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LIGHT SCATTERING BY TWO DIMENSIONALELECTRON SYSTEMS IN SEMICONDUCTORS

A. Pinczuk

To cite this version:A. Pinczuk. LIGHT SCATTERING BY TWO DIMENSIONAL ELECTRON SYSTEMSIN SEMICONDUCTORS. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-477-C5-487.�10.1051/jphyscol:1984572�. �jpa-00224193�

JOURNAL DE PHYSIQUE

Colloque C5, suppl6ment au n04, Tome 45, avril 1984 page C5-477

LIGHT SCATTERING BY TWO DIMENSIONAL ELECTRON SYSTEMS IN SEMICONDUCTORS

A . Pinczuk

A.T.&T. - Bell Laboratories, ~oZrndeZ, New Jersey 07733, U.S.A.

Resum6 - Cette contribution prgsente une revue des expe/riences utilisant la diffusion inglastique de la lumisre pour Qtudier les syst5mes d'blectronsbidimensionnels aux interfaces des semi- conducteurs. Les rhultats obtenus sur des h6ti3rostructures b modulation de dopage b base de GaAs-(ARGa)As sont dgalement considgrgs.

Abstract - This contribution reviews resonant inelastic light scattering studies of two dimensional electron systems at semi- conductor interfaces. The experimental results from modulation- doped GaAs-(ARGa)As heterostructures are considered.

1. INTRODUCTION

This paper is concerned with the spectroscopy of the elementary exci- tations of two dimensional electron gases that occur at interfaces of semiconductor micro-structures. In these systems, energy band dis- continuities and space-charge electric fields quantize the electron motion normal to the interfaces into discrete energy levels. Since these electrons are free to move in a plane parallel to the interface, each of the discrete energy levels gives rise to a two dimensional subband. Electrons in subband states have many properties in common with those of an idealized 2D electron gas /I/. Experimental and theoretical research of 2D electron systems at semiconductor inter- faces has been extensive. The recent observations of the quantized Hall effect /2,3/ and the anomalous magneto - transport behavior in the extreme quantum limit (fractional quantization) /4,5/ are among the most exciting new developments in solid state physics.

In 1978 Burstein, et al., /6/ proposed resonant inelastic light scat- tering as a method for the investigation of the elementary excitations of 2D electron systems in semiconductors. It was pointed out that with the resonant enhancements of the light scattering cross-section the method has the sensitivity required to observe the elementary excitations of systems with areal densities n 2 5~1011cm-2. It was also suggested that it should be possible to measure separate spectra of single particle and collective excitations. This feature was expected to lead to determinations of energy levels and collective electron - electron interactions. The proposal was almost immediately followed by the first observations of light scattering by intersubband excitations, between the discrete energy levels of electrons confined at GaAs-(ARGa)As heterostructures /7,8/.

During the last four years light scattering by the 2D electron systems at GaAs-(ARGa)As heterostructures has been studied extensively. Intersubband spectroscopy has confirmed the predictions of Burstein

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984572

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et al. /6/ and gave considerable information on the behavior of free electrons in the heterostructures. They have revealed energy level structures and resonant enhancement of scattering intensities / 7 , 8 / , collective Coulomb interactions /9/, correlations with electron mobilities /lo/ and the effects of photoexcitation at high intensities /11,12/. Excitations associated with in-plane motion, transitions between Landau level /13/ and plasma oscillations /14/, have also been observed in the GaAs-(ARGa)As heterostructures.

Resonant light scattering observations of excitations of 2D electron systems have been reported in several other semiconductor interfaces. Among them are the accumulation layers at MOS or MIS structures of InAs /15,16/, Si /17/ and InP /18/. Photoexcited systems have been studied in doping superlattices of GaAs /19/. Light scattering from electrons in doped Ge-GaAs heterostructures has also been reported /20/. In this review, I will consider only selected results from GaAs-(ARGa)As heterostructures. They represent examples of the physics of 2D electron systems of semiconductors that is accesible to investi- gation by light scattering. A more comprehensive review will soon become available /21/.

2. MECHANISIIS, SELECTION RULES AND KINEMATICS

Within the effective-mass approximation the mechanisms and selection rules for resonant light scattering by a 2D semiconductor plasma are similar to those of 3D systems /6,22/. Resonant enhancements, essential to these experiments, are expected for photon energies near optical gaps that involve the free electrons. In the case of direct gap zincblende semiconductors the relevant resonances occur at the fundamental, Eo, and spin-orbit split-off, EO + Ao, gaps near the r point of the Brillouin zone. In n-type Ge the resonance occurs at the L point, near the El and El + A1 optical gaps. In p-type Si it is at the ~b gap near the r point.

Two types of spectra are measured. The polarized spectra, obtained with parallel incident and scattered polarizations, are assigned to collective charge density excitations of the electrons. Depolarized spectra, in which the two polarizations are orthogonal, are assigned to single-particle spin-density excitations. These assignments have explained the results obtained near the EO + A0 resonance /9,23/. Breakdown of these selection rules has been observed in spectra obtained with photon energies near the fundamental Eo gaps /24/.

Figure 1 - Typical nearly backscattering geometry used in the experiments described here.

I SCATTERED LIGHT

The backscattering geometry shown in Figure 1 is frequently used. The in-plane and normal components of the scattering wavevector are given by

2 Tr k l l = (sin 8 - cos 9 ) (11

where X is the laser wavelength and r- is the refractive index. k l I can be varied from a small value of -104cm-1, at 8 45', up to a maximum of % 105cm-I. Typical values of kl are 7 ~ 1 0 ~ c m - ~ near the E g + A0 energy gap at n 1.9 eV; and kL = 5x13~:-I at the funda- mental gaps (Eo N 1.5 eV) . 3. INTERSUBBAND SPECTROSCOPY

The first light scattering experiments were carried out in doped GaAs-(ARGa)As heterostructures fabricated by molecular beam epitaxy. In this work the spectral features of intersubband excitations are broad, with lineshapes that reflect the sample quality at the time. The most revealing studies of intersubband excitations have been carried out in modulation-doped /25/ quantum-wells. The heterojunc- tions have the structure shown in Fig. 2.

UNDOPED (AIGal As SPACER

Figure 2 - Sequence of layers and relative position of the conduction band edge in

I I I I 4 I

I I modulation-doped multiple quantum-well G ~ A S (ARGa) AS

ES heterostructures /26/.

The undoped' (ARGa)As spacer, with thickness d3, separates further the mobile electrons in the GaAs layers from the ionized donors. This results in a substantial enhancement of the electron mobility /26/.

In the spectra of intersubband excitations shown in Fig. 3, z and z are the propagation directions of the incident and scattered light. They are close to (001) and (001) crystal axes. x' and y1 are (110) and (1fO) in-plane directions of polarization of light. These spectra, reported by Pinczuk et al. /9/, are the first in which there is clear separation between single particle and collective excitations. The depolarized z(y'x1)5 spectrum shows a single band assigned to the lowest, single-particle, spin-density intersubband excitation. The spectra of Fig. 3 were obtained with 8 45'. Thus, kl 1 is small and the intersubband transitions can be assumed to be vertlcal in the 2D wavevector space of the subbands. Therefore, E01 represents an energy close to the spacing between the lowest conduction subbands. In the polarized z (x'xl spectrum the peaks labeled I+ and I-, at E+ and E-, are assigned to the coupled collective intersubband -LO phonon excita- tions considered by Burstein et al. /22/. Abstreiter et al. /27/ have observed similar spectra.

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Figure 3 - Spectra from a modulation doped quantum-well heterostructures /9/. The inset shows calculated energies of the lowest quantum well states, the band bending and the Fermi energy. The LO, and LO2 peaks are £;om the

lo 45 20 25 30 35 40 45 50 (Ago. 18Ga0. 88) AS layers. ENERGY SHIFT ImeV)

Single particle intersubband energies were measured in depolarized spectra from several samples /7,8,9,23,27/. In all cases there is good agreement with calculated subbands spacings /28,29/. The dif- ferences in energies of single particle and collective excitations are direct evidence of the macroscopic electric fields (or depolarization fields) due to resonant screening phenomena that are associated with intersubband transitions. The effects were first predicted by Chen et al. /30/, and were invoked in the interpretation of subband opti- cal absorption /31,32/. The separate measurement of light scattering spectra by single particle and collective intersubband excitations have for the first time made possible simple determinations of depolar- ization field effects.

Collective intersubband excitations were first considered by Dahl and Sham /33/. Burstein et al. /22/ have included the coupling to opti- cal phonons that occurs in polar semiconductors. These treatments have been extended to incorporate the specific conditions that exist in quantum-well heterostructures /9,23/. The I+ and I- doublet seen in the spectrum of Fig. 3 can be interpreted in terms of simple coupled electron -LO phonon modes that are described by /9,23/

where ET = 33.6 meV and EL = 36.7 meV are the energies of TO and LO phonons of GaAs. Ep is a plasma energy that describes depolarization field effects. It can be written as /23/

Where = 11.1 is the background dielectric constant and LO1 is the intersubband Coulomb matrix element given by

r m LZ 1

where Co(z) and Sl(z) are the envelope functions of the ground and first excited subbands. The value of Lo1 is the only adjustable parameter of the analysis. The values of Lo1 determined from these experiments are in good agreement with the ones evaluated with sub- band envelope functions of simple model calculations /9,23,29/.

G O A S - I A P ~ , ~ ~ G ~ ~ ~ ~ ) A S ; T ~ ~ ~ K

z l y ' r ' ) l , hwLASER = 1.900eV

E02

1+) Eo

m z 5

1 SWPLE 0 dl = 2 5 5 i , d 2 = 3 0 2 i

> t d3 ' 0 U) Z Y

SAMPLE 1

u

SAYPLE 3 dl =25OA, d2=292A

d,=151A

Figure 4 - Depolarized light scattering spectra from three modulation doped quantum-well heterostructures. The inset shows the assignments of the intersubband transitions /lo/.

I : I I 0 20 4 0 6 0 8 0

ENERGY SHIFT I meV)

One of the intriguing questions in the spectroscopy of intersubband excitations is the relation to transport properties of the 2D electron system. Pinczuk et al. /lo/ have identified a correlation between lineshapes in resonant light scattering spectra and the in-plane mobilities. These studies have been carried out in four similar modulation-doped quantum-well GaAs-(AR0-12Ga0.88)As heterostructures in which the electron mobilities are directly related to dj, the thickness of the undoped (AR0.12Ga0.88)As spacer (see Fig. 2) /26/. Figure 4 shows spectra obtained from three of these samples. The spectral features are assigned to single particle intersubband transi- tions as indicated in the figure. The widths of the three bands can be seen to decrease dramatically with increasing Hall mobility and d3. The correlation is also associated with an unexpected dependence of the spectral bandwidths on incident photon energy in the range of the EO + A0 optical gap /10,23/. This effect is striking in samples with electron mobilities lower than ~25,000 cm2/vsec.

In the interpretation of these results it was proposed /lo/ that resonant inelastic light scattering is subject to wavevector relaxa- tion processes related to those that limit the electron mobility. These processes, due to scattering of electrons by the Coulomb poten- tial of the ionized donors, depend on d3 in modulation-doped structures. Under these circumstances wavevector conservation breaks down for resonant light scattering. Thus, non-vertical intersubband, with an in-plane wavevector transfer q # kl become active in the nearly backscattering spectra. The bands o i non-vertical transitions have widths of hqvF (where vF is the Fermi velocity). Therefore, as a consequence of breakdown of wavevector conservation the spectra are expected to show considerable width or even appear as a quasi- continuum. The stronger resonant behavior of light scattering with breakdown of wavevector conservation /21,34/ explains the dependence of spectral width on photon energy.

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Measurements of resonant enhancements of l i g h t s c a t t e r i n g by i n t e r - subband e x c i t a t i o n s have been repor ted i n s i n g l e /7,35/ and mu l t i p l e /8,36/ G ~ A S - ( a G a ) A s h e t e r o s t r u c t u r e s . The r e s u l t s , ob ta ined with photon ene rg i e s ac ros s t h e E + A o p t i c a l gaps of t h e GaAs l a y e r s ,

0 d i sp l ay t h e gene ra l behavior a n t i g i p a t e d by Burs te in e t a l . /6,22/. The most r ecen t work has been c a r r i e d i n t h e sample with t h e l a r g e s t d3 va lue of t h e group whose s p e c t r a a r e shown i n Fig. 4 (unpublished r e s u l t s by t h e a u t h o r ) . The i n t e g r a t e d i n t e n s i t i e s of t h e even p a r i t y t r a n s i t i o n Eo2 a r e much l a r g e r than those of t h e odd p a r i t y t r a n s i t i o n E01: S c a t t e r i n g by t h e odd p a r i t y e x c i t a t i o n , t h a t breaks down t h e p a r l t y s e l e c t i o n r u l e f o r quantum-wells /36/, has been assigned t o t h e k - dependent con t r ibu t ions t o t h e o p t i c a l mat r ix element /21/.

INPUT

-LASER BEAM

Figure 5 - Depolarized l i g h t s c a t t e r i n g spectrum of a modulation-doped quantum-well h e t e r o s t r u c t u r e under a magnetic f i e l d normal t o t h e 2D e l e c t r o n system /13/. THe i n s e t shows t h e sample arrangement i n t h e magnet.

STOKES ENERGY SHIFT lmeV1

4. LANDAU LEVEL EXCITATIOIgS

Resonant i n e l a s t i c l i g h t s c a t t e r i n g by t h e 2D e l e c t r o n systems i n GaAs-(ARGa)As h e t e r o s t r u c t u r e s under magnetic f i e l d s normal t o t h e l a y e r s has been inves t iga t ed /13,37/. For t h i s o r i e n t a t i o n of magnetic f i e l d t h e e l e c t r o n motion i n t h e p lane of t h e l a y e r s i s quantized i n t o Landau l e v e l s and t h e energy spectrum of t h e 2D system i s f u l l y d i s c r e t e /I/.

F igure 6 - P l o t of t h e energ ies of AR = 1 Landau l e v e l t r a n s i - t i o n s a s a func t ion of magnetic f i e l d . Resul t s from s i x d i f - f e r e n t modulation doped quantum we l l h e t e r o s t r u c t u r e s a r e shown /w.

2 4 6 8 10 12 14

MAGNETIC FIELD ITESLAI

These experiments were a l s o c a r r i e d ou t i n , n-type, modulation-doped quantum we l l s . Typical depolar ized Z ( ~ ' X ' ) ~ s p e c t r a can be seen i n Fig. 5. The photon energ ies a r e c l o s e t o t h e Eo + A 0 o p t i c a l gaps of t h e GaAs l a y e r s . The band l abe l ed % w has been assigned t o AL = 1 Landau l e v e l e x c i t a t i o n s of s i n g l e p a g t i c l e cha rac t e r . S imi la r spec- t r a have been obtained from s e v e r a l samples having a range of f r e e e l e c t r o n d e n s i t i e s and l a y e r th icknesses . Figure 6 shows t h e va lues of %wc a s a func t ion of magnetic f i e l d . The l i n e a r behavior of %wc suppor ts t h e assignment a s Al? = 1 Landau l e v e l t r a n s i t i o n s . The s lope of t h e s t r a i g h t l i n e g ives an e l ec t ron e f f e c t i v e mass of m* = 0.068 k0.003. Light s c a t t e r i n g by t h e AR = 1 e x c i t a t i o n s , having odd pari ty, i s forbidden i n t h e d ipo le approximation. Breakdown of t h i s approxi- mation has been invoked t o expla in t hese experiments /38/.

5. PLASMA OSCILLATIONS

Light s c a t t e r i n g by plasma o s c i l l a t i o n s i n mu l t i p l e quantum wel l GaAs- (ALGa)As h e t e r o s t r u c t u r e s has been observed by Olego e t a l . / 1 4 / . These o s c i l l a t i o n s a r e c o l l e c t i v e modes a s soc i a t ed wi th t h e 2D in - p lane motion of e l e c t r o n s confined i n t h e GaAs l a y e r s . Theexperiments were c a r r i e d o u t i n modulation-doped samples with t h e s t r u c t u r e shown i n Pig. 2. The photon ene rg i e s (a1.58 eV) a r e r e l a t i v e l y c l o s e t o t h e fundamental o p t i c a l gap. This allowed r e l a t i v e l y l a r g e pene t r a t i on depths whi le r e t a i n i n g t h e b e n e f i t s of resonant enhancements and r e l a - t i v e l y low luminescence.

Figure 7 - ( a ) Nearly backscat- t e r i n g spectrum showing t h e peak due t o a plasmon of a modulation- doped m u l t i p l e quantum-well he te ro- s t r u c t u r e . (b) Plasma l i n e f o r d i f f e r e n t angels of inc idence / 1 4 / .

The charge dens i ty o s c i l l a t i o n of semiconductor s u p e r l a t t i c e s have been descr ibed /39,40/ a s plasmons of an i d e a l i z e d layered 2D e l e c t r o n gas / 4 1 / . Such modes have l a r g e d i s p e r s i v e e f f e c t s of t h e form/39,41/

where

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i s t h e 2D plasma frequency a s soc i a t ed wi th t h e f r e e charge i n each l a y e r and EM i s t h e background d i e l e c t r i c cons t an t . The s t r u c t u r e f a c t o r of t he s u p e r l a t t i c e is given by

s inh k d

""1 1 fkird) = cosh k l - I bas kid

where d = dA + d2 + .d3 . For k l d < 1, Eqs. (6 ) - (8) p r e d i c t t h e a c o u s t i c - l i e be a v l o r

where

2 1 1/2

2rne d = [ 1 - cos kid 1

An important f e a t u r e of t h e plasmon descr ibed by Eqs. (9) and (10) i s t h e abscence of Landau damping s i n c e v > vF.

Figure 7 ( a ) shows aospectrum from a GaAs- (ARo. ZGao 8 ) A s sample wi th d l = 262% d2 = 317A, d = 163%, and n = 7 . 3 x 1 0 ~ ~ c m - * . The EO1 and E- bands a r e a s soc i a t ea with s i n g l e p a r t i c l e and c o l l e c t i v e I n t e r - sthband e x c i t a t i o n s . There is a l s o a lower energy peak a t 3 .5 meV. Its most remarkable c h a r a c t e r i s t i c , d i sp layed i n Fig. 7 ( b ) , i s i ts dependence on t h e angle of incidence. I t fol lows from Eq. (1) t h a t t h i s i n d i c a t e s a dependence on k . This d i s p e r s i v e behavior l e d Olego, e t a l . /14/ t o a s s ign the l iower energy peak t o a plasma o s c i l - l a t i o n of t h e f r e e c a r r i e r s i n t h e quantum we l l s .

F igure 8 - Dispersion of t h e plasma frequency measured i n two samples.

F igure 8 shows t h e plasma d i spe r s ion measured i n two samples, These r e s u l t s were i n t e r p r e t e d wi th Eqs. ( 6 ) - ( 1 0 ) . The f u l l l i n e s a r e t h e eva lua t ions wi th t h e s t r u c t u r e f a c t o r s of t h e s u p e r l a t t i c e s , while t h e dashed l i n e s a r e t h e ex t r apo la t i ons of t h e l i n e a r behavior . The agreement between measured and ca l cu l a t ed d i spe r s ion i s very good.

The s m a l l d i f f e r e n c e s c a n be e x p l a i n e d by u n c e r t a i n t i e s i n t h e v a l u e s of t h e sample pa ramete r s . The o b s e r v a t i o n o f t h e plasma o s c i l l a t i o n s and d e t e r m i n a t i o n o f i t s d i s p e r s i o n a r e an example of t h e a p p l i c a t i o n o f t h e l i g h t s c a t t e r i n g method t o s t u d i e s of t h e i n - p l a n e motion o f 2D e l e c t r o n g a s e s .

6 . CONCLUDING REMARKS

The i n t e n s e a c t i v i t y of t h e l a s t f o u r y e a r s h a s shown t h a t i n e l a s t i c l i g h t s c a t t e r i n g i s one of t h e most r e s o u r c e f u l methods f o r t h e spec- t roscopy of e lementa ry e x c i t a t i o n s of 2D e l e c t r o n sys tems a t semicon- d u c t o r i n t e r f a c e s . F u r t h e r i n t e r e s t can b e a n t i c i p a t e d , w i t h empahsis i n h e t e r o s t r u c t u r e s , spec t roscopy of in -p lane motion and lower dimen- s i o n a l i t y phenomena i n g e n e r a l .

The work reviewed h e r e i s p a r t of a j o i n t e f f o r t a t B e l l L a b o r a t o r i e s . I am g r a t e f u l t o R. D ing le , A. C . Gossard, D. Olego, J. Shah, W . Wiegmann, and 3. M. Worlock. I a l s o have t h e p l e a s u r e t o acknowledge s t i m u l a t i n g d i s c u s s i o n s w i t h G . A b s t r e i t e r , E. B u r s t e i n , R. M e r l i n , D. L. M i l l s , and P. A. Wolff.

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