The Knight field and the local nuclear dipole-dipole field in an (In,Ga)As quantum dot ensemble
T. Auer, R. Oulton, A. Bauschulte, D. R. Yakovlev, M. Bayer, S. Yu. Verbin, R. V. Cherbunin, D. Reuter, A. D. Wieck
aa r X i v : . [ c ond - m a t . o t h e r] S e p The Knight field and the nuclear dipole-dipole field in an InGaAs/GaAs quantum dotensemble
T. Auer ⋆ , R. Oulton † , A. Bauschulte, D. R. Yakovlev ‡ , and M. Bayer Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany
S. Yu. Verbin and R. V. Cherbunin
Institute of Physics, St. Petersburg State University, St. Petersburg 198504, Russia
D. Reuter and A. D. Wieck
Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany (Dated: November 12, 2018)We present a comprehensive investigation of the electron-nuclear system of negatively chargedInGaAs/GaAs self-assembled quantum dots under the influence of weak external magnetic fields (upto 2 mT). We demonstrate that, in contrast to conventional semiconductor systems, these small fieldshave a profound influence on the electron spin dynamics, via the hyperfine interaction. Quantumdots, with their comparatively limited number of nuclei, present electron-nuclear behavior that isunique to low-dimensional systems. We show that the conventional Hanle effect used to measureelectron spin relaxation times, for example, cannot be used in these systems when the spin lifetimesare long. An individual nucleus in the QD is subject to milli-Tesla effective fields, arising from theinteraction with its nearest-neighbors and with the electronic Knight field. The alignment of eachnucleus is influenced by application of external fields of the same magnitude. A polarized nuclearsystem, which may have an effective field strength of several Tesla, may easily be influenced by thesemilli-Tesla fields. This in turn has a dramatic effect on the electron spin dynamics, and we use thistechnique to gain a measure of both the dipole-dipole field and the maximum Knight field in oursystem, thus allowing us to estimate the maximum Overhauser field that may be generated at zeroexternal magnetic field. We also show that one may fine-tune the angle which the Overhauser fieldmakes with the optical axis.
PACS numbers: 42.25.Kb, 78.55.Cr, 78.67.Hc
I. INTRODUCTION
The expectation that the electron spin in semiconduc-tor quantum dots (QDs) could serve as a building blockfor quantum computing applications has drawn renewedattention to the role of the quantum dot nuclei. In theseventies it was shown that for bulk semiconductors, theinterplay between an electron spin (at that time, of adonor trapped electron) and the nuclear spins in its vicin-ity leads to a wide variety of effects and often exhibitsunexpected behavior [1, 3, 4, 5, 6, 7, 8, 10, 11]. In QDsthe hyperfine coupling between electron and nuclear spinsis further enhanced by the strong localization of the elec-tron in the dot, giving rise to complex dynamics [9].Nuclear spins can be polarized by transfer of angularmomentum from optically oriented electrons, in a processknown as the Overhauser effect [1, 2, 12]. It was shownthat nuclear polarization obtained in this way leads toan effective magnetic field of the order Tesla for the QDelectron [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Most ofthese experiments exploited the Overhauser energy shift[2, 13] of the electron Zeeman levels split in an externalfield where the nuclear field was of the same order ofmagnitude as the external field.In this paper we report experimental studies of In-GaAs/GaAs quantum dots. We monitor the polarizationof the QD ground state photoluminescence (PL) in the presence of weak external magnetic fields. The magni-tude of these fields (a few mT or less) are shown to befar too small to have any direct effect on the dynamicsof the electron in the QD itself. Milli-Tesla fields exertedonto electron spins only affect the spin precession dynam-ics over timescales greater than tens of nanoseconds. Itis only recently that electron spin coherence times longerthan this (several microseconds) have been observed inQDs [23]. One might imagine that milli-Tesla magneticfields may be used in this long coherence time regime toprobe and manipulate spin dynamics, and in fact this hasbeen reported [24, 25]. However, as we will demonstrate,considerable caution needs to be exercised in the inter-pretation of such data, as interactions are usually presentwhich screen the direct effect of this external field on theelectron.For the above reasons, the influence of milli-Tesla fieldsonto the electron is usually negligible in semiconductors.It may come as a surprise, therefore, that we observe verydramatic effects on the electron polarization when apply-ing these fields. This occurs in our system due to the factthat the electron dynamics are governed by the magni-tude and direction of an effective nuclear, or Overhauserfield that is exerted onto the electron from ∼ nucleiin the QD. By polarizing a significant fraction of these ∼ nuclear spins in the same direction, one may gen-erate Tesla-strength Overhauser fields that completelydominate the electron spin dynamics in the system. TheTesla strength of these Overhauser fields is however de-ceptive: the interaction is not a real magnetic field, butan effective field that acts on the electron only.While the sum of the interaction from all of the nuclearspins onto the electron is large, each nucleus itself is sub-ject only to very small effective fields. Considered fromthe point of view of a single nucleus in the QD, the nu-cleus experiences three effective fields (i) from its nearestneighbors (the dipole-dipole field), of the order of magni-tude of ∼ . x -direction) and parallel (lon-gitudinal, z -direction) to the optical axis, allowing theKnight field feature to become more visible. In SectionV.C we estimate theoretically the maximum degree ofnuclear polarization we are able to obtain in our sample.Here we use the two important values we have measured,the Knight and dipole-dipole fields, the two quantitieswhich govern spin diffusion in the system after the nu-clear spins have been polarized. The Knight field actsto hold the polarization, whereas the dipole-dipole fieldallows spin diffusion. The ratio of these effective fieldsgoverns the nuclear polarization theoretically obtainable in our sample. We demonstrate that in principle, a nu-clear polarization of up to >
98 % may be generated.
II. SAMPLES AND EXPERIMENT
The sample studied is a 20 layer InGaAs/GaAs self as-sembled QD ensemble with a sheet dot density of 10 cm − . 20 nm below each QD layer a Silicon δ -dopinglayer is located with a doping density about equal thedot density. Thus each QD is permanently occupiedwith on average one “resident electron”, as confirmed bypump-probe Faraday rotation measurements [23]. TheInAs/GaAs QD heterostructure was grown by molecularbeam epitaxy on a (100) GaAs substrate. After growthit was thermally annealed for 30 seconds at 900 ‰ . Thisleads to interdiffusion of Ga ions into the QDs, whichshifts the ground state emission to 1.34 eV [27].The measurements were performed at a temperatureof T = 2 K with the sample installed in an optical bathcryostat which was placed between three orthogonal pairsof Helmholtz coils allowing application of external mag-netic fields of a few mT in all directions. The coils wereused to compensate parasitic magnetic fields, of e. g. ge-omagnetic origin as well as to apply fields up to 3 mTparallel or perpendicular to the optical axis (longitudinal, z direction or transverse, x direction, respectively).The optical excitation was performed using a modelocked Ti:Sapphire laser with a pulse duration of 1.5 psand a repetition rate of 75.6 MHz (pulses separated by13.2 ns). The excitation energy was 1.459 eV which cor-responds to the low energy flank of the wetting layer.The helicity of the exciting light could be modulated bymeans of an electro-optical modulator and a λ/ µs and 500 ms of σ + or σ − polarization wereformed.The beam was focused on the sample with a 10 cmfocal length lens which was simultaneously used to col-lect the PL. The photoluminescence was dispersed with a0.5 m monochromator and detected circular polarizationresolved with a silicon avalanche photo-diode which wasread out using a gated two-channel photon counter. Inorder to ensure a homogeneous excitation of the QDs thePL was collected from the center of the laser spot. III. OPTICAL ORIENTATION OF ELECTRONSAND NEGATIVE CIRCULAR POLARIZATION
The QDs studied contain on average one “resident”electron. After excitation of an electron-hole pair intothe wetting layer and subsequent capture of the carriersinto the QD, a trion is formed in these singly chargedQDs. The trion ground state consists of two electrons inthe conduction band s-shell with antiparallel spins and asingle hole in the valence band s-shell. The helicity of thelight emitted after the trion decays is therefore governed (b) P L c i r c u l a r po l a r i z a t i on ( % ) Excitation power (P/P ) B = 0 TT = 2 K (a) -30-20-100102030 C i r c . po l a r i z a t i on ( % ) exc +det + I n t en s i t y ( a r b . un i t s . ) Energy (eV)exc +det -
FIG. 1: (a) (color online) PL spectra of the QD sample stud-ied (lower panel). Excitation was σ + at 1.459 eV. The PLintensity of σ − emission is greater than that of σ + emission,thus the circular polarization degree is negative (upper panel).(b) Power dependence of the circular polarization degree ρ c . ρ c rises with power and finally saturates indicating that amemory of the spin orientation of the resident electrons iskept until the following excitation cycle (see text). P = 2 . . by the spin orientation of the hole. As a consequence,the helicity of the photon emitted directly determinesthe spin orientation of the resident electron left in theQD after radiative recombination.It is an established phenomenon for singly n-dopedQDs under non-resonant excitation and at zero magneticfield that the circular polarization degree ρ c of the emis-sion has the opposite helicity to the excitation (known asnegative circular polarization effect, NCP). Here we usethe standard definition: ρ c = ( I ++ − I + − ) / ( I ++ + I + − ) , (1)with I ++ denoting the intensity of PL having the samehelicity as the excitation ( σ + ) and I + − the intensity ofPL polarized oppositely to the excitation.Figure 1 (a) depicts PL spectra of the QD ensemblestudied under σ + excitation in the wetting layer, with de-tection either co-polarized ( σ + ) or cross-polarized ( σ − )to the excitation. Both PL spectra show two peaks cor-responding to the inhomogeneously broadened groundstate emission at ∼ .
34 eV (s-shell), and to the firstexcited state emission at ∼ .
37 eV (p-shell). The po-larization of the PL is negative throughout the emissionfrom the s-shell (Fig. 1 (a), upper panel).Different mechanisms explaining the NCP effect havebeen suggested [28, 29, 30, 31, 32, 33, 34]. In Ref. [29]a mechanism was proposed whereby the anisotropic ex-change between an excited state electron and the holeinduces a spin exchange or “flip-flop” process, in orderto overcome Pauli blocking of the ground state electrons with parallel spins. In Ref. [31], it was suggested thatdark excitons are preferentially captured from the wet-ting layer by the QDs. We note, however, that all of thesemechanisms imply that the electron remaining after trionrecombination accumulates spin polarization. This is theimportant feature we exploit in our experiments. Themechanism leading to this accumulation of electron spinpolarization may still be subject to discussions, but isnot decisive for our studies.As in our previous work [22], we make use of this fact.We excite the system with 75.6 MHz pulsed circularly po-larized excitation, allowing pumping of the electron spinpopulation to occur. The electron spin polarization levelreached in the sample is governed by the competitionbetween the optical pumping rate and the decay dynam-ics of the electron spins. During our measurements, wealways choose an excitation intensity well into the sat-uration regime. This ensures that when measuring thechange in electron spin polarization, the effects we ob-serve are due to changes in the decay rate of the electronspins, and not the optical pumping rate.Figure 1 (b) shows the PL polarization as a function ofexcitation density. As expected, ρ c is strongly excitationpower dependent. This dependence reflects the efficiencyof the optical excitation of the dots, i. e. the average timebetween two excitation events. As the excitation poweris increased, the electron spins become more polarizedand ρ c increases until it reaches the saturation value of-27 to -30 %. The negative circular polarization is lim-ited by the fraction of loaded QDs in the ensemble andthe spin memory of the photoexcited electrons upon re-laxation. Neutral excitons and biexcitons may also becreated in the ensemble and spin preservation during re-laxation from the wetting layer may not be perfect. Boththese facts will reduce the ρ c value. Therefore, even withfull resident electron spin polarization, the value of ρ c will not reach -100 %. In the NCP effect, both the pho-toexcited electrons that retain their polarization, and thepolarized resident electron contribute to the negative po-larization. One may estimate the circular polarization ρ c in a simple model: ρ c = F x [ P + S z ( I )], where P is theaverage polarization of the photoinjected electron spins,independent of excitation intensity and S z ( I ) is the av-erage polarization of the resident electron spin along thez-axis, and is dependent on excitation power. F is thefraction of QDs in the sample that are singly chargedand x is the fraction of negatively circularly polarizedphotons emitted when either the photoexcited or the res-ident electron is polarized. Depending on the position onthe sample, F may be as large as 0.5 so that the electronspin polarization in the singly charged dots may actu-ally be considerably larger, as the NCP measured. Thuswe see that there is a linear dependence between NCPand resident electron spin polarization where the polar-ization of the photogenerated electron spins merely addsan offset. This is not crucial in our case as we solely dis-cuss changes in NCP and thus electron spin polarization,respectively. IV. ORIENTATION OF NUCLEAR SPINS ANDTHE ELECTRON-NUCLEAR SPIN SYSTEMA. Electron spin precession in the Overhauser field
In this section, we discuss in general terms the inter-action between an electron in a QD and its constituentnuclei at B ext = 0. Recent theoretical [35, 36, 37, 38, 39]and experimental [16, 21, 40, 41, 42] work has demon-strated that this is the key electron spin relaxation mech-anism in QDs. In addition, strong Overhauser fields inQDs have been directly measured [13, 19] and inferred[22]. We consider first of all the hyperfine interaction inQDs, and consider the effect of an Overhauser field ontothe electron spin system under different optical orienta-tion regimes. We then discuss the factors which influencethe magnitude of the Overhauser field, before discussingthe experimental results in milli-Tesla fields.As an electron inside a QD is strongly localized, theinteraction between electron spin and a nuclear spin isenhanced in comparison to bulk semiconductors. A sin-gle electron populating the conduction band of a QD hasa Bloch wavefunction with s-symmetry, leading to a highelectron density at the nuclear site. The envelope wave-function has an overlap with about 10 QD nuclei. Thiscan be estimated by considering an approximately diskshaped QD 20 nm in diameter and 5 nm in height. Theirspins interact with the electron spin via the hyperfineinteraction, described by the Fermi contact Hamiltonian[43] ˆ H hf = X i A i | ψ ( R i ) | ˆ S · ˆ I i . (2)Here | ψ ( R i ) | is the probability density of the electron atthe location R i of the i th nucleus, A i is the hyperfine in-teraction constant and ˆ S , ˆ I i the operators of the electronspin and the nuclear spin, respectively.As we see from Eq. (2), the total interaction energy isdependent on the electron spin, S , and the orientationof each of the nuclear spins, I i . The interaction energytherefore crucially depends on the alignment of the nu-clear spins in the QD. One may consider that the nucleiexert an effective magnetic field B N onto the electron asgiven by X i A i | ψ ( R i ) | I i ! · S = g e µ B B N · S . (3)where g e is the electron g-factor, µ B the Bohr magneton.In light of the assumption that it is the hyperfine inter-action that causes electron spin decay, we neglect otherspin dephasing mechanisms (such as phonon interactions)at low temperatures and consider what happens to anelectron spin in this nuclear magnetic field. The exci-tation pulse induces formation of a trion, which radia-tively recombines to leave behind an electron spin po-larized along the ± z direction (the optical axis). The B S S z S z B S S av q FIG. 2: Schematic of electron spin precession in a nuclearfield oriented at angle θ to the z-axis. The average electronspin vector is tilted at θ . The polarization measured, howeveris given by the projection of this average electron spin ontothe z -direction, such that S z = S cos θ . electron spin dynamics are then governed by the effec-tive field B N . In general, the motion of a spin S ( t ) in afixed magnetic field B is described by S ( t ) = ( S · b ) b + [ S − ( S · b ) b ] cos ωt +[( S − ( S · b ) b ) × b ] sin ωt, (4)where S is the initial electron spin, b = B /B is the unitvector in direction of the magnetic field acting on theelectron and ω is the Larmor precession frequency givenby ω = | g e | µ B B/ ~ . The dynamics of the electron spin at B ext = 0 is clearly dependent, therefore, on the directionof the Overhauser field, which may have any orientationin space. If the Overhauser field is aligned along theoptical ( z ) axis, the electron spin will be static, and nopolarization will be lost. For any B N field not alignedalong z , however, the electron spin evolves in time. Inthe classical analogue, the electron spin precesses aboutthe B N field.In the measurements presented here, the electron spinprecession period in B ≈
25 mT (the order of magnitudeof the frozen nuclear spin fluctuation field, discussed be-low) is ≈ . Here, we assume that theQD in which the electron is confined is excited byevery pulse due to the high excitation intensity .By the time it is sampled by the next excitation pulse inPL, we measure a time average of this electron spin inthe ensemble. If we average the z component of S ( t ) overtime we obtain S z ( B ) = S B z B = S cos ( θ ( B )) . (5)The expression for S z is obviously analogous to the zcomponent of the projection of the initial electron spin S on the precession axis defined by the magnetic fieldwhere θ is the angle between the precession axis and the z axis (see Fig. 2).Let us now consider what occurs in our system. Fig-ure 2 shows the precession dynamics of the electron spin.The circularly polarized excitation results in spin popu-lation, giving an average initial electron spin S ( t = 0) = S . The electron precesses in the Overhauser field severaltimes and the ensemble reaches a steady state of polariza-tion S cos θ before the next pulse arrives 13.2 ns later.This next pulse reads out the average projection of theelectron onto the z -axis, as discussed in Section III. Thusthe PL polarization is dependent on the initial electronspin S and θ ( B ). S is governed by the electron spinretained during energy relaxation from the wetting layerto the QD ground state. This is constant in our exper-iments, as the excitation conditions are kept the same.The angle of the Overhauser field to the z -axis, θ ( B ), isthe factor that changes dramatically in these measure-ments, and we consider now what governs the magnitudeand orientation of the Overhauser field. B. The nuclear magnetic field in the absence ofoptical orientation
In this Section we discuss the electron spin dynamicsfor the case where the nuclear spins are given no par-ticular orientation. One might expect that in a systemof randomly oriented nuclear spins, the nuclear magneticfield would be zero, and the electrons thus would be unaf-fected by the presence of the nuclei. Generally, however,the magnetic field generated by the sum of the nuclearspins is never exactly zero. The QD contains a large butnevertheless finite number of nuclei ( N = 10 ), whichmeans that statistically, the number of spins parallel andantiparallel in any given direction will not be equal, butdiffer by a value p N/ B f , ori-entated in a random direction in 3D space, about whichthe electrons precess. The magnitude of B f can be es-timated by B f = b N / √ N with b N being the maximumnuclear magnetic field for 100 % nuclear polarization. Weestimate below a value of b N = 8 . N ≈ one thus obtains B f ≈
26 mT with an in-planecomponent of B f,xy ≈
20 mT. Experimental values of 10to 30 mT for B f agree well with this estimate [44].How this B f field affects electron spin dephasing de-pends crucially on the timescale of reorientation of thenuclear spins compared to the precession period of theelectron in B f (5 – 6 ns for B f ≈
25 mT). Nuclear spindynamics tend to be much slower than electron spin dy-namics: the nuclear spin fluctuation field changes on atimescale of 10 − s [41, 52] due to the precession of thenuclear spins in the milli-Tesla magnetic field generatedby the electron spin [36] (see Table I for an overviewabout the relevant timescales). This means that overtimescales less than 1 µ s, the electron is exposed to a”snapshot” of B f , where the nuclear spin configurationremains “frozen”. In the absence of an external magneticfield, only the internal field B = B f acts on the electron.The direction and the magnitude of this frozen nuclearspin fluctuation will vary from dot to dot which leads toa rapid decay of the average electron spin orientation inthe ensemble (note that this is also true for single dots TABLE I: Typical timescales occuring in the electron-nuclearspin system assuming g e = 0 . at zero external field [8, 10,36].precession of electron spin in ∼
10 mT B f field ∼ − sprecession of nuclear spins in Knight field ∼ − srelaxation of nuclear spins in dipole-dipole field ∼ − spolarization of nuclear spins using NCP ∼ − s when the electron polarization is measured as an aver-age over many excitation cycles exceeding the nuclearfluctuation time). Despite the fact that the B f field israndomly oriented at any given time, the average elec-tron spin measured over the ensemble does not decay tozero. Assuming that the nuclear spins are randomly dis-tributed, B f,x = B f,y = B f,z and thus from Eq. (5): θ = arccos 1 √ b = 54 . ° ,S z = S . (6)For a randomly oriented nuclear spin system, the averageangle of B f is θ = 54 . ° , and the electron spin polariza-tion hence quickly decays to about 1/3 of its initial valuedue to the frozen nuclear field. Total decay then followson a microsecond timescale due to continuous change indirection of this nuclear field [36, 41]. The value of 1/3obviously arises from the fact that the projection onto alldirections in 3D space is equal. The initial orientation ofthe electron is nevertheless important. In this system theelectron starts with orientation along the z -axis but withno preferential direction in the x - y plane, so that, whenensemble averaging, a residual projection onto the z -axisis retained, but no preferential direction exists in the x - y plane. C. Optical orientation of the nuclear spins
Strong optical pumping of the system with circularlypolarized light leads to a continuous transfer of angularmomentum from the photons to the electron spins. Itis well-known that via spin flip-flops with polarized elec-trons, orientation of the nuclear spins along the axis ofexcitation may occur (Overhauser effect). For this pro-cess the respective orientation of the B f field in a QD atthe point of time of the trion decay may be important.For QDs containing a B f field predominantly orientedalong z the electron spin’s z projection stays large andthus enough time is given to flip a nuclear spin. How-ever, in QDs where the B f field is by chance predomi-nantly transverse, the electron spin precesses and is notable to polarize nuclear spins. While some B f configura-tions may inhibit electron spin preservation, the nuclearspin system changes on a microsecond timescale, so even-tually most QDs will experience some nuclear polariza-tion. Continuous optical pumping realigns the electron -32-30-28-26-24-22-20-200 -100 0 100 200 modulationT m = 18 m s Magnetic field B z (mT) C i r c u l a r po l a r i z a t i on r c ( % ) no modulation S S B B S z S z qq ´(a) (b) FIG. 3: (a) Dependence of the negative circular PL polariza-tion on a longitudinal magnetic field. For fast modulation ofthe excitation polarization (full circles) no significant nuclearpolarization can build up and at B z = 0 the nuclear fluc-tuation field lead to depolarization of the ensemble average S z . When B z becomes large enough to suppress the nuclearfluctuation field B f the polarization is maintained. Withoutmodulation (open circles) nuclear polarization in z directionbuilds up and the resultant B N field plays the same role asthe external field, also at B z = 0. (b) Schematic for preces-sion of the electron spin in the fluctuation field: (top) as inFig. 2; and (bottom) for a total field at a shallower angle tothe z -axis. For precession about the axis with θ ′ < θ more of S z is conserved. after angular momentum transfer to a nucleus, such thatmany nuclear spins become oriented. Without opticalorientation, the nuclear fluctuation fields in every QD ofthe ensemble are evenly distributed in all three dimen-sions. With optical orientation, an additional field, theOverhauser field B N , is generated along the z axis, thatmay be much larger than the in-plane component B f,xy : B N > B f,xy . (7)The electrons now precess about a nuclear field whose z component dominates, resulting in an increase of averageelectron spin polarization S z in comparison to the case ofa totally randomly oriented nuclear system. The angle θ in Eq. (5) decreases and S z increases.A significant nuclear polarization obtained by strongpumping and sufficiently long illumination of the sampleleads to a nuclear field B N ≫ B f,xy parallel to the z axisand a marked reduction of the influence of the nuclearfluctuation field, maximally restoring electron spin align-ment in z direction [44]. Note that at the very edges ofthe QDs nuclear spin diffusion out of the dot may occur,depending on the nuclear species. In the core of the QDs,however, where the Knight field is strong, spin diffusionis suppressed, and it is here that the nuclear spins maybecome polarized in our case.Figure 3 (a) illustrates the effect of either an externallyapplied field or an internal field generated by nuclearpolarization. Two curves are shown displaying the dependence of the PL polarization on an external fieldin z direction. One of them was obtained for excitationhelicity modulation between σ + and σ − with period T m = 18 µs which is about two orders of magnitudefaster than the nuclear spins need to be polarized (withmeasurements performed during the σ + cycle only). Theother one was recorded with unmodulated excitation,i. e. constant σ + excitation. Let us first consider themodulated excitation case. Here, the net angular mo-mentum flux into the system averaged over time is zeroand no significant nuclear polarization B N builds up (wewill demonstrate in Sec. V.A that significant nuclearpolarization requires tens of milliseconds pumpingtime). In this case, the B f field reduces the electronpolarization at external field B z = 0 to about −
21 %.When B z is increased, the resultant field B = B z + B f isat an angle closer to the z axis ( θ ′ < θ ). The projectionof the electron polarization, given by Eq. (5), increases,and ρ c goes from ≈ −
21 % to ≈ −
30 %. For values B z ≫ B f , θ ′ ≈ ≈ −
31 %.We now compare this to the case where a nuclear fieldis allowed to accumulate. Unmodulated excitation causesoptical orientation of the nuclear spins even at B z = 0and a nuclear field B N in z direction builds up. Thisnuclear field plays exactly the same role as an externalfield in increasing the projection of the electron spin ontothe z axis. The resultant field onto the electron, given by B f + B N , is, again, closer to the z axis. In this case, B N dominates over B f . The polarization reaches ≈ −
29 %already at B z = 0, demonstrating that for almost allQDs, a significant nuclear polarization must occur. Notethat the small dip which is still apparent for unmodulatedexcitation in Fig. 3 is caused by the ensemble distributionand the distribution of the B f field. While a large frac-tion of the QDs contains a polarized nuclear spin systemthere may still be some with less nuclear polarization.For higher external magnetic field, almost all of the ofQDs house a strongly polarized nuclear spin system.Note that in this work, the relative strength of the un-derlying fluctuation field, B f , and the optically generatedOverhauser field, B N , hold the key to the electron spindynamics. In fact, it is the presence of the significant fluc-tuation field as well as an optically generated Overhauserfield that are unique to QDs in semiconductor systems.Figure 4 summarizes all the different influences be-tween the resident electron spin, a single nuclear spinand the nuclear spin ensemble. D. The influence of Knight field and dipole-dipoleinteraction on nuclear spin polarization
We have discussed the effect of the nuclei on the elec-tron spin until this point by considering them en masse.This approach is sufficient for explaining the optical ori-entation of the nuclear system as a whole, and the sub- single electronspin
LASER angularmomentum - NSF field- Overhauser field nuclear spinensemble
Knight field hyperfineinteraction
FIG. 4: Summary of the interactions in the electron-nuclearspin system: A net angular momentum flux into the systemis provided by the circularly polarized laser light. On a singlespin level the electron and the nuclei interact via the hyperfineinteraction. The action of the entirety of the nuclear spinsupon the electron spin can be described by the Overhauserand the B f field. The nuclear spins of the ensemble relax dueto the dipole-dipole interaction between them. sequent effect on the electron spin. The electron-nuclearsystem is, however, a highly interdependent coupled sys-tem, and to understand the electron spin dynamics onemust also gain a detailed understanding of the nuclearspin dynamics. In this section, therefore, we considerwhat happens to a single nuclear spin as it interacts withits nearest neighbors, the electron and an external field.The nuclear spin system is relatively isolated, such thatin the absence of external magnetic fields, only two in-teractions dominate. First is the nearest-neighbor inter-action. In an unpolarized nuclear ensemble, each nucleusexperiences an effective magnetic field from the neighbor-ing nuclear spins. This field (also known as the dipole-dipole field), denoted as ˜ B L , is of the order of 0 . − seconds [8]. As we will see in theexperimental results in Section V.A., the timescale fordynamic nuclear polarization is 10 − seconds, two ordersof magnitude slower than the dipole-dipole interaction. Ithas nevertheless been observed that even in zero appliedexternal field conditions, significant nuclear polarizationmay occur in QDs [26, 45]. The dipole-dipole interactionis not spin conserving, which leads to the question of whynuclear polarization may occur at all, and brings us ontothe second interaction.The second important interaction that a nuclear spinhas with its environment is that with the electron (thehyperfine interaction). Equation 1 may be expanded tobe expressed as:ˆ H hf = X i A i | ψ ( R i ) | [ ˆ S z · ˆ I iz +( ˆ S + · ˆ I i − + ˆ S − · ˆ I i + ) / . (8)The second term expresses the electron-nuclear spin flip-flop interaction responsible for the optical orientation ofthe nuclear spins, as described before. The first term,on the other hand, may be re-expressed as an effectivemagnetic field from the electron, B ie , acting onto the i th nuclear spin:ˆ H hf = − ~ µ i X i ˆ B ie · ˆ I iz + X i A i | ψ ( R i ) | ( ˆ S + · ˆ I i − + ˆ S − · ˆ I i + ) / , (9) where B ie = A i | ψ ( R i ) | ˆ S z . (10)This effective field, the “Knight field”, was first identi-fied in nuclear magnetic resonance (NMR) experiments[46] as a shift in frequency of the characteristic NMRresonance for particular metals. This shift occurs dueto a paramagnetic effect from the presence of conductionband electrons, with the magnitude of this shift in energybeing equal to the hyperfine splitting of the ground stateof the atom. In our experiments, we cannot measure theKnight shift directly without NMR techniques, and so,as we will see in Section V, we apply a magnetic fieldequal and opposite to this Knight field and investigatethe back-action effect on the Overhauser field.From Eq. 10 we see that each nuclear spin experiencesa Knight field that depends on (i) the location of thenucleus in the QD (the Knight field will be strongestfor a nucleus in the centre of the QD, where | ψ ( R i ) | islargest) and (ii) S z , the projection of the electron spinonto the z -axis. We have already discussed in SectionIV.A that the electron precesses on a timescale of lessthan 10 ns, whereas the interaction of each nucleus ismuch slower than this. We may therefore use the time-averaged electron spin projection S z = S cos θ , as givenin Eq. 5.In the absence of an external field, the value of theKnight field is an important quantity in determiningwhether nuclear polarization occurs. The effective fieldgenerated by the electron acts to screen the dipole-dipoleinteraction, and inhibits nuclear spin diffusion [8, 26]. Ifone makes the assumption that the maximum nuclear po-larization may be achieved as long as dipole-dipole diffu-sion is completely suppressed, one may express the com-petition between the Knight field and the dipole-dipolefield as [8, 47]: B N b N ≈ B e B e + ˜ B L . (11)where b N is the maximum achievable nuclear field for agiven alloy system. Thus, the value B e is an importantone: a Knight field value significantly larger than thedipole-dipole field will allow nuclear polarization to oc-cur. Due to Knight field variation, nuclear polarizationwill obviously vary across the QD.Going back to Eq. 10 we see that the Knight field isalso dependent on the electron spin polarization, S z . Wewill see later that the experimentally determined valueof the Knight field is dependent on the electron spin po-larization, but it is also useful to define the maximumKnight field, b ie for a given nucleus, such that: B ie = − b ie h S i S . (12)where b ie is the maximum Knight field at a particularnuclear site [8, 10]. The maximum Knight field is, infact, the more important quantity to be determined in aparticular QD system. As long as a strong nuclear fieldis generated ( B N ≫ B f ) an electron will remain alignedalong the z -axis, and the Knight field at all the nuclei willbe the maximum possible in that system. As we will seelater, in our experiments the only technique we have toprobe the Knight field is to apply a field equal and oppo-site to it in order to depolarize the nuclei. At this pointthe average electron spin decreases significantly, and thismust be taken into account when determining the exper-imental value of the Knight field value that we measure.However, although we measure B e , we may extrapolate b e because we also have a value for S z .Finally, it is also important to note that in our exper-iment we determine a ”weighted average” value of theKnight field. The Knight field value will vary betweeneach nucleus, going from a maximum value in the centre( b maxe ) to zero outside the QD. Making the approxima-tion that | ψ ( r ) | = exp( − r/a ), where a is the radiusof the QD and r is the distance from the QD centre, onemay estimate that the average value is equal to half themaximum Knight field b maxe / B e ,that is an average of the entire nuclear spin ensemble inthe QD, and is also dependent on S z . V. RESULTS AND DISCUSSIONA. Hanle measurements and the dipole-dipole field
In this Section we discuss the application of a purelytransverse magnetic field. We discuss the often-used in-terpretation of this type of experiment for determiningthe electron spin relaxation time, and demonstrate ex-perimentally that for a QD system, this interpretationdoes not hold, and that in general it cannot be used forHanle curve widths < z -axis) at time t = 0 and monitoring thedecrease of PL polarization as a transverse (known asVoigt geometry) magnetic field is applied [8]. An electronspin will precess in the z - y plane, such that, according toEq. (4), the dynamics will be given by S z ( t ) = S cos( ωt ).When integrating over many cycles, the projection of thespin onto the z -axis is clearly zero for integration timesconsiderably smaller than the spin lifetime. For a finiteelectron spin lifetime, T S , however, the dynamics is givenby S z ( t ) = S exp( − t/T S ) cos( ωt ).The Hanle effect [49] may be used in a quantitativemanner to determine the spin lifetime. The PL polariza-tion is monitored as a Voigt geometry magnetic field isincreased. Monitoring S z via the PL polarization whilstsweeping the transverse field yields a Lorentzian curve, -2 -1 0 1 2-22-24-26-28-30 C i r c u l a r po l a r i z a t i on c ( % ) Magnetic field B x (mT) B = 0.22 mT FIG. 5: Dependence of the PL circular polarization degree ona transverse magnetic field B x under fixed circularly polarizedexcitation (full circles: measured data, solid line: Lorentz fit).When the external field is strong enough to overcome ˜ B L thenuclear spins precess about the external field leading to areorientation of the Overhauser field along the external field.The electron spins become depolarized by precession aboutthe nuclear field. In practice, however, for B x > ˜ B L nuclearpolarization is no longer generated and the electron spin isleft with the fluctuation field (see text). the width of which is inversely proportional to the T S time of the electron: T S = ~ | g e,x | µ B / , (13)with the electron g factor g e,x along x , and the measuredhalf width at half maximum B / of the Lorentz curveobtained.Figure 5 shows a graph obtained from a Hanle mea-surement under unmodulated excitation, with pulses of75.6 MHz repetition rate on our QD sample. A transversemagnetic field B x was swept and the PL polarization foreach field value was recorded. The PL polarization dropssharply from its maximum value at B x = 0 the half-widthof the peak being ∼ T S ∼
57 ns assuming g e,x = 0 . σ + and σ − with period T m thatwe are able to vary over a range of milliseconds. Fig. 6shows Hanle curves under excitation modulated in suchway (note that because the signal could only be mea-sured during the σ + pulse, the measurement is inherentlymore noisy). The narrow peak appearing with unmod-ulated excitation at B x = 0 gradually disappears whenthe modulation frequency is increased. It almost com-pletely vanishes for a modulation period of T m = 1 ms.This value is in complete disagreement with the valueobtained from Eq. (13): if the electron spin relaxationtime is of the order of T ⋆ ∼
57 ns, the slower 1 ms mod- -28-26-24-22-20 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 no modulation T m = 50 ms8.0 ms1.0 ms Magnetic field B x (mT) C i r c u l a r po l a r i z a t i on ρ c ( % ) FIG. 6: (color online) (a) Hanle curves for excitation modu-lated between σ + and σ − with period T m (smoothed). Thenarrow peak gradually vanishes for faster modulation and isalmost absent for T m = 1 ms. ulation should not inhibit it. In general, as the dynamicsof the electron spin takes place on a nanosecond (preces-sion) to microsecond (coherence) timescale (see Table I),this millisecond effect is far too long to be of electronicorigin. Although transverse fields also lead to the depo-larization of polarized electron spins in the QDs understudy, the situation obviously fundamentally differs fromthe one underlying the original Hanle effect.When considering the discussion in Section IV.B, itbecomes immediately clear that the Hanle effect as dis-cussed above should never be observed in this low fieldregime. In fact, it appears at first surprising that a dra-matic change in the PL polarization is observed at all.As discussed in Section IV.B, the fluctuation field fromthe nuclei, B f ∼ | B f + B N | , which willalways screen any mT applied field. Thus, care has to betaken when the width of the Hanle peak is used to deter-mine the spin lifetime of the electron in QDs as has beensuggested in the literature [24, 25]. This works only fornuclear fluctuation fields B f ≪
10 mT. At fields lowerthan several tens of milli-Tesla the Overhauser field orthe frozen fluctuation field will always dominate the spindynamics in QDs.The fact that an effect is only observed when tensof milliseconds excitation is used implies that the effectarises from dynamic nuclear polarization which is knownto occur on these timescales [14, 21]. In the following wewill explain the mechanisms leading to the specific shapeof the curve in Fig. 6.For modulated excitation with T m < ρ c ∼ −
21% irrespective of the appliedfield. For unmodulated excitation on the other hand, anuclear magnetic field builds up along the z axis when B x = 0. The nuclear field adds to the frozen fluctuationfield B f,z and reduces the angle θ between the z axis andthe total nuclear magnetic field B N,tot = B N + B f asdiscussed in Sec. IV. The PL polarization, ρ c ∝ S z in-creases from ∼ −
21 % to ∼ −
26 % [53], similar to thebehavior shown in Fig. 3.Now let us consider what happens when a transversefield B x of a few mT is switched on. The electron isnot sensitive to this field as it is screened by the muchstronger nuclear fields (the fluctuation field or the Over-hauser field), and so at first, it will continue to polarizethe nuclear spins in the z -direction. The nuclear spins,however, are sensitive to this transverse field. As dis-cussed in Section IV.D, the nuclei feel three fields: theexternal field B x , the Knight field B e and the dipole-dipole field ˜ B L , all of which are of the same order ofmagnitude. If the external field dominates over the othertwo, then over the timescale of µ s, the nuclear spins beginto precess about this field.This situation has been investigated in detail inRef. [10] for donors in bulk GaAs. In this work, it was dis-cussed that application of a magnetic field in an obliquedirection results in the optically oriented nuclear spinsprecessing about this external field effectively aligningthe Overhauser field along it. The electron still experi-ences this Overhauser field which is, however, now ori-ented along the external field: the Overhauser field ef-fectively magnifies the external field by several orders ofmagnitude. This was described in reference [8, 10] as: B N = κ ( S , B ext ) B ext , (14)where κ is known as the multiplication factor. Note thatthe polarization of the nuclear spins only occurs due tooptical orientation via the spin polarized resident elec-tron, whereas the external field solely directs the opti-cally generated Overhauser field. If an Overhauser fieldof several Tesla is generated and realigned to the exter-nal milli-Tesla magnetic field, κ can reach values of 10 or more. This extraordinary effect is the reason why suchsmall external fields can have such a dramatic effect onthe electron spin system.For a purely transverse field, the polarized nuclearspins will precess about the x -direction in a few µ s. Thishas the subsequent effect of destroying the electron spinpolarization: it will precess about the x -axis and all pro-jection onto the z -axis is lost. At this point, no furthernuclear polarization via the electron spin can occur. Thenuclear polarization already present will diffuse. Thuswe see that dynamic nuclear polarization cannot occurin the steady state when a B x field is applied.We observe though that a finite applied field is requiredto reduce the ρ c . In Fig. 5 the PL value drops from ∼ -29% to ∼ -24% steadily over the range of ∼ B L .0For external magnetic fields B x < ˜ B L the external fieldis screened by ˜ B L . In order to realign the nuclear spins, B x has to dominate over the dipole-dipole field.For measurements in a purely transverse magnetic fieldthe x component of the Knight field is zero, thus theonly transverse field experienced by the nuclear spins isthe external field. The width of the Hanle peak is hencesolely determined by the competition between ˜ B L and B ext . This is discussed further in [8, 10]. In this regimethe width of the depolarization peak is a measure for thedipole-dipole field ˜ B L as discussed above. The averagepeak half width from several measurements correspondsto a dipole-dipole field of˜ B L = 0 . ± .
02 mT . (15)We shall now examine what happens when B x is in-creased above the magnitude of ˜ B L and why the polar-ization remains at a constant level for B x & B N relative to the fluctua-tion field B f . The field B N will decrease as B x increases:this is due to the fact that B N is dependent on S z anddecreases with decreasing electron spin because then theability of the electron spin to polarize the nuclear spins isreduced. Thus, at a sufficiently large B x , the Overhauserfield is close to zero, and only the fluctuation field B f re-mains. The fluctuation field then dominates the electronspin dynamics, and in Fig. 5 the ρ c value reaches ≈ -23%, the value found at 0 T in Fig. 3 (a) for modulatedexcitation (i.e. with no nuclear polarization). B. The Knight field
We have seen from Sec. V.A that the nuclei are sen-sitive to extremely small fields. In Section IV we alsodiscussed the importance of the Knight field in allowingnuclear polarization. The Knight field ˜ B e is the effec-tive magnetic field felt by each nucleus from the residentelectron. ˜ B e is antiparallel to the electron spin, and inour scheme, is thus parallel to the z -axis. At B ext = 0the Knight field screens the effect of the dipole-dipolefield and allows dynamic nuclear polarization to occur;however, ˜ B e must be stronger than ˜ B L [10, 26].The magnitude of the Knight field is a quantity whichvaries not only between different QDs, but also betweenindividual nuclei in a single QD, as its magnitude is pro-portional to the density of the electron wavefunction at aparticular nuclear site. For QDs, the Knight field may bean order of magnitude stronger than in bulk, due to theincreased electron density over fewer nuclei in the QD.This is why dynamic polarization in QDs at B ext = 0can be much stronger than in bulk material. It is there-fore of great interest to gain a measure of the strength ofthis effective field. -2 -1 0 1 2-27-28-29-30-29-28-27 Magnetic field B z (mT) C i r c u l a r po l a r i z a t i on c ( % ) + excitation - B e FIG. 7: (color online) B z dependence of the PL polarization( ρ c ) for B x = 0 and fixed excitation polarization (full circles:measured data, solid line: smoothed). The dip at B eff = 0 isshifted from B z = 0 by the value of the Knight field ˜ B e . Notethat the sign of ρ c is opposite to the circular polarization ofthe excitation, see Eq. 1. An approximate measure of the Knight field in QDswas determined for the first time by Lai et al. [26]. In thismeasurement, the PL polarization of a single QD excitonstate was measured as a milli-Tesla field was swept alongthe z direction. It was found that a dip in the polarizationwas visible at ∼ . B z = − ˜ B e .A nucleus in the QD then experiences an approximatecancelation of the Knight field with the external field.Without the Knight field, dipole-dipole depolarization ofthe nuclear spins occurs quickly, and dynamic nuclearpolarization does not build up.An identical measurement is performed in our systemin the strong pumping regime, with the results shown inFig. 7. Here, the B z dependence of the polarization isdepicted for both excitation helicities. The PL polariza-tion exhibits a barely discernible dip which is offset from B z = 0 and whose position is reversible with helicity. Asin Ref. [26], the shift corresponds to the external fieldwhich is needed to compensate the Knight field. How-ever, the effect is very small. This is due to the fact thatin an ensemble the Knight field is fairly inhomogeneous,1 -2 -1 0 1 2-20-22-24-26-28 C i r c u l a r po l a r i z a t i on c ( % ) Magnetic field B z (mT) -28-26-24-22-20 exc + B x = 2.27 mT (b) B x = 1.70 mTB x = 1.13 mTB x = 2.27 mTB x = 1.70 mT B x = 1.13 mT (a) exc - FIG. 8: (color online) B z dependence of the PL polarizationfor different values of B x : 1.13 mT, 1.7 mT and 2.27 mT.Excitation σ − (a) and σ + (b). The constant shift of the peakin the middle of the W-shaped structure (indicated by arrows)corresponds to the magnitude of the Knight field. Solid linein (b): Polarization behavior expected when the Knight fieldand the fluctuation field are neglected. and it is very difficult to depolarize all of the nuclearspins at the same time.In our previous work [22] we presented evidence thatwe achieve extremely large Overhauser fields ( > z axis, the polarization will be independent ofthe magnitude of B N : as discussed in Section IV.C. Aslong as B N ≫ B f and aligned along the z -axis, the elec-trons do not depolarize. In order to see a visible effectwhen applying B z = − ˜ B e , one must reduce B N to beof the same order of magnitude as B f . For an Over-hauser field of a few Tesla, this means that ∼
99% of thenuclei contributing to this field must be depolarized si-multaneously. We therefore use a method to measure theKnight field that was first reported in 1977 for electronson donors [8, 10], and present this method as ideal onefor investigating the Knight field in QDs.In order to make the Knight field more visible, we ad-ditionally apply a constant transverse magnetic field B x , and again step the B z field. This technique, used exten-sively in [10], allows one to access nuclear polarizationregimes where B N is smaller, without significantly chang-ing the electron spin polarization generated (as wouldbe the case, for example, when decreasing the excitationpower). In this technique, a transverse field is appliedof magnitude 1.13 to 2.27 mT. The z -field is then swept,and the polarization is measured. By keeping the B x field constant and sweeping B z , one effectively sweepsthe angle, θ ext that the total field B tot = B x + B z makeswith the z -axis, given by θ ext = arctan( B x /B z ). In fact,we will demonstrate that by using this technique, theangle that the B N field makes with the z -axis may befinely tuned, and even become closer to the x -axis than θ = 54 ° , the angle at which the ensemble averaged frozenfluctuation field is generally tilted from the z axis.Let us consider what should be expected as the B z fieldis swept in the presence of a B x field. By sweeping the B z field, the angle of the total applied field is swept from90 ° for B z = 0 to close to 0 ° when B z ≫ B x . As in therest of the work presented here, the applied field has nodirect effect on the electron, but each of the nuclei willrespond to this field, and in the absence of other effects,precess about the axis at an angle θ .Now let us assume that strong dynamic nuclear polar-ization is generated, where B N ≫ B f . As we have dis-cussed previously, the electron spin polarization is gov-erned by this field: S z = S cos θ ( B N ). Thus sweepingthe field should reveal a change in circular polarizationfrom 0 to − B N field is created. In Fig. 8, B z dependencies for various applied transverse fields areshown for both σ + and σ − excitation. We do not observethe change from 0 to -100 %, despite the fact that clearchanges in PL polarization occur on sweeping the field.A clear asymmetry is present, however, that is reversedupon reversal of the excitation polarization.It is not surprising that the PL polarization does notdrop to zero for low B z values. As in Fig. 5, for S ⊥ B ,no nuclear polarization can occur. The electron spin,however, is still exposed to the nuclear fluctuation fieldwhich does not fully depolarize the electron spin. Thevalue ρ c ∼ -23 %, is the same value as for B x ≫ .
22 mTin Fig. 5, as we expect.The solid line in Fig. 8 (b) shows the polarization be-havior expected when the Knight field and the fluctuationfield are neglected, and if we were to assume that B N isparallel to B ext . The B N field direction would vary from θ = 90 ° at B z = 0 T to θ = 0 ° at B z ≫ B x , and the timeaveraged electron spin h S i would follow it also. In fact,as the B N field angle moves away from the z axis, itsmagnitude decreases and the fluctuation field B f beginsto dominate. For this reason the data do not follow thesolid line.The PL polarization exhibits a pronounced asymmet-ric W-shaped behavior on sweeping the longitudinal field,that is inverted on changing the excitation helicity. Letus consider first the points indicated by arrows in Fig. 8,corresponding to local turning points in the curve. These2occur at ∼ ± . ρ c reaches a value of ≈ −
23 % for all the curves, andmoreover, that these points, approximately correspond inmagnitude and sign to the Knight field values observedin Fig. 7. At this point, the B z field approximately com-pensates the Knight field for many of the QD nuclei. Dueto the cancelation of the Knight field the nuclear spinsexperience a purely transverse magnetic field. In this ge-ometry nuclear polarization is not allowed to built up as B ext ⊥ S . Thus the ρ c value measured corresponds tothe fluctuation field value of ≈ −
23 %.The marked asymmetry in the curves that are invertedwhen changing helicity allow easy determination of thecompensation point, and this is indicated by the arrowsin each figure. As soon as the B z field is swept away fromthe compensation point, nuclear polarization may beginto occur. Moving away from the compensation point,the magnitude of generated nuclear field B N increases.However, the magnitude of the B N field with respect tothe fluctuation field B f and the orientation of B N has acomplex effect on the resultant electron spin orientation,which we attempt to clarify in the next section. C. Tuning the angle of the Overhauser field withmilli-Tesla external fields
The experiment shown in Fig. 8 involves choosing anexternal B x field, which is kept constant, and sweep-ing another B z field, from negative to positive values,through zero. We therefore effectively sweep the angleof the external field from along the x -axis to along the z -axis, as explained previously. To further elucidate ourdata we have taken the same data shown in Fig. 8 (b)and replotted it, not as a function of B z , but as a functionof the angle of the total external field, as shown in Fig. 9.This angle is given by θ ext = arctan( B x /B z ). Note that θ ext = 90 ° and 0 ° corresponds to the external field alongthe x and z axes, respectively. Upon replotting the data,it is evident that ρ c is dependent on θ ext and not on theabsolute magnitudes of B x or B z . The curves for B x =1.13, 1.70 and 2.27 mT obviously coincide, and show thesame asymmetry as well as the W-shaped feature. Thevalue of ρ c at the Knight field compensation point is ∼ -23 % (indicated by the arrows). Moving away from thispoint a reduction in ρ c is observed, until a turning pointis reached, and ρ c then increases sharply. We now explainthis behavior qualitatively.In this experiment, the magnitude of the Overhauserfield B N generated, is small, unless the applied B z field isvery large. Therefore we are always in the regime wherethe Overhauser field generated is of the same order ofmagnitude as the fluctuation field ( B N ∼ B f ), and thetwo are in direct competition. At the Knight field com-pensation point, the magnitude of B N is at its lowest,and the electron sees a pure B f field. The electron pre-cesses around this fluctuation field, at θ f = 54 . ° (firstpanel from bottom in Fig 9(b)). -28-26-24-22-20 30 60 90 120 150 B f B f B f B f (b) θ ~ 0 o θ < 54.7 o θ = 54.7 o o < θ < 90 o B TOT = B f B N B x = 1.13 mT B x = 1.70 mT B x = 2.27 mT θ ext = 125.3° Angle θ ext (degree) C i r c u l a r po l a r i z a t i on ρ c ( % ) θ ext = 54.7° B f θ f B TOT B N B TOT B N B TOT B N B TOT θ = 90 o (a) FIG. 9: (a) Data from Fig.8 (b) replotted as a function ofangle of applied field, θ ext = arctan( B x /B z ). Arrow showsapproximate position of the Knight field compensation point.(b) series of schematics showing the magnitude and direc-tion of the relevant nuclear fields for important regions of theapplied field angle. Blue arrow: fluctuation field, B f ; Red ar-row: optically generated Overhauser field B N ; Black arrow:resultant total nuclear field B tot = B f + B N Now let us consider what happens as we move awayfrom θ ext = 90 ° towards the value 54.7 ° . We see that thepolarization decreases, indicating that electron spin pro-jection onto z decreases. This appears counterintuitive.If the Overhauser field angle θ ( B N ) of the polarized nu-clear field is becoming closer to the z -axis, the electronpolarization should increase. The second panel revealswhy S z decreases in this region. The nuclear field gener-ated for θ ext > . ° is much smaller in magnitude than B f , but as θ ext is decreased, the magnitude of B N in-creases (due to the fact that S z is larger) and begins tocompete more strongly with B f . It is clear from the sec-ond panel in Fig. 9(b) that for θ ext > . ° , the total field B tot = B f + B N is at a larger angle than B f . This meansthat the stronger the Overhauser field B N generated, themore the electron depolarizes.At θ = 54 . ° a turning point is reached. At this point, B f and B N are at the same angle (see the third panel ofFig. 9(b)), and increasing B N is no longer detrimental.Upon increasing θ ext further, any increase in B N leads toa total field B tot which is always at an angle smaller than54 . ° . This has a positive feedback effect: the electronspin is preserved, and therefore may polarize more nuclei.As θ ext is decreased further, B N , and hence S z increasequickly, as depicted in the final panel at the top.We have described the behavior shown in Fig. 8 in aquantitative way only. A qualitative description wouldrequire detailed knowledge of B N as a function of angle,a value which is likely to be non-linear, and is beyond thescope of this paper. However, it shows clear evidence that3these small external fields may be used to accurately fine-tune the angle of the Overhauser field generated. Withimprovements in nuclear pumping rate one may be ableto control this angle over even wider ranges. The Over-hauser field may therefore replace a strong external fieldused to manipulate electron spins, ranging from the Voigtto the Faraday geometry. D. Evaluation of Knight field and nuclear field
Finally, in order to evaluate the Knight field we de-termine in Fig. 8 the magnetic field at which the Knightfield was compensated. This compensation point has aposition of | ˜ B e | = 0 . ± . . (16)This agrees well with the value of 0.6 mT which wasmeasured in single QD experiments [26].From Eq. (12) in Section IV.D, it was discussed thateach i th nucleus in the QD has an unique Knight field, b ie . In this experiment, a weighted average Knight fieldis measured: as discussed in Section IV.D, one may ap-proximate this weighted average to be ˜ B e ∼ B e,max / S z is reduced.Similarly to Eq. (12) ˜B e = − ˜ b e h S i S . (17)Let us now consider the compensation points ˜ B e = − B z carefully again (arrows in Fig 8). At this point, theexternal field and internal Knight fields cancel, and gen-eration of nuclear polarization is suppressed. The onlyfield from the nuclei is now the fluctuation field B f . Asdiscussed before, the electron precesses about this fieldat θ = 54 . ° , and thus, from Eq. (6), S z = S /
3. FromEq. (17), ˜ B e = ˜ b e S z /S , it follows that:˜ b e ∼ . ± . . (18)The value ˜ b e gives the maximum Knight field onto thesystem if no depolarization of the electron spin occurs.The information we have gained about the magnitudeof the Knight field and the dipole-dipole field now enableus to estimate the magnitude of the maximum achievablenuclear magnetic field using Eq. (11). In the calculationsa leakage factor f accounting for phenomenological lossesof nuclear spin polarization not explicitly discussed here,was set to one so that the results should be understoodas the maximum polarization which may be generatedwith the measured values ˜ B e = 0 . B L = 0 . -3 -2 -1 0 1 2 30.00.51.0 ~ ~~ B N / b N Magnetic field B z (mT) (B z + B e ) (B z + B e ) + B L2 ~ B e = 1.5 mT FIG. 10: ( B z + ˜ B e ) / (( B z + ˜ B e ) + ˜ B L ) which is a measure forthe competition between ˜ B L and B e concerning the ability ofan Overhauser field to be generated. Values ˜ B e = 0, 0.5 and1.5 mT are shown. The most important point to consider isthe value at exactly B z = 0 (vertical dashed line), where noexternal field supports nuclear polarization. mT. Simply taking these values, one obtains a value forthe estimated maximum field of: B N b N ≈ ˜ B e ˜ B e + ˜ B L = 0 . . (19)Thus we observe that even a moderate Knight field effec-tively dominates over the dipole-dipole field, and up to84% nuclear polarization may be obtained in the absenceof any other leakage.One may calculate b N in the case that 100% of thenuclear spins are polarized. For In . Ga . As QDs withelectron g-factor g e = 0 . b N,In . ≈ − . T,b
N,Ga . ≈ − . T,b
N,As ≈ − . T. (20)With these values we obtain a maximum nuclear field of b N,max = P i b N,i = − . B N = 6 . B f field and, as we have shown, the residualelectron spin polarization along z gives rise to a Knightfield of ∼ . z -direction thatdominates over the fluctuating field, the electron doesnot precess about an oblique field, and both the electron4spin projection and the Knight field reach their maximumvalues. We have already shown this maximum value tobe ˜ b e ∼ . B ext = 0 at least a moderate Overhauserfield builds up in many of our QDs, we may use the max-imum value of the Knight field ( B e = b e )calculated fromEq. (18) to be B N b N ≈ ˜ b e ˜ b e + ˜ B L = 0 . . (21)Thus, we observe that as soon as a significant Overhauserfield builds up in the QDs, the Knight field magnitudeis at a maximum, and one should theoretically be ableto obtain almost 100 % nuclear spin alignment (and anOverhauser field of B N = 8 . ρ c value. The ρ c value is also governed by theprobability of electron spin-flip from the wetting layer tothe ground state during relaxation. Thus, the nuclearspins in a particular QD may be prepared with a strongalignment in the z direction. An electron in the groundstate therefore will not precess and lose its spin projec-tion onto the z -axis. This is regardless of whether it hasspin up or down. Thus while the sign of the Knight fieldwill change if the electron is spin-flipped, the magnitudestays the same.By way of illustration, Fig. 10. demonstrates the ef-fect that different values of the Knight field have on themaximum obtainable nuclear polarization. In this figure,the function B N b N ≈ ( B z + ˜B e ) ( B z + ˜B e ) + ˜ B L . (22)is plotted as a function of B z for Knight field values of˜ B e = 0 , . , . B L is taken to be 0 .
22 mTfrom Fig. 5. The vertical dashed line indicates the value B z = 0. It is clear here that for no Knight field, nonuclear polarization will occur according to this simplemodel, but for values of ˜ B e determined in this work, thenuclear polarization should reach large values.In our previous work on the same sample [22] strongevidence was found for high Overhauser fields, that allowthe formation of a nuclear spin polaron state. The mea-surements here demonstrate that almost 100 % alignmentshould indeed be possible in these QDs. Note that wemake the assumption that without dipole-dipole depolar-ization, 100 % alignment would be achieved. Clearly, themaximum nuclear polarization achievable is dependenton several factors, of which the Knight field magnitudeis just one. VI. SUMMARY AND CONCLUSIONS
We have demonstrated that the effect of negative cir-cular polarization may be used both to polarize the spins of the resident electrons in n-doped QDs and to opticallyorient the nuclear spins in the QDs via spin transfer fromthe spin oriented electrons. Furthermore, the spin polar-ization of the resident electrons may be read out by mea-suring the circular polarization of the photoluminescenceupon circularly polarized non-resonant excitation of theQDs.The electron spin polarization at the same time servesas a sensitive detector for the state of the nuclear spinsystem. Milli-Tesla external magnetic fields may be usedto manipulate the nuclear spins which in turn amplifythe external field by orders of magnitude making it pos-sible to detect their action via the electron polarization.Exploiting this co-dependence of electron and nuclearspins, we studied Hanle curves for excitation modulatedbetween σ + and σ − helicity with different modulationperiods. We were able to show that it takes tens of mil-liseconds to maximally polarize the nuclear spin systemin the QDs using our polarization method. It became ob-vious that one has to examine thoroughly whether Hanlemeasurements in a specific case may be used to determinespin dephasing times of the electron. Even when nuclearpolarization is suppressed by modulated excitation, therandom frozen fluctuation nuclear field is still present,dominating the dynamics of the electron spins. Thus onemay conclude that determining the electron spin lifetimeusing the Hanle effect for magnetic fields less than a fewtens of milli-Tesla is incorrect due to screening eitherfrom the fluctuation field or the dynamic polarization:an alternative method must be used in QDs.The Hanle measurements, however, allowed us to de-termine the dipole-dipole field ˜ B L ≈ .
22 mT. Further-more, the magnetic field dependence of the PL polariza-tion in a combination of Faraday and Voigt geometriescould be used to obtain an accurate determination of themagnitude of the Knight field ˜ b e ≈ . B N tothe z -axis.After having determined the values of the dipole-dipoleand Knight fields for this one system, the maximum nu-clear field achievable may be calculated. It was foundthat, neglecting losses, the nuclear field at zero exter-nally applied field may be as high as ≈ . >
98 % for a fully polarized electron spin, asthe maximum Knight field value of ≈ T = 2 K, as theoretically predicted[51] and for which some experimental evidence alreadyhas been provided [22, 45].The fact that we use a QD ensemble for our studiesmay be considered a disadvantage because of the ensem-ble inhomogeneities. However, the variation of the pa-rameters of the electron-nuclear spin system we measureis not necessarily primarily due to the distribution in theensemble but vary for a single QD due to the inhomo-5geneity over the dot volume. For a single QD, the nu-clear configuration may be very different each time it isprobed, and because the nuclear spins may remain frozenfor microseconds to seconds, one must integrate over verylong times to ensure true averaging effects. The fact thatwe do see collective effects in our sample is a proof thatthe ensemble broadening is relatively small concerningthe parameters of the electron-nuclear spin system. Onthe contrary, it is a remarkable finding that the ensemblereacts collectively yielding the pronounced features wehave observed.To summarize, it is clear that the electron-nuclear sys-tem may be manipulated with just a few milli-Tesla, instark contrast to conventional semiconductor systems.The dynamics of this complex system is only beginningto be understood, but clearly holds the key to achieving long electron spin qubit coherence times for use in appli-cations such as quantum information processing, whilstthe Knight field plays a crucial role in novel schemes forthe use of QD nuclear spins as a quantum memory[54]. Acknowledgments
This work was supported by the Deutsche Forschungs-gemeinschaft (grant BA 1549/12-1) and the BMBFresearch program “Nanoquit”. S. Yu. Verbin andR. V. Cherbunin acknowledge support of the RussianMinistry of Science and Education (grant RNP.2.1.1.362)and the Russian Foundation for Basic Research, R. Oul-ton thanks the Alexander von Humboldt Foundation. [ ⋆ ] e-mail: [email protected] [ † ] Present address: Centre for Quantum Photonics, Univer-sity of Bristol, Bristol BS8 1UB, United Kingdom.e-mail: [email protected] [ ‡ ] Also in: A. F. Ioffe Physico-Technical Institute, RussianAcademy of Sciences, 194021 St. Petersburg, Russia[1] G. Lampel, Phys. Rev. Lett. , 491 (1968).[2] A. W. Overhauser, Phys. Rev. , 411 (1953).[3] A. I. Ekimov and V. I. Safarov, ZhETF Pis. Red. , 453(1972) [JETP Lett. , 319 (1972)].[4] V. L. Berkovits, A. I. Ekimov, and V. I. Safarov,Zh. Eksp. Teor. Fiz. , 346 (1973) [Sov. Phys. JETP , 169 (1974)].[5] M. I. D’yakonov, V. I. Perel’, V. L. Berkovits, andV. I. Safarov, Zh. Eksp. Teor. Fiz. , 1912 (1974)[Sov. Phys. JETP , 950 (1975)].[6] M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz. ,362 (1973) [Sov. Phys. JETP , 177 (1974)].[7] V. A. Novikov and V. G. Fleisher, Pis’maZh. Tekh. Fiz. , 935 (1975).[8] F. Meier and B. P. Zakharchenya (editors), Optical Ori-entation , Modern Problems in Condensed Matter Sci-ences, Vol. 8 (North-Holland, Amsterdam, 1984).[9] I. A. Merkulov, G. Alvarez, D. R. Yakovlev andT. C. Schulthess, arXiv:0907.2661 (2009).[10] D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov,Phys. Rev. B , 5780 (1977).[11] A. I. Ekimov and V. I. Safarov, ZhETF Pis. Red. , 257(1972) [JETP Lett. , 179 (1972)].[12] V. L. Berkovits, C. Hermann, G. Lampel, A. Nakamura,and V. I. Safarov, Phys. Rev. B , 1767 (1978).[13] S. W. Brown, T. A. Kennedy, D. Gammon, andE. S. Snow, Phys. Rev. B , R17339 (1996).[14] D. Gammon, Al. L. Efros, T.A. Kennedy, M. Rosen,D.S. Katzer, and D. Park, Phys. Rev. Lett. , 5176(2001).[15] F.H.L. Koppens, J.A. Folk, J.M. Elzerman, R. Han-son, L.H. Willems van Beveren, I.T. Vink, H.P. Tranitz,W. Wegscheider, L.P. Kouwenhoven, and L.M.K. Van-dersypen, Science , 1346 (2005)[16] I. A. Akimov, D. H. Feng, and F. Henneberger,Phys. Rev. Lett. , 056602 (2006). [17] B. Eble, O. Krebs, A. Lematre, K. Kowalik, A. Kudel-ski, P. Voisin, B. Urbaszek, X. Marie, and T. Amand,Phys. Rev. B , 081306 (2006).[18] P.-F. Braun, B. Urbaszek, T. Amand, X. Marie,O. Krebs, B. Eble, A. Lemaitre, and P. Voisin,Phys. Rev. B , 245306 (2006).[19] A. I. Tartakovskii, T. Wright, A. Russell, V. I. Fal’ko,A. B. Van’kov, J. Skiba-Szymanska, I. Drouzas,1R. S. Kolodka, M. S. Skolnick, P. W. Fry, A. Tahraoui,H.-Y. Liu, and M. Hopkinson, Phys. Rev. Lett. ,026806 (2007).[20] D. H. Feng, I. A. Akimov, and F. Henneberger,Phys. Rev. Lett. , 036604 (2007).[21] P. Maletinsky, A. Badolato, and A. Imamoglu,Phys. Rev. Lett. , 056804 (2007).[22] R. Oulton, A. Greilich, S. Yu. Verbin, R. V. Cherbunin,T. Auer, D. R. Yakovlev, M. Bayer, I. A. Merkulov,V. Stavarache, D. Reuter, and A. D. Wieck ,Phys. Rev. Lett. , 107401 (2007).[23] A. Greilich, D. R. Yakovlev, A. Shabaev, Al. L. Efros,I.A. Yugova, R. Oulton, V. Stavarache, D. Reuter,A. Wieck, and M. Bayer, Science , 341 (2006).[24] R. J. Epstein, D. T. Fuchs, W. V. Schoen-feld, P. M. Petroff, and D. D. Awschalom,Appl. Phys. Lett. , 733 (2001).[25] A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware,J. G. Tischler, A. Shabaev, Al. L. Efros, D. Park,D. Gershoni, V. L. Korenev, and I. A. Merkulov,Phys. Rev. Lett. , 047402 (2005).[26] C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu,Phys. Rev. Lett. , 167403 (2006).[27] S. Fafard and C. Allen, Appl. Phys. Lett. , 2374 (1999).[28] S. Cortez, O. Krebs, S. Laurent, M. Senes, X. Marie,P. Voisin, R. Ferreira, G. Bastard, J-M. Grard, andT. Amand, Phys. Rev. Lett. , 207401 (2002).[29] S. Laurent, M. Senes, O. Krebs, V. K. Kalevich, B. Ur-baszek, X. Marie, T. Amand, and P. Voisin, Phys. Rev. B , 235302 (2006).[30] M. Ikezawa, B. Pal, Y. Masumoto, I. V. Ignatiev,S. Yu. Verbin, and I. Ya. Gerlovin, Phys. Rev. B ,153302 (2005).[31] M. E. Ware, E. A. Stinaff, D. Gammon, M. F. Doty, A. S. Bracker, D. Gershoni, V. L. Korenev,S. C. Badescu, Y. Lyanda-Geller, and T. L. Reinecke,Phys. Rev. Lett. , 177403 (2005).[32] Y. Masumoto, S. Oguchi, B. Pal, and M. Ikezawa,Phys. Rev. B , 205332 (2006).[33] V. K. Kalevich, I. A. Merkulov, A. Yu. Shiryaev,K. V. Kavokin, M. Ikezawa, T. Okuno, P. N. Brunkov,A. E. Zhukov, V. M. Ustinov, and Y. Masumoto,Phys. Rev. B , 045325 (2005).[34] I. A. Akimov, K. V. Kavokin, A. Hundt, and F. Hen-neberger, Phys. Rev. B , 075326 (2005).[35] A. V. Khaetskii and Y. V. Nazarov Phys. Rev. B ,12639 (2000).[36] I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B , 205309 (2002).[37] A. Imamoglu, E. Knill, L. Tian, and P. Zoller,Phys. Rev. Lett. , 017402 (2003).[38] D. Stepanenko, G. Burkard, G. Giedke, andA. Imamoglu, Phys. Rev. Lett. , 136401 (2006).[39] C. Deng, and X. Hu, Phys. Rev. B , 241303(R) (2006).[40] R. I. Dzhioev, B. P. Zakharchenya, V. L. Korenev, andM. V. Lazarev, Phys. Solid State , 2014 (1999).[41] P.-F. Braun, X. Marie, L. Lombez, B. Urbaszek,T. Amand, P. Renucci, V. K. Kalevich, K. V. Ka-vokin, O. Krebs, P. Voisin, and Y. Masumoto,Phys. Rev. Lett. , 116601 (2005).[42] A. Greilich, A. Shabaev, D.R. Yakovlev, Al. L. Efros,I.A. Yugova, D. Reuter, A.D. Wieck, and M. Bayer, Sci-ence , 1896 (2007).[43] A. Abragam, The Principles of Nuclear Magnetism ,(Clarendon Press, Oxford, 1983)[44] R. V. Cherbunin, S. Yu. Verbin, T. Auer, D. R. Yakovlev,D. Reuter, A. D. Wieck, I. Ya. Gerlovin, I. V. Ignatiev,D. V. Vishnevsky and M. Bayer, Phys. Rev. , 035326(2009).[45] R. Oulton, S. Yu. Verbin, T. Auer, R. V. Cherbunin,A. Greilich, D. R. Yakovlev, M. Bayer, D. Reuter, andA. Wieck, Phys. Stat. Sol. (b) , 3922 (2006).[46] W. D. Knight, Phys. Rev. , 1259 (1949). [47] T. Auer, PhD. Thesis: “The electron-nuclear spin systemin (In,Ga)As quantum dots” (Sierke Verlag, Goettingen,2008).[48] This weighted average reflects the fact that, while for anucleus at a distance r = a / e − , the contribution this nucleus makes to the Over-hauser field is e − of that of a nucleus at the centre. Thusthe average Knight field value measured is fairly narrowand weighted heavily towards the nuclei in the QD cen-ter.[49] W. Hanle, Z. Phys. , 93 (1924).[50] M. Gueron, Phys. Rev. , A200 (1964).[51] I. A. Merkulov, Phys. Solid State , 930 (1998).[52] The dominant mechanism for the evolution of the nu-clear spins over this timescale is still under debate: bothdirect precession of the nuclear spins in the electronicfield and electron mediated nuclear-nuclear spin-flip pro-cesses are thought to play a role. In any case, it is infact the electron itself that causes nuclear spin evolution[21]: without an electron present only the dipole-dipoleinteraction is important.[53] Note that the maximum and minimum values of ρ c varyslightly between Figs. 5 and 6, with the maximum vary-ing from − to −
30 %. This is because the measure-ments are taken on different measurement runs. As wehave demonstrated in reference [22], slight temperaturevariations in the bath cryostat give rise to different val-ues of nuclear polarization decay over very long times.The behavior of the changes in polarization in this paperwhen a magnetic field is swept are not strongly dependenton temperature, however. Additionally, slight changes inexcitation wavelength result in a change in electron spinmemory preserved during energy relaxation from the wet-ting layer. Again, this slight variation does not affect theelectron-nuclear spin dynamics here.[54] Z. Kurucz, M. W. Sorensen, J. M. Taylor, M. D. Lukin,M. Fleischhauer, Phys. Rev. Lett.103