PHYSICAL REVIEW B 70, 165101 (2004)
Theory of resonant Raman scattering in one-dimensional electronic systems
D.-W. Wang,1 A. J. Millis,2 and S. Das Sarma3 1Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA 2Department of Physics, Columbia University, New York, New York 10027, USA 3Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA (Received 19 May 2004; revised manuscript received 15 July 2004; published 11 October 2004)
A theory of resonant Raman scattering spectroscopy of one-dimensional electronic systems is developed on the assumptions that (i) the excitations of the one-dimensional electronic system are described by the Luttinger- liquid model, (ii) Raman processes involve virtual excitations from a filled valence band to an empty state of the one-dimensional electronic system, and (iii) excitonic interactions between the valence and conduction bands may be neglected. Closed form analytic expressions are obtained for the Raman scattering cross sections, and are evaluated analytically and numerically for scattering in the polarized channel, revealing a “double- peak” structure with the lower peak involving multispinon excitations with total spin S=0 and the higher peak being the conventional plasmon. A key feature of our results is a nontrivial power-law dependence, involving the Luttinger-liquid exponents, of the dependence of the Raman cross sections on the difference of the laser frequency from resonance. We find that near resonance the calculated ratio of intensity in the lower energy feature to the intensity in the higher energy feature saturates at a value of the order of unity (times a factor of the ratio of the velocities of the two modes). We explicate the differences between the “Luttinger-liquid” and “Fermi-liquid” calculations of resonant Raman spectroscopy (RRS) spectra and argue that excitonic effects, neglected in all treatments so far, are essential for explaining the intensity ratios observed in quantum wires. We also discuss other Luttinger-liquid features which may be observed in future RRS experiments.
DOI: 10.1103/PhysRevB.70.165101 PACS number(s): 71.10.Pm, 78.30.Ϫj, 73.20.Mf, 73.21.Hb
I. INTRODUCTION and then emitted at another frequency and momentum via the One-dimensional (1D) physics has become increasingly recombination of the hole in the valence band with another important to condensed-matter science in recent decades be- electron in the conduction band (step 2). The final state of the cause many systems, including, for example, single wall car- system contains one or more particle-hole pairs excited in the bon nanotubes,1 organic conductors,2 superconducting conduction band. The dispersion of these particle-hole pair nanowires,3 spin chains,4 and even ultracold atoms in highly states may be inferred from the energy and momentum dif- anisotropic magneto-optical traps5 have been developed to ference between the incident and the scattered photons levels that allow experimental studies of hitherto unprec- (Stokes shift), while the nature [collective mode (plasmon) edented extent and precision. Among these 1D (or quasi-1D) and multipair excitation] may be inferred from the depen- systems, the semiconductor quantum wire structure (QWR) dence of the scattering amplitude on the polarization of the is one of the most important and widely studied, because of incoming and outgoing photons and on the energy of the its simple band structure and highly tunable doping density.6 incident photon. Also, great progress in microfabrication techniques has made The standard theory of Raman spectroscopy in high quality samples available.7 semiconductors15–18 is based on the Fermi liquid quasiparti- Theoretically, one-dimensional systems are very well un- derstood. The pioneering work of Tomonaga,8 Luttinger,9 and Haldane10 along with many other studies has produced an essentially complete understanding of the low-energy physics. However, the relation between the theoretical results and experimental data is not as clear and direct as one would like. For example, only a few unambiguous observations of the fundamental concept of spin-charge separation have been reported.11–13 Resonant Raman spectroscopy (RRS) has become a pow- erful tool for studying the elementary excitations of electrons in many different systems. Applications to low-dimensional doped semiconductor nanostructures (two-dimensional quan- tum wells or one-dimensional quantum wires) have been par- FIG. 1. (a) Schematic representation of RRS process in the di- 6,14 ticularly prominent. In the usual RRS experiment, sche- rect gap two band model of electron doped GaAs nanostructures. i matically represented in Fig. 1(a), external photons are ab- and f are the initial and final frequencies of the external photons. sorbed at one frequency and momentum by exciting an elec- (b) and (c) are the three-leg and four-leg scattering vertices for tron from the valence band to the conduction band (step 1) electron-photon interaction.
1098-0121/2004/70(16)/165101(17)/$22.5070 165101-1 ©2004 The American Physical Society WANG, MILLIS, AND DAS SARMA PHYSICAL REVIEW B 70, 165101 (2004) cle picture and neglects resonance effects. The neglect of Luttinger liquid. In the Coulomb case we predict an asym- resonance effects means that in the unpolarized geometry metric broadening of the spectral peak in the higher energy (identical polarization of incoming and outgoing light) the side arising from the curvature of the plasmon dispersion. Raman process couples to the electron-density operator so We find that most aspects of the RRS spectra are similar to the Raman cross section is proportional to the dynamical those predicted by the Fermi-liquid approach. As noted in 18,19 structure factor of the conduction-band electrons. Within our previous work,25 characteristic Luttinger effects are re- Fermi liquid theory and in the long wavelength limit relevant vealed in the dependence of the intensities on the difference ␦ to Raman scattering, the electron structure factor has a of the laser frequency from the resonance condition. Going function peak at the plasmon (CDE) energy and a much beyond the bosonic expansion developed in our earlier weaker feature at a lower energy associated with incoherent work,25 we present explicit results for the total spectral particle-hole pairs. The lower energy feature is referred to as weights of the charge boson and spin boson excitations in the the single-particle excitation SPE peak. The calculated ( ) polarized RRS channel at zero temperature. Far-from reso- RRS intensity therefore has strong spectral peaks when the =q , energy difference between the incident and outgoing photons nance we find that the spin-singlet mode at energy, vF coincides with the collective CDE mode frequencies at the has spectral weight much smaller than that of the charge wave vector defined by the experimental geometry, while the boson (i.e., plasmon); however, as resonance is approached intensity at the SPE energy is much weaker than that at the the weights in the two contributions become comparable. We CDE energy (the ratio is of the order of the square of the explain the difference between the Luttinger-liquid results ratio of momentum transferred by the light to the Fermi mo- presented here and the Fermi-liquid results presented 15–18,23,30 mentum of the electron gas; for typical experimental relevant previously. Our results remain inconsistent with parameters, about three orders of magnitude). This theoreti- present experimental data, so we argue that Luttinger-liquid cal result, however, is in qualitative disagreement with the effects have not yet been unambiguously detected in RRS experimental data,20–22 in which the polarized spectrum ex- experiments. One possibility is that excitonic effects, ne- hibits a “double peak” structure with comparable-intensity glected in the present and previous treatments, are important. peaks at both CDE and SPE modes. This puzzling The paper is organized as follows: In Sec. II we develop a feature20–22 of a ubiquitous strong SPE peak in addition to general theory of resonant Raman scattering in Luttinger liq- the expected CDE peak occurs in one, two, and even in uids. We then apply our theory to calculate the spectrum for three-dimensional doped semiconductor nanostructures, for short- and long-ranged electron-electron interaction by using both intrasubband and intersubband excitations. Many theo- bosonic expansion method in Sec. III. In Sec. IV, we go retical proposals23–28 have been made to explain this two- beyond the bosonic expansion and calculate the full spectral peak RRS puzzle. However, two of us have recently argued29 weights of the charge boson and spin-singlet excitations in that within the standard theory, which neglects resonance the polarized channel. We compare our calculation to the effects, none of the above proposed modified mechanisms’ previous Fermi liquid calculations in Sec. V. We critically theory can even qualitatively explain the experimental data. discuss the assumptions and resonance effects of our results But the situation changes when resonance effects are in- and compare them with the present experimental data in Sec. cluded. In an important paper, Sassetti and Kramer24 (SK) VI. Finally we summarize our results in Sec. VII. first proposed that in 1D systems, the prominent lower en- ergy SPE peak is due to “spinon” excitations whose coupling II. RESONANT RAMAN SCATTERING CROSS SECTION to light is enhanced by resonance effects. The qualitative OF A LUTTINGER LIQUID idea that resonance effects can strongly affect the relative absorption cross sections of different modes is indeed impor- In this section we present a derivation of the RRS scatter- tant. Unfortunately, as we have recently noted, the theory ing cross section for a Luttinger liquid. We consider an ide- presented in Ref. 24 suffers from two technical flaws. First, it alized model of a quasi-one-dimensional system, in which is not self-consistent: it uses a Fermi-liquid-based expression the electron motion in the transverse directions (y and z) is to account for the resonance effects, but a Luttinger-liquid- assumed to be completely frozen due to a strong confinement based expression to account for the conduction-band dynam- potential in both conduction and valence band (in other ics. As we shall show below (and have already mentioned in words, we consider only the lowest conduction and highest a previous brief communication25) Luttinger-liquid physics valence subband). Our results are likely to apply also to the affects the matrix element in a crucial way. Second, while the case where the resonance occurs via some other intermediate SK calculation correctly notes that as resonance is ap- state, but we have not considered this case explicitly. proached the coupling becomes long ranged in space, it The longitudinal (x direction) motion is assumed to be omits the equally important fact that the coupling becomes free without defects or disorder. An important feature of long ranged also in time. quantum wires is the Coulomb interaction, which is typically In this paper, which amplifies and extends our previous unscreened and leads to a long-ranged interaction with a short communication,25 we derive and present a complete, well-known characteristic form25 involving the transverse closed-form expression for the RRS scattering amplitudes in wave function.30 The scale dependence of the unscreened the Luttinger-liquid model and calculate the resulting RRS Coulomb interaction complicates considerably the analysis spectra in different resonance conditions at zero temperature. of Luttinger-liquid formulas, but as will be seen also leads to We treat both the analytically tractable cases of short-ranged the appearance of additional structures in the predicted RRS interactions and the physically relevant case of the Coulomb spectra. In this section we derive an expression for the RRS
165101-2 THEORY OF RESONANT RAMAN SCATTERING IN ONE-… PHYSICAL REVIEW B 70, 165101 (2004)
cross section of a general model with arbitrary electron in- duction band. In Eq. (3), the charge (spin) boson energy ͑͒ ͑ ͒ 32 teraction. Subsequent sections present results for short- p p is related to the interaction V q via ranged interactions (where the analysis can be carried through in considerable analytical detail) and the more V͑p͒ = ͉p͉ ͱ1+ , ͑4͒ physically relevant long-ranged interaction, which requires p vF vF additional approximations. The appropriate theoretical starting point for analysis of ͉ ͉ ͑ ͒ Raman scattering is the following general Hamiltonian: p = p vF, 5 H = H + H + H + Hk, , ͑1͒ where we have assumed that electron-electron interaction is tot V LL cv int spin independent so that the spin boson velocity is the same where HV and HLL are the Hamiltonians for the valence and as noninteracting electron Fermi velocity. k, conduction bands, respectively, and Hint gives the coupling Finally, for the electron-photon interaction Hamiltonian of these carriers to externally applied radiation. Hcv describes Hint we consider only the three-leg vertex for which photon the excitonic interaction between conduction-band electrons number is not conserved [see Fig. 1(b) and Refs. 30, 33, and and valence-band holes, and may be important in certain 34], and neglect the four-leg vertex where photon number is conditions.31 We will assume excitonic effects are irrelevant conserved [see Fig. 1(c)], because the contribution of the 30 in the energy regime of interest and will therefore set Hcv latter does not give rise to resonance effects and is thus =0 throughout this paper. In this case, the RRS process in- much smaller than the contribution of the three-leg vertex in volves only one electron excited out of the valence band, so near-resonance conditions. We also represent the radiation by that interactions in the valence band can be also neglected. a classical field, so that We consider a single one-dimensional conduction subband k, ͑ ␦ ␦ ͕͓͒ † ͑ ͒ ͑ ͒ and a valence subband, linearize the dispersion about the Hint = ͚ g1 s,sЈ + g2 s,−sЈ cr,p,s t vr,p−k,sЈ t Fermi level and include an (at this stage arbitrary) interaction r,p,s,sЈ between conduction-band electrons. These considerations † ͑ ͒ ͑ ͔͒ −it ͓ † ͑ ͒ ͑ ͒ imply + vr,p,sЈ t cr,p−k,s t e + cr,p,s t vr,p+k,sЈ t † ͑ ͒ ͑ ͔͒ it͖ ͑ ͒ V † ͑ ͒ + v t cr,p+k,s t e , 6 HV = ͚ Er,pvr,p,svr,p,s, 2 r,p,sЈ r,p,s where and k are energy and wave vector of the photon 1 interacting with electrons. g1 and g2 are coupling constants H = ͚ v ͑rp − k ͒c† c + ͚ ͚ V͑q͒ for non-spin-flip and spin-flip scattering, respectively, and LL F F r,p,s r,p,s 2L r,p,s r1,r2,s1,s2 q,p1,p2 their actual values are not important in our study. In the ϫ † † ͑ ͒ remainder of this section, we present an explicit derivation of c c cr ,p −q/2,s cr ,p +q/2,s , 3 r1,p1−q/2,s1 r2,p2+q/2,s2 2 2 2 1 1 1 expressions for the non-spin-flip (polarized spectrum) RRS where the conduction-band Hamiltonian can be bosonized as cross section following from Eqs. (2), (3), and (6). in the standard Luttinger-liquid theory32 yielding Following our earlier work (Refs. 25 and 30),weuse second-order time-dependent perturbation theory to calculate ͑ † † ͒ k, HLL = ͚ pbpbp + p p p . the rate at which Hint causes transitions from the ground p state ͉0͘ to some state ͉n͘ in which the valence band is filled V and the conduction-band state has changed. One finds We approximate the valence-band energy Er,p by a linear dispersion about the Fermi wave vector of electrons in the 1 T/2 t1 conduction band, ͑ ⍀͒ ͚ ͯ͵ ͵ W q, ; = lim dt1 dt2 T→ϱ T V Ϸ ⍀ V͑ ͒ n −T/2 −T/2 Er,p − rrs − vF rp − kF , 2 where ⍀ ϵEc +EV +E is the RRS resonance energy see ϫ͗ ͉ q/2,⍀+/2͑ ͒ −q/2,⍀−/2͑ ͉͒ ͯ͘ ͑ ͒ rrs F F g [ n Hint t1 Hint t2 0 , 7 Fig. 1(a)]. Note that in this expression it is assumed that the valence band is also one dimensional. If transverse motion in where q and are photon wave vector and frequency shifts the valence band is important, these degrees of freedom ⍀ should be integrated out, which will broaden the valence- after Raman scattering and is the mean frequency of inci- dent and scattered photons during the process. We choose the band propagator. (Note that the structure of the quantum wire system means that momentum transverse to the wire need backward scattering channel so that all wave vectors in- volved are along the wire direction for simplicity. Because not be conserved in an optical transition.) c† and v† are r,p,s r,p,s the valence band is filled in the ground state, the part of the creation operators of electrons of chirality r=±1 (left/right correlator involving valence electrons is simple: moving), wave vector p, and spin s, in conduction band and † † valence band, respectively; bp and p are charge boson and ͗ † ͑ Ј͒ ͑ Ј͒ † ͑ ͒ ͑ ͒͘ v t vr ,p ,s t v t1 vr ,p −q/2,s t2 0 spin boson creation operators in Luttinger-liquid theory (see r1,p1−q/2,s1 2 2 2 2 1 r3,p3,s3 4 4 4 ͑ ͒ Refs. 10 and 32 for a general review). vF kF is the Fermi = ␦ ␦ ␦ ␦ ␦ ␦ r1,r2 r3,r4 p1,p2+q/2 p3,p4−q/2 s1,s2 s3,s4 velocity (wave vector) of conduction-band electrons. V͑q͒ is V ͑ Ј Ј͒ V ͑ ͒ iE t −t iE t1−t2 the effective 1D electron-electron interaction within the con- ϫe r1,p1 2 1 e r3,p3 . ͑8͒
165101-3 WANG, MILLIS, AND DAS SARMA PHYSICAL REVIEW B 70, 165101 (2004)
Using the space-time translational symmetry, Eq. (7) can Substituting Eq. (12) into Eq. (10) and using the following be further simplified by representing fermion operators in identity for linear boson operators, A and B (valid when coordinate space: ͓A,B͔ commutes with both A and B, the colons denote nor- ͗ ͘ mal ordering and · 0 denotes expectation value with respect L ϱ to the ground state of HLL): ͑ ⍀͒ ͵ ͵ i͑T−qR͒͗ ˆ †͑ ͒ ˆ ͑ ͒͘ W q, ; = lim dR dT e O R,T O 0,0 0. L→ϱ 0 −ϱ A B ͗ A B͘ A+B ͑9͒ e e = e e 0:e :,
͗ ͘ we can rewrite O˜ as Here ¯ 0 are the expectation values on the ground-state wave function and ϱ ͑⍀ ⍀ ͒ V ϱ ˆ ͑ ͒ ͵ i − rrs t ͑ ͒ L O R,T = L͚ dt e Gr,s − rvFt,t ˆ ͑ ͒ ͵ ͵ ͑ ͒ ͑ ͒ r,s 0 O R,T = ͚ dx dt x,t r,s R + x/2,T + t/2 r,s 0 0 ⌽ ͑ V ͒ ⌽ ͑ V ͒ :e rs, R,−rvFt;T,t ϻ e rs, R,−rvFt;T,t :, ϫ† ͑ ͒ ͑ ͒ r,s R − x/2,T − t/2 , 10 where we have separately normal ordered the charge ͑ ͒ and and spin boson operators and Gr x,t ͑ ␣͒−1͗ ͑ ͒† ͑ ͒͘ =lim␣→0 2 r x,t r 0,0 is the electron Green’s ⌽ i⍀t function in the conduction band. The phase operators, e ͑ V ͒ ͑⍀ ⍀ V ͒ ͑ ͒ i Ept−px i − rrs+vFkF t␦͑ V ͒ and ⌽ are, respectively, x,t = ͚ e = e x + rvFt . L p ͑11͒ ⌽ ͑ ͒ −␣p/2ͱ rs, R,x;T,t =2͚ e pϾ0 pL Equations (9)–(11) are our fundamental results. They show ϫ͕− sinh sin͓p͑rx + v t͒/2͔ that the RRS process creates an electron-hole pair separated p p † ͑ ͒ in space by x and in time by t, with the amplitude for a given ϫ͓b eip rR+vpT + H.c.͔ space-time separation controlled by the function ͑x,t͒, −rp ͓ ͑ ͒ ͔ which is essentially the propagator for the valence-ban hole. + cosh p sin p rx − vpt /2 V Far-from resonance ͉͑⍀−⍀ ͉ӷv k ͒,͑x,t͒ is short ranged ͑ ͒ rrs F F ϫ͓ † −ip rR−vpT ͔͖ ͑ ͒ brpe + H.c. , 13 in both x and t, so that Oˆ becomes similar to the ordinary density operator and W becomes a density-density correla- 29,30 tion function. As the mean photon energy is tuned closer −␣p/2ͱ V ⌽ ͑R,x;T,t͒ =2s ͚ e ͕sin͓p͑rx − t͒/2͔ to the resonance condition, becomes longer ranged, so that rs, vF pϾ0 pL Oˆ becomes nonlocal in both space and time. This nonlocality ͑ V ͒ ϫ͓† −ip rR−vFT ͔͖ ͑ ͒ will be seen to give rise to the interesting resonance effects, rpe + H.c. . 14 by allowing the light to couple to something other than the To calculate the expectation value of a product of normal dynamical structure factor. We also observe that up to this orderings, we use two additional identities for the linear point we have not made use of the one dimensionality in any bosonic operators [valid under the same condition as Eq. important way: the equations may easily be generalized to (12)], two and three dimensions. Previous theories15–18,23–26,28,29,33 of the RRS process have been based on similar expressions ͗eAeB͘ = ͗eA+B͘ e͓A,B͔/2 ͑15͒ but with a function which essentially forces t=0 (i.e., no 0 0 retardation effects). We shall see below that the time depen- and dence is very important. We now incorporate the special features of one- ͗ 2͘ ͗eA͘ = e A 0/2, ͑16͒ dimensional physics by using the standard32 bosonization 0 representation of electron operators in Eq. (10) by assuming so that zero temperature, ͗ 2͘ ͗ 2͘ irk x ͗ A ϻ B ͘ − A 0/2 − B 0/2͗ A B͘ e F :e e : 0 = e e e e 0 ͑x,t͒ = expͩi ͚ ͱ ͕e−p␣/2+irpx͓cosh b ͑t͒ r,s ͱ p rp ͗ 2͘ ͗ 2͘ ͓ ͔ ͗ ͘ 2␣ Ͼ pL − A 0/2 − B 0/2͗ A+B͘ A,B /2 AB 0 p 0 = e e e 0e = e . ͑17͒ † ͑ ͒ ͑ ͔͒ ͖ͪ ͑ ͒ − sinh pb−rp t + s rp t + H.c. , 12 Combining all the results from Eqs. (9)–(17) and defining the ͑ ͒ ͱ ͑ ͒ average photon frequency relative to the resonance fre- where exp −2 p = p /pvF = 1+V p / vF is the momentum + ⍀ ⍀˜ ⍀ ⍀ dependent LL exponent, and ␣→0 is a convergence factor. quency rrs as = − rrs, we obtain
165101-4 THEORY OF RESONANT RAMAN SCATTERING IN ONE-… PHYSICAL REVIEW B 70, 165101 (2004)
W͑q,;⍀˜ ͒ = ͚ ͚ Wr1,r2͑q,;⍀˜ ͒, s1,s2 r1,r2 s1,s2
ϱ ϱ ϱ Wr1,r2͑q,;⍀͒ = ͵ dT ͵ dR ei T−iqR͵ dt f* ͑t ;⍀˜ ͒͵ dt f ͑t ;⍀˜ ͒ s1,s2 1 r1s1 1 2 r2s2 2 −ϱ 0 0 V V ϫ͕exp͓͗⌽ ͑R,− r v t ;T,t ͒⌽ ͑0,− r v t ;0,t ͒͘ ͔ r1s1, 1 F 1 1 r2s2, 2 F 2 2 0 V V ϫexp͓͗⌽ ͑R,− r v t ;T,t ͒⌽ ͑0,− r v t ;0,t ͒͘ ͔ −1͖, ͑18͒ r1s1, 1 F 1 1 r2s2, 2 F 2 2 0
where expansion is finite because the sine function in Eqs. (20) and (21) removes the 1/p divergence in small p region and, by ͑ ⍀˜ ͒ i⍀˜ t ͑ V ͒ ͑ ͒ frs t; = e Grs − rvFt,t . 19 oscillating, ensures the convergence at large p. The expan- sion approach has a simple physical interpretation: the term The −1 in the last line of Eq. (18) arises from normal of order n in the expansion corresponds to a final state with ordering.10 n excited bosons. The charge and spin part of the expectation values in Eq. Section III B shows results obtained on the assumption of (18) can be calculated, respectively, to be (at zero tempera- short-ranged interaction between electrons in conduction ture) ͑ ͒ band, i.e., V p =V0 with an appropriate momentum cutoff. ͗⌽ ͑R,x ,T,t ͒⌽ ͑0,x ,0,t ͒͘ The well-known analytical methods of Luttinger-liquid (LL) r1s1, 1 1 r2s2, 2 2 0 ϱ theory can then be applied, yielding a physical understanding dp ␣ of the features of RRS spectrum. Section III C presents re- ͵ − p/2 −ipvpT͕ ͓ ͑ ͒ ͔ =2r1r2 e e sin p x1 − r1vpt1 /2 0 p sults obtained for the unscreened Coulomb interaction. ϫsin͓p͑x − r v t ͒/2͔C C eipR 2 2 p 2 r1,p r2,p B. Results for short-ranged interaction ͓ ͑ ͒ ͔ + sin p x1 + r1vpt1 /2 For simplicity here we assume that the valence-band elec- ϫsin͓p͑x + r v t ͒/2͔C C e−ipR͖, ͑20͒ tron velocity at the Fermi momentum of the conduction 2 2 p 2 −r1,p −r2,p V band, vF, is much less than the conduction-band velocity vF ϵ ͑ ͒ ͑ ͒ where Cr,p cosh p sinh p for r=+1 −1 , and and can be neglected. In this approximation (which does not affect any essential results), ͗⌽ ͑R,x ,T,t ͒⌽ ͑0,x ,0,t ͒͘ r1s1, 1 1 r2s2, 2 2 0 i⍀˜ t ˜ e 1 ϱ ͑ ⍀˜ ͒ i⍀t ͑ ͒ϳ ͑ ͒ dp ␣ f t, = e Grs x =0,t ␣ , 22 =2s s ␦ ͵ e− p/2 sin͓p͑x − v t ͒/2͔ ͱ t͑iE t͒ LL 1 2 r1,r2 1 F 1 2 i vvF 0 0 p V where ϫ ͓ ͑ ͒ ͔ ir1pR−ipvFT ͑ ͒ sin p x2 − vFt2 /2 e . 21 1 v vF 2 Equations (18)–(21) represent a complete solution to the ␣ = ͩ + −2ͪ = sinh ͑23͒ 4 v v RRS cross section of a Luttinger liquid in the absence of F excitonic effects. We will evaluate them analytically and nu- is the Luttinger exponent and E0 is an energy cutoff above merically in the following sections. which the Luttinger-liquid model ceases to describe the physics. [Note that here we follow the standard convention in the literature32 and this ␣ has nothing to do with the con- III. LEADING-ORDER EFFECTS IN BOSONIC vergent factor used in Eq. (11).] E0 is expected to be roughly EXPANSION c 32 of the order of the conduction-band Fermi energy EF. The A. Overview time integrals can now be reduced to ␥ functions, and the expansion evaluated order by order. Although Eq. (18) is a complete closed-form expression for the RRS cross section, it requires a numerical evaluation We first show the results for the one-boson contribution in which is not simple because final results are obtained by the non-spin-flip process. Here the spin boson term is can- cancellation of rapidly oscillating terms. It is instructive to celed by the spin sum as required for the conservation of consider the analytical approximation obtained by expanding angular momentum. We have the exponential factors in Eq. (18) in a Taylor series in ⌫2͑− ␣͒ n ͑ ⍀˜ ͒ ͉ ͑⍀˜ ␣͉͒2 2␦͑ ͒ ͑ ͒ ͗⌽⌽͘ (n is an integer) and then evaluating the series term W1 q, ; = A1 ,q, e − qv , 24 2qv v by term. Note that in the present problem (unlike in the F evaluation of the electron Green’s function) each term in the where the function A1 is
165101-5 WANG, MILLIS, AND DAS SARMA PHYSICAL REVIEW B 70, 165101 (2004)
␣ ␣ ⍀˜ − qv /2 ⍀˜ + qv /2 ͑⍀˜ ␣͒ ͩ ͪ ͩ ͪ ͑ ͒ A1 ,p, = − . 25 E0 E0 Thus the one-boson term is a ␦ function at the plasmon en- ergy, i.e., it gives the conventional CDE contribution to the scattering cross section. The amplitude for excitation de- pends on the difference of the mean photon frequency from the resonance condition and on the transferred momentum q. Ӷ͉⍀˜ ͉ ϰ͉⍀˜ ͉2␣−2 Far-from resonance, q =qv and W1 , while at ⍀˜ ϰ 2͑␣ ͒ resonance =0 and W1 sin /2 . Thus LL effects enter the CDE part ͑ϳqv͒ of the spectrum in two ways (for short-ranged interaction): first, far-from resonance, they change the frequency dependence of spectral weights from ⍀˜ −2, the noninteracting result, to ⍀˜ −2+2␣. Second, near- resonance ͑⍀˜ Ͻvq/2͒ Luttinger-liquid effects resolve the singularities which are found in the standard Fermi-liquid expressions for the matrix elements: see Sec. V for a more detailed discussion. For the second-order (i.e., two-boson) contribution, we obtain 2 q ⌫ ͑− ␣͒ dp W ͑q,;⍀˜ ͒ = ͭ͵ ͉A ͑⍀˜ ,p,q − p,␣͉͒2 2 2 ͑ ͒ 2 vF 0 p q − p 2 ϫcosh ͑2͒␦͑ − qv͒ ϱ dp +2͵ ͉A ͑⍀˜ ,p,p − q,␣͉͒2 ͑ ͒ 2 q p p − q 2 FIG. 2. Polarized RRS spectra calculated via the bosonic expan- ϫsinh ͑2͒␦ ͓ − pv − ͑p − q͒v͔ ⍀˜ ⍀ ⍀ q sion method for various resonance conditions, = − rrs. One- dp + ͵ ͉A ͑⍀˜ ,p,q − p,␣͉͒2 ␦͑ − qv ͒ͮ and two-boson contributions have been plotted separately in order ͑ ͒ 2 F 0 p q − p to show their relative contributions (see text). Finite broadening factor ␥ is introduced to express the ␦ function Ref. 35 . Note that ͑ ͒ ( ) 26 the overall spectral weights decrease dramatically off-resonance, as with ͑␣=,͒ indicated by the individual scale factors on the right-hand side of each plot. ˜ ␣ ␣ ⍀ − p v␣ /2 − p v␣ /2 A ͑⍀˜ ,p ,p ,␣͒ = ͩ 1 2 ͪ 2 1 2 E 0 be excited via the two spinon ͗,͘ process giving rise to ␣ ⍀˜ ͩ + p1v␣ /2 − p2v␣ /2ͪ the so-called SPE mode at =qvF. − In Fig. 2, we show the perturbatively calculated LL polar- E0 ␣ ized RRS spectra including one- and two-boson contribu- ⍀˜ tions to the Raman intensity, for a particular choice of Lut- ͩ − p1v␣ /2 + p2v␣ /2ͪ − tinger exponent The data are plotted as a function of energy E0 ␣ transferred at fixed momentum transfer q; the different ⍀˜ + p v␣ /2 + p v␣ /2 graphs show results for different values of the laser fre- + ͩ 1 2 ͪ . ͑27͒ E quency relative to the resonance condition. One observes that 0 (i) the overall spectral weight decays very fast as ⍀ is moved ⍀˜ ϳ⍀˜ −4+2␣ Note that for large (far-from resonance) W2 so off-resonance, and (ii) the SPE peak is only noticeable near that in this limit the two-boson term is small compared to the resonance. (iii) The continuum structure arising from the sec- one-boson term, confirming the validity of the expansion in ond term in Eq. (26) is visible but not very distinct. We also the far-from resonance region. The first term in Eq. (26) remark that the three-boson term in the expansion (not cal- gives a renormalization of the strength of the CDE pole. The culated in this paper) will produce an additional continuum ͑ ͒ second and third terms produce new effects appearing at sec- structure lying between the SPE =qvF and the plasmon ond order: first near the CDE peak, branch mixing [second ͑=qv͒ energies, due to the coupling between charge and line of Eq. (26)] processes appear, leading to a continuum spin bosons via the RRS process. In our above calculation, ␥ absorption beginning at the CDE threshold, =qv. Second we have introduced a broadening factor, =0.05EF, to real- the spin-singlet combination of spin bosons (third line) can ize the ␦-function peak.35
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according to the resonance condition. Off resonance, the two-boson-singlet-spinon peak is much weaker than the charge boson plasmon peak, while their total spectral weight is also very small compared with the near-resonance result. Near resonance, an additional feature appears: the two- charge-boson contribution gives another resonance peak at Ͼ =2 q/2 q, arising from the nonlinear q dependence of q due to the long-ranged Coulomb interaction. The result- ing curvature acts to broaden the spectral weight on the higher energy side. Next order bosonic terms will contribute the spin-charge mixing weights from ͗,͘-type higher order correlations. This will introduce a continuum of rela- tively weak spectral weight which is located in the energy region between the spinon and the plasmon energies, as noted previously. Finite temperature and homogeneous broadening effect usually smear out the singularity and cause a asymmetric continuum about the resonance peaks.35
IV. BEYOND THE BOSONIC EXPANSION We now consider effects beyond the bosonic expansion in the analytically tractable finite range interaction model. We begin with the exact expression, Eqs. (20) and (21). The values of mean position R and time T are expected to be FIG. 3. Polarized RRS spectra calculated via bosonic expansion roughly inversely proportional to the momentum q and fre- method for various resonance conditions in the LL model with quency transferred to the system. If and q are small long-ranged Coulomb interaction. One- and two-boson contribu- compared to the basic scales (EF and kF) we may use a long tions have been plotted separately in order to show their relative wavelength approximation to evaluate the phase factors in contributions (see the text). the standard way obtaining ϱ ϱ C. Long-ranged Coulomb interaction Wr1,r2͑q,;⍀˜ ͒ = ͵ dt f* ͑t ;⍀˜ ͒͵ dt f ͑t ;⍀˜ ͒ s1,s2 1 r1s1 1 2 r2s2 2 We now consider the modifications arising in the physi- 0 0 cally relevant case of a long-range, unscreened Coulomb in- ϱ ϱ iT −iqR teraction. As discussed elsewhere36 this corresponds to a ϫ͵ dT e ͵ dRe ␣ −ϱ −ϱ scale dependent Luttinger interaction parameter, q ␣ 1/2͑ ͒͑ Ӷ ͒ ϫ͕͓ ͑ ͔͒␣ = 0 ln q0 /q for q q0 and the charge mode plasmon Fr ,r r1R − vT;t1,t2 energy,36 1 1 ␣͑1͒ ϫ͓ ͑ ͔͒ F R − vT;t1,t2 ϳ ␣ 1/2͑ ͒ ͑ ͒ r1,r2 q 4 0vFq ln q0/q . 28 ␣͑2͒ ϳ ϫ͓F ͑− ͑R + vT͒;t ,t ͔͒ −1͖, Here q0 2.5/a (a is the typical width of QWR) is the mo- r1,r2 1 2 mentum scale below which Coulomb effects become impor- ͑29͒ ␣ ͱ 2 ⑀ tant and 0 = e /2 0vF (Ref. 36) for typical QWR struc- 1 ͑1͒ 1 where the exponents are ␣ = ␦ s s ,␣ = C C , and ture. In the GaAs QWR material which was recently 2 r1,r2 1 2 2 r1 r2 studied,37,38 typical LL parameters are ␣ ϳ0.389 and q ␣͑2͒ 1 0 0 = 2 C−r C−r . The F function is ϳ1.4ϫ106 cm−1. 1 2 ͑ a a ͒2 ͑ ␣͒2 The scale dependence of the interaction prevents analytic v t1 + v t2 − X + i a ͑ ͒ r1 r2 ͑ ͒ F X;t1,t2 = , 30 evaluation of the Luttinger-liquid formulas, but the bosonic r1,r2 ͑ a a ͒2 ͑ ␣͒2 v t1 − v t2 − X + i expansion can still be carried out, although both the integrals r1 r2 ͑ ⍀˜ ͒ ⌽ a ͑ V͒ ͑ ͒ defining f t, and those defining must be evaluated nu- where vr = va +rvF /2 for a= , v =vF . merically. We have not attempted a direct evaluation of this expres- In Fig. 3 we show the one- and two-boson contributions sion. Instead, we note that the spectrum is expected on gen- to the polarized RRS intensity of 1D QWR using Coulomb eral grounds to consist of two ␦ functions, at the CDE and interaction parameters corresponding to the system studied in SPE energies, and two continua. The ␦ functions are the Ref. 37. The basic features of the results are similar to the features most easily observable experimentally, and it is pos- short-ranged interaction, having one weak singlet-spinon sible to obtain convenient expressions for their weights. We ␦ peak at the SPE spinon energy =qv =qvF arising from the note that terms giving rise to functions at the CDE energy two coupled spin bosons, and one strong charge boson peak or SPE energy must be functions only of R−vT and R at the plasmon energy =qvq. Their relative strength varies −vT, respectively. To isolate these terms we apply the iden-
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͑ ͒͑ ͒͑ ͒ ͑ ͒͑ ͒ ͑ ͒͑ ϱ ϱ tity ABC−1= A−1 B−1 C−1 + A−1 B−1 + A−1 C Ã s1s2͑ ⍀˜ ͒ ͵ ͑ ͒͵ ͒ ͑ ͒͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ W q, =2 dt f t dt f −1 + B−1 C−1 + A−1 + B−1 + C−1 to Eq. (29). 1 r1s1 1 2 r2s2 Only terms involving just one of the three F functions can 0 0 ␦ Ͼ ϱ contribute to the function, so we find that for ,q 0 the Ј ϫ͑ ͒͵ Ј −ir1qR ͕͓ ͑ ͔͒s1s2/2 ͖ intensities may be written t2 dR e F1,1 R;t1,t2 −1 . −ϱ ͑⍀˜ ͒ ͑⍀˜ ͒␦͑ ͒ ͑ ͒ ͑ ͒ ICDE = W − vq + ¯ , 31 34
͑⍀˜ ͒ s1,s2͑⍀˜ ͒ ␦͑ ͒ ͑ ͒ ISPE = ͩ ͚ s1s2W ͪ − vq + ¯ , 32 These formulas have an involved analytic structure, which s1s2=±1 includes a discontinuity when either the incoming or the out- where the ellipses denote the continuum terms and the ␦ going laser frequency is precisely in resonance with the function coefficients are, respectively, valence-band to conduction-band Fermi level energy differ- ϱ ϱ ence. For large frequencies, the correct asymptotic behavior ͑ ⍀˜ ͒ ͵ ͑ ͒͵ Ã arises from cancellation among oscillating terms. To make W q, =8 dt f t dt f 1 r1s1 1 2 r2s2 0 0 the structure manifest and obtain forms which are convenient ϱ for numerical evaluation we employ the further series of ϫ͑ ͒͵ Ј −iqRЈ͕͓ ͑ Ј ͔͒␣ ͖ transformations given in the Appendix, leading for t2 dR e F1,1 R ;t1,t2 −1 , −ϱ ͉⍀˜ ͉Ͼ v,q/2 to ͑33͒
͑1+␣͒ ͑3/2͒+␣ ͩ ͪ⌫͑ ␣͒ 2 sin 1−2 2 /4 /2 cos2͑cot ͒␣ ͑ ͉⍀˜ ͉ Ͼ ͒ ͵ ͵ W q, vq/2 = d d ␣ ͱ ⍀˜ 1−2 0 0 1 + cos 2 cos 2 1 1−2␣ 1 1−2␣ ϫ − , vq 1 + cos 2 cos 2 v hoq 1 + cos 2 cos 2 ΄ͩcos − ͱ ͪ ͩcos + r ͱ ͪ ΅ 2⍀˜ 2 2⍀˜ 2 ͑35͒
2͑1/2͒+␣⌫͑1−2␣͒ /4 /2 cos 2 ͑ ͉⍀˜ ͉ Ͼ ͒ ͵ ͵ W q, vq/2 = d d ␣ ͱ ⍀˜ 1−2 0 0 1 + cos 2 cos 2 1 1−2␣ 1 1−2␣ ϫ − . vq 1 + cos 2 cos 2 vq 1 + cos 2 cos 2 ΄ͩcos − ͱ ͪ ͩcos + ͱ ͪ ΅ 2⍀˜ 2 2⍀˜ 2 ͑36͒
͉⍀˜ ͉Ͻ For v,q/2 we obtain
2͑5/2͒−␣ 1+␣ /4 /2 cos2 ͑cot ͒␣ ͑ ͉⍀˜ ͉ Ͻ ͒ ͩ ͪ⌫͑ ␣͒͵ ͵ W q, vq/2 = sin 1−2 d d ͑ ͒1−2␣ ͱ vq 2 0 0 1 − cos 2 cos 2 1 1 ϫ ͑ ͒ 1−2␣ + 1−2␣ 37 ͩͱ1 − cos 2 cos 2 2⍀˜ ͪ ͩͱ1 − cos 2 cos 2 2⍀˜ ͪ + sin − sin 2 vq 2 vq
and
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2͑3/2͒−␣ /4 /2 cos 2 ͑ ͉⍀˜ ͉ Ͻ ͒ ⌫͑ ␣͒͵ ͵ W q, vq/2 = 1−2 d d ͑ ͒1−2␣ ͱ vq 0 0 1 − cos 2 cos 2 1 1 ϫ ͑ ͒ 1−2␣ − 1−2␣ . 38 ͩͱ1 − cos 2 cos 2 2⍀˜ ͪ ͩͱ1 − cos 2 cos 2 2⍀˜ ͪ + sin − sin 2 vq 2 vq
The change in analytic structure occurring at ͉⍀˜ ͉ results calculated from Eqs. (35)–(38). For clarity of presen- tation we have removed the prefactors ⌫͑1 =v,q/2 is manifest in these expressions by the ͱ ␣͒ ͑ ͒1−2␣ factor of 1±cos 2 cos 2 in the denominator. In the −2 / v,q and have expressed the dependence on la- ͉⍀˜ ͉Ͻ ͱ ser frequency via the ratio of the laser frequency measured v,q/2 regime 1−cos 2 cos 2 and the ͱ 1−2␣ from resonance ⍀˜ to the mode frequency (note that at fixed ͓ 1−cos 2 cos 2/2±͑2⍀˜ /v q͒sin ͔ have singulari- , momentum transfer q the SPE and CDE mode frequencies ties at the same point, leading to a strong (but still integrable) will differ . Far-from resonance, the CDE intensity is seen to ͉⍀˜ ͉Ͼ ) singularity at = =0, which is absent for v,q/2. To be much larger than the SPE intensity, but close to reso- understand the singularity one may expand for small , , nance, the two are comparable. The main panel of Fig. 5 finding the leading behavior (see the Appendix), shows the ratio of the SPE to the CDE absorption. We see that the ratio is only appreciable if the incident photon en- ⍀˜ ergy is essentially on resonance. The ratio exhibits a deriva- 1 ͩ 2 ͪ ˜ tive discontinuity (seen more clearly in the inset) at ͉⍀͉ W ϳ J␣ ͉͑⍀͉ Ͻ v q/2͒͑39͒ , ␣ , v,q = /2 (here =v,q represents the resonance energy for charge and spin modes, respectively); the on-resonance ratio with J␣ a function with an additional integrable divergence at is about 0.5 independent of the Luttinger exponent. (Of ͉⍀˜ ͉ course the factors of velocity which have been removed from =v,q/2. The extra contribution arising from Eq. (39) leads to steps in the CDE and SPE intensities as the fre- the results will lead to an additional dependence on ␣.) quency is moved across resonance, with magnitude diverging in the noninteracting limit ␣→0. Figure 4 shows, on a logarithmic scale the CDE and SPE V. RPA CALCULATION In order to compare our calculation to previous literature we present the analog, for the model considered here, of the
FIG. 4. Logarithm of polarized-channel Raman scattering cross section (normalized to appropriate mode velocity), IRRS, plotted against incident laser frequency (normalized to mode excitation fre- quency =v,q and measured from the average of incident photon resonance energy and outgoing photon resonance energy with re- FIG. 5. Comparison of ratio of ␦-coefficients for CDE and SPE ⍀ spect to the resonance energy rrs), for two different values of the absorption for Luttinger (solid and long-dashed lines) and RPA- Luttinger exponent. ␣. The plotted intensities are coefficients of ␦ Fermi liquid (dash-dot and short-dashed lines) model, normalized to functions describing coherent charge (CDE) and spin (SPE) final appropriate mode frequencies and plotted against incident laser fre- states and are computed via Luttinger-liquid methods as described quency (scale as in Fig. 4). Inset: same ratio, displayed in linear in the text. scale.
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⌳.35 We follow this procedure here, but emphasize that be- cause the behavior in the region ͉⍀˜ ͉Ͻ/2 is determined en- tirely by this phenomenological parameter, our results in this region have, strictly speaking, no meaning. We obtain
⍀˜ + /2 ⍀˜ − /2 tan−1ͩ ͪ − tan−1ͩ ͪ ⌳ ⌳ D1=⌸͑,q͒ . ⌳ ͑42͒
Here FIG. 6. Feynman diagrams considered in evaluation of RPA to 1 1 1 unpolarized RRS cross section. Diagram 1 gives the dominant con- ⌸͑,q͒ = ͩ + ͪ ␦ ␦ tribution to the SPE absorption. Excitonic interactions (such as 2 vF − vFq − i + vFq − i those shown as dash-dot lines in the lower left corner of this dia- 1 gram) are not considered in this paper. Diagram 2 gives the domi- = . ͑43͒ 2 2 2 nant contribution to the CDE absorption. vF − vFq The analytical expression corresponding to the diagram Fermi-liquid calculation discussed extensively in previous labeled 2 is works.15–18,23,30 The previous calculations neglect back- scattering entirely, and treat the forward scattering part of the V D2=͑D22͒2 ͑44͒ interaction in the random-phase approximation (RPA);we 1+V make the same approximations here. We obtain analytical expressions apparently not given in previous literature. We with consider here a model with short-ranged interaction, param- etrized by a constant amplitude V. The CDE feature is then a dp ͵ ͑ ⍀ ͒ zero-sound collective mode with a velocity shifted from the D22 = − T͚ Gd i + i /2 + i n,p + q/2 2 Fermi velocity by the interaction strength. n ϫ ͑ ͒ ͑ ͒ ͑ ͒ We evaluate the diagrams shown in Fig. 6. The analytic Gc p + q,i n + i Gc p,i n , 45 expression corresponding to the diagram labeled 1 is
dp ͵ ͑ ͒ ͑ ͒ = T͚ Gc p + q,i n + i Gc p,i n dp 2 2 ͵ ͑ ⍀ ͒ n D1=−T͚ Gd i + i /2 + i n,p + q/2 2 n ͑ ͒2 vFq 1 ϫ ͑ ͒ ͑ ͒ ͑ ͒ = . ͑46͒ Gc p + q,i n + i Gc p,i n . 40 2 2 2 vF − vFq
Setting the valence-band velocity to zero, analytically con- The arguments leading to Eq. (42) may be repeated here. The tinuing on the laser frequency ⍀ and measuring it from reso- divergence is weaker; indeed the obtained expressions are nance and performing the frequency sum yields finite everywhere except at ͉⍀˜ ͉=/2, where there is a loga- rithmic divergence. Introducing the same broadening as above yields dp 1 D1=−͵ 2 i + p − p+q f −1 f −1 dp 1 f p −1 ϫ p p+q ͑ ͒ D22 = − ͵ ͩ ͩ − ͪ, 41 2 i + − ͑⍀˜ ͒ ͑⍀˜ ͒2 ͑⍀˜ ͒2 p p+q + i /2 + p + i /2 + p + i /2 + p+q f p+q −1 where f p is Fermi distribution function. We next linearize the − ͪ ͑⍀˜ ͒ conduction-band dispersion, perform the p integral (bearing − i /2 + p+q in mind that we must take the principal value), drop terms of ˜ 2 2 order vq/E , and analytically continue on . Unlike the 1 ͑⍀ + /2͒ + ⌳ F = ⌸͑q,͒ lnͫ ͬ. ͑47͒ Luttinger-liquid expressions obtained in previous sections, 2 ͑⍀˜ − /2͒2 + ⌳2 this expression as it stands is infinite in the range ͉⍀˜ ͉Ͻq/2. Previous works23,30 resolved this divergence by introducing a The total unpolarized RRS cross section, in this approxima- phenomenological broadening parametrized by a quantity tion, is thus
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͑⍀˜ ͒2 ⌳2 ⍀˜ + /2 ⍀˜ − /2 1 2 + /2 + −1 −1 ln ͫ ͬV⌸ tan ͩ ͪ − tan ͩ ͪ ΄ ⌳ ⌳ 4 ͑⍀˜ − /2͒2 + ⌳2 ΅ I = ⌸͑q,͒ + RRS ⌳ 1+V ͑⍀˜ ͒2 ⌳2 ⍀˜ + /2 ⍀˜ − /2 1 2 + /2 + −1 −1 ln ͫ ͬ tan ͩ ͪ − tan ͩ ͪ 1 ΄ ⌳ ⌳ V 4 ͑⍀˜ − /2͒2 + ⌳2 ΅ = + ͑48͒ 2 2 2 ⌳ 2 2 2 vF − vFq vF − vq
ͱ ͑ ͒ with v =vF 1+ V/ vF . Observe that if the phenomeno- tent to which a Raman scattering has observed, or can be logical broadening parameter ⌳ is set to 0 the RPA approxi- expected to observe, the evidence for Luttinger-liquid behav- mation to the RRS intensity diverges when ⍀˜ =±/2, in ior in quasi-one-dimensional systems. other words when the incoming or the outgoing photon is in resonance with a transition of the system. A. Assumptions From these formulas the SPE and CDE intensities may easily be obtained. We have We note that in the calculation presented in this paper both Luttinger-liquid and Fermi-liquid calculations require 2 2 2 1 v ͑⍀˜ + /2͒ + ⌳ three assumptions. ͩ ͪ 2 ␦͑ ͒ I = ln ͫ ͬ − qv , CDE (i) The photon energy and momentum q transferred to 8v v ͑⍀˜ ͒2 ⌳2 F F − /2 + the one-dimensional electron system are small in comparison ͑49͒ with the Fermi energy and Fermi momentum of the one- dimensional electron system. (ii) The Raman process involves excitation from a one- ⍀˜ + /2 ⍀˜ − /2 tan−1ͩ ͪ − tan−1ͩ ͪ dimensional valence band to an empty state in a one- 1 ⌳ ⌳ dimensional conduction band, followed by decay of an elec- I = ΄ SPE ⌳ 2vF tron from a filled state in the conduction band into the hole state in the valence band. (iii) The excitonic correlations between conduction-band 1 ͑⍀˜ + /2͒2 + ⌳2 states and the intermediate state (one hole in the valence − ln2ͩ ͪ΅␦͑ − qv ͒. ͑50͒ band and one additional electron in the conduction band F ) 4 ͑⍀˜ − /2͒2 + ⌳2 may be neglected. Assumption i is essentially a condition that the experi- The CDE/SPE ratio is plotted in the inset to Fig. 5 for ⌳ ( ) mental resolution is sufficient to reveal the low-energy phys- =0.1. We see from this inset that the ratio is generically ics of the system of interest. From the theoretical point of very small (smaller than the corresponding ratio in a Lut- view it could easily be relaxed at the expense of introducing ͉⍀˜ ͉Ͻ tinger liquid), but that for all v,q/2, the divergence of a more complicated description of the conduction band, i.e., the SPE term (cut off here by a small value of the phenom- considering the curvature of band structure about the Fermi ⌳ enological parameter ) yields a large value of the ratio. surface. Assumption (ii) is clearly applicable to strictly one- dimensional systems such as carbon nanotubes where the VI. DISCUSSION ( valence band is clearly one dimensional) but may not be In this paper we have presented a complete analytical applicable to quantum wire structures created by lithographic theory of resonant Raman scattering in Luttinger liquids, and or molecular-beam epitaxy (MBE) techniques on a three- for comparison we have also given the analogous results for dimensional substrate. In this situation, valence-band carriers what is referred to in the literature as the Fermi-liquid ap- may be able to move transverse to the wire because momen- proximation. In this section we discuss the implications of tum transverse to the wire is not conserved in the optical our results. In Sec. VI A we outline the essential assumptions absorption process. In this case the function ͑x,t͒ in Eq. underlying the calculation and discuss how these can affect (11) must be integrated over a range of transverse momenta, the results and if they may be relaxed. In Sec. VI B we leading to a broadening of the resonance. However, assump- compare the excitation spectra of the Luttinger-liquid model tion (ii) may also be relaxed, if only the intermediate state and the standard Fermi-liquid model, and show why they (of whatever origin) disperses only along the wire. lead to qualitatively similar Raman spectra in many one- The crucial assumption is (iii), neglect of excitonic corre- dimensional systems. lations. These are likely to play a crucial role in the QWR Finally, in Sec. VI C we compare the near-resonance RRS structures by keeping the valence hole near the wire and spectrum calculated in these two models and discuss the ex- therefore allowing a sharp resonance to occur. Since we have
165101-11 WANG, MILLIS, AND DAS SARMA PHYSICAL REVIEW B 70, 165101 (2004) neglected these excitonic correlations in the theory presented =v͑q͉͒q͉ with velocity v͑q͒ greater than the Fermi velocity ͑ → ͒ in this paper at several points, it cannot be trivially modified vF in the long wavelength limit q 0 . to include them in a fully self-consistent way. Therefore con- The other class of states, referred to in the Raman litera- struction of a self-consistent theory including excitonic ef- ture as single-particle excitation (SPE), are the particle-hole fects in the Raman scattering process is an important open continuum states. In two- and three-dimensional systems, ഛഛ issue in the field. We will briefly discuss the possible exci- these excitations exist in the range 0 vFq;ind=1 they ͉ ͉ ͑ 2 ͒ഛ͉͉ഛ ͉ ͉ tonic effects to the Raman scattering spectroscopy in Sec. exist only in the range vF q − vFq /2p0 vF q p2͉ ͔͒ being a measureץ/ ⑀2ץ͑/ VI C. +͑v q2 /2p ͒ with p ϵ͓v F 0 0 F p p=pF By combining the three assumptions given above with of the curvature of the quasiparticle dispersion. In any di- standard second-order perturbation methods, we obtained re- mension, the SPE or continuum excitations make an impor- sults, presented in Sec. IV and Eqs. (35)–(38), for the reso- tant contribution to the specific heat and to most response nant Raman scattering intensities of a one-dimensional elec- functions. However, the contribution of the continuum exci- tronic system described by the Luttinger-liquid model. For tations to the structure factor (density response function) is comparison, we also presented analytical expressions for the typically much smaller than that of the CDE (plasmon) Raman intensities of the previously discussed Fermi-liquid mode. In higher dimension ͑dϾ1͒ the CDE contribution to model (in which the only electron-electron interaction con- the structure factor is larger than the SPE contribution by a sidered is forward scattering in the density channel, treated in factor of the dimensionless long wavelength interaction the RPA). In the far-from resonance regime, we showed that strength, which for a charged Fermi liquid involves factors of ͑ ͒d−1 a perturbative treatment could be applied, and this was used qTF/q arising from the long-ranged nature of the Cou- to provide a detailed characterization of the different features lomb interaction (here qTF is the Thomas-Fermi screening of the spectrum, including CDE and SPE peaks and a con- length). tinuum absorption and the dependence of the intensity of In one-dimensional system the factor arising from the these features on the difference of the laser frequency from Coulomb interaction is only logarithmic; however, the spe- resonance. We also obtained simple, easily evaluated expres- cial kinematics of one-dimensional systems lead to addi- sions for the integrated intensities in the SPE and CDE peaks tional constraints. In dജ2 the low-energy physics of nonor- for all values of the laser frequency, and from these we ob- dered Fermi systems is described by Fermi-liquid theory, tained results for the ratio of SPE to CDE intensities. In the which is essentially the RPA approximation but with interac- following two sections, we will respectively discuss the re- tion vertices renormalized by finite amounts by high-energy sults and physical interpretation of our theory applied in the processes. In d=1 these renormalizations destabilize the two different regions of interest: far-from resonance region Fermi-liquid description entirely. However, the RPA may be and near-resonance region. a reasonable intermediate energy-scale approximation, and as explained in Sec. V has been used successfully to model Raman scattering. In this context the terms RPA and Fermi B. Excitation and Raman spectra in Luttinger liquid are synonymous. Within the RPA the SPE contribution and Fermi liquids to the structure factor is smaller than the CDE contribution ͑ ͒2 The Raman process creates a particle and a hole in the by a factor of q/p0 . Therefore, in the linearized dispersion ͑ →ϱ͒ conduction band, at a space-time separation controlled by the limit p0 , the SPE contribution to the d=1 structure difference of the laser frequency from a resonance condition. factor vanishes exactly within the RPA. Indeed this vanishing The particle-hole pair generated by Raman scattering is in can be shown via a Ward Identity39 to occur to all orders in general not an eigenstate of the conduction-band Hamil- the interaction. We emphasize, however, that in the RPA to tonian, but decays into the true eigenstates. The important the one-dimensional electron gas, the SPE excitations still question, therefore, is “what properties of the eigenstates are exist, and can be revealed either by considering response revealed by Raman scattering.” In the one-dimensional con- functions other than the structure factor, or from the specific text one may sharpen this question to “what aspects of the heat. A specific example of a response function other than the Raman spectrum reveal characteristic features of the structure factor is provided by Raman scattering: far-from Luttinger-liquid physics expected to occur in one- resonance the leading contribution to the Raman vertex is the dimensional systems?” In order to answer this question pre- density operator, but corrections, whose magnitude depends cisely, it is necessary to discuss the eigenstates of one- on the difference of photon frequency from resonance, in- dimensional electronic systems. volve other operators. A detailed discussion of these opera- We are concerned here with polarized channel Raman tors may be found above in the context of Eq. (11). scattering in systems with negligible spin-orbit coupling and Now let us consider the Raman scattering spectrum in therefore must consider excitations which do not change the Luttinger-liquid model, where two classes of elementary ex- total spin of the system. In a Fermi liquid with repulsive citation exist: charge and spin bosons. States of total spin interactions, there are two kinds of relevant states. One is the zero may be constructed from states of two or more spin zero sound (or plasmon) mode, referred to in the Raman bosons, even if no charge bosons are present, and these may literature as the charge density excitation (CDE) mode. (In make important contributions to many response functions principle there are other collective modes, but they are rarely and to the specific heat. In the linearized dispersion limit, important in practice.) The CDE excitation typically has a states involving only spin bosons do not appear in the struc- well-defined energy-momentum dispersion relation ture factor, because it couples only to the charge sector.30,40
165101-12 THEORY OF RESONANT RAMAN SCATTERING IN ONE-… PHYSICAL REVIEW B 70, 165101 (2004)
However, curvature in the dispersion leads to charge-spin tion. As this occurs, operators other than the structure factor coupling and in particular to terms coupling a charge boson begin to contribute to the Raman spectrum, and the nature to two spin bosons. Thus, just as in the Fermi-liquid case, if and coefficients of these operators may be used to distinguish nonlinearity in the underlying dispersion is neglected, only between the two different theoretical models. the CDE mode is visible in the structure factor but if band- In discussing the near-resonance behavior it is helpful to energy curvature is included, SPE contributions (two or more refer to Fig. 6 which shows the diagrams commonly consid- ͑ ͒2 ered in the Fermi-liquid treatment of the RRS process. From spin bosons at energy =qvF) appear at order q/p0 .Of course such multispinon contributions also appear in re- these diagrams one sees that in the Fermi-liquid approxima- sponse functions other than the structure factor. Therefore tion, the states created by the Raman process have nonvan- ishing overlap with the two sorts of exact eigenstates CDE even though the interpretations are different, the off- ( and SPE of the system. Diagram 1 gives the dominant con- resonance Raman scattering spectra calculated within Fermi- ) tribution to the probability of creating a SPE excitation. The liquid model and Luttinger model are basically the same, for direct overlap between the state created by the Raman pro- the spin-unpolarized one-dimensional electron gas. cess and the exact eigenstate of the system means that in the However, the situation changes when considering a 1D absence of excitonic correlations between the particle-hole spin-polarized electronic system. This could be realized in pair and the intermediate resonant state (sketched as dash-dot practice by applying a magnetic field large enough to fully lines in diagram 1 of Fig. 6), this diagram diverges strongly. spin polarize the conduction band. It is believed that a cor- (The diagram might also diverge strongly even in the pres- rect treatment (for example, via the bosonization method of ence of excitonic effects: this point has not been investi- the Luttinger model) would predict that the only elementary gated.) Indeed one sees from Eq. (42) that very near reso- excitation is a boson, which is roughly equivalent to the zero nance the term diverges as ͑⍀˜ ±/2͒−1 and (in the absence of sound or plasmon mode excitation of a Fermi liquid. Thus in excitonic correlations or of the phenomenological broaden- the spinless Luttinger-liquid case, in all response functions ing ⌳) is infinite for ͉⍀˜ ͉Ͻ/2. On the other hand, one sees and also in the specific heat, only superpositions of plasmons from diagram 2 that the coupling of the Raman process to the would be observed, whereas in the so-called Fermi liquid CDE excitation goes through a range of virtual states (repre- approximation one would observe features at both energies sented by the triangles in diagram 2); this broadens the reso- as in the spinful case—one zero sound (or collective) mode nance effect so that the divergence as resonance is ap- and one electron-hole continuum. In the language of Raman proached is only logarithmic and the result is finite for scattering experiment, only the CDE mode would be visible ͉⍀˜ ͉Ͻ/2. However, the one-dimensional kinematics lead for in the Luttinger-liquid case, no matter how closely the sys- ͉⍀˜ ͉Ͼ tem is tuned to resonance, whereas for a hypothetical 1D /2 to a much greater CDE amplitude than a SPE am- Fermi liquid, excitation at the SPE energy would become plitude, so that the SPE/CDE ratio only becomes of order unity very close to the resonance condition. Further, for visible as the laser frequency is tuned to resonance (due to ͉⍀˜ ͉Ͻ the interband scattering matrix element). /2 the Fermi-liquid approximation to the SPE inten- To summarize this section: as long as the electron number sity is, strictly speaking, infinite, and must be regularized by is not changed in the conduction band during the Raman consideration of processes so far omitted from the approxi- scattering process (i.e., off-resonance regime), the apparently mation. At present there is no theory of this regularization, profound differences between a Luttinger and a Fermi liquid which has been parametrized by a phenomenological broad- ening ⌳. Because they are entirely determined by the regu- produce only minor and quantitative differences in the exci- larization parameter, the Fermi-liquid results presented in tation spectrum of a spin-unpolarized system. In both mod- this paper in the near-resonance regime are probably mean- els, one has two classes of excitation, which may be labeled ingless. We suspect that this issue also arises in the higher CDE and SPE, respectively, in the Raman scattering experi- dimensional calculations performed so far. ͑ ͒2 ment, where the latter is smaller by a factor of q/p0 due to In the Luttinger-liquid approximation, the particle-hole the weak nonlinearity of the band structure. However, for a pair created in the Raman process has vanishing overlap with spin-polarized electron gas, there is an important difference any exact eigenstate. This has three consequences: first, a between a Luttinger and a Fermi liquid: in the Fermi-liquid continuum absorption arising from multiboson excitations is approximation a SPE branch of excitations still exists, predicted to lie in between the SPE and CDE peaks (actually, whereas in a Luttinger liquid it does not. this continuum would exist also in a Fermi liquid, but would be much weaker). Also, away from resonance, the SPE/CDE C. Physics of RRS spectrum—near resonance ratio is larger than in the Fermi-liquid approximation, and it varies with a power related to the Luttinger liquid exponent The previous sections have shown that in the polarized ␣. Finally, in the on-resonance regime ͉⍀˜ ͉Ͻ/2 both SPE channel of RRS experiment (i.e., no spin flipping of final and CDE contributions are predicted to be finite, although electron configuration , the differences between the predicted ) ͉⍀˜ ͉ Raman spectra of a Luttinger-liquid and Fermi-liquid ap- both exhibit a discontinuity when = /2. Fermi-liquid be- ͑␣ proximations to the one-dimensional electron gas are quan- havior can be recovered by taking noninteracting limit → ͒ ␣−1 titative, not qualitative, as long as the photon energy is off- 0 because the magnitude of the discontinuity is and resonance. However, a significant distinction between these hence diverges in the noninteracting ͑␣→0͒ limit. two models may arise from the Raman spectrum changes as We suggest that in the on-resonance regime ͉⍀˜ ͉ϳ/2 nei- the laser frequency is brought closer to a resonance condi- ther the Fermi-liquid nor the Luttinger-liquid calculations are
165101-13 WANG, MILLIS, AND DAS SARMA PHYSICAL REVIEW B 70, 165101 (2004) likely to be quantitatively reliable, because excitonic corre- was supported (D.W.W. and S.D.S.) by US-ONR, US-ARO, lations (neglected here) are likely to be important: near reso- and DARPA, and by NSF-DMR-0338376 (A.J.M.). nance the intermediate state lasts so long that it must interact with the conduction-band excitations, and these interactions APPENDIX: EVALUATION OF ␦-FUNCTION are likely to have a nontrivial interplay with the nonanalyt- CONTRIBUTIONS ͉⍀˜ ͉ icities arising when = /2. Several qualitative features of 1. Overview such excitonic effects can be expected: First, the attractive interaction between the excited electrons above the Fermi This appendix presents the additional transformations surface of conduction band and the hole in the valence band needed to put Eqs. (33) and (34) into forms convenient for can form a metastable intermediate state of excitons, which further discussion and numerical evaluation. We have to deal should have strong binding energy due to the geometrically with an expression of the form (note that for simplicity, we tight confinement potential in one-dimensional quantum wire define ⍀ to be the light frequency with respect to the reso- system. Therefore the assumption (ii) discussed in Sec. VI A nance energy throughout this appendix) can be further relaxed and one may treat the valence-band ϱ ͑⍀ ͒ ϱ ͑⍀ ͒ dt ei +i t1 dt e−i −i t2 ͵ 1 ͵ 2 ͑ ͒͑͒ hole to be a 1D quasiparticle as what we assumed in this W = 1+␣ 1+␣ I t1,t2 A1 paper. Second, the existence of excitonic metastable interme- 0 t1 0 t2 diate state may change the position of resonance since the with →0+ being an infinitesinal converging factor and binding energy of a 1D exciton can be of order of several meV.41 A recent RRS experiment31 in a two-dimensional ϱ ͑ ͒ −iqR ͑ ͒͑͒ quantum well system apparently observed two resonances. I t1,t2 = ͵ dR e F t1,t2,R A2 ϱ Finally, the scattering matrix element have to be calculated − by considering the overlap between the exciton state and the and F given by electron-hole continuum state, which will certainly broaden v2͑t + t ͒2  the resonance effects near the resonance condition. The spec- 1 2 − ͑R + i͒2 tral weight distribution as a function of photon energy trans- 4 F = −1, ͑A3͒ 2͑ ͒2 fer at a given momentum transfer q will also be changed v t1 − t2 − ͑R + i͒2 due to such inhomogeneous broadening. But so far we could 4 not speculate if the Luttinger-liquid properties of a one- dimensional electronic system can be unambiguously ob- where for the charge case v=v and =͑1+␣͒/2, while in  served in the RRS spectrum even after including the full the spin case, v=v =vF and =1/2. As in Sec. III we have excitonic effects. A more complete theory to include such assumed that the valence-band velocity is much smaller than nontrivial effects is necessary for the future study. the Fermi velocity in the conduction band. The analytic structure of the F function means that we can write (note we VII. SUMMARY have also rescaled R) In this paper we have used the full Luttinger-liquid model B qR to analytically and numerically calculate the RRS spectrum ͑ ͒ ͑͒͵ ͩ ͪ ͑ ͒͑͒ I t1,t2 = 2 sin dR sin F2 B,A,R A4 for both polarized and depolarized spectroscopy, and have A 2 presented for comparison a Fermi-liquid calculation. We ob- with tained results for both short-ranged (screened Coulomb) in- teractions and for the more physically relevant case of the B2 − R2  F ͑t ,t ,R͒ = ͩ ͪ ͑A5͒ long-ranged Coulomb interaction. We clarified the difference 2 1 2 R2 − A2 between the Fermi-liquid and Luttinger-liquid results, and ͉ ͉ argued that proper treatment of excitonic interactions (ne- and B=t1 +t2 and A= t1 −t2 . glected in all treatments so far) is essential for obtaining By making the following changes of variables: R ͱ 2 ͱ 2 2 reliable results. Our results in Figs. 2 and 3 show that the → u;u→v+A ;v→ B −A sin and restoring the explicit RRS spectra of both short-ranged and long-ranged interac- forms of B and A we obtain tions are qualitatively similar, except that in the unscreened /2 ͑ ͒ ͑͒ ͵ Coulomb case the momentum dependent charge velocity I t1,t2 = 2 sin t1t2 d makes the phase space of multiboson excitations to be highly 0 restricted, which shift the multiboson excitations to higher   vq energy than the plasmon energy =qv , causing such un- ͑cos ͒1+2 ͑sin ͒1−2 sin ͱt2 + t2 −2t t cos 2 q 2 1 2 1 2 usual broadening. The SPE-CDE ratio is generically larger in ϫ . ͱ 2 2 a Luttinger liquid than in a Fermi liquid, except on- t1 + t2 −2t1t2 cos 2 resonance, where excitonic effects probably invalidate either ͑A6͒ calculation.
ACKNOWLEDGMENTS 2. Evaluation, large ⍀ We thank A. Pinczuk and P. Calleja for a useful discussion If ⍀Ͼvq/2 then it is convenient to rotate the t integrals about the details of Raman scattering experiments. This work via
165101-14 THEORY OF RESONANT RAMAN SCATTERING IN ONE-… PHYSICAL REVIEW B 70, 165101 (2004)
t → i , ͑A7͒ 2͑1/2͒+␣⌫͑1−2␣͒ /4 /2 cos 2 1 1 ͵ ͵ W = d d ⍀1−2␣ ͱ 0 0 1 + cos 2 cos 2 t → − i , ͑A8͒ 2 2 1 1−2␣ ϫ getting q 1 + cos 2 cos 2 ΄ vF ͱ ͩcos − ͪ ϱ ⍀ ϱ ⍀ 2⍀ 2 − 1 − 2 d 1 e d 2 e W = ͵ ͵ I͑ , ͒͑A9͒ 1−2␣ 1+␣ 1+␣ 1 2 1 0 1 0 2 − . q 1 + cos 2 cos 2 vF ͱ ΅ ͩcos + ͪ with 2⍀ 2 ͑ ͒ ͑A15͒ I 1, 2 /2 This formula has integrable singularities at ==/2 and ͑͒ ͵ = 2 sin 1 2 d = /2 and is convenient for numerical evaluation. 0 vq ͑cos ͒1+2͑sin ͒1−2 sinh ͱ2 + 2 +2 cos 2 3. Evaluation, small ⍀ 2 1 2 1 2 ϫ . If ⍀Ͻvq/2 then the rotation cannot be made and we ͱ2 2 1 + 2 +2 1 2 cos 2 should instead define ͑ ͒ A10 ͑ ͒ t1 = t cos , A16 It is then useful to define ͑ ͒ t2 = t sin . A17 = cos , ͑A11͒ 1 Transformations similar to those leading to Eq. (A13) then yield ͑ ͒ 2 = sin , A12 2͑5/2͒−␣ 1+␣ ͑⍀͒ ͩ ͪ⌫͑ ␣͒ W = 1−2␣ sin 1−2 so that ͑vq͒ 2 ϱ /4 /2 2 ͑ ͒␣ /2 /2 d cos cot ͑͒͵ ͵ ͵ −⍀͑cos +sin ͒ ϫ͵ d͵ d W = 2 sin d d ␣ e ͱ 2 0 0 1 − cos 2 cos 2 0 0 0 1   vq ͑ ͒1+2 ͑ ͒1−2 ͱ ϫ ␣ cos sin sinh 1 + sin 2 cos 2 1 − cos 2 cos 2 2⍀ 1−2 2 ͱ ϫ . ͩ + sin ͪ ͱ1 + sin 2 cos 2 2 vq ͑A13͒ 1 + ␣ 1 − cos 2 cos 2 2⍀ 1−2 ͱ The integral may now be done. It is convenient to shift the ͩ − sin ͪ origin of the integral by /4 and symmetrize in and to 2 vq combine the two terms arising in the spin term, getting ͑A18͒
͑1+␣͒ and 2͑3/2͒+␣ sinͩ ͪ⌫͑1−2␣͒ 2͑3/2͒−␣ 2 ͑⍀͒ ⌫͑ ␣͒ W = 1−2␣ W = 1−2␣ 1−2 ⍀ ͑vq͒ /4 /2 cos2 ͑cot ͒␣ /4 /2 cos 2 ϫ͵ d͵ dͱ ϫ͵ d͵ dͱ 0 0 1 + cos 2 cos 2 0 0 1 − cos 2 cos 2 1 1−2␣ 1 ϫ ϫ 1−2␣ q 1 + cos 2 cos 2 1 − cos 2 cos 2 2⍀ ΄ v ͱ ͩͱ ͪ ͩcos − ͪ + sin 2⍀ 2 2 vq 1 1−2␣ 1 − , − 1−2␣ . q 1 + cos 2 cos 2 1 − cos 2 cos 2 2⍀ v ͱ ΅ ͩͱ ͪ ͩcos + ͪ − sin 2⍀ 2 2 vq ͑A14͒ ͑A19͒
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We observe that unlike Eqs. (A14) and (A15) these expres- 1 ͵ ͵ sions have a double singularity at small , and indeed lead d d Ich = d d ␣ ␣ ͱ22− ͑cos ͒ to RRS spectra of quite different magnitudes. To see this point more clearly, consider the integrand, Ich of Eq. (A18) as 1 → ϫ , 0. We obtain 2⍀ 1−2␣ ͩ ͪ ␣ 1+ sin 1 1 1 vq I = ͩ ͪ ch ͱ ͱ2 2 2⍀ 1−2␣ 2 + ͱ2 2 1 ͩ + + ͪ + . ͑A21͒ vq 2⍀ 1−2␣ ͩ1− sin ͪ 1 vq + . ͑A20͒ 2⍀ 1−2␣ ͱ2 2 ͩ + − ͪ vq Changing variables to = cos and = sin we have Integration over gives an answer proportional to 1/␣.
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36 D. W. Wang, A. J. Millis, and S. Das Sarma, Phys. Rev. B 64, 39 I. E. Dzyaloshinsky and A. I. Larkin, Zh. Eksp. Teor. Fiz. 65,411 193307 (2001). (1973)[Sov. Phys. JETP 38, 202 (1974)]. 37 A. R. Goñi, A. Pinczuk, J. S. Weiner, J. M. Calleja, B. S. Dennis, 40 D.-W. Wang, A. J. Millis, and S. Das Sarma, Solid State L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 67, 3298 Commun. 131, 637 (2004). 1991 . ( ) 41 S. Das Sarma and D. W. Wang, Phys. Rev. Lett. 84, 2010 (2000); 38 J. Rubio, J. M. Calleja, A. Pinczuk, B. S. Dennis, L. N. Pfeiffer, D. W. Wang and S. Das Sarma, Phys. Rev. B 64, 195313 (2001). and K. W. West, Solid State Commun. 125, 149 (2003).
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