Ž .Applied Surface Science 166 2000 273–277www.elsevier.nlrlocaterapsusc

Simulation of excitonic spectra in electric field to characterize thequality of low dimensional structures

O.L. Lazarenkova), A.N. PikhtinSt. Petersburg Electrotechnical UniÕersity, Professor PopoÕ str. 5, St. Petersburg 197376, Russian Federation

Abstract

In order to characterize the real low dimensional structures, the simulation of excitonic spectra has been done. AbsorptionŽ .spectrum of multiple quantum well MQW in graded electric field, the effect of macroscopic quantum well width and depth

fluctuations, and the influence of surface roughness have been considered. The simulated spectra are in a good agreementwith experimental data. q 2000 Published by Elsevier Science B.V.

PACS: 73.40 and 78.20; 71.35; S7.12; S7.15Keywords: Quantum well; Electric field; Inhomogeneity; Modulation optical spectroscopy

A lot of modern devices of nanoelectronics andoptoelectronics are based on specific properties oflow dimensional structures. Their characteristics arevery dependent on the quality of ultra thin epitaxiallayers.

Therefore, it is necessary to develop the tech-niques for investigation of quantum-size structuresafter the formation as during the technology process.

Ž . Ž .Photoreflectance PR and phototransmittance PTas contactless techniques of modulation spectroscopy

Ž .are the major tools for quantum well QW charac-terization. Their essence is to record the change of a

Ž . Ž .reflected PR or transmitted PT probe light beamunder the periodic modulation of the sample charac-

) Corresponding author. Tel.: q7-812-234-31-64; fax: q7-812-234-31-64.

Ž .E-mail address: omk.meeeltech.ru O.L. Lazarenkova .

teristics by the second light beam, which changes thebuilt-in electric field. Thus, the relatively sharp fea-tures are observed in the spectra, even at roomtemperature. However, some difficulties are cur-rently encountered in analyzing the observed spectraof QW structures. The excitonic effects have to betaken into account on principle.

In this paper, we develop an approach previouslyw xproposed by ourselves 1 to simulate PR and PT

excitonic spectra for QWs with different kinds ofheterogeneity. To solve this problem, one shouldknow the effect of electric field on reflection andabsorption of QW structure.

w xIn Ref. 1 we have demonstrated that for simula-tion of optical spectra in electric field in the approxi-

w xmation of weakly interacting quantum states 2 onecan use formulas that deal with optical transitionsbetween quasi bound electron and hole states with

Ž .finite broadening. The energy of the electron holestate corresponds to the real part of the resolvent

0169-4332r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0169-4332 00 00405-0

( )O.L. LazarenkoÕa, A.N. PikhtinrApplied Surface Science 166 2000 273–277274

poles of the one-electron Hamiltonian, and the field-induced homogeneous broadening of the states corre-sponds to the imaginary part of the resolvent poles.The effect of electric field on the energy and homo-

Ž .geneous broadening of electron hole state has beenw xdiscussed in detail in Ref. 2 . The probabilities of

interband optical transitions may have a very intri-cate field dependence as a consequence of the elec-tron and hole envelope wave function symmetry

w xtransformation in an electric field 1 .

1. Absorption spectrum of multiple quantum well( )MQW in graded electric field

In real heterostructures, the electric field may beuniform or graded as in optical modulators based onMQW structures placed in the i-area of a p-i-n diodeŽ .see Fig. 1a .

In MQW structures, it is impossible to neglect theelectric field gradient inside the structures as it ispossible for an ultra thin layer of a single quantum

Ž .well SQW . The interference of signals of QWsinfluenced by different values of electric field maybe the cause of some changes of optical spectra. InFig. 1 we compare the calculated spectra of GaAsr

Ž . ŽGa Al As SQW dotted line and MQW solid0.32 0.68.line structures for different values of the mean

electric field F in the active region. The width of the˚well was the same in both structures: Ls95 A,

˚barrier width is L s98 A, the quantity of QWs isB

Ns50. The electric field in the active region differson about 25 kVrcm. We have not used any fittingparameters in our calculations. One can see that theeffect of the electric field gradient varies for differ-ent mean values of electric fields. It is not equivalentto the similar broadening of each excitonic resonancebecause of complicated field-dependence of the pa-rameters that define absorption.

The comparison of calculated spectra with experi-Ž . w xmental data circles in Fig. 1 reported in Ref. 3 has

demonstrated a good agreement. One can follow theredistribution in electric field of the different exci-tonic transitions intensity both in experimental andcalculated spectra. The fact that the experimentalexcitonic peaks are wider than the calculated ones

Fig. 1. The distribution of electric field in real optical modulatorŽ .a and the effect of gradient electric field on the absorption

Ž .spectra of GaAsrGa Al As SQW dotted line and MQW0.32 0.68Ž . Ž .solid line structures for different mean electric field: b Fs10

Ž . Ž .kVrcm, c Fs47 kVrcm, d Fs73 kVrcm. The width ofwell is the same in both cases: Ls9.5 nm, barrier width isL s9.8 nm, the number of QWs is Ns50. The increase of theB

electric field in the active region is about 25 kVrcm. TheŽ . w xexperimental data circles have been taken from Ref. 3 . The

measurements and simulations refer to room temperature spectra.

convinces that one should take into account inhomo-geneous broadening caused by well width and depthfluctuations.

2. The effect of QW width fluctuation

In real structures the island-like fluctuation of theepilayer thickness may exist. It results in QW widthfluctuation. The character of exciton-fluctuation in-teraction depends on the ratio of the exciton Bohrradius and the lateral extension of islands. We con-sider the macroscopic fluctuations when there is noeffective localization of excitons. Such a situation is

( )O.L. LazarenkoÕa, A.N. PikhtinrApplied Surface Science 166 2000 273–277 275

realized in heterostructures that grow according toisland mechanisms.

In the case of macroscopic fluctuations, it isnecessary to convolute the initial spectrum with aGaussian which width is proportional to the value offluctuations and the absolute value of the first deriva-tive of the exciton energy with respect to the fluctu-ating parameter:

exEE FŽ .nmd LG F s k dL.Ž .nm d L

EL

The parameter k depends on the fluctuation lateraldL

extension and is about 1. Here

EEex EEe EEhnm n m

( q ,EL EL EL

that is, the field dependence of the inhomogeneousbroadening due to well width fluctuation is definedby the transformation of electron spectrum. The cor-responding families of dimensionless field depen-

w xdences have been presented in Ref. 2 . Note that thederivatives for electron and hole quantum levels mayhave different signs at some field values, that is, theexciton peak inhomogeneous broadening may be less

Ž .than for the electron hole one.

3. The effect of QW depth fluctuation

We shall now examine, in a similar manner, theeffect of QW depth fluctuation due to the fluctuationof alloy composition. There are two different situa-

Ž .tions: 1 the solid solution is the material of theŽ .barriers, and 2 the solid solution is the material of

the QW. Both of them are equivalent to band offsetfluctuation.

In the first case, the energy gap of the QWmaterial is constant and inhomogeneous broadeningof the excitonic resonances

exEE FŽ .nmdD EgG F s k dD EŽ .nm dD E ggED Eg

is defined only by electron and hole energy spectrumtransformation:

EEe x EEe EEhnm n m

s q ,ED E ED E ED Eg g g

w xwhich have been considered in Ref. 2 .In the second case, it is necessary to take into

account the heterogeneity of the QW material energygap. Therefore,

EEex EEe EEhnm n m

s q y1.ED E ED E ED Eg g g

Note again that the difference in signs complicatesthe field dependence of the total broadening of exci-tonic peak.

The field-induced changes in the reflection spec-tra of QWs with different fluctuation parameters ofthe same relative values are compared in Fig. 2. For

Fig. 2. The field-induced changes in the reflection spectra of QWwith the following parameters: E s0.985 eV, V s120 meV,g e

V s80 meV, m s0.06Pm , m s0.5Pm , m s0.07Pm , Lh e 0 h h 0 lh 0

s20 nm, for different fluctuated parameters of the same relative˚Ž . Ž . Ž . Ž .values: a dLrLs5% 10 A ; b dDE rDE s5% 10 meV .g g

( )O.L. LazarenkoÕa, A.N. PikhtinrApplied Surface Science 166 2000 273–277276

our calculations, we used the typical parameters ofŽ .Ž .the real In,Ga As,P rInP SQW: E s0.935 eV,g

˚V s0.12 eV, V s0.08 eV, Ls200 A, ks12.5,c Õ

m s0.06Pm , m s0.5Pm , m s0.07Pm , ande 0 hh 0 lh 0

G s2.79 meV. At the zero electric field, this QWT

contains three electrons, three light holes, and sevenheavy holes confined states. Thus, the structure ofthe reflection spectra is very complicated. One cansee that the QW depth fluctuations determine thetotal broadening, if the solid solution is the materialof the QW. One can see that it is impossible toconsider that the exciton peaks are monotonouslybroadened in electric field, if there are any fluctua-tions of the QW parameters. When the electric fieldincreases, the changes of the exciton resonanceswidth may be very complicated. If you carry out aseries of measurements at different values of electricfield, you may obtain the main cause of the inhomo-

w xgeneous broadening using presented in 2 dimen-sionless dependences.

4. The top layer effect

The reflected light from the QW interferes withthe light reflected from the sample surface. Theinterference depends on the numbers of wavelengthconsisting the top layer. The form of the absorptionspectrum periodically changes. It may look like asingle resonance, antiresonance or antisymmetricalstructure. In real samples, the surface may be veryrough. In such samples, the top layer width has anuncertainty d d . Thus, the symmetry of the reflection1

spectrum near the excitonic resonance distorts:

Rd d1 sR qdRd d1 .nm nm nm

This distortion periodically depends on the top layerwidth and for the normal incidence may be calcu-lated as

16 k k y1 k "G( (ER ž /0 0 1 0nmd d1dR s dd sddnm 1 1 3Ed1 k q1(ž /0

` E qD E y"v cos 2k d y "G q"G sin 2k dŽ . Ž . Ž . Ž .i synm 0 nm 1 1 nm 0 nm 1 1y3= 2 iy1 .Ž .Ý 2 2E qD E y"v q "G q"GŽ . Ž .is1 i synm 0 nm nm 0 nm

Here, k denotes the effective dielectric constant of0

the structure, k sv k rc, G is the radiative(1 0 0 nm

broadening of excitonic resonance, E is thei sy nm

energy of excitonic resonance, DE is the renor-0 nm

malization of the exciton resonance energy, and d1

denotes the width of the top layer.Such distortion of the absorption spectrum sym-

metry may be the cause of misunderstandings in theinterpretation of experimental data for samples withan appreciably rough surface.

5. Characterization of low dimensional structuresby photomodulation spectroscopy

The results obtained for the influence of electricfield on absorption and reflection spectra of QWstructure makes it possible to simulate PR and PTspectra of real low dimensional systems.

In order to illustrate our approach to simulateoptical spectra in an electric field, the calculated and

Ž .experimental PR Fig. 3 and PT spectra of the sameŽ .Ž .laser structure In,Ga As,P rInP have been com-

pared. For our calculations, we used the same param-eters of SQW as in Fig. 2.

Ž . Ž .Fig. 3. Theoretical solid line and experimental squares PRspectra of laser structure based on SQW.

( )O.L. LazarenkoÕa, A.N. PikhtinrApplied Surface Science 166 2000 273–277 277

The best fitting of calculated and experimentaldata is achieved at the mean value of built-in electricfield in the area of the QW equals to 16 kVrcm, thereduction of the electric field due to the laser pumpequals to 1 kVrcm, and the inhomogeneous broad-ening is 20 meV. It is necessary to make experimentswith different F to determine the cause of the broad-ening. Note that the evaluated values of these param-eters are the same as for PR and PT spectra.

6. Conclusion

Thus, we have demonstrated that the electric fieldmodifies not only excitonic spectrum of QW but alsoits transformation due to different tapes of inhomo-

geneity of real structures. The effect of QW widthand depth fluctuations, the top layer effect and theeffect of gradient of electric field in MQW on opticalabsorption, reflection, and PR spectra have beencalculated for real semiconductor heterostructuresused in lasers and optical modulators.

References

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w x Ž .2 O.L. Lazarenkova, A.N. Pikhtin, Semiconductors 32 1998992.

w x3 D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W.Ž .Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 1985

1043.