same time to electron spin–flop from Sz = −1/2
the first twelve energy levels produced by Zeeman splitting to Sz =1/2.
and electron–Mn exchange interaction; (b) expected value of The transition time between the two states with the
theMn spin along the direction of an applied magnetic field. sameF isthereforealsotherelaxationtimeforMnspin–
z
ThesystemisconsideredwithoutSOinteractionandtheMn flipfromM =F +1/2toM =F −1/2. Onthe other
z z z z
atom is placed at R=(7.4,6.9,1.5) nm. hand,toobtaintheelectronspinrelaxationtime,onehas
to takethe averageoverallpossibletransitionsthat lead
to an electron spin–flip or spin–flop.
of electron–Mn exchange interaction leads to a splitting
The dependence of Mn and electron spin relaxation
of the twelve–folddegenerate ground level into two new,
times on magnetic field in the case of a Mn atom placed
seven–foldandfive-folddegenerateones. Inthe caseofa
near the center of the dot [at R = (7.4,6.9,1.5) nm] at
finite magneticfieldB, onlyF remainsagoodquantum
z room temperature is shown in Figs. 2(a) and 2(b).
number. TheZeemantermintheHamiltonianeliminates
External field affects the degree of Zeeman splitting,
alldegeneraciesandtwelveseparatenondegeneratelevels
which as a consequence determines the energy of the
emerge.
phononthroughwhich the systemcanrelax. Since a lin-
A typical state of this system is a superposition of all
ear dispersion relation connects the phonon energy and
basis vectors with the same total spin projection num-
its wave vector, the external field impact on spin re-
ber F . Because electron–Mn spin coupling is relatively
z laxation times comes mostly from scattering matrix ele-
weak, there are only few dominant states in this linear
ments[definedinEqs.(11)-(13)]. Forrelaxationthrough
combination. The eigenstate wave function |Ψi can be
deformation potential, the scattering matrix element is
represented as
∼ q, while for piezo–field it is ∼ 1/q. Along with the q2
factorthatcomesfromtheintegrationoverqinEq.(14),
|Ψi=α|0,+,M i+β|0,−,M i+..., (16)
− + this gives an overall ∼ 1/q3 dependence for relaxation
where the basis vectors are ordered by the strength of time through deformation potential and ∼ 1/q depen-
their contribution to the eigenstate. In Eq. (16), |0i de- dence for the relaxation time through piezo–field. As a
notes the ground orbital state, |+i and |−i are the elec- consequence, spin relaxation becomes more probable as
tronspinupanddownstates,whiletheMnspinquantum theexternalfieldincreases,ascanbeseenfromFig.2(a).
number is related to F via M =F ±1/2, where F = The above mentioned ∼ 1/q3 and ∼ 1/q dependences
z ± z z
−2,...,2. The next term in Eq. (16) depends mostly on are only approximately followed because the form factor
the Mn–electron coupling strength J (R), i.e. on the [Eq.(14)] also depends onq but this dependence is rela-
0j
position of the Mn atom. tivelyweakintherangeofmagneticfieldsofourinterest.
For the study of Mn (and electron) spin relaxation, Fig. 2(a) shows that the results for all possible tran-
thenextpropertythatweshouldturnourattentiontois sitions are very similar, ranging from microseconds to
the expected value of the Mn spin projection, M . Its nanoseconds when the magnetic field varies. This comes
z
dependence on magnetic field for first ten levels with from the fact that phonon energies for each transition
F = −2,...,2 is presented in Fig. 1(b). For most val- are very similar. At zero magnetic field, total spin F (in
z
ues of the magnetic field the expected values of Mz are addition to Fz) becomes a good quantum number. Due
very close to the corresponding half integer values from to independence of electron–phonon interaction Hamil-
the interval −5/2to 5/2,suggestingthat Mnspinis well tonian on spin, the transitions between the states with
defined for a given eigenstate. This corresponds to the either different F or Fz become forbidden. This leads to
case where one of the α and β coefficients in Eq. (16) is infinite relaxation times at B =0 in our model.
much larger than the other. In such a case, the electron Besides the Mn spin, the electron spin relaxation also
spin projection S = F −M is also well defined. M occurs. As mentioned above, a consequence of the spin
z z z z
and S cease to be well defined only for certain fields conserving Hamiltonian is the connection between these
z
and for some states where energy level crossings occur two. Eachtime aMnatommakesaflip to aneighboring
5
largernumberoforbitalstatesinthecalculationandthe
1ms 1ms
needforanumerical,ratherthananalytical,approachto
T
dl the problem.
T
pl
T
T pt
1(cid:181)s 1(cid:181)s C. The role of SO interaction

The electron(Mn)spinrelaxationprocessescausedby
spin-independent electron–phonon interaction are only
(a) possible due to the presence of terms in Hˆ that mix
1ns 1ns the states of different spin. Such a term inoHˆ is the
o
0 2 4 6 8 10 electron–Mn exchange interaction Hˆ , which mixes
B ( T ) m−el
both the states of different electron spin and the states
1ms Tdl Tdl of different Mn spin.
1ks
Tpl Tpl Another term which leads to mixing of the states of
opposite electron spin is the SO interaction [Eq. (7)].
T Tpt Tpt T The results presented in Sec. IIIB were obtained while
1(cid:181)s
1s neglecting this term and the goal here is to assess the
accuracy of such an approximation.
In the presence of SO interaction, F is no longer a
(b) (c) z
1ns good quantum number. However, it turns out that the
0 2 4 6 8 10 0 2 4 6 8 10 mean value of F for an eigenstate is very close to the
z
B ( T ) B ( T ) value of Fz in the absence of SO interaction. Therefore,
theeigenstatesofthe Hamiltoniancanstillbelabeledby
Figure 2. Spin relaxation times for a Mn atom placed
the mean value of F for a state.
at R = (7.4,6.9,1.5) nm at room temperature (T=295K), z
causedbydeformationpotentialinteractionwithLAphonons In Figs. 3(a) and (c) we compare the electron spin re-
(dl), piezoelectric potential interaction with LA (pl) and TA laxationtimesinthepresenceandabsenceofSOinterac-
phonons(pt): (a)FivetransitionscorrespondingtoMnatom tion. The values appear to be rather similar which sug-
spin-flip. (b) Electron spin relaxation time obtained by av- gests a weak effect of SO interaction on spin relaxation
eraging. Emptysymbolsrepresent thedatafor spin–down to times.
spin–up process, while the filled ones represent the opposite. AqualitativedifferencethatSOinteractionintroduces
(c) The same as (b) but for the calculation performed using
is that it allows for transitions between the states with
an inaccurate approximation.
different values of F , since F is no longer a goodquan-
z z
tum number. The calculated transition rates for such
transitions from the lowest state with F = 2 to other
z
spin state, the electron makes a flop and therefore the statesareshowninFig.3(b). Thesetransitionsappearto
electron spin relaxation time is obtained by averaging be much weaker than the transitions between the states
overallpossibletransitions. Thesimilaritybetweenthese with the same F .
z
two can be seen from Fig. 2(b). There are two types of As a conclusion to this section, we may say that the
processesforelectrons: fromspin–downtospin–upstate, changes in spin relaxation times due to SO interaction
and the opposite one. In general there is a difference are very small and for all practical purposes SO interac-
between the relaxation times for these two but only in tion can be neglected. Therefore, the rest of the results
the case of sufficiently low temperatures. presented in this paper will not include SO interaction.
As mentioned earlier, one, or at most two, coefficients
in Eq. (16) give the dominant contribution to an eigen-
state. Bothofthesecoefficientscorrespondtotheground D. Mn position and temperature
orbital state. It is therefore very tempting to introduce
an approximation in which we would reduce the Hilbert The position of Mn atom determines the electron–Mn
space of the system to basis states that originate from exchange coupling constants J (R) [Eq. (5)], which is
ij
orbital ground state only. However, this is not appro- theonlyplacewhereitappearsintheHamiltonian. Since
priate since such an approximation would yield infinite this is the only term in the Hamiltonian that mixes the
relaxationtimes. Alessdrasticapproximation,wherethe states of different electron or Mn spin, one may expect
eigenstates are calculated accurately, but only first two that it has a significant effect on spin relaxation times.
termsinEq.(16)arekeptforthecalculationoftransition To understand the role of the Mn atom position on spin
rates, is also highly inaccurate. As shown in Fig. 2(c), relaxationtimes,wehaveperformedacalculationfortwo
it gives unrealistically large spin relaxation times. The positionsofMnatom. Position1isnearthecenterofthe
resultsobtainedfromthese tempting, butinaccurateap- dotatR=(7.4,6.9,1.5)nm,whileposition2waschosen
proximations, demonstrate the necessity of including a to maximize the J (R) coupling constant and is given
01
6
1ms
T
FZ=-2 dl
T
FZ=0 pl
T T
1(cid:181)s FZ=2 pt

(b)
(a) (c)
1ns
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
B ( T ) B ( T ) B ( T )
Figure 3. The influence of SO interaction on spin relaxation times for a Mn atom placed at R=(7.4,6.9,1.5) nm. Magnetic
fielddependenceofrelaxationtimecausedbydeformationpotentialinteractionwithLAphononsfor: (a)thetransitionbetween
two levels with the same mean value of Fz in the presence (full symbols) and absence (empty symbols) of SO interaction; (b)
the transitions between the level (F = 3, Fz =2) and other levels. Full line represents the data for the transition to (F =2,
Fz =2),whiledashed,dottedanddash–dottedlinescorrespondtotransitionsto(F =3,Fz =1),(F =2,Fz =1)and(F =3,
Fz =3)respectively. Thetransitionstootherlevelsthatarenotshownhaverelaxationtimeslongerthan1ms. Thelevelswere
labeledaccordingtothequantumnumbersF andFz whicharegoodquantumnumbersatzeromagneticfieldintheabsenceof
SOinteraction. (c) Magnetic field dependenceof average electron spin–up tospin–down relaxation timein thepresence(filled
symbols) and absence (empty symbols) of SO interaction. The labels dl, pl and pt havethe same meaning as in Fig. 2.
as R = (4.5,7,1.5) nm. The obtained spin relaxation Spinrelaxationtimesattemperaturesof3Kand295K
times are shown in Fig. 4. At position 1, the J , J are also shown in Fig. 4. When an individual transi-
01 02
and J coupling constants are nearly zero. As a conse- tionis concerned,the temperatureappearsinourtheory
03
quence,thestatesofinteresthavethefirstlargestcontri- only through the phonon occupation number and con-
bution in the linear combination [Eq. (16)] other than α sequently its effect is easily predictable – higher tem-
andβ fromcoefficients correspondingto orbitalstate |4i peratures lead to shorter spin relaxation times. When
[ca coefficients in the notation of Eq. (9)]. On the average spin relaxation times are concerned, the tem-
4SzMz
other hand, at position 2, the first largest contribution perature appears in the theory also through the thermal
other than α and β comes from ca . Since state |1i weightingfactors[Eq.(15)]. Thisbecomesespeciallyim-
1SzMz
has a lowerenergy than state |4i, the coefficients ca portant at higher magnetic fields, where the state with
1SzMz
for position 2 are larger than ca coefficients in the F = 3 becomes the ground state. The transition prob-
4SzMz z
case of position 1. As a consequence, the relevant states ability from this state to other states is small but this
at position 2 exhibit a stronger mixing of basis states of statehasahighweightingfactor. Theaveragetransition
differentspin,whichtranslatesintofasterspinrelaxation rateisthensignificantlydifferentthanthetransitionrate
at position 2, as can be seen from Fig. 4. for individual transitions between states with the same
F (F = −2,...,2). As a consequence of all the men-
z z
tionedtemperatureeffects,coolingdownthesystemfrom
room temperature to cryogenic temperature leads to an
1s 1s increase in spin relaxationtimes by three orders of mag-
case 1. nitude.
case 2.
1ms case 3. 1ms
T
E. Quantum dot dimensions

1(cid:181)s 1(cid:181)s
The dependence of spin relaxation times on quantum
dot dimension L is shown in Fig. 5(a). The relative
z
1ns 1ns
position of an Mn atom inside the dot is kept during
0 2 4 6 8 10 this change of dimensions. The electron orbital states
B ( T ) [Eq. (3)] in the range of energies that is of interest here
Figure4. TheinfluenceofMnatompositionandtemperature all have quantum number nz = 1. As a consequence,
on electron spin relaxation times. The dependence of spin their energies do not change when Lz is changed (upto
relaxationtimeforthetransitionfromSz =1/2toSz =−1/2 an irrelevant constant). Therefore, the change in Lz af-
on magnetic field for different temperatures T and Mn ion fects only the electron wave functions and consequently
positions R – case 1: T = 295 K, R = (7.4,6.9,1.5) nm; the electron–Mn exchange coupling constants J (R).
ij
case 2: T =3 K, R =(7.4,6.9,1.5) nm; case 3: T =295 K, When the dot dimensions and the confinement volume
R=(4.5,7.0,1.5) nm.
increases, the probability of finding an electron near the
7
Mn atom, due to normalization of the wave function, SO coupling which is a consequence of bulk inversion
decreases. As a consequence, J (R) constants decrease. asymmetry. Realistic quantum dots also exhibit Rashba
ij
Thisleadstoweakermixingofstatesofdifferentspinand SO coupling49 as a consequence of structural inversion
therefore spin relaxation times increase, when quantum asymmetry. Ourmodelquantumdotissymmetric,there-
dot dimensions increase, as can be seen in Fig. 5(a). fore Rashba SO coupling is not present. However, real-
istic quantum dots certainly exhibit a certain degree of
asymmetry and consequently the Rashba SO coupling.
The interplay of Dresselhaus and Rashba SO coupling
in quantum dots has been studied in Ref. 39. It was
100ns
foundthatbothoftheseareofsimilarstrengthandcause
similar spin relaxation times. Moreover, in Ref. 32 the
T Tdl Tdl authors found that Dresselhaus SO coupling provides a
10ns Tpl Tpl biggeradmixtureofstatesofdifferentspin. Sincewehave
Tpt Tpt foundthatDresselhausSOcouplinghaspracticallynoef-
fectonspinrelaxationratesinoursystem,itis expected
1ns (a) (b) that the same will be the case for Rashba SO coupling.
Spin relaxation by direct electron spin–phonon cou-
20 30 40 100 150 200 pling in GaAs quantum dots was also considered in

LZ( ¯ ) LX ( ¯ ) Refs. 32 and 50. A conclusion was reached that it is by
far less effective than the relaxationcaused by SO inter-
Figure 5. The influence of quantum dot dimensions on elec-
action induced mixing and electron–phonon interaction.
tron spin relaxation times at a temperature of T = 295 K
Spin relaxation due to hyperfine interaction between
and magnetic field of B = 5 T. Filled symbols represent
the electron spin and the spin of the nuclei was investi-
the data for spin-up to spin-down process, while the empty
gated in Ref. 33. The hyperfine interaction was found
ones represent the opposite. The labels dl, pl and pt have
the same meaning as in Fig. 2. The Mn atom is placed at to be the dominant mechanism responsible for spin re-
R=(4.5,7.0,1.5) nm for the default dot dimensions and its laxation rates at magnetic fields which are low enough.
relative position is kept constant when dot dimensions are Sinceourmodelgivesrelaxationratesthattendtozeroat
varied. (a) The dependence of electron spin relaxation time vanishing magnetic field, one may expect that hyperfine
on the dot dimension in z direction; (b) The dependence of interactionwillbecomerelevantatlowmagneticfieldsin
electronspinrelaxationtimeonthedotsizeinxyplane–the our system too.
Lx/Ly ratio and Lz are keptconstant, while Lx is varied. Direct coupling of Mn spin to phonons was considered
in Ref. 20 as a potential mechanism of Mn spin relax-
For the same reasons [weaker Jij(R) for larger dots], ation. The correspondingrelaxationtimes werefoundto
one may expect that the relaxation times will increase be at least of the order of 1 ms, being much longer than
when the dot dimensions in xy plane increase. How- the onesoriginatingfromothermechanismsinour work.
ever, the trend obtained in Fig. 5(b) is somewhat differ-
Based on the discussion above, we may say that the
ent. The reason is that when L and L increase, the
x y spinrelaxationmechanismconsideredinourworkis cer-
distance between electron orbital energies [Eq. (4)] de-
tainly the most relevant mechanism in the system stud-
creases. This leads to stronger contributions from basis
ied.
states that originate from electron orbitals states other
Next, we compare the trends that we obtain for the
than |0i [the missing terms in Eq. (16)]. i.e. to stronger
dependence of relaxationtimes on variousparametersto
mixing of states of different spin in the expansion from
the ones obtained in quantum dots without Mn atoms.
Eq. (16), which leads to shorter spin relaxation times.
InRef.36,spinrelaxationinGaAsquantumdotscaused
Tosummarize,anincreaseofdotdimensions inxy plane
by the presence of SO interaction and electron–acoustic
leadsonthe onehandto adecreaseofJ (R)andonthe
ij phonon interaction was investigated. In that situation,
other hand to an increase in mixing in Eq. (16). As a
spinrelaxationtakesplace due to the transitionbetween
consequence of these two opposite trends, one obtains a
thefirsttwoZeemansublevelsofoppositespin. Theonly
nonmonotonous dependence of spin relaxation times on
spin mixing mechanism in such a system is the SO in-
L .
x teraction. Spin relaxation times were found to decrease
with the increase in magnetic field and were found to
be mainly determined by the value of the phonon wave
IV. DISCUSSION
vector responsible for the transition. These conclusions,
as one might have expected, are the same in our work.
Inthis section,wediscussthe relevanceofotherpossi- Interestingly, it was found on the other hand in Ref. 36
ble spin relaxation mechanisms which were not included that the dominant electron–phonon interaction mecha-
in our model and compare our results to other relevant nism that causes spin relaxation is piezoelectric interac-
results from the literature. tion with TA phonons. In our system, piezoelectric in-
In this work, we have only considered the Dresselhaus teractionwithTAphononsiscomparabletodeformation
8
potential interaction with LA phonons at low magnetic V. CONCLUSION
fields, whereas athigher magnetic fields deformationpo-
tential interaction with LA phonons becomes the dom- Inconclusion,we foundthatthe interactionsresponsi-
inant relaxation mechanism. When the dependence of ble for electronand Mn spin relaxationin quantum dots
relaxation times on dot dimensions in xy plane is con- doped with a single Mn atom are the electron–Mn spin
cerned, it was found in Ref. 36 that these decrease when exchange interaction and the electron–phonon interac-
dotdimensionsincrease. Thepresenceofexchangeinter- tion. The former provides mixing of the states with dif-
action between the electron and Mn spin, as the domi- ferent spin within an eigenstate and allows for the spin-
nant spin mixing mechanism in our system, causes a dif- flip transition caused by the latter. SO interaction has
ferenttrendinourcase,asseeninSec.IIIE.Itwasfound a negligible effect on spin relaxation times, in contrast
that spin relaxation times increase with the increase of to conventional quantum dots where SO interaction is
dot dimensions in the z direction both in our work and the only mechanism which allows for mixing of states
in Ref. 36, albeit for a different reason. In our case, this of different spin within an eigenstate and the relaxation
is caused by the decrease of electron–Mn spin exchange throughtheinteractionwithphonons. Amongthediffer-
coupling constant when the dot volume increases (see entelectron–phononinteractionmechanisms,electronin-
Sec. IIID), while in the work ofRef. 36 this is causedby teractionwithLAphononsturnsouttobethe dominant
the decrease of SO interaction. mechanism responsible for spin relaxation. Spin relax-
ation times decrease with the increase in magnetic field.
This dependence is mainly determined by an increase in
energylevelsplittingandconsequentlytheincreaseinthe
wavevectorof the phonon responsiblefor the transition.
The position of Mn ion within the dot determines the
strength of electron–Mn spin exchange interaction and
Finally, we compare the electron spin relaxationtimes therefore significantly alters the spin relaxation times.
that we obtained to the limits on their values in- We find that spin relaxation times in our electron–Mn
ferred from Mn spin optical orientation experiments.17 spin system are longer than in similar hole–Mn spin or
In Ref. 19, a theoretical analysis of Mn spin optical ori- exciton–Mn spin systems. This suggests that potential
entationexperiments17 wasperformedtounderstandthe spintronic or quantum computing devices based on the
physical origin of the observed Mn spin orientation. It interaction between charge carriers and Mn spin should
was found that optical orientation that occurs on the use electrons as charge carriers.
∼10ns timescale17 can be explained if hole spin relax-
ation times of the order of ∼10ns are assumed. Similar
hole spin relaxation times were obtained from a micro- VI. ACKNOWLEDGMENTS
scopic theory in Ref. 20. In the same time, the electron
spin relaxationtime needs to be longer than that.19 Our This work was supported by European Community
theoreticalresultsatalowtemperatureandlowmagnetic FP7 Marie Curie CareerIntegrationGrant(ELECTRO-
field (B <1 T) (Fig. 4) indicate electron spin relaxation MAT), Serbian Ministry of Science (project ON171017)
times longer than tens of microseconds, which is fully and FP7 projects PRACE-1IP, PRACE-2IP, HP-SEE
consistent with the conclusion obtained in Ref. 19. and EGI-InSPIRE.
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