Raman Scattering by a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Raman scattering in a two-dimensional Fermi liquid with spin-orbit coupling
Saurabh Maiti and Dmitrii L. Maslov
Department of Physics, University of Florida, Gainesville, FL 32611 (Dated: September 2, 2018)We present a microscopic theory of Raman scattering in a two-dimensional Fermi liquid (FL) withRashba and Dresselhaus types of spin-orbit coupling, and subject to an in-plane magnetic field ( ~B ).In the long-wavelength limit, the Raman spectrum probes the collective modes of such a FL: thechiral spin waves. The characteristic features of these modes are a linear-in- q term in the dispersionand the dependence of the mode frequency on the directions of both ~q and ~B . All of these featureshave been observed in recent Raman experiments on Cd − x Mn x Te quantum wells.
I. INTRODUCTION
Raman scattering is an inelastic light scattering pro-cess that allows to study dynamics of elementary exci-tations in solids both in space and time and to probeboth single-particle and collective properties of elec-tron systems. It helps to understand a variety of phe-nomena: from pairing mechanisms in high-temperaturesuperconductors to properties of spin waves in spin-tronic devices. The latter requires probing dynamics ofelectron spins, which can be done if the incident andscattered light are polarized perpendicular to each other(cross-polarized geometry). Non-trivial spin dynamics is encountered in systemswith spontaneous magnetic order, or in an external mag-netic field, or else in the presence of spin-orbit cou-pling (SOC). In this paper, we develop a microscopictheory of Raman scattering in a two-dimensional (2D)Fermi liquid (FL) in the presence of an in-plane mag-netic field and both Rashba and Dresselhaus types ofSOC. Breaking spin conservation leads to a substantialmodification of the Raman spectrum already at the non-interacting level. Unlike in the SU (2)-invariant case, thecross-section of Raman scattering cannot be expressedsolely in terms of charge and spin susceptibilities. Asa result, it is not a priori clear how single-particle andcollective spin-excitations in systems with SOC manifestthemselves in a Raman scattering process. We show herethat the scattering cross-section, in the cross-polarizedgeometry and in the long-wavelength limit, is parameter-ized by particular components of the spin susceptibilitytensor. These components contain poles at frequenciesthat correspond to a new type of collective modes–chiralspin waves (CSW) –arising from an interplay betweenelectron-electron interaction ( eei ) and SOC.In the absence of SOC, the dispersion of a collectivemode in the spin channel must start with a q term byanalyticity. Recently, a dispersing peak was observed inresonant Raman scattering on magnetically doped CdTequantum wells in the presence of an in-plane magneticfield. The unusual features observed in these exper-iments were a linear-in- q term in the dispersion and a π -periodic modulation of the spectrum as the magneticfield is rotated in the plane of 2D electron gas (2DEG).We argue that the observed peak is actually one of the long sought after CSW (in the presence of a field). TheCSWs are collective oscillations of the magnetization ( ~M )that exist even in the absence of the magnetic field. Inzero field, there are three such modes, which are mas-sive, i.e., their frequencies are finite at q →
0, and dis-perse with q on a characteristic scale set by spin-orbitsplitting. The modes are linearly polarized, with ~M be-ing in the 2DEG plane for two of the modes and alongthe normal for the third one. If an in-plane magneticfield ( ~B ) is applied, the mode with ~M || ~B remains lin-early polarized while the other two modes with ~M ⊥ ~B become elliptically polarized. Figure 1a depicts the evo-lution of the excitation spectrum with B at q = 0. Asthe field increases, two, out of the three, modes run intothe continuum of spin-flip excitations (SFE). The thirdmode merges with the continuum at B = B c , when thespin-split Fermi surfaces (FSs) become degenerate, andre-emerges to the right of this point. As the field is in-creased further, this mode transforms gradually into thespin wave [Silin-Leggett (SL) mode] of a partially polar-ized FL. In the experimental setup of Refs. 14–16,the effective Zeeman energy is larger than both Rashbaand Dresselhaus splittings which, according to Fig. 1 a ,allows us to focus on the case of a single CSW adiabati-cally connected to the SL mode. We show that, at small q , the dispersion of this mode can be written asΩ( ~q, ~B ) = Ω ( θ ~B ) + w ( θ ~q , θ ~B ) q + Aq , (1)where θ ~x is the azimuthal angle of ~x . We show, first bya symmetry argument and then by an explicit calcula-tion, that the Rashba and Dresselhaus types of SOC con-tribute sin( θ ~q − θ ~B ) and cos( θ ~q + θ ~B ) terms to w ( θ ~q , θ ~B ),correspondingly, while the mass term, Ω ( θ ~B ) and theboundaries of the SFE continuum are π -periodic func-tion of θ ~B . Our theory describes quantitatively the re-sults of Refs. 14–16, where a single Raman peak was ob-served, and predicts that up to three peaks can be ob-served at lower magnetic fields in systems with highermobility and/or stronger SOC.The rest of the paper is organized as follows. In Sec.II, we discuss the general theory of Raman response ina Fermi-liquid with SOC. In Sec. III, we present sym-metry arguments for the form of the CSW dispersion inthe presence of a magnetic field. In Sec. IV, we presentresults of an explicit calculation for the CSW dispersion. θ B /2 π Ω / ∆ D η =10 η =5 η =0.6 η =0 (b) 0 0.2 0.4 0.6 0.8 100.20.40.60.811.2 φ /2 π Ω ( m e V ) (c) FIG. 1: Color online: a) Schematically: excitation spectrum (at q = 0) of a 2D Fermi liquid with spin-orbit couplingand subject to an in-plane magnetic field ~B . Shaded region: continuum of spin-flip excitations; lines: chiral spin waves. b)Continuum as a function of the angle θ ~B between ~B and the (100) direction. ∆ R / ∆ D = 0 . η = ∆ Z / ∆ D , where ∆ D/R/Z isDresselhaus/Rashba/Zeeman splitting. c) Continuum (shaded) and collective-mode frequency at q = 0 (line) as a function of φ = θ ~B − π/ − x Mn x Te quantum well.
In Sec. V we compare our theory to the experimental re-sults of Refs. 14–16. In Sec. VI, we summarize our mainresults. Details of the calculations are given in Appendix.
II. GENERAL FORMALISM
In general, Raman scattering probes both charge andspin excitations. However, if the polarizations of the in-cident ( ~ e I ) and scattered ( ~ e S ) light are perpendicular toeach other, the Raman vertex couples directly to the elec-tron spin. The Feynman diagrams for the differentialscattering cross-section in such a geometry are shown inFig. 2. The first diagram on the right-hand side in thetop row is a pure spin part, while the rest is a spin-chargepart, which is non-zero due to SOC, and is represented bya sum of bubbles connected by the Coulomb interaction(dashed line).
FIG. 2: Diagrams for the differential cross-section of Ramanscattering in cross-polarized geometry. Shaded bubbles de-note vertex corrections due to the exchange part of eei , calcu-lated in the ladder approximation. V ( q ) is the bare Coulombpotential. Shading of inner parts of the bubbles in Fig. 2 denotesrenormalizations due to the exchange part of eei . If thisrenormalization is neglected, the differential (per unit en-ergy and per unit solid angle) scattering cross-section isreduced, upon projection onto the conduction band, to the result of Ref. 6: d A d Ω d O ∝ X µµ ′ | γ µµ ′ | Im L µµ ′ − πe q Im Z ¯ Zǫ ( q, Ω) , (2a) γ µµ ′ = i h ~n · ˆ ~σe i~q · ~r i µµ ′ , (2b) L µµ ′ = f ( ε µ ) − f ( ε µ ′ )Ω + ε µ − ε µ ′ + iδ , (2c) Z = X µµ ′ γ µµ ′ h e − i~q · ~r i µµ ′ L µµ ′ ; (2d)¯ Z = X µµ ′ γ ∗ µµ ′ h e i~q · ~r i µµ ′ L µµ ′ (2e) ǫ ( q, Ω) = ǫ + 2 πe q X µµ ′ |h e i~q.~r i µµ ′ | L µµ ′ . (2f)Here, ǫ is the background dielectric constant, µ and µ ′ refer to the quantum numbers of electrons which in-clude the momentum and spin/chirality: µ = { ~k, ν } , µ ′ = { ~k ′ , ν ′ } with ν, ν ′ = ± ε µ is the energy of astate with quantum number µ , f ( ε ) is the Fermi function, h X i µµ ′ ≡ R d~rψ † µ ( ~r ) X ( ~r ) ψ µ ′ ( ~r ) is the matrix element of X ( ~r ) between states µ and µ ′ , and ~n = ~e I × ~e S . Thematrix elements are computed with respect to the eigen-vectors of the non-interacting Hamiltonian for a (001)quantum well with Rashba and Dresselhaus SOC andin the presence of an in-plane magnetic field. In whatfollows, we will be assuming that the energy scales asso-ciated with the magnetic field and SOC are much smallerthan the Fermi energy. In this case, it suffices to use alow-energy version of the Hamiltonian H ~k = v F ( k − k F )ˆ σ + X i =1 , s i ˆ σ i , (3)where ˆ σ is the identity matrix, v F and k F are the Fermivelocity and momentum in the absence of SOC and mag-netic field, correspondingly, and vector ~s parametrizesthe effects of SOC and magnetic field: s = 12 (cid:0) ∆ R sin θ ~k + ∆ D cos θ ~k + ∆ Z cos θ ~B (cid:1) ,s = − (cid:0) ∆ R cos θ ~k + ∆ D sin θ ~k − ∆ Z sin θ ~B (cid:1) , (4)where ∆ R , ∆ D , and ∆ Z are the Rashba, Dresselhaus,and Zeeman energy splittings, correspondingly. The x axis is chosen along the (100), and x , and x axes arechosen along the (010) and (001) directions, correspond-ingly. The Green’s function of Eq. (3) is given byˆ G K = g + + g − σ + g + − g − | s | X i =1 , s i ˆ σ i ,g ± = 1 iω n − ε ~k, ± , ε ~k, ± = v F ( k − k F ) ± | s | , (5)where | s | = p s + s .Using the eigenvectors of Hamiltonian (3) ψ µ ( ~r ) = e i~k · ~r (cid:18) iνe − iφ ~k (cid:19) , (6)where tan φ ~k = − s /s , we obtain for the matrix ele-ments of Raman scattering γ µµ ′ = iδ ~k − ~k ′ − ~q h n n iν ′ e − iφ ~k + ~q − iνe iφ ~k o + n n − ν ′ e − iφ ~k + ~q − νe iφ ~k o + n n νν ′ e − i ( φ ~k + ~q − φ ~k ) − oi , h e i~q.~r i µµ ′ = δ ~k − ~k ′ − ~q h νν ′ e − i ( φ ~k + ~q − φ ~k ) + 1 i . (7) We are interested in the small- q limit, when the q -dependent terms in the dispersions of CSWs correctionsto the frequencies of these waves at q = 0, which impliesthat v F q ≪ Ω. At the same time, both the magnetic fieldand SOC are assumed to be weak compared to the Fermienergy (in appropriate units), i.e., Ω ≪ v F k F . Combin-ing the two inequalities above, we obtain q ≪ Ω /v F ≪ k F . (8)The first part of this inequality ensures that the diagonalelements of L µµ ′ in Eq. (2a) are small by charge conserva-tion: L ~k, ± ; ~k + ~q, ± ∝ q . Therefore, the main contributionto the cross-section in this limit comes from off-diagonalterms with νν ′ = −
1, which correspond to processes thatflip spin/chirality. The second part of Eq. (8) impliesthat φ ~k + ~q can be approximated by φ k , upon which theoff-diagonal matrix elements are simplified to γ µµ ′ ≈ − iδ ~k − ~k ′ − ~q × (cid:2) in ν cos φ ~k + + in ν sin φ ~k + n (cid:3) , (9a) | γ µµ ′ | ≈ δ ~k − ~k ′ − ~q × (cid:2) ( n cos φ ~k + n sin φ ~k ) + n (cid:3) , (9b) h e i~q · ~r i µµ ′ ≈ νν ′ + 1 = 0 . (9c)By the same argument, the diagonal terms in Z and ¯ Z in Eq. (2a) are small as q , while the off-diagonal terms aresmall because they contain off-diagonal matrix elements h e i~q · ~r i µµ ′ , which are small by Eq. (9c). Therefore, thesecond term in Eq. (2a) can be neglected compared tothe first one, and the cross-section is reduced to d A d Ω d O ∝ X µµ ′ | γ µµ ′ | Im L µµ ′ = Z ~k (cid:2) n cos φ ~k + n sin φ ~k + n n sin 2 φ ~k + n (cid:3) Z ω ( g + ˜ g − + g + ˜ g − ) , (10)where R ~k ≡ R d k/ (2 π ) ; R ω ≡ dω/ π ; g ν = 1 / ( iω − ε ~k,ν )and ˜ g ν = 1 / ( i { ω + Ω } − ε ~k + ~q,ν ). The various terms inthe equation above can be expressed via the componentsof the spin-spin correlation function in the chiral basis: χ ( Q ) = − Z ~k Z ω ( g + ˜ g − + g + ˜ g − ) cos φ ~k ,χ ( Q ) = − Z ~k Z ω ( g + ˜ g − + g + ˜ g − ) sin φ ~k ,χ ( Q ) = − Z ~k Z ω ( g + ˜ g − + g + ˜ g − ) sin φ ~k cos φ ~k ,χ ( Q ) = χ ( Q ) ,χ ( Q ) = − Z ~k Z ω ( g + ˜ g − + g + ˜ g − ) , (11) Making use of the definitions, we re-write Eq. (10) as d A d Ω d O ∝ n χ + n χ + n n ( χ + χ ) + n χ , (12)Accounting for the exchange part of eei amounts sim-ply to replacing the free-electron spin-spin correlationfunction by the renormalized ones, i.e., to replacing thebare bubbles in Fig. 2 by the shaded ones: d A d Ω d O ∝ n χ + n χ + n n ( χ + χ ) + n χ . (13)The poles of the renormalized components of the spinsusceptibility tensor correspond to CSWs. This is themain point of departure from the theory of Ref. 6, whichaccounts for plasmons but not for CSWs. On a techni-cal level, renormalized χ ij can be found either from theequations of motion of the FL theory or from Ran-dom Phase Approximation (RPA) in the spin channel. Equation (13) is valid at finite q provided that q ≪ k F .Suppose for a moment that SOC is absent while themagnetic field is applied along the x -axis. In this case,the q = 0 susceptibility components satisfy χ = χ and χ = χ = 0, due to conservation of spin com-ponent along the direction of the field. [At small butfinite q , χ and χ are also finite but small (in pro-portion to q ) and can be neglected.] The Raman cross-section is then proportional to ( n + n ) χ . Becauseof spin-rotational symmetry in the absence of the field, χ is proportional to the transverse susceptibility χ ⊥ = χ + − + χ − + . The pole of χ ⊥ corresponds to the SLmode. The direction of ~n = ~ e I × ~ e S affects only themagnitude of the Raman signal but not its profile.In the presence of SOC, however, the result for the Ra-man cross-section does not reduced to χ ⊥ . Instead, thepartial components of χ lm contribute to the cross-sectionwith weights determined by the direction of ~n . The polesof χ lm now correspond to CSWs shown in Fig. 1 a , one ofthem being adiabatically connected to the SL mode. Ingeneral, equations of motion for the three components ofmagnetization are coupled to each other, and hence allthe components of ˆ χ have poles at the same frequenciesbut with different residues. The profile of the spectrumis determined by the residues as well as by the relativeweights of the geometrical factors n i n j as given in Eq.(13). III. SYMMETRY ARGUMENTS FORDISPERSION OF CHIRAL SPIN WAVES
In this section, we show that the general form of theCSW dispersion can be determined solely by symmetryand dimensional analysys. The role of the microscopictheory, presented in Sec. IV, amounts then to fixing thedimensionless functions of the exchange interaction pa-rameter(s), which enter this form.As SOC and magnetic field are weak, the interactionof two quasiparticles with momenta ~k and ~k ′ can be de-scribed by an SU (2)-invariant Landau functionˆ F ( ϑ ) = ˆ σ ˆ σ ′ F s ( ϑ ) + ˆ ~σ · ˆ ~σ ′ F a ( ϑ ) , (14)where ϑ = θ ~k − θ ~k ′ . Within this approximation, thecharge and spin sectors of the theory are decoupled, andwe focus on the exchange interaction parameterized by F a ( ϑ ).To set the stage, we discuss the SL mode in the ab-sence of SOC. Kohn’s theorem protects the q = 0 termin the dispersion from renormalization by eei ; there-fore, Ω ( θ ~B ) = ∆ Z in Eq. (1). Now we consider thedispersion at small but finite q : q ≪ ∆ Z /v F . In the ab-sence of SOC, rotations in the orbital and spin spaces are independent, which excludes the dot product ~q · ~B andany power of it thereof. Therefore, the dispersion startswith a q term. Combining the symmetry argumentswith dimensional analysis, we findΩ( ~q, ~B ) = ∆ Z + a ( { F a } ) ( v F q ) ∆ Z , (15)where a is a dimensionless function which dependson the angular harmonics of F a ( ϑ ): a ( { F a } ) = a ( F a , F a , F a . . . ). A precise form of a is determinedby the microscopic theory. We now apply the same reasoning to CSWs. If onlyRashba SOC is present, there is no preferred in-planedirection, hence a linear-in- q term is absent while thequadratic term is isotropic. There are three CSWs in thiscase, whose dispersions for q ≪ ∆ R /v F can be writtenas Ω α ( ~q,~
0) = ˜ a α ( { F a } ) | ∆ R | + ˜ a α ( { F a } ) ( v F q ) | ∆ R | , (16)where α = 1 . . . a α , are some other dimensionlessfunctions of the FL parameters. Since Kohn’s theoremdoes not apply in the presence of SOC, ˜ a α = 1. Explicitforms of the functions ˜ a α , were found in Refs. 8–10. Ifonly Dresselhaus SOC is present, the Hamiltonian canbe transformed to the Rashba form by replacing θ ~k → π/ − θ ~k . Therefore, the CSWs are the same in both casesalthough the Hamiltonians have different symmetries.If both Rashba and Dresselhaus types of SOC arepresent, the q = 0 term in the dispersion remainsisotropic because in the absence of the magnetic field,the limit of q = 0 can be reached along any direction inthe plane), while a linear-in- q term is still not allowedby symmetry. Indeed, since the dispersion must be ana-lytic in q , a term of the type | ~q | is not allowed, and thelinear term could only be of the form c q + c q with c , = const. However, such a form does not obey thesymmetries of the D d group (rotation by π and reflec-tion about the diagonal plane). The quadratic term canhave an anisotropic part of the form q q ∝ sin 2 θ ~q , butwe already know that such a term is absent if only Dres-selhaus SOC is present. Therefore, such a term can onlyarise due to interplay between Rashba and Dresselhaustypes of SOC, and its should be proportional to the prod-uct ∆ R ∆ D . Hence the anisotropic part is small comparedto the isotropic, q part.Finally, let both types of SOC and the magnetic fieldbe present but, in agreement with the experimental con-ditions of Refs. 14–16, we focus on the case whenthe Zeeman energy is largest scale in the problem, i.e.,∆ Z ≫ ∆ R , ∆ D . The q = 0 term in the dispersion thendepends on the orientation of ~B in the plane; the D d -symmetry forces this dependence to be of the sin 2 θ ~B form. Since the anisotropic part of Ω ( θ ~B ) is non-zeroonly if both types of SOC are present, the coefficient ofthe sin 2 θ ~B term must be on the order of ∆ R ∆ D / ∆ Z . Inaddition to the anisotropic part, there are also isotropiccorrections of order ∆ R / ∆ Z and ∆ D / ∆ Z .To lowest order in SOC, the form of the linear termis determined by the symmetries of the Rashba (group C ∞ v ) and Dresselhaus (group D d ) types of SOC. In bothcases, we need to form a scalar (Ω) out of a polar vector( ~q ) and a pseudovector ( ~B ). In the C ∞ v group, this isonly possible by forming the Rashba invariant B q − B q ∝ sin( θ ~q − θ ~B ), which is the same term as in theoriginal Rashba Hamiltonian with ˆ ~σ → ~B . Likewise, theonly possible scalar in the D d group is the Dresselhausinvariant B q − B q ∝ cos( θ ~q + θ ~B ). The quadraticterm in the high-field limit can be taken the same as thequadratic term in the SL mode, Eq. (15). Combining together all the arguments given above, wearrive at the following form of the coefficients in Eq. (1)Ω ( θ ~B ) = ∆ Z + a ( { F a } ) ∆ R + ∆ D ∆ Z + ˜ a ( { F a } ) ∆ R ∆ D ∆ Z sin 2 θ ~B ,w ( θ ~q , θ ~B ) = v F a ( { F a } ) (cid:20) ∆ R ∆ Z sin( θ ~q − θ ~B )+ ∆ D ∆ Z cos( θ ~q + θ ~B ) (cid:21) ,A = a ( { F a } ) v F ∆ Z , (17)where a , ˜ a , a and a are dimensionless functions. Wewill now perform a microscopic calculation to confirm theabove from for the dispersion of collective modes. Thelinear-in- q term of the dispersion is the most interest-ing one, as it is the only term that breaks the symmetrybetween ∆ R and ∆ D . This allows one to extract sepa-rately the Rashba and Dresselhaus components of SOCfrom the spectrum of the collective modes. Note thatthis information cannot be deduced from the q = 0 termalone. IV. MICROSCOPIC CALCULATION OFCHIRAL SPIN WAVE DISPERSION
In this section, we confirm Eq. (17) by an explicitcalculation within the s -wave approximation, in whichthe Landau function contains only the zeroth harmonic: F a ( θ ) = F a . In this case, the FL theory is equivalentto the Random Phase Approximation (RPA) in the spinchannel, and we adopt the RPA method for conve-nience. In this scheme, the shaded bubbles in Fig. 2 areapproximated by the ladder series of diagrams. Sum-ming up this series, we obtain for the tensor of the spinsusceptibilityˆ χ ( Q ) = ( gµ B ) χ ∗ ( Q ) (cid:20) ˆ I + F a N F ˆ χ ∗ ( Q ) (cid:21) − , (18)where Q = ( ~q, i Ω n ), g is the effective Land´e factor, µ B is the Bohr magneton, N F is the density of states at the Fermi surface, ˆ I is the 3 × χ ∗ ij ( Q ) = − R K Tr h ˆ σ i ˆ G ∗ K ˆ σ j ˆ G ∗ K + Q i , K ≡ ( ~k, iω m ) and G ∗ K differsfrom G K in Eq. (5) only in that the bare Zeeman energyis replaced by the renormalized one, ∆ ∗ Z = ∆ Z / (1 + F a ),while ∆ R and ∆ D are not renormalized in the s -waveapproximation. The collective modes correspond tothe poles of ˆ χ and can be found as roots of the equationDet (cid:20) ˆ I + F a N F ˆ χ ∗ ( ~q, i Ω n → Ω + i + ) (cid:21) = 0 . (19)The details of solving this equation are purely mathemat-ical in nature and are presented in Appendices A 1, A 2,and A 3. Here, we only state that the results indeed coin-cide with those in Eq. (17) and give explicit expressionsfor the dimensionless functions, which read˜ a ( F a ) = − (1 + F a )(2 + F a )2 F a ,a ( F a ) = (1 + F a )(2 + 3 F a )4 F a ,a ( F a ) = − (1 + F a ) (cid:2) (4 + F a )(1 + F a ) + ( F a ) (cid:3) F a (2 + F a ) ,a ( F a ) = (1 + F a ) F a . (20)[The forms of a and a coincide with the results ofRefs. 12 and 24, respectively, in the s -wave approxima-tion.] For repulsive eei , F a < a ( F a )] is positive. However, the overall sign of the lin-ear term depends on the signs of the Rashba and Dres-selhaus couplings, as well as on the orientations of ~q and ~B .The continuum of SFE corresponds to interband tran-sitions and is given by a set of Ω that satisfy Ω = | ε ~k, + − ε ~k, − | for all states on the FS. Because ε ~k, ± varyaround the FS, Ω varies between the minimum (Ω min )and maximum (Ω max ) values, which determine a finitewidth of the continuum even at q = 0 (see Fig. 1a). Inthe presence of ~B , both Ω min and Ω max depend on θ ~B .The anisotropic part of ε ~k, ± is given by ∆ R ∆ D sin 2 θ ~k +∆ R ∆ ∗ Z sin( θ ~k − θ ~B )+∆ D ∆ ∗ Z cos( θ ~k + θ ~B ). As one cansee, changing θ ~B → θ ~B + π can be compensated by θ ~k → θ ~k + π , and thus Ω min , max ( θ ~B ) have a period of π . Figure 1b shows Ω min , max ( θ ~B ) for a range of the mag-netic fields. V. COMPARISON WITH EXPERIMENT
We are now in a position to apply our results to re-cent Raman data on a Cd − x Mn x Te quantum well.
In these experiments, ~q and ~B are chosen to be perpen-dicular to each other, i.e., θ ~B − θ ~q = ± π/
2. Accordingly,the dispersion in Eqs. (1) and (17) is simplified toΩ( ~q, ~B ) = ∆ Z (cid:2) a ( r + d ) + ˜ a rd sin 2 θ ~B ± a (cid:0) r − d sin 2 θ ~B (cid:1) v F q ∆ Z + a ( v F q ) ∆ Z (cid:21) . (21)where r = ∆ R / ∆ Z and d = ∆ D / ∆ Z . Up to the angulardependence of the mode mass, this form was conjecturedin Refs. 15,16 on phenomenological grounds. The mea-sured frequency of the mode at q = 0 gives ∆ Z = 0 . B = 2 T. For the number density of n = 2 . × cm − and effective mass of m ∗ = 0 . m e ( m e is the bare elec-tron mass), k F = 1 . × − ˚A − and v F = 1 . · ˚A. Therange of v F q is from 0 to 0 . ( θ ~B ), andspin-wave velocity, w ( ± π/ θ ~B , θ ~B ) [cf. Eq. (1)] areplotted in Fig. 3 a . Using the parameters specified above,the best agreement with the data is achieved by choosing F a = − .
41, ∆ R ≈ .
05 meV, and ∆ D ≈ . α = 1 . · ˚A and β = 3 . · ˚A. The valuesof α and β are very close to those found in Ref. 16. Anestimate for F a , obtained using a screened Coulomb in-teraction with a dielectric constant ǫ = 10 . yields F a ≈ − .
35, which is pretty closeto the number that reproduces the experimental data.The calculated q -dependence of the mode (Fig. 3b) alsoreproduces the experimentally observed one very well.Based on this agreement between the theory and ex-periment, we argue that the collective mode observed inRefs. 14–16 is, in fact, one of the sought-after CSWs, probed in the regime of a strong magnetic field. Thesame type of experiments performed at lower fields onsystems with stronger SOC and/or higher mobilities, e.g.,on GaAs/GaAlAs or InGaAs/InAlAs, should reveal thewhole spectrum shown in Fig. 1a. φ /2 π w ( µ e V µ m ) (a) φ /2 π Ω ( q = ) ( m e V ) v F q (meV) Ω ( m e V ) (b)B=2TB=−2T φ = π /4 FIG. 3: a) Variation of the CSW velocity w , as defined inEq. (1), with the angle φ = θ ~B − π/
2. Inset: variation ofthe frequency at q = 0 with φ . b) Dispersion of a CSW inthe CdMnTe quantum well in the magnetic field of 2 T for φ = π/
4. To be consistent with experimental geometry, wefixed ~B ⊥ ~q . The negative sign of the field implies flipping itsdirection. Comparison with other theoretical approaches is nowin order. The phenomenological model of Refs. 15 and 16describes the data assuming very strong (up to a factor of 6 .
5) renormalization of SOC by many-body effects, whichis not consistent with the moderate ( <
2) values of pa-rameter r s in Cd − x Mn x Te. Our microscopic theory ex-plains the data without such an assumption. In addition,it is argued in Ref. 15 and 16 the entire q -dependence ofa collective-mode spectrum in the presence of SOC canbe obtained by a linear shift of the quadratic term in theSL dispersion: Aq → A | ~q + ~q | . We see from Eq. (20)that this would require | a | = 2 | a | . This is reproducedonly in the weak coupling limit | F a | ≪ Note.
When this manuscript was almost completed, asubset of authors of Refs. 15 and 16 announced a first-principles study in Ref. 29, in which they also identifiedthe π -periodic modulation of the mode frequency at q = 0as a second-order effect in SOC. VI. CONCLUSIONS
In conclusion, we developed a microscopic theory ofRaman scattering in a two-dimensional Fermi liquid inthe presence of both Rashba and Dresselhaus spin-orbitcouplings, and subject to an in-plane magnetic field. In-terplay between an exchange part of the electron-electroninteraction and spin-orbit coupling leads to resonancepeaks at frequencies corresponding to novel collectivemodes–chiral spin waves. We derived the polarizationdependence of the Raman signal and showed that theRaman spectrum can be used to uniquely determine dif-ferent components of spin-orbit coupling by measuringa characteristic linear-in- q term in the dispersion of chi-ral spin waves. Our theory describes quantitatively allthe features of the Raman signal observed recently on aCdTe quantum well. The formalism developed herecan be readily extended to other two-dimensional sys-tems with broken inversion symmetry, such as grapheneon transition-metal-dichalcogenide substrates and sur-face states of topological insulators/superconductors.
Acknowledgments
The authors are grateful to Y. Gallais, A. Kumar, I.Paul, F. Perez, and C. A. Ullrich for stimulating discus-sions, and to M. Imran for his help at the initial stage ofthis work.
Appendix A: Computational details
The general form of the dispersion a CSW is estab-lished in Eqs. (1) and (17) using symmetry and dimen-sional analysis. Here, we confirm this form by an explicitcalculation which yields the forms of functions a , ˜ a , a ,and a . We start from Eq. (19) and find the expansions ofthe spin-spin correlation functions, χ ∗ ab ( Q ), to order q .Unless stated otherwise, all frequencies in the Appendixare the Matsubara ones. Whenever it does not lead to aconfusion, the Matsubara index n will be suppressed andthe frequencies will be denoted simply as i Ω. Analyticcontinuation to real frequencies will be done at the finalstep. We follow the general scheme developed in Refs. 10and 12 to find the dispersions of the collective modes.For brevity, we will switch to notations where F a ≡ − u and χ ∗ ab ≡ − Π ab .
1. Spin-charge correlation functions in thepresence of magnetic field and spin-orbit coupling
In this Appendix, we consider the general propertiesof the spin-charge correlation function defined asΠ ab ( Q ) = Z ~k Z ω Tr h ˆ σ a ˆ G ∗ K + Q ˆ σ b ˆ G ∗ K i , (A1) where, as before, K = ( ~k, iω m ), etc., a, b ∈ { . . . } withthe 0 th component corresponding to the charge and the1 st . . . rd components corresponding to three spin pro-jections. The Green’s function G ∗ P is obtained from G P in Eq. (5) by replacing ∆ Z → ∆ ∗ Z = ∆ Z / (1 − u ). Sub-stituting Eq. (5) into Eq. (A1), we obtainΠ ab ( Q ) = Z ~k Z ω (cid:20) g + ˜ g + + g − ˜ g − (cid:26) δ ab + Tr[ˆ σ a ˆ σ i ˆ σ b ˆ σ j ] s i ˜ s j | s || ˜ s | (cid:27) + g + ˜ g − + g − ˜ g + (cid:26) δ ab − Tr[ˆ σ a ˆ σ i ˆ σ b ˆ σ j ] s i ˜ s j | s || ˜ s | (cid:27) + g + ˜ g + − g − ˜ g − (cid:26) Tr[ˆ σ b ˆ σ a ˆ σ j ] s j | s | + Tr[ˆ σ a ˆ σ b ˆ σ j ] ˜ s j | ˜ s | (cid:27) + g + ˜ g − − g − ˜ g + (cid:26) Tr[ˆ σ b ˆ σ a ˆ σ j ] s j | s | − Tr[ˆ σ a ˆ σ b ˆ σ j ] ˜ s j | ˜ s | (cid:27)(cid:21) , (A2)where summation over repeated indices i, j ∈ { , } is im-plied and a tilde above a quantity means that its (2 + 1)momentum is shifted by Q with respect to the momen-tum of the corresponding quantity without a tilde, i.e.,˜ g + ≡ g + ( K + Q ), etc. Since s l ( l = 1 ,
2) depend onlyon the spatial momentum, it is understood that in thiscase ˜ s l ≡ s l ( ~k + ~q ). Because the magnetic field andSOC are assumed to be weak, integration over the actual(anisotropic) FS can be replaced by that over a circularFS of radius k F . Accordingly, Z ~k · · · = N F Z dθ ~k π Z dξ ~k · · · ≡ N F Z θ Z ξ ~k . . . (A3)where N F = k F / πv F is the 2D density of states and ξ ~k = v F ( k − k F ). In the same approximation, ε ~k,ν = ξ ~k + ν | s | , where | s | = p s + s is evaluated at ξ = 0 but doesdepend on the azimuthal angle θ ~k of ~k . Furthermore, wewill be neglecting the difference between ˜ s l (evaluatedat ~k + ~q ) and s l (evaluated at ~k ): keeping such termswould amount to higher order corrections. Accordingly,we approximate ε ~k + ~q,ν = ξ ~k + ~q + ν | ˜ s | ≈ ξ ~k + ~v F · ~q + ν | s | ,where we also neglected the O ( q ) term as being higherorder in q/k F .The chiral Green’s functions at momentum ~k + ~q needto be expanded to second order in ~q . Within the same approximations as specified above,˜ g ν ≡ g ν ( ~k + ~q, iω + i Ω)= ¯ g ν + ∂ j ¯ g ν q j + 12 ∂ j ∂ j ′ ¯ g ν q j q j ′ ≈ ¯ g ν + ¯ g ν ~v F · ~q + ¯ g ν ( ~v F · ~q ) , (A4)where ¯ g ν ≡ g ν ( ~k, iω + i Ω) and ∂ l ≡ ∂/∂k l .In what follows, we will need the following integrals Z ξ ~k Z ω g ν ¯ g ν ′ = − ( ν − ν ′ ) | s | i Ω + ( ν − ν ′ ) | s | , Z ξ ~k Z ω g ν ¯ g ν ′ = i Ω( i Ω + ( ν − ν ′ ) | s | ) , Z ξ ~k Z ω g ν ¯ g ν ′ = i Ω( i Ω + ( ν − ν ′ ) | s | ) , (A5)where R ξ ~k ≡ R dξ ~k . Using these integrals, we find Z ξ ~k Z ω g + ˜ g + = ~v F · ~qi Ω + ( ~v F · ~q ) ( i Ω) (A6a) Z ξ ~k Z ω g − ˜ g − = Z ξ ~k Z ω g + ˜ g + , (A6b) Z ξ ~k Z ω g ± ˜ g ∓ = ∓ | s | i Ω ± | s | + i Ω( i Ω ± | s | ) ~v F · ~q + i Ω( i Ω ± | s | ) ( ~v F · ~q ) . (A6c)Since we put ˜ s l = s l in Eq. (A2), the spin-charge com-ponents of the polarization operator, Π a ( Q ) with a = 0,are reduced to a combination Π a ∝ R ξ ~k R ω ( g + ˜ g + − g − ˜ g − ) s a , which is equal to zero by Eq. (A6b). Thusthe spin sector is decoupled form the charge sector in thelimit of q/k F →
0. Restricting to a, b ∈ { , , } , weevaluate the traces occurring in Eq. (A2) asTr[ˆ σ a ˆ σ b ˆ σ j ] s j = iλ abj s j Tr[ˆ σ a ˆ σ i ˆ σ b ˆ σ j ] s i s j = − δ ab s + 4 s a s b , (A7) where λ abc is the Levi-Civita tensor and it is understoodthat s = 0. We thus obtain a compact form of the spin-spin correlation function with a, b ∈ , , ab ( Q ) = N F Z θ ~k Z ξ ~k Z ω (cid:20) ( g + ˜ g + + g − ˜ g − ) s a s b | s | + ( g + ˜ g − + g − ˜ g + ) (cid:18) δ ab − s a s b | s | (cid:19) − i ( g + ˜ g − − g − ˜ g + ) λ abc s c | s | (cid:21) , (A8)where the integrals over ω and ξ ~k are to be substituted from Eqs. (A6a-A6b). For further convenience, we also listexplicit formulas for s a and related quantities: s = 12 (cid:0) ∆ R sin θ ~k + ∆ D cos θ ~k − ∆ ∗ Z cos θ ~B (cid:1) ,s = 12 (cid:0) − ∆ R cos θ ~k − ∆ D sin θ ~k − ∆ ∗ Z sin θ ~B (cid:1) , s = ∆ R sin θ ~k + ∆ D cos θ ~k + (∆ ∗ Z ) cos θ ~B + ∆ R ∆ D sin 2 θ ~k − R ∆ ∗ Z sin θ ~k cos θ ~B − D ∆ ∗ Z cos θ ~k cos θ ~B , s = ∆ R cos θ ~k + ∆ D sin θ ~k + (∆ ∗ Z ) sin θ ~B + ∆ R ∆ D sin 2 θ ~k + 2∆ R ∆ ∗ Z cos θ ~k sin θ ~B + 2∆ D ∆ ∗ Z sin θ ~k sin θ ~B , s = 4( s + s ) = ∆ R + ∆ D + (∆ ∗ Z ) + 2∆ R ∆ D sin 2 θ ~k − R ∆ ∗ Z sin( θ ~k − θ ~B ) − D ∆ ∗ Z cos( θ ~k + θ ~B ) , s s = − (∆ R + ∆ D ) sin θ ~k cos θ ~k + (∆ ∗ Z ) sin θ ~B cos θ ~B − ∆ R ∆ D + ∆ R ∆ ∗ Z cos( θ ~k + θ ~B ) + ∆ D ∆ ∗ Z sin( θ ~k − θ ~B ) , (A9)where θ ~B is the angle between ~B and the x axis.We can now apply the general result, Eq. (A8) to particular situations. In what follows, we will make v F q andΩ dimensionless by rescaling to ∆ ∗ Z = ∆ Z / (1 − u ) and define, for the use in Appendices only, r ≡ ∆ R / ∆ ∗ Z and d ≡ ∆ D / ∆ ∗ Z . Note that these definitions differ from those in the main text, where ∆ R and ∆ D are rescaled to ∆ Z .
2. Silin-Leggett mode
To test our general formula, we apply it first to a simple case of the Silin-Leggett mode, the dispersion of whichis known. The Silin-Leggett mode is the collective mode in the absence of SOC ( r = d = 0) and in the presenceof ~B . Although the coordinate system in this case is in fact defined by the direction of ~B , we choose ~B to point inan arbitrary direction with respect to a fixed coordinate system. This will useful for a more general case, when thespin-rotational symmetry is broken by SOC. In this case, s = ∆ ∗ Z is independent of angle θ ~B . Using Eq. (A8), wefind: Π ( Q ) − N F = ( v F q ) + sin θ ~B Ω + 1 (cid:18) − + 3Ω + 1Ω (Ω + 1) ( v F q ) (cid:19) , Π ( Q ) − N F = ( v F q ) + cos θ ~B Ω + 1 (cid:18) − + 3Ω + 1Ω (Ω + 1) ( v F q ) (cid:19) , Π ( Q ) − N F = 1Ω + 1 (cid:18) − Ω (Ω + 1) ( v F q ) (cid:19) , Π ( Q ) − N F = − sin θ ~B cos θ ~B Ω + 1 (cid:18) − + 3Ω + 1Ω (Ω + 1) ( v F q ) (cid:19) , Π ( Q ) − N F = − Ω sin θ ~B Ω + 1 (cid:18) − − + 1) ( v F q ) (cid:19) , Π ( Q ) − N F = Ω cos θ ~B Ω + 1 (cid:18) − + 1(Ω + 1) ( v F q ) (cid:19) . (A10)Π ab can be written more compactly in a matrix form asˆΠ( Q ) − N F = κ a + ( B + κ b ) sin θ ~B − ( B + κ b ) sin θ ~B cos θ ~B ( C + κ c ) sin θ ~B − ( B + κ b ) sin θ ~B cos θ ~B κ a + ( B + κ b ) cos θ ~B − ( C + κ c ) cos θ ~B − ( C + κ c ) sin θ ~B ( C + κ c ) cos θ ~B ( B + κ g ) , (A11)where B = 1Ω + 1 C = − ΩΩ + 1 . (A12)The other definitions are apparent from a term-by-termcomparison of expressions in Eq. (A10) and the corre-sponding entries in Eq. (A11). All the κ ’s are correc-tions that are small in v F q . At v F q = 0, the eigen-mode equation Det(1 + u ˆΠ / N F ) = 0 has a solutionΩ + 1 = s ≡ u − u such that Ω = − (1 − u ) .Restoring the dimensional frequency and continuing an-alytically to real frequencies, we find that the frequencyat the q = 0 is simply given by Ω = ∆ Z , in agreementwith Kohn’s theorem. To find corrections to this resultat small but finite v F q , we look for a solution of the formΩ + 1 = s + κ . Expanding the eigenmode equation tolinear order in all the κ ’s, we obtain1 − B u +( B + C ) u = u [(1 − B u )( κ a + κ b + κ g ) − u C κ c ] . (A13)Solving for κ , we obtain κ = 2(1 − u ) u ( v F q ) . (A14) Restoring the units and continuing analytically, we ob-tain the frequency of the mode at finite q asΩ = ∆ Z − (1 − u ) u ( v F q ) ∆ Z . (A15)Relabeling u → − F a , we obtain the coefficient a , asgiven by Eq. (20).
3. Chiral spin wave in the high-field limit a. Frequency of a chiral spin wave at q = 0 In this section, we derive the frequency of the chiralspin wave at q = 0 and in the high-field limit, when∆ R , ∆ D ≪ ∆ Z or r, d ≪ u is very close 1, i.e., the system is close to a fer-romagnetic transition, there is no real difference betweenconditions ∆ R , ∆ D ≪ ∆ Z and ∆ R , ∆ D ≪ ∆ ∗ Z .] WithΠ ab ( i Ω) ≡ Π ab ( i Ω , ~q = 0), we obtain from Eq. (A8)Π ( i Ω) − N F = sin θ ~B Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) + 12 r + d Ω + 1 + 2 rd sin 2 θ ~B (Ω + 1) , Π ( i Ω) − N F = cos θ ~B Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) + 12 r + d Ω + 1 + 2 rd sin 2 θ ~B (Ω + 1) , Π ( i Ω) − N F = 1Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) + r + d Ω + 1 + 4 rd sin 2 θ ~B (Ω + 1) , Π ( i Ω) − N F = − sin θ ~B cos θ ~B Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) + rd Ω + 1 − rd (Ω + 1) , Π ( i Ω) − N F = − Ω sin θ ~B Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) − rd Ω cos θ ~B (Ω + 1) , Π ( i Ω) − N F = Ω cos θ ~B Ω + 1 (cid:20) − r + d Ω + 1 + 2 r + d − rd sin 2 θ ~B (Ω + 1) (cid:21) + 2 rd Ω sin θ ~B (Ω + 1) . (A16)In a matrix form,ˆΠ( i Ω) − N F = κ a + ( B + κ b ) sin θ ~B κ f − ( B + κ b ) sin θ ~B cos θ ~B ( C + κ c ) sin θ ~B + κ d cos θ ~B κ f − ( B + κ b ) sin θ ~B cos θ ~B κ a + ( B + κ b ) cos θ ~B − ( C + κ c ) cos θ ~B − κ d sin θ ~B − ( C + κ c ) sin θ ~B − κ d cos θ ~B ( C + κ c ) cos θ ~B + κ d sin θ ~B κ a + ( B + κ b ) , (A17)0where B and C are the same as in Eq. (A12) and the κ ’s, which are small in r and d , are again defined by a term-by-term comparison of Eq. (A16) and the entries in Eq. (A17). We seek a solution of the form Ω + 1 = s + κ , where κ is also small in r and d . To linear order in κ ’s, the eigenmode equation reads1 − B u + ( B + C ) u = u (cid:2) (1 − B u )(2 κ b + 3 κ a − κ f sin 2 θ ~B ) − u C ( κ c + κ d sin 2 θ ~B ) (cid:3) . (A18)Solving for κ we get: κ = ( r + d ) (cid:26) (1 − u )(2 − u )2 u (cid:27) − rd sin 2 θ ~B (cid:26) (1 − u )(2 − u ) u (cid:27) . (A19)Restoring the units and continuing analytically to real frequencies, we obtain the coefficients a and ˜ a in Eq. (20). b. Linear-in- q term in the dispersion Now we are interested in all terms to linear order in r , d , and v F q . Accordingly, we need to expand the quantitiesin Eq. (A9) to linear order in these variables:2 s = ∆ ∗ Z (cid:0) − cos θ ~B + r sin θ ~k + d cos θ ~k (cid:1) , s = ∆ ∗ Z (cid:0) − sin θ ~B − r cos θ ~k − d sin θ ~k (cid:1) , s = (∆ ∗ Z ) (cid:0) cos θ ~B − r sin θθ ~k cos θ ~B − d cos θ ~k cos θ ~B (cid:1) , s = (∆ ∗ Z ) (cid:0) sin θ ~B + 2 r cos θ ~k sin θ ~B + 2 d sin θ ~k sin θ ~B (cid:1) , s = (∆ ∗ Z ) (cid:0) − r sin( θ ~k − θ ~B ) − d cos( θ ~k + θ ~B ) (cid:1) , s s = (∆ ∗ Z ) (cid:2) sin θ ~B cos θ ~B + r cos( θ ~k + θ ~B ) + d sin( θ ~k − θ ~B ) (cid:3) . (A20)Furthermore, ~v F · ~q = v F q cos( θ ~k − θ ~q ) Z ξ ~k Z ω ( g + ˜ g + + g − ˜ g − ) = 2 v F qi Ω cos( θ ~k − θ ~q ) , Z ξ ~k Z ω ( g + ˜ g − + g − ˜ g + ) = − + 1 " − r θ ~k Ω Ω + 1 − i Ω v F q (Ω + 1) " Ω − r θ ~k (3Ω − + 1 cos( θ ~k − θ ~q ) , Z ξ ~k Z ω ( g + ˜ g − − g − ˜ g + ) = 2 i ΩΩ + 1 " − ˜ r θ ~k (Ω − + 1 + 4Ω v F q (Ω + 1) " − ˜ r θ ~k (Ω − + 1 cos( θ ~k − θ ~q ) , (A21)where ˜ r ϑ ≡ r sin( ϑ − θ ~B ) + d cos( ϑ + θ ~B ). Let’s further introduce r ,ϑ ≡ r sin ϑ + d cos ϑ,r ,ϑ ≡ r cos ϑ + d sin ϑ,r ,ϑ ≡ r ,ϑ sin θ ~B + r ,ϑ cos θ ~B (A22)1such that r ,ϑ cos θ ~B − r ,ϑ sin θ ~B = ˜ r ϑ . Note that the pairs (˜ r ϑ , r ,ϑ ) and ( r ,ϑ , r ,ϑ ) are related by a θ ~B rotation.This leads to: Π ( Q ) − N F = sin θ ~B Ω + 1 (cid:20) i Ω(3Ω − + 1) v F q ˜ r θ ~q (cid:21) + v F qi Ω Ω (3Ω + 1)2(Ω + 1) r ,θ ~q sin 2 θ ~B , (A23)Π ( Q ) − N F = cos θ ~B Ω + 1 (cid:20) i Ω(3Ω − + 1) v F q ˜ r θ ~q (cid:21) − v F qi Ω Ω (3Ω + 1)2(Ω + 1) r ,θ ~q sin 2 θ ~B , (A24)Π ( Q ) − N F = 1Ω + 1 (cid:20) i Ω(3Ω − + 1) v F q ˜ r θ ~q (cid:21) , (A25)Π ( Q ) − N F = − sin θ ~B cos θ ~B Ω + 1 (cid:20) i Ω(3Ω − + 1) v F q ˜ r θ ~q (cid:21) − v F qi Ω Ω (3Ω + 1)2(Ω + 1) r ,θ ~q cos 2 θ ~B , (A26)Π ( Q ) − N F = − Ω sin θ ~B Ω + 1 (cid:20) i Ω(Ω − + 1) v F q ˜ r θ ~q (cid:21) − v F qi Ω Ω r ,θ ~q cos θ ~B (Ω + 1) , (A27)Π ( Q ) − N F = Ω cos θ ~B Ω + 1 (cid:20) i Ω(Ω − + 1) v F q ˜ r θ ~q (cid:21) − v F qi Ω Ω r ,θ ~q sin θ ~B (Ω + 1) . (A28)In a matrix form,ˆΠ( Q ) − N F = κ a sin 2 θ ~B + ( B + κ b ) sin θ ~B − κ a cos 2 θ ~B − ( B + κ b ) sin θ ~B cos θ ~B − ( C + κ c ) sin θ ~B − κ d cos θ ~B − κ a cos 2 θ ~B − ( B + κ b ) sin θ ~B cos θ ~B − κ a sin 2 θ ~B + ( B + κ b ) cos θ ~B ( C + κ c ) cos θ ~B − κ d sin θ ~B ( C + κ c ) sin θ ~B + κ d cos θ ~B − ( C + κ c ) cos θ ~B + κ d sin θ ~B ( B + κ b ) . (A29)Once again B and C are the same as before and the κ ’s are obtained by a term-term comparison of Eqs. (A23-A28)and Eq. (A29).We look for a solution of the form Ω + 1 = s + κ . To linear order in κ ’s, the eigenmode equation is now of thefollowing form 1 − B u + ( B + C ) u = 2 u { (1 − B u ) κ b − u C κ c } (A30)Solving for κ , we get κ = − − u ) [(4 − u )(1 − u ) + u ] u (2 − u ) v F q ˜ r θ ~q = − − u ) [(4 − u )(1 − u ) + u ] u (2 − u ) v F q (cid:0) r sin( θ ~q − θ ~B ) + d cos( θ ~q + θ ~B ) (cid:1) . (A31)From here, one can read the coefficient a as given by Eq. (20). T. P. Devereaux and R. Hackl,“Inelastic light scatter-ing from correlated electrons”, Rev. Mod. Phys. , 175(2007). Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.Takanashi, S. Maekawa, and E. Saitoh, “Transmission ofelectrical signals by spin-wave interconversion in a mag-netic insulator”, Nature , 262 (2010). W. Hayes and R. Loudon, Scattering of Light by Crystals(John Wiley and Sons, 1978). E. I. Rashba and V. I. Sheka, “Symmetry of Energy Bandsin Crystals of Wurtzite Type II. Symmetry of Bands withSpin-Orbit Interaction Included”, Sov. Phys. Solid State ,1257 (1961); Yu. Bychkov and E. I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, JETPLett. , 79 (1984). G. Dresselhaus, “Spin-Orbit Coupling Effects in ZincBlende Structures”, Phys. Rev. , 580 (1955). A.V. Chaplik, L.I. Magarill and R.Z. Vitlina, “Inelasticlight scattering by 2D electron system with SO interac-tion”, Nanoscale Research Letters , 537 (2012). A. Shekhter, M. Khodas, and A. M. Finkelstein, “Chiralspin resonance and spin-Hall conductivity in the presenceof the electron-electron interactions”, Phys. Rev. B ,165329 (2005). A. Ashrafi and D. L. Maslov, “Chiral Spin Waves in FermiLiquids with Spin-Orbit Coupling”, Phys. Rev. Lett. ,227201 (2012). S.-S. Zhang, X.-L. Yu, J. Ye, and W.-Mi. Liu, “Collectivemodes of spin-orbit-coupled Fermi gases in the repulsiveregime”, Phys. Rev. A , 063623 (2013). S. Maiti, V. A. Zyuzin, and D. L. Maslov, “Collectivemodes in two- and three-dimensional electron systems withRashba spin-orbit coupling”, Phys. Rev. B , 035106(2015). S. Maiti and D. L. Maslov, “Intrinsic Damping of Col-lective Spin Modes in a Two-Dimensional Fermi Liquidwith Spin-Orbit Coupling”, Phys. Rev. Lett. , 156803(2015). S. Maiti, M. Imran, and D. L. Maslov, “Electron spin res-onance in a two-dimensional Fermi Liquid with spin-orbitcoupling”, Phys. Rev. B , 045134 (2016). A. Kumar and D. L. Maslov, “Effective lattice model forcollective modes in a Fermi liquid with spin-orbit cou-pling”, arXiv:1701.02781. F. Baboux, F. Perez, C. A. Ullrich, I. D’Amico, G. Kar-czewski, and T. Wojtowicz, “Coulomb-driven organizationand enhancement of spin-orbit fields in collective spin ex-citations”, Phys. Rev. B , 121303(R) (2013). F. Baboux, F. Perez, C. A. Ullrich, G. Karczewski, and T.Wojtowicz, “Electron density magnification of the collec-tive spin-orbit field in quantum wells”, Phys. Rev. B ,125307 (2015). F. Perez, F. Baboux, C. A. Ullrich, I. DAmico, G. Vignale,G. Karczewski, and T. Wojtowicz, “Spin-Orbit TwistedSpin Waves: Group Velocity Control”, Phys. Rev. Lett. , 137204, (2016). V. P. Silin, “Oscillations of a Fermi liquid in a magneticfield”, Sov. Phys. JETP , 945 (1958). A. J. Leggett, “Spin diffusion and spin echoes in liquid Heat low temperature”, J. Phys. C: Solid State Phys. , 448(1970). E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics PartII (Pergamon Press, New York, 1980). includes only thestates within either the conduction or valence band. S.D. Sarma and D.W. Wang, “Resonant Raman Scatter-ing by Elementary Electronic Excitations in Semiconduc-tor Structures”, Phys. Rev. Lett. , 816 (1999). F. Perez, “Spin-polarized two-dimensional electron gas em-bedded in a semimagnetic quantum well: Ground state,spin responses, spin excitations, and Raman spectrum”,Phys. Rev. B , 045306 (2009). W. Kohn, “Cyclotron Resonance and de Haas-van AlphenOscillations of an Interacting Electron Gas”, Phys. Rev. , 1242 (1961). Y. Yafet, “ g -factors and spin-lattice relaxation”, in SolidState Physics, Vol. 14, edited by F. Seitz and D. Turnbull(Academic, New York, 1963), p. 92. See V. P. Mineev, “Transverse spin dynamics in a spin-polarized Fermi liquid”, Phys. Rev. B , 144429 (2004),and references therein. Strictly speaking, the coefficients of the linear invariantsmay depend on the direction of ~B . This effect, however,occurs only to order ∆ R ∆ D and can thus be neglected atsmall q . By the same token, the correction to the q termin SL mode is on the order ∆ R ∆ D / ∆ Z and can also beneglected. G.-H. Chen and M. E. Raikh, “Exchange-induced enhance-ment of spin-orbit coupling in two-dimensional electronicsystems”, Phys. Rev. B , 4826 (1999). D. S. Saraga and D. Loss, “Fermi liquid parameters in twodimensions with spin-orbit interaction”, Phys. Rev. B ,195319 (2005). I. Strzalkowski, S. Joshi, and C. R. Crowell, “Dielectricconstant and its temperature dependence for GaAs, CdTe,and ZnSe”, Appl. Phys. Lett. , 350 (1976).29