Photoreflectance and the Seraphin coefficients in quantum well structures

X.L.Zheng D.Heiman and B.LaxFrancis Bitter National Magnet Laboratory, Massachusetts Institute of Technology

Cambridge, Massachusetts 02139

F.A.ChambersAmoco Corporation, Amoco Research Center

Naperville, Illinois 60566

ABSTRACT

The lineshape of photoreflectance spectra from quantum well structures remains an unsolvedproblem, despite previous attemps to develop an adequate model using Seraphin coefficients for a bulksemiconductor. We show that the Seraphin coefficients in quantum well structures are substantiallydifferent from those in the bulk. We obtain analytical forms of the coefficients for a single quantumwell, which show that the interference phase depends on the geometrical structure of the quantum well.The predictions are confirmed by our experimental photoreflectance data from quantum well samples.Our study suggests a new approach to the study of the lineshape problem of modulation reflectance.

1. INTRODUCTION.

Photoreflectance (PR) is an important modulation spectroscopy technique., with a unique role inthe characterization of semiconductor materials. It is generally accepted that the modulation in PR forbulk materials originates from modulation of the surface electric field due to the charge carriersgenerated by optical pumping.[1 -4] This spectroscopic method has been successfully used on quantumwell structures in recent years.[5,6] However, little is known about the mechanism in quantum wellsand the nature of the lineshapes[6,7].

In order to understand this problem, we first identified the lineshapes of the unmodulatedreflectance in a single quantum well.[8] Our study showed that the major contribution to thereflectance spectrum of a GaAs /( Ga,AI)As quantum well arises from interference terms between thereflected waves from the air /( Ga,AI)As surface and the (Ga,AI)As /GaAs interfaces. The interferenceeffect has been recognized by other authors[9,10], but its dominant role was not fully demonstratedand experimental confirmation was lacking.

In this paper we proceed from the results of our previous study on a single quantum well[8], butfor the first time, include the Seraphin coefficients which are substantially different from those inbulk. We derive analytic forms for these coefficients for a single quantum well. This importantdifference has not been used in previous interpretations of the photoreflectance lineshapes. Since theSeraphin coefficients are directly related to the lineshapes in the modulation reflectance, the newcoefficients for quantum well structures should provide a more realistic basis for studying thelineshapes of modulation reflectance spectra.

2, THEORY

The Seraphin coefficients a and ß are defined by0R 1dRAE + 1dRAeR -Rder r Rde1 i

=aAer+ f3 LSIEi, (1)where R(w) is the reflectance as the function of frequency w, and e is the dielectric function. We shouldclearly distinguish between two interpretations of the use of AR /R. In the first interpretation, used in

SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988) / 43

Photoreflectance and the Seraphin coefficients in quantum well structures

X.LZheng D.Heiman and B.LaxFrancis Bitter National Magnet Laboratory, Massachusetts Institute of Technology

Cambridge, Massachusetts 02139

FAChambersAmoco Corporation, Amoco Research Center

Naperville, Illinois 60566

ABSTRACT

The lineshape of photoreflectance spectra from quantum well structures remains an unsolved problem, despite previous attemps to develop an adequate model using Seraphin coefficients for a bulk semiconductor. We show that the Seraphin coefficients in quantum well structures are substantially different from those in the bulk. We obtain analytical forms of the coefficients for a single quantum well, which show that the interference phase depends on the geometrical structure of the quantum well. The predictions are confirmed by our experimental photoreflectance data from quantum well samples. Our study suggests a new approach to the study of the lineshape problem of modulation reflectance.

1. INTRODUCTION

Photoreflectance (PR) is an important modulation spectroscopy technique, with a unique role in the characterization of semiconductor materials. It is generally accepted that the modulation in PR for bulk materials originates from modulation of the surface electric field due to the charge carriers generated by optical pumping. [1-4] This spectroscopic method has been successfully used on quantum well structures in recent years. [5, 6] However, little is known about the mechanism in quantum wells and the nature of the lineshapes[6,7].

In order to understand this problem, we first identified the lineshapes of the unmodulated reflectance in a single quantum well. [8] Our study showed that the major contribution to the reflectance spectrum of a GaAs/(Ga,AI)As quantum well arises from interference terms between the reflected waves from the air/(Ga,AI)As surface and the (Ga,AI)As/GaAs interfaces. The interference effect has been recognized by other authors[9,10], but its dominant role was not fully demonstrated and experimental confirmation was lacking.

In this paper we proceed from the results of our previous study on a single quantum well[8], but for the first time, include the Seraphin coefficients which are substantially different from those in bulk. We derive analytic forms for these coefficients for a single quantum well. This important difference has not been used in previous interpretations of the photoreflectance lineshapes. Since the Seraphin coefficients are directly related to the lineshapes in the modulation reflectance, the new coefficients for quantum well structures should provide a more realistic basis for studying the lineshapes of modulation reflectance spectra.

2. THEORY

The Seraphin coefficients a and p are defined by AR_ldR IdR- r i

where R(co) is the reflectance as the function of frequency co, and e is the dielectric function. We should clearly distinguish between two interpretations of the use of AR/R. In the first interpretation, used in

SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988) / 43

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oc

3.3

3.2

3.1

Air /GaAsEo =1.o

Et= 13.18

y= 3.5 meVA =10000 meV2

GaAIAs/GaAs£p=1 2.24Ei= 13.18y= 3.5 meVA=10000 meV2

0

E-Eg (meV)

+10

Fig. 1. Calculated reflectance. (a) air /GaAssurface: observed reflectance lineshape isdispersive -like. (b) (Ga,AI)As /GaAs interface: thereflectance is very small, and the reflectancelineshape becomes absorptive. A simple oscillatormodel is used in the calculations.

semiconductors, can be found in the literature[11 ].We are interested in the ratio of pia. Two typicalexamples are: A) the air /GaAs surface, whereIB(a1 =0.08, and the a term dominates, so it islegitimate to make the approximation AR /R hcr,This is why the observed reflectance lineshapefrom bulk semiconductors is dispersive -like. Thereflectance in this case is about 0.3. B) For a(Ga,AI)As /GaAs interface, we can have 113(aí >_1.

Now the discontinuity in the real part of thedielectric functions is small, and the imaginarycounterpart is comparable or even larger then thereal one. Hence the reflectance is very small, onlyabout 10-3, and the reflectance lineshape becomesabsorptive. The calculations for both cases areshown in Fig. 1, where a simple oscillator model isused for e(w).

The reflectance from a quantum well structureis different from either case. Our recent studydemonstrated that the reflectance structure from asingle quantum well is dominated by twointerference terms, each term is a product of twowaves, one from the air /(Ga,AI)As surface and onefrom each of the (Ga,AI)As /GaAs interfaces:

R- ro2+2ro(1 -roe) {[rtr cos(28)- rtisin(25)]- e'2T1d [ri r cos (28+29) -ri ¡sin(28 +2cp)] },(2)

rlr+i rii =e+eo+ 2e0(e+er)

e-EO+i 2E0(e-E)

(3)

the unmodulated reflectance, AR /R is not the e = er +e?,spectrum, but gives the relative contributions due where ro, and ( ri r, ru ¡ ) are the reflectivityto changes of the real and imaginary parts of the coefficients from air /(Ga,AI)As anddielectric function. The second interpretation is (Ga,AI)As /GaAs interfaces, respectively; 8 and cpused in the modulation reflectance, where AR = are the phase angles due to the front barrier layerR(w,4)- R(w,t =0) is the true modulation spectrum,thickness and the quantum well width, I is theusing 4 as the modulation parameter. absorption coefficient in the well region of width d,

In the modulation reflectance, Aer and Aei are and art I is at least an order -of- magnitude lessthe functions of modulation parameter i;, and should than ro. This last estimation is used throughout thecorrespond to a specific modulation mechanism. calculations.The Seraphin coefficients a and ß are only related The merit of our present approach, whichto the dielectric properties of adjoining materials,

ignores terms of the order of 10-1 or less in anindependent of the modulation mechanism, and areexact full calculation, is that it provides thethe same in both interpretations. The lineshapes of

the modulation reflectance are related to both essential aspect of the problem. However, it is valid

coefficients and the changes of the dielectric only for those quantum wells with smallfunctions as in Eq. (1). discontinuity in the dielectric functions. We should

The analytic form of a and ß, for the point out that the reflected wave from the

reflectance from an interface of two (GaAl)As /GaAs substrate can not change the basiclineshapes of the quantum

44 / SPIE Vol 946 Spectroscopic Characterization Techniques for Semiconductor Technology Ill (1988)

3.3

x 3.2

OC

3.1

2.0

x 1.0

CC

0.0

Air/GaAs

Co=1.0

1= 13.18

Y= 3.5 meV

A=10000 meV2

GaAIAs/GaAs

£0=12.24

£1= 13.18

Y= 3.5 meV

A=10000 meV2

-10 +10

E-Eg (meV)

semiconductors, can be found in the literature[11]. We are interested in the ratio of p/oc. Two typical examples are: A) the air/GaAs surface, where IB/al=0.08, and the a term dominates, so it is legitimate to make the approximation AR/R ~ oAer>This is why the observed reflectance lineshape from bulk semiconductors is dispersive-like. The reflectance in this case is about 0.3. B) For a (Ga,AI)As/GaAs interface, we can have |p/a| >1 . Now the discontinuity in the real part of the dielectric functions is small, and the imaginary counterpart is comparable or even larger then the real one. Hence the reflectance is very small, only about 10~3, and the reflectance lineshape becomes absorptive. The calculations for both cases are shown in Fig. 1, where a simple oscillator model is used for e(co).

The reflectance from a quantum well structure is different from either case. Our recent study demonstrated that the reflectance structure from a single quantum well is dominated by two interference terms, each term is a product of two waves, one from the air/(Ga,AI)As surface and one from each of the (Ga,AI)As/GaAs interfaces:

Fig. 1. Calculated reflectance, (a) air/GaAs surface: observed reflectance lineshape is dispersive-like. (b) (Ga,AI)As/GaAs interface: the reflectance is very small, and the reflectance lineshape becomes absorptive. A simple oscillator model is used in the calculations.

r cos(28)-r 1 jsin(28)] [r 1r cos(28+2cp)-rijsin(28+2cp)]},(2)

_ e-e0-fiV2e0(e~er)H-———/e+e0+Y 2£0(e+er) (3)

the unmodulated reflectance, AR/R is not the spectrum, but gives the relative contributions due to changes of the real and imaginary parts of the dielectric function. The second interpretation is used in the modulation reflectance, where AR = R(co,£)-R(co,4=0) is the true modulation spectrum, using £ as the modulation parameter.

In the modulation reflectance, Aer and Aej are the functions of modulation parameter £, and should correspond to a specific modulation mechanism. The Seraphin coefficients a and p are only related to the dielectric properties of adjoining materials, independent of the modulation mechanism, and are the same in both interpretations. The lineshapes of the modulation reflectance are related to both coefficients and the changes of the dielectric functions as in Eq. (1).

The analytic form of a and p, for the reflectance from an interface of two

6 = ^ e?+6i, where ro, and ( ri r , rn) are the reflectivity coefficients from air/(Ga,AI)As and (Ga,AI)As/GaAs interfaces, respectively; 8 and q> are the phase angles due to the front barrier layer thickness and the quantum well width, TI is the absorption coefficient in the well region of width d, and In | is at least an order-of-magnitude less than ro. This last estimation is used throughout the calculations.

The merit of our present approach, which ignores terms of the order of 10' 1 or less in an exact full calculation, is that it provides the essential aspect of the problem. However, it is valid only for those quantum wells with small discontinuity in the dielectric functions. We should point out that the reflected wave from the (GaAI)As/GaAs substrate can not change the basic lineshapes of the quantum

44 / SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988)

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well transitions, because the transition energies from the fundamental band gaps of these two bulkmaterials are far from the confinement -shifted quantum well transition energies.

Choosing Er and Ei as two independent variables in Eq. (2) and (3), we derive the Seraphincoefficients for a single quantum well:

a= (2 /ro)(1- r02) {ci i [cos(2S)- e-2rldcos(2S +2cp)]+ c12[sin(25)- e-2Tld sin(25 +2q )]+ 2d e -2rld coi [ri r cos(25 +29) -ri isin(26 +2cp)] },

13= (2 /ro)(1- r02) {ci 2[cos(25)- e- 2rldcos(25 +2cp)]- Cil [sin(25)- e -2rld sin(25 +29)]+ 2d e -2rld co2 [ri r cos(25 +2q) -ri isin(25 +2cp)]

where

Erarlr arlrc11 = + ,

E De aEr

Eiar IrC12= ,

E aE

2n / E-Ercol = oE Y 2

2n Ei 1

CO2 =Xo E 2(E -Er)

Since c12/ci 1 <10 -1, coi /c1 1 <1 0 -2 and co2 /ci 1 <10-2, we have2

a- 2 1 -ro cll[cos(25) -e 2dcos (25 +29)],

-

ro

2

2 1-ro cll[sin(25)-e 2dsin(2S+2(p)] .

ro

},

(4a)

(4b)

For a 100 A quantum well and 4.1x104 cm -1, then e-2r1d =0.9. If we assume e-2rld =1, then wehave a simple relation:

ß)a = cot (28+9). (5)

Surprisingly, the ratio of the Seraphin coefficients ß)a can change from -D. to +oo, and depends on thequantum well geometric configuration, primarily on the front layer thickness. In contrast, in the bulkcase this ratio depends on the material dielectric functions only, more precisely, it depends on thecompetition at the interface -- the discontinuities of the real and imaginary parts of the dielectricfunction.

Thus, Eq. (1) and (4) provide a new basis for the lineshape analysis of the modulation reflectancefor a single quantum well. For a multiple quantum well structure, Eq. (2) and (4) become lengthy, butthe principle is the same. In two recent lineshape models for quantum wells, one utilized the bulk 3rdderivative lineshape theory[6], and the other used three terms involving first derivativemodulation[7]. Both adopted the bulk semiconductor assumption, namely, ß =0, and start

SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology Ill (1988) / 45

well transitions, because the transition energies from the fundamental band gaps of these two bulk materials are far from the confinement-shifted quantum well transition energies.

Choosing er and ej as two independent variables in Eq. (2) and (3), we derive the Seraphin coefficients for a single quantum well:

a^2/r 0 )(1-r 0 2 ){cii[cos(28)-e- 2Tl d cos(28 + 2(p)]+ Ci2[sin(2S)-e- 2Tl d sin(28+2cp)]+ 2d e' 2Tl d coi [M r cos(2S+2cp)-ri jsin(28+2cp)]}, (4a)

P=(2/r 0 )(1-r 0 2 ){ci2[cos(2S)-e- 2Tldcos(28+2(p)]- cii[sin(2S)-e- 2Tl d sin(2S+2cp)]+ 2d e' 2Tl d CQ2 [Mr cos(2S+2cp)-njsin(28+2(p)] },

where

c n =

C 02 =

Since C12/C11<10" 1 , Coi/cn<i0'2 and Co2/cn<i0"2 , we have 2

1-fn -2T|da ~ 2 ——-c n [cos(25)-e cos(28+2(p)] ,

r 0(4b)

2 1-Tn -2tld

p~- 2——-c n [sin(28)-e sin(25+2cp)] . TO

For a 100 A quantum well and TI= 4.1x104 cm ' 1 , then e" 2<n d=0.9. If we assume e'2j\ d =1, then we have a simple relation:

p/a - cot (28-f 9). (5)

Surprisingly, the ratio of the Seraphin coefficients p/a can change from -«> to +00, and depends on the quantum well geometric configuration, primarily on the front layer thickness. In contrast, in the bulk case this ratio depends on the material dielectric functions only, more precisely, it depends on the competition at the interface -- the discontinuities of the real and imaginary parts of the dielectric function.

Thus, Eq. (1) and (4) provide a new basis for the lineshape analysis of the modulation reflectance for a single quantum well. For a multiple quantum well structure, Eq. (2) and (4) become lengthy, but the principle is the same. In two recent lineshape models for quantum wells, one utilized the bulk 3rd derivative lineshape theory[6], and the other used three terms involving first derivative modulation[7]. Both adopted the bulk semiconductor assumption, namely, p =0, and start

SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988) / 45

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sample 1 (100 A) /

sample 2 (200 A)

sample 3 (500 A)

sample 4 (1000 A)w,_,

fV

1540 1560

E (meV)

sample 1 (100 A)/

i \.ísample 2 (200 A)

I OR = 0.01

sample 3 (500 A)

sample 4 (1000 A)

1 1

1540 1560 1580

v\

tr

1.Ox104

1580

E (meV)

Fig. 3. Reflectance at 4.2 K for GaAs /(Ga,AI)Assingle quantum wells. Data taken simultaneouslywith the PR shown in Fig. 2(a).

sample 1 (100 A)

sample 2 (200 A)òt

sample 3 (500 A)

sample 4 (1000 A)

\I

1.0 x 10-4

1450

1 1 1

1470 1490

E (meV)

Fig. 2. Photoreflectance of GaAs /( Ga,AI)As singlequantum wells at two different temperatures (a)4.2 K and (b) 300 K. Both temperatures showsimilar lineshapes for a given sample. Thelineshape changes periodically with differentthickness of the front ( Ga,AI)As

from I R/R - aAer. The first model used anadjustable phase parameter 0, which gives an extradegree of freedom to adjust for the properproportion of absorptive and dispersive lineshapes.A reasonable fit to the experimental data can alwaysbe achieved, however, the physics behind this extraparameter O is obscure. In the second model, themodulation is due to three different parameters:exciton bandgap, line broadening and transitionstrength. Since the first one dominates, this modelpredicts only absorptive -like lineshapes in PR.

3. EXPERIMENTS

Four samples of GaAs /( Ga,AI)As singlequantum wells were grown using the molecularbeam epitaxial technique. The sample were allgrown under same conditions, each contained a 100A quantum well, but with varying front ( Ga,AI)Aslayer thicknesses of 100, 200, 500 and 1000 A

46 / SPIE Vol. 946 Spectroscopic Characterization Techniques forSemiconductor Technology Ill (1988)

sample 1 (100 A)A/V-^

sample 2 (200 A)r ^ _

W

DC a

sample 3 (500 A)

sample 4 (1000 A)

J____L _L1540 1560 1580

E (meV)

sample 1 (100 A)

/ sample 2 (200 A)

sample 1 (100 A)

| sample 2 (200 A)

sample 3 (500 A)

sample 4 (1000 A)

1450 1490

E (meV)

Fig. 2. Photoreflectance of GaAs/(Ga,AI)As single quantum wells at two different temperatures (a) 4.2 K and (b) 300 K. Both temperatures show similar lineshapes for a given sample. The lineshape changes periodically with different thickness of the front (Ga,AI)As

uc

0)cc

AR = 0.01

sample 3 (500 A)

sample 4 (1000 A)

1540 1560

E (meV)

Fig. 3. Reflectance at 4.2 K for GaAs/(Ga,AI)As single quantum wells. Data taken simultaneously with the PR shown in Fig. 2(a).

from AR/R - aAer . The first model used an adjustable phase parameter e, which gives an extra degree of freedom to adjust for the proper proportion of absorptive and dispersive lineshapes. A reasonable fit to the experimental data can always be achieved, however, the physics behind this extra parameter e is obscure. In the second model, the modulation is due to three different parameters: exciton bandgap, line broadening and transition strength. Since the first one dominates, this model predicts only absorptive-like lineshapes in PR.

3. EXPERIMENTS

Four samples of GaAs/(Ga,AI)As single quantum wells were grown using the molecular beam epitaxial technique. The sample were all grown under same conditions, each contained a 100 A quantum well, but with varying front (Ga,AI)As layer thicknesses of 100, 200, 500 and 1000 A

46 / SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988)

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(samples 1,2,3 and 4). PR and reflectance were performed simultaneously in an optical fiber system. The

results of the reflectance data on the same samples were reported previously[8]. The reflectancelineshape changes with the thickness of the front layer and was shown to be periodic with a period of1100 A.

Fig. 2 show the PR data at two different temperature 4.2 K (Fig. 2a) and 300 K (Fig. 2b). Bothshow similar types of lineshapes for a given sample. The periodic changes of the PR lineshapes, fromdispersive -like(sample 2), absorptive -like(sample 3) to inverse absorptive- like(sample 4), areconsistent with our predictions based on Eq. (4) and (5). This verifies our Seraphin coefficients for asingle quantum well. The lineshape phase from sample 1 has a noticeable deviation from the prediction,but is not apparent in the reflectance data. It is conceivable that the surface electric field in such ashort region (100 A) may produce a different effect. Low temperature data from sample 2 shows adouble structure with an unknown origin, but should not cause a problem in the lineshape analysis.

Comparisons of the lineshapes for PR (Fig.2a) and reflectance (Fig. 3) on the same samples showsimilarities for sample 2, 3 and 4, but a 90 degree difference for sample 1. We applied the firstderivative relation fore er (4) and a Ei (i;), and calculated the PR lineshape by combining Eq.(1) and(4), where a simple harmonic oscillator model (Lorentzian line shape) is used.[8] A satisfactory fitto samples 2 -4 has not been achieved. This also can be seen by comparing the data from the samesamples in Fig. 2a and Fig. 3 -- both PR and reflectance have similar lineshapes for samples 2 -4, sothey should not be related by the first derivative relation. However, for some other samples, thisrelation appears to hold and may be accidental.[12] Therefore we suggest that the modulationmechanism, i.e. how the modulation parameter 4 affects e er (4) and A Ei (e), is still unclear.

4. ACKNOWLEDGMENT

We thank R.L.Aggarwal for the helpful discussion.

5. REFERENCE

1. E.Y.Wang, W.A.Albers, and C.E.Bleil, in II-VI Semiconducting Compounds, ed. D.G.Thomas,(Benjamin,1967) p.136.2. R.E.Nahory, and J.L.Shay, Phys. Rev. Lett. 21,1569 (1968)3. J.L.Shay, Phys. Rev. Z, 803 (1970)4. D.E.Aspnes, Sol. State Commun. $, 267 (1970)5. O.J.Glembocki, B.V.Shanabrook, N.Bottka, W.T.Beard, and J.Comas, Appl. Phys. Lett. 41, 970(1985).6. H.Shen, P.Parayanthal, F.H.Pollak, M.Tomiewiez, T.J.Drummond, and J.N.Schulman, Appl. Phys.Lett. 4$,, 653 (1986).7. B.V.Shanabrook, O.J.Glembocki, and W.T.Beard, Phys. Rev. Bue,, 2540 (1987).8. X.L.Zheng, D.Heiman, B.Lax, and F.A.Chambers, Appl. Phys. Lett. 52, 287 (1988).9. L.Schutheis, and K.Ploog, Phys. Rev. BaQ, 1090 (1984).10. P.C.Klipstein and N.Apsley, J. Phys. C19., 6461 (1986).11. For example, see R.L.Aggarwal, in Semiconductors and Semimetals, Vol. 9. eds. Willardson andBeer, (Academic,1972) p.157.12. X.L.Zheng, D.Heiman, B.Lax, and F.A.Chambers, Superlat. & Microstructure. x, xxx (1988).

SP /E Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988) / 47

(samples 1,2,3 and 4). PR and reflectance were performed simultaneously in an optical fiber system. The results of the reflectance data on the same samples were reported previously[8j. The reflectance lineshape changes with the thickness of the front layer and was shown to be periodic with a period of 1100 A.

Fig. 2 show the PR data at two different temperature 4.2 K (Fig. 2a) and 300 K (Fig. 2b). Both show similar types of lineshapes for a given sample. The periodic changes of the PR lineshapes, from dispersive-like(sample 2), absorptive-like(sample 3) to inverse absorptive-like(sample 4), are consistent with our predictions based on Eq. (4) and (5). This verifies our Seraphin coefficients for a single quantum well. The lineshape phase from sample 1 has a noticeable deviation from the prediction, but is not apparent in the reflectance data. It is conceivable that the surface electric field in such a short region (100 A) may produce a different effect. Low temperature data from sample 2 shows a double structure with an unknown origin, but should not cause a problem in the lineshape analysis.

Comparisons of the lineshapes for PR (Fig.2a) and reflectance (Fig. 3) on the same samples show similarities for sample 2, 3 and 4, but a 90 degree difference for sample 1. We applied the first derivative relation for A er (£) and A ej (£), and calculated the PR lineshape by combining Eq.(1) and (4), where a simple harmonic oscillator model (Lorentzian line shape) is used.[8] A satisfactory fit to samples 2-4 has not been achieved. This also can be seen by comparing the data from the same samples in Fig. 2a and Fig. 3 -- both PR and reflectance have similar lineshapes for samples 2-4, so they should not be related by the first derivative relation. However, for some other samples, this relation appears to hold and may be accidental.[12] Therefore we suggest that the modulation mechanism, i.e. how the modulation parameter § affects A er (4) and A ej (£), is still unclear.

4. ACKNOWLEDGMENT

We thank R.L.Aggarwal for the helpful discussion.

5. REFERENCE

1. E.Y.Wang, W.A.AIbers, and C.E.BIeil, in //-V7 Semiconducting Compounds, ed. D.G.Thomas, (Benjamin,1967) p.136.2. R.E.Nahory, and J.L.Shay, Phys. Rev. Lett. 21,1569 (1968)3. J.L.Shay, Phys. Rev. 2., 803 (1970)4. D.E.Aspnes, Sol. State Commun. fl, 267 (1970)5. O.J.GIembocki, B.V.Shanabrook, N.Bottka, W.T.Beard, and J.Comas, Appl. Phys. Lett. 4£, 970 (1985).6. H.Shen, P.Parayanthal, F.H.Pollak, M.Tomiewiez, T.J.Drummond, and J.N.Schulman, Appl. Phys. Lett. 48. 653 (1986).7. B.V.Shanabrook, O.J.GIembocki, and W.T.Beard, Phys. Rev. B3£, 2540 (1987).8. X.L.Zheng, D.Heiman, B.Lax, and F.A.Chambers, Appl. Phys. Lett. 52, 287 (1988).9. LSchutheis, and K.PIoog, Phys. Rev. B2Q, 1090 (1984).10. P.C.KIipstein and N.Apsley, J. Phys. CJJL 6461 (1986).11. For example, see R.L.Aggarwal, in Semiconductors and Semimetals, Vol. 9. eds. Willardson and Beer, (Academic,1972) p.157.12. X.L.Zheng, D.Heiman, B.Lax, and F.A.Chambers, Superlat. & Microstructure. *, xxx (1988).

SPIE Vol. 946 Spectroscopic Characterization Techniques for Semiconductor Technology III (1988) / 47

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