Spin transport and spin manipulation in GaAs (110) and (111) quantum wells

Authors


Abstract

Spin dephasing via the spin–orbit interaction is a major mechanism limiting the electron spin lifetime in zincblende III–V quantum wells (QWs). QWs grown along the non-conventional crystallographic directions [111] and [110] offer new interesting perspectives for the control of spin–orbit (SO) related spin dephasing mechanisms due to the special symmetries of the SO fields in these structures. In this contribution, we show that the combination of such special symmetries with the transport of carriers by the type II modulation accompanying a surface acoustic wave allows the transport of spin polarized carriers over distances of tens of math formulam in GaAs(110) QWs. In the case of GaAs(111), the Rashba contribution to the SO field generated by an electric field perpendicular to the QW plane is used to compensate the Dresselhaus contribution at low temperatures, leading to spin lifetimes of up to 100 ns. The compensation mechanism is less effective at high temperatures due to nonlinear terms of the Dresselhaus contribution. Perspectives to overcome this limitation via the combination of (111) structures with the transport of spins by surface acoustic waves are discussed.

1 Introduction

The manipulation of spins in semiconductor materials has become an active area of investigation, in particular after different proposals for spin-based electronic information processing have been put forward [1-3]. Device functionalities based on electron spins require processes for the generation, storage, detection, and transport of spins as well as interaction mechanisms to efficiently manipulate the spin vector. Advanced applications further presuppose the interaction between two spins in order to realize spin–spin gates. In this respect, two main challenges are (i) the enhancement of the electron spin coherence times in intrinsic III–V bulk semiconductors, which is typically of only one ns, thus restricting the number of spin manipulation steps that can be realized before decoherence effects set in, and (ii) the development of spin manipulation techniques that do not compromise the spin lifetime.

While isolated electron spins can be efficiently manipulated using a magnetic field, electron spins in a crystal interact with other spins from electrons (the exchange interaction) or nuclei (the hyperfine interaction) as well as with the lattice potential. These interactions can be used for the realization of spin control gates, but at the same time they may also lead to spin relaxation if not appropriately controlled. The electron–hole exchange interaction leads to the so-called Bir–Aronov–Pikus (BAP) spin dephasing mechanism ([4]), which is particularly important for excitons ([5]) as well as in highly p-type doped III–V semiconductors at low temperatures ([6]). Spin scattering via the hyperfine-interaction is normally negligible for free carriers, but becomes important for localized electrons in quantum dots.

In this contribution, we investigate spin transport and manipulation in III–V semiconductor quantum well (QW) structures grown along non-conventional crystallographic directions, such as the [110] and [111] directions. The motivation for these studies arises from the special characteristics of the interaction of spins with the lattice potential in these structures, the well-known spin–orbit (SO) interaction. The SO-interaction arises from the fact that an electron experiences a varying electric field while moving through a noncentrosymmetric crystal (such as the zinc-blende semiconductors). The latter translates into a momentum (math formula) dependent effective magnetic field math formula in the electron reference frame, which acts on its spin. The SO-coupling can be controlled by external electric [1, 7] and strain fields ([8]). As a result, it provides an interesting mechanism for spin generation and detection (i.e., optical orientation ([9])) as well as for spin manipulation without the application of magnetic fields. Examples are the generation of non-equilibrium spin populations using electric currents [10-14], as well as the spin-galvanic [15, 16] and spin Hall effects [13, 17, 18].

The SO-interaction can also lead to spin dephasing in an electron ensemble, since electrons with different math formulas will experience SO-fields of different strengths and directions. Their spins will then precess at different rates, leading to a reduction of the resulting spin ensemble, an effect known as the Dyakonov–Perel (DP) spin dephasing mechanism [19-21]. The SO-interaction can also couple different spin states during electron scattering processes. One example is the Elliott-Yafet spin dephasing mechanism ([22]), which describes spin-flip transitions induced during electron scattering by impurities or phonons and is expected to be influential in highly doped, low-bandgap materials. Finally, the SO-coupling is also behind the intersubband spin relaxation mechanism ([23]) in QW structures, where electron scattering between two subbands is accompanied by a spin flip.

Two main approaches have been proposed to control the relaxation and to manipulate electron spins in semiconductors. The first, which will be discussed in detail in Section 'Symmetry effects on the spin–orbit interaction', exploits the dependence of the SO-interaction on the symmetry of QW structures. The second relies on the use of confinement potentials of quantum wires and dots (QDs). Confinement isolates spins from the semiconductor matrix, thus ensuring the long spin coherence times required for the coherent manipulation by external fields [24, 25]. Confinement can also be efficiently employed to control SO-effects via motional narrowing. The latter relies on the fact that the DP spin scattering rate is inversely proportional to the momentum scattering rate ([19]): frequent momentum scattering at the borders of the confinement potential randomizes the effective SO-field math formula experienced by the spins, thus leading to longer lifetimes [26, 27]. The required confinement dimensions are dictated by the SO length math formula, which is defined as the typical distance required for a one rad spin precession under math formula. math formula can reach a few math formulam in wide GaAs QWs, thus making it possible to employ mesoscopic confinement potentials for spin control. Finally, no DP relaxation is expected in one-dimensional systems (1D) since the axis of the fictive SO-field math formula is fixed and its direction reversible with k [28, 29].

The highest degree of spin control and manipulation has so far been achieved in QDs defined either by Stranski–Krastanov growth or by metal gates ([25]). In these systems, single electron spins in QDs have been manipulated using a variety of mechanisms including the electrical control of the exchange interaction ([30]) as well as electron spin resonance (ESR) induced by a varying magnetic field induced by a strip-line ([31]) or by a periodically varying math formula ([32]). In the last case, the effective magnetic field associated with the SO-interaction can be varied through an oscillatory spin motion induced by an electric field [33, 34]. Proposals for coherent spin control using ESR generated through carrier motion in a magnetic field gradient are also available [35, 36].

A further challenge toward spin-based information processing schemes is the realization of scalable systems consisting of several spin subsystems. These processes normally require the transport of spins, which cannot be easily combined with 3D-confinement. In this contribution, we explore the use of moving piezoelectric potentials created by surface acoustic waves (SAWs) for spin transport and manipulation. These moving fields provide a way of combining the advantage of confinement with the long range transport required to couple remote spin systems.

In this work, we review recent results on the control of the SO-interaction and acoustic transport in GaAs(110) and (111) QWs. We start in Section 'Symmetry effects on the spin–orbit interaction' with a theoretical description of SO-effects arising from the symmetry of these QWs, and how they can be controlled using spatial confinement or external electric and strain fields. In (110) QWs, the lifetime enhancement associated with the crystallographic symmetry is restricted to spins aligned with the growth direction [23, 37]. Higher lifetimes for other orientations can be achieved by lateral confinement. (111) QWs are particularly interesting because SO-effects can be suppressed for all spin orientations via the application of a vertical (i.e., perpendicular to the QW plane) electric field [38, 39]. Experimental exploitation of the non-conventional QW orientations requires the development of epitaxial growth procedures to create (Al,Ga)As structures with quality similar to the conventional (001) ones. Recent results about the growth via MBE of high quality QWs as well as QWs embedded in microcavities are presented in Section 'Epitaxial growth of GaAs(110) and GaAs(111)B quantum wells'.

The experimental studies of the spin dynamics in the QWs were carried out by combining spectroscopic techniques with spin transport by SAWs. The use of SAWs for the transport of carriers and spins in (Al,Ga)As(001) structures is reviewed in Section 'Acoustic transport of carriers and spins'. The application of SAWs for the transport of spins in (110) QWs is discussed in Section 'Acoustic spin transport in GaAs(110) QWs'. Emphasis is placed on the combination of lateral confinement with acoustic transport in order to achieve efficient spin transport at temperatures above 100 K. Section 'Suppression of the spin–orbit interaction in GaAs(111)B quantum wells' addresses the spin dynamics in GaAs(111) QWs. Here, we demonstrated that an electric field applied across the QW can suppress SO-effects, leading to lifetimes exceeding 100 ns. Approaches for acoustic spin transport are discussed in Section 'Conclusions and future perspectives', which also summarizes the main conclusions of this work.

Table 1. Spin precession frequencies math formula for electron spins in III–V (001), (110), and (111) QWs with in-plane wave vector math formula induced by the bulk inversion asymmetry (Dresselhaus contribution ([43]), math formula) and by structural inversion anisotropies created by the application of a vertical electric field math formula (Rashba contribution ([44]), math formula). Also listed are the SIA contributions due to the biaxial strain induced by lattice mismatch (math formula) ([41]) and to the strain field of a Rayleigh SAW (math formula). The strain-induced contribution of BIA-type ([41]), which is considered to be small, is not taken into account. math formula denotes the effective QW thickness, while math formula, math formula, and math formula are constants describing the dependence of the spin splittings due to the Dresselhaus, Rashba, and strain contributions. The parameter math formula relates the splitting between the energetic spin eigenstates in the bulk III–V material to the electron wave vector (cf. Ref. ([45])). The parameter math formula for the Rashba term follows the convention in Winkler ([46]). For the biaxial strain, the non-vanishing strain components are assumed to be math formula, math formula, and math formula. The SAW calculations assume that electrons propagate with a wave vector math formula (where math formula and math formula denote the electron mass and the SAW propagation velocity, respectively) under the strain field of a Rayleigh SAW, which have non-vanishing strain components math formula, math formula, math formula (see Section 'Acoustic transport of carriers and spins' for details)
structure(001) QW(110) QW(111) QW
  1. math formula Calculated for a biaxial strain with in-plane component math formula and out-of-plane component math formula;
  2. math formula Calculated including only the SIA-type strain contribution [41, 42];
  3. math formula The SAW propagates along the x direction (cf. Section 'Acoustic transport of carriers and spins').
x-axismath formulamath formulamath formula
y-axismath formulamath formulamath formula
z-axismath formulamath formulamath formula
    
Dresselhaus (math formula)math formulamath formulamath formula
Rashba (math formula)math formulamath formulamath formula
biaxialmath formula strainmath formula (math formula)math formulamath formulamath formula
SAWmath formula strainmath formula (math formula)math formulamath formulamath formula
    

2 Symmetry effects on the spin–orbit interaction

2.1 The Dresselhaus contribution

The SO-interaction in III–V QWs is governed by two major contributions. The first one is of intrinsic nature and associated with the bulk inversion asymmetry (BIA) of the III–V zinc-blende lattice. The effective SO-magnetic field math formula associated with this contribution, which is normally denoted as the Dresselhaus term, can be expressed in terms of a wave vector-dependent spin-splitting energy math formula. Here, math formula is the Bohr magneton and g the electron g-factor. Expressions for math formula are summarized for QWs with different growth directions in Table 1. math formula depends on the spatial extent math formula of the electronic wave function along the growth direction, as well as on the spin-splitting parameter math formula for the QW material. The expressions listed in Table 1 assume the axes listed in the first row of the table (where x and y lie in the QW plane and z is along the growth direction) and apply for small k-vectors (i.e., math formula).

Figure 1.

Orientations of the spin–orbit magnetic fields associated with the bulk inversion asymmetry [Dresselhaus contribution, math formula] and with the structural inversion asymmetry [Rashba contribution, math formula] fields in III–V quantum wells (QWs) with (001), (110), and (111) orientations.

The first row of plots in Fig. 1 sketches the orientation of math formula for QWs grown along different crystallographic directions. math formula lies in the QW plane for (001) and (111) QWs, its amplitude and orientation depending on k. For (110) QWs, in contrast, math formula has a single component along z that only depends on math formula. As a result, DP relaxation does not affect z-oriented spins, thus leading to long spin lifetimes [23, 37, 40]. The math formula component along z leads, however, to short relaxation times for spins oriented in the xy plane, thus preventing efficient spin manipulation. The lifetime of these spins can be enhanced by using lateral confinement to force the carriers to move along a single direction. Note, in particular, that math formula only depends on the math formula component of the carrier momentum and vanishes for math formula. As a result, the SO-field vanishes for transport channels along the math formula direction. This effect will be explored for spin transport at high temperature to be described in Section 'High temperature transport in GaAs (110) microcavity structures'.

2.2 The Rashba contribution

The second important contribution to the SO-interaction arises from structural inversion asymmetries (SIA) in the QW potential introduced by an external field along z. In most cases, the SIA is generated by an electric field math formula perpendicular to the QW plane, leading to the so-called Rashba SO-contribution ([44]). Expressions for this contribution to the SO-precession frequency, math formula, in QWs of different symmetry are listed in the 3rd row of Table 1. math formula lies always in the QW plane and perpendicular to math formula. Since its strength can be electrically controlled, the Rashba effect provides a powerful approach for the dynamic manipulation of moving spins ([1]).

In (110) QWs, the Rashba effect can be used to rotate spins around axes perpendicular to z ([47]). For the other orientations, the Rashba field can also compensate the BIA contributions. In math formula QWs, math formula vanishes for spins propagating along the x-direction (math formula in cartesian coordinates) provided that ([48])

display math(1)

A similar expression, but with a negative sign, applies for math formula. Under compensation, long-living spin eigenstates in the form of a persistent spin helix exist, as recently verified by different groups [49-51].

(111) QWs are particularly interesting for the investigation of SO-effects because the Rashba and Dresselhaus contributions have the same symmetry (cf. Fig. 1). As a result, the total SO-interaction precession field math formula vanishes for all (small) k-vectors by selecting an electric field given by

display math(2)

This compensation mechanism was originally proposed in the theoretical works by Cartoixa et al. ([38]) and Vurgarfmann and Meyer ([39]). Numerical calculations have also been carried out for doped QWs ([52]). Recent experimental verification has been provided by Ballocchi et al. ([53]) and Hernández-Mínguez et al. ([54]) (see also Refs. [55-57]). These experiments will be reviewed in Section 'Suppression of the spin–orbit interaction in GaAs(111)B quantum wells'.

2.3 Strain effects

In addition to electric fields [1, 7], strain fields also affect the QW symmetry, thereby introducing a SIA contribution to the SO-coupling.1 Epitaxial layered structures are normally subjected to biaxial strain induced by the mismatch between the lattice parameters of the individual layers. Theoretical predictions for the dynamics of spins in strained QWs have been presented in Refs. [42, 60] –experimental results demonstrating spin precession induced by a strain field in (001) QWs are given in Refs. [8, 59, 61].

Expressions for biaxial strain contribution (math formula) in QWs with different orientations are summarized in the 4th row of Table 1. As in the case of the Dresselhaus and Rashba mechanisms, the expressions are only valid for small electron wave vectors math formula. In this approximation, math formula vanishes in (001) QWs. In (110) and (111) QWs, in contrast, math formula has the same symmetry as the Dresselhaus contribution ([42]). Lattice strain induced effects are expected to play a prominent role in (In,Ga)As structures grown on GaAs substrates.

2.4 Higher order terms in k

The expressions for the Dresselhaus contribution in Table 1 are valid in the low temperature regime, where the thermal expectation values of math formula. Outside this regime, one has to take into account contributions of higher order in k in the Dresselhaus term. In (110) QWs and in the absence of many-body effects, these contributions do not change the symmetry properties of the SO-interaction ([62]). We will examine here only the case of (111) QWs, where the following correction up to third order terms in k has to be added to math formula ([38]):

display math(3)

being math formula.

An additional SO-contribution arises from the fact that the GaAs QWs are normally grown on surfaces slightly tilted away from the the math formula direction (see Section 'MBE growth of (Al,In,Ga)As(110) QWs'). By calculating the Dresselhaus contribution for this new orientation, one can show that the tilting, math formula, introduces an extra SO-term given by:

display math(4)

Note that this contribution lifts the degeneracy between the x and y components of the precession frequency given by the other Dresselhaus contributions.

The lifetimes math formula for spins oriented along the axis i (math formula) can be estimated from the total precession frequency math formula according to:

display math(5)

Here math formula depends on the momentum scattering time. The thermal averages indicated by math formula can be calculated by taking into account that math formula, where math formula. However, a better approximation is obtained integrating the previous equation for math formula, where the electron energy E is assumed to follow a Boltzmann distribution.

Equation (5) yields the lifetime of spins oriented along a certain crystallographic direction. For spins precessing around an in-plane magnetic field with a short precession period compared with the spin lifetime, the average lifetime math formula becomes ([23])

display math(6)

where the magnetic field is oriented along the y direction. Since math formula, the same expression also applies for spins precessing around x.

3 Epitaxial growth of GaAs(110) and GaAs(111)B quantum wells

As was pointed out in the introduction, QWs grown in other than the most commonly used [001] direction are predicted to be advantageous for electron spin control. From the growth point of view, molecular beam epitaxy (MBE) on GaAs(110) and GaAs(111) appears very interesting and different from growth on GaAs(001). Already a comparison of the surface energies for different substrate orientations suggests that epitaxial growth on (111)B and (110) surfaces is distinctly different from epitaxy on common (001) substrates. The surface energy of the (001) surface can be estimated to be 2.9 J mmath formula and is thus much larger than the surface energies derived for the (110) and (111) surfaces, which amount to 2.1 and 1.7 J mmath formula, respectively ([63]). The surface energy can be assumed to be inversely proportional to the lifetime of Ga adatoms on the growing GaAs surface ([64]). Therefore, Ga adatoms migrating on the (001) surface are expected to be much more efficiently incorporated into the crystal than on (110) and (111) surfaces.

The previous growth model ignores the distinct surface reconstructions occurring during growth on different GaAs surfaces. Growth on GaAs(001) is usually accomplished under As-rich condition on a 2 math formula 4 reconstructed surface, where As dimers are typically arranged in parallel rows along [math formula], separated by left-out trenches in between [65, 66]. The nucleation process starts on a ridge of As dimers when a trapped Ga atom bonds to another As dimer. This nucleus grows then as a two-dimensional island by capturing further Ga atoms and As dimers, thus filling also the trenches. When this island is big enough, the uppermost As dimers split to form the stable 2 math formula 4 reconstructed surface once again ([67]). Contrary, the nucleation behavior on (110) and (111) surfaces can be described by conventional two-dimensional growing islands without the need to consider any site specificity ([67]). Thus, the widespread growth on GaAs(001) has to be considered as a well-studied but rather unique scenario, which does not necessarily apply to other substrate orientations.

3.1 MBE growth of (Al,In,Ga)As(110) QWs

The GaAs(110) surface is very unique in the sense, that it is the only stable surface of GaAs that does not show any surface reconstruction. It is further characterized by the same number of Ga and As atoms in each monolayer. The Ga-As bonds run symmetrically to the (110) plane leading to the non-polar nature of this surface. When homo-epitaxial growth is accomplished on GaAs(110) under the conditions typically used for growth on GaAs(001), strong facetting is observed leading primarily to elongated triangular features with math formula100math formula side-facets ([68]).

During the hetero-epitaxial growth required for the fabrication of QW structures on GaAs(110), effective strain relaxation by dislocations moving on slip planes is restricted to the [001] direction. In contrast, this mechanism is possible along both orthogonal math formula surface directions of hetero-epitaxial layers grown on GaAs(001) ([69]). Furthermore, the elastic properties of GaAs(110) and GaAs(001) are different, resulting in a strongly reduced critical thickness for hetero-epitaxial growth on GaAs(110) compared to growth on GaAs(001) ([70]).

Figure 2.

Nomarski micrographs of (Al,Ga)As(110) microcavity structures grown under different Asmath formula background pressures math formula: (a) math formula mbar, (b) math formula mbar, and (c) math formulambar. The scale bar and the direction markers apply for all three images.

The (110) samples used in this study were grown at a relatively low temperature of 490°C under high Asmath formula pressure in order to suppress facetting and to ensure a mirror-like surface. This is illustrated in Fig. 2, where three Nomarski micrographs of the surfaces of nominally identical (Al,Ga)As(110) microcavity structures are shown. The only varied parameter was the Asmath formula background pressure (math formula), which was increased from (a) math formulambar to (c) math formulambar. Only the sample grown under the highest math formula shows a mirror-like surface. The reduced mobility of the group-III adatoms due to the low growth temperature and the high As flux seems to promote smooth surfaces on GaAs(110) based heterostructures. Under these conditions, QWs with good optical qualities can be grown. Figure 3 shows a photoluminescence (PL) spectrum taken at math formula K on a sample containing one GaAs QW and three (In,Ga)As/(Al,Ga)As QWs. The thickness of each QW is 20 nm (the In contents x and the full width at half maximum, FWHM, of the lines are indicated in the plot). The PL line of the GaAs QW is characterized by a narrow linewidth of only 0.5 meV, which is not much broader than the best values of about 0.16 meV (cf. Fig. 6) of comparable QWs grown on GaAs(001).

To increase the efficiency of photon to electron–hole pair interconversion during optical probing of the spin dynamics, (110) single QWs were grown embedded in a microcavity structure (cf. Section 'High temperature transport in GaAs (110) microcavity structures'). In such cases, all math formula layers of thick mirror layers were grown as short-period-super-lattices (SPSLs), as it has been shown that the numerous interfaces of SPSLs hinders the propagation of misfit dislocation through the epilayers ([70]). The matching of microcavity resonance wavelength and QW emission wavelength was ensured by applying in situ continuous spectral reflectivity measurements ([71]).

Figure 3.

PL spectrum taken at math formula K of a (110) sample containing one GaAs and three Inmath formulaGamath formulaAs QWs. All QWs are 20 nm thick. The In contents x and the FWHM linewidths are indicated in the plot.

3.2 MBE growth of (Al,Ga)As(111) QWs

The GaAs(111) surface is characterized by its threefold symmetry with alternating math formula and math formula directions separated by 30°. Unfortunately, this threefold symmetry leads to the formation of pyramidal structures on the surface under usual growth conditions ([72]). However, mirror-like morphologies without any pyramidal features can be achieved when substrates with a slight off-orientation (≤ 5°) are used ([73]). Epilayers with surfaces free of pyramids can also be deposited on exactly oriented GaAs(111)B(= math formula) substrates if the V/III flux ratio and growth temperature are accurately chosen ([74]). Under these conditions, a static math formula R 23° surface reconstruction can be observed [72, 74-77]. Correspondingly, the (111)B samples used in this study were grown at a temperature of 600°C on math formula R 23° reconstructed surfaces of slightly off-oriented (1–3°) GaAs(111)B substrates to ensure mirror-like surfaces without pyramidal features.

Due to the threefold symmetry of the (111) surface, opposing directions are not equivalent, an important point to take into account when choosing the direction of the off-orientation. The enormous impact of the direction of the off-orientation on the epitaxial growth is illustrated in Fig. 4. Nomarski micrographs display the morphologies of two nominally identical (Al,Ga)As microcavity structures grown on (111)B substrates with a 3° off-orientation toward (a) math formula and (b) math formula. Whereas in the latter case a mirror-like surface is obtained (panel b), an off-orientation toward the opposing direction leads to zig-zag shaped growth steps, which evolve to extended triangular features (as depicted in panel a) when thick structures are grown.

Figure 4.

Nomarski micrographs of nominally identical (Al,Ga)As(111)B microcavity structures grown on substrates with an off-orientation of 3° toward (a) math formula and (b) math formula.

Even when the growth parameters are chosen to yield mirror-like surfaces on slightly off-oriented (111)B substrates, the next challenge is then to adjust the level of step bunching according to the desired application. Although Ren and Nishinaga ([78]) demonstrated a clear correlation of step bunching with growth temperature, the controlled tuning of step bunching level is experimentally difficult to realize. This is illustrated in Fig. 5, which displays three atomic force micrographs taken at different positions of a 2 substrate. In our growth setup, there is a slight lateral temperature gradient from the centre of the wafer (highest temperature) to the edge of the wafer (lowest temperature). Thus, for the highest temperature (panel a) we observe a rather regular step bunching with step heights of about eight monolayers (MLs) and terrace widths on the order of 70–80 nm. In the area midway between centre and edge (medium temperature, panel b) we see some long range step bunching with step heights on the order of about 30 MLs and large terrace widths of about 300 nm. Finally, in the area near to the wafer edge (coldest region, panel c) the surface is characterized by double ML steps, thus indicating that step bunching was strongly suppressed in this area. Depending on the intended application, step bunching might or might not be favorable and, correspondingly, a tight control of growth temperature can play an important role.

Figure 5.

Atomic Force Micrographs of the surface of an (Al,Ga)As heterostructure grown at a nominal temperature of T = 600°C on a GaAs(111)B substrate with an off-orientation of 2° toward math formula. The images were taken on the same 2 wafer at different positions: (a) at the center, (b) midway between center and edge, and (c) near the edge of the wafer. The scalebar and the direction markers are valid for all three images.

Although there is a good understanding of the epitaxial growth processes on GaAs(111)B substrates, the optical quality of QWs are still inferior compared to QWs grown on the GaAs(001) surface. This is illustrated in Fig. 6, where we compare our best PL spectra obtained from GaAs(001) and GaAs(111)B QWs. Both spectra were recorded at a temperature of math formulaK under comparable conditions. Whereas the PL signal of the 20 nm thick GaAs(001) QW (black, full line) appears as a very narrow line with a FWHM of only 0.16 meV, the PL line of the 25 nm thick GaAs(111)B sample (red, broken line) is much broader and reveals a FWHM of 2.1 meV. The spectrum of the (111)B SQW sample seems to consist of two lines and might indicate the appearance of neutral and charged excitons. However, this assumption has to be clarified in further experiments.

Figure 6.

PL spectra taken at math formulaK of a 25 nm thick GaAs(111)B SQW structure (red, broken line) and of a 20 nm thick GaAs(001) QW structure (black, full line) for comparison. The FWHM linewidths are indicated in the plot.

4 Acoustic transport of carriers and spins

This section reviews recent results on carrier and spin transport by SAWs in III–V semiconductor nanostructures. SAWs are elastic vibrations propagating along a surface. In a piezoelectric material (such as the III–V compounds), these waves are accompanied by a piezoelectric potential math formula, which enables their electric generation by interdigital transducers (IDTs) placed on the surface of the material (cf. Fig. 7). As illustrated in the inset of the figure, math formula creates a moving modulation of the conduction and valence band edges, which can capture electrons and holes and transport them with the acoustic velocity.

Figure 7.

Transport and manipulation of spins in GaAs QW structures using surface acoustic waves. Electrons and holes generated by light pulses are spatially separated and transported by the type II modulation of the SAW field (cf. inset). During the acoustic transport, the spins interact with magnetic and electric fields created by gate electrodes. Spin polarization is detected either by Kerr rotation of the polarization axis of a second laser pulse, or by the carrier recombination induced by metal stripes (M) placed beyond the interaction region, where the SAW potential quenches. The polarization of the luminescence is used to determine the spin state.

The initial investigations of the acoustic carrier transport in semiconductors date back to the seventies, when acoustically induced electron transport [79, 80], as well as the role of photogenerated carriers in the SAW attenuation was established ([81]). By reducing the lateral dimensions of the potentials, the acoustic transport of single-electrons [82-84] has been demonstrated. The type II potential modulation can trap and transport photoexcited electrons and holes over hundreds of math formulam ([85]). This approach has also been used to create acoustically pumped single-photon sources ([86]).

The mobile character of the piezoelectric potential is specially suitable for the transport and manipulation of photoexcited spins using the scheme illustrated in Fig. 7. Here, optically oriented electron spins are captured by math formula and transported along the SAW propagation direction. Magnetic fields or electric gates based on the SO-interaction can then be used to control the spin orientation during transport. The spin state can be probed during transport via PL spectrometry [87, 88] or Kerr reflectometry [89, 90]. After transport, the electrons and holes can be forced to recombine, leading to the emission of circularly polarized light. The acoustic potentials have the following favorable properties:

  1. the spatial separation of electrons and holes by the type II potential increases the recombination lifetime and suppresses spin dephasing via the electron–hole exchange interaction (the excitonic or BAP mechanism) [5, 87, 88];
  2. motional narrowing effects induced by mesoscopic (math formulam-sized) confinement potentials reduce spin dephasing within an electron spin ensemble. Very long spin lifetimes (math formulans) and coherent transport lengths (math formulam) have been measured during acoustic transport by mobile potential dots (dynamic quantum dots, DQDs) produced by acoustic fields in GaAs (001) QWs [88, 91];
  3. the spins are transported with a well-defined average momentum math formula along the SAW propagation direction, where math formula is the electron effective mass and math formula the acoustic propagation velocity. This is specially interesting for studies of SO-effects in view of the dependence of math formula on electron momentum. The controlled precession of spins during acoustic transport has been demonstrated in Ref. ([88]). The dependence of the precession frequency on QW thickness has been used for the direct determination of the SO-splitting constant in GaAs QWs (see Section 'Symmetry effects on the spin–orbit interaction') [88, 91, 92]. Finally, Sanada et al. ([93]) have recently demonstrated the full control of the spin vector via the SO-interaction by tailoring the shape of the acoustic transport channel;
  4. the dynamic strain field of a SAW can also be used to induce a SO-field for spin control [90, 94, 95]. We briefly consider here the effects of the strain field of a Rayleigh SAW. For a Rayleigh SAW propagating along x, the strain has three non-vanishing components with amplitudes math formula, math formula, math formula, which vary in time and space with the acoustic frequency (math formula) and wavelength (math formula), respectively. If the carriers are transported close to the minima of the electronic piezoelectric energy, they experience a constant strain field during transport. The last row of Table 1 summarizes the k-dependence of the SAW strain contribution math formula for QWs with different orientations. For a SAW along x (i.e., for math formula), (math formula) is proportional to the Rashba contribution and can, therefore, be compensated by an applied electric field. This becomes particularly interesting for the acoustic transport of long-living z-oriented spins in (110) QWs, where precession due to the strain field can be avoided by the application of an electric field.

Most of the previously mentioned investigations have been carried out in (Al,Ga)As(001) QW structures. In the following sections, we review recent results on spin transport in (110) QWs as well as future perspectives for the spin transport and manipulation in (111) structures.

5 Acoustic spin transport in GaAs(110) QWs

This section reviews results on the acoustic transport of electron spins in GaAs(110) structures. We first analyze spin transport by SAWs along a single QW at low temperature and compare the transport dynamics to the predictions of Section 'Symmetry effects on the spin–orbit interaction'. The long lifetimes of spins in these QWs make them good candidates for spintronic devices also at higher temperatures, where spin dephasing is more pronounced. In the second part of this section, we address acoustic transport at higher temperatures in piezoelectrically defined channels created in GaAs(110) microcavity structures. In these channels, the electron propagation is confined within mesoscopic 1D channels, which, according to the discussion in Section 'The Dresselhaus contribution', provides further suppression of the spin dephasing mechanisms. This, together with the enhanced optical generation and detection of electron spin polarization in QWs embedded in microcavities, allows for spin transport over long distances at temperatures exceeding math formulaK.

5.1 Low temperature spin transport

The studies were carried out on a 20 nm-thick undoped GaAs(110) QW with Almath formulaGamath formulaAs barriers located 400 nm below the surface. In order to enhance the SAW piezoelectric field, the samples were coated with a 424 nm thick piezoelectric ZnO film. SAWs propagating along the math formula surface direction were generated by IDTs for an acoustic wavelength math formulam fabricated by optical lithography ([40]).

The experiments for the optical detection of spin transport were carried at 20 K in a cold finger cryostat with an optical window and coaxial connections for the application of radio-frequency (rf) signals to the IDTs. An external coil applies in-plane magnetic fields, math formula, of up to 150 mT. The acoustic transport of spins was studied in this sample by spatial- and time-resolved PL as well as magneto-optic Kerr reflectometry [96, 97]. This last technique has been successfully applied for spin transport studies in GaAs QWs involving a single acoustic wave ([89]), as well as moving potential dots generated by the interference of two perpendicular SAW waves ([90]). Here, a circularly polarized pump laser pulse focused on the SAW path (15math formulam diameter spot) optically generates out-of-plane polarized spins. They are then probed during transport by measuring the rotation of the polarization angle, math formula, of a weaker, linearly polarized probe pulse reflected on the sample (spot of typically 10math formulam diameter). math formula is proportional to the projection of the average spin vector along the direction perpendicular to the QW plane (the z direction). The probe pulse can be displaced by math formula along the SAW path and time-delayed, math formula, with respect to the pump pulse. Both pump and probe energies are tuned to the electron heavy-hole exciton energy on the QW (math formulanm). Further details about the experimental setup can be found at Ref. ([89]).

The SAWs were generated by applying a radio-frequency signal of math formulaMHz with a nominal power math formuladBm to the IDT. Although the cryostat nominal temperature was 20 K, additional heating induced by the rf power increases it to math formula53 K, as estimated from the energetic shifts of the electron heavy-hole emission line.

Figure 8.

Electron out-of-plane spin polarization as a function of the delay time (math formula) and distance (math formula) between pump and probe pulses under the following conditions: (a) SAWs off and math formula, (b) SAWs off and math formulamT, (c) SAWs on and math formula, and (d) SAWs on and math formulamT. The slope of the dash-dotted line in panels (c) and (d) is equal to the SAW velocity.

The two-dimensional plots of Fig. 8 compare the spatial-temporal evolution of the out-of-plane spin polarization generated by the pump in the absence and presence of a SAW (left and right panels) under in-plane magnetic fields math formula and math formulamT (top and bottom panels, respectively). Except for carrier diffusion shortly after the pump pulse (i.e., for times <0.1 ns), no transport takes place in the absence of SAWs. math formula in this case evolves according to:

display math(7)
display math(8)

where h is Planck's constant, math formula the Larmor precession frequency, and math formula the effective spin relaxation time ([23]). By fitting the data along the line math formula (dashed line) in Fig. 8a to Eq. (7), we obtained math formulans for non-precessing spins. As the direction of the optically excited spins is parallel to the effective magnetic field associated with the SO-interaction, DP spin dephasing is not expected in this case. The spin lifetime is mainly limited by the BAP scattering ([4]) induced by the high carrier density created by the pump pulse. Under precession around math formula (Fig. 8b), math formula reduces to 0.78 ns. The latter is attributed to the activation of DP dephasing for rotating spins moving with math formula (cf. Table 1).

In our undoped samples, math formula is related both to the intrinsic spin lifetime math formula and to the carrier recombination time, math formula, by the expression

display math(9)

We have estimated math formula from the time dependence of the probe reflectance on the sample surface, which yields 1.84 ns in the absence of a SAW. The situation changes dramatically when SAWs are applied to the QW. In this case, the photo-excited electrons and holes are stored and transported at different phases of the SAW field (cf. upper inset of Fig. 7), thus reducing the overlap of their wave functions and increasing math formula to values above 40 ns. The electrons maintain the spin polarization over transport distances of more than 15math formulam, as illustrated in Fig. 8c and d. The oscillations in math formula along the dot-dashed line in panel (d) are attributed to the precession around math formula of the photoexcited electron spins moving with math formula. Due to the constant transport velocity, the time-dependence of the spin polarization is mapped into a spatial dependence. Fits to Eq. (7) show that math formula is enhanced to 3.78 and 2.15 ns for math formula and 65 mT, respectively. The longer recombination lifetime, together with the reduction of BAP scattering due to the spatial separation of electron–hole pairs by the SAW field, enhances math formula by about three times with respect to the values in absence of SAWs.

Contrary to math formula, the Larmor precession frequency of the spin ensemble has the same value math formulaMHz both with (panel d) and without (panel b) acoustic excitation. Note, in particular, that the Dresselhaus component math formula (cf. Table 1) vanishes for transport along the math formula direction. The g-factor, math formula, obtained from math formula agrees well with reported values for electron spins in similar (110) QW structures [23, 40]. This result is also consistent with a short hole spin relaxation time (in the tens of ps range) ([98]), which makes the polarization dynamics during acoustic transport entirely determined by the electron spins. Finally, the fact that math formula increases under a SAW indicates that the contribution math formula from the SAW strain (cf. Table 1, last row) is negligible for the present experimental conditions.

5.2 High temperature transport in GaAs (110) microcavity structures

The efficiency of the optical processes for the generation and detection of spins, as well as the spin lifetimes, normally reduce at high temperatures. The reduction in optical efficiency can be partially compensated by embedding the QWs within optical microcavities. Spin scattering can be minimized by combining the special properties of (110) QWs with lateral confinement.

We argued in Section 'The Dresselhaus contribution' that electron transport along 1D channels fixes the rotation axis of the SO-field. If it is further assumed that these fields depend only linearly on k, spin dephasing by means of the DP mechanism becomes totally suppressed [28, 99, 100]. The enhancement of the spin lifetimes of electrons in etched (In,Ga)As channels [on GaAs(001)] with decreasing channel widths has been experimentally observed in experiments at math formulaK by Holleitner et al. ([29]). Here, we use a similar technique to transport spins in (110) QWs embedded in microcavities at temperatures above 100 K.

Instead of defining the channels by chemical etching, which might induce spin dephasing by electron scattering at the rough sidewalls (i.e., via the Elliot-Yafet spin dephasing mechanism ([22])), our approach employs piezoelectrically defined acoustic transport channels. The experimental setup is depicted in Fig. 9a: the transport channel is defined by a thin metal layer containing a narrow slit (3–5math formulam-wide) oriented along the [001] surface direction of a (110) QW structure grown on an (Al,Ga)As Bragg mirror stack (Fig. 9b). The whole structure is then coated by a piezoelectric ZnO/a-SiOmath formula layer stack, which simultaneously acts as the upper Bragg mirror of an optical microcavity. A narrow (approx. math formulam wide) SAW beam with wavelength math formulam (corresponding to a frequency math formulaMHz) is then launched along the slit direction by a focusing IDT. The metal layer screens the piezoelectric potential generated by the SAW everywhere in the QW plane except underneath the slit. The latter then defines a narrow channel for the ambipolar acoustic transport of electrons and holes. The transport is blocked at the end of the slit, where the quenching of math formula by the metal forces the recombination of the carriers.

The lower Bragg mirror and the cavity layers (including the 20 nm-thick GaAs QW) in Fig. 9b were grown by MBE. The 12 nm thick Ti layer was evaporated and photolithographically patterned to form slits with nominal widths (math formula) of 3 and 4math formulam. The upper ZnO/SiOmath formula Bragg mirror was rf-sputtered on top of the patterned Ti layer. The vertical distance between metallic layer and QW is only 110 nm, thus enhancing the intensity of the stray fields created by the piezoelectric layers.

The composition and thicknesses of the layers were optimized to increase the SAW field close to the QW structure. SAWs result from the confinement of the acoustic fields due to the lower acoustic velocity near the surface. This waveguiding effect reduces if a GaAs substrate is overgrown with an Almath formulaGamath formulaAs layer, since the average acoustic velocity near the surface increases with the Al content x. The latter has been minimized in Fig. 9b by reducing the average Al content (math formula56%) of the Bragg mirror layers and by the insertion of 3/4 math formula layers with low x in the lower Bragg mirror. For the same reason, the upper Bragg mirror consists of only two mirror layer pairs on top of a math formula ZnO layer, leading to a cavity quality factor (Q) of only about 100. A significantly enhanced QW PL has nevertheless been observed when the temperature dependent cavity resonance wavelength matches the QW heavy hole exciton wavelength at math formulaK (not shown here).

Figure 9.

(a) Hybrid microcavity with an acoustic transport channel formed by the structured semitransparent Ti layer. The dashed line illustrates the lateral profile of the piezoelectric energy math formula. (b) Detailed layer structure of the hybrid microcavity. The optical layer thicknesses are given in units of the cavity resonance wavelength math formula= 820 nm at math formulaK.

Figure 10.

Calculated snapshot of the moving profile of the piezoelectric energy math formula at the depth of the QW in the hybrid cavity structure depicted in Fig. 9. Calculations were carried out for a nominal channel width (math formula) of 3 math formulam, an acoustic wavelength math formula = 5.6math formulam and an acoustic linear power density of math formula = 200 W mmath formula. The spatially separated electrons and holes are symbolized by red and blue dots, respectively. The orange lines mark the borders of the semitransparent Ti layer.

Figure 10 shows a calculated snapshot of the moving profile of the piezoelectric energy, math formula, at the depth of the QW in the structure depicted in Fig. 9. This calculation was carried out for math formula= 3 math formulam, math formula= 5.6math formulam and an acoustic linear power density (math formula) of 200 W mmath formula. Due to the type II like modulation of the piezoelectric potential, electrons and holes (symbolized by red and blue dots, respectively) are transported spatially separated from each other. The orange lines mark the borders of the semitransparent Ti layer. The amplitude of math formula is also plotted in Fig. 9a as a dashed line. Based on this calculation, the acoustic transport takes place with the carriers confined within a 1–2math formulam wide channel. Furthermore, the efficient screening of math formula at the end of the slit should stop the acoustic transport and induce recombination.

Figure 11.

(a) Schematic illustration of the scanning method used to monitor the acoustic transport of electron spins. (b) PL intensity images taken at math formulaK corresponding to acoustic transport distances of 96, 58, and 26 math formulam (top to bottom).

The optical detection of spin transport was carried out using the scheme displayed in Fig. 11a. A circular polarized laser (denoted math formula in the figure) focused onto the SAW channel generates carriers in the QW with out-of-plane polarized spins at varying distances math formula from the end of the channel (distances from top to bottom panel: math formula, 58, and 26 math formulam). The carriers are then transported by the SAW toward the end of the channel, where the screening of math formula terminates the transport and provokes radiative recombination. The spin polarization (math formula) is determined from the PL intensities with left (math formula) and right (math formula) circular polarization emitted at the end of the slit according to

display math(10)

The two-dimensional intensity images of the total PL (math formula) for different math formula are depicted in Fig. 11b. At the generation point, the high concentration of photo-generated charge carriers partially screens math formula, thus reducing the transport efficiency. As a result, some of the carriers recombine radiatively giving rise to a weak PL signal. Away from the generation point, the electrons and holes are efficiently captured and transported by the SAW field, thereby making the recombination (e.g., via trapping sites) negligible. In contrast, at the end of the channel, where math formula is screened by the Ti layer, considerable PL intensities are recorded and the electron spin polarization can be analyzed. By scanning math formula in small steps (approx. 2math formulam) one can determine the electron spin polarization as a function of the transport length [math formula]. Taking into account the well-defined SAW velocity math formula, the electron spin lifetime math formula can be determined from the decay math formula. The propagation delay t and distance math formula are related to each other by math formula.

Figure 12.

Out-of-plane spin lifetimes (math formula) of electrons transported acoustically in a GaAs(110) QW at math formulaK as a function of the applied acoustic power (math formula). math formula indicates the nominal channel width.

Figure 12 displays math formula measured in samples with nominal channel widths of 3 and 4 math formulam under varying acoustic power levels. Considering the large range of applied acoustic power values, math formula can be regarded as power independent. However, electron spin lifetimes measured in the narrower channel (math formula 10–11 ns) are clearly larger than the ones deduced for the wider channel (math formula 7–8 ns). This enhancement of electron spin lifetimes is expected as the channel widths of the investigated samples are comparable to the SO precession length, which is of approx. math formulam for a 20 nm thick GaAs QW.

6 Suppression of the spin–orbit interaction in GaAs(111)B quantum wells

The special characteristics of the SO-fields in GaAs(111) QWs makes them very good candidates for spintronics. As mentioned in Section 'Symmetry effects on the spin–orbit interaction', an electric field applied across a (111) QW can suppress spin dephasing mechanisms associated with the SO-interaction. The studies of the spin dynamics in GaAs(111) QWs as a function of electric and magnetic fields, as well as temperature presented in this section provide experimental evidence for the electric suppression of spin dephasing.

Figure 13.

Spectroscopically resolved PL as a function of the bias voltage, math formula, applied to the SQW sample and the corresponding vertical electric field (math formula) estimated from the fitting of the observed Stark shift (white broken line). The inset shows a scheme of the n–i–p structure in which the SQW is embedded.

6.1 Electric control of the spin lifetime

The experiments were carried out in QWs embedded in the intrinsic region of a n–i–p structure (cf. inset of Fig. 13). A bias voltage (math formula) applied between the top n-doped layer and the p-doped substrate generates the vertical electric field (math formula) necessary for SO-compensation. Two kinds of samples were studied: a GaAs single quantum well (SQW) with a thickness of 25 nm and a multiple quantum well (MQW) composed of 20 QWs, each 25 nm-thick, separated by (Al,Ga)As barriers. The position of the QWs between the n-doped layer and the p-doped substrate is the same in both samples. To confine the applied electric field along the z-direction, the n-doped layer and part of the top (Al,Ga)As spacer at the intrinsic region were chemically etched into mesa structures with a diameter of 300math formulam.

We determined the field applied to the QWs by measuring the PL under different voltage biases (cf. Fig. 13 for the SQW sample, note that forward and reverse biases correspond to math formula and math formula, respectively). The quantum confined Stark effect (QCSE) induced by math formula modifies the energy, line width, and intensity of the PL line. By comparing the energy shift induced by math formula under reverse bias with numerical simulations of the field distribution, we determined the relationship between math formula and math formula indicated in the upper and lower horizontal axis of the figure.

The n–i–p structures were designed to reach SO-compensation under a reverse bias given by Eq. (2). For low reverse bias (math formulaV), the PL intensity is relatively high (cf. Fig.  13) and the spin polarization math formula can be determined from PL circular polarization using Eq. (10). The dependence of math formula on reverse bias obtained in the MQW sample using this approach is illustrated by the profiles in Fig. 14a, while the corresponding spin lifetimes are plotted as black dots in Fig. 14c. The spin lifetime increases with the amplitude of the reverse bias from barely 1 ns for math formula to up to 60 ns for math formulaV. Control experiments carried out in a p–i–n structure on GaAs(111)B, where the reverse electric field has the opposite orientation relative to the [111] axis, have shown the opposite behavior [53, 57]. Since in both cases the bias leads to the spatial separation of the carriers along the growth direction, the bias dependence of math formula cannot be accounted for by the BAP mechanism.

For reverse voltages math formulaV, the overlap between the electron and hole wave functions in the QW as well as the PL reduce significantly, thereby hindering the detection of the spin dynamics from the PL polarization ([57]). We overcome this limitation by carrying out experiments under pulsed reverse bias ([54]). Here, laser and bias pulses with the same repetition frequency are synchronized with each other so that the laser pulse hits the sample at a time instant math formula shortly after the application of a reverse bias pulse of amplitude math formula. The photoexcited carriers are then driven toward opposite interfaces of the QW, where they remain stored until the pulse is removed. The stored carriers quickly recombine when the bias pulse is removed giving rise to a short PL pulse.

Figure 15 shows time-resolved PL traces of the MQW sample submitted to math formulans long bias pulses with a repetition period of 80 ns. For math formulaV, the QCSE prevents the recombination of the photoexcited carriers during the bias pulse. Electrons and holes remain then stored in the QWs during a time math formula. Information about the stored carrier density and spin polarization is extracted at the end of the bias pulse, when a small forward bias (math formulaV) is applied to induce carrier recombination. As shown in Fig. 15, the amplitude of the PL pulses after the bias pulse increases for math formulaV, since for these biases the carrier lifetime exceeds the pulse width. The PL rise time is determined by the falling time of the bias pulses of 2 ns. The time-integrated PL after bias pulses with math formulaV corresponds to 60% of the PL emitted right after the laser pulse under a bias of 0 V (dark blue curve in Fig. 15). This means that the QWs can efficiently store a high density of carriers over long times. The reduction of the retrieved PL intensities for pulse voltages math formulaV is attributed to field-induced carrier extraction to the contacts during the storage time, as this bias range also coincides with the onset of the breakdown current of the structures.

Figure 14.

(a) Time-resolved spin polarization during reverse bias of different amplitudes math formula. (b) math formula and math formula traces for bias pulses with math formula V and durations of 20 ns (black), 30 ns (red), 40 ns (green), 50 ns (blue), and 60 ns (cyan). (c) Spin lifetimes as a function of math formula obtained from the measurements on panels (a) [black dots] and (b) [open squares].

Figure 15.

Time-resolved PL of the MQW sample under math formulans bias pulses with a repetition period of 80 ns. The value of math formula during the pulse goes from a forward bias of 0.6 to a −3.6 V reverse bias. Between pulses, math formulaV, so that recombination of the carriers stored during the pulse is allowed.

The spin dynamics for large reverse biases (math formulaV) can be obtained from the circular polarization of the PL pulses emitted at the end of the bias pulses. Figure 14b shows math formula and math formula for bias pulses of math formulaV and different lengths. The remarkable difference in intensity of math formula and math formula after a time delay of more than 20 ns attests to the conservation of the spin polarization during the long charge storage.

We have determined the out-of-plane spin lifetimes math formula for each math formula by measuring math formula at the end of voltage pulses of different durations. The results for the MQW sample, indicated by the open squares in Fig. 14c, cover the range of large reverse bias and are thus complementary to those obtained for lower fields from Fig. 14a [black dots]. The spin lifetime increases with reverse bias until it reaches a maximum of about 100 ns for a bias math formulaV, which corresponds to a vertical electric field of about math formulakV cmmath formula (the compensation field). Beyond this value the spin lifetime decays. The observation of a maximum in Fig. 14 unambiguously establishes the BIA/SIA compensation as the mechanism for bias-induced spin lifetime enhancement in (111) QWs. Furthermore, the peak spin lifetimes exceeding 100 ns are among the longest values reported for undoped GaAs structures. Finally, it is important to note that the Rashba contribution in the n–i–p structures can be electrically increased to values up to approximately 1.5–2 times the Dresselhaus contribution. This range is limited by the onset of field-induced carrier extraction from the QW for math formulaV.

Figure 16.

(a) math formula and math formula traces measured at math formulaV and an in-plane math formulamT. (b) Time-dependent spin polarization extracted from traces in (a). The pink broken line shows the result of the fitting. (c) Spin lifetime as a function of math formula for different in-plane magnetic fields. The solid lines are guide to the eye, while the broken lines correspond to the values extracted from the model described in the text. The inset shows the spin lifetime as a function of the in-plane magnetic field for a fixed math formulaV. All measurements are performed at math formulaK.

6.2 Spin lifetime under an external magnetic field

The reverse field is also expected to increase the lifetime math formula of spins precessing around an externally applied in-plane magnetic field math formula. In order to avoid perturbations related to the non-uniformity of the vertical field distribution across the QWs in MQW samples, the investigations under math formula were performed on the SQW sample. Figure 16a shows math formula and math formula under math formulaV and math formulamT for this sample. Quantum beats accompanying the PL intensity decay are clearly observed, which are attributed to the Larmor precession of the electron spins around the external magnetic field ([101]). Figure 16b shows the spin polarization, math formula, obtained from Eq. (10). Due to the fact that the temporal width of the laser pulse (600 ps) is comparable to the spin precession period (3 ns), the initial spin polarization is smaller than the theoretically expected value of approx. 0.25. The g-factor obtained from a fitting to an exponentially decaying cosinus function (pink broken line) is math formula, in good agreement with the expected value for such a wide QW. As the bias voltage increases, math formula shifts toward smaller values at a rate math formulaVmath formula.

Figure 16c compares the electric field dependence of the spin lifetime for out-of-plane (black squares) and precessing (red circles and green triangles) spins in the SQW. The electric field dependence of math formula and math formula can be divided into two regions: for electric fields away from math formula, math formula is longer than math formula. This behavior is due to the fact that, at low temperatures (that is, for small k-vectors), math formula (cf. Section 'Symmetry effects on the spin–orbit interaction'). Under this condition, spins along one of the in-plane directions (e.g., x) only feel the dephasing SO-field along the orthogonal in-plane direction (i.e., math formula). Spins along z, in contrast, will see two dephasing fields (associated with math formula and math formula) and decay faster. Therefore, math formula and, according to Eq. (6), math formula.

The situation reverses close to math formula: here, the in-plane linear Dresselhaus contribution can be compensated by the Rashba term (cf. Table 1), but the z-component math formula (cf. Eqs. (3) and (4)) not, thus leading to math formula. A reduced lifetime for precessing spins (relative to z-oriented ones) has been observed over a wide range of magnetic fields (cf. inset of Fig. 16c, with data measured for math formulaV). Finally, at the intersection of the two regions all three components of math formula are equal, and the spin lifetime becomes isotropic.

The measured spin lifetimes math formula and math formula can be compared to the expected values from the model explained in Section 'Symmetry effects on the spin–orbit interaction'. According to the model, the ratio between the Dresselhaus and Rashba constants, math formula, and the momentum scattering time math formula, determine the compensation field math formula (cf. Eq. (2)) and the spin lifetime at compensation (cf. Eq. (5)). These two parameters were varied in Eq. (5) to fit the measured results. The broken lines in Fig. 16c show the result of our calculations for math formulaVÅ, and math formulaps. The model predicts a much stronger field dependence than the measured one. The reason for these discrepancies will be further discussed below. The qualitative features, however, are very well reproduced, including the reduction of the spin lifetime around math formula for precessing spins as well as opposite behavior away from the compensation field.

6.3 Spin lifetime dependence on temperature

According to the theoretical model, the Rashba term compensates the Dresselhaus one when the following condition is fulfilled:

display math(11)

At very low temperatures, the math formula term, which comes from nonlinear terms of the Dresselhaus contribution (cf. Eq. (3)), can be neglected. In this case, compensation takes place at a vertical electric field fully determined by math formula and math formula (cf. Eq. (2)). When the temperature increases, however, a larger range of electron k-space states above the conduction band minimum are occupied, and math formula in Eq. (11) cannot be neglected any more. As a consequence, SO-compensation does not occur simultaneously for all wave vectors at a fixed external electric field. The inclusion of high order terms in k leads to a temperature dependence of the maximum spin lifetime, as well as of math formula.

Figure 17 shows the dependence of math formula on math formula and temperature measured in the SQW sample. As expected, the math formula around math formula decreases from 50 ns to barely 10 ns as T increases from 33 to 60 K. The inset shows the values of math formula obtained for the measured temperatures. The decrease of math formula with T is a clear confirmation that, in this temperature range, the cubic terms in the Dresselhaus contribution cannot be neglected. The broken lines in Fig. 17 show the results of our model for a ratio math formulaVÅ, and assuming the same momentum scattering time (math formulaps) for all temperatures. As in the case of the magnetic field dependence, the simulations reproduce reasonably well the decrease of the maximum spin lifetime and the shift of math formula with increasing temperature. They predict, however, a much stronger dependence of the spin lifetime on the electric field amplitude and on temperature. This discrepancy is due to the fact that we suppose the same value of math formula for all electrons, independently of their energy state and temperature.

Figure 17.

Dependence of the out-of-plane spin lifetime math formula on math formula and temperature, obtained from the decay of the spin polarization in a SQW after optical excitation with a laser pulse. The solid lines are a guide to the eye. The dashed lines are spin lifetimes calculated at the same temperatures using the model described in the text. The inset shows the value of the compensation point, math formula, extracted from the experimental values for each temperature.

Finally, we briefly discuss additional possible mechanisms for the different field dependences of the calculated and measured profiles. One obvious mechanism is the non-uniformity of the field distribution. This mechanism can be discarded for the SQW sample: within the range of reverse bias around the compensation field, the PL shifts in Fig. 13 vary almost linearly with reverse field with a proportionality coefficient of math formulanm kVmath formulacmmath formula. The line width of the PL line of 2.05 nm measured in this voltage range then yields an uncertainty in field of math formulakV cmmath formula, which is much smaller than the width of the lifetime peak in Figs. 16 and 17.

An alternative mechanism to account for the discrepancy between calculation and measurements are SO-contributions induced by strain fields (cf. Section 'Strain effects'). The biaxial strain due to the difference between the elastic constant of GaAs and (Al,Ga)As (which was estimated to have levels lower than 3math formula), is expected to have negligible effects on the spins. In addition, this type of strain induces a SO-contribution in GaAs(111) QWs with the same symmetry as the Dresselhaus term in Table 1, and can thus be compensated by the applied bias. Another possibility are uniaxial strain fields induced, for instance, by sample mounting.

7 Conclusions and future perspectives

The special symmetry of the SO-interaction in III–V semiconductor QWs grown along the non-conventional crystallographic directions [110] and [111] allows the detection of electron spins with long decoherence times. In (110) QWs, the combination of such SO-symmetries with the special properties of SAWs leads to acoustic transport of electron spins with out-of-plane polarization along distances of several micrometers at low temperatures. Acoustic spin transport also reduces BAP spin dephasing via the spatial separation of electrons and holes by the type II modulation potential. Furthermore, by embedding these QWs within optical microcavities compatible with SAW generation, we have demonstrated acoustic spin transport within narrow piezoelectric channels over distances of several tenths of math formulam at temperatures above 100 K. Further developments of these structures should enable spin transport also at higher temperatures.

In the case of GaAs(111) QWs, the application of vertical electric fields enables the efficient suppression of the SO-interaction at low temperatures. As a result, spin lifetimes up to 100 ns have been reached. These lifetimes arise from the compensation of terms of the SO-interactions that are linear in electron momentum math formula. At higher temperatures, higher orders terms in k become important and prevent the compensation effect that leads to such long lifetimes. According to the results obtained from their (110) counterparts, acoustic spin transport within narrow channels appear to be a promising candidate to overcome this limitation: the well-defined velocity and direction of the acoustic wave allows the spatial confinement of spins and their transport along special crystallographic directions, for which the high order terms of the SO-fields in (111) QWs are minimized. Figure 18 shows a device proposal combining SAWs with vertical electric fields. This type of devices was designed based on our experience with (110) QW structures for acoustic transport (see, for instance, Fig. 9). The QW is embedded within a piezoelectric ZnO/SiOmath formula microcavity for operation at relatively high temperatures. The gate placed along the SAW path allows the application of vertical electric fields for the suppression of the SO-interaction in the QW. In addition, the narrow channel defined by the gate enhances the piezoelectric potential math formula underneath it, thereby laterally confining the electrons and holes moving in the SAW field. This confinement effect, which is discussed in detail in Ref. ([102]) and illustrated by the calculated depth profiles for math formula in Fig. 18c, arises from the fact that math formula near the interface between the ZnO/SiOmath formula stack and the GaAs structure increases when the surface of the samples is coated with a conductive layer. The metal structure placed underneath the ZnO/SiOmath formula DBR at the end of the path (named ”stopper” in Fig. 18) quenches the SAW piezoelectric field at the end of the transport path, thereby forcing the carrier recombination required for the optical retrieval of the spin information. The realization of a device of such characteristics is expected to be one of our main activities during the next years.

Figure 18.

(a) Top view and (b) cross-section along the transport path of a QW inserted within a hybrid microcavity for acoustic spin transport. The (111) QW is sandwiched between two distributed Bragg reflectors (Bragg mirrors DBRmath formula and DBRmath formula). The stopper and the gate for the application of the vertical field math formula consist of a semitransparent metal layer. (c) Depth profile for the piezoelectric potential (math formula) calculated underneath the gate (red curve, cf. vertical dashed line in (b)) and on the free surface (green curve). Note the larger amplitudes for math formula underneath the gate ([102]). These profiles were calculated for a Rayleigh SAW with math formulam and linear power density of 38.6 W mmath formula propagating along the math formula surface direction of a math formula structure. In the diagram, the microcavity is grown on a 1math formulam thick GaAs spacer deposited on the doped GaAs subtrate.

Note from the authors

After submission of this manuscript two new articles appeared in the literature regarding spin dynamics in GaAs(111) QWs. The first one deals with the estimation of the Dresselhaus and Rashba coefficients by spin grating experiments ([103]), while the second one shows a model for the DP mechanism which takes the collision processes explicitly into account and allows math formula to change with temperature ([104]).

Acknowledgments

We thank F. Iikawa for discussions, as well as M. Höricke, W. Seidel, B. Drescher, S. Rauwerdink, and A. Tahraoui for sample processing. Financial support by the DFG (Priority Program SPP1285) is thankfully acknowledged.

  1. math formula Calculated for a biaxial strain with in-plane component math formula and out-of-plane component math formula;