Nuclear Spin Effects in Semiconductor Quantum Dots

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Optical and electrical manipulation of single electron and valence band hole spins in semiconductor quantum dots (QDs) has now become possible, owing to the progress in fabrication and new experimental techniques 1-8 , potentially enabling realization of spin qubits for quantum information processing (QIP) 9 .
The spin of the confined electron in a QD experiences the hyperfine interaction with 10 4 -10 6 nuclear spins [10][11][12][13][14] . This interaction is usually quantified using an effective Overhauser magnetic field, B nuc , reaching in some cases up to a few Tesla for a highly polarized nuclear spin system 12 , with a statistical fluctuation σ Bnuc of a few mT 13,14 . The presence and dynamic properties of the Overhauser field thus have a significant impact on the behavior of the electron spin, and accordingly have received close attention in the quest for realization of a QD spin qubit. Beyond the active research into a quiescent and controllable magnetic environment in solids, nuclear spins themselves have been suggested as a resource with extended coherence (potentially in the ms range) useful for QIP 15 . Very high nuclear polarization degrees now routinely achievable in QDs have also enabled nuclear magnetic resonance (NMR) in single QDs to be realized 12, [16][17][18] , which can be applied for non-invasive probing of chemical composition and strain in the volume occupied by the confined electron 19 .
Here we will review the most pronounced manifestations of nuclear magnetism with a focus on experiments in individual epitaxial and lithographic III-V semiconductor QDs. We will discuss ways to detect and manipulate nuclear spins both in optical and electrical measurements, with one of the important goals of reducing the randomness in the nuclear field. We also discuss how the effect of the hyperfine interaction differs for the case of valence band holes compared to electrons. Furthermore, we review NMR experiments in small ensembles of nuclear spins in single QDs. Finally, we will briefly outline imminent future directions in nuclear magnetism research in semiconductor nano-structures.

Hyperfine interaction and detection of nuclear spin polarization
We start from a brief introduction to electron-nuclear spin interaction, description of typical QD structures, and ways to detect nuclear spin polarization by optical means and electrical probing.
Hyperfine interaction. The dominant contribution to the electron-nuclear hyperfine interaction (HI) originates from the contact Fermi interaction 13,14 (the hole-nuclear spin interaction is dipole-dipole in nature as described below). The electron-nuclear HI results in a static effect contributing to the energies of the two spin systems, which is usually described in terms of effective magnetic fields: an Overhauser field, B nuc , acting on the electron, a result of interaction with a large number of nuclear spins 13,14,20 , and a Knight field experienced by individual nuclear spins as a result of interaction with the spin of the localized electron 13,14,16,21 . Importantly, HI also leads to a dynamical effect responsible for the transfer of spin between the two systems 13,14 .
The nuclear field B nuc fluctuates around its average as a result of the redistribution of nuclear spin polarization due to dipolar coupling or via virtual excitations of the electron spin. In the limit of large N , where N is the effective number of nuclei, this can be described by a Gaussian distribution 10,11 with the standard deviation σ Bnuc = B max nuc / √ N , where B max nuc is the maximum Overhauser field of the order of a few Tesla [22][23][24][25] . For an electron confined in a GaAs quantum dot and interacting with a typical number of 10 6 spin-3/2 nuclei this results in σ Bnuc ∼ 6 mT. σ Bnuc can exceed 20 mT in small self-assembled In(Ga)As dots with high concentration of spin-9/2 In. In an experiment with a large number of identical measurements, electron spins initialized in the same state will exhibit different dynamics as they will evolve in a slightly different effective magnetic field. When averaging over many measurements, this will effectively result in spin dephasing on the scale of a typical precession period of the electron in the field of the order of σ Bnuc : the dephasing time T * 2 is of the order of 15 ns for σ Bnuc ≈6 mT 1, 5,10,11,26,27 . Much of this dephasing due to the random nuclear field can be unwound using spin-echo techniques, since the nuclear field evolves slowly on the timescale of the electron spin dynamics. The remaining decay of the electron spin coherence, with characteristic timescale T 2 , gives information on the timescale of the nuclear field fluctuations 1,5,26,28-30 .
Detection of nuclear spin polarization in epitaxial quantum dots. We first discuss semiconductor QDs fabricated directly by crystal growth techniques using molecular beam epitaxy (MBE) and metal-organic vapor phase epitaxy (MOVPE) 31 . Structures with typical in-plane dimensions of 20 to 80 nm and heights of 2 to 10 nm are formed (Fig.1b), providing strong electron and hole confinement of tens of meV. Of particular interest for spin manipulation in optical experiments are neutral (uncharged) and singly-charged QDs, possible to obtain in charge-tunable devices 32

Detection of nuclear spin polarization in gate-defined dots.
A lithographic QD is formed in a two-dimensional electron gas (2DEG) hosted by a GaAs/AlGaAs heterostructure (Fig.2a) 20,42,43 .
Surface gates on top of the heterostructure are used to locally deplete the 2DEG, which makes it possible to control the electron number in the formed QDs, the tunnel coupling between neighboring QDs, and the tunnel coupling between the QDs and reservoirs. Typical dimensions of these dots are 40 nm in the plane and 10 nm in the growth direction. Gate-defined dots are probed electrically by either measuring electron transport through the QD (or through several dots in series), or by directly probing the charge state of the QD using a nearby charge detector 20,42 .
Analogously to the case of optical measurements, the nuclear polarization along the external magnetic field B ext can be probed by measuring the shift induced by B nuc in the total electron Zeeman splitting,

Dynamic nuclear polarization
The HI enables not only sensing of the nuclear magnetic field through measurement of the electron spins, but also manipulation of the nuclear spins via the electron spins: The transverse terms of the HI mediate electron-nuclear flip-flops 13,14 in which the electron changes its spin by ±1 with a simultaneous of ways to overcome the energy mismatch and to induce a preferential pumping direction, both by optical and electrical means. There have been many attempts to find an efficient way of achieving DNP [21][22][23][24][25]33,34,49,52,53 , with the main motivation to achieve a 100% polarization of nuclear spins, which would prevent nuclear-nuclear flip-flops, strongly suppressing the randomness in the nuclear field and concomitant electron spin decoherence 10 .

Dynamic nuclear polarization in optically pumped quantum dots. Most experiments on
optical pumping of nuclear spins in QDs are performed at temperatures below 50 K (normally below 10 K). In most cases DNP occurs following the spin transfer from optically pumped or resident electrons.
Electron Overhauser shifts in excess of 100 μeV can be obtained using optical pumping [22][23][24][25]33,34 , and B nuc up to 3 T have been reported [23][24][25] . Using rough estimates of the dot composition, degrees of nuclear polarization up to 60% are now routinely obtained.  Compared to these transport measurements, experiments with specially designed gate voltage pulse cycles offer a more controlled way to realize a DNP pump scheme. In Ref. 49 Fig. 2e. DNP pulses were applied between successive measurements. They increase or decrease ΔBnuc depending on whether the DNP cycle starts from an S (green) or T+ state (black). Data from Ref. 44 and 4c) transfers up to one unit of angular momentum into the nuclear spin bath. Finally, one electron is pushed out of the double dot and the next cycle begins. Spin transfer in the opposite direction and thus full bidirectional control (Fig. 4c) was demonstrated 53 by initialization of a T + (1,1) state followed by a similar slow passage through the S-T + degeneracy point. These pump cycles can be extended to reduce fluctuations of the hyperfine field 44,45 .
Rather than exploiting level degeneracies, the energy for electron-nuclear flip-flops can be provided by a resonant ac electric field (inset to Fig. 4b). The electric field modulates the hyperfine coupling constant of each nucleus to the electron and therefore the transverse term of the hyperfine coupling.
We will now discuss nuclear spin dynamics in gate-defined QDs, which can be measured accurately The diffusive long-time behavior has been probed by directly measuring the fluctuations of the Overhauser field 75,76 using methods discussed above (see Fig.5a). At low magnetic fields ( 20 mT), one finds an about tenfold speedup of spin diffusion 75 that likely reflects the activation of additional diffusion channels by the reduced Zeeman energy mismatch, such as electron mediated spin transfer between nuclei. The electron mediated diffusion also leads to a dependence of the decay rate of an induced polarization on the occupancy of the dot 76 . Note, that gate-defined QDs are usually made of unstrained GaAs, therefore quadrupole effects are weak and were neglected in the discussion above.  Fig. 5b,c). The monotonic decay of the Hahn-echo signal with characteristic evolution time, τ , at high fields is a result of the diffusive dynamics of B z nuc due to dipolar coupling. The resulting spectral diffusion is predicted to cause a exp(−(τ/T SD ) 4 ) decay of the echo 78,79 with a characteristic time constant T SD of a few tens of μs.
The oscillations found at lower fields, which eventually turn into full collapses and revivals, were first predicted based on a fully quantum mechanical treatment 73 , but can also be understood with a semiclassical model. It is based on the electronic Zeeman energy splitting being proportional to the (Fig. 5c top left). nuclear field, B ⊥ nuc , is a vector sum of contributions from the three nuclear species 69 Ga, 71 Ga and 75 As (Fig. 5c, top right). Due to the different precession rates of these species, B ⊥ nuc 2 thus oscillates at the three relative Larmor frequencies (Fig. 5c, bottom). The amplitude and phase of the oscillating nuclear fields fluctuate over the course of many repetitions, thus leading to randomization of the resulting phase and suppression of the echo signal. However, if the precession interval is approximately a multiple of all three Larmor periods, the oscillations imprint no net phase on the electron spin and the echo amplitude revives. A quantitative model treating the components of B ⊥ nuc as classical variables also explains the faster decay of the echo envelope at low fields (Fig. 5b) in terms of dephasing of the nuclear spins themselves. Unlike electrons having s-type atomic wavefunctions, the hole has a wavefunction constructed predominantly from p-orbitals with zero density at the nuclear site. This leads to a vanishing contact part of the HI, which combined with extended hole spin life-times in QDs 81 presents holes as a potentially viable alternative to electrons for implementation of spin qubits 81,82 . Recent theory predicts that the hole HI, dipole-dipole in nature, can be as large as 10% of that of the electron, and is strongly anisotropic [83][84][85][86] .

Interaction of valence band holes with nuclear spins
Furthermore, heavy-hole (hh) states with pure p-symmetry couple only to the nuclear field along z, i.e.
exhibit an Ising-type interaction with nuclear spins and slow decoherence 84,85 . On the other hand, it has been shown theoretically that the HI leads to efficient decoherence of the pure hh states having an admixture of d-orbitals in the wave-function, estimated to be considerable (e. g. ∼ 20% for Ga) from recent experiments 87 . Another decoherence mechanism arises from heavy-light hole mixing, as light-hole (lh) states couple to all nuclear spin components 84,86 . However, in the majority of studied QDs hh-lh mixing is very small 81,82 , so this decoherence mechanism should in principle manifest itself in rare cases 84 techniques revealed that the |C/A| ratio can reach as high as 0.15-0.2, and that C > 0 for anions (As, P ) and C < 0 for cations (In, Ga) 87 . The sign difference was explained by the contribution of atomic d orbitals to the cationic hole Bloch wavefunction, whereas for anions the wavefunctions is purely p-type.
It must be noted, that understanding of the hole spin decoherence and the role of the hole hyperfine interaction may still be incomplete. There is a rather large spread of measured T * 2 for the hole spin: >100 ns using coherent population trapping 3 and 2 to 20 ns in ultra-fast optical measurements of the hole spin Ramsey fringes [6][7][8] . It is also an emerging paradigm that electrical noise in the diodes comprising hole-charged QDs may be a factor strongly limiting the T * 2 values 6,7 . On the other hand, the more fundamental property, such as the admixture of d-orbitals in the hole Bloch function 87 important for hole spin decoherence via the HI, may also vary from dot-to-dot leading to variation in T * 2 values.

Narrowing of nuclear field distribution using 'closed-loop' spin pumping
The Xu et al. 27 observed enhancement of the electron T * 2 using coherent dark-state spectroscopy carried out on a single electron-charged dot. This effect was explained by suppression of nuclear spin fluctuations under the Overhauser field locking similar to the line-dragging in Ref. 38 . A marked enhancement of the electron T * 2 by a factor of several hundred, arising as a result of suppressed nuclear spin fluctuations, was observed.
Narrowing in gate defined dots. In gate-defined dots, several quite distinct approaches to suppress nuclear field fluctuations have been successfully used. The conceptually simplest possibility for suppressing the randomness of the Overhauser field (or its gradient) is to rapidly measure it and to use DNP to restore its desired value. For S-T 0 qubits, this approach 44 permitted a reduction of the rmsfluctuations of the hyperfine field gradient, σ ΔBnuc , by about a factor of 2. A more powerful approach relies on directly conditioning the spin transfer from the electrons to the nuclei on the current value of the hyperfine field, thus letting the electron spin itself act as a complete feedback loop not requiring external intervention. This approach was used to control both the hyperfine field in the individual halves of a double quantum dot using an ESR-based spin transfer technique 45 , and the field gradient ΔB nuc between the two dots of an S-T 0 qubit via exchange mediated spin transfer 44 .
Such feedback schemes can be understood based on the so called pumping curve, which provides the polarization rate as a function of the current value of the hyperfine field. Figs. 7(a),(c) show pumping curves for both feedback methods. A stable fixed point is obtained whenever the pumping curve crosses zero with a negative slope so that fluctuations away from the fixed point are corrected by an opposing pump effect.
The ESR pumping curve (Fig.7a) emerges from the resonance condition of the microwave excitation with the Zeeman field B ext +B nuc seen by the electron, with the overall negative background slope arising from relaxation of the nuclear spin polarization 45 . The narrowing effect was inferred from dragging and locking of the ESR resonance frequency in response to changes of the externally applied field (Fig.7b), which were found to be compensated by nuclear polarization such that the total field remained constant 45 .
The oscillatory behavior of the corresponding pumping curve for the S-T 0 qubit (Fig.7c)  Note that it was reported earlier that the same type of DNP without feedback could extend T * 2 to beyond 1 μs 90 . Although theoretical scenarios 91 have been proposed to explain such an effect 90 , it later turned out that another interpretation of the data is much more plausible 92 .

Nuclear magnetic resonance in single quantum dots
Direct manipulation of nuclear spins using nuclear magnetic resonance (NMR) is desirable for several reasons. As in the previous sections, this provides new insights in the spin properties of QD electrons and holes. NMR measurements provide information on the nuclear spin coherence, an important insight in the properties of the magnetic environment of the electron and hole spin-qubits. Pulsed NMR may also serve as a tool for fast redirection of the large Overhauser fields inside the dot, an additional tool for qubit control 17 . Finally, NMR can be used to reveal the structural properties of the dot to provide direct correlations with its electronic properties and feedback for QD fabrication.
The Hamiltonian for a nuclear spin I having a gyromagnetic ratio γ can be written as 13,66 : where ν L = γB z /(2π) is the nuclear Larmor frequency, I z the z-projection of the nuclear spin, and H Q describes the interaction of the nuclear quadrupole moment with the electric field gradient. H Q arises in quantum dots as a result of strain or alloy fluctuations, and is particularly pronounced in self-assembled QDs. In a magnetic resonance experiment transitions between spin states with ΔI z = ±1 are induced with a transverse magnetic field oscillating at a radio-frequency (rf) close to ν L . The corresponding changes in the nuclear spin state populations are detected using optical or electrical methods from changes in the electron Overhauser shift in QDs.
First NMR in QDs was carried out in optical measurements on single GaAs/AlGaAs interface dots 12 , where quadrupole effects were weak. The discrete exciton energy structure in QDs was successfully uti- lized: changes in the electron Overhauser shift induced by rf excitation could be measured with accuracy of a few μeV. NMR spectra were measured by simply stepping rf frequency through the resonance 12,16 .
Further advancement of the nuclear spin control was made possible by employing pulsed NMR measured optically 17 in a single GaAs/AlGaAs QD and electrically 18  Recently, high-resolution optically detected NMR has been carried out in single self-assembled InGaAs/GaAs and InP/GaInP QDs by employing novel spin population transfer techniques 19 . Instead of using a wide rf band in order to increase the number of affected nuclear spin transitions, an "inverted" rf spectrum was employed having two very broad bands (≈10 MHz) with a gap in between. This approach led to signal enhancement more than 100 for 9/2 spins compared to the standard saturation techniques, and allowed measurements with resolution down to ≈10 kHz 19 . These techniques reveal a wealth of structural information such as chemical composition and strain distribution in the volume of the wavefunction of the confined electron 19 , and present a powerful microscopy tool for non-invasive structural analysis of single QDs. In order to gain an additional enhancement in spatial resolution of NMR, the use of an effective magnetic (Knight) field of the photo-excited electron may be possible 16 .
The spatial distribution of the Knight field follows that of the strongly localized electron wavefunction: Knight field gradients of the order of 10 3 T/cm can be achieved, potentially enabling determination of the nuclear spins position with resolution of 1 nm inside a single QD 16 .

Future directions and other materials
The above sections present the state-of-the-art in nuclear magnetism in semiconductor QDs. Below we comment on future developments in this field. We also briefly outline other classes of materials where electron-nuclear spin interactions have been investigated.

Control of nuclear spins for realization of coherent spin qubits.
Efforts to achieve quiescent nuclear spins for improving coherence of the central spin (spin qubit) may continue in several directions.
From analysis of QD composition using NMR 19 , it is now clear that polarization degrees of 90% or above may be accessible in optically pumped dots 19,33 . The effect of this on the coherence of electron or hole spin qubit needs to be verified. Alternatively, approaches achieving stabilized nuclear spin distributions are very attractive as they do not require very high polarization degrees.
Another way to achieve suppression of the nuclear spin fluctuations is to realize an ordered nuclear spin state 96 . This in principle can be achieved by cooling nuclear spins to ultra-low sub-μK temperatures using adiabatic demagnetization (AD), although first attempts in self-assembled dots experienced difficulties owing to strong quadrupole effects 65 . In future, similar experiments could be attempted in unstrained GaAs dots.
Understanding of nuclear spin coherence in strained structures is another direction aimed at achieving quiescent magnetic environment. Recent initial studies showed more robust nuclear spin coherence in structures with strain 97 , also naturally present in self-assembled dots and some nanowires.
Holes remain rather attractive as a spin qubit due to the significantly weaker HI compared to that of electrons. Recent studies open the way for engineering of the hole-nuclear spin interaction by appropriate choice of QD composition 87 . In this way, improved hole spin qubit control may be obtained, a subject of further studies.
Nuclear spins: beyond the semi-classical approximation. In the coming years, we anticipate a new direction in research on quantum dots which takes nuclear spin control into the quantum regime.
This is the regime where the nuclear spin state can no longer be captured in terms of a classical nuclear field or probability distribution of nuclear fields, which have been used to describe current experiments.
Creating quantum states of the nuclear spin bath can be done using the coupling Hamiltonian between electron and nuclear spins, which implies that the electron will influence the dynamics of the nuclear bath via some quantum back action. It is thus very interesting to explore if there is an experimentally detectable deviation from classical models that can be unambiguously attributed to the back action effect.
As a first example that this may be possible, the creation of squeezed states of the nuclear spin bath in quantum dots was recently proposed, using microwave irradiation 98,99 . In spin squeezing, the uncertainty of one component of the (total) spin is reduced below the uncertainty limit at the expense of increased uncertainty in an orthogonal component 100 . Interestingly, it was shown theoretically that sufficiently strong spin squeezing implies entanglement in the spin bath 101 . As another example, although harder to achieve with current techniques, proposals exist for coherent exchange of a qubit state between the quantum dot electron spin and a collective degree of freedom of the nuclear spin bath 15 . If realized, this would mean that the nuclear spin system can be used as a long-lived quantum memory, since even simple Hahn echo decay times of nuclear spins in quantum dots are about 1 ms 17,18 . Such a coherent information transfer would require special "dark" nuclear states with a reduced transverse hyperfine field, which in principle can be created via fast DNP.
However, these states are highly sensitive to dephasing of the nuclear spins, and are subject to a fragile balance between hyperfine mediated spin transfer and dephasing due to the Knight shift 102 .
A first step would thus be to establish whether such states, which would manifest themselves in a saturation of the nuclear polarization rate, can indeed be created. Another example is the creation of superradiance effects giving strongly enhanced electron-nuclear flip-flop rates, which could be observed in transport measurements 103 as well as optical spectroscopy 104 . Common to all these examples is the collective effect of a large number of nuclear spins coherently interacting with a single (central) electron spin. Finally, it remains to be seen how much narrowing procedures can be improved, and whether they will eventually permit access to probe some form of intrinsic free induction decay that arises from the bath dynamics rather than ensemble averaging, as studied theoretically in Refs. 74,79 .
Other material systems. While this review focuses on III-V semiconducting quantum dots, there have been a handful of other material systems where the interplay between a central electron spin and the surrounding nuclear and even electron spin baths have been investigated. Prominent examples include carbon nanotubes 105 , both natural (with 1% of 13 C) and 100% 13 C , phosphorus spins in Si 106,107 and Si quantum dots 108 , and diamond NV centers 109 . While much of the physics discussed in this review is applicable to these systems, there are a few notable differences. For example the spin echo response of phosphorus spins in Si is due to the 29 Si host atoms and is theoretically and experimentally shown to have a time dependence given by exp[−(t/T 2 ) 2.3 ] (Refs. 78,106 ). The resulting exponent of 2.3 as opposed to 4 in GaAs 78,106 is a result of the detailed envelope wavefunctions associated with each system and is therefore not a universal exponent. A second example is recurrences in the electron spin echo signal as seen in Fig.5b. In GaAs such recurrences are a result of commensurate evolution of the nuclear spins of different species. A similar phenomenon is also seen in NV centers in diamond due to the 13 C nuclear spins. However, unlike the GaAs case where multiple species are required in order to see recurrences, in diamond, since the dominant interaction between the central spin and the nuclear spins is dipolar, a single nuclear spin species is sufficient.