Photoreflectance study of hexagonal CdSe

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J. Phys. D: Appl. Phys.32 (1999) 2337–2341. Printed in the UK PII: S0022-3727(99)04695-1

Photoreflectance study of hexagonalCdSe

Akio Kaneta and Sadao Adachi

Department of Electronic Engineering, Faculty of Engineering, Gunma University, Kiryu-shi,Gunma 376-8515, Japan

Received 1 June 1999

Abstract. Photoreflectance (PR) spectra have been measured to determine thefundamental-absorption-edge (E0) structure of hexagonal CdSe single crystal for bothE ⊥ candE ‖ c polarizations at temperaturesT between 77 and 300 K. The measured PR spectraare found to be successfully explained by an excitonic line shape for temperaturesT below∼200 K and by a three-dimensional one-electron line shape forT > 200 K. No clearevidence of the exciton interference effect has been observed within our experimentaltemperature rangeT = 77–300 K. The temperature dependence of theE0-exciton parameters(exciton energy, amplitude and broadening parameter) has been determined and analysedusing the Varshni equation and an empirical expression of Bose–Einstein type.

1. Introduction

Modulation spectroscopy is popular means of analysingcritical-point (CP) structures in semiconductors [1].Photoreflectance (PR) is one of a class of electro-modulationand is known to be useful non-destructive and contactlesstechnique for the characterization of semiconductors andtheir microstructures. It can yield sharp CP structures andalso be sensitive to surface or interface [2].

Cadmium selenide (CdSe) is a member of the II–VIsemiconducting compounds. At normal temperature andpressure it crystallizes in the hexagonal, wurtzite lattice(space group= P63mc (C4

6v)). Measurements of the opticalproperties of CdSe with polarized light have shown strongdifference for the electric vector perpendicular (E ⊥ c)and parallel to its hexagonal axisc (E ‖ c). Theseproperties have been studied using various techniques,such as far-infrared spectroscopy [3–5], absorption [6–9],reflectivity [7–11], photoluminescence [12], refractive index[6, 13, 14] and spectroellipsometry (SE) [15–17]. Somemodulation techniques, such as electroreflectance (ER) [18],thermoreflectance [19, 20] and piezoreflectance [21], havealso been employed to investigate the optical properties ofCdSe and, consequently, its electronic energy-band structure.To our knowledge, however, there has been no reported dataon the PR of CdSe.

In this paper we measure the PR spectra of CdSe forboth E ⊥ c andE ‖ c in the 1.5–2.4 eV (E0) photon-energy range at temperaturesT in the range 77–300 K.The measured spectra are fitted by a standard critical-point(SCP) line shape [22]. Temperature variations of the CPparameters (CP energy, amplitude, broadening constant, etc)are of considerable interest both from the technological andthe academic points of view. The dependence of the CP orexciton energy onT is commonly described by the empiricalequation proposed by Varshni [23]. An alternative approachhas been suggested by Vina et al [24], who considered an

empirical expression of Bose–Einstein type. TheE0-excitonparameters for CdSe obtained in this study are analysed usingthese empirical expressions.

2. Experimental details

The hexagonal CdSe single crystal used in this study wasgrown by a vapour-phase method, it was not intentionallydoped and the surface orientation was parallel to thec-axis. The sample was mechanically polished with diamondpaste and degreased in organic solvents. It was thenchemomechanically polished with a solution of 0.01 vol.%Br2 in methanol, followed by rinsing with deionized waterand, further, with organic solvents [17].

The experimental set-up for the PR measurement wasessentially the same as that described in the literature [25].The sample was mounted in a metal liquid N2 cryostatequipped with quartz glass window. The 488.0 nm line of a100 mW Ar+ laser (NEC GLG3110) chopped at 525 Hz wasused as the pumping light. The probe light from a 150 Wxenon lamp was irradiated near to normal on the samplesurface. The spectra were taken in the 1.5–2.4 eV photon-energy range using a grating spectrometer (JASCO CT-25C)and a photomultiplier tube (Hamamatsu R375). A cut-offfilter to pass only long wavelengths above 520 nm was usedto block the light scattered from the source.

3. Results and discussion

Modulation spectroscopy, such as PR and ER, measures thechange in the complex dielectric function,ε(E) = ε1(E) +iε2(E). The effect on reflectivityR of the changes1ε1 and1ε2 induced by an external modulation can be written as[1, 26]

1R

R= α1ε1 + β1ε2. (1)

0022-3727/99/182337+05$30.00 © 1999 IOP Publishing Ltd 2337

A Kaneta and S Adachi

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Figure 1. ReflectivityR and PR spectra1R/R for CdSemeasured atT = 300 K forE ⊥ c andE ‖ c. Thenormal-incidence reflectivityR is obtained from experimentalε(E) data [17] using (2).

In (1), the fractional coefficientsα andβ are functions ofthe photon energyE and their sign and relative magnitudedetermine the results of analysis in the different spectralregions [1, 26].

Figure 1 shows the PR spectra of CdSe measured atT = 300 K for bothE ⊥ c andE ‖ c polarizations.Also shown in the figure are the normal-incidence reflectivityspectraR for CdSe obtained from experimentalε(E) datausing

R = (ε21 + ε2

2)1/2 − [2ε1 + 2(ε2

1 + ε22)

1/2] + 1

(ε21 + ε2

2)1/2 + [2ε1 + 2(ε2

1 + ε22)

1/2] + 1. (2)

The experimentalε(E) data used in (2) were measured bySE using the same sample as in this study [17].

As shown schematically in figure 2, the valence bandmaxima of wurtzite can be obtained from that of zinc blendeby the action of a small hexagonal crystal field [27]. Thecombined effect of spin–orbit (1so) and hexagonal crystal-field (1cr ) perturbations in the wurtzite lattice splits the015

valence band into09(A), 07(B) and07(C) valence bands.The hexagonal, wurtzite CdSe crystal thus exhibits threeexcitonic structures,E0A, E0B andE0C , at theE0 edge. Thedipole-transition selection rule suggests that the structuresof 07 symmetry (E0B andE0C) can be observed for bothE ⊥ c andE ‖ c, while that of09 symmetry (E0A) canonly be observed forE ⊥ c. This result is known as the‘quasi-cubic’ model [28, 29].

As seen in figure 1, the dominant peaks in theR spectraare theE0A andE0B structures. The PR technique is foundto strongly enhance the changes in reflectivityR. Theresultant PR spectra reveal additional weak CP structureE0C at the higher spectral region (∼2.2 eV). The derivativenature given by (1) is the reason why PR spectra are sharperand more richly structured than the conventionally measuredreflectivity spectraR.

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Figure 2. Relations between the valence-band splittings in thezinc blende and wurtzite lattices and between the irreduciblerepresentations of the bands at the0 point (k = 0). (b) shows theactual splitting found in the zinc blende crystals, while (c) givesthat found in the wurtzite crystals.

We analysed our measured PR spectra using the SCPmodel. This model is given by the following equation [22]:

1R

R= Re

[ p∑j=1

Cj eiθj (E − Egj + i0j )−nj]

(3)

wherep is the number of the spectral functions to be fittedandC, θ , Eg and0 are, respectively, the amplitude, phaseangle, CP energy and lifetime broadening parameter.n refersto the type of optical transitions in question:n = 2, 2.5and 3 for an excitonic transition, a three-dimensional (3D)one-electron transition and a two-dimensional one-electrontransition, respectively. The phase angleθ accounts for (1) aswell as for the influence on a non-uniform modulation field.

Figures 3 and 4 show the PR spectra for CdSe measuredat three different temperatures, 90, 150 and 300 K, forE ⊥ c(figure 3) andE ‖ c (figure 4). We can see that the line shapesof these spectra are quite different for temperatures between90 (150) and 300 K. The CP structures observed below∼200 K can be characterized by two extrema, suggestingan origin of the excitonic transition. On the other hand, thePR line shapes forT > 200 K should be characterized simplyby a deep dip.

The full curves in figures 3 and 4 (T = 90 and 150 K)represent the theoretical fits using (3) and assuming anexcitonic model (n = 2). The quasi-cubic model yields anexpression for the CP amplitude ratio in terms of the bandadmixture coefficientsαB andαC [17]

C0A⊥:C0B⊥:C0C⊥:C0A‖:C0B‖:C0C‖= 1:α2

B :α2C :0:2α2

C : 2α2B. (4)

In [17], values ofαB andαC for CdSe are reported to be 0.620and 0.785, respectively. One can thus obtain the followingamplitude ratio for CdSe

C0A⊥:C0B⊥:C0C⊥:C0A‖:C0B‖:C0C‖= 1:0.38:0.62:0:1.23:0.77. (5)

TheCj ’s in (3) are constrained to satisfy this amplitude ratioin the present study. In figures 3 and 4 the exciton amplitude

2338

Photoreflectance study of hexagonal CdSe

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Figure 3. PR spectra for CdSe measured at three differenttemperatures,T = 90, 150 and 300 K, forE ⊥ c. The full curvesrepresent the best fits of the data to (3) assuming an excitonicmodel (n = 2; T = 90 and 150 K) and a 3D one-electron model(n = 2.5; T = 300 K). The exciton (n = 2) and 3D one-electron(n = 2.5) amplitude parameters determined here areC0A⊥ ∼ 1.3× 10−6 eV2 and∼ 5.0× 10−8 eV2.5, respectively. Thebroken curve (T = 300 K) is obtained by assumingn = 2(excitonic model) andC0A⊥ ∼ 1.3× 10−6 eV2 but properlychangingθ and0 values in (3). The vertical arrows indicate theE0-exciton positions as evaluated from the SCP analysis.

of C0A⊥ ∼ 1.3× 10−6 eV2 is fit determined forT < 200 K(see also figure 6 below).

The peculiar PR spectra observed forT > 200 K areconsidered to have an origin of the 3D one-electron transition.This is confirmed by the fits to the spectra assuming the 3Done-electron expression (n = 2.5), as shown in figures 3and 4 by the full curves (T = 300 K). The amplitudeof the 3D one-electron transition is fit determined to beC0A⊥ ∼ 5.0× 10−8 eV2.5 (see also figure 6 below).

It should be noted that PR and ER spectra at theE0 or higher energy region of some II–VI and III–Vsemiconductors are strongly influenced by an excitoninterference effect [30–35]. This effect is a consequence ofthe different dielectric-susceptibility mediums present at thesurface. Hopfield and Thomas [30] suggested that in the caseof the Wannier–Mott excitons, there should exist an exciton-free (dead) layer at the surface whose thickness is roughlyequal to the exciton Bohr radius. This model successfullyexplained the existence of a spike in reflectivity spectra ofsome samples [30, 31].

We do not suppose that the PR spectral change observedfor T > 200 K is due to the exciton interference effect. Thereason for this is as follows. First, this effect should disappearat temperatures above∼20 K in CdSe as a result of the excitonlifetime broadening [15]. Secondly, the effect should promisethat the spectra can be explained in a continuous manner asthe temperature is increased from∼200 K, i.e. they can bebest fitted by only changingθ and0 values in (3) (keepingn = 2 andC0A⊥ ∼ 1.3 × 10−6 eV2 (constant)). This isnot true, as demonstrated by the broken curves in figures 3

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Figure 4. As figure 3, but forE ‖ c.

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Figure 5. Temperature dependence of theE0-exciton energies,E0A, E0B andE0C , for CdSe determined by the SCP analysis (fullcircles). The open circles represent experimental data determinedby Logothetidiset al [15] using SE. The broken and full curves areobtained from (6) and (7), respectively. The fit-determinedparameters are listed in table 1.

and 4 (T = 300 K). We therefore consider that the changein the PR spectra observed above∼200 K is not due to theexciton interference effect, but is due to the 3D one-electrontransition.

Figure 5 plots theE0-exciton energies,E0A, E0B andE0C , againstT determined from the SCP fits (full circles).The open circles represent experimental data determined byLogothetidiset al [15] using SE. Note that the data forT > 210 K were plotted after subtracting an exciton Bohrenergy of 15 meV [9] from the fit-determined values ofEgα(α = A, B andC).

2339

A Kaneta and S Adachi

Table 1. Values ofEg(0), α andβ in (6) and those ofEB , αB and2B in (7), together with the temperature coefficients∂Eg/∂T(T = 300 K) obtained from (8) and (9).

Eg(0) α β ∂Eg/∂T EB αB 2B ∂Eg/∂T

(eV) (10−4 eV K−1) (K) (10−4 eV K−1) (eV) (meV) (K) (10−4 eV K−1)

E0A 1.830 6.4 230−5.19 1.887 60 230−4.97E0B 1.858 6.7 230−5.44 1.917 63 230−5.22E0C 2.261 6.1 230−4.95 2.318 59 230−4.89

Traditionally, temperature variation of the band-gapenergies is discussed in terms of Varshni’s formula [23]

Eg(T ) = Eg(0)− αT 2

T + β(6)

whereEg(0) is the band-gap energy atT = 0 K, α is ineV K−1 andβ is proportional to the Debye temperature ofthe material (in K). The broken curves in figure 5 show thefitted results of the data to (6). The fit-determined parametersare listed in table 1.

Vi na et al [24] proposed a new expression for thetemperature dependence of the band-gap energies by takinginto account the Bose–Einstein occupation factor

Eg(T ) = EB − αB(

1 +2

e2B/T − 1

)(7)

where the parameter2B describes the mean frequency ofthe phonons involved andαB is the strength of the electron–phonon interaction. This expression is more palatable than(6) from the theoretical point of view. The full curves infigure 5 represent the results of the data fitted to (7). Thefit-determined parameter values are listed in table 1. It isfound that (6) and (7) show an equally good agreement withthe experimental data.

Differentiating (6) and (7) with respect toT , we obtain

∂Eg(T )

∂T= −α

[2T

T + β− T 2

(T + β)2

](8)

and∂Eg(T )

∂T= −2αB

2B

T 2

e2B/T

(e2B/T − 1)2. (9)

The temperature coefficients∂Eg/∂T for CdSe atT = 300 Kobtained from (8) and (9) are also listed in table 1. Thesevalues are slightly larger than those reported in the literature(−2.8× 10−4 to−4.1× 10−4 eV K−1) [36].

The effect of temperature provides not only the shift ofthe CP energies, but also the broadening of the CP structures.The line width of the CP structure againstT can be commonlydescribed with an expression similar to (7) [24]

0(T ) = 01 +

(00

e20/T − 1

)(10)

where 01 represents the broadening due to temperature-independent mechanisms, such as the Auger process,electron–electron interaction, crystalline imperfections andsurface scattering. This equation is based on the assumptionof broadening by interband scattering via phonon absorptionand holds only for the lowest direct gapE0.

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Figure 6. Temperature dependence of theE0-exciton (CP)amplitude,C0A⊥, and broadening parameters,00A, 00B and00C ,for CdSe determined by the SCP fit. The full curves are obtainedfrom (10). The fit-determined parameters are listed in table 2.

Table 2. Values of01, 00 and20 in (10).

01 (meV) 00 (eV) 20 (K)

E0A T < 200 K 20 1.40 750T > 200 K 48 0.08 750

E0B T < 200 K 23 1.55 750T > 200 K 51 0.10 750

E0C T < 200 K 28 1.50 750T > 200 K 57 0.18 750

We plot in figure 6 the temperature dependence of theamplitude parameter,C0A⊥, and broadening parameters,00A,00B and00C , for CdSe. The amplitude parameterC0A⊥ may,in principle, be of the order of the thermal expansion of thematerial and may, thus, be safely assumed to be independentof T . Indeed, our SCP analysis suggests no strong variationof C0A⊥ with T , as clearly seen in figure 6. The0 values, onthe other hand, showed remarkable dependence onT . Thefull curves in figure 6 represent the results of the data fittedto (10). The fitting parameters determined here are shownin table 2. Due to the experimental uncertainties, we haveforced0 to have the same values between forE ⊥ c andE ‖ c polarizations. The kink seen atT ∼ 200 K is dueto change in the SCP model assumed (i.e.n = 2 (excitonictransition) forT below∼200 K andn = 2.5 (3D one-electrontransition) forT > 200 K, in (3)).

2340

Photoreflectance study of hexagonal CdSe

4. Conclusions

We have measured the PR spectra of hexagonal CdSe forE ⊥ c andE ‖ c in the 1.5–2.4 eV spectral range attemperaturesT between 77–300 K. The measured spectraindicate the presence of two clear structures, one at∼1.75–1.85 eV (E0A andE0B) and the other at∼2.15–2.25 eV (E0C).These structures have shown strong difference forT belowand above∼200 K. The SCP analysis suggests that the PRline shape below∼200 K is due to excitonic effect and thatabove∼200 K is due to the 3D one-electron transition. Wenote that the change in the PR spectra above∼200 K couldnever be explained by the exciton interference effect. Thetemperature dependence of theE0-exciton parameters hasbeen determined and analysed using the Varshni equationand an empirical expression of Bose–Einstein type.

Acknowledgments

We would like to express our appreciation to Dr K Era forkindly supplying the CdSe single crystal used in this study.The authors would also like to thank Drs T Miyazaki andS Ozaki for their kind advice concerning this work.

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