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Electronic excitations in narrow quantum wells via intersubband Raman scattering: Theoretical considerations夽 Leonarde N. Rodrigues ∗ , A. Arantes, M.J.V. Bell, V. Anjos ∗ Laboratório de Espectroscopia de Materiais, Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil

a r t i c l e

i n f o

Article history: Received 10 May 2016 Received in revised form 12 September 2016 Accepted 12 September 2016 Available online 17 October 2016 Keywords: Raman scattering Electron gas Collective excitations Single-particle excitations Narrow quantum wells

a b s t r a c t In this work a generalized self-consistent ﬁeld theory was applied to investigate the elementary excitations of two-dimensional electron gas formed from narrow quantum wells via resonant intersubband Raman scattering. The developed model considers the existence of equally coupled and degenerated excitations of the electron gas and allows to observe that in extreme resonance regime the plasma oscillations splits into two contributions: a set of renormalized collective excitations (plasmons) and unrenormalized electronic transitions (single-particle excitations). Our results show that the asymmetries which appear in the Raman proﬁle of doped narrow quantum wells can be interpreted as the entrance or exit of resonance of collective modes overlapped with single-particle transitions. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The inelastic light scattering is a powerful tool that has been widely used in the study of the semiconductor materials and it has become an indispensable technique for the understanding of fundamental physical processes [1–3]. The investigation of quantized electronic systems as quantum wells, wires and quantum dots via Raman spectroscopy is quite attractive since it allows to obtain information concerning many-body effects of interacting particles. By means of electronic Raman scattering is possible to grasp the nature of collective excitations in quasi-2D systems which are known as Charge Density Excitations (CDE) and Spin Density Excitations (SDE). CDE are plasmonic oscillations arising from the coupling between charge ﬂuctuations via Coulombian and exchange-correlation interactions while SDE occurs only when exchange-correlation effects are present. Both depend on selection rules that are associated with polarizations of the light [4]. The CDE is active when the laser energy is resonant with a semiconductor optical gap and the incoming and outgoing light polarizations are parallel to each other (polarized spectra). The SDE is active when

夽 Selected paper from for IV Encontro Brasileiro de Espectroscopia Raman (EnBraER), in Juiz de Fora, December 06–09, 2015. ∗ Corresponding authors. E-mail addresses: leonarde @yahoo.com.br (L.N. Rodrigues), [email protected]ﬁsica.ufjf.br (V. Anjos). http://dx.doi.org/10.1016/j.vibspec.2016.09.008 0924-2031/© 2016 Elsevier B.V. All rights reserved.

the laser energy is resonant with a semiconductor optical gap and the incoming and outgoing light polarizations are perpendicular to each other (depolarized spectra). Nevertheless, when the laser matches interband transitions energy of the material (extreme resonance regime), in addition to the collective excitations, emerges transitions of the electron gas noninteracting-like known as SingleParticle Excitations (SPE). It is largely accepted that the observation of the SPE are related to extreme resonance regime irrespective of the dimensionality of the electron gas system [4–6], which makes then a fascinating phenomenon. However, the physics of such transitions is still not completely understood [7]. On the other hand, an approach developed in previous works has shown in a complete and well-founded way that the SPE resides, in fact, in unrenormalized collective excitations [8–11]. In addition, in Ref. [11], an analogy is presented between resonant electronic Raman scattering and the forced coupled harmonic oscillators problem, as well as, a correspondence with the formation of the superconducting state in BCS theory of normal metals. This article aims to provide a theoretical interpretation on the behavior of the electronic Raman results found in Ref. [1], where for GaAs narrow quantum wells formed from the semiconductor sequence AlGaAs/AlAs/GaAs/AlAs/ AlGaAs, the Raman spectra are more inﬂuenced by SPE. Two wells were studied: one with 10 nm and the other with 17 nm GaAs width. For the 10 nm wide GaAs single quantum well (extreme case) only the SPE are seen in Raman spectra because collective excitations (CDE and SDE) are too small and overlapped by the SPE peak. A qualitative comparison between

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theoretical and experimental data is performed which allows to interpret the asymmetries of Raman proﬁle lines as the entrance or exit of resonance of collective modes overlapped with singleparticle transitions. 2. Generalized self-consistent ﬁeld theory To calculate the response of a nonuniform electron gas submitted to the action of an external potential we use a formalism based on time dependent local density approximation (TDLDA) [12,13]. The fundamental idea of the self-consistent ﬁeld theory approximation is to assume that the system of many-electrons responds to an effective ﬁeld as a system of independent particles.1 Therefore, an external potential acting on the system induces a charge density ﬂuctuation as response to the applied ﬁeld. This induced ﬂuctuation produces an induced potential. This new potential acts on the system again producing a new total potential (effective potential, Veff ), and so on ad inﬁnitum. In other words, this process is self-consistent and can be represented by Eq. (1), V eff (r, t) = V ext (r, t) + V ind (r, t),

(1)

where Vext (r, t) is the external potential (laser). The induced potential is described by [8,12,14]

V

ind

(r, t) =

e2 + Uexc (r)ı(r − r ) ın(r , t)dr . εl (ω)|r − r |

(2)

The ﬁrst term in Eq. (2) corresponds to the direct term (Coulomb interactions), the second include many-body effects 2 )/(ω2 − ω2 ) is the (exchange and correlation). εl (ω) = ε∞ (ω2 − ωLO TO GaAs frequency-dependent lattice dielectric function that contains the bulk frequencies of the longitudinal (ωLO ) and transverse (ωTO ) optical phonons (the phonon lifetime was neglected) with dielectric constant ε∞ . The induced density ﬂuctuation in the timecoordinate representation is given by, ın(r, t) = ˆ † (r, t) ˆ (r, t)t =

∗ ˛ (r)

†

c˛ cˆˇ t , ˇ (r)ˆ

(3)

˛ˇ † where ˆ † and ˆ are ﬁeld operators. The coefﬁcients and cˆ˛ (ˆcˇ ) in Eq. (3) are wave functions and fermion creation (destruction) operators of single-particle conduction subband states. Taking the Fourier transform of Eq. (1) and solving the Heisenberg equation of † motion for expectation values ˆc˛ cˆˇ t in Eq. (3) [15,16], eff

Vji = Vjiext +

eff

Cij,mn 0mn Vmn ,

(4)

mn

2e2 Akz εl (ω) +

and 0mn =

k,

1 A

dzdz ij (z )e−kz |z−z | mn (z)

CDE dz[−Uexc (z)]mn (z)

(5)

fm (k) − fn (k + q) . m (k) − n (k + q) + ω

(6)

2Nnm ωnm lim (0mn + 0nm ) = . 2 ) 2 (ω2 − ωnm q→0

Calculations developed here follow the approach used in Refs. [8,11].

(7)

T → 0K Eq. (7) takes into account both upward and downward transitions. m ) is the bare electronic transition energy, 0mm = ωnm = (n − 0 and Nnm ≡ k, fm (k) − fn (k) = k, 1 is the number of electrons that contributed to each transition m → n. The energy transferred to the electron system by the light is ω = (ωL − ωS ) where ωL and ωS corresponds to the incident (laser) and scattered photon energies. The idea employed in Ref. [8] is to map the inelastic light scattering of an electron gas into a problem of a set of forced damped harmonic oscillators. The damping of the transitions (i.e., scattering by impurities) is considered when we replace ω → ω + i . Thus, 2 (ω + i )2 (ω)2 + i ω, where ≡ 2 is the damping related to each transition. Therefore, we associate a harmonic coordinate to each transition pair deﬁned as

xji ≡

2Nji ωji

eff

2 (ω2 − ωji2 + iji ω)

Vji .

(8)

In this way, Eq. (4) can be rewritten in the following way, eff

Vji = Vjiext +

eff

Cij,mn 0nm Vnm

mn n>m

2 (ω2 + iji ω)xji =

2Nji ωji Vjiext +

Uij,mn (ω)xnm ,

(9)

mn

where

4Nji Nnm ωji ωnm + (ωji )2 ıij,mn .

(10)

From Eqs. (5) and (10) one can see that the matrix U is real and symmetric. Therefore, its eigenvectors constitute a base which can be used to solve the equation of x via LU decomposition. The expression for the inelastic light scattering which connects experiment and theory is the differential scattering cross section [18] given by 2

Cij,mn is the matrix element of the Fourier transform which couples the charge density ﬂuctuations between the subbands ij and mn. The ﬁrst term in Eq. (5) represents contributions of the direct Coulombian interactions (Hartree term) and the second represents the exchange-correlation effects for the CDE. For the SDE, only the

1

0nm ≡

Uij,mn (ω) ≡ Cij,mn (ω)

where Cij,mn =

second term on the right side in Eq. (5) should be considered. The CDE and U SDE explicit expressions for the functional derivatives Uexc exc are obtained from [17]. In Eq. (5), A is the area and the wave functions that describe the conﬁnement in the z direction (written in terms of the envelope wave functions) are ij (z ) = i (z ) j (z ). 0mn is the response function of the noninteracting electronic system, fm(n) (k) is the Fermi–Dirac distribution and (k) is the dispersion relation in the parabolic band approximation and k, q ∈ kx ky . In this work, we considered only intersubband transitions where there is no lateral momentum transferred by light (q → 0). In this way, we can rewrite Eq. (6) as

∂ = r02 ∂ ∂ω

ω S

ωL

S(ω),

(11)

where r02 = e2 /mc 2 is the classical electron radius and S(ω) =

F

ˆ eff |I|2 ı(EF − EI − ω) |F|M

,

(12)

I

ˆ eff is the effective operator is known as dynamic structure factor. M for a transition between the many-body state |I with energy EI to the ﬁnal state |F with energy EF and I an average over the initial

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195

state. The dynamic structure factor is proportional to imaginary part of the response function, i.e., S(ω) = −

ˆ † ω . ImM eff

(13)

Eq. (13) is the application of the dissipation-ﬂuctuation theorem [19]. Through time dependent perturbation theory, cross sections may be expressed in terms of Fourier transform of a correlation ˆ † ω is the Fourier amplitude at −ω of the function [11]. Note that M eff expectation value of the effective operator. The effective operator of the light scattering can be expressed as [11], ˆ eff = M

†

ext Vˇ˛ ˆcˇ cˆ˛ ,

(14)

˛ˇ

where ext Vˇ˛ =

1 ˇ|p · AL |hh|p · AS |˛ m Eˇ − Eh − ωL

(15)

h

is a resonant factor [20]. In Eq. (15), |h represent single-particle intermediate states (valence band), E˛,h are state energies, AL and AS are vector potentials of incident (laser) and scattered light, p is the momentum of the electron. The light-matter interaction is given via p · AL = eˆ L peikL · r and p · AS = eˆ S pei(−kS ) · r where eˆ L(S) are incident (scattered) radiation polarizations and kL(S) are incident (scattered) light wave vector. The inelastic light scattering process given by conduction band |˛ → |ˇ transitions are described by one-electron states which include the lattice periodic Bloch functions in Kane model (u (r)), the envelope functions ( i (z)) of the quantum well in the growth direction as well as plane waves (eik·r ) normalized by sample area that take into account the free electron motion in the xy √ plane. For example, ˛ (r) = r|˛ = r|i, k, = eik · r i (z)u (r)/ A. Using |˛ = |i, k, , |ˇ = |j, k + q, , |h = |j , k , and considering only vertical transitions (q → 0) we can rewrite Eq. (15) in the following way [11],

ext Vˇ˛ = (ˆeL · eˆ S )ı˛ ,ˇ + i(ˆeL × eˆ S )˛ ||ˇ

×

2 j|eikL · r |j j |e−ikS · r |i Pcv ı , 3m Eg + ˇ + h − ωL + i k k

(16)

j k

where now, in Eq. (16), |i, |j and |j states are plane waves and envelope functions. The spin states in conduction band are represented by | ˛ and | ˇ , = ( x , y , z ) is the Pauli spin matrix vector, Pcv are the interband matrix elements, m is the bare electron mass, Eg is an optical gap of GaAs. The states have dispersion relation given by ˇ = j + 2 k2 /2me , h = j + 2 k2 /2mso where me (mso ) is the electron (split-off) mass. Eq. (16) contains the selection rules about the light polarizations as explained previously. The ﬁrst term in Eq. (16) corresponds to the polarized spectra and the second term the depolarized ones. We will consider a backscattering geometry and |kL | = |kS | = kz . Only vertical intersubband processes will be taken into account which make them degenerate transitions and equally coupled by Cij,mn (Eq. (5)) as shown in Fig. 1(a). In addition, spinﬂip excitations in this work are neglected. Inserting both terms of Eq. (16) in Eq. (14) and using the z component of the Pauli spin matrix vector ( z ), the effective scattering operator is given by ˆ eff = M

†

†

Vjiext ˆcj↑ cˆi↑ ± cˆj↓ cˆi↓ ,

(17)

ij

From Eq. (17) it can be observed that the action of the external potential is to produce an induced ﬂuctuation of charge to † each pair of creation (annihilation) fermion operators cˆj↑ (ˆci↑ ) with spin up (↑) or down (↓). The sign +(−) in Eq. (17) refers to the CDE mechanism (electronic oscillations in phase) and SDE mechanism (electronic oscillations out of phase), respectively. Note that, in Eq. (18), ∗−1 = m∗−1 + m∗−1 e so is the reduced mass between the ext in Eq. (15) includes the resoconduction and valence band. Vˇ˛ nant denominator that is responsible for the scattering amplitude which has two resonance conditions. If the laser energy matches an optical gap of the semiconductor material, we have Egap = Eˇ − Eh . Such condition is called near resonance and it allows eliminating the intermediate states of the valence band via closure relation in Eq. (15) replacing the resonant denominator by an average denominator plus a phenomenological damping. In this situation, the net effect, is to produce only collective excitations (CDE and SDE). However, if the laser energy is resonant with interband transitions of the semiconductor material, the exact denominator of Eq. (18) should be taken into account (Fig. 1(b)). This condition is called extreme resonance and only in this regime, besides of the collective contributions of the electron gas, emerges in the Raman spectra, single-particle transitions (SPE). The electron gas responses in both regimes are shown in Fig. 1(c) and (d). † Using Eqs. (17) and (14) and the equation of motion for ˆci cˆj t , the expectation value of the effective operator is given by †

ˆ ω = M eff

ij

where Vjiext

Fig. 1. (a) Schematic illustration of second order inelastic light scattering mechanism where in the ﬁrst step n photons promote N electrons to empty states in the conduction band (CB). In the second step N electrons from the Fermi sea recombines with the holes left in the valence band (VB) emitting radiation. (b) Schematic illustration of a single quantum well conduction and valence band potential proﬁle and its corresponding energy levels. (c), (d) Schematic diagram showing the elementary excitations of the electron gas in the conduction band in (c) near resonance regime exhibiting the collective plasmon mode and in (d) extreme resonance regime exhibiting collective and single-particle excitations.

∗

2Nji ωji 2 (ω2 − ωji2 + iji ω)

eff

Vji .

(19)

j>i

[P 2 ] j|eikz z |j j |eikz z |i = cv , 3m0 Eres − ωL + i

j

and Eresonant = Eg + j + j + 2 k2 /2∗ .

[Vjiext ]

(18)

In order to solve Eq. (19) a harmonic coordinate xji deﬁned in Eq. (8) will be used. To understand the nature of the SPE, the induced eff eff potential will be replaced via transformation Vji → Vji + Vjiext

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¯ and Eq. (19) rewritten as, − Vjiext

†

ˆ ω = M eff

⎡ ⎣

ij

¯ |2 − |V ext ¯ |2 2Nji ωji |Vjiext ji

2 (ω2 − ωji2 + iji ω)

j>i

+

∗

¯ ] x 2Nji ωji [Vjiext ji

.

(20)

This allows to obtain the cross sections in Eq. (11). The ﬁrst term on the right hand of Eq. (20) represents unrenormalized collective modes because it has poles on the bare frequencies transitions. Therefore, it is responsible by singleparticle excitations. The second term on the right hand of Eq. (20) represents renormalized transitions because it has poles in the eigenfrequencies of the coupled system. Therefore, it is responsible by collective excitations in the Raman spectra (CDE or SDE). Note that the numerator in Eq. (20) is expressed as a variance. Hence, the origin of SPE is revealed as the result of a deviation from the mean value of the external potential. Only in the extreme reso¯ |2 − |V ext ¯ |2 = nance regime, |V ext / 0. On the other hand, in the near ji

ji

¯ | |Vjiext

|Vjiext |

= and only renormalized collective resonance regime excitations are present. In the condition of near resonance all oscillator strengths are transferred to the collective part and, therefore, the single-particle features are completely screened [8]. The developments presented in this article provide a better understanding about elementary excitations of the electron gas in semiconductor nanostructures and furnish in a more clear manner the essence of single-particle excitations. 3. Results and discussion In order to have a resonance regime the incoming laser energy should match an optical gap of the semiconductor. When the laser matches interband transitions energy of the material (transitions involving valence and conduction states plus gap), we have an extreme resonance regime. The gap to be used can be fundamental or split-off gap. The fundamental gap brings an additional complicating factor, i.e., problems involving intrasubband processes in the valence band related for example to heavy hole-light hole transitions. This is described by third order Raman processes in perturbation theory. On the other hand, in the split-off gap the Raman scattering is described by second order processes. These facts were explored in Ref. [11]. As stated in this reference, it does not matter if the laser energies are resonant with the fundamental or split-off gap, the Raman proﬁles are the same. In fact, the difference between second and third order processes are revealed only by Raman efﬁciency measurements (intensity of a Raman peak in function of the incoming or outgoing laser light). Also notice that we do not take into account intraband transitions. All the parameters used here were taken from [21]. Due to the discussion above, the incoming laser energy used is resonant with the split-off gap of GaAs. This is not the case in Ref. [1] where the laser energy is resonant with the fundamental gap of GaAs. However, the inferences derivated by this article are still valid. Another point is that we found better agreement of the electronic transitions using a 17 nm wide single quantum well instead of 18 nm as given in Ref. [1]. This difference can be induced by the limited information of growth parameters of the samples or because we did not take into account the coupling between the valence and electronic conduction band states. Therefore, we employ a 17 nm quantum well, which produces a resonant Raman

Fig. 2. Calculated polarized (CDE) and depolarized (SDE) Raman spectra and singleparticle excitations of a L = 17 nm wide GaAs/AlAs single quantum well, total electronic density Ns = 6.3 × 1011 cm−2 and incident laser energy Einc = 1880 meV.

proﬁle more similar with the experimental one. With the 10 nm quantum well we did not observe this discrepancy in energy. The reason of this may be that the states are more separated making the coupling between them weak. In Fig. 2 the calculated polarized and depolarized Raman spectra are presented for a 17 nm wide single quantum well with total electronic density of 6.3 × 1011 cm−2 . The CDE and SDE of the electron gas were found in 57.1 meV and 45.5 meV, respectively. The unrenormalized transitions (SPE) were found in 48.4 meV. In a later work [22], for the same single quantum well, the authors also showed one collective mode for the polarized Raman spectra around 35 meV. These CDE can be interpreted from the point of view of the coupling between plasmons and phonons. There is a longitudinal macroscopic electric ﬁeld associated with the charge density ﬂuctuations (plasmons). Such ﬁeld couples with the LO phonon of the GaAs quantum well at 36.6 meV. This coupling effect is taken into account through the dielectric function εl (ω) in Eq. (5) [11]. On the other hand, if this coupling is not considered then the dielectric function will be constant, which will give rise to two plasmon modes in 54.9 meV and 135 meV (not shown). A full plasmon-phonon dependence in εl (ω) in our case provides two coupled CDE-phonon modes (not shown) in 35 meV and 57.1 meV as well as 36.5 meV and 135.5 meV. They are phononlike and plasmon-like modes, respectively. The 36.5 meV mode is very weak because the CDE energy is much greater than LO phonon energy, which results in a very weak coupling that prevents its experimental observation. We must also take into consideration the large background due to strong luminescence caused when the laser pump matches the fundamental gap of the GaAs as used by the authors in [1]. Figs. 3 and 4 show the extreme case where for a 10 nm wide single quantum well and total electronic density of 1.2 × 1012 cm−2 the collective excitations (CDE and SDE) become small and only SPE prevail in the Raman spectra. In our calculations we have found one CDE collective mode in 144.1 meV and SDE one in 131.5 meV. In Ref. [1], the authors did not observe these collective modes via Raman spectroscopy. However, in a later work [22], for the same 10 nm quantum well, one mode was found in 142.2 meV and it was interpreted as a CDE by intersubband infrared absorption measurements. This result differs by 1.9 meV from our calculations as

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Fig. 3. (a) Calculated polarized (CDE) and depolarized (SDE) Raman spectra and single-particle excitations of a L = 10 nm wide GaAs/AlAs single quantum well, total electronic density Ns = 1.2 × 1012 cm−2 and incident laser energy Einc = 2000 meV. (b) The same spectra as shown in (a) with a damping factor consistent with the Raman experimental widths in Ref. [1].

shown in Fig. 3(a). The SPE are found in 134.0 meV by [1]. In Fig. 3, this single-particle transition is located in 135.6 meV. In Fig. 3(b) a phenomenological damping factor was inserted. With this line width, collective excitations are overlapped by the broader SPE peak. Such behavior is followed by the experimental data based on Ref. [1].

Fig. 4 shows the calculated polarized and depolarized Raman spectra for the 10 nm quantum well in different resonance conditions. In such situation the Raman spectra have an asymmetric behavior. In Ref. [1], the authors attributed the observed asymmetry of the spectra to a possible relaxation of the in-plane wave-vector conservation rule. On the contrary, the results

Fig. 4. (a) Calculated polarized and depolarized Raman spectra and single-particle excitations of a L = 10 nm wide GaAs/AlAs single quantum well and total electronic density Ns = 1.2 × 1012 cm−2 for different incident laser energies Einc .

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reported here indicate that varying the laser energy regardless the existence of non-vertical intersubband transitions (q = / 0) will produce theoretical Raman lines which exhibit such asymmetries. They are originated by the entrance or exit of resonance of the collective modes overlapped by SPE which oscillator strengths affect minimally the spectral proﬁles as shown in Fig. 4(a) and (b). The shifts from SPE peak occur by many-body Coulomb effects. In general, SDE are shifted to lower energies from SPE due to exchange-correlation effects (redshifted) and CDE are shifted to higher energies from SPE due to direct as well as exchangecorrelation effects (blueshifted). They are known as excitonic shift in the SDE case (ˇ) and depolarization shift in the CDE case (˛). For the 17 nm quantum well (Fig. 2), ˛ = 8.7 meV and ˇ = 2.9 meV. The value of ˛ differs by 0.7 meV and ˇ differs by 0.3 meV from experimental Raman values found in Ref. [1] and corroborate with the theoretical values in Ref. [22]. For the 10 nm quantum well (Fig. 3), ˛ = 8.5 meV and ˇ = 4.1 meV. The value ˛ differs by 0.5 meV from experimental values and ˇ corroborates with the theoretical value in Ref. [22]. Now, calculations of the polarized and depolarized Raman cross sections are compared with experimental data obtained from combination between intersubband infrared absorption [22] and inelastic light scattering measures found in Ref. [1]. 4. Conclusion In summary, this article has investigated the resonant electronic Raman scattering of narrow quantum wells. As the quantum well width is reduced, single-particle excitations become more important than collectives ones in Raman spectra. This is the case for a 10 nm n-doped GaAs quantum well where elementary excitations of the electron gas exhibit a distinctive Raman proﬁle such that only single-particle excitations were observed in the Raman spectra. In addition, the spectra show an asymmetric behavior when changing the incident laser energy. Our results show that the asymmetries can be interpreted as the entrance or exit of resonance of the collective modes merged in single-particle transitions. These results provide a different interpretation from the experimental ones presented by Unuma et al. [1] which consider these asymmetries as relaxation of the wave-vector conservation rule. Our results reinforce the idea that single-particle excitations in Raman spectra of electron gas of low dimensional semiconductors structures have their origin well-grounded under three conditions: extreme resonance regime with interband transitions, existence of degenerate intersubband excitations and degenerate interactions between pairs of excitations (i.e., Coulomb and/or exchange-correlation). From the above considerations, the Raman spectra of quantized electron gas in the extreme resonance regime are basically constituted by a set of renormalized excitations, i.e., charge or spin density excitations and a non-renormalized one with respect to the bare electronic transitions called single-particle excitations.

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