Dynamical formation and active control of persistent spin helices in III-V and II-VI quantum wells

The Dirac equation predicts a coupling of the spin angular momentum of an electron to its orbital motion which is referred to as SO coupling. The coupling leads to the fine structure of atomic levels and the splitting of the valence band to the Γ₈ band (with the j = 3/2 total angular momentum) and the split-off Γ₇ band (with j = 1/2) in zinc-blende semiconductors like GaAs, see figure 1(a).

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In semiconductor structures lacking a center of space inversion, the remaining spin degeneracy in the conduction and valence bands is lifted apart from some high-symmetry points in the Brillouin zone. This splitting can be described by a Zeeman-like contribution $-{\boldsymbol{\mu }}\cdot {{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}})$ to the Hamiltonian, where ${\boldsymbol{\mu }}$ is the operator of magnetic moment and ${{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}})$ is an effective magnetic field dependent of the wave vector ${\boldsymbol{k}}$. Generally, the effective magnetic field may originate from SO coupling in the electric fields of either atoms constituting a non-centrosymmetric crystal lattice (Dresselhaus term) or imposed by an asymmetric heterostructure (Rashba term) [9, 10]. The effective magnetic field is odd in the wave vector ${\boldsymbol{k}}$, which follows from time reversal symmetry and ensures the fulfillment of Kramer's degeneracy ${E}_{\uparrow }({\boldsymbol{k}})={E}_{\downarrow }(-{\boldsymbol{k}})$. We note that, in a centrosymmetric system, space inversion additionally imposes ${E}_{\uparrow \downarrow }({\boldsymbol{k}})={E}_{\uparrow \downarrow }(-{\boldsymbol{k}})$. Together with Kramer's degeneracy, it provides ${E}_{\uparrow }({\boldsymbol{k}})={E}_{\downarrow }({\boldsymbol{k}})$, i.e., the spin degeneracy holds across the entire Brillouin zone. Therefore, the observation of the spin splitting in the absence of an external magnetic field, which is crucial for the formation of PSH, is restricted to non-centrosymmetric structures. This requirement is met, among other systems, in GaAs and CdTe crystals of the T_d point group and QWs based on them [35, 36].

The Dresselhaus Hamiltonian, that describes SO splitting of the conduction band due to bulk inversion asymmetry (BIA), is given by

$$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{BIA}} & = & {\gamma }_{D}\left[{\sigma }_{x^{\prime} }{k}_{x^{\prime} }({k}_{y^{\prime} }^{2}-{k}_{z}^{2})\right.\\ & & \left.+{\sigma }_{y^{\prime} }{k}_{y^{\prime} }({k}_{z}^{2}-{k}_{x^{\prime} }^{2})+{\sigma }_{z}{k}_{z}({k}_{x^{\prime} }^{2}-{k}_{y^{\prime} }^{2})\right],\end{array}\end{eqnarray} \tag{ 1 }$$

where ${\gamma }_{D}$ is the material dependent bulk Dresselhaus parameter, σ_j are the Pauli matrices, and $x^{\prime} | | [100],y^{\prime} | | [010]$, and $z| | [001]$ are the cubic axes. Microscopically, γ_D can be calculated in the framework of the multiband ${\boldsymbol{k}}\cdot {\boldsymbol{p}}$ theory using the Löwdin partitioning technique [35, 36] or the tight-binding approach [37]. The results of the calculations for a number of III-V compounds can be found in [37].

For a QW grown along z, the expectation value $\langle {k}_{z}\rangle$ vanishes while $\langle {k}_{z}^{2}\rangle$ does not and one obtains the effective 2D Hamiltonian [17]

$$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{BIA}} & = & {\gamma }_{D}\langle {k}_{z}^{2}\rangle ({\sigma }_{y^{\prime} }{k}_{y^{\prime} }-{\sigma }_{x^{\prime} }{k}_{x^{\prime} })\\ & & +{\gamma }_{D}({\sigma }_{x^{\prime} }{k}_{x^{\prime} }{k}_{y^{\prime} }^{2}-{\sigma }_{y^{\prime} }{k}_{y^{\prime} }{k}_{x^{\prime} }^{2}).\end{array}\end{eqnarray} \tag{ 2 }$$

We note that additional SO contributions with the same symmetry properties may arise from interface inversion asymmetry and strain in lattice-mismatched heterostructures [21].

In the following, we discuss the spin dynamics in the in-plane coordinate system $x\parallel [1\bar{1}0]$ and y∥[110] which is natural for (001)-oriented QWs [38]. The Hamiltonian (2) in the new coordinate frame takes the form

$$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{BIA}} & = & -{\gamma }_{D}\langle {k}_{z}^{2}\rangle ({\sigma }_{x}{k}_{y}+{\sigma }_{y}{k}_{x})\\ & & +\displaystyle \frac{{\gamma }_{D}}{2}({\sigma }_{y}{k}_{x}-{\sigma }_{x}{k}_{y})({k}_{x}^{2}-{k}_{y}^{2}).\end{array}\end{eqnarray} \tag{ 3 }$$

It can also be written in the equivalent form

$$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{BIA}} & = & \beta k({\sigma }_{x}\sin \theta +{\sigma }_{y}\cos \theta )\\ & & +{\beta }_{3}{k}^{3}({\sigma }_{x}\sin 3\theta -{\sigma }_{y}\cos 3\theta ),\end{array}\end{eqnarray} \tag{ 4 }$$

where θ is the polar angle of the wave vector ${\boldsymbol{k}}$. ${\boldsymbol{k}}=k(\cos \theta ,\sin \theta )$, β = β₁ − β₃ is the Dresselhaus parameter which combines

$$\begin{eqnarray}&&{\beta }_{1}=-{\gamma }_{D}\langle {k}_{z}^{2}\rangle \qquad \mathrm{and}\qquad {\beta }_{3}=-{\gamma }_{D}\displaystyle \frac{{k}^{2}}{4}.\end{eqnarray} \tag{ 5 }$$

In the case of a degenerate electron gas, k should be replaced by the Fermi wave vector k_F. The Hamiltonian (4) contains the first and third angular harmonics. The contribution of the third angular harmonic is typically small, since β₃ ≪ β₁, and can be often neglected.

The Rashba Hamiltonian arising from the structure inversion asymmetry (SIA) is given by

$$\begin{eqnarray}&&{H}_{\mathrm{SIA}}=\alpha ({\sigma }_{x}{k}_{y}-{\sigma }_{y}{k}_{x})=\alpha k({\sigma }_{x}\sin \theta -{\sigma }_{y}\cos \theta ).\end{eqnarray} \tag{ 6 }$$

In the case of SIA produced by an external and/or built-in electric field ${ \mathcal E }$, the Rashba parameter α can be related to the electric field by $\alpha ={\gamma }_{R}{ \mathcal E }$. The parameter γ_R depends on the QW material and design [35, 36]. The ${\boldsymbol{k}}\cdot {\boldsymbol{p}}$ theory yields γ_R = 5.206 eŲ and γ_R = 6.930 eŲ for bulk crystals GaAs and CdTe, respectively, [35].

The total effective Hamiltonian of a free electron in a QW, including the spin-independent kinetic energy ${{\hslash }}^{2}{k}^{2}/(2{m}^{* })$ with the effective mass ${m}^{* }$, the Dresselhaus and Rashba contributions, H_BIA and H_SIA, respectively, can be conveniently presented in the form

$$\begin{eqnarray}&&H=\displaystyle \frac{{{\hslash }}^{2}{k}^{2}}{2{m}^{* }}+\displaystyle \frac{1}{2}g{\mu }_{B}{\boldsymbol{\sigma }}\cdot {{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}}),\end{eqnarray} \tag{ 7 }$$

where g is the effective g-factor, μ_B is the Bohr magneton, and ${{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}})$ is the effective magnetic field caused by SO coupling. The field is given by

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}})=\mathop{\underbrace{\displaystyle \frac{2k}{g{\mu }_{{\rm{B}}}}\left[\begin{array}{c}(\beta +\alpha )\sin \theta \\ (\beta -\alpha )\cos \theta \end{array}\right]}}\limits_{{{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}}+\mathop{\underbrace{\displaystyle \frac{2k}{g{\mu }_{{\rm{B}}}}\left[\begin{array}{c}{\beta }_{3}\sin 3\theta \\ -{\beta }_{3}\cos 3\theta \end{array}\right]}}\limits_{{{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}}\end{eqnarray} \tag{ 8 }$$

and contains two contributions ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ and ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ describing the first and the third angular harmonics, respectively. Most remarkably, the effective magnetic field may be strongly anisotropic in ${\boldsymbol{k}}$-space. This is crucial for the PSH evolution and it will be a major aspect of our work to investigate how the Rashba and Dresselhaus parameters can be extracted and manipulated by indirectly monitoring ${{\boldsymbol{B}}}_{\mathrm{so}}$.

Neglecting the small contribution ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$, which is cubic in ${\boldsymbol{k}}$ and has a minor influence on the PSH formation, and solving the Schrödinger equation with the Hamiltonian (7) one obtains the energy dispersion

$$\begin{eqnarray}&&{E}_{\uparrow \downarrow }({\boldsymbol{k}})=\displaystyle \frac{{\hslash }{k}^{2}}{2{m}^{* }}\pm k\sqrt{{\alpha }^{2}+{\beta }^{2}-2\alpha \beta \cos 2\theta }\end{eqnarray} \tag{ 9 }$$

which features a linear dependence of the spin splitting on the electron wave vector ${\boldsymbol{k}}$.

A cross-section of the dispersion given by equation (9) is depicted in figure 1(c). For comparison, the dispersion in the degenerate case and in the case of Zeeman splitting are sketched in figures 1(a) and (b), respectively. Specifically, the dispersion with the Rashba and Dresselhaus SO coupling implies a lateral shift of the 'spin-up' and 'spin-down' subbands in ${\boldsymbol{k}}$-space. Note that the dispersion exhibits an anisotropy in ${\boldsymbol{k}}$-space if both α and β are non-zero. The lateral separation of the spin parabolas in the dispersion cross-section implies the linear increase of the splitting with the wave vector and represents a peculiar characteristic of the Rashba and Dresselhaus SO coupling. The Zeeman splitting in a real magnetic field, on the contrary, corresponds to the spin branches that are vertically separated as depicted in figure 1(b). In figure 1(a), the band structure of a zinc-blende bulk crystal is shown in the absence of external fields and without the Rashba and Dresselaus SO coupling. In this case, the conduction band as well as the valence bands are spin-degenerate.

The PSH appears as a unidirectional wave of electron spin polarization under the specific condition that the Rashba and Dresselhaus parameters are equal in strength, $| \alpha | =| \beta |$ at the Fermi level. Here, we consider the experimental situation of an electron spin ensemble created by an ultrashort focused laser pulse. The pulse is circularly polarized and excites electrons with the spins oriented along the QW normal (z). The spin polarized electrons diffuse in the QW plane from the small spot where they have been created. Consequently, a coupling of the electron spins to the effective magnetic fields arises from non-zero ${\boldsymbol{k}}$-values during diffusion. Thus, the diffusing spins undergo precession while they spread in the QW plane. Each electron sees a sequence of individual effective magnetic fields depending on its individual diffusion trajectory. Nevertheless, the overall spin coherence is maintained in the special regime of the PSH, see figure 2(e). This manifests in the characteristic spin texture which emerges due to the combined spin precession and diffusion.

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From this observation the question arises of how the spin coherence is stabilized although the spins of individual electrons are affected by different effective magnetic fields during their random walks. To understand it, we consider the structure of the effective field ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ at α = β. In this regime, the y-component of the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ vector vanishes and the field ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ becomes unidirectional and dependent of k_y only. As a consequence, only a momentum along the y-axis results in precession while a propagation along the x-axis does not create any effective magnetic field and, thus, does not affect the spin. Moreover, for ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ proportional to k_y in the first power, the spin rotation angle is fully determined by the electron displacement along y being independent of its particular trajectory. This allows one to eliminate the spin-orbit interaction by a unitary transformation of the Hamiltonian and can be also interpreted in terms of the emerging SU(2) spin rotation symmetry.

A schematic distribution of the effective magnetic field on the Fermi circle is depicted in figures 2(a)–(d) for different combinations of α and β. For pure Rashba SO coupling in panel (b) the distribution forms a vortex while for pure Dresselhaus SO coupling in panel (a) it has an anti-vortex structure. Only in the perfectly balanced regime, the field along the k_x axis is entirely gone, see panel (c). The mixed regime in panel (d), with β/α = 0.45 , results in a slightly distorted symmetry of the balanced regime.

As a result of the diffusion-driven spin precession in the effective field ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$, the initially isotropic spot of spin polarization S_z created by a focused laser pulse evolves into a standing unidirectional helical wave of spin density where the z-component of the spin polarization oscillates along the y-axis. The determined precession length λ_0,y can be then used to extract the sum of the Rashba and Dresselhaus parameters since λ_0,y in the PSH regime is given by

$$\begin{eqnarray}&&{\lambda }_{0,{\rm{y}}}=\displaystyle \frac{\pi {{\hslash }}^{2}}{{m}^{* }}{\left(\alpha +\beta \right)}^{-1}.\end{eqnarray} \tag{ 10 }$$

The excitation of the spin helix mode is efficient only if the spatial width of the initially created spin polarization is smaller than the wavelength λ_0,y.

Beyond the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ contribution, the effective magnetic field contains the small ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ term. The dependence of ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ on the direction of ${\boldsymbol{k}}$ is described by the third angular harmonic. Therefore, the accumulated rotation angle in the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ field depends on individual electron trajectories and averages out in the diffusion regime. It means that the precession around ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ does not contribute to the formation of the PHS but rather is a source of the PSH relaxation [22]. An electron drift induced by an externally applied electric field can change the spatio-temporal spin dynamics in such a way that the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ component cannot be neglected anymore. However, this occurs only in the non-linear regime in the electric field when the drift velocity approaches the Fermi velocity. Therefore, we disregard ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ in this paper unless stated otherwise.

An important consequence of the SU(2) spin rotation symmetry for the balanced Rashba and Dresselhaus regime is a significant enhancement of the lifetime of the spin helix [11, 13, 39, 40]. The reason is the suppression of the Dyakonov-Perel (DP) spin relaxation mechanism as the dominant source for spin decoherence. The DP mechanism is based on the precession of individual electron spins in the effective magnetic fields ${{\boldsymbol{B}}}_{\mathrm{so}}({\boldsymbol{k}})$ which change in time following the time dependence of the electron wave vectors [16, 17]. In the regime of the PSH, the spin rotation angle is one-to-one related to the electron displacement in the QW plane independent of its particular pathway between the initial and final points and the spin decoherence in the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ field is avoided. In reality, the SU(2) symmetry is not perfectly realized and the spin lifetime remains finite because of the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ contribution, remaining imbalance of α and β, and other sources of spin decoherence.

The Faraday effect describes a rotation of the polarization plane for a linearly polarized light beam which propagates through a transparent medium exposed to an external magnetic field [41, 42]. An equivalent effect was observed in reflection geometry by John Kerr and is therefore referred to as MOKE schematically depicted in figure 3 [42, 43]. Following the approach of [44], the role of the external magnetic field can be replaced by any macroscopic magnetization ${\boldsymbol{M}}$ in the sample generated, e.g., by optical orientation of magnetic moments μ_s. The resulting interaction of magnetization and the electromagnetic light wave is described by Maxwell's equations. Here we briefly derive the spectral dependence of the Kerr rotation (KR) angle Θ_K in vicinity of an optical resonance. This regime serves as a model for the detection of electron spin polarization in the conduction band of semiconductors.

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The MOKE can be described by Maxwell's equations in a non-magnetic material, i.e., the magnetic permeability tensor is set to the vacuum case. Consequently, only the dielectric permittivity tensor $\epsilon (\omega )$, remains to mediate the light–matter interaction. If the magnetization and the propagation direction of the light is oriented along the z-axis, ${\boldsymbol{\epsilon }}$ of an otherwise isotropic material becomes an asymmetric tensor with complex elements _ij

$$\begin{eqnarray}\epsilon (\omega ,{\boldsymbol{M}})=\left(\begin{array}{ccc}{\epsilon }_{\mathrm{xx}} & {\epsilon }_{\mathrm{xy}} & 0\\ -{\epsilon }_{\mathrm{xy}} & {\epsilon }_{\mathrm{yy}} & 0\\ 0 & 0 & {\epsilon }_{\mathrm{zz}}\end{array}\right)\qquad \mathrm{with}\qquad {\epsilon }_{\mathrm{ij}}={\epsilon }_{\mathrm{ij}}^{{\prime} }+i\,{\epsilon }_{\mathrm{ij}}^{{\prime\prime} }.\end{eqnarray} \tag{ 11 }$$

By inserting equation (11) into the electromagnetic wave equation one obtains two normal modes of the light wave which can be identified as right- and left-circularly polarized. These modes feature different complex refractive index ${N}_{\pm }^{2}={\epsilon }_{\mathrm{xx}}\pm i\,{\epsilon }_{\mathrm{xy}}$ respectively where ± denotes right- and left-circular polarization. From Fresnel's formulas it is known that a difference in the refractive index will result in different complex reflection coefficients given by ${r}_{\pm }=({N}_{\pm }-1)/({N}_{\pm }+1)$. The complex reflection coefficient determines the amplitude and phase change for a reflected wave. This manifests either as circular birefringence for the case of a phase difference or as circular dichroism for the case of an amplitude difference. Also a combination of both effects is possible. These scenarios are visualized in figures 3(b)–(d). This figure shows the polarization components before and after the reflection from a magnetized sample. After the reflection the light beam has undergone a Kerr rotation and also exhibits Kerr ellipticity (see figure 3(d)). The former refers to a rotated plane of polarization, if the phase of the two reflected circular polarized waves has shifted. The latter describes the emergence of an elliptical polarization in case the amplitudes of the two circular components are imbalanced after the reflection. The angle of Kerr rotation (KR) Θ_K, as well as the ratio of major and minor axis η_K in the elliptical polarization, can be quantified from the off-diagonal elements of the permittivity tensor in equation (11) according to

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{{\rm{K}}}=-\displaystyle \frac{{\epsilon }_{\mathrm{xy}}^{{\prime} }}{n({n}^{2}-1)}\qquad \mathrm{and}\qquad {\eta }_{{\rm{K}}}=-\displaystyle \frac{{\epsilon }_{\mathrm{xy}}^{{\prime\prime} }}{n({n}^{2}-1)}\end{eqnarray} \tag{ 12 }$$

where $n=[{\mathfrak{R}}({N}_{+})+{\mathfrak{R}}({N}_{-})]/2$ is the averaged real part of the refractive indices ${N}_{\pm }$. As the real and the imaginary part of ${\boldsymbol{\epsilon }}$ can be expected to scale linearly with the magnetization, it can be assumed that Kerr rotation and ellipticity are useful measures to read out the spin polarization which initially gave rise to ${\boldsymbol{M}}\ne 0$.

Equation (12) suggests a pronounced frequency dependence of Θ_K in the vicinity of an optical resonance. We further evaluate this dispersion for an isolated two-level system with a transition energy ℏω₀. It is assumed that its spin degeneracy is lifted and that the energy splitting of the spin up and down sub-levels amounts to an energy Δ small compared with the transition energy. The Kubo formalism [44, 45] allows to derive the expression

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{\mathrm{xy}}^{{\prime} } & = & {{Cf}}_{12}{\rm{\Delta }}\cdot f^{\prime} ({\omega }_{0})\quad \mathrm{with}\\ f^{\prime} (x) & = & \displaystyle \frac{\partial }{\partial x}\left[\displaystyle \frac{2{\gamma }_{\mathrm{hw}}{x}^{2}}{{\left({x}^{2}-{\omega }^{2}+{\gamma }_{\mathrm{hw}}^{2}\right)}^{2}+{\left(2\omega {\gamma }_{\mathrm{hw}}\right)}^{2}}\right]\end{array}\end{eqnarray} \tag{ 13 }$$

that enables to calculate the spectral dependence of Θ_K around the resonance, where both sub-levels have the same transition strength f₁₂ and γ_hw is the spectral half-width of the transition. The line shape of equation (13) is shown in figure 4(c). It is evident that the KR is favourably measured with the probe de-tuned to the slopes of the resonance.

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The use of ultrafast lasers conveniently permits to achieve a temporal resolution in the (sub-)picosecond time domain by using pump-probe techniques. For this method, the output of a pulsed light source is split into two separate pulse trains referred to as pump and probe beams. Subsequently, the pump and probe are recombined on the sample with well-defined polarization, temporal and spatial position. Here, the probe is influenced by changes in the reflection dR caused by the pump beam. Such changes can arise from a variation of the material permittivity, due to the injection of charge carriers and/or a coherent magnetization generated by injected spins [46]. Afterwards, the probe beam is spectrally separated from the pump and directed in a detection scheme which extracts dR/R, often facilitated by lock-in amplifiers referenced to a modulation of the intensity or polarization of the pump beam. The delay time t_d between pump and probe pulses is set by a mechanical translation stage which allows for the detection of a time-resolved signal dR/R(t_d).

The experimental setup combines the MOKE with a pump-probe scheme into a Kerr microscope which is depicted schematically in figure 3.1 (a). The output of a mode-locked, femtosecond Ti-sapphire laser with 60 MHz repetition rate is used to provide the pump and probe pulse trains. For spectrally resolved measurements each pulse train is sent through a grating-based 4f-pulse shaper where the pump and probe are individually sliced out from the spectrally broad output of the mode-locked laser [47]. After the wavelength selection for both pump and probe, the temporal resolution of the system is ∼1ps, limited by the chosen spectral bandwidth of ∼0.7 nm. The pump pulse train is sent through a mechanical delay stage and its polarization state is modulated between σ⁺ and σ⁻ helicities, using an electro-optical modulator (EOM) (QioptiqLM 0202P). The probe is set to linear polarization. In order to record spatially resolved spin dynamics, the pump arm is complemented by a controllable telescope. Lateral motion of the second lens in this telescope enables movement of the pump spot along the x- and y- direction relative to the probe position (see figure 4(b)). Afterwards, both beams are collinearly focused onto the sample, using a 50×objective (Mitutoyo M-Plan APO NIR) which provides a focused spot size for the probe (pump) beams of 1 (3) μm.

In order to minimize spatial fluctuations the sample is nested in a stable helium flow cryostat (Oxford Microstat HiRes2) and cooled to ∼3.5 K. Such cryogenic temperatures suppress phonon scattering as a possible dephasing mechanism and thus help to conserve the coherence of the spin ensemble. An electromagnet positioned around the sample provides variable magnetic fields up to 230 mT in the Voigt geometry. The reflected, non-degenerate pump and probe pulse trains pass a spectrometer (Oxford Shamrock 500i) to filter out the pump radiation. The subsequent detection of KR is based on a Wollaston prism which splits the probe train into two beams with perpendicular linear polarization. These two beams are fed into a balanced detection scheme, converting the KR into an electrical signal using a lock-in amplifier. The detected KR signal depends on the direction of magnetization in the sample. Due to excitation with σ⁺ (σ⁻) polarized light the spin magnetization becomes oriented parallel (anti-parallel) to the light propagation direction. The direction of magnetization determines the sense of rotation of the KR and thus the detection of negative or positive signal on the lock-in amplifier. Accordingly, the EOM can be used to impose a temporal modulation to the circular polarization direction of the pump and thus the sign of the pump-probe signal. To this end, the corresponding modulation frequency is sent to the lock-in detection, enabling the measurement of KR angles as small as ∼10 microradians.

The present work is independently conducted in heterostructures made from two prototypical III-V and II-VI semiconductors, namely in GaAs and CdTe QWs. This twofold material choice is of great scientific interest as it allows for a comparison of the PSH characteristics in materials with different SO interaction strength. In addition, PSH physics in II-VI compounds has not been studied extensively [7, 48].

Both crystals are grown as zincblende crystals by molecular beam epitaxy (MBE). The zincblende structure lacks a center of inversion, making it particularly interesting for the study of SO interactions. In general, the strength of the atomic SO coupling is given by the SO coupling constant a ∝ Z/r_B³ where the Bohr radius itself is inversely proportional to the atomic number Z. As a result, the coupling constant can be assumed to scale with the fourth power of the atomic number a ∝ Z⁴ [49]. From this relation a much stronger SO coupling behavior is expected in the CdTe system where the constituent atoms are of higher atomic number than in GaAs:

$$\begin{eqnarray*}&&{Z}_{\mathrm{Ga}}=31\lt {Z}_{\mathrm{As}}=33\lt {Z}_{\mathrm{Cd}}=48\lt {Z}_{\mathrm{Te}}=52\end{eqnarray*}$$

The extrinsic Rashba SO coupling mechanism is not directly affected by the constituent atoms as it originates from extrinsic potential gradients given by the neighboring layers of the QW. However, the Dresselhaus SO coupling arises from asymmetrically ordered lattice atoms and is thus influenced by the atomic number. However, this dependence is not equivalent with that of the atomic SO coupling constant. A material specific impact becomes present foremost in the Dresselhaus coupling constant γ_D which can be expected to differ notably between GaAs and CdTe. This difference is backed by theoretical predictions [50] which take into account the band structures of the materials.

Both samples are fabricated as two-dimensional QWs embedded in multilayer heterostructures. The restriction to two dimensions yields a variety of advantages, e.g., a strongly enhanced optical absorption due to an increased oscillator strength of the electron-hole pairs [51]. Moreover, QWs enable a higher degree of spin polarization due to split lh- and hh-bands at the Γ-point [52]. A schematic layer composition and a sketch of the Hall bar structure are displayed in figures 5(a) and (b). In order to create a 2DEG with unbound electrons it is required to lift the Fermi energy into the conduction band, resulting in an electron occupation according to the Fermi–Dirac statistics. A common approach to achieve such a regime is modulation doping. Here, charge transfer from remote dopants into the QW ensures a spatially homogeneous Fermi energy. As a result, a 2DEG in the QW is formed.

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3.3.1. The GaAs heterostructure:

3.3.2. The CdTe heterostructure:

This section presents a microscopic theory of the spatio-temporal dynamics of a spin ensemble exposed to external electric ${\boldsymbol{ \mathcal E }}$ and magnetic fields ${\boldsymbol{B}}$. The approach used here is based on an extended version of the drift-diffusion kinetic equation [31, 54, 55]. This differential equation can be solved numerically in order to visualize the formation and decay of the spin helix and compare the results with experimental data. We also provide analytical results for some parameters of the spin density waves such as their decay rates, frequencies, and velocities. Similar results can be also obtained by means of the Green function technique [56, 57] or Monte-Carlo simulations [8].

Kinetic equation that describes the evolution of the spin distribution ${\boldsymbol{s}}({\boldsymbol{r}},{\boldsymbol{k}},t)$ in real and momentum spaces has the form

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\boldsymbol{s}}}{\partial t}+\displaystyle \frac{e}{{\hslash }}\left({\boldsymbol{ \mathcal E }}\cdot \displaystyle \frac{\partial }{\partial {\boldsymbol{k}}}\right){\boldsymbol{s}}+\displaystyle \frac{{\hslash }}{{m}^{* }}\left({\boldsymbol{k}}\cdot \displaystyle \frac{\partial }{\partial {\boldsymbol{r}}}\right){\boldsymbol{s}}\\ \quad =\,({{\boldsymbol{\Omega }}}_{L}+{{\boldsymbol{\Omega }}}_{{\boldsymbol{k}}})\times {\boldsymbol{s}}+\mathrm{St}\,{\boldsymbol{s}},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${{\boldsymbol{\Omega }}}_{L}=(g{\mu }_{B}/{\hslash }){\boldsymbol{B}}$ and ${{\boldsymbol{\Omega }}}_{{\boldsymbol{k}}}=(g{\mu }_{B}/{\hslash }){{\boldsymbol{B}}}_{\mathrm{so}}$ are the spin precession frequency in the external and spin-orbit effective magnetic fields, respectively, and $\mathrm{St}\,{\boldsymbol{s}}$ is the collision integral. The latter describes the relaxation of the distribution function due to electron scattering by impurities, phonons, and other electrons.

In the collision dominated regime, when ${{\rm{\Omega }}}_{{\boldsymbol{k}}}\tau \ll 1$ with τ being the scattering time, equation (14) yields the drift-diffusion equation for the local spin density ${\boldsymbol{S}}({\boldsymbol{r}},t)={\sum }_{{\boldsymbol{k}}}{\boldsymbol{s}}({\boldsymbol{r}},{\boldsymbol{k}},t)$ [31]

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\boldsymbol{S}}}{\partial t}+\left({{\boldsymbol{v}}}_{\mathrm{dr}}\cdot \displaystyle \frac{\partial }{\partial {\boldsymbol{r}}}\right){\boldsymbol{S}}={D}_{s}\displaystyle \frac{{\partial }^{2}{\boldsymbol{S}}}{\partial {{\boldsymbol{r}}}^{2}}-{\boldsymbol{\Gamma }}{\boldsymbol{S}}\\ \quad -\,\left({\boldsymbol{\Lambda }}\displaystyle \frac{\partial }{\partial {\boldsymbol{r}}}\right)\times {\boldsymbol{S}}+({{\boldsymbol{\Omega }}}_{L}+{{\boldsymbol{\Omega }}}_{\mathrm{dr}})\times {\boldsymbol{S}}.\end{array}\end{eqnarray} \tag{ 15 }$$

Here, ${{\boldsymbol{v}}}_{\mathrm{dr}}$ is the drift velocity of spin density perturbations, D_s is the spin diffusion coefficient, ${\boldsymbol{\Gamma }}$ is spin relaxation rate tensor [17], ${\boldsymbol{\Lambda }}$ is the tensor describing the spin precession at diffusion, and ${{\boldsymbol{\Omega }}}_{\mathrm{dr}}$ is the frequency of the spin precession caused by electron drift [58]. In the relaxation time approximation, the kinetic parameters above have the form

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{\mathrm{dr}}=\left\langle \displaystyle \frac{\partial ({\varepsilon }_{{\boldsymbol{k}}}{\tau }_{1})}{\partial {\varepsilon }_{{\boldsymbol{k}}}}\right\rangle \displaystyle \frac{e{\boldsymbol{ \mathcal E }}}{{m}^{* }},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{D}_{s}=\displaystyle \frac{1}{{m}^{* }}\langle {\varepsilon }_{{\boldsymbol{k}}}{\tau }_{1}^{* }\rangle ,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\alpha \beta }=\displaystyle \sum _{i=1,3}\left\langle \left({{{\boldsymbol{\Omega }}}_{{\boldsymbol{k}}}^{(i)}}^{2}{\delta }_{\alpha \beta }-{{\rm{\Omega }}}_{{\boldsymbol{k}},{\boldsymbol{\alpha }}}^{(i)}{{\rm{\Omega }}}_{{\boldsymbol{k}},{\boldsymbol{\alpha }}}^{(i)}\right){\tau }_{i}^{* }\right\rangle ,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\alpha \beta }=\displaystyle \frac{2{\hslash }}{{m}^{* }}\left\langle {{\rm{\Omega }}}_{{\boldsymbol{k}},{\boldsymbol{\alpha }}}^{(1)}{k}_{\beta }{\tau }_{1}^{* }\right\rangle ,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\Omega }}}_{\mathrm{dr}}=\displaystyle \frac{e{\hslash }}{{m}^{* }}\left\langle \displaystyle \frac{\partial }{\partial {\varepsilon }_{{\boldsymbol{k}}}}\left[({\boldsymbol{ \mathcal E }}\cdot {\boldsymbol{k}})\,{{\boldsymbol{\Omega }}}_{{\boldsymbol{k}}}^{(1)}{\tau }_{1}\right]\right\rangle ,\end{eqnarray} \tag{ 20 }$$

where τ₁ is the momentum relaxation time determining the electron gas mobility, ${\varepsilon }_{{\boldsymbol{k}}}={{\hslash }}^{2}{{\boldsymbol{k}}}^{2}/(2{m}^{* }),{\tau }_{1}^{* }$ and ${\tau }_{3}^{* }$ are the relaxation times of the first and third angular harmonics of the spin distribution, respectively, which are contributed and can be dominated by electron-electron scattering [59–61], ${{\boldsymbol{\Omega }}}_{{\boldsymbol{k}}}^{(i)}=(g{\mu }_{B}/{\hslash }){{\boldsymbol{B}}}_{\mathrm{so}}^{(i)}$, and the angle brackets denote averaging with the energy-derivative of the Fermi–Dirac function, $\langle {A}_{{\boldsymbol{k}}}\rangle ={\sum }_{{\boldsymbol{k}}}{A}_{{\boldsymbol{k}}}{f}_{\mathrm{FD}}^{{\prime} }/{\sum }_{{\boldsymbol{k}}}{f}_{\mathrm{FD}}^{{\prime} }$, ${f}_{\mathrm{FD}}^{{\prime} }={{df}}_{\mathrm{FD}}/d{\varepsilon }_{{\boldsymbol{k}}}$. Note that the drift velocity of spin-density perturbations ${{\boldsymbol{v}}}_{\mathrm{dr}}$ can differ from the conventional drift velocity of electrons ${{\boldsymbol{v}}}_{\mathrm{dr}}^{(e)}=\langle {\tau }_{1}\rangle e{\boldsymbol{ \mathcal E }}/{m}^{* }$ provided the momentum relaxation time τ₁ depends on energy.

For (001)-grown QWs with the effective magnetic field given by equation (8) and the degenerate electron statistics, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{\Gamma }} & = & {D}_{s}{\left(\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\right)}^{2}\left\{\left[\begin{array}{ccc}{\left(\beta -\alpha \right)}^{2} & 0 & 0\\ 0 & {\left(\alpha +\beta \right)}^{2} & 0\\ 0 & 0 & 2({\alpha }^{2}+{\beta }^{2})\end{array}\right]\right.\\ & & \left.+\displaystyle \frac{{\tau }_{3}^{* }}{{\tau }_{1}^{* }}{\beta }_{3}^{2}\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2\end{array}\right]\right\},\end{array}\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}{\boldsymbol{\Lambda }}=2{D}_{s}\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\left[\begin{array}{cc}0 & \alpha +\beta \\ \beta -\alpha & 0\\ 0 & 0\end{array}\right],\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\Omega }}}_{\mathrm{dr}}=\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\left\{\left[\begin{array}{c}(\alpha +\beta ){v}_{\mathrm{dr},y}\\ (\beta -\alpha ){v}_{\mathrm{dr},x}\\ 0\end{array}\right]-\displaystyle \frac{{\beta }_{3}}{1+(\varepsilon /{\tau }_{1})(\partial {\tau }_{1}/\partial \varepsilon )}\left[\begin{array}{c}{v}_{\mathrm{dr},y}\\ {v}_{\mathrm{dr},x}\\ 0\end{array}\right]\right\},\end{eqnarray} \tag{ 23 }$$

where all the values should be taken at the Fermi level. Note that the second term in equation (21) stems from ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ [17, 18] while the second term in equation (23) originates from the β₃ contribution to β in ${{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}$ [62].

We solve the drift-diffusion equation (15) by the Fourier transformation presenting the spin density in the form

$$\begin{eqnarray}&&S({\boldsymbol{r}},t)=\displaystyle \sum _{{\boldsymbol{q}}}{{\boldsymbol{S}}}_{{\boldsymbol{q}}}(t)\exp ({\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}),\end{eqnarray} \tag{ 24 }$$

there ${\boldsymbol{q}}$ is the wave vector of the spatial harmonic. The time evolution of the harmonic ${{\boldsymbol{S}}}_{{\boldsymbol{q}}}(t)$ is given by

$$\begin{eqnarray}&&{{\boldsymbol{S}}}_{{\boldsymbol{q}}}(t)=\exp (-{{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}t){{\boldsymbol{S}}}_{{\boldsymbol{q}}}(0),\end{eqnarray} \tag{ 25 }$$

where the tensor ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$ is defined by

$$\begin{eqnarray}\begin{array}{rcl}{\left({{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}\right)}_{\alpha \beta } & = & {{\rm{\Gamma }}}_{\alpha \beta }+D{{\boldsymbol{q}}}^{2}{\delta }_{\alpha \beta }+{\epsilon }_{\alpha \beta \gamma }{\left({{\boldsymbol{\Omega }}}_{L}+{{\boldsymbol{\Omega }}}_{\mathrm{dr}}-{\rm{i}}{\boldsymbol{\Lambda }}{\boldsymbol{q}}\right)}_{\gamma }\\ & & +{\rm{i}}{\boldsymbol{q}}\cdot {{\boldsymbol{v}}}_{\mathrm{dr}}.{\delta }_{\alpha \beta }\end{array}\end{eqnarray} \tag{ 26 }$$

For a given wave vector ${\boldsymbol{q}}$ the tensor ${{\rm{\Gamma }}}_{{\boldsymbol{q}}}$ has three eigenvalues describing the properties of eigen spin modes. The real part of an eigenvalue determines the decay rate of the mode while the imaginary part determines the mode frequency and velocity. The initial distribution of the harmonics ${{\boldsymbol{S}}}_{{\boldsymbol{q}}}(0)$ can be readily found from the initial distribution of the spin polarization in real space, ${{\boldsymbol{S}}}_{{\boldsymbol{q}}}(0)=\int S({\boldsymbol{r}},0)\exp (-{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}})d{\boldsymbol{r}}$. The integral (24) can be calculated numerically while some general results can be found analytically.

To proceed further with the analytical theory we disregard the ${{\boldsymbol{B}}}_{\mathrm{so}}^{(3)}$ contribution to the effective field and consider first the spatio-temporal spin dynamics in the absence of external fields, ${{\boldsymbol{v}}}_{\mathrm{dr}}=0$ and ${{\boldsymbol{\Omega }}}_{L}={{\boldsymbol{\Omega }}}_{\mathrm{dr}}=0$. In this case, the matrix ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$ is Hermitian. All three eigenvalues of ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$ are real and correspond to the relaxation rates of the spin modes. A minimal value of the decay rate, which corresponds to the longest lifetime, is achieved at a certain finite ${\boldsymbol{q}}$. This wave vector of the long-lived spin waves is directed along y if αβ > 0 and along x if αβ < 0. In the former case, the standing long-lived spin helix is formed from the spin waves with the wave vectors ${\boldsymbol{q}}=(0,\pm {q}_{\mathrm{SH}})$ with q_SH given by [57]

$$\begin{eqnarray}&&{q}_{\mathrm{SH}}=\displaystyle \frac{2{m}^{* }(\alpha +\beta )}{{{\hslash }}^{2}}\sqrt{1-\displaystyle \frac{1}{16}{\left(\displaystyle \frac{\alpha -\beta }{\alpha +\beta }\right)}^{4}}\,\end{eqnarray} \tag{ 27 }$$

and has the relaxation rate

$$\begin{eqnarray}&&{\gamma }_{\mathrm{SH}}={D}_{s}{\left(\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\right)}^{2}\displaystyle \frac{{\left(\alpha -\beta \right)}^{2}}{16}\left[7+\displaystyle \frac{4\alpha \beta }{{\left(\alpha +\beta \right)}^{2}}\right].\end{eqnarray} \tag{ 28 }$$

The polarization of the waves are given by the eigenvectors

$$\begin{eqnarray}&&{{\boldsymbol{s}}}_{\mathrm{SH}}=\displaystyle \frac{1}{\sqrt{2}}\left[0,\ \sqrt{1+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{\alpha -\beta }{\alpha +\beta }\right)}^{2}},\ \pm {\rm{i}}\sqrt{1-\displaystyle \frac{1}{4}{\left(\displaystyle \frac{\alpha -\beta }{\alpha +\beta }\right)}^{2}}\right]\end{eqnarray} \tag{ 29 }$$

revealing their helical structure. Interestingly, at $\alpha \approx \beta$ the lifetime of the persistent spin helix given by 1/γ_SH is twice as long as the lifetime of the x component of the spin polarization given by 1/Γ_xx.

When the terms standing for the effects of external magnetic or electric are present in equation (26), the matrix ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$ is no longer Hermitian and has complex eigenvalues. The imaginary parts of the eigenvalues describe the rotation of the spin helix in time and the related phase velocity.

The applied magnetic field induces the Larmor precession for the spin component orthogonal to the magnetic field. For weak fields, the Larmor precession leads to small imaginary parts ${\rm{i}}{\omega }_{{\boldsymbol{q}}}$ of the eigenvalues of the matrix ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$ which are given by ${\omega }_{{\boldsymbol{q}}}={\rm{i}}{{\boldsymbol{\Omega }}}_{L}\cdot [{{\boldsymbol{s}}}_{{\boldsymbol{q}}}\times {{\boldsymbol{s}}}_{{\boldsymbol{q}}}^{* }]$, where ${{\boldsymbol{s}}}_{{\boldsymbol{q}}}$ are the corresponding eigenvectors of ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$. For the long-lived spin helix with the eigenvector (29) we obtain ω_SH = q_SH v_SH,y with the phase velocity

$$\begin{eqnarray}&&{v}_{\mathrm{SH},y}=-\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{* }(\alpha +\beta )}\,{{\rm{\Omega }}}_{L,x}.\end{eqnarray} \tag{ 30 }$$

The group velocity of the spin wave is determined by ${{\boldsymbol{v}}}^{(\mathrm{gr})}=\partial {\omega }_{{\boldsymbol{q}}}/\partial {\boldsymbol{q}}$. For the long-lived spin helix, the group velocity has the form

$$\begin{eqnarray}&&{v}_{\mathrm{SH},y}^{(\mathrm{gr})}=\displaystyle \frac{1}{16}{\left(\displaystyle \frac{\alpha -\beta }{\alpha +\beta }\right)}^{4}{v}_{\mathrm{SH},y}\,\end{eqnarray} \tag{ 31 }$$

and is much smaller that the phase velocity ${v}_{\mathrm{SH}}$ since α is close to β. Therefore, the external magnetic field applied along the x-axis leads to a motion of the spin pattern with the phase velocity determined by α + β without a considerable shift of the pattern envelope.

In the case of a strong in-plane magnetic fields, the dominant term in the right-hand side of equation (26) is the one describing the spin precession with the Larmor frequency ${{\boldsymbol{\Omega }}}_{L}={{\rm{\Omega }}}_{L}(\cos \phi ,\sin \phi ,0)$, where ϕ is the polar angle of the vector ${{\boldsymbol{\Omega }}}_{L}$. There are three eigenvalues of the matrix ${{\boldsymbol{\Gamma }}}_{{\boldsymbol{q}}}$: 0 and ∓iΩ, and three corresponding eigenvectors: ${{\boldsymbol{s}}}_{{\boldsymbol{q}}}^{(0)}=[\cos \phi ,\sin \phi ,0]$ and ${{\boldsymbol{s}}}_{{\boldsymbol{q}}}^{(\pm )}=[-\sin \phi ,\cos \phi ,\pm {\rm{i}}]$. Taking into account other terms in the right-hand side of equation (26) as a perturbation and neglecting the mode mixing we obtain the dependence of the eigenvalues on ${\boldsymbol{q}}$

$$\begin{eqnarray}&&{\gamma }_{{\boldsymbol{q}}}^{(0)}={D}_{s}\left[{{\boldsymbol{q}}}^{2}+{\left(\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\right)}^{2}\left({\alpha }^{2}+{\beta }^{2}-2\alpha \beta \cos 2\phi \right)\right],\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{{\boldsymbol{q}}}^{(\pm )} & = & {D}_{s}\left\{{{\boldsymbol{q}}}^{2}\pm \displaystyle \frac{4{m}^{* }}{{{\hslash }}^{2}}\left[{q}_{y}(\beta +\alpha )\cos \phi +2{q}_{x}(\beta -\alpha )\sin \phi \right]\right.\\ & & \left.+{\left(\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\right)}^{2}\left[\displaystyle \frac{3}{2}({\alpha }^{2}+{\beta }^{2})+\alpha \beta \cos 2\phi \right]\right\}.\end{array}\end{eqnarray} \tag{ 33 }$$

The minimum of the spin relaxation rate is achieved for the modes ${{\boldsymbol{s}}}_{{\boldsymbol{q}}}^{(\pm )}$ at ${{\boldsymbol{q}}}_{\mathrm{SH}}=\mp (2{m}^{* }/{{\hslash }}^{2})[(\beta -\alpha )\sin \phi ,(\alpha +\beta )\cos \phi ]$, i.e., in the direction determined by the polar angle

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{SH}}=\displaystyle \frac{\pi }{2}+{\tan }^{-1}\left(\displaystyle \frac{\alpha -\beta }{\alpha +\beta }\tan \phi \right).\end{eqnarray} \tag{ 34 }$$

The spin helix has the phase speed

$$\begin{eqnarray}&&{v}_{\mathrm{SH}}=\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{* }}\,\displaystyle \frac{{{\rm{\Omega }}}_{L}}{\sqrt{{\alpha }^{2}+{\beta }^{2}+2\alpha \beta \cos 2\phi }}\end{eqnarray} \tag{ 35 }$$

and the relaxation rate

$$\begin{eqnarray}&&{\gamma }_{\mathrm{SH}}=\displaystyle \frac{{D}_{s}}{2}{\left(\displaystyle \frac{2{m}^{* }}{{{\hslash }}^{2}}\right)}^{2}\left({\alpha }^{2}+{\beta }^{2}-2\alpha \beta \cos 2\phi \right).\end{eqnarray} \tag{ 36 }$$

Figure 6 shows the dependence of the direction of the long-lived spin helix on the magnetic field orientation for two different ratio between α and β. When the magnetic field is rotated, the spin helix direction rotates in the same direction if $| \alpha | \gt | \beta |$ and in the opposite direction if $| \alpha | \lt | \beta |$, see the red and blue curves, respectively. The dependence Φ_SH (ϕ) is strongly nonlinear: The spin helix direction is 'pinned' to its natural axis in the absence of the magnetic field. The closer the absolute values of α and β are, the stronger the pinning is, see red and blue curves. However, the spin helix is always oriented perpendicularly to the strong magnetic field if the field is applied along the $x\parallel [1\bar{1}0]$ or $y\parallel [110]$ axes.

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Finally, we consider the effect of an external electric field on the spin helix evolution. We note that the second term in the right-hand side of the drift-diffusion equation (15) can be eliminated by switching to the reference frame moving with the drift velocity ${{\boldsymbol{v}}}_{\mathrm{dr}}$. In such a frame, the effect of the electric field is limited to the drift-induced spin precession with the frequency ${{\boldsymbol{\Omega }}}_{\mathrm{dr}}$ and, in fact, is equivalent to the action of the magnetic field. Repeating the calculations which are done above for the case of a weak magnetic field and returning back to the reference frame we obtain the phase velocity of the spin helix in the electric field

$$\begin{eqnarray}&&{v}_{\mathrm{SH},y}=\displaystyle \frac{{\beta }_{3}}{\alpha +\beta }\,\displaystyle \frac{{v}_{\mathrm{dr},y}}{1+(\varepsilon /{\tau }_{1})(\partial {\tau }_{1}/\partial \varepsilon )}.\end{eqnarray} \tag{ 37 }$$

The current-induced spin precession of quasi-stationary electron distribution has been studied in [62].

Complementary to the kinetic theory another approach for the theoretical simulation of spin dynamics is based on the Monte Carlo (MC) method. In general, the MC method is commonly used in computational algorithms as it permits to solve deterministic problems where the need for complex calculation can be bypassed through a large number of random sampling. In comparison with the analytical solving of a differential equation a MC simulation demands a much higher expense in computational calculations. However, often times MC based algorithms can be formulated with much less effort since the calculation of complicated physical effects can be avoided in favor of random sampling. To simulate the spatio-temporal dynamics of a spin ensemble, the diffusion of the spin-afflicted electrons is modeled as the result of consecutive elastic scattering events with a randomization of the electron's direction. Recent publications have shown that the MC algorithm is a suitable tool for the validation of the PSH evolution in general [63] and as well for rather minute features predicted by theory [62, 64, 65].

The specific MC algorithm employed here accumulates the iterative pathway of up to 1 × 10⁶ electrons, traveling with the a constant Fermi velocity which is given by the electron density as ${v}_{{\rm{F}}}={\hslash }\sqrt{2\pi {n}_{{\rm{e}}}}/{m}^{* }$. Assuming the electron velocity to be constant is a reasonable approach within the Fermi–Dirac regime where the occupied states for non-localized carriers are restricted to a small energy band around the Fermi energy. For increased temperatures where this energy band is smeared out a distribution of velocities, in accordance with the Boltzmann statistics, must be applied. Figure 7(a) visualizes a schematic random walk in the QW plane where the electron experiences an elastic scattering event after having traveled for the free mean path l = v_Fτ given by the scattering time. The scattering time is individually drawn from an exponential distribution $f(t)=(1/{\tau }_{1}^{* })\exp (-t/{\tau }_{1}^{* })$ where ${\tau }_{1}^{* }$ is the momentum relaxation time of individual electrons. This sampling accounts for the fact that the time inbetween two scattering events exhibits a stochastic variation. The momentum relaxation time can be obtained from the experimentally observed spin diffusion coefficient given by ${D}_{{\rm{s}}}={\tau }_{1}^{* }{v}_{{\rm{F}}}^{2}/2$. As a result of scattering, the wave vector $\vec{k}$ of the electron is rotated on the two-dimensional Fermi circle in k-space depicted in figure 7(b) and hence the electron changes its direction. The rotation of the wave vector is approximated by a random rotation angle φ taken from a uniform distribution from π to −π. The effective magnetic field, seen by the electron spin during the unperturbed propagation between two scattering events, is taken as constant. However, recalling equation (1) it has been shown that Rashba and Dresselhaus SO coupling is momentum dependent and consequently ${{\boldsymbol{B}}}_{\mathrm{so}}$ will change together with the wave vector after each event of scattering. Therefore, it is necessary to calculate the three-dimensional rotation for each spin polarization vector ${\boldsymbol{S}}=({S}_{{\rm{x}}},{S}_{{\rm{y}}},{S}_{{\rm{z}}})$ individually. The axis of rotation is given by the effective field vector ${{\boldsymbol{B}}}_{\mathrm{so}}({k}_{{\rm{x}}},{k}_{{\rm{y}}})$ that acts as a torque on the spin momentum for each iteration i as

$$\begin{eqnarray}&&{\left(\begin{array}{c}{S}_{{\rm{x}}}\\ {S}_{{\rm{y}}}\\ {S}_{{\rm{z}}}\end{array}\right)}_{({\rm{i}}+1)}={\boldsymbol{R}}\left(\displaystyle \frac{{{\boldsymbol{B}}}_{\mathrm{so},{\rm{i}}}}{| {{\boldsymbol{B}}}_{\mathrm{so},{\rm{i}}}| },{{\rm{\Omega }}}_{\mathrm{so},{\rm{i}}}\tau \right)\cdot {\left(\begin{array}{c}{S}_{{\rm{x}}}\\ {S}_{{\rm{y}}}\\ {S}_{{\rm{z}}}\end{array}\right)}_{{\rm{i}}}\end{eqnarray} \tag{ 38 }$$

where ${\boldsymbol{R}}({\boldsymbol{T}},\theta )$ is a standard three-dimensional rotation matrix which rotates a vector around the axis ${\boldsymbol{T}}$ by the angle θ = Ω_soτ. The angle of rotation is the accumulated Larmor precession between to scattering events. Within the MC simulation the resulting vector of spin polarization is tracked together with the current position and wave vector of each electron. Hence, the spatial position of each iteration step is obtained from

$$\begin{eqnarray}&&{\left(\begin{array}{c}x\\ y\end{array}\right)}_{({\rm{i}}+1)}=\displaystyle \frac{{\hslash }}{{m}^{* }}\left[\begin{array}{c}{k}_{{\rm{F}}}\sin ({\varphi }_{{\rm{i}}})\\ {k}_{{\rm{F}}}\cos ({\varphi }_{{\rm{i}}})\end{array}\right]\cdot \tau +{\left(\begin{array}{c}x\\ y\end{array}\right)}_{{\rm{i}}}.\end{eqnarray} \tag{ 39 }$$

After a certain number of scattering events N the S_z component of spin polarization and the spin position is transferred to a two-dimensional false-color plot which displays the spin polarization in dependence on the spatial position for a time t_d = N · τ after the excitation. Modeling the Gaussian laser beam profile the initial position (x₀, y₀) of each spin is randomly sampled from a Gaussian distribution around the origin with a FWHM equivalent to the spot size of the laser. Due to a large number of electrons the bins in the false-color plot are filled up with the sum of all z spin components ${\sum }_{{\rm{i}}}{S}_{{\rm{z}}}^{{\rm{i}}}$ assigned to the specific area covered by the bin.

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The section is meant to accustom the reader to the preparation of data developed to visualize the SO coupling mechanisms that drive the measured PSH spin texture. Furthermore, it is discussed how the parameters that are relevant for the PSH are extracted. The section is concluded with a first quantification of the Rashba and Dresselhaus parameters and their validation via simulations based on both the MC method and kinetic theory. Measurements of spin polarization are performed by means of the MOKE effect, allowing for the extraction of S_z from the angle of Kerr rotation. Representing the space and time resolved S_z(x, y, t) data as false-color plot has proven to be an effective way to visualize the PSH. Here, a red to blue color scale indicates the spin polarization component S_z perpendicular to the QW plane, and the in-plane axes are chosen as $x\parallel [1\bar{1}0]$ and y∥[110]. According to section 2.2, the PSH is expected to appear as an unidirectional wave of spin polarization driven by the diffusive expansion of the electron ensemble. A first prediction of the resulting pattern has been developed in figure 2(e) where a parameterization with the spatial wavelength λ_0,y has been suggested.

This behaviour can indeed be observed in the experiments as shown in figure 8 for the GaAs sample. The three false-color plots in this figure exemplify different ways of how the PSH can be observed, starting with a two-dimensional (2D) map in space depicted in figure 8(a). This 2D map is taken for a constant back-gate voltage of −1.00 V and captures the diffusive status at a fixed delay time of 840 ps. In line with the theoretical expectation the initially oriented spin polarization has not just expanded diffusively in space but has formed an oscillating pattern along the y-axis, indicating a rotation of S_z caused by ${{\boldsymbol{B}}}_{\mathrm{so}}$. The temporal evolution of the PSH can be tracked by means of the spatiotemporal false-color plots in figures 8(b) and (c) along the x- and y- axes, respectively. Such spatio-temporal maps are composed from single line scans of S_z in dependence of the spatial position. A subset of such line scans taken for different delay times are shown in figure 8(d) along the y-axis. It becomes apparent that the spin polarization expands in space while an additional decay mechanism weakens the signal for long delay times. Moreover, the initial Gaussian shape remains intact over time and no oscillating features are present along the x-axis. However, along the y-axis the initial Gaussian spin distribution, shown in figure 8(d) for the time overlap at 0 ns, develops distinct oscillations on the micrometer range. This spatial anisotropy in the pattern is readily explained by the effective magnetic field distribution in k-space

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{so}}^{(1)}({\boldsymbol{k}})=\displaystyle \frac{2}{g{\mu }_{{\rm{B}}}}\left[\begin{array}{c}(\alpha +\beta ){k}_{{\rm{y}}}\\ (\beta -\alpha ){k}_{{\rm{x}}}\end{array}\right]\end{eqnarray} \tag{ 40 }$$

adapted from equation (8). This expression implies zero magnetic field for electrons that move along the x-axis in a regime of balanced Rashba and Dresselhaus SO coupling. The observed PSH evolution is in excellent agreement with previous studies in similar structures and the line slices along y at a fixed delay time can be described by the fit function

$$\begin{eqnarray}&&{S}_{{\rm{z}}}(y)={A}_{0}\cdot {{\rm{e}}}^{-4\mathrm{ln}(2){\left(y-{y}_{{\rm{c}}}\right)}^{2}/{w}^{2}}\cdot \cos \left[\displaystyle \frac{2\pi (y-{y}_{1})}{{\lambda }_{\mathrm{so},{\rm{y}}}}\right]\end{eqnarray} \tag{ 41 }$$

which combines the Gaussian solution of the spin diffusion equation with a cosine modulation by the spin precession length ${\lambda }_{\mathrm{so},{\rm{y}}}$, corresponding to a characteristic distance for a spin to undergo one full rotation. While the position of the oscillation pattern is determined by the cosine phase parameter y₁ the spatial parameter y_c denotes the center of the Gaussian spin polarization in space. The fit function describes line scan data at a fixed delay time. All effects of spin decay, diffusion, drift and precession can be integrated into the temporal change of the fit parameters A₀, w, y_c, and y₁. The transparent color lines in figure 8(d) represent best fits of equation (41) to the individual slices of the corresponding spatio-temporal map along the y-axis. Returning attention to the 2D map in panel (a), it is seen that the blue colored PSH stripes are somewhat curved. This fact indicates that BIA and SIA are not entirely balanced. This feature is discussed below, see figure 16.

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The focus shall now be shifted towards a more quantitative description of the phenomenon with a parameterization by means of the introduced fit function for S_z(y). In order to compare SO coupling in the GaAs and CdTe QWs, extracted fit parameters for both materials are shown in figure 9 (note that the 2D data for CdTe are shown in figure 10). Figure 9(a) depicts the spin precession length as a function of the delay time. For both materials, ${\lambda }_{\mathrm{so},{\rm{y}}}$ decreases in time which is not covered by the static expression in equation (10). However, it has been shown that such a dependence originates from the gradual evolution of the spin distribution from a narrow Gaussian spin distribution at short delay times towards a single mode oscillation at long times [63]. This finite-spot size effect results in a gradual variation of ${\lambda }_{\mathrm{so},{\rm{y}}}$ to the PSH mode wavelength ${\lambda }_{0,{\rm{y}}}$, that is described by

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\mathrm{so},{\rm{y}}}({t}_{{\rm{d}}}) & = & {\lambda }_{0,{\rm{y}}}\left[1+\displaystyle \frac{{w}_{0}^{2}}{16\mathrm{ln}(2){D}_{{\rm{s}}}{t}_{{\rm{d}}}}\right]\quad \mathrm{with}\\ {\lambda }_{0,{\rm{y}}} & = & \displaystyle \frac{\pi {{\hslash }}^{2}}{{m}^{* }}{\left(\alpha +\beta \right)}^{-1}.\end{array}\end{eqnarray} \tag{ 42 }$$

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Thus, the value of ${\lambda }_{\mathrm{so},{\rm{y}}}$ observed at a specific delay time depends on the initial spot size w₀ and the diffusion coefficient D_s [48]. Figure 9(a) shows experimental data for ${\lambda }_{\mathrm{so},{\rm{y}}}$ together with a fit to this equation. It reveals the PSH mode to be λ_0,y = (5.6 ± 0.1)μm for CdTe and ${\lambda }_{0,{\rm{y}}}=(11,2\pm 0.2)\mu {\rm{m}}$ for GaAs. The corresponding values of (α + β) resulting are (3.19 ± 0.05) meVÅ for GaAs and (4.55 ± 0.08) meVÅ for CdTe. At first glance the difference in between CdTe and GaAs is not very distinct, however, a general comparison should be performed on the level of the Dresselhaus coupling constant γ_D to rule out the dependence of α and β on sample specific parameters such as the electron density and the QW thickness. This discussion is postponed to section 5.2.1 where the BIA and SIA parameters are determined individually. Furthermore, the space- and time-resolved measurements allow for the extraction of the spin diffusion coefficient D_s by tracing the squared spatial width w²(t) of the Gaussian envelope function with time. Corresponding data for both samples are shown in figure 9(b). In particular, the expected linear time dependence ${w}^{2}(t)={w}_{0}^{2}+16\mathrm{ln}(2){D}_{{\rm{s}}}t$ of the squared width is utilized to extract of D_s = (50 ± 10) cm² s⁻¹ for CdTe and (54.0 ± 0.8) cm² s⁻¹ for GaAs. Assuming the 2DEG to be degenerate at 3.5 K facilitates the calculation of the individual electron momentum relaxation time ${\tau }_{1}^{* }$ from the theoretical relation ${D}_{{\rm{s}}}={\tau }_{1}^{* }\cdot {E}_{{\rm{F}}}/{m}^{* }$. The resulting values are noted in table 2 together with other sample specific parameters. E_F depends on the carrier density and, thus, on the backgate voltage. In the regime displayed in 9(d) the carrier density varies by a factor of ∼3. The experimentally observed change of D_s depicted in figure 9(d), however, is much weaker and only a tuning by ∼30% is seen. This discrepancy can be accounted for by the fact that the 2DEG can not be treated as degenerate gas but is rather in an intermediate regime towards the non-degenerate statistics.

Table 2.  List of sample parameters extracted from the PSH characterization for both materials. For CdTe a short and a long living decay time are observed.

	${\lambda }_{0,{\rm{y}}}\,$(μm)	Ds (cm2/s)	w0 (μm)	[α + β] (eVÅ)	Ts (ps)	EF (meV)	τee (ps)
GaAs	(11,2 ± 0.2)	(54.0 ± 0.8)	3.62	(3.19 ± 0.05)	∼2200	3.05	0.6
CdTe	(5.6 ± 0.1)	(50 ± 10)	3.87	(4.55 ± 0.08)	1350/170	(9.0 ± 0.1)	0.3

Beyond the spatial parameters ${\lambda }_{\mathrm{so},{\rm{y}}}$ and D_s, the investigation of spin lifetimes is of peculiar interest. It is expected that the SU(2) symmetry, which gives rise to the PSH, creates a regime that is extraordinarily robust against spin decoherence caused by the DP mechanism. The spin lifetime of the PSH is extracted from the temporal decay of the fit function amplitude parameter A₀. It is important to distinguish the exponential loss of spin coherence and the fact that the amplitude is decreased by spins that diffuse away from the spot of excitation. A combination of these two contributions results in the expression

$$\begin{eqnarray}&&{A}_{0}({t}_{{\rm{d}}})={A}_{0}(0)\cdot \displaystyle \frac{{w}_{0}^{2}}{{w}_{0}^{2}+16\mathrm{ln}(2){D}_{{\rm{s}}}{t}_{{\rm{d}}}}\cdot {e}^{-{t}_{{\rm{d}}}/{T}_{{\rm{s}}}}\end{eqnarray} \tag{ 43 }$$

With the extracted values for D_s and ${w}_{0}^{2}$, the spin lifetime T_s is obtained from best fits of equation (43) to the delay time dependence of A₀(t_d). Figure 9(c) shows T_s for a range of back-gate voltages, with a maximum value of ∼2.2 ns seen for U_BG ≈ −0.5 V. A peak-like tuning of lifetimes around a certain gate voltage has been predicted and observed in different studies [13, 18, 19, 31, 40]. This finding reflects the variation of the Rashba parameter α that depends on the electric field ${{ \mathcal E }}_{\mathrm{QW}}$ and is thus directly influenced by U_BG. However, later figure 17(e) shows that the ratio α/β converges for back-gate voltages below −1.0V, approximately. Therefore, the decrease of lifetime on the left hand of the peak (from −0.5 V to −2.0 V) is attributed to the strong depletion of the QW as a result of the rather high, negative back-gate voltage.

A comparison of equation (40) with the k-space anisotropy in figure 2(a) shows a clear dependence of the SU(2) symmetry robustness on the level of balance between α and β. As this symmetry suppresses the DP mechanism and therefore leads to very long spin lifetimes, the lifetime peak is assumed to appear for a voltage where α is tuned as close as possible towards β. This finding justifies the choice of U_BG = −1.00 V for all following measurements. This specific voltage represents a trade-off in-between a long-living PSH regime and a strong Kerr rotation signal. The KR signal itself is altered by U_BG via a density-dependent resonance shift, affecting the optical detection (see figure 4(c)) In conclusion, the fundamental parameters of the PSH in GaAs and CdTe are now mapped out and listed in table 2.

We now turn attention to the theoretical modeling of the 2D maps based on the approaches in sections 4.1 and 4.4. Figure 10 depicts three sets of a 2D (x, y) and spatio-temporal (y, t) maps. The sets represent one measurement of the CdTe sample and two simulations that utilize the CdTe parameters from table 2. Although only the sum of Rashba and Dresselhaus parameter is known so far, the required individual parameters can be calculated with theoretical value of β in table 1. However, for the simulations given here the individual parameters are taken from a later section, see table 3, to obtain the best agreement with the experiment. Figure 10(a) shows the status of the PSH after 770 ps delay characterized by the same unidirectional spin helix wave as in GaAs, however, with a shorter spin precession length. This finding of the Larmor precession pattern, confirms the predominant appearance of B_so for electrons that move along the y direction and is validated by both simulated 2D maps. A similar strong agreement is found by visual comparison of the spatio-temporal maps generated from the measurement (a) and simulations (b, c).

While the general reproduction is successful and thus suggests the model to be accurate some smaller deviations can be identified. Firstly, the polarization stripes of the experimental 2D map are visibly more curved than in both simulations. Therefore, this curvature is not explained by an imbalance of α and β only, because such an effect is taken into account in both simulations. It therefore seems more likely that the curvature is a feature that arises from spatially varying SO coupling which will be discussed in-depth later on in section 5.3. Secondly, for short delay times in panel (a) the experimental S_z appears much larger than in the simulations in panels (b-c). This observation suggests a much faster decline of the spin polarization at the beginning of the experiment for t_d < 300 ps. Indeed, a secondary, fast exponential decay with a lifetime of 170 ps is seen in A₀(t_d). This finding cannot be attributed to thermal effects due to the fairly long time scale. However, a variety of effects can account for short-lived contributions in the present system. For instance, it cannot be ruled out that an unwanted spin polarization with short lifetime is injected into the substrate where it distorts the spin signal which comes from the QW. Another effect could arise from a high photo-carrier density at the temporal overlap, increasing inelastic scattering rates and spin-spin interactions. Such effects for short delay times are not included in the simulations and thus are a reasonable explanation of the encountered differences. Finally, a two-fold decay that is related to the finite-spot size effect is generally expected. Due to the initial transformation of a Gaussian spectrum into the single PSH mode an additional short decay arises at the beginning of the measurement. For longer delay times, the decay is dominated by the pure PSH lifetime which is discussed in the end of section 2.2.

Simulations performed with parameters for GaAs are available in [31] for a sample similar to the one used in this work and show remarkable success in reproducing the experimental data.

The focus is now shifted towards manipulation and control of the spin polarization within the PSH. Therefore, a variety of different control knobs are consecutively investigated and their effects are quantified. Starting with the application of magnetic fields, the influence of back-gate voltage is studied and the PSH is exposed to in-plane electric fields in order to introduce spin drift. The latter case is of interest as peculiar effects on the SO coupling parameters have been reported in recent literature. However, the nature of the related changes, visible in the spin precession length, is not yet fully understood. Hence, special attention is paid to the dependence of λ_0,y on the in-plane electric field strength for which contradicting trends are claimed in [30, 31, 62]. Besides the application of external magnetic and electric fields, the SO coupling is strongly affected by the injection of photo-excited carriers into the 2DEG of the CdTe QW. This finding enables an all-optical approach towards PSH control.

5.2.1. Manipulation by in-plane magnetic fields

The characteristic spin texture of the PSH is shaped by effective magnetic fields oriented along the QW plane. Here, we add an additional external magnetic field ${{\boldsymbol{B}}}_{\mathrm{ex}}=({B}_{\mathrm{ex},{\rm{x}}},{B}_{\mathrm{ex},{\rm{y}}})$ (note that again y∥[110] and $x\parallel [1\bar{1}0]$). Figures 11(a)–(c) illustrates the effect of different applied magnetic field strengths on the spatio-temporal evolution of the PSH in the GaAs sample. The spin helix without external manipulation [panel (a)], evolves symmetrically around the origin. An additional external magnetic field B_ex,x, however, imposes a temporal spin precession with a Larmor frequency of ${{\rm{\Omega }}}_{L}=g{\mu }_{{\rm{B}}}{B}_{\mathrm{ex},{\rm{x}}}/{\hslash }$. This temporal precession at a fixed spatial position is also referred to as spin beating [67]. Such spin beatings are visible in figure 12(e), where the time dependence of S_z at position (x, y) = (0, 0) is depicted in for an applied magnetic field of 230mT. A best fit with an oscillating, single exponential decay function reveals the Larmor frequency. From this transient, a g-factor of g = −0.29 is determined for the GaAs QW, consistent with the expected increase of the g-factor from the value of −0.44 in bulk GaAs [68–70]. Equivalent measurements in the CdTe sample yield a g-factor of 1.6, in line with other experiments on the same structure [48, 71].

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The spatio-temporal measurements in figures 11(b) and (c), the external magnetic field seems to tilt the PSH stripes over time. As guide to the eye, this tilt is highlighted by grey, dashed lines in the plot. From a comparison of panel (b) and (c), a steepening of the tilt is observed with increased field strength. Generally, the tilt represents an electron velocity for which the spin precession phase is constant. This is equivalent to a fixed spin orientation for electrons that travel with a certain average momentum. As a result, the effective fields have to cancel the external field. Reformulating equation (40) with ${\hslash }{k}_{y}={m}^{* }{v}_{{\rm{y}}1}$, this holds true for a phase velocity of

$$\begin{eqnarray}&&{v}_{{\rm{y}}1}=\displaystyle \frac{{{dy}}_{1}}{{dt}}=\displaystyle \frac{g{\hslash }{\mu }_{{\rm{B}}}}{2{m}^{* }(\alpha +\beta )}\cdot {B}_{\mathrm{ex},{\rm{x}}}\end{eqnarray} \tag{ 44 }$$

in agreement with equation (35) for ${{\boldsymbol{B}}}_{\mathrm{ex}}\parallel x$, i.e. ϕ = 0. The experimental values of v_y1 range up to several km/s and can be obtained from a linear fit to the delay time dependence of y₁.

This concept of PSH manipulation by external magnetic fields is sketched in a more visual manner in figure 12(f) where the magnetic field distribution in k-space is shown along the k_y-axis for the cases with (on the right) and without (on the left) external field. Without an external field, the effective magnetic fields, illustrated by the arrows, grow in magnitude along the y-direction. The photo-injected electron ensemble is centered at k = 0 and the position, where no Larmor precession occurs, is also found at k = 0. When superimposing an external magnetic field, the position in k-space for which no net field acts on the electron spins is shifted from k_y = 0 to some finite k-value. Diffusing electrons that exhibit this specific k-value will, therefore, not undergo Larmor precessions but maintain their initial spin orientation.

The application of different magnetic field strengths along the x-axis, exemplified by the two measurements in panel (b) and (c), and the subsequent extraction of v_y1 facilitates the determination of (α + β). Specifically, a linear regression based on equation (44) is used to obtain (α + β). A value of (α + β) = 2.94 meVÅ is found, in good agreement with the value in table 2 determined from the spin precession length. Moreover, measurements in the presence of external magnetic field are used to study the influence of the back-gate voltage on α and β. The acquired data is shown in figure 11(g) and reveals a clear trend for a decrease of (α + β) with increasing U_BG. An impact of of the back-gate voltage is generally expected as it modifies both β₃ via the carrier density and α via the electric field along the QW. This influence is investigated in more detail in section 5.2.3.

The proposed explanation for the observed tilt in the PSH stripes is confirmed by corresponding simulations based on the MC method. For this purpose, the Larmor precession is added to the precession in the effective magnetic field. The false-color plots in figures 11(d) and (e) agree well with the experimental observations, shown in the panels above. The entire set of measurements presented for GaAs in figure 11 is also repeated for the CdTe sample. A value for (α + β) close to the value given in table 3 is found. However, for CdTe the MC simulations show significant dissimilarities with the corresponding experimental data. We later attribute this finding due to the influence of optically induced carriers. The corresponding data are postponed to section 5.3 where this impact of optical doping is studied.

So far external magnetic fields along the x-axis have been applied. This orientation allows to weaken or enhance the effective magnetic fields seen by the electron spins that are recorded in spatio-temporal maps along the y-axis. In contrast, by applying an external field along the y-direction it is possible to access (α − β) from spatio-temporal measurements. Such data are shown in figures 12(a)–(c). A visual inspection suggests a stronger tilt of the PSH stripes compared to analogous panels of figure 11. The fit function in equation (41) is used again for the spin distribution along the x-axis. In contrast to the fit function used along the y-axis the spin precession length along the x-axis now reads

$$\begin{eqnarray}&&{\lambda }_{0,{\rm{x}}}=\displaystyle \frac{\pi {{\hslash }}^{2}}{{m}^{* }}{\left(-\alpha +\beta \right)}^{-1}.\end{eqnarray} \tag{ 45 }$$

It is determined by the difference of Rashba and Dresselhaus parameter. Accordingly, a much longer wavelength is expected since their difference is supposed to approach zero for a perfectly balanced PSH regime. For a quantitative analysis, the temporal dependence of the x₁ parameter is extracted (note that x₁ now replaces y₁ in in equation (41)) the difference (−α + β) can be extracted from the relation for the x-axis phase velocity v_x1 that, in analogy to v_y1, takes the form

$$\begin{eqnarray}&&{v}_{{\rm{x}}1}=\displaystyle \frac{{\rm{d}}{x}_{1}}{{\rm{d}}t}=\displaystyle \frac{g{\hslash }{\mu }_{{\rm{B}}}}{2{m}^{* }(-\alpha +\beta )}\cdot {B}_{\mathrm{ex},{\rm{y}}},\end{eqnarray} \tag{ 46 }$$

see equation (35) for ${\boldsymbol{B}}\parallel x$, i.e. ϕ = π/2. Figure 12(d) confirms this linear dependence on the external magnetic field. Moreover, the significantly increased velocities compared with v_y1 are in agreement with the assumption of a PSH regime where the SO coupling parameters are similar. From a best linear fit it is possible to quantify (−α + β) = 1.15 meVÅ and to calculate the individual parameters α = 0.90 meVÅ and β = 2.05 meVÅ. The ratio of α/β ∼ 0.5 points to an imperfect PSH regime. However, the results so far show that even in such case the characteristic spin helix pattern is still distinct enough. In retrospect, this unbalance is responsible for the convex curvature in the 2D map of figure 8 [57]. Values for the CdTe sample are incorporated as well in table 3 and exhibit a ratio of α/β ∼ 1.35. In summary, the level of balance in the CdTe sample is slightly higher than in GaAs.

5.2.2. Manipulation by in-plane electric fields

While the application of magnetic fields affects mostly the spins' magnetic moments, the diffusive motion of the spin carriers remains unperturbed. Within this section it is discussed how the spin texture is manipulated by in-plane electric fields along the x- and y-axis. Recently, the effect of electric fields on the PSH evolution has raised special interest due to contradicting observations with respect to the dependence of λ_0,y on the field strength. While earlier investigations of Altmann et al [62] report an unchanged spin precession length it has been seen by Kunihashi et al [30] and Anghel et al [31] that λ_0,y varies with electron drift, however, with opposite trends. Below we revisit this topic and propose an increased 2DEG temperature due to bias-induced heating as tuning mechanism for the PSH. In the same line of experiments, we address the observation by Altmann et al [62] that the spin precession angle that stems from the cubic Dresselhaus parameter is a factor of two larger for drifting electrons than for diffusing ones. As a consequence, a temporal spin precession appears even in the absence of external magnetic fields. From the related precession frequency the cubic Dresselhaus parameter β₃ becomes accessible.

When an in-plane electric field is applied, the photo-injected electron ensemble will start to drift according to the electron mobility. We refer to the emerging spin texture as a drifting persistent spin helix (DPSH). We start with a drift motion along the y-axis. A sequence of 2D maps, capturing the temporal evolution of such a DPSH, is depicted in figure 13. The formation of the characteristic PSH stripe pattern is clearly visible while the center of spin polarization drifts towards negative y-values due to the applied electric field of ${E}_{\mathrm{IP}}^{{\rm{y}}}=0.91\,{\rm{V}}\,{\mathrm{cm}}^{-1}$ (note that an inversion of the electric field leads to inverted but otherwise very similar 2D maps). Comparing the DPSH series with the snapshot of the stationary PSH in figure 8(a) it is apparent that the DPSH features a larger number of oscillations. This observation already points to enhanced diffusion which further spreads the Gaussian envelope in the case of the DPSH.

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The temporal evolution of the DPSH for a constant electric field of ${E}_{\mathrm{IP}}^{{\rm{y}}}=1.51\,{\rm{V}}\,{\mathrm{cm}}^{-1}$ is shown as spatio-temporal false-color plot in figure 14. It visualizes the formation of oscillations induced by effective magnetic fields and a continuous shift of the spin package towards negative y-values with increasing delay time. The center position of the spin distribution, given by the y_c parameter of the fit function in equation (41), is marked as black dashed line and suggests a continuous shift of ∼17 μm during the time interval of 1.8 ns. Furthermore, there is a clear tilt of the individual stripes highlighted by five parallel dashed lines in grey color. Such a tilt implies a slowly rotating spin orientation with temporal, angular frequency ω_c for a fixed spatial position. This is noteworthy, as a fundamental feature of the PSH is its constant precession phase for a certain point in space, being responsible for the formation of the unidirectional spin wave pattern. The tilt is preserved even for large delay times and thus the decrease of the spin precession length (see equation (42)) as a result of the finite-spot size effect can be ruled out as possible explanation. This tilt can, however, be modeled as a temporal precession frequency in addition to the spatial precession already described. For this purpose, the introduction of a slightly changed fit function is favorable, which reads

$$\begin{eqnarray}&&{S}_{{\rm{z}}}(y)={A}_{0}\cdot {e}^{-4\mathrm{ln}(2){\left(y-{y}_{{\rm{c}}}\right)}^{2}/{w}^{2}}\cdot \cos \left(\displaystyle \frac{2\pi }{{\lambda }_{\mathrm{so},{\rm{y}}}}\cdot y+{y}_{1}^{{\prime} }\right).\end{eqnarray} \tag{ 47 }$$

It permits to study the cosine phase ${y}_{1}^{{\prime} }$ independently from ${\lambda }_{\mathrm{so},{\rm{y}}}$ while all other parameters maintain their original meaning. Employing this fit function to DPSH data yields a promising coincidence of theory and experimental line scans, illustrated in figure 14(e) where exemplary data are given for a delay time of t_d = 1758ps. The analysis of all line scans in panel (a) reveals the trend depicted in panel (c). The center position y_c of the Gaussian envelope depends linearly on time, readily explained by a constant drift velocity ${v}_{\mathrm{dr},{\rm{y}}}={{dy}}_{{\rm{c}}}/{dt}$. The slope gives a velocity of 9.86 km s⁻¹, which is much smaller than the Fermi velocity v_F = 128.4 km s⁻¹. This drift velocity reveals an electron mobility as high as μ_e = 6.5 × 10⁵ cm²/Vs which is a bit larger than the mobility extracted from magneto-transport measurements taken at a different back-gate voltage (see table 1).

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The tilt of the DPSH stripes in the spatio-temporal map is understood as the electron motion due to both diffusion and drift. Using the expression for the phase velocity of the drifting spin helix equation (37) and the relation ${y}_{1}^{{\prime} }=-(2\pi /{\lambda }_{\mathrm{so},{\rm{y}}}){v}_{\mathrm{SH}}{t}_{{\rm{d}}}$ we obtain

$$\begin{eqnarray}&&{y}_{1}^{{\prime} }=\displaystyle \frac{2{m}^{* }{v}_{\mathrm{dr},{\rm{y}}}}{{{\hslash }}^{2}}{\beta }_{3}\cdot {t}_{{\rm{d}}}.\end{eqnarray} \tag{ 48 }$$

For spins under the simultaneous influence of drift and diffusion, the cubic Dresselhaus term leads to an effective spin-orbit interaction beyond simple superposition of these two transport phenomena. For example, when drift and diffusion contributions are parallel, the electrons have large absolute k-values on average. Cubic scaling of the effective magnetic fields with k leads to stronger spin–orbit coupling than the mere sum of drift- and diffusion-related terms As shown in [62] one outcome is that motion due to drift leads to twice the spin precession angle than for diffusion over the same distance. This phenomenon leads to a temporal spin precession at any fixed position, described by the phase velocity in equation (48). By fitting the spatio-temporal data presented in figure 14 with the fit function in equation (47), the time dependence of ${y}_{1}^{{\prime} }$ is obtained, see figure 14(d). A clear linear trend of the data is given only for a limited time period around 1ns and a linear fit is performed to this data, neglecting points in the hatched regions. As a result, a frequency of ω_c = 0.233 × 10⁹ s⁻¹ is obtained, yielding β₃ = 0.140 meVÅ. This value is in good agreement with the predicted theory value 0.135 meVÅ that is obtained by inserting the 2DEG density of 0.8 × 10¹¹ cm⁻² into equation (5).

Having gathered a full set of parameters, a MC simulation can be created and compared to the experiment. Figure 14(b) depicts MC data created for a diffusion coefficient of D_s = 100 cm² s⁻¹, yielding a scattering time of τ = 0.46 ps. This value of D_s can be extracted from the linear expansion of w²(t_d) in the spatio-temporal experiment analogous to figure 9(b). It is noteworthy, that this spatial spread occurs about twice faster than observed in measurements without electric field. Indeed, further experiments with a variable electric field strength confirm such a tuning of the diffusion coefficient, see discussion in [72]. The tilted stripes are attributed to different contributions of β₃ depending on electron motion due to either diffusion or drift. The MC simulation provides a general proof of this theoretical model as the tilt of the stripes is well reproduced. However, the value of β₃ which gives the best match with the experiment is 0.45meVÅ which is approximately 4×larger than the experimental estimate. Considering the limited data range in figure 14(d) for extracting β₃, the discrepancy arises possibly due to uncertainty in the experimental data.

The next set of experiments studies the dependence of the PSH on the strength of the in-plane electric field. Special attention is paid to the behavior of the diffusion coefficent and the SO coupling parameters. The false-color plot in figure 15(a) illustrates a set of S_z(y) line scans taken along the y-axis for a variety of moderate electric fields ${E}_{\mathrm{IP}}^{{\rm{y}}}$, oriented along the same axis. All measurements are taken for a fixed delay time of 1525ps, allowing for the best trade-off between signal strength and advanced PSH evolution. The plot reveals a clearly increasing spatial shift of the spin distribution towards negative (positive) y-values with raising the positive (negative) electric field strength. This shift is explained by the linear dependence of the drift velocity ${v}_{\mathrm{dr},{\rm{y}}}={\mu }_{{\rm{e}}}{E}_{\mathrm{IP}}^{{\rm{y}}}$ on the applied field. The false-color plot appears to be symmetric around ${E}_{\mathrm{IP}}^{{\rm{y}}}=0$. This is confirmed by the extracting the y_c parameter for each line-scan. Here, the y_c parameter represents the center of the spin ensemble and can be translated into the drift velocity via ${v}_{\mathrm{dr},{\rm{y}}}={y}_{{\rm{c}}}/(1525\,\mathrm{ps}$). The resulting data on ${v}_{\mathrm{dr},{\rm{y}}}({E}_{\mathrm{IP}}^{{\rm{y}}})$ is depicted in figure 15(e) and points to a mobility of 0.38 × 10⁶ cm²/Vs, i.e., close to those in table 1 observed at a slightly different back-gate voltage. In accordance with equation (48) the value of β₃ is accessible from the linear dependence of the cosine offset parameter ${y}_{1}^{{\prime} }$ on the drift velocity, if the delay time is fixed. Figure 15(g) presents the field dependence of ${y}_{1}^{{\prime} }$. Note that the field strength is converted into a drift velocity by using the results of panel (e). Taking into account the uncertainty of the cubic Dresselhaus parameter, encountered in the spatio-temporal measurement and simulation (see figure 14: β₃ = (0.1 − 0.45) meVÅ), the extraction of ${y}_{1}^{{\prime} }$ has to be executed with caution. The trend is linear on first sight, confirmed by a linear fit of the entire data set. However, more accurately two linear regimes can be distinguished in the plot and evolve symmetrically around zero velocity. One regime represents a strong increase in the range up to 1500 m s⁻¹ while a second regime is identified for higher velocities with a somewhat smaller slope. Two separate fits allow to estimate β₃ = (0.15 ± 0.08) meVÅ albeit with a large statistical error. Despite this, the ambiguous behavior of ${y}_{1}^{{\prime} }$ and the dissimilarity of MC simulation and experimental value in figure 14 points to inadequcies of the applied fit function. It is possible that the limited number of oscillations along the x-axis causes errors in the extraction of the ${y}_{1}^{{\prime} }$ parameter. Additionally, it is expected that the application of current to the system induces heating of the electron gas, causing a transition to a non-degenerate gas. In this case, the Fermi energy has to be replaced by a higher, average kinetic energy $\langle E\rangle$. This effect is elaborated thoroughly around formula equation (49) from where it can be seen that heating results in an increase of β₃. This finding is in contradiction with the slope in figure 15(g) that rather exhibits a decrease of β₃ for increased voltages.

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Fitting of the field resolved data performed with equation (47) traces the evolution of the width w of the drifting spin polarization. The diffusion coefficient is then estimated from the expression ${w}^{2}=16\mathrm{ln}(2){D}_{{\rm{s}}}\cdot 1525\,\mathrm{ps}+{w}_{0}^{2}$. The corresponding data for D_s is presented in figure 15(c). A strong enhancement of the diffusion with rising field strength is evident for drift along ±y. For positive fields, D_s increases monotonically to 180 cm² s⁻¹ while negative fields raise the diffusion coefficient to a maximum and fields below −2 V cm⁻¹ lead to a decrease. The general increased diffusion is in agreement with the value of D_s = 100 cm² s⁻¹ that was needed in the MC simulation of figure 14 to reproduced the experiment at 1.51 V cm⁻¹. Precise knowledge about this variation of D_s is also important for the correct extraction of (α + β) from the precession length ${\lambda }_{\mathrm{so},{\rm{y}}}$, see figure 15(d). The experimental data reveal a clear, symmetric trend towards a shorter precession length ${\lambda }_{\mathrm{so},{\rm{y}}}$ for increasing field strength. This is also seen in the false-color plot in figure 15(a) where tilted dashed grey lines are included for visual guidance. This finding suggests a variation of the Rashba and Dresselhaus parameters by fairly weak in-plane electric fields. The symmetric shape of the data around zero field implies that the trend is not an artifact that stems from cross-couplings of the various contacts. Note that figure 11(g) has already shown the impact of back-gate voltage on the SO coupling parameters. Therefore, unwanted cross-coupling of back-gate voltage and in-plane voltages appear possible. This behaviour could result in a simultaneous detuning of the gate voltage by changing the in-plane electric field. However, in such a case the wiring of the sample would lead to a non-symmetrical behavior of λ_0,y. Consequently, the decrease of the precession length has to emerge from real changes in the SO coupling parameters.

For a quantitative analysis, one has to distinguish between ${\lambda }_{\mathrm{so},{\rm{y}}}$ and ${\lambda }_{0,{\rm{y}}}$. The former parameter states the transient wavelength of the spin polarization which still develops into a PSH. The latter describes the stationary PSH mode for ${t}_{{\rm{d}}}\to \infty$. The relation of both entities is given by equation (42). In particular, the diffusion coefficient strongly influences the temporal evolution of the precession length. As D_s itself is affected by the electric field, it has to be ruled out that the observed change of ${\lambda }_{\mathrm{so},{\rm{y}}}$ is a secondary effect caused by variations of the diffusion. To this end, ${\lambda }_{0,{\rm{y}}}$ is calculated and plotted next to the experimental precession length ${\lambda }_{\mathrm{so},{\rm{y}}}$ in figure 15(d). It is apparent that the resulting ${\lambda }_{0,{\rm{y}}}$ remains strongly field-dependent. Then the sum of Rashba and Dresselhaus parameter is calculated from equation (42) and depicted in figure 15(f). A substantial increase of (α + β) from 3.35 meVÅ up to ∼4.4 meVÅ for a field strength of ∼ ± 3 V cm⁻¹ is monitored. An unwanted tuning of the back-gate voltage is again unlikely due to the symmetric increase for negative and positive fields. Consequently, it is assumed that the out-of-plane field and thereby α are constant. Therefore the remaining parameters β₁ and β₃ have to be (indirectly) affected by the applied in-plane bias. β₁ is unlikely to be altered by the weak fields. In contrast, the cubic Dresselhaus parameter β₃ is known to depend on the the Fermi energy (see equation (5)), that might undergo a significant change if the 2DEG evolves into a non-degenerate gas. In this case, the Fermi energy has to be replaced by the energy $\langle E\rangle$ averaged with the energy-derivative of the Fermi–Dirac function,

$$\begin{eqnarray}&&{\beta }_{3}={\gamma }_{{\rm{D}}}\displaystyle \frac{{m}^{* }\langle E\rangle }{2{{\hslash }}^{2}}\end{eqnarray} \tag{ 49 }$$

where $\langle E\rangle ={E}_{{\rm{F}}}$ for electron temperature T_e ≪ E_F and $\langle E\rangle ={k}_{{\rm{B}}}{T}_{{\rm{e}}}$ for E_F ≪ T_e. Such an approach has been successfully applied in a recent study [30]. However, it is incapable of explaining an increase of (α + β). From equation (49) it is obvious that heating will inevitably lead an increase of $\langle E\rangle$, resulting in a larger β₃. At the same time, an increased β₃ will always decrease the Dresselhaus parameter β = β₁ − β₃ and leads to a reduced (α + β), contrary to the data of figure 15(f). Besides this, the absolute change of β₃, needed to reproduce the variation of Δ(α + β) ≈ 1meVÅ, would require a temperature variation of several hundreds of Kelvin. In the work of Passmann et al the possibility of such an electron gas heating up to 350 K is discussed in more detail, implying a heat-induced change of Δβ₃ ≈ 1.5 meVÅ [72].

The presented experiments so far have been conducted with electric fields applied along the y-axis where λ_0,y is relatively small. The Hall-bar geometry of the GaAs sample (see figure 5) facilitates the application of in-plane bias along the x-axis as well. This is of particular interest since, so far, in this direction no significant spin precession is observed as ${\lambda }_{\mathrm{so},{\rm{x}}}$ is much larger. However, signatures of the effective magnetic field can be revealed by the application of magnetic fields, yielding (−α + β) = 1.15 meVÅ which corresponds to a spin precession length of ${\lambda }_{0,x}=32.5\,\mu {\rm{m}}$ for the x-axis, i.e., much larger than the typical diffusion lengths. In order to access information about the PSH from the x-axis it is useful to add electric fields and thus cover the distances needed to observe SO induced precessions. Figure 16(a) shows that this approach indeed reveals a spin polarization wave. In this measurement the spin ensemble is dragged towards negative x-values by a positive field of strength ${E}_{\mathrm{IP}}^{{\rm{x}}}=6.95\,{\rm{V}}\,{\mathrm{cm}}^{-1}$, imposing a drift velocity of ${v}_{\mathrm{dr},{\rm{x}}}=-22.6\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Analogous to previous y-axis measurements, ${v}_{\mathrm{dr},{\rm{x}}}$ is extracted from the dependence of the Gaussian center parameter x_c on delay time. This linear dependence is indicated as black dashed line in figure 16(a). The employed fit function for x-axis line scans is the same as shown in equation (47) for the y-axis. The trend of an enhanced diffusion with increasing field amplitude is confirmed in panel (a). Due to the relatively high fields, needed to induce spin oscillation along the x-axis, the field range for the extraction of D_s shown can be extended in comparison to figure 15. For an applied field of 6.95 V cm⁻¹ an extremely large diffusion coefficient of D_s = 374 cm² s⁻¹ is observed, corresponding to a seven-fold enhancement when compared to the case without field.

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The diffusion coefficient together with all other parameters extracted from figure 16(a) are confirmed when used to create a MC simulation. The resulting plot is depicted in figure 16(b) where a striking match with the experiment is achieved. On the right hand side of figure 16, a selection of measurements and extracted parameter in dependence of a variable field ${E}_{\mathrm{IP}}^{{\rm{x}}}$ is compiled. Figure 16(c) represents a false-color plot of the PSH at t_d = 800 ps for fields up to ±12 V cm⁻¹. This plot is equivalent to the y-axis measurement in figure 15. For the chosen delay time, oscillations do not become visible for fields lower than $| {E}_{{\rm{x}}}^{{\rm{x}}}| =1.35\,{\rm{V}}\,{\mathrm{cm}}^{-1}$. This limit is imposed by the fixed delay time which can not be arbitrarily increased because at some point the electrons start to drift beyond the detection window.

Fitting the data to equation (47) reveals the field dependence of the spatial precession length ${\lambda }_{\mathrm{so},{\rm{x}}}$. An example of successful fitting is given in the inset of figure 16(e). Note that the error of the extracted parameters suffers from the less pronounced oscillation compared to y-axis data. Similar to the observed behavior of ${\lambda }_{\mathrm{so},{\rm{y}}}$ along the y-axis (see figure 15) a downward trend of ${\lambda }_{\mathrm{so},{\rm{x}}}$ is found for increasing field values. Along the y-axis a symmetric and linear decrease of ${\lambda }_{\mathrm{so},{\rm{y}}}$ with absolute field strength is visible, starting from a maximum at zero field (see figure 15(d)). Accordingly, the corresponding trend along the x-axis is modeled with a function ${\lambda }_{\mathrm{so},{\rm{x}}}({E}_{\mathrm{IP}})={A}_{1}\cdot | {E}_{\mathrm{IP}}^{{\rm{x}}}| +{B}_{1}$, allowing for an indirect approximation of ${\lambda }_{\mathrm{so},{\rm{x}}}(0\,{\rm{V}}/\mathrm{cm})$ which can not be directly accessed. The best fit which is plotted on top of the data points in figure 16(d) gives B₁ = 39.08 μm, which transforms into ${\lambda }_{0,{\rm{x}}}=28.7\,\mu {\rm{m}}$, taking into account the diffusion coefficient for ${E}_{\mathrm{IP}}^{{\rm{x}}}=0\,{\rm{V}}\,{\mathrm{cm}}^{-1}$ from table 2. This estimation of ${\lambda }_{0,{\rm{x}}}$ is in good agreement with (−α + β) = 1.15 meVÅ determined from measurements with in-plane magnetic fields, suggesting a value of ${\lambda }_{0,{\rm{x}}}=32.5\,\mu {\rm{m}}$. Another parameter that provides valuable information about the SO coupling is the cosine offset parameterfor the x-axis presented in figure 16(e). Here, the ${E}_{\mathrm{IP}}^{{\rm{x}}}$, i.e., drift velocity dependence of ${x}_{1}^{{\prime} }$, enables the extraction of β₃. For this purpose, the same model developed for the tilt of stripes in spatio-temporal maps, caused by an unbalance in precession due to drift and diffusion, is used. In the preceding analysis the drift velocity dependence of the cosine offset parameter, given in equation (48) has been discussed. As a result, a significant discrepancy of experiment β₃ ∼ 0.14 meVÅ and MC simulation β₃ ∼ 0.45 meVÅ is found for the y-axis. The analogous data along the x-axis show a clearly linear behavior and, indeed, the extracted value of β₃ = 0.54 meVÅ is closer to the MC simulation. This improvement is attributed to a weaker influence of the finite spot size effect. This effect causes a strong alteration of ${\lambda }_{\mathrm{so},{\rm{y}}}$ with delay time while the broad eigenmode spectrum of the Gaussian distribution develops into a single mode PSH (see discussion below equation (42)). Due to the larger precession length this effect is not so distinct along the x-axis and therefore the extraction of the cosine offset is not strongly perturbed by changes of the cosine frequency which is namely ${\lambda }_{\mathrm{so},{\rm{x}}}$. In conclusion, the cubic Dresselhaus parameter remains somewhat uncertain. A combination of values generated from spin transport along both axes, the MC simulation and the theoretically calculated values provide a range of (0.29 ± 0.20) meVÅ for β₃. For the given density of n₀ = 1.3 × 10¹¹ cm⁻² a Dresselhaus constant of γ_D = (14 ± 10) eVų is calculated from the experimentally determined cubic Dresselhaus parameter. This finding is in agreement with the range of theoretically expected literature values given in table 1.

5.2.3. Manipulation via the back-gate voltage

The GaAs sample provides a back-gate that can be used to alter the electric field along the growth direction of the heterostructure and across the QW. At the same time, the gate controls the electron density n_e in the modulation doped QW by varying the ionization of the Si-δ doping layer outside of the QW. Note that we have investigated the dependence of the spin lifetime on the back-gate voltage in earlier work [39]. So far the impact of the back-gate voltage is briefly touched in figure 9(c) where a peak-like tunability of the PSH lifetime is shown. This section will further quantify the Rashba and Dresselhaus parameter in a regime with variable back-gate voltage.

It is expected that the Rashba parameter is directly affected by a change of the potential drop along the QW, entering into α. Between the two Dresselhaus contributions a differentiation must be made since β₁ is expected to be unaffected by U_BG. On the other hand, β₃ exhibits a direct dependence on the carrier density.

Figures 17(a) and (b) compare the spatio-temporal evolution of the PSH for both ends of the voltage spectrum which is applicable to the sample (U_BG = +1.0 V and U_BG = −1.75 V). The two false-color plots reveal the characteristic appearance of oscillations in the spin polarization during its diffusive expansion along the y-axis. A difference in the oscillation wavelength of a PSH is observed, see the two parallel, dashed lines and the arrows that serve as a guide to the eye. This observation at first glance is confirmed by further measurements and an upward trend is extracted for λ_0,y along the y-axis. The corresponding data points, shown in figure 17(c) as green triangles, are generated individually from a spatio-temporal measurement. Similarly, figure 17(c) shows a decrease of λ_0,x as red dots. This trend represents the spin precession length extracted perpendicular to the x-axis. For this axis smaller effective magnetic fields act on the electron spin. Both trends are phenomenologically modeled with an exponential function in order to facilitate further calculations based on the dependence of the SO coupling paramters on U_BG. The opposite behavior of λ_0,y and λ_0,x is best understood from the fact that the former precession length is determined by the sum (α + β) whereas the latter one is dictated by the difference (β − α) of the SO coupling parameters. Consequently, these sum and difference values in dependence on the back-gate voltage can be calculated by inserting the obtained exponential fit functions from panel (c) into equations (42) and (45), respectively. They are shown in figure 17(d) where a growing difference and a diminishing summed value can be identified. This behavior agrees well with the previously made assumption that a variation of α is the dominating effect when the back-gate voltage is changed. Knowing the sum and difference of α and β, the individual values can be determined and are plotted in figure 17(e). It is evident that α decreases while the back-gate voltage is increased and β remains rather constant even though a slight decrease in the blue line is visible. A 60% change of α is evidenced and is comparable to previous studies [40, 73]. For a more detailed description, values for β₃ are added to the plot in panel (c) as orange line. These values are calculated from inserting the back-gate voltage-dependent carrier density (data not shown) into equation (5). The carrier density can be directly accessed from the spectral width of the 2DEG Photoluminescence which serves as an experimental measure for the Fermi energy. The strong variation of the cubic Dresselhaus contribution is remarkable as it offers a fine tuning for the PSH. These results are confirmed by similar findings of gate control based on β₃ [25].

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For the final set of experiments we focus solely on the CdTe sample. This part reveals a so far unexplored tuning knob for the PSH, namely the locally induced carrier concentration. Here, it is discussed how local variation in the concentration due to optical excitation affects the Rashba and Dresselhaus SO coupling. In addition, a similar impact on the diffusion coefficient is found.

It is discussed in section 5.2.1 how the application of external magnetic fields manipulates the PSH helix evolution. This effect becomes clearly visible in a tilt of the spin polarization stripes seen in spatio-temporal measurements. It is attributed to a canceling of effective and external magnetic fields such that electrons with a specific momentum are locked in their spin polarization (see scheme in figure 12). Moreover, the quantification of the corresponding momentum yields the individual parameters if extracted for both axes from the respective expressions

$$\begin{eqnarray}\begin{array}{rcl}{v}_{{\rm{y}}1} & = & \displaystyle \frac{{{dy}}_{1}}{{dt}}=\displaystyle \frac{g{\hslash }{\mu }_{{\rm{B}}}}{2{m}^{* }(\alpha +\beta )}\cdot {B}_{\mathrm{ex},{\rm{x}}}\qquad \mathrm{and}\\ {v}_{{\rm{x}}1} & = & \displaystyle \frac{{{dx}}_{1}}{{dt}}=\displaystyle \frac{g{\hslash }{\mu }_{{\rm{B}}}}{2{m}^{* }(-\alpha +\beta )}\cdot {B}_{\mathrm{ex},{\rm{y}}}.\end{array}\end{eqnarray} \tag{ 50 }$$

As a result, a constant slope is predicted for the tilted stripes. The spatio-temporal maps of spin polarization depicted in figures 18(a) and (d) represent measurements along both axes with a magnetic field of 231 mT applied in the respective perpendicular direction. In comparison to the equivalent measurements in GaAs (see figures 11 and 12), a significantly higher number of oscillations is observed even though the PSH lifetime appears comparable. At the same time, the tilt of the individual stripes is much more pronounced in CdTe. These observations are explained by a g-factor of g = 1.6 determined by the spin beat frequency at position (x, y) = (0μm, 0 μm). The g-factor enters into equation (50), demanding a higher electron velocity to cancel the external magnetic field. Combining the sum and difference of Rashba and Dresselhaus parameter, gathered from figures 18(a) and (d), the individual values α = (2.35 ± 0.17) meVÅ and β = (1.73 ± 0.16) meVÅ are determined.

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Figure 18(c) shows a simulation based on kinetic theory. Generally, this simulation is in good agreement with the experiment shown on the opposite side of the figure since the increased tilt and the number of oscillating stripes are reproduced. However, upon closer inspection the attention is drawn to a phenomenological feature of the experiment missing in the simulation. This feature is the saddle-point-like appearance of the stripes in the experiment highlighted in the plot by a dotted line. While this behavior is striking along the y-axis, it is much weaker in the data along the x-axis. It is completely absent in the kinetic theory (KT) simulation as indicated by a straight, dashed line in the simulation panel. Generally, the feature is not predicted by the theoretical model which predicts a constant tilt based on spatially constant SO coupling. The observed saddle-point-like shape, on the other hand, points more at a slowing down of v_y1 with electrons, propagating away from the origin at y = 0 μm. Taking into account the formulae in equation (50) such a spatial modulation of the stripe's slope can only arise from the α and β parameters being different in the center of the plot than at the edges, since all other entities in the theory are constants. Attempting to explain a locally disturbed SO coupling, it is reasonable to consider an influence of the photo-excitation process triggering the PSH formation. This excitation creates a non-equilibrium regime that affects the PSH if sufficiently many photo-carriers are involved.

Figure 18(f) illustrates a schematic of the spatial profile picture of the carrier density composed of a locally injected density n_op and the resident 2DEG electrons with density n₀. In case that n_op and n₀ are of similar size it is plausible that the SO coupling parameters are spatially modulated. Such a modulation takes place due to screening of the electric field in the case of α and a direct renormalization of the Fermi wave vector ${k}_{{\rm{F}}}=\sqrt{2\pi ({n}_{0}+{n}_{\mathrm{op}})}$ that determines β₃. Due to the multiple influences of the carrier density on the system, a direct quantification of the above introduced hypothesis turns out to be challenging. Therefore, measurements dependent on the excitation power are performed. The peak photo-injected carrier densities are in the range of (0.5 − 2.6)n₀, i.e. comparable to n₀ = 3.4 × 10¹¹ cm⁻². Figure 19(a) depicts five line scans taken under the same conditions, varying only the optical peak intensity. For a better comparison among the individual measurements and also to the results of the simulations, only the data for positive y-values is selected from the symmetric scans. Studying individual line scans for a fixed t_d = 750 ps enables the inspection of subtle features that are much harder to recognize in full spatio-temporal maps. The normalized experimental curves undergo a strong change of shape while the intensity is raised. An increased modulation depth together with a longer ranging spin precession length are visible in the experimental data where both trends are highlighted with arrows. From the experiment it is apparent that the higher the intensity is, the further away from the origin, the signal can still be detected. This finding suggests an increased diffusion which is confirmed by individual analysis of the full spatio-temporal map for each intensity (data not shown). In accordance to the inset in figure 19(d), a linear rise of D_s with increasing intensity is confirmed. Additionally, the same inset illustrates the estimated photo-carrier density in dependence on I_op, explaining the increased diffusion coefficient as follows. The Fermi energy depends directly on the electron density in the system and a locally elevated density will result in an increased Fermi energy and velocity which follow the same distribution. Moreover, the Fermi velocity is a determining part of the diffusion coefficient given by ${D}_{{\rm{s}}}={\tau }_{1}^{* }{v}_{\mathrm{Fermi}}^{2}/2$. However, the direct calculation of a general D_s is complicated by this spatial and temporal density modulation, the latter being caused by the recombination of 2DEG electrons with holes. While spatial modulation cannot be reasonably considered, the lifetime of electrons for similar structures is known to be comparable or shorter than the spin lifetime [74, 75]. For the present sample a lifetime of T_c ∼ 1 ns is obtained from transient reflectivity measurements (data not shown).

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It has been shown in the previous section that a changed diffusion coefficient has a severe influence on the measured spin precession length ${\lambda }_{\mathrm{so},{\rm{y}}}$ due to a faster or slower transition from a multi-mode Gaussian to the single-mode PSH. For this reason it is vital to review the possibility that the variation of the diffusion itself causes the differently shaped line scans observed in the experiment. Figure 19(b) displays KT simulations for the same parameter set as encountered in the experiment. A pronounced increase in the modulation depth can be achieved. However, the simultaneous variation of the spin precession length is not reproduced, resulting in a visible deviation of the simulation from the experiment. This leads to the conclusion that, indeed, the photo-carrier density plays an important role in the SO coupling process and an according model is required to verify this assumption. For a further validation a dynamic carrier density in space and time is modeled by the expression

$$\begin{eqnarray}\begin{array}{rcl}{n}_{{\rm{e}}}(x,y,{t}_{{\rm{d}}}) & = & {n}_{0}+\displaystyle \frac{1}{2}\,{n}_{\mathrm{op}}\mathrm{ln}(2)\displaystyle \frac{{w}_{0}^{2}}{{w}^{2}(t)}\\ & & \times \ {e}^{-4\mathrm{ln}(2)({x}^{2}+{y}^{2})/{w}^{2}(t)}\cdot {e}^{-{t}_{{\rm{d}}}/{T}_{{\rm{c}}}}\end{array}\end{eqnarray} \tag{ 51 }$$

that combines intrinsic and optical doping, with n_op being the average initial photo-carrier density within the laser spot. The combination follows an approach that considers the diffusive broadening of the electron distribution as well as the temporal decay due to recombination and is schematically shown in figure 18(f). The expression in equation (51) enters directly into the calculation of the cubic Dresselhaus parameter as

$$\begin{eqnarray}&&{\beta }_{3}(x,y,{t}_{{\rm{d}}})={\gamma }_{{\rm{D}}}\pi \cdot {n}_{{\rm{e}}}(x,y,{t}_{{\rm{d}}})/2.\end{eqnarray} \tag{ 52 }$$

It imposes a novel spatio-temporal dynamics to ${\beta }_{3}$. Beyond this influence, it is reasonable to incorporate a similar dependence into α. The Rashba parameter is indirectly affected as additional charges screen the internal electric field in the QW, entering into $\alpha ={\gamma }_{{\rm{R}}}{{ \mathcal E }}_{\mathrm{QW}}$. An according model is based on the depolarization field $\delta {{ \mathcal E }}_{{\rm{z}}}$ created by a carrier density δ n_e. The depolarization field is given by

$$\begin{eqnarray}&&\delta {{ \mathcal E }}_{{\rm{z}}}=\rho \cdot \displaystyle \frac{{e}^{2}{m}^{* }{d}_{\mathrm{QW}}^{3}}{\epsilon {\hslash }}{{ \mathcal E }}_{{\rm{z}}}\delta {n}_{{\rm{e}}}\end{eqnarray} \tag{ 53 }$$

where denotes the dielectric constant and ρ is a dimensionless parameter that depends on the shape of the QW and is not strongly affected by the carrier density itself. From this consideration the impact of optical doping can be incorporated into the Rashba coefficient according to

$$\begin{eqnarray}&&\alpha (x,y,{t}_{{\rm{d}}})={\alpha }_{0}\cdot {e}^{-\rho {m}^{* }{e}^{2}{d}_{\mathrm{QW}}^{3}/(\epsilon {\hslash })[{n}_{{\rm{e}}}(x,y,{t}_{{\rm{d}}})-{n}_{0}]}\end{eqnarray} \tag{ 54 }$$

where α₀ represents the as-grown Rashba parameter, formerly refered to as α. By now, the differential equation employed for kinetic theory simulations can be equipped with the new expressions for α and β₃ and solved numerically. This yields the simulated curves in figure 19(c) where the optical density is chosen from the relation shown in the inset, matching the intensities used in the experiment. The obtained simulations reveal a strong tuning of the zero crossings for the signal whereas the distinct tuning of the modulation depth seen in the experiment is not reproduced. As a consequence, the local variation of the SO coupling alone can not be the origin of the s-shape. However, a combined approach, where an increase of the diffusion coefficient and of the optical density is considered, reproduces the observations. This combination leads to the set of simulated curves in figure 19(d). Now, these simulations show a remarkable agreement with the experiment as both the modulation strength and the shift in the precession length are qualitatively reassembled.

The agreement stands as a strong evidence for the assumed influence of optical doping on the PSH. This finding is strengthened by the proof that neither changes of D_s nor of n_op alone can explain the experiment. Moreover, the same kinetic theory model is used to investigate the origin of the unexplained s-shape that motivated the idea of photo-carrier induced fine tuning. Figure 18 contains two false-color plots in panel (b) and (e) which depict KT simulations with density-dependent SO coupling parameters and the s-shape of stripes is now clearly recognizable. This visual comparison serves as an additional validation of the photo-injected carrier density as fine tuning knob for the PSH and finally explains the saddle-point-like behavior. There is no fundamental reason that a similar effect of photoexcited carriers as discussed above should not appear in the GaAs sample. However, GaAs has a smaller absorption and in the regime analyzed here the photocarrier density remains smaller than the density of the 2DEG. Therefore, we do not observe s-shape stripe pattern in GaAs.