Breakdown of the nuclear-spin-temperature approach in quantum-dot demagnetization experiments

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 correct^2,3 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.

Main

The study of nuclear-spin physics by optical orientation experiments in bulk semiconductor materials has been an active field of research over recent decades^5,6,7. 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 semiconductors^5,8, 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 means^5,9,10,11,12. 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, ΔE_OS, 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 P_nuc (proportional to ΔE_OS) in a QD subject to an external magnetic field B_ext=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 surroundings^12,15. 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 P_nuc behaves under slow variations of external parameters and thereby study the validity of spin thermodynamics for the QD nuclear-spin system.

Figure 1: Demagnetization of QD nuclear spins.

a, Free decay of P_nuc at B_ext=2 T for an uncharged QD after optical pumping of the nuclear spins for τ_pump=600 ms. The grey curve shows an exponential decay with a time constant of 1 h for comparison. b, Theoretical prediction of nuclear-spin temperature and polarization during adiabatic demagnetization from a field B_i (red arrow) to B_f. c, Schematic diagram of the experimental procedure for adiabatic demagnetization of QD nuclear spins. The nuclei are optically pumped at B_ext=B_i (T_spin,i∼mK). Directly after the pumping pulse, the electron is ejected from the QD. B_ext is then linearly ramped at a rate γ_B to a value B_f, at which we measure P_nuc. d, Experimental (de)magnetization of QD nuclear spins. B_i=1 T as indicated by the red arrow, γ_B=10 mT s⁻¹ and ΔE_OS(B_i)=57 μeV. The grey curve is a fit according to the theoretical predictions shown in b; we find B_loc=290 mT. Blue, green and red crosses show a similar experiment, with γ_B=5, 2.5 and 0.8 mT s⁻¹, respectively (B_i=0.5 T for these data points).

If the QD nuclei were describable using a thermodynamic approach, P_nuc would be aligned with B_ext and would be described by Curie’s law γ P_nuc=B_extC/T_spin (ref. 3) (here, γ is the nuclear gyromagnetic ratio, C the Curie constant and T_spin the nuclear-spin temperature). An adiabatic lowering of B_ext from an initial value B_i to a final value B_f would conserve P_nuc and lead to a reduction of T_spin by a factor B_f/B_i. In general, cooling by adiabatic demagnetization is possible for any system where the spin entropy S is conserved and a function of B_ext/T_spin 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 B_loc. In most cases, B_loc is given by the nuclear dipolar couplings (≈0.1 mT). As soon as B_ext≈B_loc, the local fields randomize an established nuclear-spin polarization and thereby limit the efficiency of the adiabatic spin cooling to B_loc/B_i. The resulting behaviour of nuclear-spin temperature and polarization as a function of B_f is sketched in Fig. 1b: for B_ext=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 B_loc, 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 τ_pump is used to polarize the nuclear spins. After ejecting the electron from the QD, we linearly ramp B_ext from B_i to B_f with a rate γ_B=10 mT s⁻¹. At the final field B_f, the remaining degree of nuclear-spin polarization is measured using a linearly polarized ‘probe’ pulse of length τ_probe (ref. 9). This experiment is repeated at various values of B_f 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 τ_pump=300 ms at B_i=1 T and measured at B_f with a probe pulse τ_probe=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 B_f=−1 T we recover only 63% of the initial P_nuc. In addition, by measuring P_nuc(B_f) we determined the value of the local field to be B_loc=290 mT: this value is about three orders of magnitude larger than typical nuclear dipolar fields. Finally, we observe that even for B_f=0 the QD has a remanent nuclear-spin polarization P_nuc^rem. To verify that we do not induce an unwanted increase of spin entropy by sweeping B_ext too fast, we repeated our experiment for values of γ_B of 5, 2.5 and 0.8 mT s⁻¹ (crosses in Fig. 1d). Within the experimentally accessible range, γ_B 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 P_nuc(B_i) (which can be achieved by first increasing B_ext to 2.2 T (refs 12, 16)). Figure 2a shows an experiment where we then demagnetize the polarized nuclear spins starting from B_i=2.0 T to a final field B_f′=−1 T (black data points). We then reverse the sweep direction of the magnetic field and ramp B_ext back to B_f (grey data points). This experiment shows a considerable hysteresis of the nuclear-spin polarization as a function of B_f. In particular, P_nuc^rem changes sign for the two sweep directions of B_ext. Furthermore, the magnitude of P_nuc^rem, and respectively the width of the observed hysteresis curve, depends linearly on the initial degree of nuclear-spin polarization and on B_i (Fig. 2b).

Figure 2: Irreversibility and hysteresis in the demagnetization experiment.

a, Black circles, the same experiment as in Fig. 1d, with B_i=2.0 T as indicated by the red arrow (ΔE_OS(B_i)=89.5 μeV). After reaching B_f′=−1 T, we reverse the magnetic-field sweep direction and bring the nuclei back to the initial field (grey circles). b, The remanent nuclear-spin polarization P_nuc^rem (normalized to the value P_nuc(B_i) found in a) as a function of B_i. As P_nuc(B_i)∝B_i, the nuclear-spin temperature after optical pumping is roughly constant for all values of B_i in this measurement. c, After polarization of nuclear spins at B_i=1 T (red arrow), we sweep B_ext to B_f′ and then back to B_i, where P_nuc is measured. The magnetic field sweeps become partly irreversible as soon as |B_f′|≲0.3 T≈B_Q. The lines in the figures are guides to the eye.

To obtain more information about the source of irreversibility of P_nuc during magnetic-field sweeps, we performed a further experiment, where we optically orient the nuclear spins at B_i=1 T and ramp the field to a value B_f′<B_i and then back to B_i=B_f, 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 P_nuc as soon as |B_ext|≲B_loc≈300 mT.

Finally we note that the experimental observations described here do not depend on the sign of the initial nuclear-spin temperature (T_spin,i). We have repeated the demagnetization experiments for T_spin,i<0 (that is, σ⁻ laser excitation at B_i>0, not shown here) and observed values of B_loc and P_nuc^rem consistent with the measurements presented in Figs 1 and 2. These measurements are complicated by the fact that for T_spin<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 T_spin>0.

The three principal features of our experiments, the existence of P_nuc^rem, the hysteretic behaviour of P_nuc and the partial irreversibility of our demagnetization experiment, result from a violation of the nuclear (Zeeman) spin temperature approximation^1,3. 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 ν_Q (proportional to the local strain at the nuclear site) and a quadrupolar axis z′ (with corresponding unit vector e_z′ 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 ν_Q≈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 B_Q=h ν_Q/γ. For As and In, we find B_Q=388 mT and 125 mT, respectively; the corresponding mean value agrees well with our experimental estimate for B_loc.

The spectrum of a nuclear spin with quadrupolar frequency ν_Q depends strongly on the angle θ between e_z′ and the external magnetic field (directed along e_z, Fig. 3a). Figure 3b shows the eigenenergies of a nuclear spin with I=3/2, as a function of B_ext/B_Q. At B_ext=0, the spectrum is governed by , which pairs the nuclear-spin states into doublets with angular-momentum projections ±m_z′ on e_z′. The doublets are split by an energy |ℏω_mz′,m_z′+1|=(m_z′+1/2)h ν_Q, respectively. Conversely, in a high magnetic field, the spectrum is determined by with nuclear angular momentum being quantized along the axis e_z. Even at arbitrarily high fields, however, the spectrum is significantly perturbed by and never becomes perfectly harmonic.

Figure 3: Modelling of the demagnetization experiment.

Local electric-field gradients induced by strain in self-assembled QDs result in strong QI for the nuclear spins. a, Model of local strain axis distribution e_z′ within a QD. b, Spectrum of nuclear spins (I=3/2) under the influence of both and for a variety of angles θ between e_z′ and e_z. c, Simulation of QD nuclear-spin demagnetization for a particular setting ν_Q=3 MHz and θ=0.15π. Nuclear-spin populations p are represented both by line thickness and greyscale of the lines that indicate the energy of the nuclear-spin states. At B_i=1 T, the nuclei are initialized with a Boltzmann distribution over their spin states. The populations remain constant for most values of B_f. Only if a crossover of nuclear-spin transitions occurs (red markers for B_f>0) do the occupations of the involved spin states evolve to a (local) thermal equilibrium distribution (see the text). We simulate this process for a set of configurations {θ,ν_Q} and calculate the corresponding magnetization P_nuc∝〈I_z〉. d, The resulting nuclear-spin polarization as a function of B_f starting at B_i (red arrow), which qualitatively reproduces the experimental findings shown in Fig. 2.

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 B_ext=B_i (see the Methods section) and the evolution of the p_|m〉 is calculated as a function of B_ext. 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 B_ext. Varying B_ext will change the relative nuclear-spin level spacings in the nonlinear way depicted in Fig. 3c. As the p_|m〉 remain invariant while B_ext 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 B_ext (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 B_ext through zero (red box in Fig. 3c), dipolar interactions will couple the states m_z′=±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 |m_z′|>1/2 owing to conservation of energy and angular momentum. The spin states in the |m_z′|=3/2 manifold will therefore cross and in particular preserve their populations p_3/2 and p_−3/2. The imbalance between these populations (p_3/2<p_−3/2 in Fig. 3) will result in a remnant polarization P_nuc^rem, even if B_ext is strictly zero.

We averaged our model over a set of parameters θ and ν_Q 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 ν_Q 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 B_loc≈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 In_0.5Ga_0.5As. 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 τ_pump, which is used to optically orient the QD nuclear spins¹². We typically achieve an Overhauser shift of ΔE_OS=60 μeV at B_i=1 T, corresponding to nuclear-spin polarization P_nuc≈35% or T_i≈1.5 mK (for B_i=2.2 T, ΔE_OS=89.5 μeV and P_nuc≈50%). In the range of B_ext relevant to our experiment, P_nuc∝B_i (ref. 12) such that the initial nuclear-spin temperature T_i is roughly constant and of the order of few millikelvin (ref. 6) for all values of B_i.

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 τ_pump=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 B_ext=B_i. The assumption of a thermal distribution of nuclear-spin levels at B_ext=B_i 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 B_i 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 B_ext=0, we assume that the levels m_z′=±1/2 undergo an adiabatic passage through an anticrossing induced by the coupling of these two states by dipolar interactions. Spin states with m_z′=±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 ν_Q=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 ν_Q∈{−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 ν_Q 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 B_ext. Figure 3d of the main paper shows the result of our simulation in the form of the calculated evolution of P_nuc as a function of B_f.

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,ν_Q)−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 B_ext 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 B_ext.