DETERMINATION OF BINDING ENERGIES AND WAVE FUNCTIONS IN QUANTUM WELLS FROM RAMAN RESONANCE CROSS SECTION MEASUREMENTS

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DETERMINATION OF BINDING ENERGIES AND WAVE FUNCTIONS IN QUANTUM WELLS FROM

RAMAN RESONANCE CROSS SECTION MEASUREMENTS

T. Suemoto, G. Fasol, K. Ploog

To cite this version:

T. Suemoto, G. Fasol, K. Ploog. DETERMINATION OF BINDING ENERGIES AND WAVE FUNCTIONS IN QUANTUM WELLS FROM RAMAN RESONANCE CROSS SEC- TION MEASUREMENTS. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-507-C5-510.

�10.1051/jphyscol:19875108�. �jpa-00226690�

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JOURNAL DE PHYSIQUE

Colloque C5, supplbment au n a l l , Tome 48, novembre 1987

DETERMINATION OF BINDING ENERGIES AND WAVE FUNCTIONS IN QUANTUM WELLS FROM RAMAN RESONANCE CROSS SECTION MEASUREMENTS

T

.

^SUEMOTO*

^,

^G.^FASOL*^* ^and^{K. PLOOG}

Max-Planck-Institut fiir FestkBrperforschung, Heisenbergstrasse 1, 0-7000 Stuttgart 80, F.R.G.

* ~ o h o k u University, Sendai 980, Japan

.* Cavendish Laboratory, Madingley Road, GB-Cambridge CB3 UHE, Great-Britain

RQum6.

Nous Ctudions l'intensiti de l'effet Raman du phonon L 0 et de l'excitation

inter-sousbande des trous en fonction de 1'Cnergie du laser, pour des puits quantiques AlAs - GaAs - AlGaAs B dopage p modulC. Les profils de r6sonance montrent des maximums prononcCs aux Cnergies oa se trouvent les excitations Clectroniques des puits. Nous montrons que les Cnergies des rCsonances Raman sont directement liCes aux niveaux d'Cnergie des Clectrons dans les uits de la bande de conduction. Nous Ctudions des puits ayant des Cpaisseurs allant jusqulB 745

4.

^Ainsi

nous dCterminons 1'Cnergie des Ctats Clectroniques dans les puits de la bande de conduction, jusqu'k n = 18. Nous mesurons ainsi la nonparabolicit6 de la bande de conduction. Les intensitCS des signaux Raman aux maximums de ksonance sont modulbes par une fonction d'enveloppe qui est liQ B llintCgrale de couvrage entre la fonction d'onde du trou et de 1'Clectron. Ainsi nous determinons les composantes de la skies Fourier de la fonction d'onde des trous en fonction des fonctions d'onde des Clectrons.

Abstract.

We study the resonance profiles of L 0 phonon and hole intersubband Raman signals from p-type modulation doped AlAs - GaAs

-

AlGaAs quantum wells. The resonance profiles show very pronounced sharp peaks at electronic resonance energies. We show that the energies at wh.ich the Raman resonances occur are directly related to the electron subband energies in the conduction band well. We study quantum wells which are up to 745

%i

wide. We determine the binding energy of bound levelsiithe conduction band wellup to n = 18. We determine the conduction band nonvarabolicitv experimentally. The intensity at the Raman resonance peaks is modulated by an envelope func6on which is refated to the ov&lap integral of the electron and the hole wavefunction. Thus we determine the Fourier components of the hole wavefunction in terms of the conduction band wavefunctions.

1. Introduction.

The cross section of the Raman scattering by L 0 phonons and by hole intersubband excitations as a function of dye laser energy shows sharp and pronounced peaks.

These sharp resonance peaks have been studied recently in several papers and they can be used to explore the electronic structure of quantum wells [l, 2,3]. This method of exploring the subband structure of quantum wells is fundamentally different to the usual method of luminescence excitation spectroscopy, which is equivalent to selective absorption measurements. In a two-dimensional system the optical absorption, and therefore also the luminescence excitation spectrum consists essentially of a sum of step functions plus additional excitonic contributions. As the energy increases every new subband adds one new step f~nction. In addition, there is substantial enhancement of the absorption by the excitonic interaction making this type of measurement useful to study energy levels

h

quantum wells. In modulation Idoped$antum wells the excitonic interaction is screened by the carriers and luminescence excitation spectroscopy has low sensitivity. The Raman process on the other hand is a higher order process

-

the Rarnan cross section is proportional to a sum of a series of resonant denominators which are expected to diverge whenever the laser light (ingoing resonance) and/or the Raman scattered light (outgoing resonance) is close to an electronic excitation of the system. In the present work we use both the position of the Raman resonance peaks and the value of the Raman intensity at the resonance peaks to determine

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the energies of the electron binding energies in the conduction band welIs, (b) the conduction band nonparabolicity, and (c) the Fourier components of the hole wavefunction in terms of the electron wavefunctions up t o n = 18 in p-type modulation doped AlAs

-

GaAs - AlGaAs quantum wells.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875108

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JOURNAL DE PHYSIQUE

I

0 10 20 30 40 50 RAMAN SHIFT ImeV)

t

LO- PHONON Sample A

(6986)

INCIDENT PHOTON ENERGY ieV)

Fieure 1. Raman spectra of a p-type Figure 2. Sample A (well width: 745

A):

Raman modulation doped quantum well at different cross section of the L 0 phonon for a p-type laser energies (sample No. 4950) modulation doped quantum well as a function

of laser energy. Peak positions correspond to conduction band well level energies and envelope function describing heights of peaks is proportional to overlap. Insert shows schematic diagrams of Raman scattering process.

2.

Experimental.

Our samples have twenty quantum well periods grown by MBE on (100) GaAs substrates. Each period has a 450

A

wide barrier layer, consisting of a 250

A

wide layer of Be doped AlCaAs, followed by a 50

A

AlAs spacer layer, followed by the next Be-doped AlGaAs layer. The measurements are taken with an optical multichannel analyzer (OMA) using a position sensitive detector combined with a SPEX triple monochromator. The samples were mounted on a coldfinger kept at 4K in a cryostat and excited with an Argon ion laser pumped dye laser. We correct the spectra both for the reflection loss and for the response of the detection system.

Typical Raman measurements of such a structure are shown in Figure 1 for a series of dye laser wavelengths. Below about 10 meV we see a several meV broad Raman signal which we attribute to a hole intersubband excitation. We observe a GaAs like L 0 phonon and an AlAs like L 0 phonon, which is labelled LO(b) in Figure 1. In Figures 2 and 3 we show the resonance profiles of the GaAs like L 0 phonon Raman signal for sample A (745

A

well) and sample B (670

A

well). We plot the integral of the L 0 phonon Raman signal as a function of dye laser photon energy after subtraction of the background and renormalisation for the detector response and the dye laser output. Thus around 150 Raman spectra are analyzed numerically to obtain the results shown in Figures

2

and 3.

3. Electronic binding energies and nonparabolicity.

We observe that the Raman

scattering efficiency depends strongly on the dye laser energy. There are very pronounced sharp peaks for both samples. Similar peaks occur in the resonance profiles of the hole intersubband excitation. The positions of the resonance peaks correspond directly to the electronic binding energies of the subbands in the conduction band well. Indeed the separation of the peaks increases with energy. Thus the peaks n=17 and n=18 are spaced further apart than the peaks n=8 and n=9.

The dependence of the spacing on laser energy allows to uniquely identify each resonance peak with the number of the corresponding electronic level. The approximate relationship E ,,

-

ⁿ²

also allows to establish a one-to-one correspondence between the resonance peaks of t i e Rbfe intersubband excitations and those of the L 0 phonon signals and to identify the scattering mechanism. We find evidence that the Raman mechanism is by the impurity assisted Frohlich mechanism

141.

In Figure 4 we plot the binding energies determined experimentally from the positions of the Raman resonance peaks as a function of the corresponding wavevector in the conduction band well. This wavevector was determined assuming a square conduction band well.

The dashed line shows where the positions would be in the case of a parabolic conduction band.

The full line shows the calculated energies using the conduction band dispersion from a 15 ^X15 k.p calculation [5,6]. W e find very good agreement between the experimentally determined and the

-

calculated nonparabolicity. This calculation uses the bulk values for the nonparabolicity

-

thus bulk values are a good approximation to describe the nonparabolicity in a 745

A

wide well.

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Sample B ( 4 9 8 2 )

1.6 1.7 1.8 1.9

~ N C I D E N T PHOTON ENERGY (eV)

Fi ure 3. Resonance profile of the L 0 phonon Figure 4. Experimental conduction subband in sample B (well width is 670

A).

The energies (full points) as a function of squared vertical bars indicate the calculated energies wavevector k. k is obtained from solution and the dashed curve shows the calculated of the square well problem for the conduction envelope of the resonance peaks. band. The dashed line corresponds to a parabolic

band with a mass of 0.0665 and the solid curve corresponds to binding energies calculated using the bulk effective masses from 15 X 15

h.e

calculation.

4. Fourier determination of the hole wavefunction. Figures 2 and 3 show that the Raman efficiency at the resonance energies is strongly modulated by an envelope function. The intensity for Raman scattering by L 0 phonons is given by

Here, the integral <n,k'lHeLln,k"> denotes the intraband scattering of an electron by an L 0 phonon within the 2D space and it can be assumed to be independent of n. Similarly the integral

<n,klHeiln,k'>, which denotes the elastic scattering by impurities or imperfections will be assumed to be approximately independent of n. Thus the relative intensity of resonance for different n can be written as

where the matrix elements <hlklHe In k> again have been assumed to be independent of n.Thus we find that the envelope function oPthe Raman resonance profile is proportional to the fourth power of the overlap integral of the hole wavefunction with the corresponding conduction band wavefunction. Thus the heights of the Raman peaks in Figures 2 and 3 measure the Fourier components of the hole wavefunction in terms of the conduction band wavefunctions. Since the wells studied here are very wide, electron wavefunctions up to n=18 are used for this "Fourier decomposition". In principle, the hole wavefunction could be directly synthesiz~d by summing the Fourier series. There are two obstacles to a direct summation of the Fourier series here: in the part of the spectrum below n = 6 strong luminescence inhibits the measurement of the Raman resonances and secondly the phase of the "Fourier components" is not known. Thus we use additional knowledge about the system and assume the localization of the holes in triangular wells

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CS-5 10 JOURNAL DE PHYSIQUE

at the interfaces and use an Airy function with two fit parameters. The solid line in Fi re 2 shows the best fit we obtained. Thus we determine that the hole wavefunction peaks 65 - 78 y f m m the interface of the well. Less successful fits are shown as the dotted and the dash-dotted curves.

6.

Conclusions.

We have shown that the Raman efficiency of the L 0 phonon and the hole intersubband excitation in wide p-type modulation doped AlAs-GaAs-AlGaAs quantum wells shows pronounced and sharp peaks as a function of dye laser energy. From the positions of the resonance peaks we determine the level spacings of the bound electron levels in the conduction band wells up to n = 18 in the present cases and the conduction band nonparabolicity

experimentally. The Raman efficiency at the resonance peaks is modulated by an envelope function, which is proportional to the fourth power of the overlap integral of the hole wavefunction and the corresponding conduction band wavefunctions. Thus we perform a Fourier type decomposition of the hole wavefunction in terms of the electron wavefunctions.

Acknowledgements.

The authors are extremely grateful to M. Cardona for valuable

discussions, criticism and support of this work. Further we are indebted to U. Ekenberg for many helpful discussions, to A. Fischer for sample growth and to H. Hirt, M. Siemers and P. Wurster for expert technical assistance. One of the authors (T. S.) thanks the Alexander von Humboldt Foundation for a Fellowship.

References.

1.

J. E.

Zucker, A. Pinczuk, D. S. Chemla, A. Gossard and W. Wiegmann, Phys. Rev. Lett. S l , 1293 (1983)

2. J. E. Zucker, A. Pinczuk, D. S. Chemla, A. Gossard and W. Wiegmann, Phys. Rev.

2,

7065 (1984)

3. T. suemoto, G . Fasol and K. Ploog, Phys. Rev.

D,

6034 (1986) 4. J. MenCndez and M. Cardona, Phys. Rev.

m,

3696 (1985) 5.

U.

Rossler, Solid State Commun.,

e,

943 (1984)

6. G. Fasol, to be published.