Demagnetization of Quantum Dot Nuclear Spins: Breakdown of the Nuclear Spin Temperature Approach
Abstract
The physics of interacting nuclear spins arranged in a crystalline lattice is typically described using a thermodynamic framework (1): a variety of experimental studies in bulk solid-state systems have proven the concept of a spin temperature to be not only correct (2); (3) but also vital for the understanding of experimental observations (4). Using demagnetization experiments we demonstrate that the mesoscopic nuclear spin ensemble of a quantum dot (QD) can in general not be described by a spin temperature. We associate the observed deviations from a thermal spin state with the presence of strong quadrupolar interactions within the QD that 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 quadrupolar interactions render magnetic field sweeps inherently non-adiabatic, thereby causing an irreversible loss of nuclear spin polarization.
The study of nuclear spin physics by optical orientation experiments in bulk semiconductor materials has been an active field of research over the last 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 low mK regime. Combining these optical pumping schemes with nuclear adiabatic demagnetization techniques borrowed from bulk NMR experiments (3) 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 (8); (5), 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 QDs (9) to implement an “adiabatic” demagnetization experiment on the system of nuclear spins.
The mesoscopic ensemble of nuclear spins in a QD can be conveniently polarized and measured by optical means (5); (10); (11); (13); (9). To this end, we use the photoluminescence (PL) of the negatively charged exciton () under resonant excitation of an excited QD state. It has been shown previously (14) that under appropriate excitation conditions, of the QD nuclear spins can be efficiently polarized in a timescale of a few ms. The resulting dynamical nuclear spin polarization (DNSP) can then be measured through a change in the Zeeman splitting, , of the recombination line (14); this energy shift due to the spin polarized nuclei is commonly referred to as the Overhauser shift (OS) (15).
A remarkable feature of the QD nuclear spin system is the excellent isolation from its environment if the QD is uncharged. Fig. 1a shows the corresponding free evolution of (proportional to ) in a QD subject to an external magnetic field T. The nuclear spin relaxation time clearly exceeds one hour and does not vary appreciably over the magnetic field range relevant to this work (9). Since the bulk material surrounding the QD remains unpolarized during the experiment (see Methods), 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 sourrounding (13); (16). 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 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, would be aligned with the external magnetic field and would be described by Curie’s law (3) (here, is the nuclear gyromagnetic ratio, the Curie constant and the nuclear spin temperature). An adiabatic lowering of from an initial value to a final value , would conserve and lead to a reduction of by a factor . In general, cooling by adiabatic demagnetization is possible for any system where the spin entropy is conserved and a function of only. The ultimate limit to the achievable cooling is determined by nuclear spin interactions which give the dominant contribution to at low magnetic fields. The strength and nature of these interactions can be phenomenologically described by a random local magnetic field . In most cases, is given by the nuclear dipolar couplings (mT). As soon as , the local fields randomize an established nuclear spin polarization and thereby limit the efficiency of the adiabatic spin cooling to . The resulting behavior of nuclear spin temperature and polarization as a function of is sketched in Fig. 1b: for , the spin temperature remains finite and the nuclear spins are completely depolarized. Amazingly, this depolarization is a reversible process, provided that is a conserved quantity at all fields. When the spins are re-magnetized to a magnetic field exceeding , 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 is used to polarize the nuclear spins. We then linearly ramp from to with a rate mT/s. At the final field , the remaining degree of nuclear spin polarization is measured using a linearly polarized “probe” pulse of length (9). This experiment is repeated at various values of 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 ms at T and measured at with a probe pulse ms. At a rough glance, this measurement qualitatively follows the behavior depicted in Fig. 1b. A closer inspection however, reveals significant deviations: upon ramping the external field to T we only recover of the initial . In addition, by measuring we determined the value of the local field to be mT: this value is about three orders of magnitude larger than typical nuclear dipolar fields. Finally, we observe that even for , the QD has a remnant nuclear spin polarization . To verify that we do not induce an unwanted increase of spin entropy by sweeping too fast, we repeated our experiment for values of of mT/s (crosses in Fig. 1d). Within the experimentally accessible range, 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 (which can be achieved by increasing (13); (12)). Fig. 2a shows an experiment where we demagnetize the polarized nuclear spins starting from T to a final field T (black data points). We then reverse the sweep direction of the magnetic field and ramp back to (gray data points). This experiment shows a considerable hysteresis of the nuclear spin polarization as a function of . In particular, changes sign for the two sweep directions of . Furthermore, the magnitude of , resp. the width of the observed hysteresis curve depends linearly on the initial degree of nuclear spin polarization and on (Fig. 2b).
To obtain more information about the source of irreversibility of during magnetic field sweeps, we performed a further experiment, where we optically orient the nuclear spins at T, ramp the field to a value and then back to 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 as soon as mT.
Finally we note that the experimental observations described here do not depend on the sign of the initial nuclear spin temperature (). We have repeated the demagnetization experiments for (i.e. laser excitation at , not shown here) and observed values of and consistent with the measurements presented in Fig. 1 and Fig. 2. These measurements are complicated by the fact that for , nuclear spin pumping is rather inefficient (13), leading to a low degree of DNSP and therefore a smaller signal to noise ratio than for .
The three principal features of our experiments, the existence of , the hysteretic behavior of 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 interactions (QI) of the nuclear spins in a QD (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 Hamiltonian (20),
(1) |
is characterized by a nuclear quadrupolar frequency (proportional to the local strain at the nuclear site) and a quadrupolar axis (with corresponding unit vector along the main axis of the local electric field gradient tensor). is the nuclear spin angular momentum operator with quantum number and . For typical strain values of (22), we find MHz for As and MHz for In (23). 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 . For As and In, we find mT and mT, respectively; the corresponding mean value agrees well with our experimental estimate for .
The spectrum of a nuclear spin with quadrupolar frequency depends strongly on the angle between and the external magnetic field (directed along ). Figure 3b shows the Eigenenergies of a nuclear spin with , as a function of . At , the spectrum is governed by , which pairs the nuclear spin states into doublets with angular momentum projections on . The doublets are split by an energy , respectively. Conversely, in a high magnetic field, the spectrum is determined by with nuclear angular momentum being quantized along the axis . Even at arbitrarily high fields however, the spectrum is significantly perturbed by and never becomes perfectly harmonic.
We modeled our demagnetization experiment using the steady state solution of a rate equation for the populations of spin states , which are mutually coupled through dipolar interactions (Fig. 3b and c). The nuclear spins are initialized with a Boltzmann distribution at (see Methods) and the evolution of the ’s is calculated as a function of . Due to the unequal nuclear spin level spacings, only nuclear spin flip-flops that preserve () are energetically allowed in general and therefore the spin populations remain invariant as a function of . Varying will change the relative nuclear spin level spacings in the nonlinear way depicted in Fig. 3c. Since the ’s remain invariant as is reduced, the nuclear spins are driven into a state which is out of thermal equilibrium (i.e. not Boltzmann-distributed). At specific values of (red markers in Fig. 3c), transition energies between distinct pairs of nuclear spin states can coincide — a situation denoted as a “cross-over” of nuclear spin transitions (1). At those fields, the ’s are no longer constant and the nuclear spin levels involved in the cross-over 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 anti-crossing induced by the dipolar coupling between two interacting nuclear spins at their cross-over is smaller than the coupling to the environment, pure dephasing of the nuclear spin transitions will induce irreversible cross-over transitions and will increase.
Upon sweeping through zero (red box in Fig. 3c), dipolar interactions will couple the states . 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 can not couple any of the states with due to conservation of energy and angular momentum. The spin states in the -manifold will therefore cross and in particular preserve their populations and . The imbalance between these populations ( in Fig. 3) will result in a remnant polarization , even if is strictly zero.
We averaged our model over a set of parameters and to account for the strong inhomogeneity of QI over the QD (see Methods). 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 and did not affect our results significantly. Furthermore, our simulation treats the QD spin system as a pure spin- system, while for In, . A numerical treatment of the full InGaAs nuclear spin system is beyond the scope of this paper and would most likely not alter the qualitative behavior of our simulations (see Methods).
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 can not be described by a nuclear spin temperature (24). We note that if one 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 way of nuclear spin cooling in QDs without QI: possible systems include nuclear spin-1/2 systems, such as C-nanotube QDs (25), where QI is inherently absent, or strain-free semiconductor nanostructures (26), such as epitaxially grown droplet-QDs (27). There, adiabatic nuclear spin cooling would only be limited by nuclear dipolar interactions resulting in mT. Achieving nuclear spin cooling to temperatures nK should be feasible in these systems, opening ways to studying the remnants of nuclear magnetic phase transitions in the mesoscopic system of QD nuclear spins (28).
Acknowledgments
We thank A. Högele, J. Elzerman and S.D. Huber for help with the manuscript, and T. Amand and O. Krebs for fruitful discussions. We acknowledge A. Badolato for sample-growth. This work is supported by NCCR-Nanoscience.
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to P.M. and A.I. (email: patrickm@phys.ethz.ch, imamoglu@phys.ethz.ch)
Methods
Sample and experimental techniques
Individual QDs were studied using the photoluminescence (PL) of under resonant excitation of an excited QD state. The QD sample was grown by molecular beam epitaxy on a semi-insulating GaAs substrate. The approximate composition of the QDs after self-assembled growth and post-growth annealing was InGaAs. For individual optical addressing, the QDs were grown at a low density of . The QDs were spaced by nm of GaAs from a doped n-GaAs layer, followed by nm of GaAs and 29 periods of an AlAs/GaAs (nm) barrier which was capped by 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 was performed at the center of the stability plateau in gate voltage, where PL counts as well as the resulting OS were maximized (14).
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 K. The PL emitted by the QD was analyzed in a mm monochromator allowing for the determination of spectral shifts of the QD emission lines with a precision of eV (13). A combination of an optical “pump-probe” technique, together with linear ramps of the applied magnetic field were used to adiabatically demagnetize the QD nuclear spins (see Fig. 1c); technical details of the pump-probe setup are given elsewhere (9). The “pump” pulse consists of a circularly polarized laser pulse of duration which is used to optically orient the QD nuclear spins (13). We typically achieve an OS of eV at T, corresponding to nuclear spin polarization or mK (for TeV and ). In the range of relevant to our experiment, (13) such that the initial nuclear spin temperature is roughly constant and on the order of few mK (6) for all values of .
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 on the order of hours (9) (see Fig. 1a). We note that we can exclude any significant nuclear polarization of the bulk material surrounding the QD. The observation of DNSP 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 eV and located meV above the PL emission energy. The sharpness and energy of this excitation resonance makes any excitation processes which involve the creation of free electrons in the bulk very unlikely (21). Furthermore, the pumping time 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 system. The nuclear spins are initialized with a Boltzmann distribution over the spin states at . The assumption of a thermal distribution of nuclear spin levels at 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 eV (13), 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 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), 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. Upon sweeping through , we assume that the levels undergo an adiabatic passage through an anticrossing induced by the coupling of these two states by dipolar interactions. Spin states with however remain uncoupled and undergo an adiabatic level crossing which preserves their populations.
The result of our simulations is shown in Fig. 3c and d of the main text. 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 MHz, and MHz/T. The occupations of the individual levels is encoded by the thickness and gray 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 MHz over which we average our results. Since the local strain in our QDs can be both tensile and compressive, positive and negative values for 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 . Fig. 3d of the main paper shows the result of our simulation in form of the calculated evolution of as a function of .
We note that our model is a great simplification of the actual experimental situation. First, we completely neglect cross-relaxation events between nuclei of different values. Second, our calculation was performed for a spin- system for simplicity, while the actual QD nuclear spin system consists of a mixture of spin- (Ga, As) and spin- (In), which further complicates the situation. While 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 to the case of I=3/2, but these events would still be singular in the sense that for most values of , 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 .
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