Two-dimensional plasmons in homogeneous and laterally microstructured space charge layers

332

Surface Science 170 (1986) 332 345 North-Holland, Amsterdam

T W O - D I M E N S I O N A L P L A S M O N S IN H O M O G E N E O U S AND LATERALLY M I C R O S T R U C T U R E D SPACE CHARGE LAYERS Detlef H E I T M A N N lnstitut fftr Angewandte Physik, Universitht Hamburg, D-2000 Hamburg 36, Fed. Rep. of Germany Received 17 July 1985; accepted for publication 13 September 1985

Theories and experiments on plasmons in homogeneous two-dimensional (2D) systems will be reviewed. In detail we will discuss the plasmon excitation in laterally microstructured systems. Metal-oxide-semiconductor devices and A 1 G a A s - G a A s heterojunctions have been fabricated where the originally 2D charge density is spatially modulated. For periodicities of the modulation in the submicron range superlattice effects on the plasmon dispersion are observed. For plasmons in GaAs heterojunctions also nonlocal effects on the plasmon dispersion are discussed.

1. Introduction

The study of plasmons, the resonant collective excitations of an electron or hole plasma, allows a detailed investigation of many different electronic interactions. Volume plasmons in the bulk have been treated both theoretically and experimentally in great detail (e.g. refs. [1,2]). Surface plasmons, that exist at boundaries of semi-infinite plasma and related polariton type excitations, are widely studied, recently in particular in connection with the Giant Raman effect and the localization of plasmons (e.g. ref. [3]). Several monographs are devoted to this topic (e.g. refs. [4-6]). The dispersion of two-dimensional (2D) plasmons, which exist in electronic systems where the carriers are confined in very narrow potential wells, has first been derived by Ritchie [7] for the plasmon resonance in the limit of a very thin slab. 2D plasmon resonances and screening of a 2D electron gas, including nonlocal effects, have been calculated by Stern [8]. Chaplik [9] calculated the dispersion of 2D plasmons in the metal-oxide-semiconductor (MOS) system, one of the most prominent 2D systems [10]. The 2D plasmon frequency is [8,9] 2

COp- 2e(

N~e 2 q, CO)rompq"

(1)

Here N~ is the 2D charge density, q the plasmon wave vector, mp the plasmon mass which is in an isotropic parabolic system identical with the effective band 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

D. Heitmann / 2D plasmons in space charge layers

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structure mass, and e(q, ~0) an effective dielectric function which takes into account the geometry of the system. For the MOS configuration, consisting of a semiconductor with dielectric function e~, an insulator with dielectric function eo× and thickness dox and a perfectly screening gate it is [9]:

e(q,w) = ' ~( e se +eox coth qdox).

(2)

In (1) retardation effects and higher order corrections in q (see below) are neglected ( w / c << q << k v = Fermi vector). The interest in studying the plasmon excitation in 2D systems arises in particular from the possibility that the charge density - the most important quantity that governs the plasmon resonance - can be varied in these systems over several orders of magnitudes. With the charge density, many related physical properties, e.g., the Fermi energy, the Fermi vector, the mean electron distance, etc., can be varied. This allows a detailed investigation of different mechanisms that determine the plasmon resonance itself and interactions of plasmons with different types of excitations. Another beautiful feature of 2D systems - in particular of the MOS system and of GaAs heterostructures - is that the dimensionality of the electronic system can be further reduced by the use of an improved submicron technology. Electronic systems have been fabricated with novel physical properties, novel in the sense that these properties cannot be explained by scaling down large devices. Different systems have been realized: On the one hand single, very narrow channels showing one-dimensional behavior have been prepared and investigated [11-13]. Another approach to novel systems is, on the other hand, the fabrication of large area devices with lateral periodic structures of submicron periodicity. In this way electronic systems with spatially modulated charge density have been prepared. The plasmon excitation in these periodically charge density modulated systems [14-17] will be the main topic of this review. We will discuss MOS and A 1 G a A s - G a A s systems. For the latter we will also show that the dynamic charge density modulation of the plasmon is a "microstructure" by itself resulting in the observation of nonlocai effects. We will precede this topic by a brief summary of experiments and theories for 2D plasmons in homogeneous systems.

2. 2D plasmons in homogeneous systems 2D plasmons have been observed first by Grimes and Adam for electrons on the surface of liquid helium [18]. For the Si (100) MOS system 2D plasmons have been investigated by Allen, Tsui and Logan [19] and by Theis, Kotthaus and Stiles [20,21], using far-infrared (FIR) transmission spectroscopy. In ref. [20] the 2D magnetoplasmon dispersion [9] has been verified. F I R emission of 2D plasmon has been observed by Tsui, Gornik and Logan [22] and the

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D. Heitmann / 2D plasmons in space charge layers

influence of uniaxial stress on the surface band structure of Si (100) has been characterized with 2D plasmon experiments by Englert, Tsui and Logan [23]. Recently extended studies on the 2D plasmon dispersion for electrons on Si (100), (110) and (111) have been performed [24-27]. These experiments show that there are characteristic deviations of the experimental plasmon dispersion in comparison with the classical dispersion of refs. [8] and [9]. It is convenient to discuss these deviations in terms of an effective plasmon mass mp (a normalized reciprocal squared plasmon frequency):

1 N~e2q(1-(2%r) -2) m* = - o 0a~

2e(q, 0a)e 0

'

(3)

where oar is the experimental plasmon resonance position and (2~0rr) 2 takes account of the shift of the plasmon resonance due to the damping of the plasmon resonance in a real system, r is related to the halfwidth of the experimental resonance profile. From the experiments one extracts for all three surface orientations an increase of mp with decreasing charge density N~ [26], as is shown for Si (100) and Si ( 1 l l ) in fig. 1. (For Si (100) this has also been observed in ref. [19].) The onset of this mass enhancement occurs at different charge densities for the different surface orientations. On Si (100) also an increase of mp with increasing N, beyond a certain charge density has been observed, which cannot be explained only by the occupation of the E 0 subband sytem (see ref. [10]). As shown in fig. 1, an increase of m~ is observed also for increasing wave vectors q. Different possible mechanisms that might cause these deviations of the experimental plasmon dispersion are discussed in refs. [25] and [26], and also, stimulated by the observations, in recent theoretical publications [28-30] which will be discussed below. 2D plasmons have also been investigated in other systems. Many, Wagner Rosenthal, Gerstens, Goldstein, and Kirby studied the 2D plasmon excitation with electron loss spectroscopy in ZnO [31,32], a system which is characterized by the highest charge densities that have been realized in 2D systems (up to 2 x 1014 c m 2). 2D plasmons in A 1 G a A s - G a A s heterostructures have been observed by Raman spectroscopy [33], F I R emission [34] and F I R transmission spectroscopy [16]. For the electron system on liquid He mentioned above [18,35], recently also effects of a geometrical confinement on 2D plasmons due to edge effects have been reported [36]. 2D plasmons have also been observed in hole systems, in particular in hole inversion layers of Si (110) [37,38]. The experimental plasmon dispersion reflects here the strong nonparabolicity and the anisotropy of the hole surface band structure of Si (110). There are a large number of theoretical publications on different aspects of the 2D plasmon resonance. Thus, the references cited in the following are a somewhat arbitrary selection of some older and some recent publications. Nonlocal and many-body corrections for the 2D plasmon dispersion, which

D. Heitmann / 2D plasmons in space charge layers

mp mo

335

B'\ "'t~.

q=1.3 105cm -1

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q=2 05cm-, ..... +..... ÷-

(

'~,

i .o.

_

.-%" .- o-

si(lOO)

h~o"o-o,,_~.~ _ .o ~ "P:~'-- q =1 3105cm -1 0 2ul--

0.190

"

--

,

0

5 Ns (1012cm -2)

10

Fig. 1. Experimental effective plasmon mass extracted from plasmon resonance experiments on Si (100) and Si (111) with homogeneous charge density N~ [24-26]. Broken lines indicate the plasmon mass that is derived from the surface band structure. O and Lxare from measurements on different Si(100) samples. Open and full symbols stand for different plasmon wave vectors, q = 1.3 x 105 cm i and q = 2 . 6 x 1 0 5 c m - I respectively.

b e c o m e i m p o r t a n t If the p l a s m o n wave vector a p p r o a c h e s the F e r m i wave vector k F, are treated in refs. [8, 39-41]. Effects of the finite spatial extent of the 2 D wave functions n o r m a l to the interface are c o n s i d e r e d in refs. [42] a n d [43]. F o r the c o n d i t i o n s in e x p e r i m e n t s so far [19-26], the effects of n o n l o c a l ity, m a n y - b o d y corrections a n d finite thickness are small (a few percent) a n d c a n n o t be e x t r a c t e d from the e x p e r i m e n t a l r e s o n a n c e p o s i t i o n s within the accuracy of d e t e r m i n i n g all o t h e r parameters. In particular, the effects are too small to be the o n l y origin of the e x p e r i m e n t a l l y o b s e r v e d e n h a n c e m e n t of the effective p l a s m o n mass discussed a b o v e for Si (fig. 1). In a real M O S system r a n d o m p o t e n t i a l fluctuations - the same m e c h a n i s m that causes C o u l o m b a n d surface roughness scattering - couple p l a s m o n s of small wave vectors ( q << k F ) to the e l e c t r o n - h o l e c o n t i n u u m a n d to p l a s m o n s with different directions of the wave vector. G o l d [30] p e r f o r m e d calculations of the p l a s m o n d i s p e r s i o n t a k i n g into account b o t h scattering m e c h a n i s m s within a self-consisting m e m o r y function a p p r o a c h . These calculations can e x p l a i n m a n y features o f the e x p e r i m e n t a l l y o b s e r v e d p l a s m o n d i s p e r s i o n for Si (100). T h e energetical lowering of the p l a s m o n d i s p e r s i o n at low d e n s i t y is

D. Heitmann / 2D plasmons in space charge layers

336

attributed to reactive processes, the lowering of the plasmon frequency at high densities pre-dominantly to dissipative processes via roughness scattering. The q-dependence of the effective mass [25] and the results on Si (110) and Si (111) can so far not be explained by the same theory in a satisfying way. Experiments and an exact characterization of the scatterers is necessary to prove all details of this theory. A large number of theoretical publications is devoted to the magnetoplasmon excitation (e.g. refs. [44-50]), coupling of 2D plasmons to surface plasmons [51,52], piezoelectric waves [53], acoustical [54] and optical phonons [55]. (For reviews on coupling processes, see e.g. refs. [56,67].) Plasmon excitations in a multicomponent plasma, in single-heterojunctions and multi-quantum well systems with infinite or finite number of 2D layers and for many different configurations have been calculated (e.g. refs. [43, 58-68]). Also acoustical plasmons in multicomponent systems which have carriers with different charges or masses have been discussed [69,70]. Review articles on 2D plasmons are published, e.g., by Theis [71] covering results for Si (100) up to 1980, by HtSpfel and Gornik [72], with emphasis on F I R emission [73] and the specific heat of the 2D electron system [74,75], and by Chaplik [56,57]. 2D plasmons are also included in review articles on 2D systems (e.g. refs. [10, 76-78]). The electrodynamics of 2D systems are treated in refs. [79] and [80].

3. Preparation of "2D" systems with spatially modulated charge density Fig. 2 shows two microstructured MOS configurations in which a spatially modulated charge density can be induced [14-16]. In fig. 2a the thickness of the oxide is d Z in the region t 1 and is - except for the small region of the slopes - d 2 for the rest t 2 = ~ a - tl of the period a. These structures are prepared by first fabricating a periodical surface profile in a photoresist layer on top of the homogeneous oxide by holographic lithography of two superimposed coherent laser beams. This profile acts as a mask in a dry etching process. A continuous layer of NiCr (3 nm) is evaporated under varying angles onto the structures oxide. If a voltage V~ is applied between this gate and the substrate, an electron gas with a low charge density N~2 in the region t 2 and a high charge density N~ in the region t~ is induced, Nsi = e°xe° (Vg - Vti),

edox

Vti = threshold voltage, i = 1, 2.

Periodicities from 250 to 2000 nm have been prepared, d~ and d2 are, respectively, typically 20 and 50 nm. The configuration in fig. 2b consists of a Si substrate with 20 nm thermally grown oxide. On top of the oxide a photoresist, modulated periodically in

D. Heitmann / 2D plasmons in space charge layers

337

(a) I It I itlt 2 1 dl ~.

~

(b)

, , , NiCr

Si02

/At SiO2

' Fig. 2. Sketch of MOS systems with modulated insulator thickness. In (a) the thickness of the SiO 2 is periodically modulated. In (b) a homogeneous SiO 2 layer (typically 20 nm in thickness) is covered with an insulating modulated photoresist PR. Both configuration are continuously covered with a thin NiCr film of low conductivity, which is semitransparent for FIR radiation. For both configurations also samples with additional highly conducting AI stripes are prepared, as shown in

{b).

thickness, is prepared and covered continuously with a thin NiCr layer. This photoresist is found to act like an ideal insulator with a high breakthrough voltage and without hysteresis effect due to mobile ions. The thin thermally grown oxide ensures a low number of surface states at the Si-SiO 2 interface. This technique has the advantage that high aspect ratios d 2 / d 1 can be achieved. For both configurations, also structures have been fabricated where additional highly conducting AI stripes are evaporated in a shadowing process onto the surface relief, as shown schematically in fig. 2b. These stripes are asymmetric with respect to the charge density modulation. With a high quality holographic set-up, devices with large area gates (up to 8 mm in diameter) have been fabricated, where the ratios d l / d 2 and t J t 2 are perfectly constant over the whole area. This makes efficient FIR transmission spectroscopy possible.

4. Experimental techniques Whereas bulk plasmons have purely longitudinal and only electric field components, 2D plasmons have longitudinal and transversal field components and magnetic field components as well. They are directly coupled to electromagnetic fields and are, in this respect, more similar to surface plasmons at the boundaries of semi-infinite plasmas. However, since the wave vector of 2D plasmons is always larger than to~c, they are not coupled with freely propagating waves. Thus grating couplers are needed to observe 2D plasmons in FIR

D. Heitmann / 2D plasmons in space charge layers

338

spectroscopy. Gates consisting of periodical stripes of alternating high and low conductivity have been used to couple F I R radiation with plasmons [19,20]. In principle, any periodical modulation of the optical properties in the vicinity of the 2D electron plasma, in particular the modulated charge density itself, and the modulated oxide or photoresist, will spatially modulate the normally incident radiation. This induces in the near field electromagnetic field components of wave vector q, = n(2~'/a), n = 1, 2 . . . . . which are coupled with the plasmon excitation. In the experiments, excitation of plasmons is detected in the relative change of transmission for F I R radiation that is transmitted with normal angle of incidence through the samples with semitransparent gates a T _ _ T(V~) - T ( q ) T

T(Vt)

cc - R e ( a ( c o , q)),

(3)

T(Vg) and T(Vt) are, respectively, the transmissions at gate voltage V~ and conductivity threshold voltage Vt. In the small signal approximation (e.g. ref. [76]), A T / T is proportional to the real part of the dynamic 2D conductivity 0(60, q), which includes, due to the grating coupler, also contributions for wave vectors q, =n(2~r/a). The experiments discussed in the following are performed using Fourier transform spectroscopy at low temperatures T < 10 K. Details of the experimental techniques are included in ref. [81].

5. Plasmon excitation in Si MOS systems with spatially modulated charge density In fig. 3 F I R transmission spectra are shown which have been measured on Si (100) samples of the configuration sketched in fig. 2b. Superimposed on the Drude background, well pronounced plasmon resonances are observed which shift with increasing gate voltage and corresponding charge densities to higher wave numbers, roughly as is expected from the plasmon dispersion (1). Resonances co~ and co2 are indicated which are excited, respectively, via n = 1 and n = 2 reciprocal grating vectors qn = n (2~r/a). Characteristic for these charge density modulated systems is the fact that the plasmon excitations are split into two resonances, co,_ and con+. The splitting is smaller for n = 2. To discuss the origin of this splitting, we show in fig. 4 the plasmon dispersion in a charge density modulated system. The superlattice effect of the periodically modulated charge density creates Brillouin zones with boundaries at q = m ~r/a, m = __+1,_+ 2 . . . . . If we fold the plasmon dispersion back into the first BrilIouin zone, - ¢r/a < q ~ ¢r/a, then we expect a splitting of the plasmon dispersion at the zone boundaries and at the center of the Brillouin zone at q = O. The plasmon dispersion forms bands with minigaps at q = 0 and q = ¢r/a. Since we use the same grating that produces the Brillouin zones also for the

D. Heitmann / 2D plasmons in space charge layers

339

T (%

~

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I

I

I

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J

>

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I

IX

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i

~/

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50 I00 150 wave number (cm-1)

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Fig. 3. Plasmon excitation in a microstructured MOS systems of the configuration shown in fig. 2b at different gate voltages Vg. Resonances ~01 and ~o2 are indicated, corresponding to wave vectors ql = (2rr/a) and q2 = 2 (2*r/a), respectively. The resonances are split into ~0 and o~+ resonances due to the superlattice effect of the periodical charge density modulation [17]. Fig. 4. Sketch of the plasmon dispersion in a lateral superlattice. The superlattice effect of the periodical charge density modulation induces minigaps at the center and at the boundaries m(w/a) of the Brillouin zones.

c o u p l i n g p r o c e s s (q,, = n ( 2 r t / a ) ) , we c a n o n l y o b s e r v e the g a p s at m = 2, 4 . . . . . T h e r e s o n a n c e s o b s e r v e d in fig. 3 are the l o w e r a n d u p p e r b r a n c h e s o f this d i s p e r s i o n at q = 0. T h e p l a s m o n d i s p e r s i o n in a c h a r g e d e n s i t y m o d u l a t e d s y s t e m has b e e n c a l c u l a t e d b y K r a s h e n i n n i k o v a n d C h a p l i k [64]. In their p e r t u r b a t i o n t h e o r y a p p r o a c h t h e y find that the s p l i t t i n g o f the r u t h g a p is

za~0,~,

2

:

£Om + --

,,,,~,

IN,,, I

0.5(~,2++,oL

)

21N0t,

(4)

w h e r e N., are the F o u r i e r c o e f f i c i e n t s of the c h a r g e d e n s i t y F o u r i e r series: +oo

%(~)-

Y' ttl=

,.., , , , e

,~....... / .

.

(5)

-oo

T o o b s e r v e a large s p l i t t i n g of the p l a s m o n r e s o n a n c e for q = 2~r/a, a s y s t e m

340

D. Heitmann / 2D plasmons in space charge layers

with a large second Fourier component IN2 [ in the charge density distribution has to be prepared. This explains a strong influence of the sample geometry on the amount of the experimentally observed splitting and, also, the fact that the splitting observed for higher gaps (rn = 4, 6 . . . . ) is in general smaller [15,16]. Within the accuracy of knowing the charge density distribution, it is found in refs. [15] and [16] that the experimental splitting is larger than given by (4); thus one is beyond the limits of the perturbation theory approach of ref. [64]. Another characteristic feature of plasmons in a periodically modulated system is that they have the character of standing waves, where both branches have a different symmetry. Because of the symmetry, the upper branch to~ + for the configuration in fig. 2a has a radiative character, whereas the to! _ branch is more nonradiative [57]. Thus, vice versa for this configuration, FIR radiation can excite the to1 + branch with higher efficiency, as is observed in refs. [15] and [16]. For the experiments shown in fig. 3 the configuration sketched in fig. 2b is used, where additional A1 stripes also spatially modulate the FIR radiation. Since these stripes are, due to the shadowing process, asymmetric with respect to the charge density modulation, the lower branch to~ is stronger excited (see fig. 3).

6. Superlattice effects on the plasmon dispersion in AIGaAs-GaAs heterojunctions 2D plasmons in AIGaAs-GaAs heterostructure have been observed with Raman [33] and FIR emission spectroscopy [34]. We will discuss here some recent experiments performed with FIR transmission spectroscopy [16]. The sample geometry is shown in the inset of fig. 5. A modulation doped heterojunction is grown by molecular beam epitaxy. It consists of a GaAs substrate and buffer layer, an undoped Al0.3Ga0.TAS spacer layer, an n-doped Al0.aGa0.7As layer and a GaAs cladding layer. On top of this layer a grating coupler consisting of periodical Ag stripes has been prepared with a periodicity a = 1.18 ~m. The mobility is about 80000 cm2/V • s. In fig. 5, FIR transmission spectra AT/T measured at different magnetic fields B perpendicular to the surface are shown. The prominent resonance at low wave numbers is the cyclotron resonance toc = eB/m*. The resonance to! is the magnetoplasmon resonance which follows roughly the classical magnetoplasmon dispersion [9]: O~p = to~ + J ,

(6)

where top is the plasmon frequency for B = 0 (eq. (1)). For the magnetoplasmon resonance to~ a small splitting into two resonances

D. Heitmann / 2D plasmons in space charge layers

3o~| 20

341

I',

,',

3

10

0 20

40 60 80 wave number (cm -1) Fig. 5, Differential FIR transmission spectroscopy on AIGaAs-GaAs heterojunctions with a grating coupler (Ag stripes with periodicity a = 1.18 ~tm) at different magnetic fields B normal to the interfaces [16]. The dominant resonances (CR) at low wave numbers are cyclotron resonances. Resonances at high wave numbers, shown on an expanded scale (right), are magnetoplasmon resonances. Two different kinds of splittings are observed: a small splitting of the ~l branch, ~ l and ~l+, due to a charge density modulation, and a large splitting of the to I and ~u branch, due to nonlocal effects.

~01 _ a n d o~t+ is o b s e r v e d . T h i s s p l i t t i n g is a l s o f o u n d f o r B = 0 [16]. T h i s s m a l l s p l i t t i n g is a t t r i b u t e d t o a p e r i o d i c a l c h a r g e d e n s i t y m o d u l a t i o n . T h e o r i g i n o f t h i s m o d u l a t i o n is t h e f o l l o w i n g . T h e c h a r g e d e n s i t y i n t h e c h a n n e l is, if c o o l e d d o w n i n t h e d a r k , 2.5 × 10 tl c m -2. T h e c h a r g e d e n s i t y is t h e n i n c r e a s e d w i t h short light pulses from a GaAs light emitting diode (~--820 nm) via the

342

D. Heitmann / 2D plasmons in space charge layers

persistent photoeffect. Since the Ag stripes of the grating coupler are nontransparent the density of ionized deep donors in the A1GaAs and thus of the carriers in the channel will be spatially modulated. This explanation is confirmed by the observation, that (a) for low charge densities, and (b) for permanent illumination (N~ = 8.5 x 1011 cm-2), no splitting is observed. In the latter case all deep donors are ionized due to scattered light, leading, except for a possible small amount of modulated transient photoconductivity [82], to a more homogeneous charge density. A spatially modulated persistent photoeffect in A 1 G a A s - G a A s heterojunction has also been induced by holographic illumination and was detected by an anisotropic DC conductivity [83].

7. Noniocality in the 2D plasmon dispersion For B = 2.55 T in fig. 5, a second resonance occurs at ~0, -- 70 c m - a , which is not observed at higher magnetic fields. With decreasing magnetic fields this resonance approaches ~0~ and increases in intensity, whereas the resonance ~0, decreases in intensity. These spectra indicate an interaction of the magnetoplasmon mode (6) with the harmonic 2¢0~ of the cyclotron resonance. We will show in the following that this interaction is of nonlocal origin. In a homogeneous system, cyclotron resonance excitation is only allowed for ~oc = e B / m * corresponding to transition between Landau levels with difference in index An = 1. The dynamic spatial modulation of a plasmon wave induces an inhomogeneity, which allows transitions with An = 2, 3 . . . . . This leads to a strong interaction of the magnetoplasmon resonances with harmonics of the cyclotron resonance and causes a splitting of the magnetoplasmon resonance at the crossing of the plasmon dispersion with 2~0c. This splitting between the ~ot and the ~0u branches is observed in fig. 5 The amount of the splitting is governed by the parameter [46-49] ( qve/o~ c)2 = 3a'~q/&,.

(8)

o v is the Fermi velocity, a~ = aoe(q, ~ ) m o / m * is the effective Bohr radius and gv is the valley degeneracy. In previous experiments on Si [20], this parameter is small (0.005). For GaAs with its small effective mass, gv = 1 and the larger value of q = 5.7 x 104 cm 1 in the experiments here, (qUv/~Oc) 2 is 0.18. Thus a splitting can be clearly resolved in the experiment. The observed w 1 - ¢ou splitting and the experimental dependence of the amplitude on B is in excellent agreement with recent calculations of the magnetoplasmon dispersion [50] which take into account nonlocal corrections in all orders of ( q v F / W c) and also finite values of the scattering time ~-. The latter is important to describe the amplitude of the excitation.

D. Heitmann / 2D plasmons in space charge layers

343

8. Conclusions I have reviewed some experimental a n d theoretical work o n 2 D p l a s m o n s in h o m o g e n e o u s systems a n d p l a s m o n excitation in systems with charge densities that are spatially m o d u l a t e d o n a s u b m i c r o n scale. I n M O S systems this is i n d u c e d via a m o d u l a t i o n of the gate oxide thickness, in A I G a A s - G a A s h e t e r o j u n c t i o n s via a spatially m o d u l a t e d persistent photoeffect. The characteristic feature of the p l a s m o n excitation in these systems is the presence of m i n i g a p s in the p l a s m o n dispersion due to the superlattice effect of the periodical charge density m o d u l a t i o n . For 2D p l a s m o n s in G a A s heterojunctions n o n l o c a l effects can be observed.

Acknowledgements I like to t h a n k m y colleagues, J.P. Kotthaus, E. Batke, W. Beinvogl, A.V. Chaplik, W. H a n s e n , U. Mackens, E.G. Mohr, S. Oelting, K. Ploog, L. Prager, a n d A.D. Wieck who have c o n t r i b u t e d in a collaborative effort to the results discussed here. I also acknowledge support from the " S t i f t u n g Volkswagenwerk" a n d the " D e u t s c h e Forschungsgemeinschaft".

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