Abstract
The physics of interacting nuclear spins arranged on a crystalline lattice is generally described using a thermodynamic framework1 and the concept of spin temperature. In the past, experimental studies in bulk solid-state systems have proven this concept to be not only correct2,3 but also vital for the understanding of experimental observations4. Here we show, using demagnetization experiments, that the concept of spin temperature in general fails to describe the mesoscopic nuclear-spin ensemble of a quantum dot. We associate the observed deviations from a thermal spin state with the presence of strong quadrupolar interactions within the quantum dot, which cause significant anharmonicity in the spectrum of the nuclear spins. Strain-induced, inhomogeneous quadrupolar shifts also lead to a complete suppression of angular-momentum exchange between the nuclear-spin ensemble and its environment, resulting in nuclear-spin relaxation times exceeding an hour. Remarkably, the position-dependent axes of the quadrupolar interactions render magnetic-field sweeps inherently non-adiabatic, thereby causing an irreversible loss of nuclear-spin polarization.
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Main
The study of nuclear-spin physics by optical orientation experiments in bulk semiconductor materials has been an active field of research over recent decades5,6,7. These research efforts have shown that, using the electron as a mediator, it is possible to transfer angular momentum from light onto nuclei, thereby establishing a nuclear-spin polarization that is orders of magnitude higher than the equilibrium nuclear polarization at cryogenic temperatures. As a result, the effective nuclear-spin temperature in such an optically pumped system can be pushed to the microkelvin regime. Combining these optical pumping schemes with nuclear adiabatic demagnetization techniques borrowed from bulk nuclear magnetic resonance experiments3 would be a natural extension to these experiments that could lead to a significant further reduction of the nuclear-spin temperature. This approach, previously demonstrated in bulk semiconductors5,8, suffers from the fact that, in most systems where optical orientation of nuclear spins is possible, nuclear-spin relaxation is too fast to allow for a significant reduction of magnetic fields in an adiabatic way. Here, we use the exceedingly long nuclear-spin relaxation time in self-assembled quantum dots (QDs) (ref. 9) to implement an ‘adiabatic’ demagnetization experiment on the system of ∼105 nuclear spins.
The mesoscopic ensemble of nuclear spins in a QD can be conveniently polarized and measured by optical means5,9,10,11,12. To this end, we use the photoluminescence of the negatively charged exciton (X−1) under resonant excitation of an excited QD state. It has been shown previously13 that, under appropriate excitation conditions, 20–50% of the QD nuclear spins can be efficiently polarized in a timescale of a few milliseconds. The resulting dynamical nuclear-spin polarization can then be measured through a change in the Zeeman splitting, ΔEOS, of the X−1 recombination line13; this energy shift due to the spin-polarized nuclei is commonly referred to as the Overhauser shift14.
A remarkable feature of the QD nuclear-spin system is the excellent isolation from its environment if the QD is uncharged. Figure 1a shows the corresponding free evolution of the nuclear-spin polarization Pnuc (proportional to ΔEOS) in a QD subject to an external magnetic field Bext=2 T. The nuclear-spin relaxation time clearly exceeds one hour and does not vary appreciably over the magnetic-field range relevant to this work9. As the bulk material surrounding the QD remains unpolarized during the experiment (see the Methods section), the long nuclear-spin lifetime indicates that nuclear-spin diffusion between the QD and its environment is strongly suppressed. We attribute this quenching of spin diffusion to the structural and chemical mismatch between the InGaAs QD and its GaAs surroundings12,15. The very slow nuclear-spin relaxation leaves room for further manipulation of the QD nuclear-spin system after optical pumping. In particular, we can study how Pnuc behaves under slow variations of external parameters and thereby study the validity of spin thermodynamics for the QD nuclear-spin system.
If the QD nuclei were describable using a thermodynamic approach, Pnuc would be aligned with Bext and would be described by Curie’s law γ Pnuc=BextC/Tspin (ref. 3) (here, γ is the nuclear gyromagnetic ratio, C the Curie constant and Tspin the nuclear-spin temperature). An adiabatic lowering of Bext from an initial value Bi to a final value Bf would conserve Pnuc and lead to a reduction of Tspin by a factor Bf/Bi. In general, cooling by adiabatic demagnetization is possible for any system where the spin entropy S is conserved and a function of Bext/Tspin only. The ultimate limit to the achievable cooling is determined by nuclear-spin interactions, which give the dominant contribution to S at low magnetic fields. The strength and nature of these interactions can be phenomenologically described by a random local magnetic field Bloc. In most cases, Bloc is given by the nuclear dipolar couplings (≈0.1 mT). As soon as Bext≈Bloc, the local fields randomize an established nuclear-spin polarization and thereby limit the efficiency of the adiabatic spin cooling to Bloc/Bi. The resulting behaviour of nuclear-spin temperature and polarization as a function of Bf is sketched in Fig. 1b: for Bext=0, the spin temperature remains finite and the nuclear spins are completely depolarized. Amazingly, this depolarization is a reversible process, provided that S is a conserved quantity at all fields. When the spins are re-magnetized to a magnetic field exceeding Bloc, their polarization recovers along the direction of the magnetic field and in particular conserves the sign of its initial spin temperature.
To test the validity of spin thermodynamics for the QD nuclear spins and to study the possibility of adiabatic cooling in this system, we performed demagnetization experiments on a QD, as illustrated in Fig. 1c. A circularly polarized ‘pump’ pulse of length τpump is used to polarize the nuclear spins. After ejecting the electron from the QD, we linearly ramp Bext from Bi to Bf with a rate γB=10 mT s−1. At the final field Bf, the remaining degree of nuclear-spin polarization is measured using a linearly polarized ‘probe’ pulse of length τprobe (ref. 9). This experiment is repeated at various values of Bf to record the process of ‘adiabatic’ (de)magnetization.
Figure 1d shows the result of a demagnetization experiment performed on the nuclear-spin system of an individual QD. The nuclei are polarized with a pump pulse τpump=300 ms at Bi=1 T and measured at Bf with a probe pulse τprobe=5 ms. At a rough glance, this measurement qualitatively follows the behaviour depicted in Fig. 1b. A closer inspection, however, reveals significant deviations: on ramping the external field to Bf=−1 T we recover only 63% of the initial Pnuc. In addition, by measuring Pnuc(Bf) we determined the value of the local field to be Bloc=290 mT: this value is about three orders of magnitude larger than typical nuclear dipolar fields. Finally, we observe that even for Bf=0 the QD has a remanent nuclear-spin polarization Pnucrem. To verify that we do not induce an unwanted increase of spin entropy by sweeping Bext too fast, we repeated our experiment for values of γB of 5, 2.5 and 0.8 mT s−1 (crosses in Fig. 1d). Within the experimentally accessible range, γB has no influence on our observations.
The discrepancy between our experimental findings and the predictions from a thermodynamical treatment of nuclear spins becomes even more pronounced if we increase Pnuc(Bi) (which can be achieved by first increasing Bext to 2.2 T (refs 12, 16)). Figure 2a shows an experiment where we then demagnetize the polarized nuclear spins starting from Bi=2.0 T to a final field Bf′=−1 T (black data points). We then reverse the sweep direction of the magnetic field and ramp Bext back to Bf (grey data points). This experiment shows a considerable hysteresis of the nuclear-spin polarization as a function of Bf. In particular, Pnucrem changes sign for the two sweep directions of Bext. Furthermore, the magnitude of Pnucrem, and respectively the width of the observed hysteresis curve, depends linearly on the initial degree of nuclear-spin polarization and on Bi (Fig. 2b).
To obtain more information about the source of irreversibility of Pnuc during magnetic-field sweeps, we performed a further experiment, where we optically orient the nuclear spins at Bi=1 T and ramp the field to a value Bf′<Bi and then back to Bi=Bf, where we measure the remaining degree of nuclear-spin polarization. The result of this experiment (Fig. 2c) indicates that the magnetic-field sweeps start to induce irreversibilities in Pnuc as soon as |Bext|≲Bloc≈300 mT.
Finally we note that the experimental observations described here do not depend on the sign of the initial nuclear-spin temperature (Tspin,i). We have repeated the demagnetization experiments for Tspin,i<0 (that is, σ− laser excitation at Bi>0, not shown here) and observed values of Bloc and Pnucrem consistent with the measurements presented in Figs 1 and 2. These measurements are complicated by the fact that for Tspin<0 nuclear-spin pumping is rather inefficient12, leading to a low degree of dynamical nuclear-spin polarization and therefore a smaller signal-to-noise ratio than for Tspin>0.
The three principal features of our experiments, the existence of Pnucrem, the hysteretic behaviour of Pnuc and the partial irreversibility of our demagnetization experiment, result from a violation of the nuclear (Zeeman) spin temperature approximation1,3. We explain these features by taking into account the strong inhomogeneous quadrupolar interaction (QI) of the nuclear spins in a QD (refs 17, 18, 19). The self-assembled growth of InGaAs QDs is driven by a strong lattice mismatch between InGaAs and its surrounding GaAs matrix, which results in a heavily strained QD lattice. As a consequence, QD nuclei experience large electric-field gradients, which couple to the nuclear quadrupolar moment. The resulting quadrupolar Hamiltonian20,
is characterized by a nuclear quadrupolar frequency νQ (proportional to the local strain at the nuclear site) and a quadrupolar axis z′ (with corresponding unit vector ez′ along the main axis of the local electric-field-gradient tensor). is the nuclear-spin angular-momentum operator with quantum number I and . For typical strain values of 2% (ref. 21), we find νQ≈2.8 MHz for As and 1.2 MHz for In (ref. 22). For comparison of the interaction strength of with a pure nuclear Zeeman Hamiltonian , it is convenient to express the QI strength by an equivalent magnetic field BQ=h νQ/γ. For As and In, we find BQ=388 mT and 125 mT, respectively; the corresponding mean value agrees well with our experimental estimate for Bloc.
The spectrum of a nuclear spin with quadrupolar frequency νQ depends strongly on the angle θ between ez′ and the external magnetic field (directed along ez, Fig. 3a). Figure 3b shows the eigenenergies of a nuclear spin with I=3/2, as a function of Bext/BQ. At Bext=0, the spectrum is governed by , which pairs the nuclear-spin states into doublets with angular-momentum projections ±mz′ on ez′. The doublets are split by an energy |ℏωmz′,mz′+1|=(mz′+1/2)h νQ, respectively. Conversely, in a high magnetic field, the spectrum is determined by with nuclear angular momentum being quantized along the axis ez. Even at arbitrarily high fields, however, the spectrum is significantly perturbed by and never becomes perfectly harmonic.
We modelled our demagnetization experiment using the steady-state solution of a rate equation for the populations p|m〉 of spin states |m〉, which are mutually coupled through dipolar interactions (Fig. 3b and c). The nuclear spins are initialized with a Boltzmann distribution at Bext=Bi (see the Methods section) and the evolution of the p|m〉 is calculated as a function of Bext. Owing to the unequal nuclear-spin level spacings, only nuclear-spin flip-flops that preserve p|m〉 () are energetically allowed in general and therefore the spin populations remain invariant as a function of Bext. Varying Bext will change the relative nuclear-spin level spacings in the nonlinear way depicted in Fig. 3c. As the p|m〉 remain invariant while Bext is reduced, the nuclear spins are driven into a state that is out of thermal equilibrium (that is, not Boltzmann distributed). At specific values of Bext (red markers in Fig. 3c), transition energies between distinct pairs of nuclear-spin states can coincide—a situation denoted as a ‘crossover’ of nuclear-spin transitions1. At these fields, the p|m〉 are no longer constant and the nuclear-spin levels involved in the crossover can relax to a Boltzmann distribution. The irreversibility observed in our magnetic-field sweeps is a consequence of this partial relaxation of nuclear spins to thermal equilibrium. We speculate that the resulting increase of the nuclear-spin entropy is induced by an energy-conserving coupling to the environment of the nuclear spins. If the minimal energy gap of the anticrossing induced by the dipolar coupling between two interacting nuclear spins at their crossover is smaller than the coupling to the environment, pure dephasing of the nuclear-spin transitions will induce irreversible crossover transitions and S will increase.
On sweeping Bext through zero (red box in Fig. 3c), dipolar interactions will couple the states mz′=±1/2. The associated passage through the avoided crossing between these single-spin states is adiabatic and preserves the respective populations in the two lowest-lying spin states. In contrast, nuclear dipolar interaction cannot couple any of the states with |mz′|>1/2 owing to conservation of energy and angular momentum. The spin states in the |mz′|=3/2 manifold will therefore cross and in particular preserve their populations p3/2 and p−3/2. The imbalance between these populations (p3/2<p−3/2 in Fig. 3) will result in a remnant polarization Pnucrem, even if Bext is strictly zero.
We averaged our model over a set of parameters θ and νQ to account for the strong inhomogeneity of QI over the QD (see the Methods section). The result of this full simulation is shown in Fig. 3d. We highlight that the good qualitative agreement with our experimental results (Fig. 2a) is rather insensitive to the set of parameters used in our simulation. In particular, the choice of the distribution for the parameters θ and νQ did not affect our results significantly. Furthermore, our simulation treats the QD spin system as a pure spin-3/2 system, whereas for In I=9/2. A numerical treatment of the full InGaAs nuclear-spin system is beyond the scope of this paper and would most probably not alter the qualitative behaviour of our simulations (see the Methods section).
Our results show that the nuclear-spin system of a self-assembled QD provides a rare example for a solid-state nuclear-spin ensemble that cannot be described by a nuclear-spin temperature23. We note that, if we could assign a spin temperature to the QD nuclear-spin system, optical pumping combined with adiabatic demagnetization of the nuclear spins would be a novel and efficient means of nuclear-spin cooling in QDs without QI: possible systems include nuclear spin-1/2 systems, such as 13C-nanotube QDs (ref. 24), where QI is inherently absent, or strain-free semiconductor nanostructures25, such as epitaxially grown droplet QDs (ref. 26). There, adiabatic nuclear-spin cooling would be limited only by nuclear dipolar interactions resulting in Bloc≈0.1 mT. Achieving nuclear-spin cooling to temperatures of ≈100 nK should be feasible in these systems, opening ways to study the remnants of nuclear magnetic phase transitions in the mesoscopic system of QD nuclear spins27.
Methods
Sample and experimental techniques.
Individual QDs were studied using the photoluminescence of X−1 under resonant excitation of an excited QD state. The QD sample was grown by molecular-beam epitaxy on a (100) semi-insulating GaAs substrate. The approximate composition of the QDs after self-assembled growth and postgrowth annealing was In0.5Ga0.5As. For individual optical addressing, the QDs were grown at a low density of ≲0.1 μm−2. The QDs were spaced by 25 nm of GaAs from a doped n2+-GaAs layer, followed by 30 nm of GaAs and 29 periods of an AlAs/GaAs (2/2 nm) barrier, which was capped by 4 nm of GaAs. A bias voltage applied between the top Schottky and back Ohmic contacts controls the charging state of the QD. Optical pumping of QD nuclear spins was performed at the centre of the X−1 stability plateau in gate voltage, where photoluminescence counts as well as the resulting Overhauser shift were maximized13.
The QD sample was immersed in a liquid helium bath cryostat equipped with a superconducting magnet and was held at the cryostat base temperature of 1.7 K. The photoluminescence emitted by the QD was analysed in a 750 mm monochromator, allowing for the determination of spectral shifts of the QD emission lines with a precision of ∼1 μeV (ref. 12). A combination of an optical ‘pump–probe’ technique, together with linear ramps of the applied magnetic field, was used to adiabatically demagnetize the QD nuclear spins (see Fig. 1c); technical details of the pump–probe set-up are given elsewhere9. The ‘pump’ pulse consists of a circularly polarized laser pulse of duration τpump, which is used to optically orient the QD nuclear spins12. We typically achieve an Overhauser shift of ΔEOS=60 μeV at Bi=1 T, corresponding to nuclear-spin polarization Pnuc≈35% or Ti≈1.5 mK (for Bi=2.2 T, ΔEOS=89.5 μeV and Pnuc≈50%). In the range of Bext relevant to our experiment, Pnuc∝Bi (ref. 12) such that the initial nuclear-spin temperature Ti is roughly constant and of the order of few millikelvin (ref. 6) for all values of Bi.
Directly after applying the pump pulse to the QD, the gate voltage is switched to a value where the QD is charge neutral. In this regime, nuclear-spin polarization has an exceedingly long relaxation time of the order of hours9 (see Fig. 1a). We note that we can exclude any significant nuclear polarization of the bulk material surrounding the QD. The observation of dynamical nuclear-spin polarization in our experiment depends sensitively on the excitation laser energy, which we tune to an intra-dot (p-shell) excitation resonance with a width of ≈300 μeV and located ≈36 meV above the photoluminescence emission energy. The sharpness and energy of this excitation resonance makes any excitation processes that involve the creation of free electrons in the bulk very unlikely28. Furthermore, the pumping time τpump=600 ms used in our experiment is much too short to lead to a significant bulk nuclear-spin polarization, even if some free electrons were created during laser illumination.
Details of the model.
The model we developed to explain our experimental findings is based on the steady-state solution of a rate equation for the populations of a nuclear-spin I=3/2 system. The nuclear spins are initialized with a Boltzmann distribution over the spin states at Bext=Bi. The assumption of a thermal distribution of nuclear-spin levels at Bext=Bi is justified by the fact that nuclear spins are polarized by hyperfine interaction with the QD electron: optical pumping of the electron leads to a broadening of its spin states by several μeV (ref. 12), allowing for electron–nuclear flip-flops between the electron and any two given nuclear-spin states which are coupled by the hyperfine interaction. It is therefore reasonable to assume that the occupations of nuclear-spin levels at Bi follow a Boltzmann distribution.
We then change the magnetic field by keeping the populations of spin levels fixed. Only at the specific fields where cross-relaxation is permitted (Fig. 3c) do we allow for a local thermal equilibrium to be established between the spin levels involved in the cross-relaxation transitions. All other populations and the total energy of the nuclear-spin system remain constant. On sweeping through Bext=0, we assume that the levels mz′=±1/2 undergo an adiabatic passage through an anticrossing induced by the coupling of these two states by dipolar interactions. Spin states with mz′=±3/2 however remain uncoupled and undergo an adiabatic level crossing, which preserves their populations.
The result of our simulations is shown in Fig. 3c,d. We illustrate the evolution of the occupations of the individual nuclear-spin states in Fig. 3c, where we show the spectrum of a nuclear spin for the parameters νQ=3 MHz, θ=0.3π/2 and γ=10 MHz T−1. The occupations of the individual levels are encoded by the thickness and grey shade of the corresponding lines. Magnetic fields where cross-relaxation processes take place are indicated by red lines. We repeated this calculation for a set of angles and quadrupolar frequencies νQ∈{−4,−3,−2,2,3,4} MHz, over which we average our results. As the local strain in our QDs can be both tensile and compressive, positive and negative values for νQ are possible. By solving the complete Hamiltonian , we can relate the occupancies of the spin levels to our experimentally observed nuclear-spin polarization—the expectation value of the nuclear-spin polarization along the direction of Bext. Figure 3d of the main paper shows the result of our simulation in the form of the calculated evolution of Pnuc as a function of Bf.
We note that our model is a great simplification of the actual experimental situation. First, we completely ignore cross-relaxation events between nuclei of different (θ,νQ)−values. Second, our calculation was performed for a spin-3/2 system for simplicity, whereas the actual QD nuclear-spin system consists of a mixture of spin 3/2 (Ga, As) and spin 9/2 (In), which further complicates the situation. Although a numerical treatment of the full InGaAs nuclear-spin system is beyond the scope of this paper, we argue that such a treatment would not alter the physical picture conveyed by our simulation. Including I=9/2 spins would lead to a nuclear-spin spectrum similar to the one illustrated in Fig. 3b. The number of magnetic-field values where cross-relaxation events would be energetically allowed would increase compared with the case of I=3/2, but these events would still be singular in the sense that for most values of Bext the nuclear spins could not thermalize. The system would thus still be driven out of thermal equilibrium and the relaxation events during cross-relaxation would lead to an increase of nuclear-spin entropy. Including flip-flop events between In and As nuclear spins would have a similar effect: these transitions would be allowed for a subset of close nuclei and would allow for partial thermalization only at specific values of Bext.
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Acknowledgements
We thank A. Högele, J. Elzerman and S. D. Huber for help with the manuscript, and T. Amand and O. Krebs for discussions. We acknowledge A. Badolato for sample growth. This work is supported by NCCR-Nanoscience and an ERC Advanced Investigator Grant.
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Maletinsky, P., Kroner, M. & Imamoglu, A. Breakdown of the nuclear-spin-temperature approach in quantum-dot demagnetization experiments. Nature Phys 5, 407–411 (2009). https://doi.org/10.1038/nphys1273
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DOI: https://doi.org/10.1038/nphys1273
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