3.1 Spin polarization by periodic pump pulse application
Before an optical excitation starts the spins in the quantum dot ensemble are randomly oriented. The spin orientation can be achieved by applying circularly polarized pump pulses, which transfer their angular momentum to the spins, exploiting optical selection rules according to the conduction- and valence-band structure. Generally all corresponding protocols rely on this principle, irrespective of details of the implementation. We focus here on the fast orientation using the laser systems mentioned above. For any application it would most likely be required that the total time for this process is below a nanosecond, in order to be competitive with existing information technologies.
In principle, the laser oscillators could also have been adjusted such that they emit pulses with about 100 fs duration. The pulses are Fourier-transform-limited so that their spectral width scales inversely with the pulse duration. From g-factor studies it is known that the g-factors of electrons and holes vary considerably with the optical transition energy across the photoluminescence spectra shown in Fig. 1. To avoid excitation of a strongly inhomogeneous spin ensemble, we therefore used our lasers not in the femtosecond configuration, but in the ps configuration, as described in the previous section. The 2 ps pulses correspond to a spectral width of 1 meV. The spectrum of such pulses is shown in Fig. 1 by the blue trace.
We excited the quantum dots resonantly with the ground-state transition, leading to the formation of negatively or positively charged excitons, depending on the residual charge, in the s-shell of the quantum dots. We note that spin orientation is also possible for non-resonant excitation into an excited quantum dot shell, for example. However, in this case the spin initialization becomes more complex due to the relaxation of carriers into their ground states (). Moreover, the spin mode-locking signal, which is described below, is considerably weaker than for the resonant case ().
Figure 3. Dependence of the g-factors of electrons (solid circles) and holes (open circles) on the magnetic field orientation in the quantum dot plane (). The g-factors were determined from the precession frequency in ellipticity measurements according to .
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In presence of resident carriers the excitation process becomes more elaborate due to the influence of Pauli blocking. The spin component of the resident carrier that is parallel to the optically injected electron (for n-doped dots) or the optically injected hole (for p-doped dots) blocks the excitation process. Only for the counter-oriented component is excitation allowed, so that for a dot structure with a resident electron, the following electron–trion superposition state is generated:
The unexcited electron components will give rise to a net electron spin contribution with strength . From the trion there is no electron contribution, as the two contained electrons form a singlet, though there is a signal from the hole spin ().
A corresponding Faraday rotation trace of precessing spins recorded on n-doped quantum dots is shown in Fig. 4 for T. The impact of the hole spin can be seen only at short delays below 1 ns after application of the pump pulse at time zero. There also an exciton contribution from undoped quantum dots shows up, which vanishes on similar time scales as the hole signature. For the quantum dots under study the radiative decay time was measured to be about 0.5 ns for both neutral and charged excitons, explaining the disappearance of corresponding contributions of optically injected carriers after these times ().
Figure 4. Faraday rotation trace as function of the delay between probe and pump (at time zero), recorded on the n-doped quantum dot sample. Pump and probe central photon energies were detuned by 0.1 meV, to maximize the signal strength in Faraday rotation (see also related discussion in text). T.
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However, oscillations in the Faraday rotation signal are observed much longer than during the presence of excitonic complexes, evidencing that spin coherence has been initiated for the resident electrons. The precession of the component that has remained unexcited due to Pauli blocking continues beyond the trion-component decay. There are two reasons for that spin polarization: (i) assuming a zero hole g-factor, the hole-in-trion spin would remain fixed while the electrons are in a spin-singlet state corresponding to zero total spin. The non-excited component of the electron spin precesses until stochastic radiative recombination of the trion occurs. The contained electron would be returned with an orientation parallel to that of the hole spin. Relative to the already precessing spin's orientation, the parallel component of the returned electron would amplify the resident electron spin polarization. Repeating this enhancement of spin orientation through our periodic pulsed excitation protocol, the electron spin becomes fully polarized (). (ii) Taking the hole spin precession into account, the exciton can recombine with either of the two electrons in the spin-singlet state, so that after periodic excitation –and even more so in an ensemble –the trion contribution becomes randomized and only the residual spin contributes. Periodically repeated excitation reduces the component that can be excited into a trion so that also with hole spin precession electron spin polarization builds up ().
For long delays an exponential decay of the signal is observed, which beyond 1 T becomes accelerated with increasing field strength. This decay arises from the variation of the resident carrier g-factors in the optically excited quantum dot ensemble, which is still sizable despite of the narrow-band excitation by ps pulses. At lower fields below 0.5 T variations of the nuclear field are also influential. The resulting spread of the precession frequency leads to the carrier spins in the ensemble running out of phase, after their initial orientation. This dephasing is typically described by the characteristic time of an exponential damping factor.
To obtain insight into the damping dynamics, we have varied the magnetic field orientation in the quantum dot plane. The resulting dephasing times as function of the field orientation determined from corresponding Faraday rotation traces are shown in Fig. 5 for both electron and hole spins. Obviously, there is no strong dependence of the electron spin dephasing time on the field direction, while for the hole spins the variation is considerable. The behavior very much reflects the variation of the carrier g-factors, as shown in Fig. 3. This supports that the spin dephasing is indeed mostly determined by g-factor variations. In particular, the dephasing time distribution of the hole spins resembles the anisotropy of the corresponding hole g-factor. Notably, the dephasing time is shortest for the directions of small g-factors, showing that the associated variations are more prominent than the effect of variations along the large g-factor direction, for which one might expect a faster dephasing due to the faster spin dynamics. However, the relative variations for large g-factors are apparently less important than for small g-factors at the applied field of 1 T.
Figure 5. Variation of the dephasing time of the electron (upper panel) and hole spins (lower panel) with the magnetic field direction in the quantum dot plane. The data were collected on n-doped quantum dots, where one observes a mode-locked signal at negative delays. The hole spin dephasing time is taken from the positive delay signal, so that it corresponds to the signal from the optically excited negatively charged trion. This seems surprising at first hand, as the lifetime of the hole spins is limited by the radiative decay time of the trion, for which we measured about 0.5 ns. However, the spin signal decay time is found to be much shorter with values in the 100 ps range, so that this time is not given by the radiative decay, but rather by the g-factor variations. T.
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Figure 6. Panel (a): Ellipticity traces of n-doped quantum dots recorded as function of the delay time between the pump hitting the sample at time zero and the probe, recorded for different pump powers as indicated at each trace. The pump pulse separation was increased by a pulse picker to ns. T. Panel (b): Amplitude of the ellipticity signal by resident electron spins determined from the traces in the top panel by exponentially damped fits with a harmonic function versus pump power (bottom axis) and pulse area (top axis). For the fits delays after 1 ns were considered, where the excited exciton complexes are largely decayed. The line is a fit to the data assuming a Rabi-oscillation behavior (see text).
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Figure 6a shows ellipticity traces recorded for different pump powers. The signal amplitude rises starting from low excitation powers until at a certain power level a maximum is reached, after which a drop of the signal strength is observed. Panel (b) gives the ellipticity amplitude, determined from exponentially damped fits to the data with harmonic functions, versus pump power. The data are indicative for a strongly damped Rabi oscillation-like behavior, where a maximum is reached but beyond the corresponding power no distinct further oscillations with minima and maxima can be observed. The strong damping is expected from the variation of the transition matrix elements in the inhomogeneous optically excited dot ensemble. These inhomogeneities translate directly into variations of the pulse area for Rabi oscillations.
In case of spin orientation of the holes the argument is quite similar (). The superposition state generated by optical excitation is here given by
where now the spin-up hole remains unexcited for -polarized light. Thereby this component gets pumped by the repeated laser pulse application and the subsequent decay of the positively charged exciton component. Corresponding ellipticity traces are shown in Fig. 7a. The hole spin precession is recognizable from the low-frequency oscillation, which is, however, at positive delays, superimposed by exciton and electron spin precession signals. A pure hole spin precession signal is observed at negative delays, whose origin is the spin mode locking, to be described in the next section, showing the same precession frequency as after the pump. The hole spin precession amplitude is strongly damped due to the inhomogeneity of the hole in-plane g-factor inducing a fast post-pump dephasing and rephasing only shortly before the pump pulse arrives. Therefore, only up to one spin precession period is observed on both sides of zero delay ().
Figure 7. Panel (a): Ellipticity trace recorded at T as function of the delay between pump and probe, measured on quantum dots with resident holes. The pulse separation is ns. The low-frequency oscillation from hole spin precession is superimposed by strong oscillations, reflecting the electron-in-trion spin precession in charged dots as well as the exciton spin precession in non-charged dots within the ensemble. The hole is the only component contributing to the mode-locked signal at negative delays before the pump. The inset shows a series of ellipticity traces in the mode-locking delay range recorded for varying pump powers. Panel (b): Amplitude of the mode-locked hole spin signal at negative delays as function of the applied laser pump power (bottom axis), determined from the data in the inset of the upper panel. The rising part of this dependence is fitted by a theoretical formula for the strength of the mode-locking signal, from which the conversion of pulse power into pulse area at the top axis is obtained.
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The amplitude of the mode-locked hole spin precession signal scales similarly with power as the positive delay signal. This amplitude as function of pump power (bottom axis) is shown in Fig. 7b. Also, here a strongly damped Rabi-oscillation behavior is observed, where after reaching maximum spin polarization a decay towards a constant level is obtained. For fitting the data we have used Eq. (32) of Ref. (), derived by considering the specifics of the mode-locked signal formation. This analysis shows that maximum amplitude does not occur for pulse area as for the positive delay signal, but for an area of about .
3.2 Spin polarization by pump pulse doublets
Theoretical calculations show that the repeated periodic excitation of singly charged quantum dots, as described before, leads to fast generation of spin polarization close to unity, as long as the pulse separation is well below the spin relaxation times. Using the most efficient pulses with area , more than 99% polarization can be reached after application of a dozen pulses (). With the used lasers this corresponds to a time of slightly more than 150 ns, about an order of magnitude below the transverse spin coherence time ; see below. Using lasers with shorter resonator length, the pulse repetition rate can be easily increased into the GHz range, so that spin orientation times of 10 ns should be feasible.
Once polarization has been maximized by repeated excitation we would in the ideal case not expect further enhancement of the spin polarization by application of additional pulses. However, the results presented so far have shown that in an ensemble variations of the dipole transition matrix element are important. Furthermore, after spin orientation variations of the precession frequency due to g-factor dispersion lead to signal dephasing. Also, the variation of the laser power in the Gaussian-shaped laser spot leads to variations of the pump pulse area for dots with different locations in the spot area. These power variations could only be avoided if pulses with flat-top spatial profiles were used. They could be formed by spatial light modulators, for example.
Therefore, we have tested the impact of an additional laser pulse applied at a time when on one hand exciton complex decay has largely occurred, while on the other hand the effects of dephasing are not too destructive for the signal formed by the resident carriers. We concentrated on the n-doped quantum dot sample and chose delays of about 0.6–0.7 ns. For the first pump we selected a pulse area of . For the second pulse, also with area , we chose the same circular polarization and applied it in one case at a maximum of the ellipticity signal, so that the spin polarization is maximum, from which we would ideally expect no further amplification of spin coherence: at the moment when pump 2 comes in, the resident electron spins point upwards, so that absorption of the -polarized pump 2 should be Pauli-blocked. In the other case we applied the second pump in a minimum of the ellipticity signal, so that in the ideal case the second pulse should annihilate the coherence generated by pump 1. Pauli blocking is not effective in this case as the electron spins are pointing downwards, so that a pulse should be able to convert the Bloch vector completely into a trion.
The corresponding results are shown in Fig. 8. Panel (a) shows the case when pump 2 comes in at maximum ellipticity signal generated by pump 1. The two top traces show the effect of pump 1 and pump 2 alone. The comparison shows that the two spin polarizations induced by one of the pulses without action of the other one are very similar, as expected from their identical parameters. The lower trace shows the signal when both pumps are acting, in comparison to the calculated sum of the two upper signals (labeled as sum signal). The latter would be expected if the two pumps would address independent spin ensembles. The recorded signal is smaller, as expected because the pumps address nominally the same spins. On the other hand, the signal is stronger than if only one pump is used. In particular, one sees that the second pump injects exciton complexes whose signatures have disappeared only about 1 ns after its incidence. If one compares the signals at about 2 ns delay and later, still a net signal from newly generated spin coherence remains. This shows that the mentioned inhomogeneities cannot be neglected. For example, due to dephasing a fraction of quantum dots can be excited again, and in these dots the spin polarization becomes re-established. Furthermore, dots located in the periphery of the first pump spot are exposed to an excitation density below , so that the second pump is expected to enhance their polarization.
Figure 8. Effect of the application of pump pulse doublets for spin orientation in comparison to single-pump application. Panel (a) shows the case when the second pump comes at a maximum of the ellipticity signal (black trace at bottom) compared to the signals induced if only one of the two pumps acts (the top traces, dark and light blue). For comparison, the sum of the latter two signals is also shown superimposed on the signal with the two pumps on (red trace). Both pumps have polarization. Panel (b) shows the ellipticity signal when the hit point of pump 2 was changed from maximum to minimum signal generated by pump 1. The other parameters were left unchanged.
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In Fig. 8b the corresponding data are shown for the case when pump 2 comes in at an ellipticity minimum. Here the joint action of the two pumps (bottom trace) clearly leads to a weaker signal, whose strength is even below the strength of the signal that is obtained by adding the signals induced by pump 1 or pump 2 alone (the top traces), being in antiphase relative to each other. This latter fact highlights the impact of dephasing on the signals, as without dephasing the two signals should ideally exactly compensate each other.