Spin-orbit effects in GaAs quantum wells: Interplay between Rashba, Dresselhaus, and Zeeman interactions

The interplay between Rashba, Dresselhaus and Zeeman interactions in a quantum well submitted to an external magnetic field is studied by means of an accurate analytical solution of the Hamiltonian, including electron-electron interactions in a sum rule approach. This solution allows to discuss the influence of the spin-orbit coupling on some relevant quantities that have been measured in inelastic light scattering and electron-spin resonance experiments on quantum wells. In particular, we have evaluated the spin-orbit contribution to the spin splitting of the Landau levels and to the splitting of charge- and spin-density excitations. We also discuss how the spin-orbit effects change if the applied magnetic field is tilted with respect to the direction perpendicular to the quantum well.


I. INTRODUCTION
The study of spin-orbit (SO) effects in semiconductor nanostructures has been the object of many experimental and theoretical investigations in the last few years, see e.g. Refs. 1- 16 and Refs. therein. In spite of this, the extraction from measurements of the effective spinorbit coupling constant of both Dresselhaus 17 and Bychkov-Rashba 18,19 SO interactions is not a simple matter, since the SO corrections to the electron energy spectrum in a magnetic field (B) are vanishingly small because they correspond to second order effects in perturbation theory. Thus, few physical observables are sensitive enough to the SO interactions and allow for a quantitative estimate of their coupling constants. One such observable is the splitting of the cyclotron resonance (CR), which has been determined in transmission experiments with far-infrared radiation, 20 and is due to the coupling between charge-density and spin-density excitations. 21 A less clear example is the change in the Larmor frequency -spin splitting. 22 The spin splitting has been observed in electron-spin resonance 23,24 and in inelastic light scattering experiments. 25,26 In this work we extend our previous results 21, 22 by obtaining an approximate, yet very accurate, analytical solution of the quantum well SO Hamiltonian that contains both Dresselhaus and Bychkov-Rashba interactions. In the limit of high magnetic field, this solution coincides with the results of second order perturbation theory, and allows to study the SO corrections to the Landau levels in a simple way, and to study the transitions induced by an external electromagnetic field acting upon the system. This work is organized as follows. In Sec. II we present the general formalism for the single-particle (sp) Hamiltonian. These results are used in Sec. III to study the transitions caused by an external electromagnetic field. The role of the electron-electron (e-e) interaction is discussed In Sec. IV within a sum rule approach. In Sec. V we discuss the splitting of the Landau levels and the appearance of charge-and spin-density modes making, whenever possible, qualitative comparisons with the experimental results. 20,24,[27][28][29] A brief summary is presented in Sec. VI, and the generalization of some of the expressions derived in Sec. II to the case of tilted magnetic fields is presented in the Appendix.

II. SINGLE-PARTICLE STATES
In the effective mass, dielectric constant approximation, the quantum well Hamiltonian H can be written as H = H 0 + e 2 ǫ N i<j=1 1 |r i −r j | , where H 0 is the one-body Hamiltonian consisting of the kinetic, Zeeman, Rashba and Dresselhaus terms (1) m = m * m e is the effective electron mass in units of the bare electron mass m e , P ± = P x ±iP y , σ ± = σ x ± iσ y , where the σ's are the Pauli matrices, and P = −ih∇ + e c A represents the canonical momentum in terms of the vector potential A, which in the following we write in the Landau gauge, A = B(0, x, 0), with B = ∇ × A = Bẑ. The second term in Eq. (1) is the Zeeman energy, where µ B =he/(2m e c) is the Bohr magneton, and g * is the effective gyromagnetic factor. The third and fourth terms are the usual Rashba and Dresselhaus interactions, respectively. Note that for bulk GaAs, taken here as an example, g * = −0.44, m * = 0.067, and the dielectric constant is ǫ = 12.4. To simplify the expressions, in the following we shall use effective atomic unitsh = e 2 /ǫ = m = 1.

Introducing the operators
with [a − , a + ] = 1 and ω c = eB/c being the cyclotron frequency, the sp Hamiltonian h 0 can be rewritten as where ω L = |g * µ B B| is the Larmor frequency andλ R,D = λ R,D We expand φ 1 and φ 2 into oscillator states |n as φ 1 = ∞ n=0 a n |n , φ 2 = ∞ n=0 b n |n , on which a + and a − act in the usual way, i.e., 1 2 (a + a − + a − a + )|n = (n + 1 2 )|n , a + |n = √ n + 1|n+1 , a − |n = √ n|n−1 , and a − |0 = 0. This yields the infinite system of equations When only the Rashba or Dresselhaus terms are considered, Eqs. (5) can be exactly solved. 14,[30][31][32] For the sake of completeness, we give here the corresponding results. In the λ D = 0 case, combining Eqs. (5) one obtains either of which yields the energies One also obtains which together with the normalization condition |a ε ± n n−1 | 2 + |b ε ± n n | 2 = 1 solves exactly the problem [for n = 0, a −1 = 0, b 0 = 1, and ε 0 = 1 2 (1 + ω L /ω c )]. Eqs. (6) indicate that in the series expansion of the spinor |φ , only one a i and one b i coefficient appears. Specifically, In the limit of zero spin-orbit, the spinors |n d and |n u become |n 0 1 and |n 1 0 , respectively. The exact expressions for the a i and b i coefficients entering Eq. (9) are easy to work out. Expressions valid up to λ 2 R,D order are given in the next subsection. The λ R = 0 case can be worked out similarly. One obtains the secular equation which yields One also obtains which together with the normalization condition |a ε ± n n | 2 + |b ε ± n n−1 | 2 = 1 solves exactly the problem (in this case, for n = 0, b −1 = 0 and a 0 = 1).
Again, in the series expansion of the spinor |φ , only one a i and one b i coefficient appears: and the same comments as before apply.
If both terms are simultaneously considered, the SO interaction couples the states of all Landau levels, and an exact analytical solution to Eqs. (5) is unknown, and likely does not exist. We are going to find an approximate solution that in the λ 2 R,D /ω c ≪ 1 limit coincides with the results of second order perturbation theory, i.e., it is valid up toλ 2 R,D order, and it is quite accurate as compared with exact results obtained numerically. Combining Eqs.
(5), one can write and n − 1 + α − ε a n−2 − (n + 1)(n + 2) The approximate solution is obtained by taking a n−2 = a n+2 = b n−2 = b n+2 = 0 in the above equations. This means that for each level |n , the SO interaction is allowed to couple it only with the |n − 1 and |n + 1 levels. This solution, which consists of a |n d and a |n u spinor, is therefore obtained by solving first the secular, cubic equation Together with the equations and the normalization condition |a n−1 | 2 +|a n+1 | 2 +|b n | 2 = 1, they determine the |n d solution.
The solution corresponding to the |n u spinor is obtained by solving the secular equation Together with the equations and |a n | 2 + |b n−1 | 2 + |b n+1 | 2 = 1, they determine the |n d solution.
Since all the estimates available in the literature (see for example Refs. 15,21,22 and Refs. therein) yield λ 2 R,D values of the order of 10 µeV, and ω c in GaAs is of the order of the meV even at small B(∼ 1 T), it is worth to examine the above solutions in thẽ λ 2 R,D = 2λ 2 R,D /ω c ≪ 1 limit, in which the secular equations have solutions easy to interpret. To orderλ 2 R,D , the relevant solution to Eq. (16) containing both SO terms is that corresponds to the spinor |n d In the following, we will refer to this solution as to the quasi spin-down (qdown) solution, since in the zero spin-orbit coupling limit |n d becomes |n 0 1 . Analogously, Eq. (18) has the solution ε u n = n + β − 2(n + 1) that corresponds to the spinor |n u In the following, we will refer to this solution as to the quasi spin-up (qup) solution, since in the zero spin-orbit coupling limit |n u becomes |n 1 0 . When either λ R or λ D are zero, Eqs. (21) and (24) reduce to the exact Eqs. (9) and (13), respectively, and the corresponding a i and b i coefficients, valid up to order λ R,D , can be extracted from Eqs. (22) and (25). These Eqs. show that a n and b n are of order O(1), whereas a n±1 and b n±1 are of order O(λ R,D ), and a n±2 and b n±2 are of order O(λ 2 R,D ). This shows that the neglected terms in Eqs. (14) and (15) are of order O(λ 4 R,D ). The sp energies obtained from Eqs. (20) and (23), valid in the λ 2 R,D /ω c ≪ 1 limit, are Together with the structure of the associated spinors, Eqs. (21) and (24), this sp energy spectrum constitutes one of the main results of our work. By suitable differences of these energies, one may obtain the sp transition energies discussed in the next Sec.
The above sp energies coincide with the ones that can be derived from second order perturbation theory with the standard expression where |n = |n, ↑ , |n, ↓ are the spin-up and spin-down eigenstates of the sp Hamiltonian 2 )ω c + 1 2 ω L , respectively. The approximate solution Eq. (26) is very accurate in the high B limit (see below). It also carries an interesting information in the opposite limit of vanishing B. In this limit (16) and (18) yield the solutions which show that, at B ≃ 0, to order λ 2 R,D the Landau levels are not split due to the SO interaction, as one might naively infer from Eqs. (26). Another merit of the approximate solution is that it displays in a transparent way the interplay between the three spin-dependent interactions, namely Zeeman, Rashba and Dresselhaus. Such interplay has been also discussed in Ref. 22, in relation with the violation of the Larmor theorem due to the SO couplings, and in Ref. 33, where the Zeeman and SO interplay is discussed using the unitarily transformed Hamiltonian technique. Note also that in GaAs quantum wells, which are the object of application in this paper, due to the sign of g * , the lowest energy level is the qup one at the energy , containing the Rashba contribution alone, whereas the following level is the qdown one at the energy where M is a 2N × 2N matrix, while a and b are column vectors made with the sets of coefficients {a n , n = 0, . . . , N − 1} and {b n , n = 0, . . . , N − 1}, respectively. We have diagonalized M using a large enough N to ensure good convergence in the lower eigenvalues.
There is an excellent agreement between analytical and numerical results, differences starting to be visible only for strong Rashba intensities and high Landau bands. Actually, in Fig. 1  We can use the preceding results to study the sp transitions induced in the system by the interaction with a left-circular polarized electromagnetic wave propagating along the z-direction, i.e., perpendicular to the plane of motion of the electrons, whose vector potential is A(t) = 2A(cos θî + sin θĵ), with θ = ωt − qz. The sp interaction Hamiltonian where the velocity operator v ± is defined as The Hamiltonian h int can be rewritten as where the operators α + and α − acting on the spinor |φ are In the dipole approximation (q ≈ 0), the charge-density excitation operator is v ± . We note that, even in the presence of e-e interactions, this operator satisfies the f-sum rule: (33) where N is the electron number and ω n0 are the excitation energies.
We consider next several useful examples of sp matrix elements involving the operators α + , which is proportional to v + , and σ − , and the qup and qdown sp states of Eqs. (21) and (24). For the operator α + , we can write in general and have to distinguish between qup-qup, qdown-qdown, qup-qdown, and qdown-qup transitions. The qup-qup and qdown-qdown transitions represent the usual CR, and the qupqdown and qdown-qup are related to spin-flip transitions.
Let us start with the qup-qup and qdown-qdown transitions. To the order λ 2 R,D they are dominated by the transition n → n + 1 at the energies The energy splitting of the cyclotron resonance is The α + excitation operator also induces a qup-qdown transition with energy E d n − E u n and matrix element | n d |α + |n u | =λ D ω L /(ω c − ω L ). This is a spin-flip transition. In particular, when n = 0 it is related to the Larmor resonance at the energy 22 Note that the transition matrix element is linear inλ D , and that in the presence of the Rashba interaction alone, α + causes no spin-flip transition.
Other excitations that deserve some attention are those induced by the operators α + σ ± and α + σ z . They are detected in inelastic light scattering experiments as spin dipole resonances. 27 The operator α + σ z excites the same cyclotron states as α + , at the energies E d n+1 − E d n and E u n+1 − E u n , and with the same transition matrix element √ n + 1.
In contrast, the operator α + σ + mainly induces the transition from qdown to qup states at the energy E u n+1 − E d n , whereas the operator α + σ − induces the transition from qup to qdown states at the energy E d n+1 − E u n . The transition matrix elements are given by | (n + 1) u |α + σ + |n d | = | (n + 1) d |α + σ − |n u | = 2 √ n + 1. We thus see that the dipole transitions between Landau levels |n and |n + 1 at 'unperturbed' energies E n+1 − E n are split by the SO interaction, an effect that under some circumstances may be observed, as will be discussed in Sec. V.

IV. ELECTRON-ELECTRON INTERACTION AND SUM RULES
In this section our aim is to discuss the role played by the e-e interaction in the physical which tells us that, in photoabsorption experiments on quantum wells, a narrow absorption peak must appear at the cyclotron frequency ω = ω c excited by the cyclotron operator j P + j , and the Larmor theorem which states that in inelastic light scattering experiments at small transferred momentum, or in electron-spin resonance experiments, a narrow collective state must be excited by the Larmor operator S − = j σ j − at the Larmor frequency. These two modes are not influenced by the e-e interaction. Things radically change if we include in H the SO interaction. We then obtain and [H, where |0 and |φ n are the exact gs and excited states of the full Hamiltonian H (including e-e interactions), and ω n0 = E n −E 0 are the corresponding excitation energies. For k = 0 −3 we obtain Clearly, the more sum rules are known, the better knowledge of the Hamiltonian spectrum.
With the four sum rules of Eq. (42) we can obtain information only on two excited states -see below. Consequently, we will limit the analysis to the cases in which either the Rashba or Dresselhaus SO terms are present because, as one can see from Eq. (39) and Eq. (40) as well, in this case only two states would then be coupled by the corresponding SO interaction.
Let us first consider the case where F = G = i P − i , i.e., G † is the cyclotron operator. Evaluating the commutators in Eqs. (42) we have, to order λ 2 R,D , To obtain these Eqs. we have used that i P − i |0 = 0 and have assumed that the gs of the system is fully polarized, i.e., 0| i σ i z |0 = N. As such, these expressions are useless unless the left-hand side can be directly evaluated from the definition Eq. (41), and this evaluation yields a closed expression for the excitation energies and transition matrix elements. This is the case if we consider either of the λ R,D terms alone, because only two states are excited by the cyclotron operator G † = i P + i acting on the gs |0 . Dropping e.g. the λ R term, a straightforward calculation yields where π 1 and π 2 are the transition strengths to the cyclotron |φ n 1 and Larmor |φ n 2 states, π 1 = | φ n 1 | i P + i |0 | 2 and π 2 = | φ n 2 | i P + i |0 | 2 , and ω 10 , ω 20 are the respective excitation energies. This is in full agreement with the results of Sec. III, and shows that the ee interaction does not affect the frequency and transition strengths of the cyclotron and Larmor resonances.
The case λ D = 0 can be worked out similarly, and the same conclusion may be extracted.
We recall and stress again the results obtained in the previous section, namely that when λ D = 0, the Larmor state |φ n 2 is not excited by the cyclotron operator i P + i ( π 2 turns out to be zero). Alternatively, all previous calculations could have been carried out using for G † the Larmor operator, namely, F = G = i σ i + . Assuming again that 0| i σ i z |0 = N, we obtain the same results and draw the same conclusions as before. This is a consequence of the structure of Eqs. (39) and (40).
Using more sum rules, e.g. m − 4 and m + 5 , one may obtain information on other states that can be excited by the cyclotron operator i P + i . Their consideration shows that the e-e interaction does not affect, to order λ 2 R,D , neither the cyclotron nor the Larmor state, whose frequencies are the same as determined in Sec. III when both the SO terms are included in

H.
When the gs of the system has both qup and qdown occupied states, 21 the spin dipole operator i P + i σ i z entering Eq. (40) excites a state at an energy ω c (1+K) -see below, instead of ω c as it corresponds to the cyclotron (charge dipole) operator i P + i , and the results in Eq. (44) must be corrected for. This effect is not related to the SO interaction, and appears even in the absence of it. The spin dipole operator does not commute with the e-e interaction as the cyclotron operator i P + i does, and K is precisely the contribution to the spin dipole operator m + 1 sum rule arising from the e-e interaction when one takes F = G = i P − i σ i z : where K can be extracted from inelastic light scattering experiments. 27 It turns out to be zero for fully polarized ground states, and small and negative -of the order of 10 −2 -otherwise.
Similarly, the spin flip dipole operators i P + i σ i ± , whose excitations can be also measured by inelastic light scattering, do not commute with the e-e interaction, which give rise to some energy corrections. It turns out that these corrections are equal for the three spin dipole operators i P + i σ i z,± because the value of K is the same for all them. Hence, the energy splittings among these excitations are not influenced by the e-e interaction, depending only on the Zeeman and SO energies as found and discussed at the end of Sec III.
Finally, we want to comment on the consequences of the failure of the Kohn theorem due to the SO coupling using the m + 1 sum rule for F = i P − i σ i z and G = i P − i : This sum rule allows to study the interplay between charge and spin modes. If we cast it into a sum over 'spin dipole states' |m and another over 'charge dipole states' |ρ , we obtain If there is no SO coupling, Kohn's theorem holds, implying that m| i P + i |0 = 0. Thus, when the spin gs 2S z = 0| i σ i z |0 is not zero [otherwise, m + 1 = 0 from Eq. (47) Evaluating the sum rules m − 0 and m + 1 for the operators G = i P − i and F = i P − i σ i z , one easily obtains Eqs. (50) explicitly show that if 0| i σ i z |0 = 0, or if the SO coupling is neglected, the mixed strength π 2 is zero, and the spin dipole state cannot be excited in photoabsorption experiments. The strength π 2 is nonzero only at odd filling factors ν (ν = 2πℓ 2 n e , where ℓ = hc/eB is the magnetic length and n e is the electron density), for which 2S z /N = 1/ν. Besides, when the system is fully polarized at ν = 1, the operators i P − i and i P − i σ i z coincide and excite the same mode, so there is no splitting. The SO corrections O(λ 2 ) can be calculated by taking into account the occupation of the ground state, either using the sum rule approach of this Section, or the method of unitarily transforming the Hamiltonian, as described in Refs. 21,33. This calculation yields the energy splitting of the CR we discuss in the next Section.

V. COMPARISON WITH EXPERIMENTS AND DISCUSSION
An actual confrontation of the theoretical results we have obtained with the experiments is not an easy task because of the smallness of the SO effects, and because of the way they are presented in the available literature, which makes it extremely difficult to carry out a quantitative analysis of such a subtle effect. Thus, we have to satisfy ourselves with a semiquantitative analysis, or to point out that these results are compatible with fairly rough estimated values of the SO coupling constants. We present now three such examples and a possible way to increase SO effects so that they could be easier to determine.
Using unpolarized far-infrared radiation, Manger et al. 20 have measured the cyclotron resonance in GaAs quantum wells at different electron densities. The main finding of the experiment is a well resolved splitting of the CR for ν=3, 5, and 7, and no significant splitting for ν = 1 and for even filling factors. We have seen that the SO interaction couples charge-density and spin-density excitations yielding the SO splitting of the CR given in Eq. (35). However, this expression, by itself, is unable to explain the filling factor dependence of the observed splitting, for which one has to bear in mind that the SO coupling between the i P − i and i P − i σ i z operators is strongly enhanced when the spin gs is not zero, as explicitly shown in Eq. (50). We have also noted that K contributes to the splitting. Eq. (35) has to be generalized to include these features. We obtain where the factor 2S z /N takes into account the actual sp contents of the gs. This equation, together with Eq. (50), embodies the theoretical explanation of the experimental findings. 20 In particular, it gives an appreciable splitting only for odd filling factors, for which the spin ground state S z is not zero. The analysis of the experimental splittings using the Eq. (51) yields values for the quantity m|λ 2 R − λ 2 D |/h 2 of about 30µeV, in agreement with the ones recently used to reproduce the spin splitting in quantum dots 15 and wells. 22 This is, in our opinion, one of the most clear evidences of a crucial SO effect on a physical observable, because its absence would imply that the physical effect does not show up.
The spin splitting of the first three Landau levels of a GaAs quantum well has been measured in a magnetoresistivity experiment by Dobers et al. 24 We have shown in the previous sections that this splitting is not influenced by the e-e interaction, and that there is no spin splitting as B goes to zero [Eq. (28)]. Both facts are in agreement with the analysis of the experimental data, and with previous theoretical considerations 38 about the B-dependence of the gyromagnetic factor g * , whose determination was the physical motivation of the magnetoresistivity experiment presented in Ref. 24. These authors have derived a B-and ndependent g * factor g * (B, n) = g * 0 − c(n + 1 2 )B, where g * 0 and c are fitting constants that depend on the actual quantum well. The possibility of a SO shift was not considered, and their chosen law for g * implies that the spin splitting energy ∆E n does depend on the Landau level index n entering in a B 2 term, as they have ∆E n = |g * µ B B|. A B dependence in g * is crucial to explain the experimental data, and also to reproduce them theoretically. 22 For the spin splitting of the Landau levels we obtain -recall that ω L = |g * µ B B|-i.e., a splitting that increases with n because of the SO coupling.
This SO correction has been worked out for the n = 0 level in Ref. 22 using the equation of motion method. It is known that the experimental results 24 for n = 1 and 2 can be reproduced if g * depends on n and B, as already shown in that reference. We have verified that the n-dependence of g * cannot be mimicked by the n-dependence introduced by the SO interaction, Eq. (52). Recently, the analysis of g * has been extended to a wider magnetic field range using time-resolved Faraday rotation spectroscopy. 28,29 As a third example, we address the inelastic light scattering excitation of the spin dipole modes at ν = 2 as measured in Ref. 27. For this filling factor, in the absence of SO coupling the spin-density inter-Landau level spectrum is expected to be a triplet mode 34,39 excited by the three operators i P + i σ i z,± with energy splittings given by the Zeeman energy ω L . In the presence of SO interactions, we still expect a triplet mode to appear. Indeed, for ν = 2 we have S z = 0 and the cyclotron and spin dipole modes excited by the operator i P + i σ i z are decoupled, as previously discussed. Thus, for this operator only one single mode should be detected at an average energy ω = ω c (1 + K). The other operators i P + i σ i ± , yield the two other spin dipole modes at the energies The splitting is thus symmetric and depends on the SO strengths. In the experiment, triplet excitations were observed in all measured samples up to electron densities corresponding to r s = 3.3 (we recall that r s = 1/ √ πn e ). B was accordingly changed to keep the filling factor at ν = 2. Only one triplet mode spectrum at B = 2.2 T was shown. From this spectrum, we infer that there is space for a ∼ 5 − 10% SO effect on the splitting, assuming that at this fairly small magnetic field, g * is that of bulk GaAs, g * = −0.44. We have where C R,D ≡ mλ 2 R,D /h 2 , and the tilting angle θ enters the quantities V, Z, and S defined in the Appendix. Tilting effects might arise because of the 1 − |g * |m * S/2 denominator in the above equation, but sizeable effects on ∆E CR should only be expected for materials such that |g * |m * /2 is large. This is not the case for GaAs, but it is, e.g., for InAs and InSb, which have |g * |m * /2 = 0.169 and 0.355, respectively. For the latter case the dependence of ∆E CR with the in-of-well field B x , with a fixed B z , is shown in Fig. 3. Notice that ∆E CR is sharply increased when B x exceeds a given value (1T for the parameters in Fig. 3), which is proving the strong enhancement of SO effects introduced by the horizontal component of the tilted field configuration. Figure 3 also shows the comparison with the exact diagonalization data (symbols), indicating that the analytical formula, Eq. (54), is accurate up to rather large tilting angles and for varying relative weights of Rashba and Dresselhaus terms. As a matter of fact, this analytical result does not depend on B z although, for the sake of comparison with the exact diagonalization, we have used B z = 1 T in Fig. 3. The evolution with B x is not always monotonous, especially for C R > C D where we find an initial decrease of ∆E CR with increasing B x , vanishing at B x ∼ 0.8 T, and eventually increasing again.
The tilting also affects the spin splitting of the Landau levels which generalizes Eq. (52) for θ = 0. As we have commented before, in a recent experiment where spin precession frequencies in a InGaAs quantum well have been measured using electrically detected electron-spin resonances, 29 a strong dependence of the effective gyromagnetic factor g ef f on the applied tilted B has been found. In particular, at θ = 45 o g ef f exhibits oscillations with B which indicate its sensitivity to the Landau level filling, and a coupling between spin and orbital eigenstates which is explicitly present in the spin-orbit term of Eq.(55). The effective g-factor that can be extracted from this equation at θ = 45 o , by taking the ratio 2∆E n /(m * Sω c ), has the structure where the parametrization g * = g * 0 − c 1 (n + 1 2 )B of Refs. 24,29 has been introduced in Eq. (55), and the c 2 term is the SO contribution. For the smaller B values in the experiment, and for reasonable values of mλ 2 R,D /h 2 , of the order of 1-10µeV, the SO contribution is important enough and should not be neglected; under these circumstances, time-resolved Faraday rotation spectroscopy could be sensible to Rashba and/or Dresselhaus spin-orbit effects.

VI. SUMMARY
We have discussed the appearance of spin-orbit effects in magnetoresistivity and inelastic light scattering experiments on quantum wells. In particular, we have addressed SO effects on the splitting of the cyclotron resonance, on the sp Landau level spectrum, and on spindensity excitations. Our discussion has been based on the use of an analytical solution of the quantum well Hamiltonian valid up to second order in the SO coupling constants. The accuracy of this solution has been assessed comparing it with exact numerical diagonalizations.
We have carried out semi-quantitative comparisons with available experimental data, with the twofold aim of extracting the value of the SO coupling constants and of indicating possible manifestations of the SO interactions. We have also pointed out that tilting the -usually-perpendicularly applied magnetic field might enhance spin-orbit effects, making them easier to detect.