Abstract
A twodimensional electron gas (2DEG) with equalstrength Rashba and Dresselhaus spinorbit coupling sustains persistent helical spinwave states, which have remarkably long lifetimes. In the presence of an inplane magnetic field, there exist singleparticle excitations that have the character of propagating helical spin waves. For magnonlike collective excitations, the spinhelix texture reemerges as a robust feature, giving rise to a decoupling of spinorbit and electronic manybody effects. We prove that the resulting spinflip wave dispersion is the same as in a magnetized 2DEG without spinorbit coupling, apart from a shift by the spinhelix wave vector. The precessional mode about the persistent spinhelix state is shown to have an energy given by the bare Zeeman splitting, in analogy with Larmor’s theorem. We also discuss ways to observe the spinhelix Larmor mode experimentally.
Introduction
Spinflip waves are collective excitations of magnetic systems^{1,2}: rather than flipping individual magnetic moments, which causes a large exchange energy penalty, the periodic reversal of magnetic moments extends as a precessional wave over the entire system, which is energetically favorable. There has been recent interest in spin waves in ferromagnetic thin films as an information carrier, which constitutes the basis for magnonics^{3,4}. Spin waves exist in systems with localized and itinerant magnetic moments. In the latter case, the precession of the interacting spins, the charge motion, and the spinorbit coupling (SOC) due to inversion asymmetry are all interrelated and lead to novel phenomena. For example, chiral spin waves have been observed in asymmetric monolayers of iron^{5,6} and on the surface of topological insulators^{7}, helical spin waves have been predicted in a twodimensional electron gas (2DEG) subject to Rashba SOC^{8}, and twisted spin waves have been predicted and observed in magnetized 2DEGs^{9}.
The fundamental and practical aspects of spin waves in the presence of SOC have drawn interest recently in the context of spintronics^{10,11,12,13}. SOC provides the conversion of charge based information into the spin wave^{14,15}. However, the presence of both SOC and Coulomb interaction still poses interesting challenges, especially in the dynamical regime.
In this paper, we will study spin waves in a 2DEG in the presence of inplane magnetization and SOC. This system exhibits a rich interplay between Coulomb manybody effects, Rashba and Dresselhaus SOC, applied magnetic field, and electron density, which we have studied earlier^{9,16,17,18}. What we found is that the spin waves are modified by the SOC in a subtle manner: the spin waves get a boost of their group velocity whose magnitude and orientation depends on the crystallographic propagation direction in the quantum well plane. This interesting behavior of the spin waves can be understood via a transformation into a spinorbit twisted reference frame; however, in general this only holds to lowest order in SOC^{9}.
In this paper, we consider a very special case in which exact results can be proved to all orders in SOC, namely, the case of a persistent spin helix^{19,20,21,22,23,24,25,26,27}. The spin helix arises in a 2DEG in which the strengths of the Rashba and Dresselhaus fields, α and β, are equal (in this paper we only include contributions to the Dresselhaus effect which are linear in the wavevector, i.e., β = β_{1}; the cubic β_{3} coupling coefficient^{23,27} is ignored). We here consider a 2DEG embedded in a zincblende quantum well grown along the [001] direction: SU(2) symmetry is then partially restored, and a helical spin texture can be sustained along the [110] direction. This result is robust against spinindependent disorder scattering and Coulomb interactions^{20}. The main experimental signature of this state is that spin packet excitations are protected from decoherence, leading to extraordinarily long lifetimes^{22,25}.
If a magnetic field is applied in the plane of a 2DEG with α = β, spinpacket excitations can sustain longlived precessional motion^{23}. Furthermore, some interesting magnetoelectronic effects can occur in 2DEGs^{28,29,30} or quantum wires^{31,32} which reflect the special condition α = β. However, to our knowledge the spinwave dynamics under these circumstances, which involves Coulomb interactions between the electrons at finite magnetic field, has not been explicitly addressed before.
Our treatment of spin waves is based both on timedependent densityfunctional theory (TDDFT) in the linearresponse regime^{33} and on an equationofmotion approach featuring the full manybody Hamiltonian^{9,34}. We derive the exact form of the spinwave dispersion for systems with a spinhelix texture, and find that it is obtained from the dispersion without SOC by a simple wave vector shift. The spinwave stiffness S_{sw} remains unchanged.
The main result of this paper is that we identify an exact dynamical state of the 2DEG with α = β, which can be characterized as a collective precession of the 2DEG about the spinhelix state. The precession occurs at the bare Zeeman frequency, and we therefore refer to it as the spinhelix Larmor mode. This mode will be characterized by its long lifetime, and we will discuss ways in which it could be experimentally observed.
Results
2DEG with spinorbit coupling in a magnetic field: the spin helix
We consider the electronic ground state of a 2DEG in an ndoped zincblende quantum well in the presence of an inplane magnetic field and Rashba and Dresselhaus SOC. Since the magnetic field is parallel to the 2DEG, it only acts on the spin and there is no Landau level quantization (as long as the magnetic length \({l}_{B}=\sqrt{\hslash /eB}\) is not significantly smaller than the well width). We will use the effectivemass approximation and work in units where \(\hslash \) = e^{*} = m^{*} = 1, where e^{*} and m^{*} are the effective charge and mass, respectively.
Figure 1 defines two reference frames. The primed frame \( {\mathcal R} ^{\prime} \) is fixed with respect to the quantum well: the x′, y′ and z′axes point along the crystallographic [100], [010], and [001] directions, respectively; the 2DEG is in the x′ − y′ plane. The Rashba and Dresselhaus SOC fields will be introduced in \( {\mathcal R} ^{\prime} \), but it will be convenient for the discussion of spin waves to work in a coordinate system \( {\mathcal R} \) which is oriented such that its x and z axes lie in the quantum well plane, and the zaxis points along the inplane magnetic field B. As shown in Fig. 1, the xaxis is at an angle φ with respect to the x′axis.
Singleparticle states
Without magnetic field or SOC, the lowest electronic conduction subband in the quantum well is spindegenerate. For simplicity, we will treat the electronic states as purely twodimensional; however, the main results in this paper will not change qualitatively if one takes the finite well width into account. The magnetic field lifts the degeneracy and splits the lowest subband into two, which we shall denote by the index p = ±1. In the reference frame \( {\mathcal R} ^{\prime} \), the associated singleparticle states can be written as Φ′_{ p k }(r′) = e^{ik ⋅ r′}Ψ′_{ p k }, where r′ = (x′, y′), k = (k_{x′}, k_{y′}), and Ψ′_{ p k } is a twocomponent spinor of the form
Here, “spinup“ and “spindown” (↑ and ↓) refer to the spin quantization axis z′.
The states Ψ′_{ p k } are obtained from the following KohnSham singleparticle equation:
where \({\hat{\sigma }}_{\mathrm{0,}x^{\prime} ,y^{\prime} ,z^{\prime} }\) are the usual Pauli matrices. The offdiagonal parts in Eq. (2) involve
Here, Z = g^{*}μ_{ B }B is the bare Zeeman energy (μ_{ B } is the Bohr magneton, and g^{*} is the effective gfactor). The presence of Coulomb manybody effects in the interacting 2DEG gives rise to the Zeeman exchangecorrelation (xc) energy Z_{xc}, which we discuss below. α and β are the usual Rashba and Dresselhaus linear coupling parameters; we ignore contributions to the Dresselhaus effect that are cubic in the wavevector, since these tend to be much smaller than the linear contributions^{23,27}.
It is convenient to change the reference frame for the spin, and go over to reference system \( {\mathcal R} \), whose zaxis is along the magnetic field direction. We introduce two inplane vectors, q_{0} and q_{1}, given in \( {\mathcal R} ^{\prime} \) by
With this, Eq. (2) transforms into
where the scalar products k ⋅ q_{0} and k ⋅ q_{1} remain invariant under \( {\mathcal R} {^{\prime} }\) → \( {\mathcal R} \). Ψ_{ p k } is now a twocomponent spinor whose spatial coordinates and spin quantization axes are defined with respect to \( {\mathcal R} \).
Let us now discuss the xc contribution. The inplane magnetic field causes the 2DEG to become uniformly magnetized. The xc energy per particle of a homogeneous 2DEG^{35}, e_{xc}(n, ζ), can be written as a functional of the density n and the spin polarization ζ, where n = n_{↑} + n_{↓} and ζ = (n_{↑} − n_{↓})/n (↑ and ↓ are now defined with respect to \({\hat{e}}_{z}\)). The Zeeman xc energy is then given by
The renormalized Zeeman energy^{36} can now be defined as Z^{*} = Z + Z_{xc}.
The general solution of Eq. (7) has been considered elsewhere^{37}; instead, we concentrate here on the special case α = β and φ = π/4 or 5π/4. Under these circumstances, q_{1} = 0 and Eq. (7) simplifies considerably:
where
To keep the discussion a bit simpler, we will limit ourselves to the case φ = π/4 in the following, so \({\bf{Q}}\mathrm{=4}\alpha {\hat{e}}_{x}\) (the φ = 5π/4 case is essentially the same, just in the opposite direction).
The solution of Eq. (9) is straightforward. We obtain
where
The singleparticle energies (12) have the important property
which is illustrated in Fig. 2b, using the parameters Z^{*} = 0.0381 and α = 0.05. This large value of α (typical experimental values of α are about an order of magnitude smaller) was chosen for clarity of presentation.
B = 0: spinhelical singleparticle eigenstates
In the absence of external magnetic fields, i.e., for Z^{*} = 0 (see Fig. 2a), the degeneracy of the two energy branches E_{+,k+Q} and E_{−,k} gives rise to a persistent spinhelix state of the 2DEG, as illustrated by the stripelike pattern in Fig. 1. This is easy to see: Due to the degeneracy, linear combinations of the eigenstates Ψ_{−,k} and Ψ_{+,k+Q} are also solutions of Eq. (9). The singleparticle wave functions for wave vector k can therefore be written as
where a^{2} + b^{2} = 1. From the associated spindensity matrix it is straightforward to determine the magnetization in the \( {\mathcal R} \) frame. We obtain m_{ z } = a^{2} − b^{2}, and since the macroscopic magnetization must vanish (i.e., m_{ z } = 0), this implies a = b. Writing \(a,b={e}^{i{\varphi }_{a,b}}/\sqrt{2}\) and defining δ_{ ab } = ϕ_{ a } − ϕ_{ b }, we get
Accordingly, the stripes in Fig. 1 indicate a periodic rotation of the electronic spin in and out of the plane, with a wave vector \({\bf{Q}}=4\alpha {\hat{e}}_{x}\) oriented at a 45° angle with respect to the x′axis (i.e., the [110] direction). This is the spin helix pattern^{20,27}.
The magnetizations m_{ x } and m_{ y } associated with the states \({{\rm{\Phi }}}_{{\bf{k}}}^{{\boldsymbol{+}}}\) and \({{\rm{\Phi }}}_{{\bf{k}}}^{{\boldsymbol{}}}\) cancel out; therefore, adding up all occupied spinhelix states below the Fermi energy E_{ F } gives zero. This means that the ground state of the Nelectron system has no spin texture. The persistent spin helix pattern can be observed if additional quasiparticles are injected at the Fermi level, as shown in Fig. 2a. Such states will have a very long lifetime^{20,22,23}.
B ≠ 0: spinhelical singleparticle excitations
For a finite magnetic field (Z^{*} > 0), the degeneracy of the two energy branches E_{+,k+Q} and E_{−,k} is lifted, as shown in Fig. 2b. As a consequence, the spinhelix pattern (15) is not a property of the ground state anymore: instead, the spin helix becomes a nonequilibrium feature.
To see this, consider a singleparticle excitation across the Fermi surface, with wave vector transfer Q, as illustrated by the thick green arrow in Fig. 2b. Firstorder perturbation theory tells us that the timedependent wave function has the form
where \(\gamma \ll 1\) and we made use of Eq. (13). In the \( {\mathcal R} \) frame, the x and y components of the associated magnetization are given by
where \({\varphi }_{\gamma }\) is the phase of γ. This defines a forward propagating spin helix (i.e., a spinflip wave), with amplitude 2γ, wave vector Q, and group velocity Z^{*}Q/Q^{2}. Singleparticle excitations of this type, which are longlived due to the property (13), were experimentally observed by Walser et al.^{23,38} using circularly polarized optical pump pulses and imaging via timeresolved Kerr rotation microscopy. A theoretical study using a driftdiffusion model was recently carried out by Ferreira et al.^{39}.
So far, our discussion has been for singleparticle excitations, i.e., we did not consider collective excitations. In the following Section, we will present a very special case of a collective mode, which we call the spinhelix Larmor mode, which can be viewed as a coherent superposition of the left and rightpropagating singleparticle spin helices considered above. As we will see, this gives rise to a collective, standing precessional wave with wave vector \(Q\) which is undamped. Propagating collective spinflip waves and their wave vector dispersions will be considered in Methods.
Spinhelix Larmor mode
We consider a 2DEG in the presence of a uniform magnetic field \({\bf{B}}=B{\hat{e}}_{z}\), in the reference frame \( {\mathcal R} \) of Fig. 1. The manybody Hamiltonian without SOC is
We consider the spinwave operator^{9,34,40,41}
(\({\hat{\sigma }}_{+}={\hat{\sigma }}_{x}+i{\hat{\sigma }}_{y}\)), whose Heisenberg equation of motion, in the absence of SOC, is
The second term on the righthand side of Eq. (21) arises from the commutator of \({\hat{S}}_{{\bf{q}}}^{+}\) with the kinetic part of \({\hat{H}}_{0}\), where
is the transverse spincurrent operator at wave vector q. The Larmor frequency ω_{ L } is equal to the bare Zeeman energy, ω_{ L } = Z. For small values of q, the equation of motion (21) can be written as
Here, ω_{sw,0}(q) implicitly contains the coupling between the collective spin waves and the singleparticle spincurrent dynamics^{34}. The real part, \(\Re {\omega }_{{\rm{sw}},0}({\bf{q}})={\omega }_{L}+{S}_{{\rm{sw}}}{q}^{2}\mathrm{/2}\), is the spinwave dispersion, where S_{sw} is the spinwave stiffness (explicit expressions and discussion of S_{sw} are provided in the Methods section). The imaginary part of ω_{sw,0}(q) accounts for the damping of the spin wave due to electronelectron interactions^{42}.
Now let us include SOC. The SOC Hamiltonian for the spinhelix case is given by [see Eq. (9)]
and the total Hamiltonian of the system is \(\hat{H}={\hat{H}}_{0}+{\hat{H}}_{{\rm{SOC}}}\). The equation of motion for \({\hat{S}}_{{\bf{q}}}^{+}\) in the presence of SOC is given by
We now show that the SOC contribution can be transformed away. We introduce the SU(2) unitary transformation \(\hat{U}=\exp [i{\sum }_{i}{\bf{Q}}\cdot {{\bf{r}}}_{i}{\hat{\sigma }}_{z,i}\mathrm{/2]}\), which leads to
In other words, \(\hat{U}\) causes a boost of the wave vector argument of the spinwave and the spincurrent operators, q → q − Q, and transforms the momentum operator of the ith electron into \(\hat{U}{{\bf{p}}}_{i}{\hat{U}}^{\dagger }={{\bf{p}}}_{i}+{\bf{Q}}{\hat{\sigma }}_{z,i}\mathrm{/2}\). On the other hand, \(\hat{U}\) leaves the Coulomb and the Zeeman parts of \({\hat{H}}_{0}\) unchanged. Thus, we obtain
Since \({{Q}}^{2}{\hat{\sigma }}_{0}\mathrm{/8}\) commutes with \({\hat{S}}_{{\bf{q}}{\bf{Q}}}^{+}\), the transformed equation of motion (25) becomes
where
For small values of q − Q, this becomes \([{\hat{S}}_{{\bf{q}}{\bf{Q}}}^{+},{\hat{H}}_{0}]={\omega }_{{\rm{sw}}\mathrm{,0}}({\bf{q}}{\bf{Q}}){\hat{S}}_{{\bf{q}}{\bf{Q}}}^{+}\). Substituting this into Eq. (28), we find that the spin waves of the system with α = β are those of the system without SOC (governed only by \({\hat{H}}_{0}\)), but where the wave vector is shifted:
Let us now consider the important special case q = Q. It is known^{43} that, in the absence of SOC, the spinpolarized electron system carries out a collective precessional motion with \({\omega }_{{\rm{sw}}\mathrm{,0}}={\omega }_{L}\). The Larmor frequency ω_{ L } is equal to the bare Zeeman energy, ω_{ L } = Z. The Larmor mode has infinite lifetime (zero line width), because it can be represented as a superposition of two exact eigenstates of the system Hamiltonian \({\hat{H}}_{0}\): the manybody ground state 0〉_{0} (the subscript 0 indicates absence of SOC), with groundstate energy \({E}_{0}\), and the manybody eigenstate \({\hat{S}}_{0}^{+}\mathrm{0}{\rangle }_{0}\), with energy E_{0} + Z. An important feature of the Larmor’s mode is that it does not carry any spin current in the plane. This is obvious from the equation of motion (21) in the absence of SOC: for the Larmor’s mode which occurs at q = 0, no current is induced by the homogenous precession.
Coming back to the case with SOC with the full Hamiltonian \(\hat{H}\), we can now formulate the spinhelix Larmor theorem. If the spin wave has wave vector Q, commensurate with the spinhelix texture, all Coulomb manybody contributions drop out and the frequency is given by the bare Zeeman energy:
which follows directly from Eq. (30). In the presence of SOC, the Larmor’s mode occurs at q = Q and is not a homogenous mode anymore; however, one still has the property that no spincurrent is driven by the precession, since the total current term in Eqs (25) and (29) disappears when q = Q: this happens because SOC induces spin currents opposite to the spin currents induced by the motion. The spin wave then has vanishing group velocity, ∇_{ q }ω_{sw}(q)_{q = Q} = 0, which means that it is a standing wave. All electronic spins precess about their local orientation, given by the spin helix configuration, with the Larmor frequency ω_{ L } = Z. The spinhelix Larmor mode is a superposition of two manybody eigenstates of \(\hat{H}\): the ground state 0〉 and the state \({\hat{S}}_{{\bf{Q}}}^{+}\mathrm{0}\rangle \).
We illustrate the spinwave dispersions with and without SOC in Fig. 3. The left panel shows ω_{sw,0}(q) (which is independent of the direction of q), and the right panel shows ω_{sw}(q) for q parallel to \({\bf{Q}}\), i.e., along the [110] direction. For this particular case, ω_{sw}(q) is simply obtained by a horizontal shift by Q of ω_{sw,0}(q) (likewise for the particlehole continua). The spinwave dispersions plotted in Fig. 3 are obtained from the numerical solution of Eq. (42), see below. The smallwave vector expansion (44) is very close to the exact result.
Experimental Schemes
Experimental observation of the spinhelix Larmor mode should be possible in specially designed doped magnetic semiconductor quantum well samples where the α = β condition is met. The spinflip waves under an inplane magnetic field can be detected using inelastic light scattering, similar to our earlier work^{9,16,17,18}. The Larmortype precessional mode about the spinhelix state should then be recognizable by a significant narrowing of the linewidth.
Timeresolved Kerr rotation spectroscopy is another suitable technique to detect collective spin excitations in a 2DEG^{44,45}. In the experimental setup by Walser et al.^{23} however, only the spinhelical singleparticle excitations were observed; at the magnetic field strength of 1 T considered in the experiment, the collective modes were too shortlived to be seen since their separation from the spinflip continuum was too small and the mode frequency was of the same order as the linewidth.
We also propose a device design which would allow one to excite Larmor’s mode optically and probe it electronically. In the absence of SOC, the Larmor’s mode is usually probed in a paramagnetic resonance setup, by a microwave (mw) magnetic field \({{\bf{b}}}_{{\rm{mw}}}=b\,\cos (\omega t){\hat{e}}_{y}\) applied in the plane perpendicular to the quantizing field \({\bf{B}}=B{\hat{e}}_{z}\). The corresponding coupling Hamiltonian is \({\hat{H}}_{{\rm{mw}}}=\frac{i}{2}{g}_{e}{\mu }_{B}b\,\cos (\omega t)({\hat{S}}_{0}^{+}{\hat{S}}_{0}^{})\). The unitary transformation \(\hat{U}\) maps the situation with SOC to the situation without SOC; hence, the coupling Hamiltonian becomes \(\hat{U}{\hat{H}}_{{\rm{mw}}}{\hat{U}}^{\dagger }=\frac{i}{2}{g}_{e}{\mu }_{B}b\,\cos (\omega t)({\hat{S}}_{{\bf{Q}}}^{+}{\hat{S}}_{{\bf{Q}}}^{})\). This corresponds to a coupling with a standing helical magnetic field in the plane (x, y). Such a magnetic field can be generated by a device like the one sketched in Fig. 4. The idea is to deposit metal stripes on top of the sample, separated by a distance π/Q (for typical values of \(\alpha \), of order ~1 meV Å, this corresponds to a few μm). The stripes are aligned parallel to the quantizing magnetic field \({\bf{B}}=B{\hat{e}}_{z}\), perpendicular to the [010] direction, and the spacing of the stripes is commensurate with the standingwave spinhelix Larmor mode. The metal stripes follow the principle of photoconductive antennas used in timedomain THz experiments^{46}. When an infrared optical pulse hits the sample, the photogenerated carriers in the substrate create a short conductive channel closing the biased circuit of the stripes. A current pulse with frequencies in the THz range will propagate through the stripes in alternating direction. This will, by induction, generate concentric oscillating magnetic fields compatible with the spinhelix pattern as depicted in the right panel of Fig. 4. The magnetic field exerts torques on the spinpolarized 2DEG underneath. If the current pulse spectrum contains the right frequency, ω = ω_{ L }, this will trigger a helical standing spin wave which will persist after the end of the pulse. Detection of the spinhelix Larmor mode, and measurement of its lifetime, should then be possible via the currents induced in the metal stripes from the stray magnetic fields associated with the standing spin wave. Such a detection technique is common in ferromagnetic magnonics^{3,4}.
In our semiconductor testbed system^{9}, we deal with lower spin densities and higher frequencies: we expect stray fields of the order of nT, and rapidly varying, at a rate of about 50–100 GHz, with a lifetime in the 100 ps to ns range. The measurements will therefore push the limits of presentday electronics, but they should be feasible. Moreover, the physics and concepts presented here are applicable to emerging 2D conducting systems with higher doping and strong spinorbit effects. The above estimate for the lifetime of the spinhelix Larmor mode is based on spinwave linewidths of the order of 50 μeV which were measured^{9} for quantum wells with α ≠ β; the corresponding dephasing time (165 ps for a linewidth of 50 μeV) can be viewed as a lower threshold to the lifetime of the Larmor mode, since it includes electronic manybody contributions which drop out if α = β. The lifetime of the spinhelix Larmor mode is mainly determined by cubic Dresselhaus contributions and disorder.
Discussion
In this paper, we have considered the spin dynamics in a 2DEG in the presence of SOC, under the very special condition where the Rashba and Dresselhaus coupling strengths are equal (α = β) and where an inplane magnetic field is applied perpendicular to the [110] direction. Without this magnetic field, the system sustains persistent spinhelix states which have been widely studied in the literature. The magnetic field lifts the degeneracy that leads to the persistent spinhelix states; it instead leads to singleparticle excitations that have the form of propagating spin helices.
The presence of Coulomb interactions causes these singleparticle excitations to combine and form collective spin waves, which are robust against any decoherence caused by SOC. We have found that for the 2DEG with α = β the spinwave dispersion is the same as for the system without SOC, apart for a rigid wave vector shift by Q (the spinhelix wave vector). The case of q = Q thus produces the special scenario which we have termed the spinhelix Larmor mode, where all manybody effects vanish and the precession frequency is given by the bare Zeeman energy (divided by \(\hslash \)). This is a new and exact result for electronic manybody systems, which opens up new ways of manipulating and driving electronic spins by optical means.
Methods
We now discuss the spinwave dispersions in the quantum well system considered above. We have seen that in the case α = β the singleparticle states (11) are pure up and down spinors; therefore, the longitudinal and transverse spin response channels are decoupled. The transverse spindensity response equation reads
where \({\overrightarrow{n}}_{T}=({n}_{\uparrow \downarrow },{n}_{\downarrow \uparrow })\) is the transverse spindensity response, \({\overrightarrow{v}}_{T}=({v}_{\uparrow \downarrow },{v}_{\downarrow \uparrow })\) is an external perturbation, and the noninteracting transverse spin response function and the transverse xc kernel are diagonal 2 × 2 matrices in the frame \( {\mathcal R} \)^{41,47}:
The general form of the individual elements of the noninteracting spindensitymatrix response function for a 2DEG is^{33}
where σ, σ′, τ, τ′ are spin indices (↑ or ↓), and η is a positive infinitesimal (since we will be considering spin waves outside the particlehole continuum, we can drop η). The Fermi function is given by f(E_{ p k }) = θ(E_{ F } − E_{ p k }), where E_{ F } is the Fermi energy of a paramagnetic 2DEG in the presence of SOC. It can be shown that \({E}_{F}={E}_{F}^{0}{\alpha }^{2}{\beta }^{2}\), where \({E}_{F}^{0}=\pi n\) is the Fermi energy of a 2DEG without SOC, and with 2D electronic density n.
We recast Eq. (12) as \({E}_{\pm ,{\bf{k}}}=\pm {Z}^{\ast }\mathrm{/2}+\frac{1}{2}{{\bf{k}}\mp {\bf{Q}}\mathrm{/2}}^{2}2{\alpha }^{2}\). With a change of the integration variable, \({\bf{k}}\to {\bf{k}}\pm {\bf{Q}}\mathrm{/2}\), the noninteracting spinflip response functions become
where the Fermi function f_{0} indicates that E_{ F } has been replaced by \({E}_{F}^{0}\).
Looking at Eqs (35) and (36) we immediately see that the spinflip response functions of the system with SOC can be expressed in terms of the corresponding functions without SOC (denoted by the superscript 0):
This simple result only holds for the special case α = β. In TDDFT, the spinflip linearresponse xc kernel of the 2DEG is given in the adiabatic localdensity approximation (ALDA) by^{33}
The spinflip wave dispersion now follows from Eq. (32) by setting \({\overrightarrow{v}}_{T}\) to zero and finding the eigenmodes, which leads to
Upon closer inspection of Eqs (35) and (36), we find
We are only interested in positive frequencies, so the spinflip wave dispersion is obtained from Eq. (40) as the implicit solution ω_{sw}(q) of
The spinwave dispersion can be analytically determined for small wave vectors. To second order, we find
This can be rewritten as
which corresponds to spin waves propagating with group velocity v_{ g }(q) = S_{sw}(q − Q), where the spinwave stiffness of the 2DEG is
Since \({K}_{{\rm{xc}}}={Z}_{{\rm{xc}}}/(n\zeta )\) and ζ = −Z^{*}/(2πn), this can also be written as
We plot the ALDA spinwave stiffness for various values of the spin polarization ζ and as a function of the 2D WignerSeitz radius r_{ s } in Fig. 5. The stiffness has negative values for all practically relevant values of r_{ s }, and crosses over to S_{sw} > 0 for very low densities (at r_{ s } = 26.96 for ζ = 0.05 and at r_{ s } = 24.49 for ζ = 0.95). In practice, one is interested in studying collective modes in the typical metallic range of r_{ s } between about 2 and 6; this is the intermediate coupling regime where the Coulomb energy (which scales as \({r}_{s}^{1}\)) is comparable to the kinetic energy (which scales as \({r}_{s}^{2}\)), but which can still be regarded as weakly correlated. As an example, in the quantum well system considered in ref.^{37} the spinwave stiffness was S_{sw} = −27.6, for r_{ s } = 2.2 and ζ = 0.053.
Data availability
All data generated or analysed during this study are included in this article.
References
 1.
Stancil, D. D. & Prabhakar, A. Spin waves: Theory and applications. (Springer, Berlin, 2010).
 2.
Zakeri, K. Elementary spin excitations in ultrathin itinerant magnets. Phys. Rep. 545, 47–93 (2014).
 3.
Kruglyak, V. V., Demokritov, S. O. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. 43, 264001–1–14 (2010).
 4.
Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nature Phys. 11, 453–61 (2015).
 5.
Zakeri, K., Zhang, Y., Chuang, T.H. & Kirschner, J. Magnon lifetimes on the Fe(110) surface: The role of spinorbit coupling. Phys. Rev. Lett. 108, 197205–1–5 (2012).
 6.
Di, K. et al. Direct observation of the DzyaloshinskiiMoriya interaction in a Pt/Co/Ni film. Phys. Rev. Lett. 114, 047201–1–5 (2015).
 7.
Kung, H.H. et al. Chiral spin mode on the surface of a topological insulator. Phys. Rev. Lett. 119, 136802–1–6 (2017).
 8.
Ashrafi, A. & Maslov, D. L. Chiral spin waves in fermi liquids with spinorbit coupling. Phys. Rev. Lett. 109, 227201–1–5 (2012).
 9.
Perez, F. et al. Spinorbit twisted spin waves: group velocity control. Phys. Rev. Lett. 117, 137204–1–5 (2016).
 10.
Winkler, R. Spinorbit coupling effects in twodimensional electron and hole systems, vol. 191 of Springer Tracts in Modern Physics. (Springer, Berlin, 2003).
 11.
Žutić, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004).
 12.
Wu, M. W., Jiang, J. H. & Weng, M. Q. Spin dynamics in semiconductors. Phys. Rep. 493, 61–236 (2010).
 13.
Bercioux, D. & Lucignano, P. Quantum transport in Rashba spinorbit materials: a review. Rep. Prog. Phys. 78, 106001–1–31 (2015).
 14.
Liu, T. & Vignale, G. Electric control of spin currents and spinwave logic. Phys. Rev. Lett. 106, 247203–1–4 (2011).
 15.
Kajiwara, Y. et al. Transmission of electrical signals by spinwave interconversion in a magnetic insulator. Nature 464, 262–266 (2010).
 16.
Baboux, F. et al. Coulombdriven organization and enhancement of spinorbit fields in collective spin excitations. Phys. Rev. B 87, 121303–1–5 (2013).
 17.
Baboux, F., Perez, F., Ullrich, C. A., Karczewski, G. & Wojtowicz, T. Electron density magnification of the collective spinorbit field in quantum wells. Phys. Rev. B 92, 125307–1–6 (2015).
 18.
Baboux, F., Perez, F., Ullrich, C. A., Karczewski, G. & Wojtowicz, T. Spinorbit stiffness of the spinpolarized electron gas. Phys. Stat. Solidi RLL 10, 315–9 (2016).
 19.
Schliemann, J., Egues, J. C. & Loss, D. Nonballistic spinfieldeffect transistor. Phys. Rev. Lett. 90, 146801–1–4 (2003).
 20.
Bernevig, B. A., Orenstein, J. & Zhang, S.C. Exact SU(2) symmetry and persistent spin helix in a spinorbit coupled system. Phys. Rev. Lett. 97, 236601–1–4 (2006).
 21.
Liu, M.H., Chen, K.W., Chen, S.H. & Chang, C.R. Persistent spin helix in RashbaDresselhaus twodimensional electron systems. Phys. Rev. B 74, 235322–1–6 (2006).
 22.
Koralek, J. D. et al. Emergence of the persistent spin helix in semiconductor quantum wells. Nature 458, 610–3 (2009).
 23.
Walser, M. P., Reichl, C., Wegscheider, W. & Salis, G. Direct mapping of the formation of a persistent spin helix. Nature Phys. 8, 757–62 (2012).
 24.
Schönhuber, C. et al. Inelastic lightscattering from spindensity excitations in the regime of the persistent spin helix in a GaAsAlGaAs quantum well. Phys. Rev. B 89, 085406–1–6 (2014).
 25.
Sasaki, A. et al. Direct determination of spinorbit interaction coefficients and realization of the persistent spin helix symmetry. Nature Nanotech. 9, 703–9 (2014).
 26.
Kammermeier, M., Wenk, P. & Schliemann, J. Control of spin helix symmetry in semiconductor quantum wells by crystal orientation. Phys. Rev. Lett. 117, 236801–1–5 (2016).
 27.
Schliemann, J. Persistent spin textures in semiconductor nanostructures. Rev. Mod. Phys. 89, 011001–1–17 (2017).
 28.
Duckheim, M. & Loss, D. Resonant spin polarization and spin current in a twodimensional electron gas. Phys. Rev. B 75, 201305–1–4 (2007).
 29.
Nakhmedov, E. P. & Alekperov, O. Interplay of Rashba and Dresselhaus spinorbit interactions in a quasitwodimensional electron gas of a finite thickness under inplane magnetic field. Eur. Phys. J. B 85, 298–1–19 (2012).
 30.
Wilde, M. A. & Grundler, D. Alternative method for the quantitative determination of Rashba and Dresselhaus spinorbit interaction using the magnetization. New J. Phys. 15, 115013–1–18 (2013).
 31.
Nazmitdinov, R. G., Pichugin, K. N. & ValínRodríguez, M. Spin control in semiconductor quantum wires: Rashba and Dresselhaus interaction. Phys. Rev. B 79, 193303–1–4 (2009).
 32.
Micu, C. & Papp, E. Equal coupling strength approach to spin precession effects concerning Rashba and Dresselhaus spinorbit interactions in the presence of inplane magnetic fields. Superlatt. Microstruct. 51, 651–662 (2012).
 33.
Ullrich, C. A. Timedependent densityfunctional theory: concepts and applications. (Oxford University Press, Oxford, 2012).
 34.
Perez, F., Cibert, J., Vladimirova, M. & Scalbert, D. Spin waves in magnetic quantum wells with Coulomb interaction and sd exchange coupling. Phys. Rev. B 83, 075311–1–9 (2011).
 35.
Attaccalite, C., Moroni, S., GoriGiorgi, P. & Bachelet, G. B. Correlation energy and spin polarization in the 2D electron gas. Phys. Rev. Lett. 88, 256601–1–4 (2002).
 36.
Perez, F. et al. From spin flip excitations to the spin susceptibility enhancement of a twodimensional electron gas. Phys. Rev. Lett. 99, 026403–1–4 (2007).
 37.
Karimi, S. et al. Spin precession and spin waves in a chiral electron gas: beyond Larmor’s theorem. Phys. Rev. B 96, 045301–1–14 (2017).
 38.
Salis, G., Walser, M. P., Altmann, P., Reichl, C. & Wegscheider, W. Dynamics of a localized spin excitation close to the spinhelix regime. Phys. Rev. B 89, 045304–1–7 (2014).
 39.
Ferreira, G. J., Hernandez, F. G. G., Altmann, P. & Salis, G. Spin drift and diffusion in one and twosubband helical systems. Phys. Rev. B 95, 125119–1–12 (2017).
 40.
Holstein, T. & Primakoff, H. Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098–113 (1940).
 41.
Perez, F. Spinpolarized twodimensional electron gas embedded in a semimagnetic quantum well: Ground state, spin responses, spin excitations, and Raman spectrum. Phys. Rev. B 79, 045306–1–16 (2009).
 42.
Hankiewicz, E. M., Vignale, G. & Tserkovnyak, Y. Inhomogeneous Gilbert damping from impurities and electronelectron interactions. Phys. Rev. B 78, 020404–1–4 (2008).
 43.
Lipparini, E. Modern manyparticle physics. 2nd edn, (World Scientific, Singapore, 2008).
 44.
Bao, J. M., Pfeiffer, L. N., West, K. W. & Merlin, R. Ultrafast dynamic control of spin and charge density oscillations in a GaAs quantum well. Phys. Rev. Lett. 92, 236601–1–4 (2004).
 45.
Barate, P. et al. Collective nature of twodimensional electron gas spin excitations revealed by exchange interaction with magnetic ions. Phys. Rev. B 82, 075306–1–9 (2010).
 46.
Dreyhaupt, A., Winnerl, S., Dekorsy, T. & Helm, M. Highintensity terahertz radiation from a microstructured largearea photoconductor. Appl. Phys. Lett. 86, 121114–1–3 (2005).
 47.
Giuliani, G. F. & Vignale, G. Quantum Theory of the Electron Liquid. (Cambridge University Press, Cambridge, 2005).
Acknowledgements
S.K. and C.A.U. are supported by DOE Grant DEFG0205ER46213. F.P. acknowledges support from the Fondation CFM, C’NANO IDF and ANR.
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C.A.U., I.D’A. and F.P. conceived the study, developed the theory, and wrote the manuscript. S.K. carried out the linear response calculations of the spinwave dispersions, participated in the discussions and contributed to the review of the manuscript.
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Correspondence to Carsten A. Ullrich.
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Karimi, S., Ullrich, C.A., D’Amico, I. et al. Spinhelix Larmor mode. Sci Rep 8, 3470 (2018). https://doi.org/10.1038/s41598018218188
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