Engineering the spin polarization of one-dimensional electrons

There is a possibility that the observed splitting of odd-numbered focusing peaks is due to angular spreading of the injected electron (depending on the details of the wavefunction within the QPC) or disorder induced current branching [25]. The electron paths diverging due to these scenarios are not spin related and are maximized if the electrons travel along a quarter circular cyclotron orbit. To rule out such possibilities in our case, we provide further information as below.

First, the experiment was repeated with a linear focusing device fabricated from the same wafer as shown in inset of figure 1(b). With a linear geometry two electron paths (due to different injection angle) that are diverging from the injector QPC come back together after travelling in a half circular cyclotron orbit, thus the effect of angular spreading of the injected electron is minimized. It is seen that the splitting of the first peak and lack of splitting of the second peak in a linear focusing device geometry is similar to that observed with the orthogonal focusing device geometry (figure 1). A change of the focusing periodicity compared to the orthogonal focusing geometry is due to the difference in the lithographically defined separation between the injector and detector.

Second, we noticed that the first focusing peak shows a pronounced splitting when the injector conductance is smaller than 2G₀ ($G_0 = \frac{2e^2}{h}$), however, such splitting is absent at large injector conductance value (e.g. 3G₀ and 4G₀) and only a single peak is observed with the orthogonal focusing geometry as shown in figure 3(a). It is also important to emphasize that the single peak at large injector conductance value aligns with the dip between the two sub-peaks rather than one of the sub-peaks. Assuming the peak splitting with small injector conductance is due to angular spreading of the injected electron or disorder induced current branching, then with even larger angular spreading at larger injector conductance the splitting should persist (for instance, in [25] the peak splitting is observable even at 6G₀). However, this expectation contradicts the experimental result and indicates the observed peak splitting is unlikely to arise from electron branching, similarly the lack of an effect due to change of disorder.

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Additionally, the intensity of sub-peaks does not necessarily become symmetric at the conductance plateaus if the peak splitting arises from the disorder-induced electron branching as shown in [25]. In our experiment, the peak becomes symmetric at conductance plateaus in all the cases as shown in figures 2 and 3.

Similarly, the smearing out of the peak splitting at higher injector conductance also excludes the possibility that the observed peak splitting is associated with the form of wavefuction. Assuming the peak splitting correlates with the wavefunction, then the increased number of nodes of wavefunction at higher injector conductance may be expected to create more sub-peaks, which clearly disagrees with the experimental result.

Third, an experiment with the quasi-1D channel shifted laterally via an asymmetric gate biasing of the injector and detector was performed. Figures 3(b) and (c) show the result with the quasi-1D channel of the injector shifted laterally by applying an offset of  −0.2 and  −0.4 V, respectively. The asymmetrically biased injector and the symmetrically biased detector were set to a conductance value of G₀. The peak splitting is not affected by the lateral shift indicating the observation is a disorder-free effect. However, it is noticed that both the sub-peaks shift toward higher magnetic field due to a reduction in the effective separation L between the injector and the detector. According to equation (2), a change in L also affects the peak splitting [8]. This observation also highlights one of the limitations of linear focusing geometry where the injector and detector share the central gate. By increasing injector conductance (usually the detector conductance is fixed as is the gate voltage applied to the central gate), it inevitably introduces a lateral shift in the quasi-1D channel. Therefore, a change in peak position and splitting of peaks should be observed in the linear focusing device as well due to the unintentional lateral shift of the quasi-1D channel.

Four, the satellite peaks which differ from the focusing peaks in figure 1 are due to scattering centres which disappear after illuminating the device with a red LED at base temperature, however, the split in odd-numbered focusing peaks remains almost unaffected by the illumination, as shown in figure 3(d), which shows that the observed effect is disorder-free.

To confirm the split peaks in our experiment are spin related, we performed focusing measurements in a tilted magnetic field. The experiment was performed in a ³He cryostat with rotatable sample holder at a base temperature of 300 mK. When the angle Θ between the 2DEG plane and magnetic field equals 90°, the field is normal to the 2DEG and when Θ decreases towards 0 an in-plane field component is introduced.

Results shown in figure 4(a) correspond to first (left panel) and second focusing peaks (right panel) at different Θ, respectively. The injector conductance is set at 0.8G₀ at zero in-plane magnetic field. The two sub-peaks of the first focusing peak are pronounced in the whole range studied here with the peak splitting increasing as Θ decreases. It may be noted that as the in-plane component of magnetic field was gradually increased, the split in the first focusing peak was found to increase. Interestingly, the right sub-peak of the first focusing peak shows a disappearance-reappearance behaviour against increasing in-plane field and this behaviour is similar to result in hole gas [13]. The graph on the right panel of figure 4(a) shows the effect of in-plane magnetic field on the second focusing peak which splits into two when Θ is below 27° (corresponding to a total magnetic field around 0.5 T). It is important to note that the summation of heights of two sub-peaks of the second focusing peak is almost equal to its united counterpart.

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Figure 4(b) shows a plot of splitting of the first and second peaks $\Delta B_{\perp}$ against in-plane magnetic field $B_{\parallel}$. It is noted that the splitting of the first peak experiences a sharp increase from 6.5 mT to around 11.5 mT when the in-plane field is increased up to 200 mT and then gradually rises to 16.3 mT in large in-plane field regime. The splitting of the second focusing peak follows a seemingly linear trend from 10–34 mT. The in-plane magnetic field dependence confirms that our observation is spin-related. The non-linear in-plane field dependence of peak splitting (or spin splitting) is an indication of competition between SOI and the Zeeman splitting [6, 26], and the theory [6] further suggested that the splitting should vary rapidly in the small field regime (where the SOI dominates) and slowly in the large field regime (where the Zeeman splitting dominates), which agrees well with our experimental result. Moreover, the anisotropy of Fermi surface (altering from a sphere to an ellipse in momentum space) in the presence of a large in-plane magnetic field [27, 28] may also contribute to the non-linear behaviour.

We can estimate the height of the two sub-peaks by reconstructing the first focusing peak with two Lorentzian peaks as shown in figure 5(a). The asymmetry between the sub-peaks shows that the spin polarization drops to zero at the conductance plateaus and remains finite elsewhere as shown in figure 5(b) which agrees with theoretical prediction [21, 22] and is expected when the observed sub-peaks are spin related.

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The splitting of odd-numbered peaks and united single even-numbered peaks [19, 20] together with the non-linear in-plane field dependence of the peak splitting [6] are indications of the SOI as shown in figure 6. The Fermi surface splits into two in the presence of SOI for spin-up and spin-down states, respectively. For the first focusing peak, the injected electrons follow two Fermi surfaces according to their spin orientation, as a consequence the cyclotron radii are different for the two spin states and two focusing peaks are expected. However, the situation changes dramatically for the second focusing peak where a reflection at the boundary is involved. If the incident angle of such reflection is  ∼0°, which is crucial for the observation of a single second peak, the momentum k changes to  −k while the spin orientation remains preserved [19, 20]. The swap of momentum together with the preservation of spin orientation will result in hopping between inner and outer Fermi surfaces as shown in figure 6(b). The spin-down electrons will initially occupy the inner Fermi surface so that they have smaller cyclotron radius. After the reflection they take the outer Fermi surface which corresponds to a larger cyclotron radius. On the other hand, the spin-up electrons will hop from outer to inner Fermi surface after the reflection. The hopping between the two Fermi surfaces eventually results in the re-union of the two spin states at the detector. If the splitting is due to linear Rashba SOI, the peak splitting $\Delta B$ relates to the spin–orbit strength as below [19, 20],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq2} \Delta B = \frac{2\sqrt{2}m^\ast \alpha}{\hbar eL} \nonumber \end{align} \tag{ 2 }$$

where $m^\ast$ is the electron effective mass, α is the Rashba coefficient and it estimated to be $5 \times 10^{-13}$ eVm in our devices. The estimated Rashba coefficient is considerably larger than the previous report in a GaAs electron gas [10] (and references within [10]). On the other hand, if the splitting is due to k³ SOI [6], the estimated peak splitting  ∼23 μeV (with $B_{\perp} \sim 60$ mT) agrees well with theoretical prediction [6]  ∼20 μeV (with $B_{\perp} \sim 50$ mT; see figure 1 of [6]). The difference in values might be due to the smaller electron concentration considered in [6]. Since all the devices were fabricated from the same wafer, therefore, in principle all the devices should exhibit the similar SOI (peak splitting). However, the experimental result reveals that the focusing spectrum depends on the shape of the injector.

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The shape of the injector primarily affects the spin polarization which is manifested as a change in the shape of the 0.7 conductance anomaly [21, 22, 29] as shown in figure 7. Conductance plateaus are well defined for 90°, 60° and 30° QPC. A pronounced additional feature occurs around 0.5G₀ in the 90° QPC, a rather faint 0.7 structure is observed in the 60° QPC, and no feature other than the integer conductance plateaus is present in the 30° QPC. It is predicted [29] that in a longer QPC the interaction effect introduce a larger intrinsic spin polarization, so that the 0.7 conductance anomaly tends to appear at 0.5G₀. On the other hand, spin polarization generated by a short QPC is negligible and the 0.7 anomaly smears out. In the studied devices, the lithographic defined channel length remains the same, tuning the angle of QPCs dramatically affects the effective length where electrons can strongly interact, which leads to a change in the shape of the 0.7 anomaly.

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It appears that the extent to which the 1D electrons are polarized alters the magnitude of the 2D SOI. A detectable SOI (observable via the peak splitting) is triggered with injected partially spin polarized electrons while a small SOI (lack of peak splitting) is obtained with unpolarized injection, indicating the possibility of modulating the SOI in the 2DEG via injection of the different spin species. A recent theoretical work suggested the connection between the 1D wire and 2D reservior can affect the experession of the SOI [30]. It is an open question as how this modulation occurs in a heterostructure, though we speculate the dynamic nuclear polarization which is directly proportional to spin polarization of the injected electrons [31] might be a possible mechanism.