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aPhys. Status Solidi C 8, No. 9, 2589–2592 (2011) / DOI 10.1002/pssc.201084067

Collective modes of a bilayer double parabolic quantum well spin polarized electron gas Colin Campbell and Ionel Tifrea*

Department of Physics and Astronomy, California State University, Fullerton, CA 92834, USA

Received 3 October 2010, accepted 4 February 2011 Published online 6 May 2011

Keywords collective excitations, bilayer system, spin polarized electron gas

* Corresponding author: e-mail: itifrea@fullerton.edu, Phone:+1-657-2782769, Fax: +1-657-2782555

We investigate the possible electromagnetically induced collective excitation modes in a bilayer quasi-two-dimensional structure build from two adjacent parabolic quantum wells (PQW). We consider both charge and spin excitation modes within and beyond the random phase approximation method. To account for higher order ef-fects we used exchange and correlation spin-dependent

local field correction factors. We analyzed intra- and in-ter-subband excitations and provide analytic results for their excitation frequencies in the long wavelength limit. In particular, we analyze the spin polarization depend-ence of the lowest frequency spin density mode and show that its relevant frequency is in the hundred GHz region.

© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction During the last several decades, the study of elementary excitations in three (3D) and two (2D) electronic systems has received a lot of interest from the physics community both theoretically and experimentally [1]. Among different types of excitations, collective modes have often been an important probe for the system's micro-scopic properties, in particular they provided valuable in-formation on the electron-electron interaction. The specific frequencies of the collective excitations can be extracted from the poles of the system's response function to an ex-ternal electromagnetic perturbation. In general, in a spin polarized electron gas one can identify two different types of collective excitations, i.e., a charge and a spin density mode [2]. The behaviour of these modes is well known for 2D and 3D electron systems. The situation becomes more interesting when bilayer systems are considered. In such systems, two quasi-2D layers situated at a given distance d, permit electron-electron interaction both inside the same layer (intra-layer interaction) and between the two compo-nent layers (inter-layer interaction). The resulting collec-tive excitations are connected to the in-phase and out-of-phase oscillations of the electron systems in each of the two layers [3]. Accordingly, for such a system one can identify four possible collective modes, two associated with charge density oscillations and the other two with spin density oscillations. The particularities of the quasi-2D layers are important, as the electron system in each of the

layer is confined along one direction [4]. Another impor-tant parameter in the problem is the distance between the component layers. It can be proved, that the two layers should be separated by a minimum critical distance dc in order for the collective excitations to exist [3].

Here, we present an estimation of the possible longitu-dinal collective excitations in a double parabolic quantum well (PQW) layer quasi-2D electron gas. Within each of the component layers, the electron gas is subject to a con-finement potential V (z) = mω 2z 2 /2 , m being the effective electron mass and ω the characteristic frequency of the confinement potential. This type of confinement can be experimentally realized in AlxGa1-xAs systems in which the percentage of Al varies from one position to the other in-side the well [5]. Samples can be easily prepared using the standard MBE growth technique. We consider the elec-tronic system to be spin polarized, i.e., the number of elec-tronic spins oriented along one direction exceeds the num-ber of electron spins oriented in the opposite direction. To characterize this situation we introduce a new parameter ξ = (n↑ − n↓ ) /(n↑ + n↓ ) . In principle, any degree of spin polarization can be obtained by varying appropriate physi-cal parameters −1≤ξ ≤ 1. There are different ways to po-larize the electron population:, for example, one can apply an external magnetic field, or use polarized light to excite only one type of electron spins. We found that there are four characteristic frequencies for the collective excitations,

2590 C. Campbell and I. Tifrea: Collective modes of a bilayer double parabolic quantum well spin polarized electron gas

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two being associated to charge oscillations and other two with spin oscillations. Between these four modes, the low-est frequency one, and thereafter the one with the higher possibility to be excited, is the out-of-phase spin density mode.

2 The Model The effective potential experienced by a

single particle of spin σ in the α-subband of the i-th layer includes the direct interaction with the external electro-magnetic perturbation, represented by its Fourier trans-forms, ( , )qj w and ( , )b q w , the Coulomb interaction term, modified by the exchange and correlation effects, and the self-consistent magnetization. Assuming implicit q and ω dependence of all variables, the effective single-particle potential is:

[{ ]

[ ]},)1()()(4

)1()()(

)()()(2

)()(

)(

jji

ii

iI

sGnGqFqvsL

sGnGqDqvebH

αβσαβσαβαβ

αβσβ

αβσαβ

σ

σπγ

ϕσγ

Δ−Δ−+Δ⋅+

Δ−Δ−+

+−⋅=

−+

−+∑ (1)

where Δnαβ(i) and )(isαβΔ are the charge and spin density fluc-

tuations in the i-th layer caused by electron transitions be-tween minibands α and β under the effect of the perturba-tion. Eq. (1) includes the effectof the charge density fluc-tuations in both layers )(inαβΔ (i=1,2) and that of the spin density fluctuations )(isαβΔ in the same layer. The ex-change and correlation effects are included within the two factors ( , )G qs w

±

. The subband structure of the energy spectrum in each well modifies the value of the Fourier transform of the bare Coulomb interaction in 2D, v(q) = 2πe2 /εq (ε is the dielectric constant) by a form fac-tor. The general expression of the form factor is: Fiα; jβ

i'α'; j'β ' (q) =

dzdz' Zαi (z)Zβ

j (z' )e−q z−z' Zβ 'j' (z' )

−∞

∞

∫−∞

∞

∫ Zα'i' (z),

(2)

where Zαi (z) = Nα exp[−z 2 /(2z0

2)]Hα−1(z / z0) is the enve-lope function part of the total electron wave function in a PQW for subband α in the i-th layer, and it is related to the geometrical confinement provided by the PQW. In the above equation, Nα is the standard normalization factor for the electron wave factor in the α-subband of the PQW and )/( 00 ωemz = is related to the specific form of the PQW confinement potential, V (z) = mω 2z 2 /2 . The general form factor, )('';''

; qF jiji

βαβα , is related to the wave

functions overlap in the two component layers of the sys-tem. We introduce Dαβ (q) = Fiα; jβ

iα; jβ (q) for the intra-layer form factor (interactions between electrons in the same layer) and Fαβ (q) = Fiα; jβ

iα; jβ (q) for the inter-layer form fac-tor (interactions between electrons in different layers). The inclusion of the short range Coulomb interaction ef-fects is done by considering the deviations of the electron

density from mean field values, and it can be described us-ing local field corrections. The exact frequency and wavevector dependence of the local field factors,

( , )G qs w± is unknown. These factors, are a generalization of the factors introduce by Kukkonen and Overhauser, to include also spin related effects [6]. Within the linear re-sponse approximation, the charge and spin fluctuations are proportional to the effective potentials experienced by the electron population. The longitudinal fluctuations are given by ),(),(),( )()(

, ωωω σαβσσσαβ qVqq ii Π=Δ , ),( ωαβ

σσ qΠ being the appropriate polarization function. The poles of the response function, reveal the existence of four collec-tive excitations, corresponding to in-phase and out-of-phase charge and spin fluctuations in the two layers. Their characteristic frequencies are given by (assuming implicit q and ω dependence)

Ω12 (q) =

A ω↑2 +ω↓

2( )+ f (q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ]−2ω↑

2ω↓2 f (q)

×A 2−G↑

+ −G↓+( )− f (q) GL,↑

− 1−G↓+( )+ GL,↓

− 1−G↑+( )[ ]

A ω↑2 +ω↓

2( )+ f (q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ] (3) Ω2

2 (q) =

A ω↑2 +ω↓

2( )+ g(q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ]−2ω↑

2ω↓2g(q)

×A 2−G↑

+ −G↓+( )− g(q) GL,↑

− 1−G↓+( )+ GL,↓

− 1−G↑+( )[ ]

A ω↑2 +ω↓

2( )+ g(q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ] (4) Ω3

2 (q) = 2ω↑2ω↓

2 f (q)

×A 2−G↑

+ −G↓+( )− f (q) GL,↑

− 1−G↓+( )+ GL,↓

− 1−G↑+( )[ ]

A ω↑2 +ω↓

2( )+ f (q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ] (5) Ω4

2 (q) = 2ω↑2ω↓

2g(q)

×A 2−G↑

+ −G↓+( )− g(q) GL,↑

− 1−G↓+( )+ GL,↓

− 1−G↑+( )[ ]

A ω↑2 +ω↓

2( )+ g(q) ω↑2 1−G↑

+ −GL,↑−( )+ω↓

2 1−G↓+ −GL,↓

−( )[ ] (6) where ωσ

2 = 2πe2n(1−ξ sgnσ ) /εm* is the spin dependent plasma frequency in each layer (for simplicity we consider the effective mass m* to be spin independent),

11 11( ) ( ) ( ),f q D q F q= + 11 11( ) ( ) ( ),g q D q F q= - and A = (2εγ 2 /Le2)q . In the long wavelength limit, ω ≥ qvF ,

Phys. Status Solidi C 8, No. 9 (2011) 2591

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Ω1 describes the classic propagation of a 2D plasmon Ω1 ~ q , while Ω2 corresponds to the acoustic plasmon associated to the out-of-phase charge oscillations. Coupled to the charge density modes there are also two possible longitudinal spin density modes, describing the in-phase and out-of-phase magnetization fluctuations in the two layers. As one can see from the characteristic frequencies Ω3 and Ω4 , the existence of the spin density modes is conditioned by the presence of an intrinsic spin imbalance in the system A ≠ 0 or by the presence of the local field correction factors. The existence of these modes in the ab-sence of the induced magnetization A = 0 is possible only if the local field correction factors satisfy the inequality

0),(, <− ωσ qGL , or, in terms of the exchange and correla-tion local field correction factors,

),(),( ,, ωω σσ qGqG cL

xL < . In the long wavelength limit,

the in-phase oscillation, Ω3, is a regular 2D magnetoplas-mon, while the out-of-phase one, Ω4 , is an acoustic mag-netic excitation. This latter mode, due to the fact that it has the lowest characteristic frequency, should be the most im-portant one in a possible de-excitation process. In the following we will focus on the existence of this mode within the random phase approxi-mation (RPA). 3 Results and discussion All four collective mode fre-quencies depend on the inter- and intra-layer form factors. The inter-layer form factor Fαβ (q) can be calculated ana-lytically as

Fαβ (q) =

2α+β −2π (α −1)!(β −1)!Nα2Nβ

2 z02 exp −qd +

q2z02

2

⎡

⎣ ⎢

⎤

⎦ ⎥

×Lα−10 −

q2z02

2

⎛

⎝ ⎜

⎞

⎠ ⎟ Lβ −1

0 −q2z0

2

2

⎛

⎝ ⎜

⎞

⎠ ⎟

(7)

where Lnp (x) represents the Laguerre polynomial. On the

other hand, the intra-layer form factor Dαβ (q) has to be es-timated numerically. Figures 1 and 2 present the intra and inter-layer form factors for the case of first subband elec-tronic occupation in the two PQW's of the bilayer structure. The complexity of these factors arises from the natural confinement provided by the two PQW's. The intra-layer form factor is closed to unity as we expected due to the strong overlap of the electronic wave functions inside the same PQW well. On the other hand, for the inter-layer form factor the situation changes drastically, larger the dis-tance between the two component PQW's smaller the over-lap of the electronic wave functions. As a consequence the inter-layer form factor F11(q) decrease fast once the elec-tronic wave vector increases. This effect is drastic at large distances between the component wells.

Figure 1 The intra-layer form factor D11(q) as function of the wave vector q.

Figure 2 The inter-layer form factor F11(q) as function of the wave vector q for various distances between the two PQW's.

Figure 3 The frequency of the out-of-phase spin density mode as function of the electronic wave vector for different values of the electronic spin polarization ξ .

2592 C. Campbell and I. Tifrea: Collective modes of a bilayer double parabolic quantum well spin polarized electron gas

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The collective modes characteristic frequencies were obtained in an approximation that accounts for spin de-pendent local field correction factors. Unfortunately, these correction factors are difficult to estimate and their form is only known in some special limits, namely for q→ 0 or q→∞ . To get an idea about the wave vector dependence of the collective mode frequencies we can perform a lower order approximation that do not account for the correction factors, namely the RPA approximation. Within this ap-proximation we can simply evaluate the characteristic fre-quencies and analyze their position in the electromagnetic spectrum by setting 0),( =± ωσ qG . The lowest frequency, and thereafter the easiest one to observe experimentally, is obtained in the case of the out-of-phase spin density mode, Ω4 (q) . Figure 3 presents the frequency of this mode as function of the electron wave vector for different spin po-larizations inside the electronic system. The induced spin polarization in the electronic system can be determinant for frequency range attached to the collective excitation spin density mode.

In conclusion, we presented a theoretical investigation of the possible collective modes in a bilayer double-PQW system. We found out that four possible modes are present, two charge and two spin density modes. In both situations, the modes can be attributed to in-phase and out-of-phase oscillations inside each of the component layer of the sys-tem. The in-phase modes resemble collective modes char-acteristic to pure two-dimensional electronic systems, al-though the out-of-phase modes are characteristic only to bilayer systems. The lowest frequency mode, and thereaf-ter the more important one in de-excitation processes, is the out-of-phase spin density mode. We presented a de-tailed analysis of the wave vector dependence of this mode for various spin polarization of the electronic system. In the low wave vector limit q → 0, the mode is acoustic (lin-ear dependence of the frequency on the wave vector), however, at higher wave vectors the dependence becomes more complicated. The analysis was performed within the random phase approximation. The characteristic frequency of this mode is in the high GHz - low THz range.

Acknowledgements The authors will like to acknowledge financial support from the CSUF Intramural Grant Program.

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