Enhanced spin coherence via mesoscopic confinement during acoustically induced transport

Long coherence lifetimes of electron spins transported using moving potential dots are shown to result from the mesoscopic confinement of the spin vector. The confinement condition to control electron spin dephasing is governed by the relation between the characteristic spin–orbit length of the electron spins and the dimensions of the dot potential, which governs the electron spin coherence lifetime. The spin–orbit length is a sample-dependent parameter determined by the Dresselhaus contribution to the spin–orbit coupling and can be predictably controlled by varying the sample geometry. We further show that the coherence lifetime of the electron spins is independent of the local carrier densities within each potential dot, which suggests the possibility of coherent, long-range transport of single electron spins.

We examine this effect through experiments in which λ SO , which is primarily determined by Dresselhaus spin-orbit effects, is varied by changing the thickness of the GaAs QWs and, in particular, show that the experimental results reflect the l s ∝ (λ SO ) 2 dependence associated with DP spin dephasing. As a result, motional narrowing effects do not depend on carrier densities, and the ability to control spin coherence during transport down to the single spin level is possible in the SAW system.

Experiment
The DQDs are produced by the interference of two SAW beams propagating along the 110 surface directions of a GaAs QW sample [16,17]. Three single-QW samples with Al 0.3 Ga 0.7 As barriers were grown by molecular-beam epitaxy on GaAs (001) semi-insulating substrates. Two were designed with thicknesses of 12 and 20 nm and placed 390 nm below the surface, whereas the third, 30 nm-thick QW was placed 175 nm below the surface. The SAWs are excited by applying a radio-frequency signal to two aluminium split-finger interdigitated transducers deposited on the sample surface using standard optical lithography protocols, and each beam has a linear power density between 2 and 7 W m −1 . The SAWs have a wavelength λ SAW of 5.6 µm, corresponding to a frequency SAW /2π of 519 MHz at a sample temperature of 12 K and propagate with a well-defined phase velocity of v SAW = 2907 m s −1 . The type-II piezoelectric potential generated by the interference of the two plane waves confines and transports the photogenerated electrons and holes within a 120 × 120 µm 2 array of DQDs, with the diameter of each dot L DQD being approximately 1 µm. The DQD array propagates along a 100 -direction with a velocity v DQD = √ 2v SAW and has a periodicity λ DQD = √ 2λ SAW . The measurements were performed at sample temperatures of either 4.2 or 12 K. As has been previously reported, the electron spin coherence is insensitive to temperatures in this range [18].
The coherent spin transport was monitored by microscopic photoluminescence (PL) measurements (cf figure 1). A circularly polarized, 768 nm laser beam was focused onto the sample to photogenerate spin-polarized electrons and holes at a position G. The carrier densities n are estimated by n = 2π P ph e −αd QW /(E ph DQD ), where P ph is the incident light power, E ph is the photon energy, α is the absorption coefficient in the GaAs QW, d QW is the QW width, and DQD = SAW is the DQD frequency. After excitation, the carriers are spatially separated by the piezoelectric potential onto different phases of the DQD lattice and transported along well defined channels. This efficient charge separation by the acoustically induced potential strongly suppresses spin exchange scattering via the Bir-Aronov-Pikus mechanism during transport [19,20]. While some recombination occurs due to electronic traps in the DQD channel, most luminescence is observed near the edge of a semi-transparent metal strip M that partially screens the piezoelectric potential of the DQDs and allows the electrons and holes to recombine. The degree of circular polarization ρ z = (I R − I L )/(I R + I L ) of the luminescence near M can then be measured, where I R and I L are the right-and left-circular components of the PL, respectively. The dependence of ρ z on the transport distance is mapped by varying the separation between G and M. Because of the rapid scattering of hole spins in GaAs [21], ρ z correlates well with the net electron spin population. Figure 2 shows the spatial dependence of ρ z for three different electron densities ranging across an order of magnitude (from 15 to 140 electrons per DQD), which correspond to volume (area) concentrations of approximately 10 14 -10 15 cm −3 (10 8 -10 9 cm −2 ). The measured values of polarization were fitted with a function of the form ρ z (x) = ρ 0 e −x/l s cos( D L (k DQD )x/v DQD ), where ρ 0 represents the initial spin polarization at G, and l s is the spin coherence length. The oscillations in ρ z result from the precession of the electron spins around B int (k DQD ) with a frequency D L during transport. The coherent precession observed here occurs in the absence of an external magnetic field and is, for the present sample, primarily related to the B int (k DQD ) associated with the spin-orbit contribution due to the lack of bulk inversion symmetry in the zinc-blende crystal (Dresselhaus term) [22,23]. Consequently, the Larmor frequency of the electron spin precession can be described by

Experimental results and discussion
where γ is the spin-orbit parameter, k DQD = m * v DQD /h is the average momentum of the electrons within the DQDs, m * is the electron effective mass, k z describes the momentum due to the QW confinement, and d eff is the effective QW thickness including the penetration d 0 of the electron wavefunction into the Al 0.3 Ga 0.7 As barrier layer [24]. The latter was calculated using a tight-binding approach yielding a value of d 0 = 2.1 nm for each barrier. There is also a contribution to D L (k DQD ) from the Bychkov-Rashba (BR) term [25] related to a structural inversion asymmetry induced, for example, by the vertical component of the piezoelectric field, but both are small for the present experimental conditions and will be neglected [23].
For the electron densities presented in figure 2, the spin coherence lengths l s are comparable and greater than 100 µm. Likewise, the coherence times T * 2 = l s /v DQD of the electron spin microensemble within each DQD remain essentially unchanged. This is in stark contrast to lifetime measurements on unconfined systems, such as bulk GaAs [3,26] and GaAs QWs [5,6], where the spin lifetime has been shown to be strongly carrier-dependent. The long spin lifetimes observed during transport by DQDs cannot, therefore, be attributed to motional narrowing resulting from the mechanisms discussed in previous reports [3,6,26,27]. Instead, we attribute motional narrowing effects to the DQD piezoelectric confinement of the electron spins. We argue that the confinement is effective because the size of the DQD L DQD is sufficiently small to prevent large precession angles of individual spins during random thermal motion within the DQDs. The effect of confinement on quantum coherence has been previously studied experimentally [8] as well as theoretically [28], in addition to the discussion of weak localization of electrons in a stationary quantum dot with dimensions smaller than the spin-orbit length λ SO [7,9].
In the context of the enhanced, long-range transport of quantum states presented here, it is thus anticipated that λ SO has a larger spatial extent than the approximately 1 µm diameter of the DQDs (L DQD ). As mentioned above, the spin-orbit length λ SO can be intuitively characterized by the distance it takes a spin to precess 1 radian around B int (k) [2,10]. Concerning the 6 contribution to λ SO due to the Dresselhaus spin-orbit interaction, the temperatures and carrier densities for a QW system relevant to the experiment conditions allow the linear term in k to dominate over the cubic term [24]. As a result, the Larmor precession frequency associated with the random motion D L (k F ) is obtained from equation (1) by replacing of k DQD by the Fermi wavevector of the electrons k F . As discussed above, the spin-orbit contribution to the Larmor precession from the BR-term and from the induced strain are small compared with the Dresselhaus contribution and will be neglected. Consequently, this approximation results in an isotropic λ SO given by Interestingly, the Fermi wavevector k F does not appear in equation (2) demonstrating that λ SO is independent of the electron momentum k in this linear approximation. Therefore, we can experimentally extract λ SO = v DQD / D L (k DQD ) directly from the measured precession frequency of the spins.
The Larmor precession frequency of the oscillations shown in figure 2 are quite uniform with a frequency D L = 0.97 ns −1 . This is similar to the value (1.1 ns −1 ) that we have previously published for comparable DQD acoustic power densities [4]. The slight difference is accounted for by dissimilarities in the mounting of the sample in the cryostat that may have introduced a slightly different static strain of the sample during cooling to 12 K [29,30]. Using the value of D L = 0.97 ns −1 , we obtain a spin-orbit length λ SO = 4.2 µm for the 20 nm QW sample, which is expectedly larger than the DQD confinement dimensions L DQD of approximately 1 µm. As a result, the mesoscopic DQD confinement potential does indeed provide the motional narrowing required to maintain the spin coherence of the microensemble within the DQD.
The preceding demonstration of the mesoscopic confinement of the electrons spins now allows us to further explore the relationship between the spin-orbit length, the confinement dimensions and the coherence length. According to equation (1), the Larmor precession frequency of the electron spins, and hence the spin-orbit length, can be varied by changing the thickness of the QW. As discussed by Chang et al, changing the relationship between λ SO with respect to the size of the dot will have dramatic effects on spin coherence that are equivalent to changing the size of the confinement potential [11]. To exploit the λ SO dependence, we have performed spin transport measurements on samples with different QW thicknesses. Figure 3 compares ρ z for the previously discussed 20 nm QW sample with similar samples consisting of single QWs of thicknesses 12 and 30 nm; important parameters from this figure are summarized in table 1. The thinner 12 nm QW shows a dramatic increase in the Larmor precession frequency that is in good agreement with the value expected using equation (1). In fact, using the measured values of D L (k DQD ) from the 12 and 20 nm QW samples along with the well-defined DQD wavevector k DQD , the spin-orbit parameter γ is calculated to be 17 and 16 eV Å 3 , respectively, using equation (1). These are in agreement with our previously determined value of 17 ± 2 eVÅ 3 [23].
In our approximation, electron spins at the Fermi surface will experience the same increase in D L (k F ) as D L (k DQD ) when the QW thickness is reduced. As shown in table 1, this will result in an inversely proportional modification of the spin-orbit length λ SO . As λ SO ( D L ) is reduced in the 12 nm-thick QW sample, it becomes similar to the spatial dimensions of the DQDs L DQD . As a result, the effective confinement of the spins is therefore less than that for the 20 nm QW sample leading to shorter coherence lengths l s , which is approximately proportional to the square of the   (1) and a value of γ = 17 eVÅ 3 [23]. λ SO was determined using the measured values of D L . The coherence lengths l s correspond to the fitted curves in figure 3, and the values in brackets compare the 110 µm coherence length from the 20 nm QW adjusted by the change in the spin-orbit length (λ SO ) 2 . Similarly, the final column presents the values (λ SO ) 2 /l s highlighting the relationship between the spin-orbit length and the coherence length. spin-orbit length (λ SO ) 2 and is indicative of the DP dephasing becoming dominant in the spin coherence. Monolayer fluctuations in the QW thickness d eff will create additional pathways for spin dephasing, but the 1-2% variations in D L would not be sufficient to account for the dramatic changes observed in the spin coherence length. 8 For the 30 nm-thick QW sample, the measured Larmor precession frequency is larger than that expected using equation (1). This is attributed to the increasing importance of the strain components to B int (k) considering the smaller Dresselhaus term for this QW thickness (cf equation (1)) and that the QW is nearer to the surface than in the other samples-the specifics of which will be discussed in detail in a later publication. Using the experimentally determined D L , λ SO is nevertheless determined to be 5.6 µm. This larger spin-orbit length is expected to increase the spin coherence length to 194 µm (given the (λ SO ) 2 proportionality), and the measured l s = 200 ± 115 µm is consistent with this expectation. The larger error in this measurement is attributed to the fact that the measured transport range is only a small fraction of the long coherence length. However, the work does indicate that increasing the confinement during spin transport in a DQD (via a larger λ SO /L DQD ratio) will enable longer coherence lengths.
Intuitively, the enhanced electron spin lifetimes result from the ability of the mesoscopic confinement potential to rapidly scatter the electron momentum and prevent a spin from undergoing the large precession angles during its mean free path that cause DP dephasing. Our sample set suggests that the spin coherence length follows a quadratic dependence with respect to the spin-orbit length. The general relation used to describe DP spin dephasing is [1,2] Equation (3) reflects our observed quadratic dependence in λ SO ( D L ) (cf table 1) as well as the origin of the long spin coherence times: rapid momentum scattering τ p due to the constant DQD confinement potential.
The measured thickness dependence of τ s for electrons confined by DQDs is, however, quite different than that expected for free electrons in a undoped GaAs QWs. In the absence of lateral confinement, the spin dephasing will have a similar [ D L (k F )] −2 term associated with the vertical confinement. The momentum scattering term τ p , on the other hand, is not dictated by scattering from the lateral confinement potential imposed by the DQDs, but rather by the carrier mobility. In particular, the electron mobility in GaAs QWs has been shown to vary as (d QW ) n , with n ∼ 6, because of interface roughness scattering [31,32], thus leading to τ p ∼ µ ∼ (d QW ) n . Due to the strong dependence of τ p on QW width, the spin relaxation time is expected to decrease with increasing d QW , in contrast with the experimental results for spin transport via DQDs. This demonstrates, in conjunction with the very long coherence times observed in figure 2 that the spin-orbit effects on coherence cannot be attributed to simple QW effects but rather the importance of the confinement potential created by the DQDs.

Conclusions
In conclusion, we have shown that the precession frequency, the spin-orbit length and the spin coherence time can be controlled by the QW width. More importantly, we have demonstrated that the enhanced coherence of electron spins results from the mesoscopic confinement of the DQDs during transport, which does indeed parallel the behaviour observed in stationary quantum dots. Considering this work along with previous results showing single electron transport, mobile potentials generated by acoustic fields are thus anticipated to be a powerful and unique tool in the transport and manipulation of single quantum states in quantum information processing as well as spintronic applications.