Spin dynamics of quadrupole nuclei in InGaAs quantum dots

1.2 MB·

by  M. S. Kuznetsova

Spin dynamics of quadrupole nuclei in InGaAs quantum dots
M. S. Kuznetsova,1 R. V. Cherbunin,1 I. Ya. Gerlovin,2 I. V. Ignatiev,1,2
S. Yu. Verbin,1 D. R. Yakovlev,3,4 D. Reuter,5 A. D. Wieck,6 and M. Bayer3,4
1Department of Solid State Physics, Saint Petersburg State University, 198504 St. Petersburg, Russia
2Spin Optics Laboratory, Saint Petersburg State University, 198504 St. Petersburg, Russia
3Experimentelle Physik 2, Technische Universität Dortmund, D-44221 Dortmund, Germany
4Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
5Department Physik, Universität Paderborn, 33098 Paderborn, Germany
6Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: January 16, 2017)
7
1 Photoluminescencepolarizationisexperimentallystudiedforsampleswith(In,Ga)As/GaAsself-
0 assembled quantum dots in transverse magnetic field (Hanle effect) under slow modulation of the
2 excitation light polarization from fractions of Hz to tens of kHz. The polarization reflects the
evolutionofstronglycoupledelectron-nuclearspinsysteminthequantumdots. Strongmodification
n
a oftheHanlecurvesundervariationofthemodulationperiodisattributedtothepeculiaritiesofthe
J spindynamicsofquadrupolenuclei,whichstatesaresplitduetodeformationofthecrystallatticein
thequantumdots. AnalysisoftheHanlecurvesisfulfilledintheframeworkofaphenomenological
3
modelconsideringaseparatedynamicsofanuclearfieldB determinedbythe±1/2nuclearspin
1 Nd
statesandofanuclearfieldBNq determinedbythesplit-offstates±3/2,±5/2,etc. Itisfoundthat
] the characteristic relaxation time for the nuclear field BNd is of order of 0.5 s, while the relaxation
l of the field B is faster by three orders of magnitude.
l Nq
a
h PACSnumbers: 78.67.Hc,78.47.jd,76.70.Hb,73.21.La
-
s
e
m INTRODUCTION principal deformation axis and is not destroyed by the
dipole-dipole interaction.7 In this case, the stability of
.
t nuclear spin system should be determined by processes
a Hyperfineinteractionofanelectronlocalizedinaquan-
m tum dot (QD) with nuclear spins forms a strongly cou- of longitudinal spin relaxation of quadrupole nuclei with
characteristic time T >> T . Although many publica-
- pled electron-nuclear spin system.1,2 This system is con- 1 2
d tions are devoted to the nuclear spin polarization6,8–26
sidered to be promising for realization of quantum in-
n (see also review articles2,27–29), there are very few works
formation processing devices.3–5 The realization of spin
o where relaxation dynamics is studied for quadrupole nu-
qubitsassumessomestabilityofthespinsystemrequired
c clei.30–32
[ for the storage and processing of quantum information.
In this paper we report on experimental study of
InQDs,theopticallypolarizedelectrontransfersitsspin
1 spindynamicsofquadrupolenucleiinthesingly-charged
moment into the nuclear subsystem where the spin ori-
v (In,Ga)As/GaAsQDs. Thenuclearspinpolarizationwas
entation may be conserved for a long time controlled by
1
studied in optical experiments by detection of the elec-
9 the nuclear spin relaxation processes.
tron spin orientation via polarized secondary emission of
5 Themainprocessdestroyingthenuclearspinpolariza-
3 tion is believed to be the transverse relaxation in local theQDsinatransversemagneticfield(theHanleeffect).
0 Wehavefoundthat,whenthephotoluminescence(PL)of
fieldscausedbythedipole-dipoleinteractionofneighbor-
. thesamplesunderstudyisexcitedbylightwiththemod-
1 ing nuclear spins. Characteristic time of the relaxation,
ulated helicity of polarization, the Hanle curves strongly
0 T , for nuclei with spins I = 1/2 is of order of 10−4 s.1
7 2 dependonthemodulationfrequency. Wehavedeveloped
Theeffectivelocalfields,B ,areoffractionofmilliTesla
1 dd a phenomenological model based on the consideration of
and can be easily suppressed by external magnetic fields
: separate polarization dynamics of the |±1/2 > nuclear
v exceeding these local fields.
i spin doublets and of the split-off doublets, | ± 3/2 >,
X In the case of self-assembled QDs, the stabilization of |±3/2 >, etc. The analysis performed using a pseudo-
nuclear spin orientation is possible, in principle, in the
r spin approach proposed in Ref. 33 has allowed us to ex-
a absence of external magnetic field.6 Due to noticeable
tract contributions from polarization of these different
differenceinthelatticeconstantsofQDsandbarrierlay-
groups of spin doublets into the effective nuclear field
ers, some elastic stress appears in the QDs causing me-
acting on the electron spin.
chanical deformation of crystal lattice. The deformation
resultsinagradientofcrystalfieldsactingonnucleifrom
neighboring atoms and splitting the nuclear spin states
for quadrupole nuclei with I >1/2.1 Because the strain- I. EXPERIMENTAL DETAILS
inducedquadrupolesplittinginself-assembledQDstypi-
cally greatly exceeds Zeeman splitting in the local fields, We studied two samples prepared from one het-
thespinorientationofquadrupolenucleiispinnedtothe erostructure with InAs/GaAs QDs grown by Stranski-
2
Krastanovmethod. SampleAwasthenannealedattem- II. EXPERIMENTAL RESULTS AND ANALYSIS
perature 900 ◦C and sample B at temperature 980 ◦C.
Theannealinggivesrisetothediffusionofindiumatoms
A. Hanle curves at optical excitation with
into the barriers so that the indium concentration and, modulated polarization.
correspondingly,thecrystallatticedeformationsdecrease
for higher annealing temperature. Theoretical modeling
Typical Hanle curves for different modulation periods
shows34 that the deformation is of about 3 % for sam-
ofexcitationpolarizationareshowninFig.1. Asonecan
ple A and 1 % for sample B. The quadrupole splitting of
see, the Hanle curves of sample A [panel (a)] annealed
thenuclearspinstatesstronglydependsonthedeforma-
at lower temperature is considerably broader than the
tion.35 It is considerably smaller for sample B compar-
curves of sample B [panel (b)]. For both samples, the
ing to sample A. Therefore, the experimental study and
shape of Hanle curves strongly depends on the modu-
analysis of two samples allows one to highlight the role
lation period. At large periods, as well as at the excita-
of quadrupole splitting of nuclear states in the observed
tionwithafixedpolarization,awellresolvedW-structure
effects.
in the small magnetic fields is observed indicating a dy-
The QDs under study contain one resident electron namicnuclearpolarization(DNP)actingonelectronspin
per dot on average due to δ-doping of barriers by donors asaneffectivenuclearfield.43 TheW-structurebecomes
during the epilaxial growth. There are 20 layers of the smoothedandthenalmostdisappearswhenthemodula-
QDs with areal density of about 1010 cm−2 separated by tion period decreases. Besides, the Hanle curves notice-
60-nm thickGaAs barriers.36 Optical characterization of ably shrink with the period shortening. At the smallest
the samples is given in Ref. 37. The photoluminescence periods used in the experiments, the Hanle curves ac-
(PL) band in sample A corresponding to the lowest op- quire almost Lorentzian shape. Experiments also show
tical transitions in the QDs is centered at photon en- that the Hanle curve width becomes independent on the
ergy E =1.34 eV with the half width at half maximum modulation frequency at its further increase but mono-
A
(HWHM), δE = 9 meV. Similar PL band in sample tonically increases with the excitation power (not shown
A
B is shifted to the higher photon energy due to smaller here). Suchregularityistypicalfordepolarizationofelec-
indium content, E =1.42 eV with δE =7 meV. tron spins with no nuclear spin effects.1
B B
Itisimportantthat,inspiteofthelargeoverallmodifi-
Inourpresentexperiments,dependenceofcircularpo-
cationofHanlecurves, thepolarizationdegreemeasured
larizationofPLismeasuredasafunctionofthemagnetic
at zero magnetic fields is almost independent of modula-
field applied perpendicular to the optical axis. The de-
tion frequency. It approaches some value with the rise of
polarization curves (Hanle curves) are measured under
excitation power and becomes also almost independent
opticalexcitationbyacontinuouswaveTi:sapphirelaser
of the power at strong enough excitation. We assume
into the wetting layer of each sample (E = 1.459 eV
WL that this stability of polarization indicates the total po-
for sample A and E =1.481 eV for sample B). Polar-
WL larization of electron spins in the QDs at zero magnetic
izationofthelaserradiationisslowlymodulatedbetween
field. Thedeviationoftheexperimentallyobtainedvalue
σ+ and σ− by an electro-optical modulator followed by
of the polarization from unity is most probably caused
a quarter-wave plate with a frequency varied from frac-
by contribution of non-polarized PL from the neutral or
tions of Hz to several kHz. No resonant effects studied
doubly-charged QDs.37
in Refs. 37 and 38 are observed at such slow modulation
Remarkable difference in behavior of Hanle curves is
of the polarization.
observed for two samples studied, compare Fig 1(a) and
The PL is dispersed by a 0.5-m spectrometer and de- Fig 1(a). Namely, for sample B with higher annealing
tected with a silicon avalanche photodiode. The circular temperature, strong modification of Hanle curve is ob-
polarization degree, ρ = (I++ −I+−)/(I++ +I+−), is servedevenatlargemodulationperiod,T =1s,while
mod
measured using a photo-elastic modulator operating at for sample A the modification is hardly seen at the ten
a frequency of 50 kHz and a two-channel photon count- timesshorterperiod. Besides,theHanlecurvenarrowing
ing system. Here I++(I+−) is the PL intensity for co- forsampleBisfollowedbystrongincreaseofpolarization
(cross-) circular polarization relative to that of excita- farbeyondtheW-structure. Nosuchincreaseisobserved
tion. In the maximum of PL band of the QDs, the po- forsampleA.ThisdifferenceintheHanlecurvebehavior
larizationisnegativeandreflectsthemeanspinpolariza- indicates large difference in the dynamics of nuclear spin
tion of resident electrons as it was extensively discussed system in these two samples.
earlier.39,40 Hereafter we use the maximal absolute value The analysis of complex shape of the Hanle curves is
of ρ obtained at the center of PL band for each sample, the main topic of the rest part of the paper. As it is
A =max|ρ(ω)|,forquantitativecharacteristicofthe shown in Ref. 26, the W-structure and the shape of cen-
NCP
electronspinpolarization:40S =A /2,alongtheop- tralpartoftheHanlecurvesfortheQDsunderstudycan
z NCP
tical axis. Because the resident electrons are interacting bewelldescribedintheframeworkofaphenomenological
with the QD nuclei, the negative circular polarization model. The model considers the electron spin precession
(NCP) can be used as a sensitive tool to monitor the about an effective magnetic field, which is the sum of
nuclear spin state.40–42 the external magnetic field, B, an effective field of the
3
Figure 1. (Color online) Hanle curves for sample A (left panel) and sample B (right panel) measured at the optical excitation
of one circular polarization (CW) as well as with modulated polarization with periods given in the legends. Excitation power
P =14 mW for sample A and 10 mW for sample B. Diameter of laser spots on the samples, d≈60 µm. T =1.8 K.
DNP (Overhauser field),44 B , and an effective field of ied in detail in Ref. 37.
N
the nuclear spin fluctuations, B .45
f
At the presence of quadrupole splitting, behavior of
In the GaAs-based structures with no quadrupole ef-
the |±1/2 > doublet and the split-off doublets in the
fects, the regular nuclear field is developed, in Hanle ex-
magneticfieldorthogonaltodeformationaxis(transverse
periments, parallel, rather than anti-parallel, to the ex-
field) is very different. Hereafter we call these compo-
ternalmagneticfieldbecauseofthenegativesignofelec-
tron g factor.1 The W-structure, in particular, the dips nents as the dipole and quadrupole components of the
nuclear field, respectively.
inthestructure,areformedduetothelargenuclearfield,
which magnifies the effect of external magnetic field on
According to Ref. 7 the split-off nuclear states are not
the electron spin.26,43 At larger magnetic fields, i.e., at
splitted in the transverse magnetic field at the first or-
the wings of Hanle curves, the electron polarization is
der. This means that the quadrupole components of nu-
additionally suppressed by the nuclear field added to the
clear field conserve their orientation. Correspondingly,
external magnetic field. Therefore, it would be expected
the electron spin polarization is also conserved due to
that the modulation of excitation modulation suppress-
hyperfine interaction with the stabilized nuclear spins.
ing the nuclear polarization should partially restore the
Only when the Zeeman splitting becomes comparable
electron polarization at the wings of Hanle curve.
with the quadrupole splitting, the nuclear spins are no
ExperimentallyobservedevolutionoftheHanlecurves
longerpinnedtothemajoraxisoftheelectricfieldgradi-
strongly differs from this prediction. As it is seen
ent. Correspondingly, the nuclear spin orientation is de-
inFig.1,theincreaseofmodulationfrequencyisfollowed
stroyed and electron polarization decreases. That forms
byasmoothingoftheW-structurethatindicatesthede-
the wings of the Hanle curve. The fast modulation of
crease of nuclear polarization. At the same time, the
excitation suppressing the quadrupole component of nu-
width of Hanle curves decreases, rather than increases,
clear field should result in destroying the nuclear polar-
at it is predicted by the standard model.1
izationatsmallermagneticfieldsthatisinnarrowingthe
We assume that the main reason for such behavior is Hanle curve.
thequadrupoleeffectsinthenuclearspinsystem.7 Agra-
dientofcrystalfieldsplitsoffthespindoublets|±3/2>, This simplified discussion of dynamic processes in the
|±5/2>, etc, from doublet |±1/2>. In the structures electron-nuclear spin system allows one to qualitatively
understudy,thegradientismainlyinducedbythecrystal explain the experimentally observed behavior of Hanle
lattice deformation. The principal axis of this deforma- curves at different modulation frequencies. An accurate
tion is directed along the growth axis of the structures analysis of the Hanle curves allowed us to obtain valu-
that is along the optical axis in our experiments.35 The able information about both the dipole and quadrupole
quadrupole splitting caused by this deformation is stud- components of nuclear field.
4
B. Phenomenological model The electron spin precession competes with the spin re-
laxation, which can be described by an effective field
To extract information about the dynamics of nuclear Bτ =h¯/(geµBτs) where ge is the electron g factor, µB is
polarization from the Hanle curves, we generalize the theBohrmagneton,andτe istheelectronspinrelaxation
phenomenological model proposed in Ref. 26. In par- time. To include the relaxation, we should generalize
ticular, we consider two effective nuclear fields acting on Eq. (3):
the electron spin. The first one, the dipole field B ,
Nd
is determined by polarization of the ±1/2 nuclear spin B2 +B2
ρ= totz τ. (4)
states. The second one, the quadrupole field BNq, is B2 +B2
due to the polarization of the split-off states ±3/2, etc. tot τ
We should note that, in (In,Ga)As-based structures, the
For simplicity, we assume here that the relaxation time
nucleiofallchemicalelements,includingisotopesconsti-
τ does not depend on the external magnetic field. This
tuting the structure, possess quadrupole moments. e
assumption will be verified by the simulations of Hanle
Theelectronspinprecessesinthetotalfield,B ,con-
tot curves described below. Similar to Ref. 26, we assume
sisting of several contributions:
that the total field squared can be expressed as:
B =B+B +B +B , (1)
tot Nd Nq f
B2 =(B+B +B )2+(B +B )2+(cid:104)B2(cid:105). (5)
tot Ndx Nqx Ndz Nqz f
where B is an effective field of the nuclear spin fluctu-
f
ations. Due to the fast precession of electron spin about
Here we use the fact that the external magnetic field is
B , only the projection, S , is conserved:
tot Btot directed along the x-axis. We also assume that no valu-
able nuclear polarization appears along the y-axis. The
(S ·B ) B
SBtot = 0|Btott|ot =S0(cid:112)Btott2ozt. (2) nduisctlreiabrutsepdin: fluctuations are assumed to be isotropically
Here S is the electron spin polarization created along
0
the optical axis (z-axis). The electron spin polarization (cid:104)B2(cid:105)=(cid:104)B2 (cid:105)+(cid:104)B2 (cid:105)+(cid:104)B2 (cid:105)=3(cid:104)B2 (cid:105). (6)
f fx fy fz fz
measuredintheexperimentsS istheprojectionofS
z Btot
on the direction of observation (the optical axis). Corre-
The z-projection of the total field squared, B2 , is de-
spondingly, the measured degree of PL polarization is: totz
termined by similar way with taking into account only
z-components of the regular and fluctuating fields. Fi-
S B2
ρ= z = totz. (3) nally we obtain:
S B2
0 tot
ρ(B)= Be = (BNdz+BNqz)2+(cid:104)Bf2z(cid:105)+Bτ2 . (7)
B0 (B+B +B )2+(B +B )2+3(cid:104)B2 (cid:105)+B2
e Ndx Nqx Ndz Nqz fz τ
HereB =b S isthez-componentofKnightfieldacting the Zeeman splitting of the doublet in an external mag-
e e z
on the nuclei and B0 = b S is the Knight field at zero neticfieldisconsiderablysmallerthantheenergysepara-
e e 0
external magnetic field. Constant b is proportional to tionbetweenthedoubletsdeterminedbythequadrupole
e
the hyperfine interaction constant.1 It is considered as a splitting. The Zeeman splitting, δE = g βB, can be
m m
fitting parameter. described by an anisotropic nuclear g factor, g . Here β
m
Wesupposethatcomponentsofthenuclearfield,B is the nuclear magneton. The nuclear spin polarization
Ndx
andB ,B andB aredeterminedbythenuclear and, correspondingly, the nuclear field are created along
Ndz Nqx Nqz
spinprecessionaboutthetotalfieldactingonthenuclei. aneffectivemagneticfield,Bemff =gmxB+gmzBe.46 We
The field consists of the external magnetic field, B, and should stress that the direction of Bemff deviates, in gen-
of the Knight field, B . For simplicity, we neglect the x- eralcase,fromthedirectionofvectorsumoffieldsBand
e
and y-components of the Knight field because they are Be because of the anisotropy of the nuclear g factor.
much smaller than the external magnetic field. Using a simple vector model26 one can obtain general
expressions for components of nuclear field:
Evolution of the nuclear field created by nuclei with
quadrupolesplittingofspinstatescanbeanalyzedinthe
B2
framework of a pseudo-spin model proposed in Ref. 33. B =B e ,
Nmz Nm(g∗ B)2+B2
According to the model, each spin doublet with the spin mx e (8)
projection, m = ±1/2, ±3/2, ..., onto the principal (g∗ B)B
B =B mx e ,
quadrupole axis may be considered independently, while Nmx Nm(g∗ B)2+B2
mx e
5
Here gm∗x =gmx/gmz are the normalized g factors deter- 0.3
minedastheratioofg factorscharacterizinginteractions (a) Sample A T = 50 ms
mod
0.25 T=900°C
with the magnetic fields applied across and along the 0.5 ms

principal quadrupole axis, respectively. In small trans- P 0.2 fit curve
C
verse magnetic fields, the splitting of nuclear states with N
A
m= ±1/2 (the dipole states) linearly depends on the 0.15
magneticfieldandg∗ ≈2,whiletheZeemansplittingof
dx 0.1
the doublet is considerably smaller than the quadrupole
splitting. Wewillusethisapproximateequalitybecause, 0.05
as it will be seen in the next section, the dipole nuclear
0
fieldsignificantlydiffersfromzeroonlyinsmallmagnetic 0 30 60 90 120
fields. Magnetic field (mT)
The splitting of the ±3/2, ±5/2, ..., doublets is
0.3
strongly anisotropic one in the transverse magnetic field (b) Sample B
T =1 s
and nonlinearly depends on the magnetic field magni- 0.25 T=980°C mod
tude. For nuclei with I =3/2, splitting of the ±3/2 spin Tmod=10 ms
states is described by expression:47 CP 0.2 fit curve
N
A
0.15
δE = EQ[a+((cid:112)1−a+a2−(cid:112)1+a+a2)], (9)
±3/2 2 0.1
where a = 4γ¯hB/E . Here E is the quadrupole split- 0.05
Q Q
tingofthe±1/2and±3/2doubletsatzeromagneticfield
0
and γ is the gyromagnetic ratio for the nuclei. Eq. (9) 0 20 40 60 80 100 120
allowsonetoobtainanexactexpressionforthenuclearg
Magnetic field (mT)
factor. We found, however, that this complex expression
can be well fitted for all the nuclei and magnetic fields
considered here by a phenomenological formula: Figure 2. (Color online) Examples of the Hanle curve simu-
lations for samples A (a) and B (b) for different modulation
B2 periods given in the legends. Symbols are the experimental
gq∗x =kB2+B2 , (10) data and solid lines are the fits.
∆
where k and B are the fitting parameters. According
∆
to this expression, the g factor quadratically rises with lem we have first obtained approximate values of the
magnetic field at small B and then reaches a constant parameters. For this purpose we fixed one parameter,
value at B (cid:29) B∆. An analysis shows that both the pa- be = 4 mT for sample A and be = 2 mT for sample B,
rameters are strongly different for Ga and As nuclei due and obtained other parameters by simple fitting proce-
to different quadrupole splittings. Therefore, to accu- dure using Eqs. (7) and (8). Then we solved the total
rately model the nuclear field, a sum of contributions of equation using the obtained values of the parameters as
different nuclei, like those given by Eqs. (9), would be the initial ones and setting the limits for their possible
considered. The experimental results, however, do not variations. We found that there is only one root of the
contain sufficient information required for separation of equation, which satisfies the physical conditions: Sz is
different contributions. We, therefore, simplify our anal- the real and positive quantity.
ysis and suggest the simplest, linear, dependence for the Numerical solution of the equation for different mag-
g factor, neticfieldsallowedustosimulateHanlecurvesbyappro-
priate choice of the fitting parameters. We have ignored
g∗ =kB, (11) some asymmetry of Hanle curves observed experimen-
qx
tally(seeFig.1)andsimulatedonlyapartofeachHanle
to model the effective nuclear field averaged over all the
curve measures at B > 0. An analysis has shown that
nuclei. Results, described in the next subsection, show
the fitting parameters are not noticeably changed when
that this dependence allows us to explain main peculiar-
another part of Hanle curves is modeled.
ities of the Hanle curves.
Substitution of the expressions (8) into Eq. (7) gives
rise to an equation of the 9-th degree relative to Knight
field B . Solution of this equation for different magnetic C. Analysis of Hanle curves
e
fieldsgivesthefielddependenceofelectronspinpolariza-
tionthatistheHanlecurve. Comparisonofthemodeled Thephenomenologicalmodeldevelopedaboveallowed
Hanle curve with that obtained experimentally allows us us to describe well the non-trivial shape of Hanle curves
to determine fitting parameters B , b , B , B , B , measured for both samples at different modulation peri-
τ e Nd Nq fz
and k for each modulation period. To solve the prob- ods. Example of the Hanle curves obtained in the model
6
200 external magnetic fields, the dipole field almost disap-
Sample B
T= 980C B pearsandthewingsofHanlecurveismainlydetermined
150 BBNdx (mT)Nd 200 BNNddxz bfiyeldco.mApseotinteioncaonfxse-eanindFzi-gc.om3,ptohneenzt-csoomftphoenqeuntadrrauppidollye
Ndz B
decreases at large B and the x-component increases that
B
) Nqx results in relatively sharp decrease of electron spin po-
T
m BNqz 0 larization observed experimentally. So, the dipole field
(N100 0 5 10 15 forms the W-structure and the quadrupole field forms
B B (mT)
the wings of Hanle curve.
Letusnowdiscussotherparametersofthemodel. Pa-
50
rameter B is determined by time τ of the electron spin
τ e
relaxation, seecommentto Eq.(4). As itwasmentioned
above, τ depends on the excitation power but should
e
0 be independent of the modulation period. Therefore we
0 20 40 60 80 100 fixed its value, B =18 mT. This value is obtained from
τ
Magnetic field (mT)
the Hanle curve width at the fastest modulation used
when the nuclear spin effects are negligibly small.
Figure 3. (Color online) Examples of the magnetic field de- Parameter b [see Eq. (8)] characterizes the Knight
e
pendences of the longitudinal and transverse components of
field B averaged over all the nuclei interacting with
thedipoleandquadrupolenuclearfieldscalculatedforsample e
electron spin. The magnitude of this parameter is de-
B using Eq. (8). The parameters used in the calculation are
termined by the electron density on the nuclei.1 The de-
extracted from the Hanle curve measured at the modulation
scribedabovesimulationsoftheHanlecurveshaveshown
period T =300 ms. The magnetic field dependence of B
mod e
is taken from the experimentally measured Hanle curve. In- thatthisparameterhastobechangedundervariationof
set shows behavior of components of the dipole field at small the modulation period. In particular, be = 5.3 mT at
magnetic fields. At zero magnetic field, B =300 mT. slow modulation (T > 0.01 s) and b = 8 mT at fast
Ndz mod e
modulation (T < 0.01 s) for sample A. For sample
mod
B, b = 1.13 mT at slow modulation (T > 0.3 s)
e mod
are shown in Fig. 2. The good correspondence of the and be = 3.6 mT at fast modulation (Tmod < 0.1 s).
measured and simulated Hanle curves allows us to ob- We assume that this variation of be with the modulation
tain values of the fitting parameters at each modulation period is due to different rates of spin relaxation for dif-
period and, therefore, to evaluate their frequency depen- ferent nuclear states. If the relaxation of some nuclear
dence. Althoughthereareseveralfittingparameters,val- statesisslowerthanthemodulationperiod,suchnuclear
uesofmostparameterscanbedeterminedindependently states are ”switched off” from the joint electron-nuclear
because they control different features of Hanle curves. spin dynamics. Correspondingly, the Knight field should
In particular, parameters B and B , describing the be averaged over a subset of nuclear states, which are
Nd Nq
photoinduced dipole and quadrupole nuclear spin polar- not ”switched off”. Difference in the magnitudes of be for
ization,determinethecentralpartwithW-structureand sample A and B is explained by different electron densi-
the peripheral part of Hanle curves, respectively. ties on nuclei in these samples. Sample A contains QDs
Examples of magnetic field dependences of the dipole annealed at lower temperature (Tann =900 ◦C) than the
and quadrupole components of nuclear spin polarization sample B (Tann = 980 ◦C) so that the indium content
are shown in Fig. 3. As seen, the x- and z-component is larger, the electron localization volume is smaller, and
of the dipole field have large magnitude in small mag- the hyperfine interaction is stronger in sample A.34
netic fields. In particular, the dipole component B Parameterk describingnonlinearsplittingofthe±3/2
Ndz
has a maximal value at zero magnetic field and rapidly etc. doublets in magnetic field [see Eq. (11)] is found
decreases with B while component B rapidly rises in to be almost independent of modulation period for both
Ndx
the same range of magnetic field (see insert in Fig. 3). samples. Its average value is: k = 0.9×10−4 mT−1 for
As it is discussed in Ref. 26, such behavior of nuclear sample A and k = 1.8×10−4 mT−1 for sample B. The
field is responsible for the W-structure in Hanle curves. obtained values of k can be compared with those found
Subsequent decrease of the B component completes from Zeeman splittings of the ±3/2 states in different
Ndx
the W-structure. Beyond the W-structure, i.e., in large nuclei. According to the data of Ref. 37, k(Ga) = 20×
magnetic fields, the dipole component of nuclear field is 10−4 mT−1, k(As) =1.3×10−4 mT−1 for sample A and
virtually absent. k(Ga) = 40×10−4 mT−1, k(As) = 7×10−4 mT−1 for
The quadrupole field is weakly changed in small mag- sample B. As seen, these values considerably differ for
netic fields. In particular, x-component of the field is the Ga and As nuclei and are larger than those obtained
almost zero while z-component has some finite, almost from the modeling of Hanle curves.
constant,value. Itisthecomponent,whichstabilizesthe Possible reason for the difference of the k values ob-
electron spin polarization making the Hanle curve broad tainedfromthefittingofexperimentaldataandfromthe
at slow modulation of excitation polarization. At large splittings can be related to the fast phase relaxation of
7
nuclear spin polarization caused by fluctuating electron 300
(a) Sample A
spin polarization under strong optical pumping used in 250 B T= 900°C
the experiments. An analysis shows48 that this relax- Nd
)
T
ation should additionally weaken the effect of transverse m 200
magnetic field on the nuclear spin dynamics. B(f 150
Another possible reason is a contribution of the As , q B
nuclei in an asymmetric atomic configuration containing BN 100 Nq
oneorfewInneighbors. Crystalfieldgradientcausedby B, Nd 50 B
astatisticaloccupationoflatticenodesbytheInandGa f
atoms gives rise to a quadrupole splitting of spin states 0
0 10 20 30 40 50
in the As nuclei.1 The principal axis of the gradient may
T (ms)
be oriented along different crystal axes. The quadrupole mod
splitting in these nuclei is stronger, therefore the value 450 (b) STa=m 9p8l0e° BC
of k should be smaller. These nuclei can be responsi- ) CW
ble for stabilization of the electron spin polarization at mT 300
large magnetic fields and, correspondingly, for the wings (d
N 150
of Hanle curves observed experimentally. This contri- B
bution also explains the fact that the widths of Hanle 0
curves for sample A and sample B do not strongly differ 0 100 200 300 400 500
(seeFig.1)althoughthelatticedeformationinsampleA T (ms)
mod
is three times larger compared to that in sample B.37 100 (c) B CW
Finallyweshouldnote, thatthecontributionofInnu- T) Nq
m 80
cleiintotheeffectofstabilizationoftheelectronspinpo- (
larization is negligible because of wide spread of Zeeman Bf 60
splittins of different states (m=±3/2,±3/2,...,±9/2). ,Nq Bf CW
B 40
20
0
0 1000 2000 3000 4000 5000
D. Dynamics of nuclear fields
T (ms)
mod
ThesimulationofHanlecurvesdescribedaboveallows
Figure 4. (Color online) Dependences of the dipole field
us to analyze evolution of the dipole and quadrupole nu-
B , quadrupole field B , and the field of nuclear spin
clear fields at the modulation of excitation polarization. Nd Nq
fluctuations B on modulation period T for sample A
Figure 4 shows the evolution of initial (photoinduced) f mod
(a) and sample B (b, c). Symbols are the values extracted
values of nuclear fields B and B for both the sam-
Nd Nq from the analysis of experimental data. Solid lines are the
ples. The magnitudes of nuclear fields, in particular, of fits by Eqs. (12) with characteristic times: for sample A:
the dipole component, obtained in the simulations have τ = 1.6 ms, τ = 1.4 ms, τ = 0.5 ms; for sample B:
Nd Nq f
relatively large spread. As it is already discussed (see τ =191ms,τ =0.18ms,τ =328ms,τ =0.2ms,
Nd 1Nq 2Nq 1f
Fig. 3), the dipole component significantly differs from τ2f =467 ms.
zero only at small magnetic fields in the range of W-
structure of the Hanle curves. Therefore, any small in-
accuracy of experimental data in this range noticeable effect)anddrasticallyincreasesthedipolefieldachievable
affects the component. The quadrupole component is at the CW or slowly modulated excitation.
determinedinthelargermagneticfieldrangeand, there- Both the dipole and quadrupole fields decrease with
fore,itsmagnitudeisfoundwithlessuncertainty. Never- shortening the modulation period. The decrease of nu-
theless, in spite of the spread, the obtained values of the clear fields is naturally explained by some inertia of nu-
dipole and quadrupole components demonstrate certain clearspinsystem,whichdoesnotallowittobereoriented
tendency in evolution of nuclear spin polarization. during the half-period of the modulation. This effect en-
As seen from the figure 4, all the nuclear fields tend to ables to estimate the characteristic relaxation times for
gotosomestationaryvaluesatslowenoughmodulation. each nuclear field. The dependences of the nuclear field
These stationary values are very different for different amplitudes on the modulation periods for sample A can
nuclear fields and different samples. For example, as it be approximated by simple phenomenological equation:
is shown in Fig. 4(a), the dipole field B is only three
Nd
(cid:20) (cid:18) (cid:19)(cid:21)
times larger than the quadrupole field BNq in sample B =B 1−exp −Tmod , (12)
A. For the strongly annealed sample B, the dipole field N N∞ τ
N
is at least of 10 times larger than the quadrupole field
[compareFigs.4(b)and 4(c)]. So,theannealingslightly Similar equation well describes evolution of the dipole
decreases the quadrupole field (which is very expectable nuclear field for sample B. At the same time, evolution
8
of the quadrupole nuclear field in this sample is not ex- the nuclear spin system.
ponential and we have to use more complicate equation: The most exciting experimental result is the drastic
slowing down of nuclear spin relaxation at the increase
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
B =B 1−a2exp −Tmod −b2exp −Tmod , of annealing temperature from 900 ◦C for sample A to
N N∞ τ τ 980 ◦C for sample B. This decrease of relaxation rate is
N1 N2
(13) directlyseeninFig.1andthemodelallowsustoestimate
with condition a2 +b2 = 1. Here B is the value of the rate. In principle, the annealing increases the local-
N∞
nuclear field under the continuous wave excitation. ization volume for the resident electrons that may result
As seen from Fig. 4, the relaxation time of the dipole inadecreaseoftherelaxationrateviahyperfinecoupling
field for the stronger annealed sample B is larger by with electron spins.11,52 However, as it is pointed out in
more than two orders of magnitude comparing to that Ref. 34, the annealing gives rise only to the two-fold in-
forsampleA.Sodrasticdifferenceintherelaxationrates crease of the volume and, correspondingly, to the two-
pointsouthighsensitivityofthenuclearspindynamicsto fold decrease of the hyperfine interaction that certainly
quadrupole effects. We should mention also that the re- cannot explain so large difference in relaxation rates.
laxationdynamicsinbulkn-GaAs,wherethequadrupole Other result of our modeling is that the nuclear spin
splitting is very small, is further slowed down by a few dynamics is non-trivial and consists of a fast process
orders of magnitude.32,49–51 with characteristic time of order of 1 ms and a slow pro-
Dynamics of quadrupole field in sample A is charac- cess with characteristic time of about 1 s (see Fig. 4).
terized by a relaxation time, which is close to that for We assume that the slow process is the relaxation of
dipole field in this sample. However, the dynamics in the dipole nuclear field in the conditions when dynamics
sample B is characterized by two relaxation times. The of the dipole and quadrupole components is decoupled.
first one is close to that for sample A and the second These conditions is probably realized in the stronger an-
one is only several times smaller than relaxation time of nealedsampleBwherethedipolefieldisstronglymagni-
dipole field in this sample. Possible reason for such be- fied comparedto thequadrupole field[see Figs.4(b) and
haviorofquadrupolefieldinsampleBisdiscussedinthe (c)].
next section. The dynamics of nuclear fields in sample A is much
FittingoftheHanlecurvesallowedustoobtaintheef- faster and characterized by practically the same relax-
fectivefieldofnuclearspinfluctuations,Bfz. Asonecan ationrate(withinexperimentalerrors)forthedipoleand
seeinFig.4,theamplitudeoffluctuationsdecreaseswith quadrupolecomponents[seeFig.4(a)]. Suchdynamicsin
decreasingperiodofthemodulation. Thepossibleorigin this sample is possibly caused by a mixing of the dipole
of this unexpected, at the first glance, effect is discussed and quadrupole nuclear spin states due to tilting of the
in the next section. The dynamics of Bfz is similar to principalaxisoftheelectricfieldgradienttensororpres-
dynamics of the quadrupole field and is characterized by ence of some biaxiality of the tensor.35
a single relaxation time for sample A and two relaxation The relaxation of quadrupole field in sample B
times for sample B. [Fig. 4(c)] is characterized by the presence of fast and
slow components. It is also possibly due to an asymme-
try of the tensor. Another possible reason is that the
III. DISCUSSION quadrupolecomponentofthenuclearfieldinthissample
is small compared to the dipole one and even small error
Thephenomenologicalmodelusedintheprevioussec- in the separation of the components during the experi-
tionforanalysisoftheexperimentaldataisbasedonap- mentaldataprocessingmayresultinanoticeableadmix-
proximation of the well separated nuclear spin doublets. ture of the dipole component. Such admixture may be
This approximation is valid in some limited range of the responsible for the complex shape time dependence for
transverse magnetic field when the Zeeman splitting of BNq seen in Fig. 4(c).
thedoubletsisconsiderablysmallerthanthequadrupole Large difference of relaxation times for the dipole and
splitting. However the experimental data analyzed in quadrupolecomponentsofnuclearfieldrequiresseparate
the present work are measured in the relatively wide discussion. The slow relaxation in the ±1/2 nuclear spin
range of magnetic field of about ±100 mT where the system is inherent property of the system well known in
Zeeman and quadrupole splittings become comparable, the nuclear magnetic resonance.47 In this process, the
see Ref. 37. In such magnetic fields, the dipole (±1/2) angular momentum ±1 should be transferred from the
and quadrupole (±3/2,...) states are mixed that makes nucleustoaphonon. Howevertherearenosuchphonons
considerationofthedipoleandquadrupolefieldsinlarge in the phonon bath. Hyperfine interaction with resident
magnetic fields to be not applicable. A more accurate electrons accelerates this process11,52 but it still remains
microscopicmodelisrequiredforanalysisofthespindy- slow.
namics in quadrupole nuclei. To the best of our knowl- Direct relaxation between the ±3/2 states is also in-
edge, there is no such model so far. Therefore, we con- effective. However any modulation of the electric field
sider the results obtained in the framework of our model gradient should provoke an efficient relaxation ±3/2 →
asqualitative,ratherthanquantitative,characteristicsof ±1/2. In particular, the crystal field gradient can be
9
modulated by fluctuations of the carrier density.30–32 In of the dynamics of the nuclear spin fluctuation and of
the case of QDs with relatively deep potential well for thequadrupolenuclearfieldsuggeststhatthequadrupole
carriers, the fluctuations are small, at least at low sam- states mainly contribute to z-component of the nuclear
ple temperature. However, in the case of optical excita- spin fluctuations. Similar conclusion has been made in
tion of QDs, the fluctuations may be much larger due to Ref. 53.
separatecaptureofelectronsandholessothatthismech-
anismofrelaxationofthequadrupolestatesmaybecome
effective. After the ±3/2 → ±1/2 relaxation, rapid pre- IV. CONCLUSION
cession of the nuclear spins in the transverse magnetic
field mixes the ±1/2 states due to large effective nuclear Strong modification of Hanle curves observed under
g factor for this doublet. In particular, the precession modulation of excitation polarization is demonstrated
frequency is of order 104 Hz in magnetic field of 1 mT. to contain valuable information about the dynamics
Thebackwardrelaxation ±1/2→±3/2, whichoccursat of coupled electron-nuclear spin system in the studied
any moment of the precession, should give rise to effec- (In,Ga)As/GaAs QDs. To extract this information, we
tive destruction of the quadrupole field. This simplified havedevelopedasimplifiedphenomenologicalmodelcon-
picture of the relaxation process may explain the rapid sidering separate dynamics of the dipole and quadrupole
relaxation of quadrupole field observed experimentally. nuclear fields. In particular, the quadrupole field can ef-
Finally we should mention about one more effect ob- ficientlystabilizeelectronspinpolarizationinlargemag-
served at the modulation of the excitation polarization. netic fields up to 100 mT. At the same time, the rel-
Thisisthesuppressionoftheeffectivefieldofnuclearspin atively fast relaxation of the quadrupole nuclear states
fluctuations, which is observed at the shortening of the to the ±1/2 states may considerably shorten the elec-
modulationperiod[seeFig.4(c)]. Thesuppressionunam- tron spin lifetime. In the studied samples with different
biguouslyfollowsfromthefactofthestrongnarrowingof quadrupole splittings, the lifetimes differ by more than
the Hanle curve down to the purely electron peak at the two orders of magnitude.
fast enough modulation of polarization. We assume that ACKNOWLEDGMENTS
the strong optical pumping with rapidly alternating po-
larization equalizes the population of nuclear states with The authors thank K.V. Kavokin for fruitful discus-
spin down and spin up and, hence, suppresses the lon- sion. The work is supported by the Russian Foundation
gitudinal component of nuclear spin fluctuations, which for Basic Research and the Deutsche Forschungsgemein-
supports orientation of the electron spin. The transverse schaft in the frame of International Collaborative Re-
componentofthefieldofnuclearspinfluctuationswhile, searchCenterTRR160(ProjectNo. 15-52-12020). I.V.I.
ofcourse, remain, but, astheanalysis showed, theircon- acknowledges the support of the Russian Foundation for
tribution to the formation of the Hanle curve is negligi- BasicResearch(ContractNo. 16-02-00245A)andSPbSU
ble small The mentioned above almost total coincidence (Grant No. 11.38.213.2014).
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