Quantum dot orbitals

We use a flexible gate layout (Fig. 1a) to shape a nearly circularly symmetric dot and set up a cryogenic piezo-rotator to apply almost perfectly aligned in-plane fields (Supplementary Note 1) up to 14 T with arbitrary angle ϕ with respect to the [100] crystal direction (Fig. 1b). The rotator capability allows us to probe the dot orbitals and their shape in large magnetic fields using the established technique of pulsed-gate orbital excited state spectroscopy18. Figure 1c displays two excited states, shown in green and blue, for field applied along the \(\hat x\) direction. While one state clearly moves down in energy (blue) with increasing field, the other one remains unaffected (green). Since only electron motion perpendicular to the applied field is affected by it, the B-invariant energy thus corresponds to the excitation along the \(\hat x\) direction, justifying labels as shown in Fig. 1c19,20. When the sample is rotated by 90°, the excitations’ roles swap and the blue line becomes invariant (Fig. 1d). Such striking behavior, including further B-directions, is reproduced by an anisotropic harmonic oscillator model21,22, which confirms that the quantum dot main axes are well aligned with the \(\hat x\) and \(\hat y\) directions. This essential information about the dot orbitals makes possible a detailed understanding of all measurements, reproducing the measured T 1 quantitatively by numerics using a single set of parameters without phenomenological constants (see Methods for details).

Fig. 1 Quantum dot setup and orbital spectroscopy. a Scanning electron microscope image of a co-fabricated lateral, surface gate defined quantum dot. The single electron wave function is indicated by the blue ellipse (not to scale) and is tunnel coupled to the left reservoir only (no tunneling to right lead). An adjacent dot (black circle) serves as a real-time charge sensor, operated in Coulomb blockade for better sensitivity. Sub-microsecond pulses are applied on the center plunger CP. The scale bar corresponds to 200 nm. b Measurement setup with sample on a piezo-electric rotator allowing change of the direction of the in-plane magnetic field (up to 14 T) with respect to the crystal axis [100], specified by the angle ϕ. c, d Energies of the two lowest orbital excited states, E x and E y , measured with respect to the ground state, as a function of the magnetic field applied along \(\hat x\) direction (c) and \(\hat y\) direction (d). Triangles are measured data, solid curves are numerics (see text) Full size image

Spin-orbit induced spin relaxation anisotropy

With a full orbital model at hand, we now turn to spin relaxation measurements, done by cycling the dot through ionization, charge and relax, and read-out configuration, as depicted in Fig. 2a. Averaging over many thousand cycles, we obtain the spin excited state probability P e as a function of the waiting time t w , the time the electron was given to relax into the spin ground state. A few examples are plotted over four orders of magnitude in t w in Fig. 2b at a magnetic field of 4 T. All such curves fit very well to the sum of two exponentials, from which we reliably extract the spin relaxation rate W ≡ \(T_{\mathrm{1}}^{ - 1}\) (see Supplementary Note 3 for more details). A pronounced dependence of W on the direction of the magnetic field is observed, as displayed in Fig. 3a as a function of the field angle ϕ. A modulation of W by a factor of ~16 is found, with minimal relaxation rate along the \(\hat y\) direction.

Fig. 2 Spin relaxation measurement. a Three step pulse scheme, shifting dot levels with gate-voltage pulses: First, during “ionization”, the dot is emptied. Second, in “charge and relax”, an electron is loaded and if the spin is down, i.e., in the excited spin state, it relaxes with rate W during the waiting time t w . Third, spin-charge conversion is used in “read-out” to detect the spin state: the spin-down electron only will tunnel off the dot, which is detected by the charge sensor. The spin relaxation rate W is extracted from the dependence of the probability P e to find the spin in the excited (down) state as a function of t w , shown in b for a magnetic field of 4 T applied along different angles ϕ as indicated. Markers show measurements with statistical error bars, curves are fits to the formula P e (t w ) ∝ (exp(−Wt w ) − exp(−Γ in t w ))/(Γ in − W), where the tunneling-in rate Γ in is determined independently (Supplementary Note 3). W is thus extracted as the only fit parameter. Error bars are standard deviations from fitting to counts (Supplementary Note 3) Full size image

Fig. 3 Spin relaxation anisotropy. Spin relaxation rate W (triangles with error bars) for in-plane magnetic fields of a 4 T and b 1.25 T, as a function the field direction. The solid curves show the results from numerics taken into account only the SOI (red), only the HF interaction (orange), and both (dark blue). The ripples in curves from numerics are fluctuations due to finite statistics over random nuclear spin configurations. Error bars are standard deviations from fits to data as introduced in Fig. 2b Full size image

This pronounced anisotropy is rooted in a combination of the dot shape asymmetry and the interference of the Rashba and Dresselhaus SOI terms. The latter can qualitatively be understood from the dependence of the total effective spin–orbit magnetic field on the direction of the electron momentum (Supplementary Note 5). First derived for symmetric quantum dots12, the spin relaxation anisotropy due to the dot shape asymmetry was also soon included in a theoretical generalization13. The shape-induced contribution to the anisotropy of W is well known here from the orbital spectroscopy and found to be small. Thus, the anisotropy here is largely due to the SOI, and given the precisely measured orbital energies, it is possible to extract the SOI coupling strengths by fitting the model (see Methods for details). The best fit delivers a ratio α/β ~ 1.6 and a spin–orbit length l so ≈ 2.1 μm setting the overall strength of the SOI. These values are well in-line with previous reports for GaAs structures18,23,24. We note that α and β are found to have the same sign for the 2D material used. Without knowledge of the orbital energies, the SOI parameters cannot be directly determined from T 1 14,25,26.

Hyperfine-phonon spin relaxation

A very long T 1 time can be achieved by reducing the magnetic field strength and orienting the magnetic field along the crystalline axis with minimal SOI field. Therefore, we carried out the same anisotropy measurements at 1.25 T. Indeed, T 1 times longer than 1 s are obtained. Interestingly, in contrast to the measurements at 4 T, around the \(\hat y\) direction with minimal W, the measured spin relaxation rate W (black markers) is seen to be almost a factor of three larger than the calculated SOI rate (red curve, Fig. 3b). This is far beyond the error bars, and indicates an additional spin relaxation channel beyond SOI-mediated phonon emission.

Because the dot orbitals are characterized, the HF contribution can be quantified by numerics (Methods). As shown in Fig. 3a, at B = 4 T the microscopic model predicts that the HF contribution (orange curve) is 1 to 2 orders of magnitude smaller than the one due to the SOI (red curve), and is therefore not observable experimentally. In comparison, at B = 1.25 T, as shown in Fig. 3b, the SOI model alone is unable to explain the data, but fits very well when the nuclei are included (purple curve), particularly now capturing the minimum close to the \(\hat y\)-direction very well. Backed by numerics, we thus conclude that this seemingly subtle feature in the angular modulation of W actually constitutes the first evidence of the HF relaxation mechanism.

To substantiate this claim, we measure the field magnitude dependence of W. In Fig. 4a we compare two sets, for the magnetic field along the \(\hat x\) and \(\hat y\) direction, where the effects of the nuclei with respect to SOI are, respectively, maximal and minimal. The two curves indeed show pronounced differences. With the field along the \(\hat x\)-direction, the relaxation follows the B5 scaling quite well over the entire range of the measured magnetic fields. Thus, for the \(\hat x\) direction, the relaxation is dominated by the SOI for the full field range. In contrast, for fields along \(\hat y\), there is a crossover around 2 T with a change of the power law scaling from roughly B5 at high fields to B3 at low fields, corresponding to a crossover from SOI to HF dominated relaxation.

Fig. 4 Hyperfine induced spin relaxation. a Spin relaxation rate W for an in-plane magnetic field along the \(\hat x\) direction (green, along [110]) and the \(\hat y\) direction (blue, along [1\(\overline 1\)0]) as a function of the field magnitude. The data are shown as triangles with error bars. Numerics considering various terms are shown as labeled. The pure B5 scaling (red dash) and B3 scaling (orange dash) are also given as a guide to the eye. The orange band around the HF curve indicates the statistical uncertainty due to a finite number of nuclear spin configurations used in the simulation. b The relaxation anisotropy W X /W Y as a function of field magnitude. Experiment is shown as triangles with error bars, numerics with both SOI and HF as a solid curve, showing the transition to isotropic relaxation at low fields. Red dashed line is SOI theory only, orange dash at W X /W Y = 1 is the isotropic HF theory. A possible dip below the theory above ≳6 T could be due to the only remaining discrepancy between theory and experiment, occurring at hight fields (see main text). Error bars are fit errors Full size image

Some comments are in place. First, dynamic nuclear spin polarization would distort the power laws. The absence of nuclear spin polarization in our measurements is guaranteed by the slowness of electron spin transitions at low fields and is an important advantage over experiments exploiting Pauli spin blockade in double dots. Second, the only remaining discrepancy of data and model is seen at high fields (see the blue data points and theory curve in Fig. 4a for B ≳ 6 T). This saturation is predicted in perturbative calculations12,27,28 and exact numerics13,29, including our model here, but it is not observed in our data. The explanation needs further investigations. Nevertheless, the issue is irrelevant for the nuclear-induced relaxation taking place at much smaller fields and longer times. Finally, we note a T 1 time of 57 ± 15 s for a magnetic field of 0.6–0.7 T along \(\hat y\), where the range represents the error from fitting (Supplementary Note 3). To our knowledge, this is the longest T 1 time reported to date in a nanoelectronic device10,18,26.

This all being said, we stress that the simple observation of a change in the power law scaling of W ∝ B3 is not sufficient as a proof of its HF origin. It could be that the phonons as an energy dissipation channel are replaced by another bath, e.g., charge noise or an ohmic bath also leads to a B3 dependence30,31,32. The absence of deviations in the scaling of the B||\(\hat x\) data indicates that phonons are responsible for the energy dissipation throughout and the crossover in the \(\hat y\) data is not related to a specific value of W, or transition energy. Also, if the SOI remained as the mixing mechanism and the energy dissipation channel instead were to change, then the spin relaxation anisotropy, quantified by the ratio W X /W Y , would remain large at low fields. However, as shown in Fig. 4b, the anisotropy is seen to decrease from about 16 at high fields towards one at fields below 1 T. This behavior displays spin relaxation with equal speed in both principal directions, thus indicating isotropic relaxation at low fields. Together with the W ∝ B3 scaling, these observations constitute unequivocal demonstration of HF-mediated spin relaxation.